**LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies**

Pablo Ballesteros, Xinyu Shu, Wiebke Heins and Christian Bohn

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50294

### **1. Introduction**

30 Will-be-set-by-IN-TECH

[23] Köro ˘glu, H. and C. W. Scherer. 2008. LPV control for robust attenuation of non-stationary sinusoidal disturbances with measurable frequencies. *Proceedings of the*

[24] Kuo, S. M. and D. R. Morgan. 1996. *Active Noise Control Systems*. New York: Wiley. [25] Morari, M. and E. Zafiriou 1989. *Robust Process Control*. Englewood Cliffs: Prentice Hall. [26] Shu, X., P. Ballesteros and C. Bohn. 2011. Active vibration control for harmonic disturbances with time-varying frequencies through LPV gain scheduling. *Proceedings of the 23rd Chinese Control and Decision Conference*. Mianyang, China, May 2011. 728-33. [27] Stilwell, D. J. and W. J. Rugh. 1998. Interpolation of observer state feedback controllers for gain scheduling. *Proceedings of the American Control Conference* 2:1215-19. [28] Witte, J., H. M. N. K. Balini and C. W. Scherer. 2010. Experimental results with stable and unstable LPV controllers for active magnetic bearing systems. *Proceedings of the IEEE International Conference on Control Applications*. Yokohama, September 2010. 950-55.

*17th IFAC World Congress*. Korea, July 2008. 4928-33.

In this chapter, the same control problem as in the previous chapter is considered, which is the rejection of harmonic disturbances with time-varying frequencies for linear time-invariant (LTI) plants. In the previous chapter, gain-scheduled observer-based state-feedback controllers for this control problem were presented. In the present chapter, two methods for the design of general gain-scheduled output-feedback controllers are presented. As in the previous chapter, the control design is based on a description of the system in linear parameter-varying (LPV) form. One of the design methods presented is based on the polytopic linear parameter-varying (pLPV) system description (which has also been used in the previous chapter) and the other method is based on the description of an LPV system in linear fractional transformation (LPV-LFT) form. The basic idea is to use the well-established norm-optimal control framework based on the generalized plant setup shown in Fig. 1 with the generalized plant G and controller K.

In this setup, u is the control signal and y consists of all signals that will be provided to the controller. The signal w is the performance input and the signal q is the performance output in the sense that the performance requirements are expressed in terms of the "overall gain" (usually measured by the *H*<sup>∞</sup> or the *H*<sup>2</sup> norm) of the transfer function from w to q in closed

**Figure 1.** Generalized plant and controller

©2012 Bohn et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Bohn et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 2 Will-be-set-by-IN-TECH 66 Advances on Analysis and Control of Vibrations – Theory and Applications LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies <sup>3</sup>

loop. In this setup, the aim of the controller design is to satisfy performance requirements expressed as upper bounds on the norm (in case of suboptimal control) or minimize the norm (in optimal control) of the transfer function from w to q. Loosely speaking, a good controller should make the effect of w on q "small" (for suboptimal control) or "as small as possible" (for optimal control). The performance outputs usually consist of weighted versions of the controlled signal, the control error and the control effort. This is achieved by augmenting the original plant with output weighting functions. Good rejection of specific disturbances can be achieved in this framework by using a disturbance model as a weighting function in the transfer path from the performance input w to the performance output q, that is, by modeling the disturbance to be rejected as a weighted version of the performance input. This forces the maximum singular value *σ*max(G*qw*(j*ω*)) or, in the single-input single-output case, the amplitude response |*Gqw*(j*ω*)| of the open-loop transfer function to have a very high gain in the frequency regions specified by the disturbance model, or, loosely speaking, enlarges the effect of w on q in certain frequency regions. A reduction of the overall effect of w on q in closed loop will then be mostly achieved by reducing the effect in regions where it is large in open loop. From classical control arguments, it is intuitive that this requires a high loop gain in these frequency regions which in turn usually requires a high controller gain. A high loop gain will give a small sensitivity and in turn a good disturbance rejection (in specified frequency regions).

Kinney & de Callafon [14], Du & Shi [8] and Du et al. [9]. An approach based on LPV-LFT

for Active Control of Harmonic Disturbances with Time-Varying Frequencies

For a practical application, the resulting controller has to be implemented in discrete time. In applications of ANC/AVC, the plant model is often obtained through system identification. This usually gives a discrete-time plant model. If a continuous-time controller is computed, the controller has to be discretized. Since the controller is time varying, this discretization would have to be carried out at each sampling instant. An exact discretization involves the calculation of a matrix exponential, which is computationally too expensive and leads to a distortion of the frequency scale. Usually, this can be tolerated, but not for the suppression of harmonic disturbances. In this context, it is not surprising that the continuous-time design methods of Darengosse & Chevrel [7], Du et al. [9], Kinney & de Callafon [14] and Köro ˘glu & Scherer [17] are tested only in simulation studies with a very simple system as a plant and a single frequency in the disturbance signal. Exceptions are Witte et al. [19] and Balini et al. [4], who designed continuous-time controllers which then are approximately discretized. However, Witte et al. [19] use a very high sampling frequency of 40 kHz to reject a harmonic disturbance with a frequency up to 48 Hz (in fact, the authors state that they chose "the smallest [sampling time] available by the hardware") and Balini et al. [4] use a maximal sampling frequency of 50 kHz. The control design methods presented in this chapter are

The remainder of this chapter is organized as follows. In Sec. 2, pLPV systems and LPV-LFT systems are introduced and the control design for such systems is described. In Sec. 3, it is described how the control problem considered here can be transformed to a generalized plant setup. The required pLPV disturbance model for the harmonic disturbance is introduced in Sec. 3.1 and in Sec. 3.2, it is described how the generalized plant in pLPV form is obtained by combining the disturbance model, the plant and the weighting functions. In Sec. 4, the transformation of the control problem to a generalized plant in LPV-LFT form is treated in essentially the same way, by formulating an LPV-LFT disturbance model (Sec. 4.1) and building a generalized plant in LPV-LFT form (Sec. 4.2). The controller synthesis for both descriptions is described in Sec. 5. Experimental results are presented in Sec. 6 and the chapter

In this section, pLPV systems and LPV-LFT systems are introduced and the control design for

 *<sup>x</sup><sup>k</sup> uk* 

*A*(θ) = *A*<sup>0</sup> + *θ*1*A*<sup>1</sup> + *θ*2*A*<sup>2</sup> + ··· + *θ<sup>N</sup> AN*, (2)

, (1)

67

LPV Gain-Scheduled Output Feedback

control design is used by Ballesteros & Bohn [2, 3] and Shu et al. [18].

finishes with a discussion and some conclusions in Sec. 7.

such systems is described in Sec. 2.1 and 2.2, respectively.

 *xk*+<sup>1</sup> *yk*

 = *A*(θ) *B C D*

where the system matrix depends affinely on a parameter vector *θ*, that is

**2.1. Control design for pLPV systems**

realized in discrete time.

**2. Control design setup**

A pLPV system is of the form

This control design setup is used in this chapter for the rejection of harmonic disturbances with time-varying frequencies. The control design problem is based on a generalized plant obtained through the introduction of a disturbance model that describes the harmonic disturbances and the addition of output weighting functions. Descriptions of the disturbance model in pLPV and in LPV-LFT form are used and lead to generalized plant descriptions that are also in pLPV or LPV-LFT form. Corresponding design methods are then employed to obtain controllers. For a plant in pLPV form, standard *H*∞ design [11] is used to compute a set of controllers. The gain scheduling is then achieved by interpolation between these controllers. For a plant in LPV-LFT form, the design method of Apkarian & Gahinet [1] is used that directly yields a gain-scheduled controller also in LPV-LFT form.

LPV approaches for the rejection of harmonic disturbances have been used by Darengosse & Chevrel [7], Du & Shi [8], Du et al. [9], Bohn et al. [5, 6], Kinney & de Callafon [14, 15, 16], Köro ˘glu & Scherer [17], Witte et al. [19], Balini et al. [4], Heins et al. [12, 13], Ballesteros & Bohn [2, 3] and Shu et al. [18]. Darengosse & Chevrel [7], Du & Shi [8], Du et al. [9], Witte et al. [19], Balini et al. [4] suggested continuous-time LPV approaches. These approaches are tested for a single sinusoidal disturbance by Darengosse & Chevrel [7], Du et al. [9], Witte et al. [19] and Balini et al. [4]. Methods based on observer-based state-feedback controllers are presented by Bohn et al. [5, 6], Kinney & de Callafon [14, 15, 16] and Heins et al. [12, 13]. In the approach of Bohn et al. [5, 6], the observer gain is selected from a set of pre-computed gains by switching. In the other approaches of Kinney & de Callafon [16], Heins et al. [13] and in the previous chapter, the observer gain is calculated by interpolation. In the other approach presented in the previous chapter, which is also used by Kinney & de Callafon [14, 15] and Heins et al. [12], the state-feedback gain is scheduled using interpolation. A general output feedback LPV approach for the rejection of harmonic disturbances is suggested and applied in real time by Ballesteros & Bohn [2, 3] and Shu et al. [18].

The existing LPV approaches can be classified by the control design technique used to obtain the controller. Approaches based on pLPV control design are used by Heins et al. [12, 13], Kinney & de Callafon [14], Du & Shi [8] and Du et al. [9]. An approach based on LPV-LFT control design is used by Ballesteros & Bohn [2, 3] and Shu et al. [18].

For a practical application, the resulting controller has to be implemented in discrete time. In applications of ANC/AVC, the plant model is often obtained through system identification. This usually gives a discrete-time plant model. If a continuous-time controller is computed, the controller has to be discretized. Since the controller is time varying, this discretization would have to be carried out at each sampling instant. An exact discretization involves the calculation of a matrix exponential, which is computationally too expensive and leads to a distortion of the frequency scale. Usually, this can be tolerated, but not for the suppression of harmonic disturbances. In this context, it is not surprising that the continuous-time design methods of Darengosse & Chevrel [7], Du et al. [9], Kinney & de Callafon [14] and Köro ˘glu & Scherer [17] are tested only in simulation studies with a very simple system as a plant and a single frequency in the disturbance signal. Exceptions are Witte et al. [19] and Balini et al. [4], who designed continuous-time controllers which then are approximately discretized. However, Witte et al. [19] use a very high sampling frequency of 40 kHz to reject a harmonic disturbance with a frequency up to 48 Hz (in fact, the authors state that they chose "the smallest [sampling time] available by the hardware") and Balini et al. [4] use a maximal sampling frequency of 50 kHz. The control design methods presented in this chapter are realized in discrete time.

The remainder of this chapter is organized as follows. In Sec. 2, pLPV systems and LPV-LFT systems are introduced and the control design for such systems is described. In Sec. 3, it is described how the control problem considered here can be transformed to a generalized plant setup. The required pLPV disturbance model for the harmonic disturbance is introduced in Sec. 3.1 and in Sec. 3.2, it is described how the generalized plant in pLPV form is obtained by combining the disturbance model, the plant and the weighting functions. In Sec. 4, the transformation of the control problem to a generalized plant in LPV-LFT form is treated in essentially the same way, by formulating an LPV-LFT disturbance model (Sec. 4.1) and building a generalized plant in LPV-LFT form (Sec. 4.2). The controller synthesis for both descriptions is described in Sec. 5. Experimental results are presented in Sec. 6 and the chapter finishes with a discussion and some conclusions in Sec. 7.

### **2. Control design setup**

2 Will-be-set-by-IN-TECH

loop. In this setup, the aim of the controller design is to satisfy performance requirements expressed as upper bounds on the norm (in case of suboptimal control) or minimize the norm (in optimal control) of the transfer function from w to q. Loosely speaking, a good controller should make the effect of w on q "small" (for suboptimal control) or "as small as possible" (for optimal control). The performance outputs usually consist of weighted versions of the controlled signal, the control error and the control effort. This is achieved by augmenting the original plant with output weighting functions. Good rejection of specific disturbances can be achieved in this framework by using a disturbance model as a weighting function in the transfer path from the performance input w to the performance output q, that is, by modeling the disturbance to be rejected as a weighted version of the performance input. This forces the maximum singular value *σ*max(G*qw*(j*ω*)) or, in the single-input single-output case, the amplitude response |*Gqw*(j*ω*)| of the open-loop transfer function to have a very high gain in the frequency regions specified by the disturbance model, or, loosely speaking, enlarges the effect of w on q in certain frequency regions. A reduction of the overall effect of w on q in closed loop will then be mostly achieved by reducing the effect in regions where it is large in open loop. From classical control arguments, it is intuitive that this requires a high loop gain in these frequency regions which in turn usually requires a high controller gain. A high loop gain will give a small sensitivity and in turn a good disturbance rejection (in specified

This control design setup is used in this chapter for the rejection of harmonic disturbances with time-varying frequencies. The control design problem is based on a generalized plant obtained through the introduction of a disturbance model that describes the harmonic disturbances and the addition of output weighting functions. Descriptions of the disturbance model in pLPV and in LPV-LFT form are used and lead to generalized plant descriptions that are also in pLPV or LPV-LFT form. Corresponding design methods are then employed to obtain controllers. For a plant in pLPV form, standard *H*∞ design [11] is used to compute a set of controllers. The gain scheduling is then achieved by interpolation between these controllers. For a plant in LPV-LFT form, the design method of Apkarian & Gahinet [1] is

LPV approaches for the rejection of harmonic disturbances have been used by Darengosse & Chevrel [7], Du & Shi [8], Du et al. [9], Bohn et al. [5, 6], Kinney & de Callafon [14, 15, 16], Köro ˘glu & Scherer [17], Witte et al. [19], Balini et al. [4], Heins et al. [12, 13], Ballesteros & Bohn [2, 3] and Shu et al. [18]. Darengosse & Chevrel [7], Du & Shi [8], Du et al. [9], Witte et al. [19], Balini et al. [4] suggested continuous-time LPV approaches. These approaches are tested for a single sinusoidal disturbance by Darengosse & Chevrel [7], Du et al. [9], Witte et al. [19] and Balini et al. [4]. Methods based on observer-based state-feedback controllers are presented by Bohn et al. [5, 6], Kinney & de Callafon [14, 15, 16] and Heins et al. [12, 13]. In the approach of Bohn et al. [5, 6], the observer gain is selected from a set of pre-computed gains by switching. In the other approaches of Kinney & de Callafon [16], Heins et al. [13] and in the previous chapter, the observer gain is calculated by interpolation. In the other approach presented in the previous chapter, which is also used by Kinney & de Callafon [14, 15] and Heins et al. [12], the state-feedback gain is scheduled using interpolation. A general output feedback LPV approach for the rejection of harmonic disturbances is suggested and applied

The existing LPV approaches can be classified by the control design technique used to obtain the controller. Approaches based on pLPV control design are used by Heins et al. [12, 13],

used that directly yields a gain-scheduled controller also in LPV-LFT form.

in real time by Ballesteros & Bohn [2, 3] and Shu et al. [18].

frequency regions).

In this section, pLPV systems and LPV-LFT systems are introduced and the control design for such systems is described in Sec. 2.1 and 2.2, respectively.

### **2.1. Control design for pLPV systems**

A pLPV system is of the form

$$
\begin{bmatrix}
\frac{\mathbf{x}\_{k+1}}{\mathbf{y}\_k}
\end{bmatrix} = \begin{bmatrix}
\frac{A(\boldsymbol{\theta})\,\mathrm{[B]}}{\mathbf{C}}
\end{bmatrix} \begin{bmatrix}
\frac{\mathbf{x}\_k}{\mathbf{u}\_k}
\end{bmatrix},
\tag{1}
$$

where the system matrix depends affinely on a parameter vector *θ*, that is

$$A(\theta) = \mathcal{A}\_0 + \theta\_1 \mathcal{A}\_1 + \theta\_2 \mathcal{A}\_2 + \dots + \theta\_N \mathcal{A}\_{N'} \tag{2}$$

**Figure 2.** General LPV-LFT system

with constant matrices *Ai*. The parameter vector *θ* varies in a polytope **Θ** with *M* vertices *<sup>v</sup><sup>j</sup>* <sup>∈</sup> **<sup>R</sup>***N*. A point *<sup>θ</sup>* <sup>∈</sup> **<sup>Θ</sup>** can be written as a convex combination of vertices, i.e. there exists a coordinate vector *λ* = [ *λ*<sup>1</sup> ··· *λ <sup>M</sup>* ] <sup>T</sup> <sup>∈</sup> **<sup>R</sup>***<sup>M</sup>* such that *<sup>θ</sup>* can be written as

$$\boldsymbol{\theta} = \sum\_{j=1}^{M} \lambda\_{j} \,\, \boldsymbol{v}\_{j} \tag{3}$$

*ș*

*G*

*u y*

*ș*

parameters of the controller. This control design method guarantees stability through the small gain theorem. It is often conservative, since the parameter ranges covered are usually

As stated in the previous section, to calculate the controller using the pLPV control design method, the generalized plant in pLPV form is needed. In this section, the steps to obtain the generalized plant in pLPV form are discussed. The disturbance model and a representation of the disturbance model in pLPV form are obtained in Sec. 3.1. In Sec. 3.2, the generalized plant is built by combining the plant, the disturbance model in pLPV form and the weighting

A general model for a harmonic disturbance with *n*<sup>d</sup> fixed frequencies is described by � A<sup>d</sup> B<sup>d</sup> C<sup>d</sup> 0

> Ad, 1 ··· **0** . .

. ... .

⎤ ⎥ <sup>⎦</sup> , <sup>B</sup>d, *<sup>i</sup>* <sup>=</sup>

Cd, 1 ··· Cd, *<sup>n</sup>*<sup>d</sup>

**0** ··· Ad, *<sup>n</sup>*<sup>d</sup>

. . ⎤ ⎥ <sup>⎦</sup> , <sup>A</sup>d, *<sup>i</sup>* <sup>=</sup>

> � 1 1 �

A harmonic disturbance can be modeled as the output of an unforced system with system matrix A<sup>d</sup> and output matrix C<sup>d</sup> given above in (7) and (10). An input matrix is not

� and <sup>C</sup>d, *<sup>i</sup>* = �

�

� 0 1 −1 *ai*

1 0�

*ai* = 2cos(2*π fiT*), (8)

�

, (7)

. (10)

, (9)

(6)

69

LPV Gain-Scheduled Output Feedback

T

*w w*T

**Figure 3.** LPV-LFT gain-scheduling control structure

**3. Generalized plant in pLPV form**

functions.

with

**3.1. Disturbance model**

larger than the ones that may occur in the real system.

A<sup>d</sup> =

B<sup>d</sup> =

<sup>C</sup><sup>d</sup> = �

⎡ ⎢ ⎣

⎡ ⎢ ⎣

Bd, 1 . . . Bd, *<sup>n</sup>*<sup>d</sup> *K*

*q w*

T

T*<sup>q</sup> <sup>q</sup>*

for Active Control of Harmonic Disturbances with Time-Varying Frequencies

with

$$
\lambda\_j \ge 0, \sum\_{j=1}^M \lambda\_j = 1. \tag{4}
$$

Defining *A*v, *<sup>j</sup>* = *A*(*vj*) for *j* = 1, ..., *M*, the system matrix *A*(*θ*) can be represented as

$$A(\theta) = A(\lambda) = \lambda\_1 A\_{\text{V},1} + \lambda\_2 A\_{\text{V},2} + \dots + \lambda\_M A\_{\text{V},M}.\tag{5}$$

The system matrix of a pLPV system *A*(*θ*) can be calculated from the *M* vertices of the polytope **Θ** by finding the coordinate vector *λ* that fulfills the conditions of (3) and (4).

Once a representation of a system is obtained in pLPV form, it is possible to find a controller using *H*<sup>∞</sup> or *H*<sup>2</sup> techniques for each vertex of the polytope. The controller for a given *θ ∈* **Θ** can be calculated through controllers for the vertex systems. The closed-loop stability is guaranteed even for arbitrarily fast changes of the scheduling parameters if a parameter-independent Lyapunov function is used (for the whole polytope) in the control design. This approach, however, is conservative because fast variations of the scheduling parameters are considered, which might not occur in a practical application. Parameter-dependent Lyapunov functions can be used to include bounds on the rate of change of the parameters, but are not considered here.

### **2.2. Control design for LPV-LFT systems**

An LPV system in LFT form is shown in Fig. 2. It consists of a generalized plant G that includes input and output weighting functions and a parametric uncertainty block *θ* that has been "pulled out" of the system. For this general system, a gain-scheduling controller can be calculated following the method presented in Apkarian & Gahinet [1]. In this method, two sets of linear matrix inequalities (LMIs) are solved. The first set of LMIs determines the feasibility of the problem which means that a bound on the control system performance in the sense of the *H*∞ norm can be satisfied. With the second set of LMIs, the controller matrices are calculated from the solution of the first set of LMIs.

As a result of applying this control design method, the gain-scheduling control structure of Fig. 3 is obtained. The time-varying plant parameters are directly used as the gain-scheduling

**Figure 3.** LPV-LFT gain-scheduling control structure

parameters of the controller. This control design method guarantees stability through the small gain theorem. It is often conservative, since the parameter ranges covered are usually larger than the ones that may occur in the real system.

### **3. Generalized plant in pLPV form**

As stated in the previous section, to calculate the controller using the pLPV control design method, the generalized plant in pLPV form is needed. In this section, the steps to obtain the generalized plant in pLPV form are discussed. The disturbance model and a representation of the disturbance model in pLPV form are obtained in Sec. 3.1. In Sec. 3.2, the generalized plant is built by combining the plant, the disturbance model in pLPV form and the weighting functions.

### **3.1. Disturbance model**

A general model for a harmonic disturbance with *n*<sup>d</sup> fixed frequencies is described by

$$
\left[\frac{A\_{\rm d} \left| B\_{\rm d} \right|}{C\_{\rm d} \left| \right|}\right] \tag{6}
$$

with

4 Will-be-set-by-IN-TECH

*u y*

with constant matrices *Ai*. The parameter vector *θ* varies in a polytope **Θ** with *M* vertices *<sup>v</sup><sup>j</sup>* <sup>∈</sup> **<sup>R</sup>***N*. A point *<sup>θ</sup>* <sup>∈</sup> **<sup>Θ</sup>** can be written as a convex combination of vertices, i.e. there exists a

> *M* ∑ *j*=1

The system matrix of a pLPV system *A*(*θ*) can be calculated from the *M* vertices of the polytope **Θ** by finding the coordinate vector *λ* that fulfills the conditions of (3) and (4).

Once a representation of a system is obtained in pLPV form, it is possible to find a controller using *H*<sup>∞</sup> or *H*<sup>2</sup> techniques for each vertex of the polytope. The controller for a given *θ ∈* **Θ** can be calculated through controllers for the vertex systems. The closed-loop stability is guaranteed even for arbitrarily fast changes of the scheduling parameters if a parameter-independent Lyapunov function is used (for the whole polytope) in the control design. This approach, however, is conservative because fast variations of the scheduling parameters are considered, which might not occur in a practical application. Parameter-dependent Lyapunov functions can be used to include bounds on the rate of

An LPV system in LFT form is shown in Fig. 2. It consists of a generalized plant G that includes input and output weighting functions and a parametric uncertainty block *θ* that has been "pulled out" of the system. For this general system, a gain-scheduling controller can be calculated following the method presented in Apkarian & Gahinet [1]. In this method, two sets of linear matrix inequalities (LMIs) are solved. The first set of LMIs determines the feasibility of the problem which means that a bound on the control system performance in the sense of the *H*∞ norm can be satisfied. With the second set of LMIs, the controller matrices are

As a result of applying this control design method, the gain-scheduling control structure of Fig. 3 is obtained. The time-varying plant parameters are directly used as the gain-scheduling

*<sup>G</sup> <sup>w</sup> w*T

> *θ* = *M* ∑ *j*=1

*λ<sup>j</sup>* ≥ 0,

Defining *A*v, *<sup>j</sup>* = *A*(*vj*) for *j* = 1, ..., *M*, the system matrix *A*(*θ*) can be represented as

**Figure 2.** General LPV-LFT system

coordinate vector *λ* = [ *λ*<sup>1</sup> ··· *λ <sup>M</sup>* ]

change of the parameters, but are not considered here.

**2.2. Control design for LPV-LFT systems**

calculated from the solution of the first set of LMIs.

with

*ș*

T*<sup>q</sup> <sup>q</sup>*

<sup>T</sup> <sup>∈</sup> **<sup>R</sup>***<sup>M</sup>* such that *<sup>θ</sup>* can be written as

*A*(*θ*) = *A*(*λ*) = *λ*1*A*v, 1 + *λ*2*A*v, 2 + ... + *λMA*v, *<sup>M</sup>*. (5)

*λ<sup>j</sup> v<sup>j</sup>* (3)

*λ<sup>j</sup>* = 1. (4)

$$\mathbf{A}\_{\mathbf{d}} = \begin{bmatrix} \mathbf{A}\_{\mathbf{d},1} \cdot \cdots & \mathbf{0} \\ \vdots & \ddots & \vdots \\ \mathbf{0} & \cdots & \mathbf{A}\_{\mathbf{d},n\_{\mathbf{d}}} \end{bmatrix}, \mathbf{A}\_{\mathbf{d},i} = \begin{bmatrix} 0 & 1 \\ -1 & a\_i \end{bmatrix},\tag{7}$$

$$a\_i = 2\cos(2\pi f\_i T),\tag{8}$$

$$\mathbf{B}\_{\mathbf{d}} = \begin{bmatrix} \mathbf{B}\_{\mathbf{d},1} \\ \vdots \\ \mathbf{B}\_{\mathbf{d},n\_{\mathbf{d}}} \end{bmatrix}, \mathbf{B}\_{\mathbf{d},i} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \tag{9}$$

$$\mathbf{C\_{d}} = \begin{bmatrix} \mathbf{C\_{d,1}} \cdot \cdots \cdot \mathbf{C\_{d,n\_d}} \end{bmatrix} \text{ and } \mathbf{C\_{d,i}} = \begin{bmatrix} 1 \ 0 \end{bmatrix}. \tag{10}$$

A harmonic disturbance can be modeled as the output of an unforced system with system matrix A<sup>d</sup> and output matrix C<sup>d</sup> given above in (7) and (10). An input matrix is not

### 6 Will-be-set-by-IN-TECH 70 Advances on Analysis and Control of Vibrations – Theory and Applications LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies <sup>7</sup>

required. However, in the generalized plant setup, a performance input is required and the disturbance model acts as an input weighting function on the performance input. This is why the disturbance model above has been given with a nonzero input matrix B<sup>d</sup> in (9).

The frequency in (8) is fixed and denoted by *fi*. As in Sec. 4 of the previous chapter, the pLPV disturbance model for *n*<sup>d</sup> time-varying frequencies *fj*, *<sup>k</sup>* ∈ [ *f*min, *<sup>j</sup>*, *f*max, *<sup>j</sup>*], *j* = 1, 2, . . . , *n*d, is defined as

$$
\left[\frac{\mathbf{A\_d^{(p\text{LPV})}}(\theta)\Big|\mathcal{B\_d^{(p\text{LPV})}}}{\mathcal{C\_d^{(p\text{LPV})}}\Big|\begin{array}{c} \\ 0 \end{array}}\right] \tag{11}
$$

A*i*(θ) =

� B(pLPV) *<sup>w</sup>* <sup>B</sup>(pLPV) *u*

� C(pLPV) *q* C(pLPV) *y*

and

in the following section.

*w*d

(pLPV)

v,

*C* **0**

**4. Generalized plant in LPV-LFT form**

plant, harmonic disturbance and weighting functions.

*i i*

¦ *A B*

d

O

1

*i*

*M*

⎡ ⎢ ⎢ ⎢ ⎣

B(pLPV)

� = ⎡ ⎢ ⎣

*D*(pLPV) *yw <sup>D</sup>*(pLPV)

� D(pLPV) *qw* <sup>D</sup>(pLPV) *qu*

*u*<sup>p</sup> <sup>+</sup>

**Figure 4.** Plant with pLPV disturbance model and weighting functions

(pLPV)

d

d *y*

A<sup>p</sup> BpC(pLPV)

� =

*D*(pLPV)

*yu* � = ⎡ ⎢ ⎣

Once the generalized plant is obtained, the controller can be calculated using the algorithms

<sup>d</sup> 0 0

0 Bp

*Wy* 0

*Wu*

⎤ ⎥

*<sup>C</sup>* ½

*C*

*C*

(pLPV) (pLPV)

*W W*

*A B*

*W W D*

(pLPV) (pLPV) *y y*

*y y*

(pLPV) (pLPV)

*W W*

*A B*

*W W D*

(pLPV) (pLPV) *u u*

*u u*

⎤ ⎥

0 00 C(pLPV)

Cp 00 0

<sup>p</sup> *y* <sup>p</sup> <sup>p</sup>

<sup>p</sup> **0** *A B*

The same steps as in the previous section are carried out, but in this section the generalized plant in LPV-LFT form is obtained such that the control design method of Apkarian & Gahinet [1] can be used. The model of the harmonic disturbance and the generalized plant in LFT form are obtained in Sec. 4.1 and 4.2, respectively. The generalized plant is the result of combining

*Wy* 0

for Active Control of Harmonic Disturbances with Time-Varying Frequencies

*Wu*

⎤ ⎥ ⎥ ⎥ ⎤ ⎥ ⎥ ⎥

<sup>⎦</sup> , (19)

LPV Gain-Scheduled Output Feedback

71

<sup>⎦</sup> , (20)

<sup>⎦</sup> (21)

¾ ¿ *q*

<sup>⎦</sup> . (22)

0 Av, *<sup>i</sup>* 0 0

0 00 A(pLPV)

B(pLPV) <sup>d</sup> 0 0 0 0 B(pLPV) *Wu*

*Wy* <sup>C</sup><sup>p</sup> <sup>0</sup> <sup>C</sup>(pLPV)

*Wy* <sup>C</sup><sup>p</sup> <sup>0</sup> <sup>A</sup>(pLPV)

⎡ ⎢ ⎢ ⎢ ⎣

with

$$\mathcal{A}\_{\rm d}^{\rm (\rm dIV)}(\theta) = \mathcal{A}\_{\rm d,0} + \theta\_1 \mathcal{A}\_{\rm d,1} + \dots + \theta\_{\rm n\_d} \mathcal{A}\_{\rm d, n\_d}.\tag{12}$$

As in Sec. 2.1, (12) can be written in the form of

$$\mathbf{A}\_{\rm d}^{\rm (pIV)}(\theta) = \mathbf{A}\_{\rm d}^{\rm (pIV)}(\lambda) = \lambda\_1 \mathbf{A}\_{\rm v, 1} + \dots + \lambda\_M \mathbf{A}\_{\rm v, M} = \sum\_{i=1}^{M} \lambda\_i \mathbf{A}\_{\rm v, i\nu} \tag{13}$$

where the matrices <sup>A</sup>v, *<sup>i</sup>* are defined in the same way as <sup>A</sup>(pLPV) <sup>d</sup> in (7) and (8), but with *ai* evaluated for all the vertices of the polytope, with *j* = 1, 2, . . . , *n*d. The coordinate vector λ can be calculated using the method described in Sec. 4.4 of the previous chapter.

### **3.2. Generalized plant**

A state-space representation of the plant is given by

$$G\_{\rm P} = \begin{bmatrix} \frac{A\_{\rm P} \parallel B\_{\rm P}}{C\_{\rm P} \parallel D\_{\rm P}} \end{bmatrix} \tag{14}$$

and it is assumed that the disturbance is acting on the input of the plant.

The block diagram of the generalized plant with the disturbance, the plant and the weighting functions

$$\mathcal{W}\_{\mathcal{Y}}^{\text{(p1PV)}} = \left[ \frac{\mathcal{A}\_{W\_{\mathcal{Y}}}^{\text{(p1PV)}} \, \big|\, \mathcal{B}\_{W\_{\mathcal{Y}}}^{\text{(p1PV)}}}{\mathcal{C}\_{W\_{\mathcal{Y}}}^{\text{(p1PV)}} \, \big|\, \mathcal{D}\_{W\_{\mathcal{Y}}}^{\text{(p1PV)}}} \right] \,, \tag{15}$$

$$\mathcal{W}\_{\boldsymbol{\mu}}^{\text{(p1.PV)}} = \begin{bmatrix} \mathbf{A}\_{\boldsymbol{W}\_{\boldsymbol{\mu}}}^{\text{(p1.PV)}} \, \big|\, \mathbf{B}\_{\boldsymbol{W}\_{\boldsymbol{\mu}}}^{\text{(p1.PV)}} \\ \hline \mathbf{C}\_{\boldsymbol{W}\_{\boldsymbol{\mu}}}^{\text{(p1.PV)}} \, \big|\, \boldsymbol{D}\_{\boldsymbol{W}\_{\boldsymbol{\mu}}}^{\text{(p1.PV)}} \end{bmatrix} \tag{16}$$

is illustrated in Fig. 4.

For every vertex of the polytopic system, the generalized plant can be described by

$$
\begin{bmatrix} x\_{k+1} \\ \mathbf{q}\_k \\ \mathbf{y}\_k \end{bmatrix} = \begin{bmatrix} \mathbf{A}\_i(\boldsymbol{\theta}) \begin{vmatrix} \mathbf{B}\_w^{(\text{p1FV})} & \mathbf{B}\_u^{(\text{p1FV})} \\ \mathbf{C}\_q^{(\text{p1PV})} \mathbf{D}\_{qw}^{(\text{p1PV})} & \mathbf{D}\_{qu}^{(\text{p1FV})} \\ \mathbf{C}\_y^{(\text{p1PV})} \begin{vmatrix} \mathbf{D}\_{yw}^{(\text{p1PV})} & \mathbf{D}\_{yu}^{(\text{p1PV})} \end{vmatrix} \end{bmatrix} \begin{bmatrix} \mathbf{x}\_k \\ \mathbf{w}\_k \\ \boldsymbol{\mu}\_k \end{bmatrix} \tag{17}
$$

where

$$\boldsymbol{x}\_{k} = \begin{bmatrix} \boldsymbol{x}\_{\mathrm{p},k}^{\mathrm{T}} \ \boldsymbol{x}\_{\mathrm{d},k}^{\mathrm{T}} \ \boldsymbol{x}\_{\mathrm{W}\_{\mathrm{y}},k}^{\mathrm{T}} \ \boldsymbol{x}\_{\mathrm{W}\_{\mathrm{u}},k}^{\mathrm{T}} \end{bmatrix}^{\mathrm{T}} \tag{18}$$

LPV Gain-Scheduled Output Feedback

70 Advances on Analysis and Control of Vibrations – Theory and Applications LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies <sup>7</sup> 71 for Active Control of Harmonic Disturbances with Time-Varying Frequencies

$$A\_{i}(\boldsymbol{\theta}) = \begin{bmatrix} \boldsymbol{A}\_{\text{P}} & \boldsymbol{B}\_{\text{P}} \mathbf{C}\_{\text{d}}^{\text{(p1PV)}} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{A}\_{\text{V}\_{i}i} & \mathbf{0} & \mathbf{0} \\ \boldsymbol{B}\_{\text{W}\_{\text{y}}}^{\text{(p1PV)}} \mathbf{C}\_{\text{P}} & \mathbf{0} & \boldsymbol{A}\_{\text{W}\_{\text{y}}}^{\text{(p1PV)}} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \boldsymbol{A}\_{\text{W}\_{\text{y}}}^{\text{(p1PV)}} \end{bmatrix}' \tag{19}$$

$$
\begin{bmatrix}
\mathbf{B}\_{\textit{w}}^{\langle \text{p1PV} \rangle} & \mathbf{B}\_{\textit{u}}^{\langle \text{p1PV} \rangle}
\end{bmatrix} = \begin{bmatrix}
\mathbf{0} & \mathbf{B}\_{\textit{p}} \\
\mathbf{B}\_{\textit{d}}^{\langle \text{p1PV} \rangle} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{B}\_{\textit{W}\_{\textit{u}}}^{\langle \text{p1PV} \rangle}
\end{bmatrix}, \tag{20}
$$

$$
\begin{bmatrix}
\mathbf{C}\_{q}^{\langle \text{pIV} \rangle} \\
\mathbf{C}\_{y}^{\langle \text{pIV} \rangle}
\end{bmatrix} = \begin{bmatrix}
D\_{W\_{y}}^{\langle \text{pIV} \rangle} & \mathbf{C}\_{\text{P}} & \mathbf{0} & \mathbf{C}\_{W\_{y}}^{\langle \text{pIV} \rangle} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C}\_{W\_{y}}^{\langle \text{pIV} \rangle} \\
\hline
\mathbf{C}\_{\text{P}} & \mathbf{0} & \mathbf{0} & \mathbf{0}
\end{bmatrix} \tag{21}
$$

and

6 Will-be-set-by-IN-TECH

required. However, in the generalized plant setup, a performance input is required and the disturbance model acts as an input weighting function on the performance input. This is why

The frequency in (8) is fixed and denoted by *fi*. As in Sec. 4 of the previous chapter, the pLPV disturbance model for *n*<sup>d</sup> time-varying frequencies *fj*, *<sup>k</sup>* ∈ [ *f*min, *<sup>j</sup>*, *f*max, *<sup>j</sup>*], *j* = 1, 2, . . . , *n*d, is

> <sup>d</sup> (θ) <sup>B</sup>(pLPV) d

<sup>d</sup> (λ) = *λ*1Av, 1 + ··· + *λM*Av, *<sup>M</sup>* =

evaluated for all the vertices of the polytope, with *j* = 1, 2, . . . , *n*d. The coordinate vector λ

The block diagram of the generalized plant with the disturbance, the plant and the weighting

A(pLPV) *Wy* <sup>B</sup>(pLPV) *Wy*

C(pLPV) *Wy <sup>D</sup>*(pLPV) *Wy*

C(pLPV) *Wu <sup>D</sup>*(pLPV) *Wu*

<sup>A</sup>*i*(θ) <sup>B</sup>(pLPV)

d, *<sup>k</sup>* <sup>x</sup><sup>T</sup>

*<sup>w</sup>* <sup>B</sup>(pLPV) *u*

*qw* <sup>D</sup>(pLPV) *qu*

*yw <sup>D</sup>*(pLPV) *yu*

*Wy*, *<sup>k</sup>* <sup>x</sup><sup>T</sup> *Wu*, *k* �T

⎡ ⎣

> � A(pLPV) *Wu* <sup>B</sup>(pLPV) *Wu*

For every vertex of the polytopic system, the generalized plant can be described by

C(pLPV) *<sup>q</sup>* <sup>D</sup>(pLPV)

C(pLPV) *<sup>y</sup> <sup>D</sup>*(pLPV) �

⎤

�

⎤ ⎥ ⎥ ⎦

⎡ ⎣ x*k wk uk* ⎤

can be calculated using the method described in Sec. 4.4 of the previous chapter.

*G*<sup>p</sup> = � Ap Bp Cp *D*p

and it is assumed that the disturbance is acting on the input of the plant.

*W*(pLPV) *<sup>y</sup>* =

*W*(pLPV) *<sup>u</sup>* =

> ⎡ ⎢ ⎢ ⎣

�

<sup>d</sup> (θ) = *A*d, 0 + *θ*1*A*d, 1 + ··· + *θn*<sup>d</sup> *A*d, *<sup>n</sup>*<sup>d</sup> . (12)

*M* ∑ *i*=1

*λi*Av, *<sup>i</sup>*, (13)

<sup>d</sup> in (7) and (8), but with *ai*

⎦ , (15)

⎦ (17)

, (18)

(11)

(14)

(16)

the disturbance model above has been given with a nonzero input matrix B<sup>d</sup> in (9).

C(pLPV) <sup>d</sup> 0

� A(pLPV)

A(pLPV)

As in Sec. 2.1, (12) can be written in the form of

<sup>d</sup> (θ) = <sup>A</sup>(pLPV)

A state-space representation of the plant is given by

⎡ ⎣ x*k*+<sup>1</sup> q*k yk*

⎤ ⎦ =

x*<sup>k</sup>* = � xT p, *<sup>k</sup>* <sup>x</sup><sup>T</sup>

where the matrices <sup>A</sup>v, *<sup>i</sup>* are defined in the same way as <sup>A</sup>(pLPV)

A(pLPV)

**3.2. Generalized plant**

functions

where

is illustrated in Fig. 4.

defined as

with

$$
\begin{bmatrix} D\_{qw}^{\langle \text{p1PV} \rangle} D\_{q\mu}^{\langle \text{p1PV} \rangle} \\ D\_{yw}^{\langle \text{p1PV} \rangle} D\_{y\mu}^{\langle \text{p1PV} \rangle} \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 \ D\_{W\_y}^{\langle \text{p1PV} \rangle} \\ \overline{0 \ 0} & 0 \end{bmatrix} . \tag{22}
$$

Once the generalized plant is obtained, the controller can be calculated using the algorithms in the following section.

**Figure 4.** Plant with pLPV disturbance model and weighting functions

### **4. Generalized plant in LPV-LFT form**

The same steps as in the previous section are carried out, but in this section the generalized plant in LPV-LFT form is obtained such that the control design method of Apkarian & Gahinet [1] can be used. The model of the harmonic disturbance and the generalized plant in LFT form are obtained in Sec. 4.1 and 4.2, respectively. The generalized plant is the result of combining plant, harmonic disturbance and weighting functions.

### 8 Will-be-set-by-IN-TECH 72 Advances on Analysis and Control of Vibrations – Theory and Applications LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies <sup>9</sup>

### **4.1. Disturbance model**

The state-space representation of a harmonic disturbance for *n*<sup>d</sup> fixed frequencies was given by (6-10). If the frequencies of a harmonic disturbance change between minimal values *fi*, min and maximal values *fi*, max, a representation for the variations of the frequencies is given by

$$a\_i(f\_i) = 2\cos(2\pi f\_i T) = \overline{a}\_i + p\_i \overline{\theta}\_{i,k}(f\_i) \tag{23}$$

with

$$
\overline{a}\_{i} = \cos(2\pi f\_{i,\text{max}}T) + \cos(2\pi f\_{i,\text{min}}T),
\tag{24}
$$

$$p\_i = \cos(2\pi f\_{i,\max} T) - \cos(2\pi f\_{i,\min} T) \tag{25}$$

and

$$
\overline{\theta}\_{i,k} \in [-1, 1]. \tag{26}
$$

**4.2. Generalized plant**

with

⎡ ⎢ ⎢ ⎣

x*k*+<sup>1</sup> q*θ*, *<sup>k</sup>* q*k yk*

⎤ ⎥ ⎥ ⎦ =

x*<sup>k</sup>* = � xT p, *<sup>k</sup>* <sup>x</sup><sup>T</sup>

> ⎡ ⎢ ⎢ ⎢ ⎣

B(LFT)

*<sup>w</sup>* <sup>B</sup>(LFT) *u* � =

> ⎡ ⎢ ⎢ ⎢ ⎣

> > +

d *y*

*ș*

D(LFT)

A =

<sup>B</sup>*<sup>θ</sup>* <sup>B</sup>(LFT)

�

⎡ ⎢ ⎣

<sup>p</sup> *u u*

*w w*<sup>d</sup>

*w*T

C*<sup>θ</sup>* C(LFT) *q* C(LFT) *y*

⎤ ⎥ <sup>⎦</sup> <sup>=</sup>

*G*d

**Figure 5.** Plant with LPV-LFT disturbance model and weighting functions

⎡ ⎢ ⎢ ⎢ ⎣

C(LFT)

C(LFT)

The generalized plant is the result of combining the plant, the harmonic disturbance and the weighting functions and it is shown in Fig. 5. The weighting functions are defined the same way as in (15) and (16). A representation of the generalized plant in LFT form is given by

<sup>A</sup> <sup>B</sup>*<sup>θ</sup>* <sup>B</sup>(LFT)

*<sup>q</sup>* <sup>D</sup>*q<sup>θ</sup>* <sup>D</sup>(LFT)

*<sup>y</sup>* <sup>D</sup>*y<sup>θ</sup>* <sup>D</sup>(LFT)

d, *<sup>k</sup>* <sup>x</sup><sup>T</sup>

*Wy* <sup>C</sup><sup>p</sup> <sup>0</sup> <sup>A</sup>(LFT)

⎡ ⎢ ⎢ ⎣

A<sup>p</sup> BpCd, *<sup>y</sup>* 0 0 0 A<sup>d</sup> 0 0

0 00 A(LFT)

C*<sup>θ</sup>* D*θθ* D*θ<sup>w</sup>* D*θ<sup>u</sup>*

*<sup>w</sup>* <sup>B</sup>(LFT) *u*

⎤ ⎥ ⎥ ⎥ ⎦

for Active Control of Harmonic Disturbances with Time-Varying Frequencies

⎡ ⎢ ⎢ ⎣

x*k* w*θ*, *<sup>k</sup>* w*<sup>k</sup> uk*

⎤ ⎥ ⎥ ⎦

, (38)

LPV Gain-Scheduled Output Feedback

<sup>⎦</sup> , (39)

<sup>⎦</sup> , (40)

½ ° ° ¾ ° ° ¿

*q*

T*q*

*y*

(37)

73

(41)

*qw* <sup>D</sup>(LFT) *qu*

*yw* <sup>D</sup>(LFT) *yu*

*Wy* 0

0 0 Bp Bd, *<sup>θ</sup>* Bd, *<sup>w</sup>* 0 00 0 0 0 B(LFT)

0 Cd, *<sup>θ</sup>* 0 0

0 00 C(LFT)

Cp 00 0

*Wy* <sup>C</sup><sup>p</sup> <sup>0</sup> <sup>C</sup>(LFT)

*Wu*

*Wu*

*Wu*

*Wy* 0

p *y*

*G*<sup>p</sup> *Wy*

*Wu*

⎤ ⎥ ⎥ ⎥

> ⎤ ⎥ ⎥

> > ⎤ ⎥ ⎥ ⎥ ⎦

*Wy*, *<sup>k</sup>* <sup>x</sup><sup>T</sup> *Wu*, *k* �T

An LPV-LFT model of the disturbance can be written as

$$x\_{\rm d,k+1} = A\_{\rm d} x\_{\rm d,k} + B\_{\rm d,\theta} w\_{\theta,k} + B\_{\rm d,w} w\_{\rm d,k\prime} \tag{27}$$

$$\mathbf{q}\_{\theta,k} = \mathbf{C}\_{\mathbf{d},\theta} \mathbf{x}\_{\mathbf{d},k\prime} \tag{28}$$

$$y\_{\mathbf{d},k} = C\_{\mathbf{d},y} x\_{\mathbf{d},k\prime} \tag{29}$$

$$w\_{\theta,k} = \overline{\theta}\_k q\_{\theta,k} \tag{30}$$

with

$$\mathbf{A}\_{\mathrm{d}} = \begin{bmatrix} \mathbf{A}\_{\mathrm{d},1} & \cdots & \mathbf{0} \\ \vdots & \ddots & \vdots \\ & \mathbf{0} & \cdots & \mathbf{A}\_{\mathrm{d},n\_{\mathrm{d}}} \end{bmatrix}, \begin{array}{c} \mathbf{A}\_{\mathrm{d},i} = \begin{bmatrix} 0 & 1 \\ -1 & \overline{a}\_{i} \end{bmatrix}, \\\ \mathbf{0} & \cdots & \mathbf{A}\_{\mathrm{d},n\_{\mathrm{d}}} \end{array} \tag{31}$$

$$\mathbf{B}\_{\mathbf{d},\theta} = \begin{bmatrix} \mathbf{B}\_{\mathbf{d},\theta,1} & \cdots & \mathbf{0} \\ \vdots & \ddots & \vdots \\ \mathbf{0} & \cdots & \mathbf{B}\_{\mathbf{d},\theta,\eta\_{\mathbf{d}}} \end{bmatrix} \text{ } \mathbf{B}\_{\mathbf{d},\theta,i} = \begin{bmatrix} \mathbf{0} \\ p\_i \end{bmatrix} \text{ }\tag{32}$$

$$\mathbf{B}\_{\mathbf{d},w} = \begin{bmatrix} \mathbf{B}\_{\mathbf{d},w,1} \\ \vdots \\ \mathbf{B}\_{\mathbf{d},w,n\_{\mathbf{d}}} \end{bmatrix} \text{ } \mathbf{B}\_{\mathbf{d},w,i} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \text{ } \tag{33}$$

$$\mathbf{C}\_{\mathbf{d},\theta} = \begin{bmatrix} \mathbf{C}\_{\mathbf{d},\theta,1} \cdot \cdots & \mathbf{0} \\\\ \vdots & \ddots & \vdots \\\\ \mathbf{1} & \mathbf{0} \end{bmatrix} \text{ \(\mathbf{C}\_{\mathbf{d},\theta,i} = \begin{bmatrix} \mathbf{0} \ \mathbf{1} \end{bmatrix} \text{)}\tag{34}$$

$$\begin{array}{c} \mathsf{L} \quad \mathsf{0} \quad \cdots \mathsf{C}\_{\mathsf{d},\theta,n\_{\mathsf{d}}} \mathsf{L} \\ \mathsf{C}\_{\mathsf{d},y} = \left[ \mathsf{C}\_{\mathsf{d},y,1} \, \cdots \, \mathsf{C}\_{\mathsf{d},y,n\_{\mathsf{d}}} \right] \, \mathsf{C}\_{\mathsf{d},y,i} = \left[ \begin{array}{c} \mathtt{1} \ \mathsf{0} \end{array} \right] \end{array} \tag{35}$$

and

$$
\overline{\theta}\_k = \begin{bmatrix}
\overline{\theta}\_{1,k} & \cdots & 0 \\
\vdots & \ddots & \vdots \\
0 & \cdots & \overline{\theta}\_{n\_d,k}
\end{bmatrix} \tag{36}
$$

### **4.2. Generalized plant**

The generalized plant is the result of combining the plant, the harmonic disturbance and the weighting functions and it is shown in Fig. 5. The weighting functions are defined the same way as in (15) and (16). A representation of the generalized plant in LFT form is given by

$$
\begin{bmatrix} x\_{k+1} \\ \mathbf{q}\_{\theta,k} \\ \mathbf{q}\_k \\ \mathbf{y}\_k \end{bmatrix} = \begin{bmatrix} \mathbf{A} & \mathbf{B}\_{\theta} & \mathbf{B}\_{w}^{\text{(\text{.}F\text{)}}} & \mathbf{B}\_{u}^{\text{(\text{.}F\text{)}}} \\ \hline \mathbf{C}\_{\theta} & \mathbf{D}\_{\theta\theta} & \mathbf{D}\_{\theta w} & \mathbf{D}\_{\theta u} \\ \mathbf{C}\_{q}^{\text{(\text{.}F\text{)}}} & \mathbf{D}\_{q\theta} & \mathbf{D}\_{qw}^{\text{(\text{.}F\text{)}}} & \mathbf{D}\_{qu}^{\text{(\text{.}F\text{)}}} \\ \mathbf{C}\_{y}^{\text{(\text{.}F\text{)}}} & \mathbf{D}\_{y\theta} & \mathbf{D}\_{yw}^{\text{(\text{.}F\text{)}}} & \mathbf{D}\_{yu}^{\text{(\text{.}F\text{)}}} \end{bmatrix} \begin{bmatrix} x\_{k} \\ w\_{\theta,k} \\ w\_{k} \\ u\_{k} \end{bmatrix} \tag{37}
$$

with

8 Will-be-set-by-IN-TECH

The state-space representation of a harmonic disturbance for *n*<sup>d</sup> fixed frequencies was given by (6-10). If the frequencies of a harmonic disturbance change between minimal values *fi*, min and maximal values *fi*, max, a representation for the variations of the frequencies is given by

*ai*(*fi*) = 2 cos(2*π fiT*) = *ai* + *piθi*, *<sup>k</sup>*(*fi*) (23)

*ai* = cos(2*π fi*, max*T*) + cos(2*π fi*, min*T*), (24) *pi* = cos(2*π fi*, max*T*) − cos(2*π fi*, min*T*) (25)

xd, *<sup>k</sup>*+<sup>1</sup> = Adxd, *<sup>k</sup>* + Bd, *<sup>θ</sup>*w*θ*, *<sup>k</sup>* + Bd, *ww*d, *<sup>k</sup>*, (27)

*θi*, *<sup>k</sup>* ∈ [−1, 1]. (26)

q*θ*, *<sup>k</sup>* = Cd, *<sup>θ</sup>*xd, *<sup>k</sup>*, (28) *y*d, *<sup>k</sup>* = Cd, *<sup>y</sup>*xd, *<sup>k</sup>*, (29) w*θ*, *<sup>k</sup>* = θ*k*q*θ*, *<sup>k</sup>* (30)

�

� 0 *pi* �

0 1�

, (31)

, (32)

, (33)

, (34)

1 0� (35)

<sup>⎦</sup> . (36)

� 0 1 −1 *ai*

**4.1. Disturbance model**

An LPV-LFT model of the disturbance can be written as

A<sup>d</sup> =

Bd, *<sup>θ</sup>* =

Cd, *<sup>θ</sup>* =

<sup>C</sup>d, *<sup>y</sup>* = �

⎡ ⎢ ⎢ ⎢ ⎣

> ⎡ ⎢ ⎢ ⎢ ⎣

Bd, *<sup>w</sup>* =

⎡ ⎢ ⎢ ⎢ ⎣

. .

*θ<sup>k</sup>* =

. .

> . .

> > ⎡ ⎢ ⎢ ⎢ ⎣

Ad, 1 ··· **0**

. ... .

**0** ··· Ad, *<sup>n</sup>*<sup>d</sup>

Bd, *<sup>θ</sup>*, 1 ··· **0**

. ... .

**0** ··· Bd, *<sup>θ</sup>*, *<sup>n</sup>*<sup>d</sup>

Bd, *<sup>w</sup>*, 1 . . . Bd, *<sup>w</sup>*, *<sup>n</sup>*<sup>d</sup>

Cd, *<sup>θ</sup>*, 1 ··· **0**

. ... .

**0** ··· Cd, *<sup>θ</sup>*, *<sup>n</sup>*<sup>d</sup>

Cd, *<sup>y</sup>*, 1 ··· Cd, *<sup>y</sup>*, *<sup>n</sup>*<sup>d</sup>

⎡ ⎢ ⎢ ⎢ ⎣

. . . ... . . .

. . ⎤ ⎥ ⎥ ⎥ <sup>⎦</sup> , <sup>A</sup>d, *<sup>i</sup>* <sup>=</sup>

. .

⎤ ⎥ ⎥ ⎥

> . .

*θ*1, *<sup>k</sup>* ··· 0

0 ··· *θn*d, *<sup>k</sup>*

⎤ ⎥ ⎥ ⎥

<sup>⎦</sup> , <sup>B</sup>d, *<sup>w</sup>*, *<sup>i</sup>* <sup>=</sup>

⎤ ⎥ ⎥ ⎥

�

<sup>⎦</sup> , <sup>B</sup>d, *<sup>θ</sup>*, *<sup>i</sup>* <sup>=</sup>

<sup>⎦</sup> , <sup>C</sup>d, *<sup>θ</sup>*, *<sup>i</sup>* <sup>=</sup> �

, <sup>C</sup>d, *<sup>y</sup>*, *<sup>i</sup>* = �

⎤ ⎥ ⎥ ⎥ � 1 1 �

with

and

with

and

$$\boldsymbol{x}\_{k} = \begin{bmatrix} \boldsymbol{x}\_{\mathrm{p},k}^{\mathrm{T}} \ \boldsymbol{x}\_{\mathrm{d},k}^{\mathrm{T}} \ \boldsymbol{x}\_{\mathrm{W}\_{\mathrm{y}},k}^{\mathrm{T}} \ \boldsymbol{x}\_{\mathrm{W}\_{\mathrm{u}},k}^{\mathrm{T}} \end{bmatrix}^{\mathrm{T}} \tag{38}$$

$$A = \begin{bmatrix} A\_{\rm P} & B\_{\rm P} C\_{\rm d,y} & 0 & 0 \\ 0 & A\_{\rm d} & 0 & 0 \\ B\_{\rm W\_y}^{\rm (LF)} C\_{\rm P} & 0 & A\_{\rm W\_y}^{\rm (LF)} & 0 \\ 0 & 0 & 0 & A\_{\rm W\_u}^{\rm (LF)} \end{bmatrix} \tag{39}$$

$$
\begin{bmatrix}
\mathbf{B}\_{\theta} \ \mathbf{B}\_{\upsilon}^{\text{(LF)}} \ \mathbf{B}\_{\iota}^{\text{(LF)}}
\end{bmatrix} = \begin{bmatrix}
\mathbf{0} & \mathbf{0} & \mathbf{B}\_{\mathsf{P}} \\
\mathbf{B}\_{\mathsf{d},\theta} \ \mathbf{B}\_{\mathsf{d},\upsilon} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{B}\_{\mathsf{W}\_{\mathsf{u}}}^{\text{(LF)}}
\end{bmatrix},\tag{40}
$$

$$
\begin{bmatrix} C\_{\theta} \\ C\_{q}^{\text{(l.FI)}} \\ C\_{y}^{\text{(l.FI)}} \end{bmatrix} = \begin{bmatrix} \mathbf{0} & C\_{\text{d},\theta} & \mathbf{0} & \mathbf{0} \\ \overline{D\_{W\_{y}}^{\text{(l.FI)}}} \mathbf{C\_{P}} & \mathbf{0} & C\_{W\_{y}}^{\text{(l.FI)}} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & C\_{W\_{y}}^{\text{(l.FI)}} \\ \hline \hline \mathbf{C\_{P}} & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{bmatrix} \tag{41}
$$

**Figure 5.** Plant with LPV-LFT disturbance model and weighting functions

### 10 Will-be-set-by-IN-TECH 74 Advances on Analysis and Control of Vibrations – Theory and Applications LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies <sup>11</sup>

and

$$
\begin{bmatrix}
\mathbf{D}\_{\theta\theta} & \mathbf{D}\_{\theta w} & \mathbf{D}\_{\theta u} \\
\mathbf{D}\_{q\theta} & \mathbf{D}\_{q\overline{w}}^{\mathrm{(LF)}} & \mathbf{D}\_{qu}^{\mathrm{(LF)}} \\
\mathbf{D}\_{y\theta} & \mathbf{D}\_{yw}^{\mathrm{(LF)}} & \mathbf{D}\_{yu}^{\mathrm{(LF)}}
\end{bmatrix} = \begin{bmatrix}
\mathbf{0} \ \mathbf{0} & \mathbf{0} \\
\mathbf{0} \ \mathbf{0} & \mathbf{0} \\
\mathbf{0} \ \mathbf{0} & \mathbf{D}\_{W\_{u}}^{\mathrm{(LF)}} \\
\mathbf{0} \ \mathbf{0} & \mathbf{0}
\end{bmatrix}.\tag{42}
$$

B = � B(pLPV) *w* **0**

C = � C(pLPV) *<sup>q</sup>* **0** �

AT

P (pLPV)

Q(pLPV)

B = � **0** B(pLPV) *u* **I 0** �

C =

D*qu* =

D*yw* =

The state-spaces matrices of the controllers for each vertex can be extracted from

**Ω***<sup>i</sup>* = �

**Ω**(pLPV)

The implemented controller is interpolated using the coordinate vector λ in

**5.2. Controller synthesis and implementation for LPV-LFT systems**

= �

= �

<sup>−</sup>(X(pLPV))−<sup>1</sup> <sup>A</sup>*<sup>i</sup>* <sup>B</sup> **<sup>0</sup>**

**0** C D(pLPV)

*<sup>i</sup>* <sup>−</sup>X(pLPV) **<sup>0</sup>** <sup>C</sup><sup>T</sup> <sup>B</sup><sup>T</sup> **<sup>0</sup>** <sup>−</sup>*<sup>γ</sup>* (D(pLPV)

B<sup>T</sup> **0 0** D<sup>T</sup>

**0** C D*yw* **0**

�

� **0 I** C(pLPV) *<sup>y</sup>* **0**

> **0** D(pLPV) *qu* �

� **0** *D*(pLPV) *yw* �

A*Ki* B*Ki* C*Ki DKi*

= Σ*<sup>m</sup>*

In this section, the algorithm for the calculation of the *H*∞-suboptimal gain-scheduling

From the state-space representation of the generalized plant the outer factors for the LMIs that

*<sup>u</sup>* )<sup>T</sup> <sup>D</sup>(LFT)

*<sup>θ</sup><sup>u</sup>* (D(LFT)

*qu* )<sup>T</sup> **0** �

(B(LFT)

�

�

<sup>ψ</sup>*<sup>i</sup>* + (<sup>P</sup> (pLPV))T**Ω***i*Q(pLPV) + (Q(pLPV))T**Ω***i*<sup>P</sup> (pLPV)

ψ*<sup>i</sup>* =

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

the matrix

and

is calculated. The matrices

are composed with

Finally, the basic LMIs

are solved for **Ω***<sup>i</sup>* for every *i*.

controller from [1] is explained in detail.

have to be solved in the design can be calculated as

<sup>N</sup><sup>R</sup> <sup>=</sup> null �

�

, (51)

LPV Gain-Scheduled Output Feedback

, (52)

� (55)

, (56)

, (57)

, (58)

. (59)

. (61)

*<sup>i</sup>*=1*λi***Ω***i*. (62)

*<* 0 (60)

. (53)

(54)

75

(63)

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

*qw* )<sup>T</sup>

*qw* −*γ***I**

for Active Control of Harmonic Disturbances with Time-Varying Frequencies

*qu*�

### **5. Controller synthesis and implementation for LPV systems**

In this section, algorithms for the calculation of the pLPV and LPV-LFT gain-scheduling controllers are explained in detail. Suboptimal controllers using *H*∞ techniques are obtained.

### **5.1. Controller synthesis and implementation for pLPV systems**

With the generalized plant in pLPV form, an *H*∞-suboptimal controller for each vertex of the polytope can be calculated using standard *H*∞ techniques [11]. The steps to obtain them are explained here in detail.

First, two outer factors

$$N\_{\mathbf{X}} = \text{null}\begin{bmatrix} \mathbf{C}\_{y}^{\text{(pl.PV)}} & \mathbf{D}\_{yw}^{\text{(pl.PV)}} \mathbf{0} \end{bmatrix} \tag{43}$$

and

$$N\_{\mathbf{Y}} = \text{null}\left[ (\mathcal{B}\_{\mu}^{\text{(pLV)}})^{\text{T}} (\mathcal{D}\_{q\mu}^{\text{(pLV)}})^{\text{T}} 0 \right] \tag{44}$$

are defined, where null[·] denotes the basis of the null space of a matrix.

Then, the LMIs

$$\mathbf{X}\_{\mathbf{X}}^{\mathrm{T}} \begin{bmatrix} \mathbf{A}\_{i}^{\mathrm{T}} \mathbf{X}\_{1} \mathbf{A}\_{i} - \mathbf{X}\_{1} & \mathbf{A}\_{i}^{\mathrm{T}} \mathbf{X}\_{1} \mathbf{B}\_{w}^{(\mathrm{p1PV})} & (\mathbf{C}\_{q}^{(\mathrm{p1PV})})^{\mathrm{T}} \\ (\mathbf{B}\_{w}^{(\mathrm{q1PV})})^{\mathrm{T}} \mathbf{X}\_{1} \mathbf{A}\_{i} - \gamma + (\mathbf{B}\_{w}^{(\mathrm{q1PV})})^{\mathrm{T}} \mathbf{X}\_{1} \mathbf{B}\_{w}^{(\mathrm{q1PV})} (\mathbf{D}\_{q\mu}^{(\mathrm{q1PV})})^{\mathrm{T}} \\ \mathbf{C}\_{q}^{(\mathrm{p1PV})} & \mathbf{D}\_{q\mu}^{(\mathrm{q1PV})} & -\gamma \mathbf{I} \end{bmatrix} \mathbf{N} \mathbf{x} < 0,\tag{45}$$

$$\mathbf{N}\_{\mathbf{Y}}^{\mathrm{T}} \begin{bmatrix} \mathbf{A}\_{i}\mathbf{Y}\_{1}\mathbf{A}\_{i}^{\mathrm{T}} - \mathbf{Y}\_{1} & \mathbf{A}\_{i}\mathbf{Y}\_{1}(\mathbf{C}\_{q}^{\mathrm{(q1IV)}})^{\mathrm{T}} & \mathbf{B}\_{w}^{\mathrm{(p1PV)}} \\ \mathbf{C}\_{q}^{\mathrm{(q1PV)}}\mathbf{Y}\_{1}\mathbf{A}\_{i}^{\mathrm{T}} & -\gamma\mathbf{I} + \mathbf{C}\_{q}^{\mathrm{(q1PV)}}\mathbf{Y}\_{1}(\mathbf{C}\_{q}^{\mathrm{(q1PV)}})^{\mathrm{T}} \mathbf{D}\_{q\mu}^{\mathrm{(p1PV)}} \\ (\mathbf{B}\_{w}^{\mathrm{(p1PV)}})^{\mathrm{T}} & (\mathbf{D}\_{q\mu}^{\mathrm{(p1PV)}})^{\mathrm{T}} & -\gamma \end{bmatrix} \mathbf{N}\_{\mathbf{Y}} < 0,\tag{46}$$

$$
\begin{bmatrix} \mathbf{X}\_1 & \mathbf{I} \\ \mathbf{I} & \mathbf{Y}\_1 \end{bmatrix} \ge 0 \tag{47}
$$

for feasibility and optimality are solved for X<sup>1</sup> and Y<sup>1</sup> for every A*<sup>i</sup>* = A*i*(θ).

With X<sup>1</sup> and Y1, the matrices

$$\mathbf{X}\_1 - \mathbf{Y}\_1^{-1} = \mathbf{X}\_2^T \mathbf{X}\_{2\prime} \tag{48}$$

$$\mathbf{X}^{\text{(pLPV)}} = \begin{bmatrix} \mathbf{X}\_1 \ \mathbf{X}\_2 \\ \mathbf{X}\_2^\text{T} \ \mathbf{I} \end{bmatrix} \tag{49}$$

are calculated.

With

$$
\overline{A}\_{i} = \begin{bmatrix} A\_{i} \ \mathbf{0} \\ \mathbf{0} \ \mathbf{0} \end{bmatrix} \tag{50}
$$

LPV Gain-Scheduled Output Feedback

74 Advances on Analysis and Control of Vibrations – Theory and Applications LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies <sup>11</sup> 75 for Active Control of Harmonic Disturbances with Time-Varying Frequencies

$$
\overline{B} = \begin{bmatrix} B\_w^{\text{(p1.PV)}} \\ \mathbf{0} \end{bmatrix}'\tag{51}
$$

$$\overline{\mathbf{C}} = \left[ \mathbf{C}\_{q}^{\text{(p1PV)}} \; \mathbf{0} \right],\tag{52}$$

the matrix

10 Will-be-set-by-IN-TECH

⎤ ⎥ <sup>⎦</sup> <sup>=</sup>

In this section, algorithms for the calculation of the pLPV and LPV-LFT gain-scheduling controllers are explained in detail. Suboptimal controllers using *H*∞ techniques are obtained.

With the generalized plant in pLPV form, an *H*∞-suboptimal controller for each vertex of the polytope can be calculated using standard *H*∞ techniques [11]. The steps to obtain them are

> � C(pLPV) *<sup>y</sup>* <sup>D</sup>(pLPV)

⎡ ⎢ ⎢ ⎢ ⎣

*yw* **0** �

*qu* )<sup>T</sup> 0

*<sup>w</sup>* (C(pLPV)

*qu* −*γ***I**

*qu* )<sup>T</sup> <sup>−</sup>*<sup>γ</sup>*

�

*<sup>q</sup>* )<sup>T</sup> <sup>B</sup>(pLPV)

*<sup>w</sup>* (D(pLPV)

*<sup>q</sup>* )<sup>T</sup> <sup>D</sup>(pLPV) *qu*

*w*

�

*<sup>q</sup>* )<sup>T</sup>

⎤ ⎥ ⎦

*qu* )<sup>T</sup>

⎤ ⎥ ⎦

≥ 0 (47)

<sup>2</sup> X2, (48)

, (50)

*<sup>u</sup>* )<sup>T</sup> (D(pLPV)

*<sup>w</sup>* )TX1B(pLPV)

*<sup>q</sup>* <sup>Y</sup>1(C(pLPV)

�

<sup>1</sup> <sup>=</sup> <sup>X</sup><sup>T</sup>

= � X<sup>1</sup> X<sup>2</sup> X<sup>T</sup> <sup>2</sup> **I**

*<sup>i</sup>* <sup>X</sup>1B(pLPV)

⎤ ⎥ ⎥ ⎥

<sup>⎦</sup> . (42)

(43)

(44)

(49)

N<sup>X</sup> *<* 0, (45)

N<sup>Y</sup> *<* 0, (46)

⎡ ⎢ ⎣

D*θθ* D*θ<sup>w</sup>* D*θ<sup>u</sup>* <sup>D</sup>*q<sup>θ</sup>* <sup>D</sup>(LFT)

**5. Controller synthesis and implementation for LPV systems**

**5.1. Controller synthesis and implementation for pLPV systems**

N<sup>X</sup> = null

� (B(pLPV)

N<sup>Y</sup> = null

*<sup>i</sup>* <sup>X</sup>1A*<sup>i</sup>* <sup>−</sup> <sup>X</sup><sup>1</sup> <sup>A</sup><sup>T</sup>

*<sup>w</sup>* )TX1A*<sup>i</sup>* <sup>−</sup>*<sup>γ</sup>* + (B(pLPV)

*<sup>q</sup>* <sup>D</sup>(pLPV)

*<sup>i</sup>* <sup>−</sup> <sup>Y</sup><sup>1</sup> <sup>A</sup>*i*Y1(C(pLPV)

*<sup>i</sup>* <sup>−</sup>*γ***<sup>I</sup>** <sup>+</sup> <sup>C</sup>(pLPV)

� X<sup>1</sup> **I I** Y<sup>1</sup>

<sup>X</sup><sup>1</sup> <sup>−</sup> <sup>Y</sup> <sup>−</sup><sup>1</sup>

A*<sup>i</sup>* = � A*<sup>i</sup>* **0 0 0** �

X(pLPV)

*<sup>w</sup>* )<sup>T</sup> (D(pLPV)

for feasibility and optimality are solved for X<sup>1</sup> and Y<sup>1</sup> for every A*<sup>i</sup>* = A*i*(θ).

are defined, where null[·] denotes the basis of the null space of a matrix.

<sup>D</sup>*y<sup>θ</sup> <sup>D</sup>*(LFT)

*qw* <sup>D</sup>(LFT) *qu*

*yw <sup>D</sup>*(LFT) *yu*

and

and

Then, the LMIs

N<sup>T</sup> X ⎡ ⎢ ⎣

N<sup>T</sup> Y

With X<sup>1</sup> and Y1, the matrices

are calculated.

With

A<sup>T</sup>

⎡ ⎢ ⎣

(B(pLPV)

C(pLPV)

A*i*Y1A<sup>T</sup>

C(pLPV) *<sup>q</sup>* Y1A<sup>T</sup>

(B(pLPV)

explained here in detail. First, two outer factors

$$
\psi\_i = \begin{bmatrix}
\overline{A}\_i^{\text{T}} & -\mathbf{X}^{\langle \text{p1PV} \rangle} & \mathbf{0} & \overline{C}^{\text{T}} \\
\overline{B}^{\text{T}} & \mathbf{0} & -\gamma & (\mathbf{D}\_{qw}^{\langle \text{p1PV} \rangle})^{\text{T}} \\
\mathbf{0} & \overline{C} & \mathbf{D}\_{qw}^{\langle \text{p1PV} \rangle} & -\gamma \mathbf{I}
\end{bmatrix}. \tag{53}
$$

is calculated. The matrices

$$\boldsymbol{\underline{\mathbf{P}}}^{\text{(p1FV)}} = \begin{bmatrix} \underline{\mathbf{B}}^{\text{T}} \mathbf{0} \ \mathbf{0} \ \underline{\mathbf{D}}^{\text{T}}\_{\eta\mu} \end{bmatrix} \tag{54}$$

and

$$\mathbf{Q}^{\text{(plFV)}} = \begin{bmatrix} \mathbf{0} \ \underline{\mathbf{C}} \ \underline{\mathbf{D}}\_{\mathcal{Y}^{\text{w}}} \ \mathbf{0} \end{bmatrix} \tag{55}$$

are composed with

$$\underline{B} = \begin{bmatrix} \mathbf{0} \ B\_{\boldsymbol{\mu}}^{\mathrm{(p\&V)}} \\ \mathbf{I} & \mathbf{0} \end{bmatrix} \; \prime \tag{56}$$

$$\underline{\mathbf{C}} = \begin{bmatrix} \mathbf{0} & \mathbf{I} \\ \mathbf{C}\_{y}^{\text{(pIdx)}} \mathbf{0} \end{bmatrix} \, \prime \, \tag{57}$$

$$\underline{\mathbf{D}}\_{q\mu} = \left[ \underset{r}{\mathbf{0}} \; \mathbf{D}\_{q\mu}^{\text{(pt.IV)}} \right] \; \tag{58}$$

$$\underline{D}\_{\mathcal{Y}w} = \begin{bmatrix} \mathbf{0} \\ D\_{yw}^{\langle \mathtt{q} \mathtt{I} \mathtt{r} \mathtt{v} \rangle} \end{bmatrix}. \tag{59}$$

Finally, the basic LMIs

$$
\psi\_l + (\mathbf{P}^{\text{(pLPV)}})^\text{T} \Omega\_l \mathbf{Q}^{\text{(pLPV)}} + (\mathbf{Q}^{\text{(pLPV)}})^\text{T} \Omega\_l \mathbf{P}^{\text{(pLPV)}} < 0 \tag{60}
$$

are solved for **Ω***<sup>i</sup>* for every *i*.

The state-spaces matrices of the controllers for each vertex can be extracted from

$$
\boldsymbol{\Omega}\_{l} = \begin{bmatrix} \boldsymbol{A}\_{\boldsymbol{K}\_{l}} \ \boldsymbol{B}\_{\boldsymbol{K}\_{l}} \\ \boldsymbol{C}\_{\boldsymbol{K}\_{l}} \ \boldsymbol{D}\_{\boldsymbol{K}\_{l}} \end{bmatrix}. \tag{61}
$$

The implemented controller is interpolated using the coordinate vector λ in

$$
\boldsymbol{\Omega}^{\text{(p1.PV)}} = \boldsymbol{\Sigma}\_{i=1}^{m} \boldsymbol{\lambda}\_{i} \boldsymbol{\Omega}\_{i}.\tag{62}
$$

### **5.2. Controller synthesis and implementation for LPV-LFT systems**

In this section, the algorithm for the calculation of the *H*∞-suboptimal gain-scheduling controller from [1] is explained in detail.

From the state-space representation of the generalized plant the outer factors for the LMIs that have to be solved in the design can be calculated as

$$N\_R = \text{null}\left[ (\mathbf{B}\_{\mu}^{\text{(LF)}})^\text{T} \; \mathbf{D}\_{\theta u}^{\text{(LF)}} \; (\mathbf{D}\_{q\mu}^{\text{(LF)}})^\text{T} \; \mathbf{0} \right] \tag{63}$$

### 12 Will-be-set-by-IN-TECH 76 Advances on Analysis and Control of Vibrations – Theory and Applications LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies <sup>13</sup>

and

$$N\_{\mathbf{S}} = \text{null}\begin{bmatrix} \mathbf{C}\_{y}^{(\text{l.FT})} \ \mathbf{D}\_{y\theta} \ \mathbf{D}\_{yw}^{(\text{l.FT})} \ \mathbf{0} \end{bmatrix}. \tag{64}$$

Q(LFT)

, B<sup>0</sup> =

A<sup>0</sup> =

⎡ ⎢ ⎢ ⎣

C<sup>0</sup> =

L = � L<sup>1</sup> L<sup>2</sup> L<sup>T</sup> <sup>2</sup> L<sup>3</sup>

controller are extracted from

**6. Experimental results**

controller obtained is of 21st order.

the system remains stable.

and

used.

� A **0 0 0**�

**0 0** C*<sup>θ</sup>* **0** C(LFT) *<sup>q</sup>* **0**

D˜ <sup>12</sup> =

⎤ ⎥ ⎥ <sup>⎦</sup> , <sup>C</sup>˜ <sup>=</sup>

⎡ ⎢ ⎢ ⎣

�

**00I 0** D*θ<sup>u</sup>* **0 0** D(LFT) *qu* **0**

, L<sup>0</sup> =

**Ω**(LFT) = ⎡ ⎣ A(LFT) *<sup>K</sup>* <sup>B</sup>(LFT) *K*

**6.1. Experimental results for the pLPV gain-scheduled controller**

= �

�

⎡ ⎢ ⎢ ⎣ 0 C˜ D˜ <sup>21</sup> 0

⎤ ⎥ ⎥ <sup>⎦</sup> , <sup>D</sup><sup>0</sup> <sup>=</sup>

� , B˜ =

⎡ ⎢ ⎢ ⎣

**<sup>0</sup>** <sup>B</sup>*<sup>θ</sup>* <sup>B</sup>(LFT) *w*

**0 I**

**0 0** 0

C(LFT) *<sup>y</sup>* **0 0 0**

⎤ ⎥ ⎥ <sup>⎦</sup> , <sup>D</sup>˜ <sup>21</sup> <sup>=</sup>

� L **0 0** 1

�

is solved for the controller matrix **Ω**(LFT) . In the last step, the state-space matrices of the

C(LFT) *<sup>K</sup>* <sup>D</sup>(LFT) *K*

The gain-scheduled output-feedback controllers obtained through the design procedures presented in this chapter are validated with experimental results. Both controllers have been tested on the ANC and AVC systems. Results are presented for the pLPV gain-scheduled controller on the ANC system in Sec. 6.1 and for the LPV-LFT controller on the AVC test bed in Sec. 6.2. Identical hardware setup and sampling frequency as in the previous chapter are

The pLPV gain-scheduled controller is validated with experimental results on the ANC headset. The controller is designed to reject a disturbance signal which contains four harmonically related sine signals with fundamental frequency between 80 and 90 Hz. The

Amplitude frequency responses and pressure measured when the fundamental frequency rises suddenly from 80 to 90 Hz are shown in Figs. 6 and 7. An excellent disturbance rejection is achieved even for unrealistically fast variations of the disturbance frequencies. In Fig. 8, results for time-varying frequencies are shown. The performance for fast variations of the fundamental frequency is further studied in Fig. 9. As in the previous chapter, with fast changes of the fundamental frequency the disturbance attenuation performance decreases but

�

� **0** B(LFT) *<sup>u</sup>* **0 I00**�

for Active Control of Harmonic Disturbances with Time-Varying Frequencies

⎡ ⎢ ⎢ ⎣

**00 0 <sup>0</sup>** <sup>D</sup>*y<sup>θ</sup> <sup>D</sup>*(LFT) *yw*

**I0 0**

, J = L<sup>−</sup>1, J<sup>0</sup> =

⎤

**00 0 0** D*θθ* D*θ<sup>w</sup>* **<sup>0</sup>** <sup>D</sup>*q<sup>θ</sup>* <sup>D</sup>(LFT) *qw*

> ⎤ ⎥ ⎥

� J **0 0 I**�

, (75)

LPV Gain-Scheduled Output Feedback

⎤ ⎥ ⎥

⎦ . (80)

, (76)

77

<sup>⎦</sup> , (77)

, (79)

<sup>⎦</sup> , (78)

With the outer factors, a first set of LMIs corresponding to the feasibility and optimality condition is given as

N<sup>T</sup> R ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ARA<sup>T</sup> <sup>−</sup> R ARC<sup>T</sup> *<sup>θ</sup>* AR(C(LFT))<sup>T</sup> *<sup>q</sup>* B*<sup>θ</sup>* B*<sup>w</sup>* <sup>C</sup>*θ*RA<sup>T</sup> <sup>−</sup>*γ*J<sup>3</sup> <sup>+</sup> <sup>C</sup>*θ*RC<sup>T</sup> *<sup>θ</sup>* <sup>C</sup>*θ*R(C(LFT))<sup>T</sup> *<sup>q</sup>* D*θθ* D*θ<sup>w</sup>* C(LFT) *<sup>q</sup>* RA<sup>T</sup> <sup>C</sup>(LFT) *<sup>q</sup>* RC<sup>T</sup> *<sup>θ</sup>* <sup>C</sup>(LFT) *<sup>q</sup>* <sup>R</sup>(C(LFT))<sup>T</sup> *<sup>q</sup>* <sup>−</sup> *<sup>γ</sup>***<sup>I</sup>** <sup>D</sup>*q<sup>θ</sup>* <sup>D</sup>(LFT) *qw* B<sup>T</sup> *<sup>θ</sup>* <sup>D</sup><sup>T</sup> *θθ* <sup>D</sup><sup>T</sup> *<sup>q</sup><sup>θ</sup>* −*γ*L<sup>3</sup> 0 (B(LFT) *<sup>w</sup>* )<sup>T</sup> D<sup>T</sup> *<sup>θ</sup><sup>w</sup>* (D(LFT) *qw* )<sup>T</sup> <sup>0</sup> <sup>−</sup>*<sup>γ</sup>* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ N<sup>R</sup> *<* 0, (65) N<sup>T</sup> S ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ <sup>A</sup>TSA <sup>−</sup> S ATSB*<sup>θ</sup>* <sup>A</sup>TSB(LFT) *<sup>w</sup>* C<sup>T</sup> *<sup>θ</sup>* (C(LFT) *<sup>q</sup>* )<sup>T</sup> B<sup>T</sup> *<sup>θ</sup>* SA <sup>−</sup>*γ*L<sup>3</sup> <sup>+</sup> <sup>B</sup><sup>T</sup> *<sup>θ</sup>* SB*<sup>θ</sup>* <sup>B</sup><sup>T</sup> *<sup>θ</sup>* SB(LFT) *<sup>w</sup>* D<sup>T</sup> *θθ* <sup>D</sup><sup>T</sup> *qθ* (B(LFT) *<sup>w</sup>* )TSA (B(LFT) *<sup>w</sup>* )TSB*<sup>θ</sup>* (B(LFT) *<sup>w</sup>* )TS(B(LFT) *<sup>w</sup>* ) <sup>−</sup> *<sup>γ</sup>* <sup>D</sup><sup>T</sup> *<sup>θ</sup><sup>w</sup>* (D(LFT) *qw* )<sup>T</sup> C*<sup>θ</sup>* D*θθ* D*θ<sup>w</sup>* −*γ*J<sup>3</sup> 0 C(LFT) *<sup>q</sup>* <sup>D</sup>*q<sup>θ</sup>* <sup>D</sup>(LFT) *qw* 0 −*γ***I** ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ N<sup>S</sup> *<* 0, (66) � R **I I** S � ≥ 0, (67) � L<sup>3</sup> **I I** J<sup>3</sup> � ≥ 0. (68)

The scalar *γ* is an upper bound of the maximum singular value, which is given as a constraint. This set of LMIs is solved for R, S, J<sup>3</sup> and L3.

The matrices L<sup>1</sup> and L<sup>2</sup> are calculated through

$$L\_3 - J\_3^{-1} = L\_2^T L\_1^{-1} L\_{2'} \tag{69}$$

and the matrix X(LFT) is computed as

$$\mathbf{X}^{\text{(JFT)}} = \begin{bmatrix} \mathbf{S} & \mathbf{I} \\ \mathbf{N}^{\text{T}} \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{I} & \mathbf{R} \\ \mathbf{0} \ \mathbf{M}^{\text{T}} \end{bmatrix} \tag{70}$$

with M and N satisfying

$$\mathbf{I} \cdot \mathbf{M} \mathbf{N}^{\mathrm{T}} = \mathbf{I} - \mathbf{R} \mathbf{S} . \tag{71}$$

Then, the basic LMI

$$
\psi + (\mathbf{Q}^{\text{(LFT)}})^{\text{T}} (\mathbf{D}^{\text{(LFT)}})^{\text{T}} \mathbf{P}^{\text{(LFT)}} + (\mathbf{P}^{\text{(LFT)}})^{\text{T}} \mathbf{D}^{\text{(LFT)}} \mathbf{Q}^{\text{(LFT)}} < 0,\tag{72}
$$

where

$$
\psi = \begin{bmatrix}
A\_0^\mathrm{T} & -X & \mathbf{0} & C\_0^\mathrm{T} \\
B\_0^\mathrm{T} & \mathbf{0} & -\gamma L\_0 & D\_0^\mathrm{T} \\
\mathbf{0} & C\_0 & D\_0 & -\gamma J\_0
\end{bmatrix} \tag{73}
$$

$$\boldsymbol{P}^{\text{(LF)}} = \begin{bmatrix} \tilde{\boldsymbol{B}}^{\text{T}} \ \mathbf{0} \ \mathbf{0} \ \tilde{\boldsymbol{D}}\_{12}^{\text{T}} \end{bmatrix} . \tag{74}$$

LPV Gain-Scheduled Output Feedback

76 Advances on Analysis and Control of Vibrations – Theory and Applications LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies <sup>13</sup> 77 for Active Control of Harmonic Disturbances with Time-Varying Frequencies

$$\mathbf{Q}^{\text{(LF)}} = \begin{bmatrix} \mathbf{0} \ \vec{C} \ \vec{D}\_{21} \ \mathbf{0} \end{bmatrix} . \tag{75}$$

$$\begin{aligned} \mathbf{A}\_{0} = \begin{bmatrix} \mathbf{A} \ \mathbf{0} \\ \mathbf{0} \ \mathbf{0} \end{bmatrix}, \mathbf{B}\_{0} = \begin{bmatrix} \mathbf{0} \ B\_{\theta} \ B\_{w}^{(\text{lFr})} \\ \mathbf{0} \ \mathbf{0} \end{bmatrix}, \tilde{\mathbf{B}} = \begin{bmatrix} \mathbf{0} \ B\_{u}^{(\text{lFr})} \ \mathbf{0} \\ \mathbf{I} \ \mathbf{0} \ \mathbf{0} \end{bmatrix} \end{aligned} \tag{76}$$

$$\begin{aligned} ^0C\_0 = \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ ^0C\_\theta & \mathbf{0} \\ ^0C\_q & ^0\mathbf{0} \end{bmatrix}, \tilde{C} = \begin{bmatrix} \mathbf{0} & \mathbf{I} \\ C\_y^{(\text{IFT})} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix}, D\_0 = \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} \, D\_{\theta\theta} & D\_{\theta w} \\ \mathbf{0} \, D\_{q\theta} & D\_{qw}^{(\text{IFT})} \end{bmatrix} \end{aligned} \tag{77}$$

$$
\tilde{D}\_{12} = \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{I} \\ \mathbf{0} & D\_{\theta\mu} & \mathbf{0} \\ \mathbf{0} & D\_{qu}^{(\text{LFI})} & \mathbf{0} \end{bmatrix}, \tilde{D}\_{21} = \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} \ \mathbf{D}\_{y\theta} \ \mathbf{D}\_{yw}^{(\text{LFI})} \\ \mathbf{I} & \mathbf{0} & \mathbf{0} \end{bmatrix} \tag{78}
$$

and

12 Will-be-set-by-IN-TECH

With the outer factors, a first set of LMIs corresponding to the feasibility and optimality

*<sup>θ</sup>* <sup>C</sup>*θ*R(C(LFT)

*<sup>q</sup>* <sup>R</sup>(C(LFT)

*<sup>θ</sup>* SB(LFT)

*<sup>w</sup>* )TS(B(LFT)

�

The scalar *γ* is an upper bound of the maximum singular value, which is given as a constraint.

<sup>3</sup> <sup>=</sup> <sup>L</sup><sup>T</sup>

� S **I** N<sup>T</sup> **0**

)TP (LFT)

= �

2L−<sup>1</sup>

� �**I** R **0** M<sup>T</sup>

+ (P (LFT)

<sup>−</sup>X−<sup>1</sup> <sup>A</sup><sup>0</sup> <sup>B</sup><sup>0</sup> **<sup>0</sup>**

<sup>0</sup> <sup>−</sup><sup>X</sup> **<sup>0</sup>** <sup>C</sup><sup>T</sup>

<sup>0</sup> **<sup>0</sup>** <sup>−</sup>*γ*L<sup>0</sup> <sup>D</sup><sup>T</sup>

**0** C<sup>0</sup> D<sup>0</sup> −*γ*J<sup>0</sup>

B˜ <sup>T</sup> 0 0 D˜ <sup>T</sup>

�

)T**Ω**(LFT)

0

⎤ ⎥ ⎥

0

12 �

MN<sup>T</sup> <sup>=</sup> **<sup>I</sup>** <sup>−</sup> RS. (71)

Q(LFT)

*<sup>θ</sup>* AR(C(LFT)

*<sup>θ</sup>* <sup>C</sup>(LFT)

*<sup>θ</sup>* SB*<sup>θ</sup>* <sup>B</sup><sup>T</sup>

*<sup>w</sup>* )TSB*<sup>θ</sup>* (B(LFT)

*<sup>q</sup>* <sup>D</sup>*q<sup>θ</sup>* <sup>D</sup>(LFT)

*θθ* <sup>D</sup><sup>T</sup>

*<sup>θ</sup><sup>w</sup>* (D(LFT)

C*<sup>θ</sup>* D*θθ* D*θ<sup>w</sup>* −*γ*J<sup>3</sup> 0

� R **I I** S �

� L<sup>3</sup> **I I** J<sup>3</sup>

<sup>L</sup><sup>3</sup> <sup>−</sup> <sup>J</sup>−<sup>1</sup>

X(LFT) =

)T(**Ω**(LFT)

⎡ ⎢ ⎢ ⎣

A<sup>T</sup>

B<sup>T</sup>

P (LFT)

ψ =

*<sup>y</sup>* <sup>D</sup>*y<sup>θ</sup>* <sup>D</sup>(LFT)

*yw* **0** �

)T

)T

)T

*<sup>q</sup>* B*<sup>θ</sup>* B*<sup>w</sup>*

*<sup>q</sup>* D*θθ* D*θ<sup>w</sup>*

*<sup>q</sup>* <sup>−</sup> *<sup>γ</sup>***<sup>I</sup>** <sup>D</sup>*q<sup>θ</sup>* <sup>D</sup>(LFT)

*<sup>q</sup><sup>θ</sup>* −*γ*L<sup>3</sup> 0

*qw* )<sup>T</sup> <sup>0</sup> <sup>−</sup>*<sup>γ</sup>*

*qw* 0 −*γ***I**

*<sup>w</sup>* C<sup>T</sup>

*<sup>w</sup>* D<sup>T</sup>

*<sup>w</sup>* ) <sup>−</sup> *<sup>γ</sup>* <sup>D</sup><sup>T</sup>

*qw*

*<sup>θ</sup>* (C(LFT) *<sup>q</sup>* )<sup>T</sup>

*θθ* <sup>D</sup><sup>T</sup> *qθ*

*<sup>θ</sup><sup>w</sup>* (D(LFT) *qw* )<sup>T</sup>

≥ 0, (67)

≥ 0. (68)

<sup>1</sup> L2, (69)

, (70)

*<* 0, (72)

<sup>⎦</sup> , (73)

, (74)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

> ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

. (64)

N<sup>R</sup> *<* 0, (65)

N<sup>S</sup> *<* 0, (66)

� C(LFT)

N<sup>S</sup> = null

and

condition is given as

C(LFT)

B<sup>T</sup>

C(LFT)

(B(LFT)

and the matrix X(LFT)

with M and N satisfying

Then, the basic LMI

where

B<sup>T</sup>

(B(LFT)

ARA<sup>T</sup> <sup>−</sup> R ARC<sup>T</sup>

*<sup>q</sup>* RA<sup>T</sup> <sup>C</sup>(LFT)

<sup>C</sup>*θ*RA<sup>T</sup> <sup>−</sup>*γ*J<sup>3</sup> <sup>+</sup> <sup>C</sup>*θ*RC<sup>T</sup>

*<sup>θ</sup>* <sup>D</sup><sup>T</sup>

*<sup>w</sup>* )<sup>T</sup> D<sup>T</sup>

*<sup>θ</sup>* SA <sup>−</sup>*γ*L<sup>3</sup> <sup>+</sup> <sup>B</sup><sup>T</sup>

This set of LMIs is solved for R, S, J<sup>3</sup> and L3. The matrices L<sup>1</sup> and L<sup>2</sup> are calculated through

is computed as

ψ + (Q(LFT)

*<sup>w</sup>* )TSA (B(LFT)

*<sup>q</sup>* RC<sup>T</sup>

<sup>A</sup>TSA <sup>−</sup> S ATSB*<sup>θ</sup>* <sup>A</sup>TSB(LFT)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

N<sup>T</sup> R

> ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

N<sup>T</sup> S

$$L = \begin{bmatrix} L\_1 \ L\_2 \\ L\_2^T \ L\_3 \end{bmatrix}, \ L\_0 = \begin{bmatrix} L \ \mathbf{0} \\ \mathbf{0} \ \mathbf{1} \end{bmatrix}, \mathbf{J} = L^{-1}, \mathbf{J}\_0 = \begin{bmatrix} J \ \mathbf{0} \\ \mathbf{0} \ \mathbf{I} \end{bmatrix}, \tag{79}$$

is solved for the controller matrix **Ω**(LFT) . In the last step, the state-space matrices of the controller are extracted from

$$\boldsymbol{\Omega}^{\text{(LFT)}} = \begin{bmatrix} \boldsymbol{\mathcal{A}}\_{\boldsymbol{K}}^{\text{(LFT)}} \ \boldsymbol{\mathcal{B}}\_{\boldsymbol{K}}^{\text{(LFT)}} \\ \boldsymbol{C}\_{\boldsymbol{K}}^{\text{(LFT)}} \ \boldsymbol{D}\_{\boldsymbol{K}}^{\text{(LFT)}} \end{bmatrix}. \tag{80}$$

### **6. Experimental results**

The gain-scheduled output-feedback controllers obtained through the design procedures presented in this chapter are validated with experimental results. Both controllers have been tested on the ANC and AVC systems. Results are presented for the pLPV gain-scheduled controller on the ANC system in Sec. 6.1 and for the LPV-LFT controller on the AVC test bed in Sec. 6.2. Identical hardware setup and sampling frequency as in the previous chapter are used.

### **6.1. Experimental results for the pLPV gain-scheduled controller**

The pLPV gain-scheduled controller is validated with experimental results on the ANC headset. The controller is designed to reject a disturbance signal which contains four harmonically related sine signals with fundamental frequency between 80 and 90 Hz. The controller obtained is of 21st order.

Amplitude frequency responses and pressure measured when the fundamental frequency rises suddenly from 80 to 90 Hz are shown in Figs. 6 and 7. An excellent disturbance rejection is achieved even for unrealistically fast variations of the disturbance frequencies. In Fig. 8, results for time-varying frequencies are shown. The performance for fast variations of the fundamental frequency is further studied in Fig. 9. As in the previous chapter, with fast changes of the fundamental frequency the disturbance attenuation performance decreases but the system remains stable.

**Figure 6.** Open-loop (gray) and closed-loop (black) amplitude frequency responses for fixed disturbance frequencies of 80, 160, 240 and 320 Hz (left) and of 90, 180, 270 and 360 Hz (right)

0 4 8 12 16 20 24

<sup>0</sup> <sup>8</sup> <sup>16</sup> <sup>24</sup> <sup>í</sup>0.2

Time [s]

LPV Gain-Scheduled Output Feedback

79

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>0</sup>

Frequency [Hz]

í0.1

0

Pressure [Pa]

**Figure 9.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and

The AVC test bed is used to test the LFT gain-scheduled controller experimentally. The controller is designed to reject a disturbance with eight harmonic components which are selected to be uniformly distributed from 80 to 380 Hz in intervals of 20 Hz. The resulting

Amplitude frequency responses are shown in Fig. 10 and results for an experiment where the frequencies change drastically as a step function in Fig. 11. Results from experiments with time-varying frequencies are shown in Figs. 12 and 13. Excellent disturbance rejection is

10

20

Amplitude [ms

**Figure 10.** Open-loop (gray) and closed-loop (black) amplitude frequency responses for fixed disturbance frequencies of 80, 120, 160, 200, 240, 280, 320 and 360 Hz (left) and 100, 140, 180, 220, 260,

 í2/A]

30

40

0.1

0.2

for Active Control of Harmonic Disturbances with Time-Varying Frequencies

Time [s]

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>0</sup>

Frequency [Hz]

measured sound pressure (right) in open loop (gray) and closed loop (black)

**6.2. Experimental results for the LFT gain-scheduled controller**

*f* 1

controller is of 27th order.

achieved.

10

300, 340 and 380 Hz (right)

20

Amplitude [ms

 í2/A]

30

40

*f* 2

*f* 3

*f* 4

Frequency [Hz]

**Figure 7.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured sound pressure (right). The control sequence is off/on/off

**Figure 8.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured sound pressure (right) in open loop (gray) and closed loop (black)

LPV Gain-Scheduled Output Feedback

78 Advances on Analysis and Control of Vibrations – Theory and Applications LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies <sup>15</sup> 79 for Active Control of Harmonic Disturbances with Time-Varying Frequencies

**Figure 9.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured sound pressure (right) in open loop (gray) and closed loop (black)

### **6.2. Experimental results for the LFT gain-scheduled controller**

14 Will-be-set-by-IN-TECH

1

í0.07

í0.1

í0.05

Pressure [Pa]

**Figure 8.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and

0

0.05

0.1

0.15

Pressure [Pa]

**Figure 7.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and

0

0.07

0.14

**Figure 6.** Open-loop (gray) and closed-loop (black) amplitude frequency responses for fixed disturbance

frequencies of 80, 160, 240 and 320 Hz (left) and of 90, 180, 270 and 360 Hz (right)

2

3

Amplitude [Pa / V]

4

5

6

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>0</sup>

Frequency [Hz]

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>í</sup>0.14

<sup>0</sup> <sup>4</sup> <sup>8</sup> <sup>12</sup> <sup>16</sup> <sup>20</sup> <sup>24</sup> <sup>28</sup> <sup>í</sup>0.15

Time [s]

Time [s]

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>0</sup>

Frequency [Hz]

0 2 4 6 8 10

Time [s]

0 4 8 12 16 20 24 28

Time [s]

measured sound pressure (right) in open loop (gray) and closed loop (black)

measured sound pressure (right). The control sequence is off/on/off

2

*f* 1

*f* 2

*f* 3

*f* 4

Frequency [Hz]

Frequency [Hz]

4

6

Amplitude [Pa / V]

8

10

12

The AVC test bed is used to test the LFT gain-scheduled controller experimentally. The controller is designed to reject a disturbance with eight harmonic components which are selected to be uniformly distributed from 80 to 380 Hz in intervals of 20 Hz. The resulting controller is of 27th order.

Amplitude frequency responses are shown in Fig. 10 and results for an experiment where the frequencies change drastically as a step function in Fig. 11. Results from experiments with time-varying frequencies are shown in Figs. 12 and 13. Excellent disturbance rejection is achieved.

**Figure 10.** Open-loop (gray) and closed-loop (black) amplitude frequency responses for fixed disturbance frequencies of 80, 120, 160, 200, 240, 280, 320 and 360 Hz (left) and 100, 140, 180, 220, 260, 300, 340 and 380 Hz (right)

**7. Discussion and conclusion**

controllers can be quite straightforward.

**Nomenclature**

ANC Active noise control.

AVC Active vibration control.

LMI Linear matrix inequality. LPV Linear parameter varying.

LTI Linear time invariant.

(in order of appearance)

K Controller.

G Generalized plant.

LFT Linear fractional transformation.

pLPV Polytopic linear parameter varying.

u, y Control input, output signal.

*σ*max Maximum singular value.

x*k*, y*k*, u*<sup>k</sup>* State vector, output and input.

w, q Performance input, performance output.

A(θ), B, C, D State-space matrices of a pLPV system.

*Gqw* Transfer path between performance input and performance output.

**Acronyms**

**Variables**

Two discrete-time control design methods have been presented in this chapter for the rejection of time-varying frequencies. The output-feedback controllers are obtained through pLPV and LPV-LFT gain-scheduling techniques. The controllers obtained are validated experimentally on an ANC and AVC system. The experimental results show an excellent disturbance rejection

81

LPV Gain-Scheduled Output Feedback

for Active Control of Harmonic Disturbances with Time-Varying Frequencies

The control design guarantees stability even for arbitrarily fast changes of the disturbance frequencies. This is an advantage over heuristic interpolation methods or adaptive filtering,

To the best of the authors' knowledge, industrial applications of LPV controllers are rather limited. The results of this chapter show that the implementation of even high-order LPV

even for the case of eight frequency components of the disturbance.

for which none or only "approximate stability results" are available [10].

**Figure 11.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured acceleration (right). The control sequence is off/on/off

**Figure 12.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured acceleration (right) in open loop (gray) and closed loop (black)

**Figure 13.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured acceleration (right) in open loop (gray) and closed loop (black)

### **7. Discussion and conclusion**

Two discrete-time control design methods have been presented in this chapter for the rejection of time-varying frequencies. The output-feedback controllers are obtained through pLPV and LPV-LFT gain-scheduling techniques. The controllers obtained are validated experimentally on an ANC and AVC system. The experimental results show an excellent disturbance rejection even for the case of eight frequency components of the disturbance.

The control design guarantees stability even for arbitrarily fast changes of the disturbance frequencies. This is an advantage over heuristic interpolation methods or adaptive filtering, for which none or only "approximate stability results" are available [10].

To the best of the authors' knowledge, industrial applications of LPV controllers are rather limited. The results of this chapter show that the implementation of even high-order LPV controllers can be quite straightforward.

### **Nomenclature**

### **Acronyms**

16 Will-be-set-by-IN-TECH

í1

í1

í1

0

Acceleration [msí2

**Figure 13.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left)

]

1

2

0

Acceleration [msí2

**Figure 12.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left)

]

1

2

0

Acceleration [msí2

**Figure 11.** Results for a disturbance with time-varying frequencies. Variation of the frequencies (left)

]

1

2

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>í</sup><sup>2</sup>

Time [s]

<sup>0</sup> <sup>13</sup> <sup>26</sup> <sup>í</sup><sup>2</sup>

<sup>0</sup> <sup>11</sup> <sup>22</sup> <sup>í</sup><sup>2</sup>

Time [s]

Time [s]

0 2 4 6 8 10

Time [s]

0 13 26

Time [s]

0 11 22

Time [s]

and measured acceleration (right) in open loop (gray) and closed loop (black)

and measured acceleration (right) in open loop (gray) and closed loop (black)

and measured acceleration (right). The control sequence is off/on/off

Frequency [Hz]

0

Frequency [Hz]

Frequency [Hz]


### **Variables**

(in order of appearance)



*Wy*, *Wu* System representations of the weighting functions.

Ad, Bd, *<sup>θ</sup>* , Bd, *<sup>w</sup>*, Cd, *<sup>θ</sup>* , Cd, *<sup>y</sup>* State-space matrices of the LFT disturbance model.

*w*d, *y*<sup>d</sup> Input and output of the disturbance model.

*ai*, *pi* Scalar parameters for the disturbance model.

<sup>Y</sup> Outer factors to build the LMIs.

<sup>1</sup> , Solutions of the first set of LMIs.

*n* = *n*<sup>p</sup> + 2*n*<sup>d</sup> + *nWy* + *nWu* Order of matrices X<sup>1</sup> and Y1.

*<sup>i</sup>* Matrix to build the basic LMI.

P ((*n*+1)×(4*n*+3)), Q((*n*+1)×(4*n*+3)) Matrices to build the basic LMI.

*<sup>i</sup>* , Matrices to build matrix ψ*i*.

*u*p, *y*p Input and output of the plant.

*nWu* Order of the weighting function for *u*.

xp, *<sup>k</sup>*,xd, *<sup>k</sup>*, State vectors of plant, disturbance and

x*Wy*, *<sup>k</sup>*,x*Wu*, *<sup>k</sup>* weighting functions.

**0** Zero matrix.

*Wy* , State-space matrices of the weighting function for *y*.

for Active Control of Harmonic Disturbances with Time-Varying Frequencies

83

LPV Gain-Scheduled Output Feedback

*Wu* , State-space matrices of the weighting function for *u*.

*<sup>u</sup>* , State-space matrices of the pLPV generalized plant.

*<sup>u</sup>* , State-space matrices of the LFT generalized plant.

<sup>A</sup>(*nWy* <sup>×</sup>*nWy* )

A(*nWu* <sup>×</sup>*nWu* )

<sup>A</sup>*i*(θ), <sup>B</sup>(pLPV)

<sup>A</sup>, <sup>B</sup>*<sup>θ</sup>* , <sup>B</sup>(LFT)

C(LFT)

C(LFT)

X(*n*×*n*)

Y (*n*×*n*) 1

<sup>B</sup>(2*n*×1)

C*<sup>θ</sup>* , D*θθ* , D*θw*, D*θu*,

*<sup>q</sup>* , <sup>D</sup>*q<sup>θ</sup>* , <sup>D</sup>(LFT)

*<sup>y</sup>* , <sup>D</sup>*y<sup>θ</sup>* , <sup>D</sup>(LFT)

N((*n*+3)×(*n*+2))

ψ(4*n*+3)×(4*n*+3)

<sup>X</sup>(2*n*×2*n*), <sup>A</sup>(2*n*×2*n*)

, <sup>C</sup>(2×2*n*)

*<sup>q</sup>* , <sup>D</sup>(pLPV)

C(pLPV)

C(pLPV) *<sup>y</sup>* , *<sup>D</sup>*(pLPV)

C(1×*nWu* )

<sup>C</sup>(1×*nWy* )

*Wy* , <sup>B</sup>(*nWy* <sup>×</sup>1)

*Wu* , <sup>B</sup>(*nWu* <sup>×</sup>1)

*<sup>w</sup>* , <sup>B</sup>(pLPV)

*qw* , <sup>D</sup>(pLPV) *qu*

*yw* , *<sup>D</sup>*(pLPV) *qw*

*<sup>w</sup>* , <sup>B</sup>(LFT)

*qw* , <sup>D</sup>(LFT) *qu* ,

*yw* , <sup>D</sup>(LFT) *yu*

**I** Identitiy matrix.

<sup>X</sup> , <sup>N</sup>((*n*+3)×(*n*+2))

*Wy* , *<sup>D</sup>*(1×1) *Wy*

*Wu* , *<sup>D</sup>*(1×1) *Wu*

*Wy*, *Wu* System representations of the weighting functions. <sup>A</sup>(*nWy* <sup>×</sup>*nWy* ) *Wy* , <sup>B</sup>(*nWy* <sup>×</sup>1) *Wy* , State-space matrices of the weighting function for *y*. <sup>C</sup>(1×*nWy* ) *Wy* , *<sup>D</sup>*(1×1) *Wy* A(*nWu* <sup>×</sup>*nWu* ) *Wu* , <sup>B</sup>(*nWu* <sup>×</sup>1) C(1×*nWu* ) *Wu* , *<sup>D</sup>*(1×1) *Wu* x*Wy*, *<sup>k</sup>*,x*Wu*, *<sup>k</sup>* weighting functions. <sup>A</sup>*i*(θ), <sup>B</sup>(pLPV) *<sup>w</sup>* , <sup>B</sup>(pLPV) C(pLPV) *<sup>q</sup>* , <sup>D</sup>(pLPV) *qw* , <sup>D</sup>(pLPV) *qu* C(pLPV) *<sup>y</sup>* , *<sup>D</sup>*(pLPV) *yw* , *<sup>D</sup>*(pLPV) *qw* **0** Zero matrix. <sup>A</sup>, <sup>B</sup>*<sup>θ</sup>* , <sup>B</sup>(LFT) *<sup>w</sup>* , <sup>B</sup>(LFT) C*<sup>θ</sup>* , D*θθ* , D*θw*, D*θu*, C(LFT) *<sup>q</sup>* , <sup>D</sup>*q<sup>θ</sup>* , <sup>D</sup>(LFT) *qw* , <sup>D</sup>(LFT) *qu* , C(LFT) *<sup>y</sup>* , <sup>D</sup>*y<sup>θ</sup>* , <sup>D</sup>(LFT) *yw* , <sup>D</sup>(LFT) *yu* N((*n*+3)×(*n*+2)) <sup>X</sup> , <sup>N</sup>((*n*+3)×(*n*+2)) X(*n*×*n*) Y (*n*×*n*) 1

18 Will-be-set-by-IN-TECH

*A<sup>i</sup>* Constant matrices of the polytopic representation of A(θ).

w*<sup>θ</sup>* , q*<sup>θ</sup>* Output and input of the parameter block for the plant in LFT form.

w˜ *<sup>θ</sup>* , ˜q*<sup>θ</sup>* Output and input of the parameter block for the controller in LFT form.

<sup>d</sup> , State-space matrices of the disturbance model for fixed frequencies.

*θ<sup>i</sup>* The *i*-th element of the parameter vector.

*M* Number of vertices of the polytope.

*λ<sup>j</sup>* The *j*-th element of the coordinate vector.

*n*<sup>d</sup> Number of frequencies of the disturbance.

*ai* Scalar parameter for the disturbance model.

Ad(θ)(2*n*d×2*n*d), State-space matrices of the pLPV disturbance model.

*A*d, *<sup>i</sup>* Constant matrices of the polytopic representation of Ad(θ).

Ad, *<sup>i</sup>*, Bd, *<sup>i</sup>*, Cd, *<sup>i</sup>* Block matrices of Ad, B<sup>d</sup> and Cd.

*T*, *fi* Sampling time and the *i*-th frequency.

*G*p System representation of the plant.

<sup>p</sup> , State-space matrices of the plant.

*nWy* Order of the weighting function for *y*.

*n*p Order of the plant.

Av, *<sup>j</sup>*, A(*vj*) System matrix for the *j*-th vertex. θ Parametric uncertainty block.

θ Parameter vector.

**Θ** Parameter polytope.

v*<sup>j</sup>* Vertices of the polytope.

*N* Number of parameters.

λ Coordinate vector.

A(2*n*d×2*n*d)

B(2*n*d×1) <sup>d</sup> , C(1×2*n*d) d

B(2*n*d×1) <sup>d</sup> , C(1×2*n*d) d

<sup>A</sup>(*n*p×*n*p)

<sup>B</sup>(*n*p×1) <sup>p</sup> , <sup>C</sup>(1×*n*p)

<sup>p</sup> , *<sup>D</sup>*(1×1) p

*nWu* Order of the weighting function for *u*.

*Wu* , State-space matrices of the weighting function for *u*.

xp, *<sup>k</sup>*,xd, *<sup>k</sup>*, State vectors of plant, disturbance and

*<sup>u</sup>* , State-space matrices of the pLPV generalized plant.

Ad, Bd, *<sup>θ</sup>* , Bd, *<sup>w</sup>*, Cd, *<sup>θ</sup>* , Cd, *<sup>y</sup>* State-space matrices of the LFT disturbance model. *w*d, *y*<sup>d</sup> Input and output of the disturbance model. *u*p, *y*p Input and output of the plant. *ai*, *pi* Scalar parameters for the disturbance model. *<sup>u</sup>* , State-space matrices of the LFT generalized plant.

<sup>Y</sup> Outer factors to build the LMIs. <sup>1</sup> , Solutions of the first set of LMIs.

**I** Identitiy matrix.

$$\begin{aligned} n &= n\_{\mathbf{P}} + 2n\_{\mathbf{d}} + n\_{W\_y} + n\_{W\_u} \\ \psi\_i^{(4n+3)\times(4n+3)} \\ \mathbf{X}^{(2n\times2n)}, \overline{\mathbf{A}\_i^{(2n\times2n)}} \\ \overline{\mathbf{B}}^{(2n\times1)}, \overline{\mathbf{C}}^{(2\times2n)} \end{aligned}$$

*n* = *n*<sup>p</sup> + 2*n*<sup>d</sup> + *nWy* + *nWu* Order of matrices X<sup>1</sup> and Y1. *<sup>i</sup>* Matrix to build the basic LMI. *<sup>i</sup>* , Matrices to build matrix ψ*i*.

P ((*n*+1)×(4*n*+3)), Q((*n*+1)×(4*n*+3)) Matrices to build the basic LMI.

B(2*n*×(*n*+1)), C((*n*+1)×2*n*) , Matrices to obtain P (pLPV) and Q(pLPV) . D(2×(*n*+1)) *qu* , <sup>D</sup>((*n*+1)×1) *yw* **<sup>Ω</sup>**((*n*+1)×(*n*+1)) *<sup>i</sup>* Solution of the basic LMI for the *i*-th vertex. A(*n*×*n*) *Ki* , <sup>B</sup>(*n*×1) *Ki* , State-space matrices of the controller for the C(1×*n*) *Ki* , *<sup>D</sup>*(1×1) *Ki i*-th vertex. N((*n*+2*n*d+3)×(*n*+2*n*d+2)) <sup>R</sup> , Outer factors to build the LMIs. N((*n*+2*n*d+3)×(*n*+2*n*d+2)) S R(*n*×*n*), S(*n*×*n*), Solutions of the first set of LMIs. J(*n*d×*n*d) <sup>3</sup> , <sup>L</sup>(*n*d×*n*d) 3 *γ* Upper bound of the maximum singular value. M(*n*×*n*), N(*n*×*n*) Matrices calculated from R and S. L(*n*d×*n*d) <sup>1</sup> , <sup>L</sup>(*n*d×*n*d) <sup>2</sup> Matrices to build L. ψ((4*n*+4*n*d+3)×(4*n*+4*n*d+3)) Matrix to build the basic LMI. X(2*n*×2*n*), A(2*n*×2*n*) <sup>0</sup> , Matrices needed to build ψ. B(2*n*×(2*n*d+1)) <sup>0</sup> , <sup>C</sup>((2*n*d+2)×2*n*) 0 D((2*n*d+2)×(2*n*d+1)) <sup>0</sup> , <sup>J</sup>((2*n*d+2)×(2*n*d+2)) <sup>0</sup> , L((2*n*d+1)×(2*n*d+1)) <sup>0</sup> , <sup>J</sup>(2*n*d×2*n*d), L(2*n*d×2*n*d) P ((*n*+*n*d+1)×(4*n*+4*n*d+3)), Matrices to build the basic LMI. Q((*n*+*n*d+1)×(4*n*+4*n*d+3)) B˜ (2*n*×(*n*+*n*d+1)), C˜ ((*n*+*n*d+1)×2*n*), Matrices to obtain P (LFT) and Q(LFT) . D˜ ((2*n*d+2)×(*n*+*n*d+1)) <sup>12</sup> , D˜ ((*n*+*n*d+1)×(2*n*d+1)) 21 **Ω**((*n*+*n*d+1)×(*n*+*n*d+1)) Controller matrix. A(*n*×*n*) *<sup>K</sup>* , <sup>B</sup>(*n*×(*n*d+1)) *<sup>K</sup>* , State-space matrices of the controller. C((*n*d+1)×*n*) *<sup>K</sup>* , <sup>D</sup>((*n*d+1)×(*n*d+1)) *K*

**8. References**

19:1479-92.

65-70.

Francisco, June 2011. 1340-45.

March 2003. Paper No. 5049-68.

Anchorage, May 2002. 4668-69.

*Transactions on Signal Processing* 41:1518-31.

Krakow-Wojanow, Poland, June 2011.

Accepted for publication.

4788-93.

*International Journal of Robust and Nonlinear Control* 4:421-48.

*Identification*. Newcastle, Australia, March 2006. 273-78.

*Applications* 150:132-38.

*Congress*. Milan, August 2011. 7897-902.

scheduling. *Control Engineering Practice* 12:1029-39.

[1] Apkarian, P. and P. Gahinet. 1995. A convex charachterization of gain-scheduled *H*∞

for Active Control of Harmonic Disturbances with Time-Varying Frequencies

85

LPV Gain-Scheduled Output Feedback

[2] Ballesteros, P. and C. Bohn. 2011a. A frequency-tunable LPV controller for narrowband active noise and vibration control. *Proceedings of the American Control Conference*. San

[3] Ballesteros, P. and C. Bohn. 2011b. Disturbance rejection through LPV gain-scheduling control with application to active noise cancellation. *Proceedings of the IFAC World*

[4] Balini, H. M. N. K., C. W. Scherer and J. Witte. 2011. Performance enhancement for AMB systems using unstable *H*∞ controllers. *IEEE Transactions on Control Systems Technology*

[5] Bohn, C., A. Cortabarria, V. Härtel and K. Kowalczyk. 2003. Disturbance-observer-based active control of engine-induced vibrations in automotive vehicles. *Proceedings of the SPIE's 10th Annual International Symposium on Smart Structures and Materials*. San Diego,

[6] Bohn, C., A. Cortabarria, V. Härtel and K. Kowalczyk. 2004. Active control of engine-induced vibrations in automotive vehicles using disturbance observer gain

[7] Darengosse, C. and P. Chevrel. 2000. Linear parameter-varying controller design for active power filters. *Proceedings of the IFAC Control Systems Design*. Bratislava, June 2000.

[8] Du, H. and X. Shi. 2002. Gain-scheduled control for use in vibration suppression of system with harmonic excitation. *Proceedings of the American Control Conference*.

[9] Du, H., L. Zhang and X. Shi. 2003. LPV technique for the rejection of sinusoidal disturbance with time-varying frequency. *IEE Proceedings on Control Theory and*

[10] Feintuch, P. L., N. J. Bershad and A. K. Lo. 1993. A frequency-domain model for filtered LMS algorithms - Stability analysis, design, and elimination of the training mode. *IEEE*

[11] Gahinet, P. and P. Apkarian. 1994. A linear matrix inequality approach to *H*∞ control.

[12] Heins, W., P. Ballesteros and C. Bohn. 2011. Gain-scheduled state-feedback control for active cancellation of multisine disturbances with time-varying frequencies. Presented at the *10th MARDiH Conference on Active Noise and Vibration Control Methods*.

[13] Heins, W., P. Ballesteros and C. Bohn. 2012. Experimental evaluation of an LPV-gain-scheduled observer for rejecting multisine disturbances with time-varying frequencies. *Proceedings of the American Control Conference*. Montreal, June 2012.

[14] Kinney, C. E. and R. A. de Callafon. 2006a. Scheduling control for periodic disturbance attenuation. *Proceedings of the American Control Conference*. Minneapolis, June 2006.

[15] Kinney, C. E. and R. A. de Callafon. 2006b. An adaptive internal model-based controller for periodic disturbance rejection. *Proceedings of the 14th IFAC Symposium on System*

controllers. *IEEE Transactions on Automatic Control* 40:853-64.

### **Author details**

Pablo Ballesteros, Xinyu Shu, Wiebke Heins and Christian Bohn *Institute of Electrical Information Technology, Clausthal University of Technology, Clausthal-Zellerfeld, Germany*

### **8. References**

20 Will-be-set-by-IN-TECH

*Ki* , State-space matrices of the controller for the

and Q(pLPV)

and Q(LFT)

.

.

, Matrices to obtain P (pLPV)

*<sup>i</sup>* Solution of the basic LMI for the *i*-th vertex.

*γ* Upper bound of the maximum singular value.

*Ki i*-th vertex.

<sup>R</sup> , Outer factors to build the LMIs.

R(*n*×*n*), S(*n*×*n*), Solutions of the first set of LMIs.

M(*n*×*n*), N(*n*×*n*) Matrices calculated from R and S.

<sup>0</sup> , Matrices needed to build ψ.

<sup>2</sup> Matrices to build L. ψ((4*n*+4*n*d+3)×(4*n*+4*n*d+3)) Matrix to build the basic LMI.

P ((*n*+*n*d+1)×(4*n*+4*n*d+3)), Matrices to build the basic LMI.

*<sup>K</sup>* , State-space matrices of the controller.

B˜ (2*n*×(*n*+*n*d+1)), C˜ ((*n*+*n*d+1)×2*n*), Matrices to obtain P (LFT)

Pablo Ballesteros, Xinyu Shu, Wiebke Heins and Christian Bohn

*Institute of Electrical Information Technology, Clausthal University of Technology,*

**Ω**((*n*+*n*d+1)×(*n*+*n*d+1)) Controller matrix.

<sup>0</sup> ,

B(2*n*×(*n*+1)), C((*n*+1)×2*n*)

*qu* , <sup>D</sup>((*n*+1)×1) *yw*

N((*n*+2*n*d+3)×(*n*+2*n*d+2))

N((*n*+2*n*d+3)×(*n*+2*n*d+2))

<sup>3</sup> , <sup>L</sup>(*n*d×*n*d) 3

<sup>1</sup> , <sup>L</sup>(*n*d×*n*d)

X(2*n*×2*n*), A(2*n*×2*n*)

D((2*n*d+2)×(2*n*d+1))

L((2*n*d+1)×(2*n*d+1))

Q((*n*+*n*d+1)×(4*n*+4*n*d+3))

D˜ ((2*n*d+2)×(*n*+*n*d+1)) <sup>12</sup> , D˜ ((*n*+*n*d+1)×(2*n*d+1))

*<sup>K</sup>* , <sup>B</sup>(*n*×(*n*d+1))

*<sup>K</sup>* , <sup>D</sup>((*n*d+1)×(*n*d+1)) *K*

*Clausthal-Zellerfeld, Germany*

<sup>0</sup> , <sup>C</sup>((2*n*d+2)×2*n*) 0

<sup>0</sup> , <sup>J</sup>(2*n*d×2*n*d),

<sup>0</sup> , <sup>J</sup>((2*n*d+2)×(2*n*d+2))

B(2*n*×(2*n*d+1))

L(2*n*d×2*n*d)

21

A(*n*×*n*)

C((*n*d+1)×*n*)

**Author details**

D(2×(*n*+1))

A(*n*×*n*)

C(1×*n*)

S

J(*n*d×*n*d)

L(*n*d×*n*d)

**<sup>Ω</sup>**((*n*+1)×(*n*+1))

*Ki* , <sup>B</sup>(*n*×1)

*Ki* , *<sup>D</sup>*(1×1)


	- [16] Kinney, C. E. and R. A. de Callafon. 2007. A comparison of fixed point designs and time-varying observers for scheduling repetitive controllers. *Proceedings of the 46th IEEE Conference on Decision and Control*. New Orleans, December 2007. 2844-49.

**Active Vibration Control Using a Kautz Filter**

**Chapter 4**

Impulse response functions (IRFs) have been largely used in experimental modal analysis in order to extract the modal parameters (natural frequencies, damping factors and modal forms) in different areas. IRFs occupy a prominent place in applications of aeronautical, machinery and automobile industries, mainly when the system has coupled modes. Additionally, IRFs

• For very complex systems, they can be determined by experimental tests, or using data of input and output measured by load cells or accelerometers, or directly with an impact

• Normally a finite impulse response (FIR) model of the structure is employed. Thus, the stability can be warranted a priori. Additionally, the many adaptive controlers are based

In general, IRFs can be identified by impact tests with an instrumented hammer or by using numerical algorithms implemented in commercial software. IRFs can be determined with those algorithms through different methods, e. g., the covariance method based on the sum of convolutions of the measured input forces. However, there is an over parametrization that is a drawback when the lag memory is high. Fortunately, an expansion of the IRFs into orthonormal basis functions can enhance the procedure of reducing the number of parameters [15]. For describing mechanical vibrating systems, Kautz filters are interesting orthogonal functions set in Hilbert space [21] that include a priori knowledge about the dominant poles. The eigenvalues associated to vibrating mechanical systems are conjugated complex poles, so, the IRFs can be expanded in orthonormal basis functions with those conditions. Kautz filters are orthogonal funcions that can be used for this purpose. These filters can decrease the computational cost and accelerate the convergence rate providing a good estimate of the IRFs

> ©2012 Lopes Junior et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2012 Lopes Junior et al., licensee InTech. This is a paper distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Samuel da Silva, Vicente Lopes Junior and Michael J. Brennan

have practical advantages for use in control theory for many reasons, e.g.:

Additional information is available at the end of the chapter

• The identified model is essentially nonparametric.

on an FIR structure and it is easy to perform a recursive estimation.

cited.

http://dx.doi.org/10.5772/50966

**1. Introduction**

hammer.

[14].


## **Active Vibration Control Using a Kautz Filter**

Samuel da Silva, Vicente Lopes Junior and Michael J. Brennan

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50966

**1. Introduction**

22 Will-be-set-by-IN-TECH

[16] Kinney, C. E. and R. A. de Callafon. 2007. A comparison of fixed point designs and time-varying observers for scheduling repetitive controllers. *Proceedings of the 46th IEEE*

[17] Köro ˘glu, H. and C. W. Scherer. 2008. LPV control for robust attenuation of non-stationary sinusoidal disturbances with measurable frequencies. *Proceedings of the*

[18] Shu, X., P. Ballesteros and C. Bohn. 2011. Active vibration control for harmonic disturbances with time-varying frequencies through LPV gain scheduling. *Proceedings of the 23rd Chinese Control and Decision Conference*. Mianyang, China, May 2011. 728-33. [19] Witte, J., H. M. N. K. Balini and C. W. Scherer. 2010. Experimental results with stable and unstable LPV controllers for active magnetic bearing systems. *Proceedings of the IEEE International Conference on Control Applications*. Yokohama, September 2010. 950-55.

*Conference on Decision and Control*. New Orleans, December 2007. 2844-49.

*17th IFAC World Congress*. Korea, July 2008. 4928-33.

Impulse response functions (IRFs) have been largely used in experimental modal analysis in order to extract the modal parameters (natural frequencies, damping factors and modal forms) in different areas. IRFs occupy a prominent place in applications of aeronautical, machinery and automobile industries, mainly when the system has coupled modes. Additionally, IRFs have practical advantages for use in control theory for many reasons, e.g.:


perform a recursive estimation.

In general, IRFs can be identified by impact tests with an instrumented hammer or by using numerical algorithms implemented in commercial software. IRFs can be determined with those algorithms through different methods, e. g., the covariance method based on the sum of convolutions of the measured input forces. However, there is an over parametrization that is a drawback when the lag memory is high. Fortunately, an expansion of the IRFs into orthonormal basis functions can enhance the procedure of reducing the number of parameters [15]. For describing mechanical vibrating systems, Kautz filters are interesting orthogonal functions set in Hilbert space [21] that include a priori knowledge about the dominant poles. The eigenvalues associated to vibrating mechanical systems are conjugated complex poles, so, the IRFs can be expanded in orthonormal basis functions with those conditions. Kautz filters are orthogonal funcions that can be used for this purpose. These filters can decrease the computational cost and accelerate the convergence rate providing a good estimate of the IRFs [14].

©2012 Lopes Junior et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Lopes Junior et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 2 Will-be-set-by-IN-TECH 88 Advances on Analysis and Control of Vibrations – Theory and Applications Active Vibration Control Using a Kautz Filter <sup>3</sup>

Kautz filters have found several applications, e.g., acoustic and audio [20], circuit theory [17], experimental modal analysis in mechanical systems [2, 12–14], vibration control [6], model reduction [4], robust control [18], predictive control [19], general system identification [5, 16, 22, 25], non-linear system identification with Volterra models [7–11], etc. Although it may seem that the mathematical and theoretical aspects of Kautz filters are more interesting for academic purposed, some practical applications can be found in the literature. For example, the flight testing certification of aircrafts for aeroelastic stability was completely charecterized through a series connection of Kautz filters in [1]. The application used a simulated nonlinear prototypical two-dimensional wing section and F/A-18 active aeroelastic wing ground vibration test data.

(IRF). The measured output signal is given by *y*(*k*) = *y*˜(*k*) + *w*(*k*), where *w*(*k*) is a white or colored noise. The eq. (1) represents a sum of convolution between the input signal *u*(*k*) and the IRF *h*(*k*). In mechanical and vibrating systems applications, the IRF can be obtained by impact tests with a hammer or by using numerical algorithms based on time or frequency

Normally, to obtain the IRF, eq. (1), is truncated in *N* terms by considering |*h*(*k*)| *< �*, ∀*j > N*,

The approach in eq. (2) changes an infinite impulse response model (IIR) into a finite impulse response model (FIR). The most common method to identify the *h*(*k*) is by using the correlation functions due to the robustness to noise issues yielding to the classical

*h*(*i*)*u*(*k* − *i*) (2)

Active Vibration Control Using a Kautz Filter 89

*h*(*i*)*Ruu*(*k* − *i*) (3)

*αjψj*(*k*), *k* = 0, 1, . . . , *N* (4)

*N* ∑ *i*=0

*N* ∑ *i*=0

• the stability of the identified model is guaranteed a priori, since the model is FIR.

• the model is linear in the parameters, hence the LS approach can be performed.

**3. Covariance method expanded in orthonormal basis functions**

*J* ∑ *j*=0

where the correlation function *Ruu*(*k*) and cross-correlation function *Ruy*(*k*) can be estimated experimentally. Based on eq. (3), a least-square (LS) identification method can be performed to estimate the expansion coefficients in the time-series that describes the FIR model *h*(*k*). This approach for estimating an IRF has some advantages over other estimators, for instance:

• the model is assumed to be described only for arbitrary zeros and poles at the origin of the

However, this identification technique often leads to conservative results because a common vibration system is hardly ever represented by a FIR model. Thus, the practical drawback is that a large number of parameters *h*(*k*) must be considered in order to obtain a good approach in eq. (3). In order to overcome this drawback, a set of orthonormal basis functions can be employed to expand the covariance method and reduces the number of parameters. Next

The IRF *h*(*k*) can alternatively be written using *αj*, *j* = 0, 1, . . . , *J*, as expansion coefficients

*y*(*k*) ≈

*Ruy*(*k*) ≈

where *�* is a residue. In this case, eq. (1) can be given as:

measured signals.

Wiener-Hopf equation:

complex plane.

described by *z*-function Ψ*j*(*z*):

section provides some considerations in this sense.

*h*(*k*) =

In specific control applications with Kautz filters, the strategies are, normally, based on active noise control using feedforward compensation, e. g. as performed in [26]. It is well-known that Wiener theory can be used to describe internal model control to change the control architecture from feedforward to feedback [3]. However, feedback compensation can also be directly implemented. Thus, the goal of the present chapter is to apply Kautz filters for active vibration control. The main steps and characteristics involved in this procedure are described. Specifically, this chapter emphasises the following:


The chapter is organized as follows. First, the IRF identification and covariance method is reviewed briefly, followed by the Kautz filter with multiple poles for expansion of impulse response. After, a vibration control strategy is described and example applications involving single-input-single-output vibrating systems are used to illustrate the approach. Finally, the results are discussed and suggestions for a non-linear identification procedure are proposed.

### **2. Impulse response function**

The output *y*˜(*k*) of a linear discrete-time and invariant system can be written as:

$$\tilde{y}(k) = \sum\_{i=0}^{\infty} h(i)u(k-i) \tag{1}$$

where the sequences {*u*(*k*), *k* = 0, 1, . . . , *Tf* } and {*y*˜(*k*), *k* = 0, 1 . . . , *Tf* } are the sampled input and output signals, respectively; *Tf* is the final time, and *h*(*k*) is the impulse response function (IRF). The measured output signal is given by *y*(*k*) = *y*˜(*k*) + *w*(*k*), where *w*(*k*) is a white or colored noise. The eq. (1) represents a sum of convolution between the input signal *u*(*k*) and the IRF *h*(*k*). In mechanical and vibrating systems applications, the IRF can be obtained by impact tests with a hammer or by using numerical algorithms based on time or frequency measured signals.

2 Will-be-set-by-IN-TECH

Kautz filters have found several applications, e.g., acoustic and audio [20], circuit theory [17], experimental modal analysis in mechanical systems [2, 12–14], vibration control [6], model reduction [4], robust control [18], predictive control [19], general system identification [5, 16, 22, 25], non-linear system identification with Volterra models [7–11], etc. Although it may seem that the mathematical and theoretical aspects of Kautz filters are more interesting for academic purposed, some practical applications can be found in the literature. For example, the flight testing certification of aircrafts for aeroelastic stability was completely charecterized through a series connection of Kautz filters in [1]. The application used a simulated nonlinear prototypical two-dimensional wing section and F/A-18 active aeroelastic

In specific control applications with Kautz filters, the strategies are, normally, based on active noise control using feedforward compensation, e. g. as performed in [26]. It is well-known that Wiener theory can be used to describe internal model control to change the control architecture from feedforward to feedback [3]. However, feedback compensation can also be directly implemented. Thus, the goal of the present chapter is to apply Kautz filters for active vibration control. The main steps and characteristics involved in this procedure are described.

• It is not necessary to have a complete mathematical model and the controller is designed

• The control method is based on experimental IRFs (nonparametric) and in orthonormal basis functions. Thus, the method is grey-box because prior knowledge of the mechanical vibrating system treated is assumed (poles of Kautz filter to represent the system).

• An example of a single-degree of freedom mechanical model is used to illustrate the main

• Additionally, an experimental example by using a clamped beam with PZT actuator and

The chapter is organized as follows. First, the IRF identification and covariance method is reviewed briefly, followed by the Kautz filter with multiple poles for expansion of impulse response. After, a vibration control strategy is described and example applications involving single-input-single-output vibrating systems are used to illustrate the approach. Finally, the results are discussed and suggestions for a non-linear identification procedure are proposed.

The output *y*˜(*k*) of a linear discrete-time and invariant system can be written as:

∞ ∑ *i*=0

where the sequences {*u*(*k*), *k* = 0, 1, . . . , *Tf* } and {*y*˜(*k*), *k* = 0, 1 . . . , *Tf* } are the sampled input and output signals, respectively; *Tf* is the final time, and *h*(*k*) is the impulse response function

*h*(*i*)*u*(*k* − *i*) (1)

*y*˜(*k*) =

wing ground vibration test data.

steps.

PVDF sensor is presented.

**2. Impulse response function**

Specifically, this chapter emphasises the following:

• Feedback control, considering dynamic canceling.

directly in the digital domain for fast practical implementation.

Additionally, complex vibration system can be controlled.

Normally, to obtain the IRF, eq. (1), is truncated in *N* terms by considering |*h*(*k*)| *< �*, ∀*j > N*, where *�* is a residue. In this case, eq. (1) can be given as:

$$y(k) \approx \sum\_{i=0}^{N} h(i)u(k-i) \tag{2}$$

The approach in eq. (2) changes an infinite impulse response model (IIR) into a finite impulse response model (FIR). The most common method to identify the *h*(*k*) is by using the correlation functions due to the robustness to noise issues yielding to the classical Wiener-Hopf equation:

*Ruy*(*k*) ≈ *N* ∑ *i*=0 *h*(*i*)*Ruu*(*k* − *i*) (3)

where the correlation function *Ruu*(*k*) and cross-correlation function *Ruy*(*k*) can be estimated experimentally. Based on eq. (3), a least-square (LS) identification method can be performed to estimate the expansion coefficients in the time-series that describes the FIR model *h*(*k*). This approach for estimating an IRF has some advantages over other estimators, for instance:


However, this identification technique often leads to conservative results because a common vibration system is hardly ever represented by a FIR model. Thus, the practical drawback is that a large number of parameters *h*(*k*) must be considered in order to obtain a good approach in eq. (3). In order to overcome this drawback, a set of orthonormal basis functions can be employed to expand the covariance method and reduces the number of parameters. Next section provides some considerations in this sense.

### **3. Covariance method expanded in orthonormal basis functions**

The IRF *h*(*k*) can alternatively be written using *αj*, *j* = 0, 1, . . . , *J*, as expansion coefficients described by *z*-function Ψ*j*(*z*):

$$h(k) = \sum\_{j=0}^{f} a\_j \psi\_j(k), \qquad k = 0, 1, \dots, N \tag{4}$$

where *ψj*(*k*) is the IRF of the transfer function Ψ*j*(*z*). The *z* transform of eq. (4) is given by a linear combination of the functions Ψ*j*(*z*):

$$H(z) \approx a\_0 \Psi\_0(z) + a\_1 \Psi\_1(z) + \dots + a\_I \Psi\_I(z) = \sum\_{j=0}^I a\_j \Psi\_j(z) \tag{5}$$

**4. Kautz filter**

The Kautz filters can be given by [16, 22, 24]:

Ψ2*n*(*z*) =

<sup>Ψ</sup>2*n*−1(*z*) =

filter through the relations:

error signal can be written by:

defined by the poles *β<sup>j</sup>* and *β*<sup>∗</sup>

in this point can be found in [12].

<sup>1</sup> Considering *h*(*k*) is a causal sequence.

**5. Active vibration control strategy**

*H*(*z*) =

+∞ ∑ *n*=0

where *y*ˆ(*k*) is the predicted output signal by the IRF ˆ

(<sup>1</sup> <sup>−</sup> *<sup>c</sup>*2)(<sup>1</sup> <sup>−</sup> *<sup>b</sup>*2)*<sup>z</sup> <sup>z</sup>*<sup>2</sup> + *<sup>b</sup>*(*<sup>c</sup>* − <sup>1</sup>)*<sup>z</sup>* − *<sup>c</sup>*

<sup>1</sup> − *<sup>c</sup>*2*z*(*<sup>z</sup>* − *<sup>b</sup>*) *<sup>z</sup>*<sup>2</sup> + *<sup>b</sup>*(*<sup>c</sup>* − <sup>1</sup>)*<sup>z</sup>* − *<sup>c</sup>*

√

where the constants *b* and *c* are relative to the poles *β* = *σ* + *jω* and *β*<sup>∗</sup> = *σ* − *jω* in the *j*-th

(1 + *βjβ*<sup>∗</sup> *j* )

A sequence of filters is utilized with different poles in each section describing the modal behavior in the frequency range of interest. A question is relative for choosing the poles and the IRFs iteratively based on application of eq. (2) and output experimental signal *ye*(*k*). An

*j* )

*<sup>b</sup>* <sup>=</sup> (*β<sup>j</sup>* <sup>+</sup> *<sup>β</sup>*<sup>∗</sup>

*c* = −*ββ*<sup>∗</sup>

*<sup>j</sup>* in the *z*-domain:

*y*ˆ(*k*) =

in *<sup>z</sup>*-domain can be described by applying the *<sup>z</sup>*<sup>−</sup> transform in the IRF *<sup>h</sup>*(*k*)1:

*N* ∑ *i*=0 ˆ

The optimization problem can be described by objective function that employs an Euclidean norm and the Kautz poles are functions of the frequencies and damping factors that are the optimization parameters. These parameters can be restricted in a range searching. This optimization problem can be solved by several classical approaches. A detailed explanation

If an IRF is well identified through covariance method expanded with Kautz filters, a model

<sup>−</sup>*cz*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*(*<sup>c</sup>* <sup>−</sup> <sup>1</sup>)*<sup>z</sup>* <sup>+</sup> <sup>1</sup> *<sup>z</sup>*<sup>2</sup> + *<sup>b</sup>*(*<sup>c</sup>* − <sup>1</sup>)*<sup>z</sup>* − *<sup>c</sup>*

<sup>−</sup>*cz*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*(*<sup>c</sup>* <sup>−</sup> <sup>1</sup>)*<sup>z</sup>* <sup>+</sup> <sup>1</sup> *<sup>z</sup>*<sup>2</sup> + *<sup>b</sup>*(*<sup>c</sup>* − <sup>1</sup>)*<sup>z</sup>* − *<sup>c</sup>*

*<sup>n</sup>*−<sup>1</sup>

Active Vibration Control Using a Kautz Filter 91

*<sup>n</sup>*−<sup>1</sup>

, (12)

*h*(*k*) estimated considering Kautz basis

*h*(*i*)*u*(*k* − *i*) (15)

*<sup>j</sup>* (13)

*e*(*k*) = *y*ˆ(*k*) − *ye*(*k*) (14)

*<sup>h</sup>*(*n*)*z*−*<sup>n</sup>* <sup>≈</sup> *<sup>α</sup>*0Ψ0(*z*) + *<sup>α</sup>*1Ψ1(*z*) + ··· <sup>+</sup> *<sup>α</sup>J*Ψ*J*(*z*) (16)

(10)

(11)

The convergence of Ψ*j*(*z*) is related to the completeness properties of these subsets of functions. If the functions Ψ*j*(*z*) are properly chosen (poles placement), the order *J << N*. Thus, it is easier to identify the coefficient *α<sup>j</sup>* using eq. (4) [2, 12, 14, 15, 22, 24], which can be written in a matrix form:

$$\begin{Bmatrix} h(0) \\ h(1) \\ \vdots \\ h(N) \end{Bmatrix} = \begin{bmatrix} \psi\_0(0) & \psi\_1(0) & \cdots & \psi\_I(0) \\ \psi\_0(1) & \psi\_1(1) & \cdots & \psi\_I(1) \\ \vdots & \vdots & \ddots & \vdots \\ \psi\_0(N) & \psi\_1(N) & \cdots & \psi\_I(N) \end{bmatrix} \begin{Bmatrix} \alpha\_0 \\ \alpha\_1 \\ \vdots \\ \alpha\_I \end{Bmatrix} \tag{6}$$

By incorporating the eq. (4) into Wiener-Hopf equation, eq. (3), one can obtain:

$$R\_{\text{ldy}}(k) \approx \sum\_{i=0}^{N} h(i) R\_{\text{ldu}}(k-i) \equiv \sum\_{i=0}^{N} \sum\_{j=0}^{I} a\_j \psi\_j(i) R\_{\text{luu}}(k-i)$$

$$= \sum\_{j=0}^{I} a\_j \sum\_{i=0}^{N} \psi\_j(i) R\_{\text{luu}}(k-i) = \sum\_{j=0}^{I} a\_j v\_j(k) \tag{7}$$

where *vj*(*k*), *k* = 0, ··· , *N* is the input signal *Ruu*(*k*) processed by each element of the discrete-time function *ψj*(*k*), *j* = 0, 1, . . . , *J*, which forms the approximation base and is the IRF of the orthogonal function:

$$w\_{\bar{\jmath}}(k) = \sum\_{i=0}^{N} \psi\_{\bar{\jmath}}(i) \mathbb{R}\_{\mu u}(k - i) \tag{8}$$

Eq. (8) is basically a filtering of the input signal *Ruu*(*k*) by a set of filter *ψj*(*k*). Finally, the eq. (7) is used to describe:

$$\begin{Bmatrix} R\_{\mathcal{U}}(0) \\ R\_{\mathcal{u}\mathcal{y}}(1) \\ \vdots \\ \vdots \\ R\_{\mathcal{u}\mathcal{y}}(N) \end{Bmatrix} = \begin{bmatrix} v\_0(0) & v\_1(0) & \cdots & v\_I(0) \\ v\_0(1) & v\_1(1) & \cdots & v\_I(1) \\ \vdots & \vdots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \vdots \\ v\_0(N) & v\_1(N) & \cdots & v\_I(N) \end{bmatrix} \begin{Bmatrix} \boldsymbol{a}\_0 \\ \boldsymbol{a}\_1 \\ \vdots \\ \boldsymbol{a}\_I \end{Bmatrix} \tag{9}$$

The effectiveness of the model is limited by the choice of the filters Ψ*j*(*z*). Thus, the choice of the basis functions is very important. For describing mechanical vibration and flexible systems, the Kautz functions have been demonstrated to provide a good generalization by including complex poles in the *z*-domain [2, 14].

### **4. Kautz filter**

4 Will-be-set-by-IN-TECH

where *ψj*(*k*) is the IRF of the transfer function Ψ*j*(*z*). The *z* transform of eq. (4) is given by a

The convergence of Ψ*j*(*z*) is related to the completeness properties of these subsets of functions. If the functions Ψ*j*(*z*) are properly chosen (poles placement), the order *J << N*. Thus, it is easier to identify the coefficient *α<sup>j</sup>* using eq. (4) [2, 12, 14, 15, 22, 24], which can be

> *ψ*0(0) *ψ*1(0) ··· *ψJ*(0) *ψ*0(1) *ψ*1(1) ··· *ψJ*(1)

*ψ*0(*N*) *ψ*1(*N*) ··· *ψJ*(*N*)

*N* ∑ *i*=0

where *vj*(*k*), *k* = 0, ··· , *N* is the input signal *Ruu*(*k*) processed by each element of the discrete-time function *ψj*(*k*), *j* = 0, 1, . . . , *J*, which forms the approximation base and is the

Eq. (8) is basically a filtering of the input signal *Ruu*(*k*) by a set of filter *ψj*(*k*). Finally, the eq.

*v*0(0) *v*1(0) ··· *vJ*(0) *v*0(1) *v*1(1) ··· *vJ*(1)

*v*0(*N*) *v*1(*N*) ··· *vJ*(*N*)

The effectiveness of the model is limited by the choice of the filters Ψ*j*(*z*). Thus, the choice of the basis functions is very important. For describing mechanical vibration and flexible systems, the Kautz functions have been demonstrated to provide a good generalization by

. ... .

. . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎧ ⎪⎪⎪⎪⎪⎨

*α*0 *α*1 . . . *αJ* ⎫ ⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⎪⎪⎪⎪⎪⎩

*N* ∑ *i*=0

> . . . . .

*J* ∑ *j*=0

*ψj*(*i*)*Ruu*(*k* − *i*) =

. ... .

. . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎧ ⎪⎪⎪⎪⎪⎨

*α*0 *α*1 . . . *αJ*

⎫ ⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⎪⎪⎪⎪⎪⎩

*αjψj*(*i*)*Ruu*(*k* − *i*)

*J* ∑ *j*=0

*ψj*(*i*)*Ruu*(*k* − *i*) (8)

*J* ∑ *j*=0

*αj*Ψ*j*(*z*) (5)

*αjvj*(*k*) (7)

(6)

(9)

*H*(*z*) ≈ *α*0Ψ0(*z*) + *α*1Ψ1(*z*) + ··· + *αJ*Ψ*J*(*z*) =

. . . . .

By incorporating the eq. (4) into Wiener-Hopf equation, eq. (3), one can obtain:

*h*(*i*)*Ruu*(*k* − *i*) ≡

linear combination of the functions Ψ*j*(*z*):

⎧ ⎪⎪⎪⎪⎪⎪⎨

*h*(0) *h*(1) . . . *h*(*N*) ⎫ ⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

*N* ∑ *i*=0 =

= *J* ∑ *j*=0 *αj N* ∑ *i*=0

*vj*(*k*) =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎪⎪⎪⎪⎪⎪⎩

*Ruy*(*k*) ≈

⎧ ⎪⎪⎪⎪⎪⎪⎨

*Ruy*(0) *Ruy*(1) . . . *Ruy*(*N*) ⎫ ⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

=

⎪⎪⎪⎪⎪⎪⎩

including complex poles in the *z*-domain [2, 14].

IRF of the orthogonal function:

(7) is used to describe:

written in a matrix form:

The Kautz filters can be given by [16, 22, 24]:

$$\Psi\_{2n}(z) = \frac{\sqrt{(1-c^2)(1-b^2)z}}{z^2 + b(c-1)z - c} \left[ \frac{-cz^2 + b(c-1)z + 1}{z^2 + b(c-1)z - c} \right]^{n-1} \tag{10}$$

$$\Psi\_{2n-1}(z) = \frac{\sqrt{1-c^2}z(z-b)}{z^2 + b(c-1)z - c} \left[ \frac{-cz^2 + b(c-1)z + 1}{z^2 + b(c-1)z - c} \right]^{n-1} \tag{11}$$

where the constants *b* and *c* are relative to the poles *β* = *σ* + *jω* and *β*<sup>∗</sup> = *σ* − *jω* in the *j*-th filter through the relations:

$$b = \frac{(\beta\_{\dot{j}} + \beta\_{\dot{j}}^{\*})}{(1 + \beta\_{\dot{j}}\beta\_{\dot{j}}^{\*})'} \tag{12}$$

$$\mathfrak{c} = -\mathfrak{B}\mathfrak{F}\_{\mathfrak{j}}^{\*} \tag{13}$$

A sequence of filters is utilized with different poles in each section describing the modal behavior in the frequency range of interest. A question is relative for choosing the poles and the IRFs iteratively based on application of eq. (2) and output experimental signal *ye*(*k*). An error signal can be written by:

$$
\varepsilon(k) = \mathfrak{Y}(k) - y\_{\varepsilon}(k) \tag{14}
$$

where *y*ˆ(*k*) is the predicted output signal by the IRF ˆ *h*(*k*) estimated considering Kautz basis defined by the poles *β<sup>j</sup>* and *β*<sup>∗</sup> *<sup>j</sup>* in the *z*-domain:

$$\mathcal{Y}(k) = \sum\_{i=0}^{N} \hat{h}(i)\mu(k-i) \tag{15}$$

The optimization problem can be described by objective function that employs an Euclidean norm and the Kautz poles are functions of the frequencies and damping factors that are the optimization parameters. These parameters can be restricted in a range searching. This optimization problem can be solved by several classical approaches. A detailed explanation in this point can be found in [12].

### **5. Active vibration control strategy**

If an IRF is well identified through covariance method expanded with Kautz filters, a model in *<sup>z</sup>*-domain can be described by applying the *<sup>z</sup>*<sup>−</sup> transform in the IRF *<sup>h</sup>*(*k*)1:

$$H(z) = \sum\_{n=0}^{+\infty} h(n)z^{-n} \approx a\_0 \Psi\_0(z) + a\_1 \Psi\_1(z) + \dots + a\_I \Psi\_I(z) \tag{16}$$

<sup>1</sup> Considering *h*(*k*) is a causal sequence.

### 6 Will-be-set-by-IN-TECH 92 Advances on Analysis and Control of Vibrations – Theory and Applications Active Vibration Control Using a Kautz Filter <sup>7</sup>

A controller can be inserted in the direct branch of the control loop to try to reject the disturbance. This controller *G*(*z*) has a digital structure given by:

$$G(z) = L(z)H^{-1}(z)\tag{17}$$

**<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>−</sup><sup>4</sup>**

(a) Input force (disturbance).

**Figure 1.** Response of the system for the uncontrolled case.

**10<sup>−</sup><sup>12</sup> 10<sup>−</sup><sup>11</sup> 10<sup>−</sup><sup>10</sup> 10<sup>−</sup><sup>9</sup> 10<sup>−</sup><sup>8</sup> 10<sup>−</sup><sup>7</sup> 10<sup>−</sup><sup>6</sup> 10<sup>−</sup><sup>5</sup>**

with Hanning window, 25 % of overlap and two sections.

**Y [m2/Hz]**

**Time [s]**

**<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>−</sup><sup>3</sup>**

(b) Output displacement.

**Time [s]**

Active Vibration Control Using a Kautz Filter 93

*<sup>z</sup>*<sup>2</sup> <sup>−</sup> 1.608*<sup>z</sup>* <sup>+</sup> 0.9875 (19)

*<sup>z</sup>*<sup>2</sup> <sup>−</sup> 1.608*<sup>z</sup>* <sup>+</sup> 0.9875 (20)

**−2 −1 0 1 2**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> <sup>45</sup> <sup>50</sup> <sup>10</sup><sup>−</sup><sup>13</sup>**

**Figure 2.** Power spectral density of the output signal (displacement) estimated using Welch method

and the eqs. (10) and (11) are utilized to construct the Kautz filter given by:

<sup>Ψ</sup>0(*z*) = 0.0926

<sup>Ψ</sup>1(*z*) = 0.1575*<sup>z</sup>* <sup>−</sup> 0.1275

The impulse response of the two sections of the Kautz filter are used to process the correlation function of the input signal *f*(*t*), through eq. (8). Equation (9) is solved by LS approach in

obtain the discrete Kautz poles given by *β* = 0.8040 + 0.5841*j* and *β*∗ = 0.8040 + 0.5841*j*. Once the system is a SISO and with only one degree of freedom, only one section of Kautz filter is employed, *J* = 1 (2 terms), and *N* = 600 samples are considered to be enough to complete description of the memory lag. The constants *b* and *c* are computed through eq. (12) and (13)

**Frequency [Hz]**

**Displacement [m]**

**<sup>3</sup> x 10<sup>−</sup><sup>3</sup>**

**Force [N]**

where *H*−1(*z*) is the inverse of the identified transfer function of the system and *L*(*z*) has the desirable dynamic. The compensator *L*(*z*) can have a second order structure or any format with a damping ratio *ξ<sup>c</sup>* bigger than the uncontrolled damping ratio. The control project is to find a gain and the *G*(*z*) formed to reduce the damping of the system. For practical implementation, these equations can be programmed directly in the discrete-time domain by using the mathematical convolution operator.

It is worth to point out that one consider only the control of stable systems described by *H*(*z*) experimentally identified. Consequently, the transfer function *H*(*z*) has all poles within the unitary circle because *H*(*z*) is identified using the Kautz poles that are set to be stable. The auxiliary function *L*(*z*) is proposed to warrant stability and the required performance in the closed-loop system

Two examples are used to show the approach proposed. The first one is a single-degree-of-freedom model that is a simple and easy example for the interested reader reproduce it. The second one is based on active vibration control in a smart structure with PZT actuator and PVDF sensor for presenting its use employing experimental data.

The results are illustrated in a single-degree-of-freedom model given by:

$$
\ddot{\mathbf{x}}(t) + 2\mathbf{\tilde{g}}\omega\_n \mathbf{\dot{x}}(t) + \omega\_n^2 \mathbf{x}(t) = f(t) \tag{18}
$$

where *x*(*t*) is the displacement vector, the over dot is the time derivative, *ξ* is the damping factor, *ω<sup>n</sup>* is the natural frequency in rad/s and *f*(*t*) is the excitation force. To simulate the uncontrolled responses, it were used the values of *ξ* = 0.01 and *ω<sup>n</sup>* = 62.83 rad/s that correspond to 10 Hz. The motion equation from eq. (18) is solved numerically through the Runge-Kutta method with a sampling rate of 100 Hz, that corresponds to a time sample of *dt* = 0.01 s, with 2048 samples. The force used was a white noise random with level of amplitude of the 3 N. The fig. (1) shows the input and output signal simulated for uncontrolled condition.

An important step to identify the IRFs is the choice of Kautz poles that need to reflect adequately the dominant dynamics of the vibrating systems. In real-world application the choice of the poles is a complicated problem. However, a simple power spectral density of the output signal (in our example the displacement) can give an orientation to help in the selection. If the system is more complicated, an optimization procedure could be used [12]. Figure (2) shows the power spectral density of the displacement. Clearly, it seems a peak value close to 10 Hz that is a possible candidate of natural frequency. The frequency response function (FRF) experimental is also estimated through spectral analysis only to compare the values of the natural frequency and damping factor, fig. (3).

Based on the frequency of 10 Hz, a continuous pole in *s*−domain given by *s*1,2 = −0.6283 ± 62.82*j*, where *j* is the imaginary unit, is set. Kautz filter is described in discrete-time domain, so, it is necessary to convert the pole to *<sup>z</sup>*−domain. The relationship *<sup>β</sup>* <sup>=</sup> *<sup>e</sup>sdt* can be used to

**Figure 1.** Response of the system for the uncontrolled case.

6 Will-be-set-by-IN-TECH

A controller can be inserted in the direct branch of the control loop to try to reject the

where *H*−1(*z*) is the inverse of the identified transfer function of the system and *L*(*z*) has the desirable dynamic. The compensator *L*(*z*) can have a second order structure or any format with a damping ratio *ξ<sup>c</sup>* bigger than the uncontrolled damping ratio. The control project is to find a gain and the *G*(*z*) formed to reduce the damping of the system. For practical implementation, these equations can be programmed directly in the discrete-time domain by

It is worth to point out that one consider only the control of stable systems described by *H*(*z*) experimentally identified. Consequently, the transfer function *H*(*z*) has all poles within the unitary circle because *H*(*z*) is identified using the Kautz poles that are set to be stable. The auxiliary function *L*(*z*) is proposed to warrant stability and the required performance in the

Two examples are used to show the approach proposed. The first one is a single-degree-of-freedom model that is a simple and easy example for the interested reader reproduce it. The second one is based on active vibration control in a smart structure with

where *x*(*t*) is the displacement vector, the over dot is the time derivative, *ξ* is the damping factor, *ω<sup>n</sup>* is the natural frequency in rad/s and *f*(*t*) is the excitation force. To simulate the uncontrolled responses, it were used the values of *ξ* = 0.01 and *ω<sup>n</sup>* = 62.83 rad/s that correspond to 10 Hz. The motion equation from eq. (18) is solved numerically through the Runge-Kutta method with a sampling rate of 100 Hz, that corresponds to a time sample of *dt* = 0.01 s, with 2048 samples. The force used was a white noise random with level of amplitude of the 3 N. The fig. (1) shows the input and output signal simulated for

An important step to identify the IRFs is the choice of Kautz poles that need to reflect adequately the dominant dynamics of the vibrating systems. In real-world application the choice of the poles is a complicated problem. However, a simple power spectral density of the output signal (in our example the displacement) can give an orientation to help in the selection. If the system is more complicated, an optimization procedure could be used [12]. Figure (2) shows the power spectral density of the displacement. Clearly, it seems a peak value close to 10 Hz that is a possible candidate of natural frequency. The frequency response function (FRF) experimental is also estimated through spectral analysis only to compare the

Based on the frequency of 10 Hz, a continuous pole in *s*−domain given by *s*1,2 = −0.6283 ± 62.82*j*, where *j* is the imaginary unit, is set. Kautz filter is described in discrete-time domain, so, it is necessary to convert the pole to *<sup>z</sup>*−domain. The relationship *<sup>β</sup>* <sup>=</sup> *<sup>e</sup>sdt* can be used to

PZT actuator and PVDF sensor for presenting its use employing experimental data.

*x*¨(*t*) + 2*ξωnx*˙(*t*) + *ω*<sup>2</sup>

The results are illustrated in a single-degree-of-freedom model given by:

values of the natural frequency and damping factor, fig. (3).

*G*(*z*) = *L*(*z*)*H*−1(*z*) (17)

*nx*(*t*) = *f*(*t*) (18)

disturbance. This controller *G*(*z*) has a digital structure given by:

using the mathematical convolution operator.

closed-loop system

uncontrolled condition.

**Figure 2.** Power spectral density of the output signal (displacement) estimated using Welch method with Hanning window, 25 % of overlap and two sections.

obtain the discrete Kautz poles given by *β* = 0.8040 + 0.5841*j* and *β*∗ = 0.8040 + 0.5841*j*. Once the system is a SISO and with only one degree of freedom, only one section of Kautz filter is employed, *J* = 1 (2 terms), and *N* = 600 samples are considered to be enough to complete description of the memory lag. The constants *b* and *c* are computed through eq. (12) and (13) and the eqs. (10) and (11) are utilized to construct the Kautz filter given by:

$$\Psi\_0(z) = \frac{0.0926}{z^2 - 1.608z + 0.9875} \tag{19}$$

$$\Psi\_1(z) = \frac{0.1575z - 0.1275}{z^2 - 1.608z + 0.9875} \tag{20}$$

The impulse response of the two sections of the Kautz filter are used to process the correlation function of the input signal *f*(*t*), through eq. (8). Equation (9) is solved by LS approach in

where *H*−1(*z*) is the inverse of the transfer function of the system identified experimentally,

and *L*(*z*) is a desirable dynamic to the system. The controller used has the following structure

where *ξ<sup>c</sup>* is the damping factor of the controlled system and *K* is a control gain. The structure in eq. (23) is continuous in the *s*−domain, and for application in a digital format is necessary to use a bilinear transform (Tustin's method). It is chosen a gain of *<sup>K</sup>* <sup>=</sup> <sup>3</sup><sup>×</sup> <sup>10</sup>−<sup>4</sup> and *<sup>ξ</sup><sup>c</sup>* <sup>=</sup> 0.08. These values are chosen based on the adequate behavior for the controlled system in the closed loop and with a low level of control actuator force required. The natural frequency in closed loop is maintained the same of the uncontrolled system. Thus, the digital compensator *L*(*z*)

*<sup>L</sup>*(*z*) = <sup>10</sup>−<sup>5</sup> 5.543*<sup>z</sup>* <sup>+</sup> 5.358

*<sup>M</sup>*(*z*) = *<sup>L</sup>*(*z*)*H*−1(*z*)*H*(*z*)

*<sup>M</sup>*(*z*) = <sup>10</sup>−<sup>5</sup> 5.543*z*<sup>3</sup> <sup>−</sup> 3.184*z*<sup>2</sup> <sup>−</sup> 3.244*<sup>z</sup>* <sup>+</sup> 4.846

Clearly the effectiveness of the controller depends on the correct identification of the *H*(*z*) to allow a perfect cancelation. Figure (5) shows the frequency response function comparison between uncontrolled and controlled system where it is seen that the peak decrease by increase actively the damping with the digital compensator. Figure (6) shows the output displacement without and with control. The disturbance force is considered with the same

A cantilever aluminium beam with a PZT actuator patch and a piezoelectric sensor (PVDF) symmetrically bonded to both sides of the beam is used to illustrate the process of IRF identification and design of a digital controller for active vibration reduction. The PZT and PVDF are bonded attached collocated near to the clamped end of the beam, as seen in fig. (7). The PZT patch is the model QP10N from ACX with size of 50 × 20 × 0.254 mm of length, width and thickness, respectively. The PVDF has dimensions of 30 × 10 × 0.205 mm of length, width and thickness, respectively, and it is bonded with a distance of 5 mm of the clamped

A white noise signal is generated in the computer, converted to analogic domain with a D/A converter and pre-processed by a voltage amplifier with gain of 20 V/V before application in the PZT actuator. The output signal is measured with the PVDF and linked directly with the

Finally, the feedback transfer function *M*(*z*) is given by:

level and type of the tests used in the uncontrolled condition.

end. The complete experimental setup is shown in figs. (7) and (8).

*n s*<sup>2</sup> + 2*ξcωns* + *ω*<sup>2</sup>

*<sup>L</sup>*(*s*) = *<sup>K</sup> <sup>ω</sup>*<sup>2</sup>

*H*(*z*) ≈ *α*0Ψ0(*z*) + *α*1Ψ1(*z*) (22)

Active Vibration Control Using a Kautz Filter 95

*<sup>z</sup>*<sup>2</sup> <sup>−</sup> 1.541*<sup>z</sup>* <sup>+</sup> 0.9044 (24)

<sup>1</sup> <sup>+</sup> *<sup>L</sup>*(*z*)*H*−1(*z*)*H*(*z*) (25)

*<sup>z</sup>*<sup>4</sup> <sup>−</sup> 3.082*z*<sup>3</sup> <sup>+</sup> 4.183*z*<sup>2</sup> <sup>−</sup> 2.787*<sup>z</sup>* <sup>+</sup> 0.8179 (26)

(23)

*n*

*H*(*z*), described by:

is given by:

that corresponds to:

of a second order system:

**Figure 3.** Frequency response function identified using spectral estimate *H*<sup>1</sup> through Welch method with Hanning window, 25 % of overlap and two sections.

order to identify the expansion coefficients *α*<sup>0</sup> and *α*1. With these values, eq. (6) is used to identify the IRF. Figure (4) presents the result of the identification process and compare with the analytical IRF. It is observed a good concordance between the experimental identified and the theoretical IRF.

**Figure 4.** Impulse response function comparison between analytical and identified by Kautz filters.

Once the IRF is identified, an experimental FIR model representative of the system is now known. This *H*(*z*) model is used to represent a controller *G*(*z*) inserted in the direct branch of the control loop with unitary feedback, by using the following expression:

$$\mathbf{G}(z) = L(z)H^{-1}(z) \tag{21}$$

where *H*−1(*z*) is the inverse of the transfer function of the system identified experimentally, *H*(*z*), described by:

$$H(z) \approx \mathfrak{a}\_0 \Psi\_0(z) + \mathfrak{a}\_1 \Psi\_1(z) \tag{22}$$

and *L*(*z*) is a desirable dynamic to the system. The controller used has the following structure of a second order system:

$$L(s) = K \frac{\omega\_n^2}{s^2 + 2\xi\_c \omega\_n s + \omega\_n^2} \tag{23}$$

where *ξ<sup>c</sup>* is the damping factor of the controlled system and *K* is a control gain. The structure in eq. (23) is continuous in the *s*−domain, and for application in a digital format is necessary to use a bilinear transform (Tustin's method). It is chosen a gain of *<sup>K</sup>* <sup>=</sup> <sup>3</sup><sup>×</sup> <sup>10</sup>−<sup>4</sup> and *<sup>ξ</sup><sup>c</sup>* <sup>=</sup> 0.08. These values are chosen based on the adequate behavior for the controlled system in the closed loop and with a low level of control actuator force required. The natural frequency in closed loop is maintained the same of the uncontrolled system. Thus, the digital compensator *L*(*z*) is given by:

$$L(z) = 10^{-5} \frac{5.543z + 5.358}{z^2 - 1.541z + 0.9044} \tag{24}$$

Finally, the feedback transfer function *M*(*z*) is given by:

$$M(z) = \frac{L(z)H^{-1}(z)H(z)}{1 + L(z)H^{-1}(z)H(z)}\tag{25}$$

that corresponds to:

8 Will-be-set-by-IN-TECH

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> <sup>45</sup> <sup>50</sup> <sup>10</sup><sup>−</sup><sup>6</sup>**

**Figure 3.** Frequency response function identified using spectral estimate *H*<sup>1</sup> through Welch method with

order to identify the expansion coefficients *α*<sup>0</sup> and *α*1. With these values, eq. (6) is used to identify the IRF. Figure (4) presents the result of the identification process and compare with the analytical IRF. It is observed a good concordance between the experimental identified and

> **Theoretical Identified**

*G*(*z*) = *L*(*z*)*H*−1(*z*) (21)

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>−</sup>0.02**

**Figure 4.** Impulse response function comparison between analytical and identified by Kautz filters.

the control loop with unitary feedback, by using the following expression:

Once the IRF is identified, an experimental FIR model representative of the system is now known. This *H*(*z*) model is used to represent a controller *G*(*z*) inserted in the direct branch of

**Time [s]**

**Frequency [Hz]**

**10<sup>−</sup><sup>5</sup>**

Hanning window, 25 % of overlap and two sections.

**−0.015 −0.01 −0.005**

**0 0.005 0.01 0.015 0.02**

**IRF**

the theoretical IRF.

**10<sup>−</sup><sup>4</sup>**

**Y [m2/N2**

**]**

**10<sup>−</sup><sup>3</sup>**

**10<sup>−</sup><sup>2</sup>**

$$M(z) = 10^{-5} \frac{5.543z^3 - 3.184z^2 - 3.244z + 4.846}{z^4 - 3.082z^3 + 4.183z^2 - 2.787z + 0.8179} \tag{26}$$

Clearly the effectiveness of the controller depends on the correct identification of the *H*(*z*) to allow a perfect cancelation. Figure (5) shows the frequency response function comparison between uncontrolled and controlled system where it is seen that the peak decrease by increase actively the damping with the digital compensator. Figure (6) shows the output displacement without and with control. The disturbance force is considered with the same level and type of the tests used in the uncontrolled condition.

A cantilever aluminium beam with a PZT actuator patch and a piezoelectric sensor (PVDF) symmetrically bonded to both sides of the beam is used to illustrate the process of IRF identification and design of a digital controller for active vibration reduction. The PZT and PVDF are bonded attached collocated near to the clamped end of the beam, as seen in fig. (7). The PZT patch is the model QP10N from ACX with size of 50 × 20 × 0.254 mm of length, width and thickness, respectively. The PVDF has dimensions of 30 × 10 × 0.205 mm of length, width and thickness, respectively, and it is bonded with a distance of 5 mm of the clamped end. The complete experimental setup is shown in figs. (7) and (8).

A white noise signal is generated in the computer, converted to analogic domain with a D/A converter and pre-processed by a voltage amplifier with gain of 20 V/V before application in the PZT actuator. The output signal is measured with the PVDF and linked directly with the

**Figure 5.** Frequency response function comparison between uncontrolled and controlled condition.

(a) Overall experimental setup.

Active Vibration Control Using a Kautz Filter 97

(b) Detail of the PVDF Sensor.

the frequency response function (FRF) experimental is estimated through spectral analysis to

Based on the spectral analysis one must choose the continuous poles candidates given by

the damping factors. Several trial and error tests were performed until to reach an adequate

*<sup>i</sup>* , *i* = 1, 2, 3, 4, 5. The most difficult parameters to be identified are

observe the values of the natural frequencies and damping factors, fig. (11).

**Figure 7.** View of the experimental setup.

<sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>2</sup>

*si* <sup>=</sup> <sup>−</sup>*ξiωni* <sup>±</sup> *<sup>j</sup>ωni*

**Figure 6.** Output displacement comparison between uncontrolled and controlled condition.

charge amplifier and pre-processed with a A/D converter. All experimental signals are saved and processed with a *dSPACE* 1104 acquisition board with a sample rate of 1 kHz and with 5 seconds of test duration. Figure (9) shows the time series signals of PZT actuator (input) and PVDF sensor (output) for uncontrolled system.

The first step in this approach is the choosing an adequate set of poles for the Kautz Filters. As the mathematical model is unknown, one needs to start by availing the power spectral density of the PVDF sensor (output) as suggested in the first example. Figure (10) presents the power spectral density of the output signal (PVDF) estimated using Welch method with Hanning window, 50 % of overlap and 5 sections. The peaks in frequencies of 13, 78, 211, 355 and 434 Hz can be considered candidates for natural frequencies. For comparison purposes, 96 Advances on Analysis and Control of Vibrations – Theory and Applications Active Vibration Control Using a Kautz Filter <sup>11</sup> Active Vibration Control Using a Kautz Filter 97

(a) Overall experimental setup.

(b) Detail of the PVDF Sensor.

**Figure 7.** View of the experimental setup.

10 Will-be-set-by-IN-TECH

**Without Control With Control**

**Without Control With Control**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> <sup>45</sup> <sup>50</sup> <sup>10</sup><sup>−</sup><sup>7</sup>**

**<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>−</sup><sup>3</sup>**

charge amplifier and pre-processed with a A/D converter. All experimental signals are saved and processed with a *dSPACE* 1104 acquisition board with a sample rate of 1 kHz and with 5 seconds of test duration. Figure (9) shows the time series signals of PZT actuator (input) and

The first step in this approach is the choosing an adequate set of poles for the Kautz Filters. As the mathematical model is unknown, one needs to start by availing the power spectral density of the PVDF sensor (output) as suggested in the first example. Figure (10) presents the power spectral density of the output signal (PVDF) estimated using Welch method with Hanning window, 50 % of overlap and 5 sections. The peaks in frequencies of 13, 78, 211, 355 and 434 Hz can be considered candidates for natural frequencies. For comparison purposes,

**Figure 6.** Output displacement comparison between uncontrolled and controlled condition.

**Time [s]**

**Figure 5.** Frequency response function comparison between uncontrolled and controlled condition.

**Frequency [Hz]**

**10<sup>−</sup><sup>6</sup>**

**−2**

PVDF sensor (output) for uncontrolled system.

**−1**

**0**

**Output Amplitude [m]**

**1**

**2**

**<sup>3</sup> x 10<sup>−</sup><sup>3</sup>**

**10<sup>−</sup><sup>5</sup>**

**FRF [m2/N2**

**]**

**10<sup>−</sup><sup>4</sup>**

**10<sup>−</sup><sup>3</sup>**

**10<sup>−</sup><sup>2</sup>**

the frequency response function (FRF) experimental is estimated through spectral analysis to observe the values of the natural frequencies and damping factors, fig. (11).

Based on the spectral analysis one must choose the continuous poles candidates given by *si* <sup>=</sup> <sup>−</sup>*ξiωni* <sup>±</sup> *<sup>j</sup>ωni* <sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>2</sup> *<sup>i</sup>* , *i* = 1, 2, 3, 4, 5. The most difficult parameters to be identified are the damping factors. Several trial and error tests were performed until to reach an adequate

**Figure 8.** Experimental setup.

**<sup>101</sup> <sup>102</sup> <sup>10</sup><sup>−</sup><sup>4</sup>**

result. A reasonable identification were reached based on the parameters given by:

**Figure 11.** Frequency response function identified using spectral estimate *H*<sup>1</sup> through Welch method

**Frequency [Hz]**

*ωn*<sup>1</sup> = 81.68 rad/s *ξ*<sup>1</sup> = 0.04 *s*<sup>1</sup> = −3.2673 ± 81.6160*j* (27) *ωn*<sup>2</sup> = 490.08 rad/s *ξ*<sup>2</sup> = 0.019 *s*<sup>2</sup> = −9.3117 ± 490*j* (28)

> *β*<sup>1</sup> = 0.9934 ± 0.0813*j* (32) *β*<sup>2</sup> = 0.8742 ± 0.4663*j* (33) *β*<sup>3</sup> = 0.2365 ± 0.9447*j* (34) *β*<sup>4</sup> = −0.4833 ± 0.6376*j* (35) *β*<sup>5</sup> = −0.6925 ± 0.3163*j* (36)

Active Vibration Control Using a Kautz Filter 99

*<sup>ω</sup>n*<sup>3</sup> <sup>=</sup> 1.3258 <sup>×</sup> 103 rad/s *<sup>ξ</sup>*<sup>3</sup> <sup>=</sup> 0.02 *<sup>s</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>26.515 <sup>±</sup> 1325.5*<sup>j</sup>* (29) *<sup>ω</sup>n*<sup>4</sup> <sup>=</sup> 2.23 <sup>×</sup> 103 rad/s *<sup>ξ</sup>*<sup>4</sup> <sup>=</sup> 0.1 *<sup>s</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>223.05 <sup>±</sup> 2219.4*<sup>j</sup>* (30) *<sup>ω</sup>n*<sup>5</sup> <sup>=</sup> 2.72 <sup>×</sup> 103 rad/s *<sup>ξ</sup>*<sup>5</sup> <sup>=</sup> 0.1 *<sup>s</sup>*<sup>5</sup> <sup>=</sup> <sup>−</sup>276.7 <sup>±</sup> 2713.2*<sup>j</sup>* (31)

Once the fourth and fifth modes are apparently well damped by analysing the frequency response the correspond poles are also considered well damped (not dominants). The Kautz filter is described in the discrete-time domain. So, it is necessary to convert to *z*−domain. The relationship *β<sup>i</sup>* = *esidt* can be used to obtain the five pair of complex discrete Kautz poles given

The cantilever beam is a SISO system, but with apparent five modes in the frequency range computed of interest. So, they are used 5 sections of Kautz filters, *J* = 4 and *N* = 1200 samples that are considered to be enough to complete the view of the memory lag. The constants *b* and

Figure (12) shows the comparison between the IFFT of the FRF from *H*<sup>1</sup> estimated and the IRF

*c* are computed and the eqs. (10) and (11) are utilized to construct the Kautz filters.

**10<sup>−</sup><sup>3</sup>**

with Hanning window, 50 % of overlap and 5 sections.

by:

identified through Kautz filter.

**Amplitude [V2/V2**

**]**

**10<sup>−</sup><sup>2</sup>**

**Figure 9.** Response of the experimental tests in the time domain for the uncontrolled case.

**Figure 10.** Power spectral density of the output signal (PVDF) estimated using Welch method with Hanning window, 50 % of overlap and 5 sections.

12 Will-be-set-by-IN-TECH

**−0.6 −0.4 −0.2 0 0.2 0.4 0.6**

**<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>400</sup> <sup>450</sup> <sup>500</sup> <sup>10</sup><sup>−</sup><sup>7</sup>**

**Figure 10.** Power spectral density of the output signal (PVDF) estimated using Welch method with

**Frequency [Hz]**

**Figure 9.** Response of the experimental tests in the time domain for the uncontrolled case.

**Output − PVDF [V]**

**0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5**

**Time [s]**

(b) PVDF Sensor.

**Figure 8.** Experimental setup.

**Input − PZT [V]**

**<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> <sup>−</sup><sup>150</sup>**

(a) PZT Actuator.

**10<sup>−</sup><sup>6</sup>**

**10<sup>−</sup><sup>5</sup>**

**PVDF − Amplitude [V2/Hz]**

Hanning window, 50 % of overlap and 5 sections.

**10<sup>−</sup><sup>4</sup>**

**10<sup>−</sup><sup>3</sup>**

**10<sup>−</sup><sup>2</sup>**

**Time [s]**

**Figure 11.** Frequency response function identified using spectral estimate *H*<sup>1</sup> through Welch method with Hanning window, 50 % of overlap and 5 sections.

result. A reasonable identification were reached based on the parameters given by:

$$
\omega\_{\rm n1} = 81.68 \quad \text{rad/s} \qquad \xi\_1 = 0.04 \qquad s\_1 = -3.2673 \pm 81.6160 \text{j} \tag{27}
$$

$$
\omega\_{n2} = 490.08 \quad \text{rad/s} \qquad \xi\_2 = 0.019 \qquad s\_2 = -9.3117 \pm 490 \text{j} \tag{28}
$$

$$
\omega\_{\text{n3}} = 1.3258 \times 10^3 \quad \text{rad/s} \qquad \xi\_3 = 0.02 \qquad \varsigma\_3 = -26.515 \pm 1325.5 \text{j} \tag{29}
$$

$$\omega\_{n4} = 2.23 \times 10^3 \quad \text{rad/s} \qquad \xi\_4 = 0.1 \qquad s\_4 = -223.05 \pm 22194 \text{j} \tag{30}$$

$$
\omega\_{n5} = 2.72 \times 10^3 \quad \text{rad/s} \qquad \text{f}\_5 = 0.1 \qquad \text{s}\_5 = -276.7 \pm 2713.2 \text{j} \tag{31}
$$

Once the fourth and fifth modes are apparently well damped by analysing the frequency response the correspond poles are also considered well damped (not dominants). The Kautz filter is described in the discrete-time domain. So, it is necessary to convert to *z*−domain. The relationship *β<sup>i</sup>* = *esidt* can be used to obtain the five pair of complex discrete Kautz poles given by:

$$
\beta\_1 = 0.9934 \pm 0.0813 \text{j} \tag{32}
$$

$$
\beta\_2 = 0.8742 \pm 0.4663j \tag{33}
$$

$$
\beta\_3 = 0.2365 \pm 0.9447 \text{j} \tag{34}
$$

$$
\beta\_4 = -0.4833 \pm 0.6376j \tag{35}
$$

$$
\beta\_5 = -0.6925 \pm 0.3163j \tag{36}
$$

The cantilever beam is a SISO system, but with apparent five modes in the frequency range computed of interest. So, they are used 5 sections of Kautz filters, *J* = 4 and *N* = 1200 samples that are considered to be enough to complete the view of the memory lag. The constants *b* and *c* are computed and the eqs. (10) and (11) are utilized to construct the Kautz filters.

Figure (12) shows the comparison between the IFFT of the FRF from *H*<sup>1</sup> estimated and the IRF identified through Kautz filter.

**Figure 12.** Impulse response function comparison between IFFT of the estimated FRF and identified by Kautz filters.

**0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5**

**Time [s]**

The controller is designed based on the inverse of the identified system described by eq. (16), called by *H*−1(*z*), in series with a compensator *L*(*z*). The *L*(*z*) is chosen by combination of 3

*<sup>L</sup>*1(*z*) = 0.003316*<sup>z</sup>* <sup>+</sup> 0.003298

*<sup>L</sup>*2(*z*) = 0.1122*<sup>z</sup>* <sup>+</sup> 0.1068

*<sup>L</sup>*3(*z*) = 0.6691*<sup>z</sup>* <sup>+</sup> 0.5813

*<sup>L</sup>*(*z*) = 0.001177*z*<sup>5</sup> <sup>−</sup> 0.003011*z*<sup>4</sup> <sup>+</sup> 0.001994*z*<sup>3</sup> <sup>+</sup> 0.001218*z*<sup>2</sup> <sup>−</sup> 0.002219*<sup>z</sup>* <sup>+</sup> 0.0008492

• Additionally, these modes are not well identified by the Kautz filter. One included the damping factor in these modes with these values shown above in order to correct

It was decided to control only the 3 first modes for two main reasons:

*<sup>z</sup>*<sup>6</sup> <sup>−</sup> 4.043*z*<sup>5</sup> <sup>+</sup> 7.297*z*<sup>4</sup> <sup>−</sup> 7.907*z*<sup>3</sup> <sup>+</sup> 5.676*z*<sup>2</sup> <sup>−</sup> 2.592*<sup>z</sup>* <sup>+</sup> 0.5706 (41)

It is important to observe that the three compensators, *L*1(*z*), *L*2(*z*) and *L*3(*z*) have the natural frequencies corresponding to the first three modes of the systems, but with an increase in the level of damping factor for reducing the vibration level in the closed-loop system. The

where *<sup>K</sup>* <sup>=</sup> 1.5 <sup>×</sup> <sup>10</sup>−<sup>3</sup> is a controller gain and the transfer functions are defined by:

*L*(*z*) = *K* (*L*1(*z*) + *L*2(*z*) + *L*3(*z*)) (37)

*<sup>z</sup>*<sup>2</sup> <sup>−</sup> 1.977*<sup>z</sup>* <sup>+</sup> 0.9838 (38)

Active Vibration Control Using a Kautz Filter 101

*<sup>z</sup>*<sup>2</sup> <sup>−</sup> 1.644*<sup>z</sup>* <sup>+</sup> 0.8633 (39)

*<sup>z</sup>*<sup>2</sup> <sup>−</sup> 0.4215*<sup>z</sup>* <sup>+</sup> 0.6718 (40)

**−0.6**

second-order system realized in parallel structure:

compensator *L*(*z*) in its final form is given by:

• The fourth and fifth modes are not dominant.

identification the anti-ressonance region.

**Figure 14.** PVDF output estimated by IRF identified with Kautz filters.

**−0.4**

**−0.2**

**0**

**Estimated Output − PVDF [V]**

**0.2**

**0.4**

**0.6**

**Figure 13.** FRF comparison between estimated FRF through *H*<sup>1</sup> spectral estimate and identified by Kautz filters.

Although, it seems that are not a complete visual agreement between the curves, the FRF seen in figure (13) presents a good agreement. It is worth to comments that with the same experimental data, [23] identified a state-space model through Eigensystem Realization Algorithms (ERA) combined with Observer/Kalman filter Identification (OKID). The results presented with Kautz filter allowed a better identification in this frequency range comparing than with ERA/OKID.

Figure (14) shows the output response of th PVDF estimated by a convolution between the IRF identified by Kautz filter with the input excitation from PZT actuator. The estimated output can be compared with the experimental measured response (see fig. 9(b)).

**Figure 14.** PVDF output estimated by IRF identified with Kautz filters.

14 Will-be-set-by-IN-TECH

**IFFT from FRF Kautz**

**Experimental FRF Identified by Kautz filter**

**<sup>0</sup> 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 <sup>−</sup><sup>1</sup>**

**Figure 12.** Impulse response function comparison between IFFT of the estimated FRF and identified by

**<sup>101</sup> <sup>102</sup> <sup>10</sup><sup>−</sup><sup>4</sup>**

**Figure 13.** FRF comparison between estimated FRF through *H*<sup>1</sup> spectral estimate and identified by

Although, it seems that are not a complete visual agreement between the curves, the FRF seen in figure (13) presents a good agreement. It is worth to comments that with the same experimental data, [23] identified a state-space model through Eigensystem Realization Algorithms (ERA) combined with Observer/Kalman filter Identification (OKID). The results presented with Kautz filter allowed a better identification in this frequency range comparing

Figure (14) shows the output response of th PVDF estimated by a convolution between the IRF identified by Kautz filter with the input excitation from PZT actuator. The estimated output

can be compared with the experimental measured response (see fig. 9(b)).

**Frequency [Hz]**

**Time [s]**

**−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1**

**10<sup>−</sup><sup>3</sup>**

**Amplitude [V2/V2**

**]**

**10<sup>−</sup><sup>2</sup>**

**10<sup>−</sup><sup>1</sup>**

Kautz filters.

Kautz filters.

than with ERA/OKID.

**IRF**

The controller is designed based on the inverse of the identified system described by eq. (16), called by *H*−1(*z*), in series with a compensator *L*(*z*). The *L*(*z*) is chosen by combination of 3 second-order system realized in parallel structure:

$$L(z) = K\left(L\_1(z) + L\_2(z) + L\_3(z)\right) \tag{37}$$

where *<sup>K</sup>* <sup>=</sup> 1.5 <sup>×</sup> <sup>10</sup>−<sup>3</sup> is a controller gain and the transfer functions are defined by:

$$L\_1(z) = \frac{0.003316z + 0.003298}{z^2 - 1.977z + 0.9838} \tag{38}$$

$$L\_2(z) = \frac{0.1122z + 0.1068}{z^2 - 1.644z + 0.8633} \tag{39}$$

$$L\_3(z) = \frac{0.6691z + 0.5813}{z^2 - 0.4215z + 0.6718} \tag{40}$$

It is important to observe that the three compensators, *L*1(*z*), *L*2(*z*) and *L*3(*z*) have the natural frequencies corresponding to the first three modes of the systems, but with an increase in the level of damping factor for reducing the vibration level in the closed-loop system. The compensator *L*(*z*) in its final form is given by:

$$L(z) = \frac{0.001177z^5 - 0.003011z^4 + 0.001994z^3 + 0.001218z^2 - 0.002219z + 0.0008492}{z^6 - 4.043z^5 + 7.297z^4 - 7.907z^3 + 5.676z^2 - 2.592z + 0.5706} \tag{41}$$

It was decided to control only the 3 first modes for two main reasons:


### 16 Will-be-set-by-IN-TECH 102 Advances on Analysis and Control of Vibrations – Theory and Applications Active Vibration Control Using a Kautz Filter <sup>17</sup>

Figure (15) shows the FRF comparison between the uncontrolled (estimated by Kautz filter) and controlled system where is possible to observe the reduction in the resonance peak caused by the controller implemented.

**Author details**

**6. References**

28(5): 1056–1064.

*Systems Technology* 9(5): 696–707.

1st edn, Birkhauser Boston.

*Controle & Automação* 18(3): 301–321.

*Journal of Control* 81(6): 962–975.

*Processing* 24(1): 52–58.

Samuel da Silva, Vicente Lopes Junior and Michael J. Brennan

*Engenharia Mecânica, Av. Brasil 56, Centro, Ilha Solteira, SP, Brasil*

*UNESP - Univ Estadual Paulista, Faculdade de Engenharia de Ilha Solteira, Departamento de*

Active Vibration Control Using a Kautz Filter 103

[1] Baldelli, D. H., Lind, R. & Brenner, M. [2005]. Nonlinear aeroelastic/aeroservoelastic modeling by block-oriented identification, *Journal of Guidance, Control and Dynamics*

[2] Baldelli, D. H., Mazzaro, M. C. & Peña, R. S. S. [2001]. Robust identification of lightly damped flexible structures by means of orthonormal bases, *IEEE Transactions on Control*

[3] Brennan, M. J. & Kim, S. M. [2001]. Feedforward and feedback control of sound and vibration - a Wiener filter approach, *Journal of Sound and Vibration* 246(2): 281–296. [4] Brinker, A. C. & Belt, H. J. W. [1998]. Using kautz models in model reduction, *in* A. Prochazka, J. Uhlír, P. J. W. Rayner & N. Kingsbury (eds), *Signal Analysis and prediction*,

[5] Campello, R. J. G. B., Oliveira, G. H. C. & Amaral, W. C. [2007]. Identificação e controle de processos via desenvolvimento em séries ortonormais. parte a: Identificação, *Revista*

[6] D. Mayer, S. H. H. H. [2001]. Application of Kautz models for adaptive vibration control, *in* IMECE (ed.), *American Society of Mechanical Engineers (Veranst.)*, ASME International

[7] da Rosa, A., Campello, R. J. G. B. & do Amaral, W. C. [2006]. Desenvolvimento de modelos de Volterra usando funções de Kautz e sua aplicação à modelagem de um

[8] da Rosa, A., Campello, R. J. G. B. & do Amaral, W. C. [2007]. Choice of free parameters of discrete-time Volterra models using Kautz functions, *Automatica* 43(6): 1084–1091. [9] da Rosa, A., Campello, R. J. G. B. & do Amaral, W. C. [2008]. An optimal expansion of Volterra models using independent Kautz bases for each kernel dimension, *International*

[10] da Rosa, A., Campello, R. J. G. B. & do Amaral, W. C. [2009]. Exact search directions for optimization of linear and nonlinear models based on generalized orthonormal

[11] da Silva, S. [2011a]. Non-linear model updating of a three-dimensional portal frame based on Wiener series, *International Journal of Non-linear Mechanics* 46: 312–320. [12] da Silva, S. [2011b]. Non-parametric identification of mechanical systems by Kautz filter with multiple poles, *Mechanical Systems and Signal Processing* 25(4): 1103–1111. [13] da Silva, S., Cogan, S. & Foltête, E. [2010]. Nonlinear identification in estructural dynamics based on Wiener series and Kautz filter., *Mechanical Systems and Signal*

[14] da Silva, S., Dias Júnior, M. & Lopes Junior, V. [2009]. Identification of mechanical

systems through Kautz filter, *Journal of Vibration and Control* 15(6): 849–865.

Mechanical Engineering Congress and Exposition, NewYork.

sistema de levitaçãoo magnética, *XVI Congresso Brasileiro de Automática*.

functions, *IEEE Transactions os Automatic Control* 54(12): 2757–2772.

**Figure 15.** FRF comparison between uncontrolled and controlled condition. Input: PZT actuator - Output: PVDF sensor.

Another advantage of this procedure face to state-feedback approaches is relative to the controlability and observability conditions. If one use procedures identification for obtaining a state-space realization, e. g. ERA/OKID as made by [23], is necessary to verify a prior the observability and controlability conditions. In some situations some modes are not controllable and observable adequately with a specific realization. Once the technique used in this chapter is not described in state-space variables and it is based on input/output variables with non-parametric IRF model, these kinds of drawbacks are avoided.

This chapter has described a procedure for non-parametric system identification of an impulse response function (IRF) based on input and output experimental data. Orthogonal functions are used to reduce the number of samples to be identified. A simple active vibration control procedure with a digital compensator that seeks to cancel the plant dynamic is also described. Once the IRF in the uncontrolled condition is well estimated by Kautz filters, the control strategy presented can increase the damping in a satisfactory level with low actuator requirements. Single-input-single-output vibrating systems have been used to illustrate the performance and the main aspects for practical implementation. This procedure can also be extended for nonlinear systems using Hammerstein or Wiener block-oriented models.

### **Acknowledgements**

The authors are thankful for the financial support provided by National Council for Scientific and Technological Development (CNPq/Brasil), INCT and São Paulo Research Foundation (FAPESP). The authors acknowledge the helpful suggestions of the Editor. The authors also are thankful the help of Prof. Dr. Gustavo Luiz Chagas ManhNes de Abreu and Sanderson ´ Manoel da Conceição for providing the experimental data in the clamped beam.

### **Author details**

16 Will-be-set-by-IN-TECH

Figure (15) shows the FRF comparison between the uncontrolled (estimated by Kautz filter) and controlled system where is possible to observe the reduction in the resonance peak caused

> **Without Control With Control**

**<sup>101</sup> <sup>102</sup> <sup>10</sup><sup>−</sup><sup>4</sup>**

**Figure 15.** FRF comparison between uncontrolled and controlled condition. Input: PZT actuator -

with non-parametric IRF model, these kinds of drawbacks are avoided.

Another advantage of this procedure face to state-feedback approaches is relative to the controlability and observability conditions. If one use procedures identification for obtaining a state-space realization, e. g. ERA/OKID as made by [23], is necessary to verify a prior the observability and controlability conditions. In some situations some modes are not controllable and observable adequately with a specific realization. Once the technique used in this chapter is not described in state-space variables and it is based on input/output variables

This chapter has described a procedure for non-parametric system identification of an impulse response function (IRF) based on input and output experimental data. Orthogonal functions are used to reduce the number of samples to be identified. A simple active vibration control procedure with a digital compensator that seeks to cancel the plant dynamic is also described. Once the IRF in the uncontrolled condition is well estimated by Kautz filters, the control strategy presented can increase the damping in a satisfactory level with low actuator requirements. Single-input-single-output vibrating systems have been used to illustrate the performance and the main aspects for practical implementation. This procedure can also be extended for nonlinear systems using Hammerstein or Wiener block-oriented models.

The authors are thankful for the financial support provided by National Council for Scientific and Technological Development (CNPq/Brasil), INCT and São Paulo Research Foundation (FAPESP). The authors acknowledge the helpful suggestions of the Editor. The authors also are thankful the help of Prof. Dr. Gustavo Luiz Chagas ManhNes de Abreu and Sanderson ´

Manoel da Conceição for providing the experimental data in the clamped beam.

**Frequency [Hz]**

by the controller implemented.

Output: PVDF sensor.

**Acknowledgements**

**10<sup>−</sup><sup>3</sup>**

**FRF [V2/V2**

**]**

**10<sup>−</sup><sup>2</sup>**

**10<sup>−</sup><sup>1</sup>**

Samuel da Silva, Vicente Lopes Junior and Michael J. Brennan

*UNESP - Univ Estadual Paulista, Faculdade de Engenharia de Ilha Solteira, Departamento de Engenharia Mecânica, Av. Brasil 56, Centro, Ilha Solteira, SP, Brasil*

### **6. References**

	- [15] Heuberger, P. S. C., Van Den Hof, P. M. J. & Wahlberg, B. [2005]. *Modelling and Identification with Rational Orthogonal Basis Functions*, 1st edn, Springer.
	- [16] Heuberger, P., Van Den Hof, P. & Bosgra, O. H. [1995]. A generalized orthonormal basis of linear dynamical systems, *IEE Transactions on Automatic Control* 40(3): 451–465.
	- [17] Kautz, W. H. [1954]. Transient synthesis in the time domain, *IRE Transactions on Circuit Theory* 1(1): 29 – 39.
	- [18] Oliveira, G. H. C., Amaral, W. C., Favier, G. & Dumont, G. A. [2000]. Constrained robust predictive controller for uncertain processes modeled by orthonormal series functions, *Automatica* 36(4): 563–571.
	- [19] Oliveira, G. H. C., Campello, R. J. G. B. & Amaral, W. C. [2007]. Identificação e controle de processor via desenvolvimento em séries ortonormais. parte b: Controle preditivo, *Revista Controle & Automática* 18(3): 322–336.
	- [20] Paetero, T. & Karjalainen, M. [2003]. Kautz filters and generalized frequency resolution: Theory and audio aplications, *Audio Engineering Society* 51(1/2): 27–44.
	- [21] Sansone, G. [1958]. *Orthogonal Functions*, Vol. 9, Dover Publications.
	- [22] Van Den Hof, P. M. J., Heuberger, P. S. & Bokor, J. [1995]. System identification with generalized orthonormal basis functions, *Automatica* 31(12): 1821–1834.
	- [23] Vasques, C. H., Conceição, S. M., Abreu, G. L. C. M., Lopes Jr., V. & Brennan, M. J. [2011]. Identification and control of systems submitted to mechanical vibration, *21st International Congress of Mechanical Engineering - COBEM 2011*, Natal, RN, Brasil.
	- [24] Wahlberg, B. [1994]. System identification using Kautz models, *IEEE Transactions on Automatic Control* 39(6): 1276 – 1282.
	- [25] Wahlberg, B. & Makila, P. M. [1996]. On approximation of stable linear dynamical systems using Laguerre and Kautz functions, *Automatica* 32(5): 693–708.
	- [26] Zeng, J. & de Callafon, R. [2005]. Filters parametrized by orthonormal basis functions for active noise control, *ASME International Mechanical Engineering Congress and Exposition - IMECE 2005*.

© 2012 Tora, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Tora, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The work of operators in heavy machinery requires constant attention to gather information about the machine's surroundings, its current status and the operations performed. Operators have to analyse the received information on the continuous basis and make decisions accordingly, to have them implemented via the control system and to perform the scheduled tasks in the optimal manner. The more powerful the machine, the more serious the consequence of errors committed by operators. The typical frequency range of vibration of machines and their equipment is determined based on testing done on heavy machines

Machine vibrations are induced by the drives' action, movements of the equipment, variable loading and machine ride. The ride of heavy machines, tractors, forestry vehicles over a rough terrain lead to cyclic tilting of the machines, which can be regarded as low-frequency (up to several Hz) and high-amplitude (about 10 degrees) vibration of the machine. The angular motions of the frame are transmitted onto the cab, and the higher the cab position, the larger the amplitude range of linear vibration of the point SIP (about 70 cm). Vibrations negatively impact on the machine structure, control processes, performance quality and the operator's comfort. Growing ergonomic concerns and competition on the market have

Cab suspensions are now incorporated in the machine structure as a new solution. The active suspension is a system whose components are based on existing vibration reduction solutions. Early vehicles were also provided with suspension systems to suppress vibrations due to the ride in the rough terrain. At first these were passive suspension systems, in which the characteristics of the components i.e. elastic and damping elements are fixed. Suspensions incorporating semiactive elements perform better as vibration isolation systems since their characteristics can be varied according to the adopted control strategy. Active suspensions

prompted the design of machines ensuring the better comfort for the operator.

**The Active Suspension** 

Grzegorz Tora

**1. Introduction** 

http://dx.doi.org/10.5772/45831

**of a Cab in a Heavy Machine** 

Additional information is available at the end of the chapter

used in Europe [1] and is found to be 0.5- 80 Hz.

## **The Active Suspension of a Cab in a Heavy Machine**

Grzegorz Tora

18 Will-be-set-by-IN-TECH

[15] Heuberger, P. S. C., Van Den Hof, P. M. J. & Wahlberg, B. [2005]. *Modelling and*

[16] Heuberger, P., Van Den Hof, P. & Bosgra, O. H. [1995]. A generalized orthonormal basis of linear dynamical systems, *IEE Transactions on Automatic Control* 40(3): 451–465. [17] Kautz, W. H. [1954]. Transient synthesis in the time domain, *IRE Transactions on Circuit*

[18] Oliveira, G. H. C., Amaral, W. C., Favier, G. & Dumont, G. A. [2000]. Constrained robust predictive controller for uncertain processes modeled by orthonormal series functions,

[19] Oliveira, G. H. C., Campello, R. J. G. B. & Amaral, W. C. [2007]. Identificação e controle de processor via desenvolvimento em séries ortonormais. parte b: Controle preditivo,

[20] Paetero, T. & Karjalainen, M. [2003]. Kautz filters and generalized frequency resolution:

[22] Van Den Hof, P. M. J., Heuberger, P. S. & Bokor, J. [1995]. System identification with

[23] Vasques, C. H., Conceição, S. M., Abreu, G. L. C. M., Lopes Jr., V. & Brennan, M. J. [2011]. Identification and control of systems submitted to mechanical vibration, *21st International*

[24] Wahlberg, B. [1994]. System identification using Kautz models, *IEEE Transactions on*

[25] Wahlberg, B. & Makila, P. M. [1996]. On approximation of stable linear dynamical

[26] Zeng, J. & de Callafon, R. [2005]. Filters parametrized by orthonormal basis functions for active noise control, *ASME International Mechanical Engineering Congress and Exposition -*

Theory and audio aplications, *Audio Engineering Society* 51(1/2): 27–44.

generalized orthonormal basis functions, *Automatica* 31(12): 1821–1834.

systems using Laguerre and Kautz functions, *Automatica* 32(5): 693–708.

*Congress of Mechanical Engineering - COBEM 2011*, Natal, RN, Brasil.

[21] Sansone, G. [1958]. *Orthogonal Functions*, Vol. 9, Dover Publications.

*Identification with Rational Orthogonal Basis Functions*, 1st edn, Springer.

*Theory* 1(1): 29 – 39.

*Automatica* 36(4): 563–571.

*Revista Controle & Automática* 18(3): 322–336.

*Automatic Control* 39(6): 1276 – 1282.

*IMECE 2005*.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/45831

## **1. Introduction**

The work of operators in heavy machinery requires constant attention to gather information about the machine's surroundings, its current status and the operations performed. Operators have to analyse the received information on the continuous basis and make decisions accordingly, to have them implemented via the control system and to perform the scheduled tasks in the optimal manner. The more powerful the machine, the more serious the consequence of errors committed by operators. The typical frequency range of vibration of machines and their equipment is determined based on testing done on heavy machines used in Europe [1] and is found to be 0.5- 80 Hz.

Machine vibrations are induced by the drives' action, movements of the equipment, variable loading and machine ride. The ride of heavy machines, tractors, forestry vehicles over a rough terrain lead to cyclic tilting of the machines, which can be regarded as low-frequency (up to several Hz) and high-amplitude (about 10 degrees) vibration of the machine. The angular motions of the frame are transmitted onto the cab, and the higher the cab position, the larger the amplitude range of linear vibration of the point SIP (about 70 cm). Vibrations negatively impact on the machine structure, control processes, performance quality and the operator's comfort. Growing ergonomic concerns and competition on the market have prompted the design of machines ensuring the better comfort for the operator.

Cab suspensions are now incorporated in the machine structure as a new solution. The active suspension is a system whose components are based on existing vibration reduction solutions. Early vehicles were also provided with suspension systems to suppress vibrations due to the ride in the rough terrain. At first these were passive suspension systems, in which the characteristics of the components i.e. elastic and damping elements are fixed. Suspensions incorporating semiactive elements perform better as vibration isolation systems since their characteristics can be varied according to the adopted control strategy. Active suspensions

© 2012 Tora, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Tora, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

lend a new quality to control of low-frequency vibrations in vehicles. The operating ranges of passive, semiactive and active suspensions, given as velocity-force characteristics, are shown in Fig 1.

The Active Suspension of a Cab in a Heavy Machine 107

most effective in the case of low-frequency vibrations [5], about 5 Hz and are often added to passive systems which well handle high-frequency vibrations. High efficiency of active systems, however, comes at the cost of high power demand. Active suspension systems typically utilise small hydraulic cylinders to achieve high accelerations of masses subjected to considerable loads. Pneumatic cylinders are capable of delivering higher velocity than hydraulic ones, yet the acceleration is strongly dependent on the external load applied. There are also hydro-pneumatic cylinders, displaying the advantages of the two previous types. Active suspension systems in truck cabs utilise electric cylinders DC and AC-servo.

The new developments of the operator's seats are in line with the advancements in vehicle suspensions. In the simplest solution where the seat is fixed rigid to the machine frame or to the cab floor, vibrations are transmitted from the cab attachment point onto the operator. The mobile seat support allows it to be moved only in the vertical so the passive, semiactive or active vibration reduction system can be added along this direction only. Typically, a shearing mechanism is used as seat support [6], though other solutions with a greater number of DOFs are reported as well [7]. On account of small mass of the seat together with the operator, the upper frequency limit is reached and the active support becomes more effective than a vehicle suspension. Active suspension systems are now incorporated in agricultural and forestry machinery, where typical excitations are in the form of low-

The correct control strategy is a key element in active and semiactive vibration reduction systems, resulting in a good compromise between numerous and sometimes mutually excluding requirements. Optimal control techniques that handle this problem include the linear quadratic regulator LQR [9] and the linear Gaussian regulator LQG [10]. In simpler cases the PID control can be applied. Active control systems may also use regulators based on neural networks [11] and fuzzy logics [12]. The control schemes that are commonly used for semiactive suspension systems include the 'sky-hook' control strategy [13] where the damping force is related to the absolute velocity of the vehicle body. The *H∞* control is insensitive to uncertain input quantities, encumbered with major errors (for instant: time constant, reduced mass, damping force, acceleration, velocity, damping ratio for the tire [14]. The *H∞* control scheme concurrently executes the mutlicriterial optimisation and in active and semiactive suspension systems it takes into account the acceleration of sprung mass, peak accelerations, jerks of the front and rear suspension, road holding, forces acting upon the relevant masses, deflection of the tire and of the suspension [15]. The work [16] focused on *H<sup>∞</sup>* control in an active suspension of vehicle investigates the influence of the time delay on stability of the control process. It is shown that the delay time (i.e. time before the cylinder in the active system is activated), the reduced mass and power ratings are major determinants of the frequency limit for effective operation of these type of suspensions. The stability condition is formulated for the predetermined range of time delay and system parameters. Extensive expertise prompts the use of filters, beside robust control for the purpose of estimation. The Kalman filter, reported in literature on the subject [17], is now widely used. To improve the operator's comfort, an active suspension of a cab can be incorporated in the machine structure (Fig 2), to reduce the cab's vibration. The active suspension system

frequency and high-amplitude vibrations [8].

comprises several sub-systems:

**Figure 1.** The range of velocity-force characteristics of passive, semiactive and active suspensions [2]

Passive suspensions are described with a damper characteristics with fixed parameters (broke line). These suspensions typically comprise elastic elements featuring a linear or nonlinear elasticity and damping elements with nonlinear characteristics. On one hand, stability of parameters of a passive suspension system is considered an advantage as its construction can be made simple, but on the other hand the vehicle suspension will not perform optimally in response to inputs other than average. Steel and rubber connectors used as joint components in vehicle suspensions play a major role in damping higher-frequency vibrations.

The areas in the first and third quadrant have relevance to the family of semiactive characteristics. A semiactive element comprises a damping element whose damping ratio can be varied through the real-time control process. A semiactive suspension does not generate any active force, hence the power demand remains on a low level. The damping ratio can be varied using throttling valves or through application of rheological fluids whose stiffness and viscosity depend on electric or magnetic field intensity. The work [3] investigates the potential applications of dry friction in semiactive dampers. Another solution uses a lever system wherein the attachment point of the spring can be varied and its elasticity controllable [4]. The upper frequency limit for the effective performance of semiactive suspension systems is about 100 Hz. Because of their low power demand, semiactive suspensions systems are being vigorously researched and widely implemented in vehicles.

The operation of active suspension systems is revealed in characteristics in all four quadrants, whereas the second and fourth quadrant capture the conditions where the actuator requires an external energy source, otherwise it acts as a passive element, dissipating energy. Active suspension systems incorporate a force actuator, either independent, or connected in parallel to a damper or a spring. Active suspensions prove most effective in the case of low-frequency vibrations [5], about 5 Hz and are often added to passive systems which well handle high-frequency vibrations. High efficiency of active systems, however, comes at the cost of high power demand. Active suspension systems typically utilise small hydraulic cylinders to achieve high accelerations of masses subjected to considerable loads. Pneumatic cylinders are capable of delivering higher velocity than hydraulic ones, yet the acceleration is strongly dependent on the external load applied. There are also hydro-pneumatic cylinders, displaying the advantages of the two previous types. Active suspension systems in truck cabs utilise electric cylinders DC and AC-servo.

106 Advances on Analysis and Control of Vibrations – Theory and Applications

in Fig 1.

Active damping

Semiactive and active damping

lend a new quality to control of low-frequency vibrations in vehicles. The operating ranges of passive, semiactive and active suspensions, given as velocity-force characteristics, are shown

Force

**Figure 1.** The range of velocity-force characteristics of passive, semiactive and active suspensions [2]

in vehicle suspensions play a major role in damping higher-frequency vibrations.

vigorously researched and widely implemented in vehicles.

Passive suspensions are described with a damper characteristics with fixed parameters (broke line). These suspensions typically comprise elastic elements featuring a linear or nonlinear elasticity and damping elements with nonlinear characteristics. On one hand, stability of parameters of a passive suspension system is considered an advantage as its construction can be made simple, but on the other hand the vehicle suspension will not perform optimally in response to inputs other than average. Steel and rubber connectors used as joint components

Velocity

Passive damping

The areas in the first and third quadrant have relevance to the family of semiactive characteristics. A semiactive element comprises a damping element whose damping ratio can be varied through the real-time control process. A semiactive suspension does not generate any active force, hence the power demand remains on a low level. The damping ratio can be varied using throttling valves or through application of rheological fluids whose stiffness and viscosity depend on electric or magnetic field intensity. The work [3] investigates the potential applications of dry friction in semiactive dampers. Another solution uses a lever system wherein the attachment point of the spring can be varied and its elasticity controllable [4]. The upper frequency limit for the effective performance of semiactive suspension systems is about 100 Hz. Because of their low power demand, semiactive suspensions systems are being

The operation of active suspension systems is revealed in characteristics in all four quadrants, whereas the second and fourth quadrant capture the conditions where the actuator requires an external energy source, otherwise it acts as a passive element, dissipating energy. Active suspension systems incorporate a force actuator, either independent, or connected in parallel to a damper or a spring. Active suspensions prove The new developments of the operator's seats are in line with the advancements in vehicle suspensions. In the simplest solution where the seat is fixed rigid to the machine frame or to the cab floor, vibrations are transmitted from the cab attachment point onto the operator. The mobile seat support allows it to be moved only in the vertical so the passive, semiactive or active vibration reduction system can be added along this direction only. Typically, a shearing mechanism is used as seat support [6], though other solutions with a greater number of DOFs are reported as well [7]. On account of small mass of the seat together with the operator, the upper frequency limit is reached and the active support becomes more effective than a vehicle suspension. Active suspension systems are now incorporated in agricultural and forestry machinery, where typical excitations are in the form of lowfrequency and high-amplitude vibrations [8].

The correct control strategy is a key element in active and semiactive vibration reduction systems, resulting in a good compromise between numerous and sometimes mutually excluding requirements. Optimal control techniques that handle this problem include the linear quadratic regulator LQR [9] and the linear Gaussian regulator LQG [10]. In simpler cases the PID control can be applied. Active control systems may also use regulators based on neural networks [11] and fuzzy logics [12]. The control schemes that are commonly used for semiactive suspension systems include the 'sky-hook' control strategy [13] where the damping force is related to the absolute velocity of the vehicle body. The *H∞* control is insensitive to uncertain input quantities, encumbered with major errors (for instant: time constant, reduced mass, damping force, acceleration, velocity, damping ratio for the tire [14]. The *H∞* control scheme concurrently executes the mutlicriterial optimisation and in active and semiactive suspension systems it takes into account the acceleration of sprung mass, peak accelerations, jerks of the front and rear suspension, road holding, forces acting upon the relevant masses, deflection of the tire and of the suspension [15]. The work [16] focused on *H<sup>∞</sup>* control in an active suspension of vehicle investigates the influence of the time delay on stability of the control process. It is shown that the delay time (i.e. time before the cylinder in the active system is activated), the reduced mass and power ratings are major determinants of the frequency limit for effective operation of these type of suspensions. The stability condition is formulated for the predetermined range of time delay and system parameters. Extensive expertise prompts the use of filters, beside robust control for the purpose of estimation. The Kalman filter, reported in literature on the subject [17], is now widely used.

To improve the operator's comfort, an active suspension of a cab can be incorporated in the machine structure (Fig 2), to reduce the cab's vibration. The active suspension system comprises several sub-systems:

The Active Suspension of a Cab in a Heavy Machine 109

*A*<sup>3</sup>

*B*<sup>1</sup>

.

*A*<sup>1</sup>

.

1

*r* 

*B*<sup>3</sup>

.

3*C*

*yr* 

three passive links, set in motion by two linear drives. The separate seat suspension mechanism reduces the vibrations along the axis *zr*. The main function of the active suspension system is to stabilise the cab such that the correct control of the drives 1 and 4 should enable its vertical movement in the direction of the gravity force. The active suspension system comprises a platform *p* suspended on three limbs with spherical pairs having the centres *B*1, *B*2, *B*3. The two limbs are rocker arms 2 and 3, connected to the machine frame *r* via a revolving pair. The third limb is the actuator l with the length *s*1. The cylinder in the actuator 1*c* is connected to the frame *r* via a cross pair with the point *A*1. The piston in the actuator 4*t* is connected to the rocker arm via a spherical joint with the point *C*4. The length of the actuator 4 equals *s*4. The part of the active suspension system comprising the rocker arms 2 and 3, a actuator and the platform *p* along the line segment *B*2*B*3 can be treated as a planar mechanism where the points *A*4, *C*4, *A*2, *B*2 and *A*3 are on the plane *yrzr* and the axes of joints *A*2 and *A*3 are parallel to *xr*. The structure of the mechanism is such that the actuators 1 and 4, when in their middle position, do not carry the cab's gravity load and when in their extreme positions, the load due to the gravity force is carried mostly by the joints *A*2, *A*3. Besides, the performance of the mechanism is affected by manufacturing

**Figure 3.** Platform mechanism stabilising the cab in the vertical: *r* - machine frame, *p* - platform, 2,3 rocker arms,4*w* - forks in cross joints, 1*c*, 4*c* - cylinders in actuators, 1*t*, 4*t* - pistons in cylinders

The cab is rigidly attached to the platform *p*. The centre of gravity (c.o.g) of the cab is at the point *Qk*. The reference systems associated with the platform {*Qpxpypzp*} and with the cab {*Qkxkykzk*} are parallel and immobile with respect to one another. Inside the cab there is a movable operator's seat *f*, which can be moved with respect to the cab, along its vertical axis *zk*. The seat suspension mechanism with the operator is not the subject matter of the present

2

*fp* 

*xp* 

*Op* 

*yp* 

5

. .

*zp* 

*Of* 

*Ok x y*<sup>k</sup>*<sup>k</sup>*

.

*zk* 

imprecision, though this influence is found to be negligible.

*C*<sup>4</sup>

.

*p*

*r*

*xr*

. *A*4

4

*Or= A2*

.

*zr*

*B*2

.

**Figure 2.** Block diagram of the active suspension of a cab


### **2. Actuator mechanism in the active suspension system**

The active suspension mechanism, shown schematically in Fig 3 has been engineered specifically for the purpose of modelling and simulations and its design involves a certain trade-off between functionality and simplicity. The presented active suspension mechanism is capable of reducing the amplitudes of the cab's linear vibrations in the direction *yr* and its angular vibrations around the axes *xr*, *yr*. The active suspension mechanism comprises just three passive links, set in motion by two linear drives. The separate seat suspension mechanism reduces the vibrations along the axis *zr*. The main function of the active suspension system is to stabilise the cab such that the correct control of the drives 1 and 4 should enable its vertical movement in the direction of the gravity force. The active suspension system comprises a platform *p* suspended on three limbs with spherical pairs having the centres *B*1, *B*2, *B*3. The two limbs are rocker arms 2 and 3, connected to the machine frame *r* via a revolving pair. The third limb is the actuator l with the length *s*1. The cylinder in the actuator 1*c* is connected to the frame *r* via a cross pair with the point *A*1. The piston in the actuator 4*t* is connected to the rocker arm via a spherical joint with the point *C*4. The length of the actuator 4 equals *s*4. The part of the active suspension system comprising the rocker arms 2 and 3, a actuator and the platform *p* along the line segment *B*2*B*3 can be treated as a planar mechanism where the points *A*4, *C*4, *A*2, *B*2 and *A*3 are on the plane *yrzr* and the axes of joints *A*2 and *A*3 are parallel to *xr*. The structure of the mechanism is such that the actuators 1 and 4, when in their middle position, do not carry the cab's gravity load and when in their extreme positions, the load due to the gravity force is carried mostly by the joints *A*2, *A*3. Besides, the performance of the mechanism is affected by manufacturing imprecision, though this influence is found to be negligible.

108 Advances on Analysis and Control of Vibrations – Theory and Applications

**Figure 2.** Block diagram of the active suspension of a cab

Suspension mechanism

Cab

Drives

Machine frame

spatial movement, measured with a set of sensors.

anticipated loads and the required drive velocities.

**2. Actuator mechanism in the active suspension system** 

the control sub-system.

1. The actuator mechanism, connected to the machine frame and the cab is placed in between. The main element (link) of the mechanism is a mobile platform to which the cab is attached. The platform is suspended or supported on the frame and depending on the

Parameters of the frame motion

Active suspension of the cab

Parameters of the drives' motion

The required drive velocities

Measuring sub-system

Control sub-system

3. Measuring sub-system. Displacement and velocity are chosen as control quantities for the active suspension system. Directly measured data yield the error signal to be used in the control process. The machine frame subjected to kinematic excitations executes a

4. Control sub-system. Errors of drive positions and their derivatives are going to be used in the feedback control of the active suspension system. Basing on the frame motion measurements, the control sub-system performs the real time calculation of the

The active suspension mechanism, shown schematically in Fig 3 has been engineered specifically for the purpose of modelling and simulations and its design involves a certain trade-off between functionality and simplicity. The presented active suspension mechanism is capable of reducing the amplitudes of the cab's linear vibrations in the direction *yr* and its angular vibrations around the axes *xr*, *yr*. The active suspension mechanism comprises just

mechanism's mobility, it can move with respect to the frame in the selected DOFs. 2. The drives set in motion the passive links in the active suspension mechanism. On account of the stiffness requirements and availability of the given type of energy, and to ensure fast response the control signals the hydraulic drives are going to be used. The drives are provided with actuators to capture the instantaneous velocities, derived in

**Figure 3.** Platform mechanism stabilising the cab in the vertical: *r* - machine frame, *p* - platform, 2,3 rocker arms,4*w* - forks in cross joints, 1*c*, 4*c* - cylinders in actuators, 1*t*, 4*t* - pistons in cylinders

The cab is rigidly attached to the platform *p*. The centre of gravity (c.o.g) of the cab is at the point *Qk*. The reference systems associated with the platform {*Qpxpypzp*} and with the cab {*Qkxkykzk*} are parallel and immobile with respect to one another. Inside the cab there is a movable operator's seat *f*, which can be moved with respect to the cab, along its vertical axis *zk*. The seat suspension mechanism with the operator is not the subject matter of the present study. The centre of gravity of the seat and the operator is at the point *Qf*, whose vertical coordinate in the reference system associated with the platform is controlled by the drive 5, implementing seat elevation.

The Active Suspension of a Cab in a Heavy Machine 111

 *<sup>x</sup>* , *<sup>y</sup>* 

(2)

(3)

(4)

*<sup>g</sup>* **z** given in the reference system {*Qhxhyhzh*} should be

*Px v* , whilst their velocity components in the direction of *yg*

*<sup>g</sup> h g* **z Rz** (1)

*o r ho*

 

**R** - the transition matrix from the

 

 

Underlying the simulation procedure is the model of the kinematic excitation applied to the

The machine frame is represented by a front bridge (*PP*, *PT*), a longitudinal frame (*P*, *T*) and a joint at the point *Or*, where the active suspension mechanism is connected to the frame. The rear bridge (*TL, TP*) is connected to the longitudinal frame via a revolving pair *T*. It is assumed that:



The vertical displacement of the centre of the front right wheel is governed by a harmonic

*h Px*

, then vertical displacement of the centre of the front left wheel is taken to be

*h Px*

*<sup>l</sup> t t* , the vertical displacement of the centre of the rear right wheel is taken to be

( ) 1 cos 2 *<sup>m</sup>*

*h*

*g*

*g v tt*

*Px l*

*v tt*

*L*

*g*

*L*

*L* 

, vertical displacement of the centre of the front left wheel is expressed

1 cos 2 *<sup>P</sup>*

*v t z hh*

( ) 1 cos 2 *<sup>L</sup>*

*<sup>l</sup> t t* , the vertical displacement is expressed as harmonic:

*P og*

*P og*

*z hh*

*P*

*T og*

*z hh*

cos 0 sin

*y y*

*xy x x y*

system {*Qhxhyhzh*} to {*Qrxryrzr*}, derived basing on the frame deflection angles

cos sin sin cos cos

 

 

( , ) sin sin cos sin cos

 

*h xy x y x x y*


versor of the gravity force [0,0, 1] *h o <sup>T</sup>*

measured in the measuring sub-system.

machine frame, shown in Fig 4.

are identical and equal to *<sup>h</sup>*

wheels during the ride in rough terrain.

are negligible,

function:

When *t t*

When *<sup>m</sup>*

*LP o z h* . When *t t*

*PT o z h* . When *<sup>m</sup>*

as the harmonic:

expressed in the system {*Qrxryrzr*}:

where:

*r*

 

### **3. Model of the kinematic excitation of the machine frame motion**

To implement the control, we need to know the angles of deflection of the vertical frame axis in the direction of the gravity force (Fig 4). The first measured angle defines the frame rotation around the longitudinal axis *<sup>x</sup>* , the other angle defines the frame rotation round the lateral axis *<sup>y</sup>* .

**Figure 4.** Model of the kinematic excitation inducing the motion of the machine frame

The model of the active suspension mechanism uses several reference systems. The immobile system {*Qgxgygzg*} is associated with the road travelled by the machine. This system is used to define the function of road profile on which the machine travels. The mobile system {*Qrxryrzr*} is associated with the machine frame (Fig 3, Fig 4). Its origin *Or*=*A*2 is one of the attachments points of the active suspension mechanism to the frame. This system is recalled to define all kinematic and dynamic quantities (no superscript on the left). The system {*Qhxhyhzh*} (Fig 4) is intermediate between the inertial system associated with the road and the mobile system associated with the frame. The origins of the reference systems {*Qhxhyhzh*}and {*Qrxryrzr*} will coincide: *Oh*=*Or*. The axes *zh* and *zg* are parallel and the plane determined by *xhzh* contains the axis *xr*. In the model of the active suspension, the directional versor of the gravity force [0,0, 1] *h o <sup>T</sup> <sup>g</sup>* **z** given in the reference system {*Qhxhyhzh*} should be expressed in the system {*Qrxryrzr*}:

$$\mathbf{z}\_{\mathcal{g}}^{o} = {}\_{h}^{r} \mathbf{R}^{\ \ h} \mathbf{z}\_{\mathcal{g}}^{o} \tag{1}$$

where: cos 0 sin ( , ) sin sin cos sin cos cos sin sin cos cos *y y r h xy x y x x y xy x x y* **R** - the transition matrix from the

system {*Qhxhyhzh*} to {*Qrxryrzr*}, derived basing on the frame deflection angles *<sup>x</sup>* , *<sup>y</sup>* measured in the measuring sub-system.

Underlying the simulation procedure is the model of the kinematic excitation applied to the machine frame, shown in Fig 4.

The machine frame is represented by a front bridge (*PP*, *PT*), a longitudinal frame (*P*, *T*) and a joint at the point *Or*, where the active suspension mechanism is connected to the frame. The rear bridge (*TL, TP*) is connected to the longitudinal frame via a revolving pair *T*. It is assumed that:


110 Advances on Analysis and Control of Vibrations – Theory and Applications

implementing seat elevation.

the lateral axis *<sup>y</sup>*

*zg* 

*g o ho <sup>g</sup> <sup>g</sup> z z*

rotation around the longitudinal axis

*yg* 

.

*TL* 

*LTz*

study. The centre of gravity of the seat and the operator is at the point *Qf*, whose vertical coordinate in the reference system associated with the platform is controlled by the drive 5,

To implement the control, we need to know the angles of deflection of the vertical frame axis in the direction of the gravity force (Fig 4). The first measured angle defines the frame

*zh* 

*yr zr* 

*<sup>P</sup> <sup>T</sup>*

*Tz Pz*

*Lg* 

*xh* 

*<sup>r</sup> PO* **<sup>r</sup>**

*xr* 

*PL* 

*LPz*

*lm* 

*wm* 

*yh* 

*y*

*<sup>x</sup>* , the other angle defines the frame rotation round

*x*

*xg* 

*hg* 

*PP* 

*PPz*

*h Px v*

**3. Model of the kinematic excitation of the machine frame motion** 

*Or=Oh* 

*TP* 

*PTz*

**Figure 4.** Model of the kinematic excitation inducing the motion of the machine frame

The model of the active suspension mechanism uses several reference systems. The immobile system {*Qgxgygzg*} is associated with the road travelled by the machine. This system is used to define the function of road profile on which the machine travels. The mobile system {*Qrxryrzr*} is associated with the machine frame (Fig 3, Fig 4). Its origin *Or*=*A*2 is one of the attachments points of the active suspension mechanism to the frame. This system is recalled to define all kinematic and dynamic quantities (no superscript on the left). The system {*Qhxhyhzh*} (Fig 4) is intermediate between the inertial system associated with the road and the mobile system associated with the frame. The origins of the reference systems {*Qhxhyhzh*}and {*Qrxryrzr*} will coincide: *Oh*=*Or*. The axes *zh* and *zg* are parallel and the plane determined by *xhzh* contains the axis *xr*. In the model of the active suspension, the directional


The vertical displacement of the centre of the front right wheel is governed by a harmonic function:

$$z\_{p\_p} = h\_o + h\_g \left[ 1 - \cos \left( 2\pi \frac{h\_p v\_{p\_x} t}{L\_g} \right) \right] \tag{2}$$

When *t t* , then vertical displacement of the centre of the front left wheel is taken to be *LP o z h* . When *t t* , vertical displacement of the centre of the front left wheel is expressed as the harmonic:

$$z\_{P\_{\perp}} = h\_o + h\_g \left[ 1 - \cos \left( 2 \pi \frac{^h v\_{Px}(t - t\_{\phi})}{L\_g} \right) \right] \tag{3}$$

When *<sup>m</sup> <sup>l</sup> t t* , the vertical displacement of the centre of the rear right wheel is taken to be *PT o z h* . When *<sup>m</sup> <sup>l</sup> t t* , the vertical displacement is expressed as harmonic:

$$\mathbf{z}\_{T\_P} = h\_o + h\_g \left[ 1 - \cos \left( 2\pi \frac{^h \upsilon\_{P\_X} (t - t\_{l\_m})}{L\_g} \right) \right] \tag{4}$$

When *<sup>m</sup> <sup>l</sup> tt t* , the vertical displacement of the centre of the rear left wheel is taken to be *LT o z h* . When *<sup>m</sup> <sup>l</sup> tt t*, the vertical displacement is expressed as harmonic:

$$\mathbf{z}\_{T\_L} = h\_o + h\_g \left[ 1 - \cos \left( 2\pi \frac{^h \upsilon\_{P\_X} (t - t\_{l\_m} - t\_g)}{L\_g} \right) \right] \tag{5}$$

The Active Suspension of a Cab in a Heavy Machine 113

*<sup>p</sup>* **<sup>x</sup>** , *<sup>o</sup> <sup>p</sup>* **<sup>y</sup>** , *<sup>o</sup>*

*y z* **<sup>a</sup>** *a a* and [1,0,0] *o T*

*<sup>p</sup>* **z** . The cab

*<sup>r</sup>* **x** :

**4.1. Direct kinematics problem of links position** 

The versor 2

The versor 4

The versor 2

The versor 6

where: 2 2

The versor 3

Solving the direct problem consists in finding the cab orientation and position of its centre of gravity *Qk* and of the point *Qf* - the centre of gravity of the seat-operator system with respect to the reference system associated with the machine frame. The cab orientation is determined by

position (and of the platform and the seat) and *rk* **r** - the radius vector of the of the cab's c.o.g with respect to the reference system associated with the frame depend on variable lengths of actuators *s*1, *s*4 and the known constant dimensions of links in the active suspension mechanism. The radius vector of the c.o.g of the seat-operator system; *rf* **r** is controlled by the lengths of three actuators *s*1, *s*4, *s*5 and the fixed dimensions of links in the active suspension mechanism. The solution of the simple problem involving the link position is explicit and consists in determining versors on the basis of two already known or already established ones.

> <sup>2</sup> <sup>222</sup> <sup>222</sup> 244 244

**c a a x** (6)

**s ac** (7)

*<sup>r</sup>* **d c xc** *c c* (8)

**s dy** (9)

 **d s s x** (10)

4 2 4 2 ( )1 2 2

*r csa csa a c a c* 

directional versors in the reference system associated with the platform *<sup>o</sup>*

*<sup>o</sup>* **<sup>c</sup>** is derived basing on two known versors 4 44 [0, , ] *o o oT*

*<sup>o</sup>* **<sup>a</sup>** and 2

*<sup>r</sup>* **x** and 2

*<sup>o</sup>* **c** :

4 2 4 42 4 4 *o oo a c s s*

*<sup>o</sup>* **c** :

*<sup>o</sup>* **d** and [0,1,0] *o T*

6 2

*<sup>o</sup>* **s** i and *<sup>o</sup>*

2 3

*d a s s*

6 6 *o oo*

*<sup>r</sup>* **x** :

<sup>2</sup> 22 2 22 2 3 23 6 3 23 6

6 3 6 3 1

*r db s db s s d s d* 

2 2

2 2 22 2 22 2 1( ) *o o o o*

*<sup>r</sup>* **y** :

*r*

2 4 4

where: 22 2 2 2 2 cos ( , ) *o o oo <sup>c</sup>* **c d cd** , 2 2 (, ) *o o* **c d** - a known fixed angle.

6 2 3 23 2 3 2( ) *o o s d a da* **d a** , 2 2 *<sup>r</sup> d OB* , 3 3 *<sup>r</sup> a OA* .

3 6 6

*o o o o*

where: : 4 44 *s AC* , 2 4 *<sup>r</sup> c OC* , 4 4 *<sup>r</sup> a OA* .

*<sup>o</sup>* **<sup>s</sup>** is obtained basing on 4

*<sup>o</sup>* **d** is obtained basing on *<sup>o</sup>*

*<sup>o</sup>* **<sup>s</sup>** is obtained basing on 2

*<sup>o</sup>* **<sup>d</sup>** is obtained basing on 6

*o o o o*

where: 2 *g h Px L t v* - phase shift time between the left and right hand side of the machine,

*m m l Px l t <sup>v</sup>* - phase shift time between the front and rear part of the machine,

*Lg* - distance corresponding to the full wave, *hg*- amplitude, *wm* - width of the front and rear bridge,

*lm* -distance between the front and rear bridge, - phase shift angle of the road profile between the left and right-hand side of the machine.

### **4. Kinematic model of the active suspension mechanism**

To determine the influence of the active suspension system on the cab motion, the kinematic model is developed based on vector calculus. Versors used to define the positions of the active suspension mechanism links are shown in Fig 5.

**Figure 5.** Versors in the kinematic model of cab stabilisation in the vertical

### **4.1. Direct kinematics problem of links position**

112 Advances on Analysis and Control of Vibrations – Theory and Applications

*L*

*T og*

*<sup>v</sup>* - phase shift time between the front and rear part of the machine,

**4. Kinematic model of the active suspension mechanism** 

**Figure 5.** Versors in the kinematic model of cab stabilisation in the vertical

**.**

*Or=A2*

*B*2

**.**

2 *o c*

.

*zr* 

*o r z*

*z hh*

, the vertical displacement of the centre of the rear left wheel is taken to be

( ) 1 cos 2 *<sup>m</sup>*

*Px l*

*v tt t*

*L*

*g*

 (5)


*A*<sup>3</sup>

*B*<sup>3</sup>

**.** 

3 *o d*

*yr* 

**.**

*A*<sup>1</sup>

*B*<sup>1</sup>

1 *<sup>o</sup> s* <sup>8</sup> *o s*

*h*


, the vertical displacement is expressed as harmonic:

*Lg* - distance corresponding to the full wave, *hg*- amplitude, *wm* - width of the front and rear

To determine the influence of the active suspension system on the cab motion, the kinematic model is developed based on vector calculus. Versors used to define the positions of the

**. . .** 

**.** *xr* 

*o <sup>r</sup> <sup>y</sup>* <sup>2</sup> *o d*

*xp* 

*zp* 

*Ok* 

*o g z*

**. .***Of* 

*rf r*

*rk r*

*Op* 

**.**

*o p z o <sup>p</sup> <sup>x</sup> <sup>o</sup> p y*

*yp* 

7 *o s*

1 *o a*

6 *o s*

When *<sup>m</sup>*

where:

*m*

bridge,

*l*

*t*

*t*

*m*

*Px l*

*A*<sup>4</sup>

*<sup>l</sup> tt t*

*LT o z h* . When *<sup>m</sup>*

2

*g h Px*

*v*

*L*

*<sup>l</sup> tt t*

*lm* -distance between the front and rear bridge,

between the left and right-hand side of the machine.

active suspension mechanism links are shown in Fig 5.

*C4* 

4 *o s*

> 4 *o a*

*o r x*

Solving the direct problem consists in finding the cab orientation and position of its centre of gravity *Qk* and of the point *Qf* - the centre of gravity of the seat-operator system with respect to the reference system associated with the machine frame. The cab orientation is determined by directional versors in the reference system associated with the platform *<sup>o</sup> <sup>p</sup>* **<sup>x</sup>** , *<sup>o</sup> <sup>p</sup>* **<sup>y</sup>** , *<sup>o</sup> <sup>p</sup>* **z** . The cab position (and of the platform and the seat) and *rk* **r** - the radius vector of the of the cab's c.o.g with respect to the reference system associated with the frame depend on variable lengths of actuators *s*1, *s*4 and the known constant dimensions of links in the active suspension mechanism. The radius vector of the c.o.g of the seat-operator system; *rf* **r** is controlled by the lengths of three actuators *s*1, *s*4, *s*5 and the fixed dimensions of links in the active suspension mechanism. The solution of the simple problem involving the link position is explicit and consists in determining versors on the basis of two already known or already established ones.

The versor 2 *<sup>o</sup>* **<sup>c</sup>** is derived basing on two known versors 4 44 [0, , ] *o o oT y z* **<sup>a</sup>** *a a* and [1,0,0] *o T <sup>r</sup>* **x** :

$$\mathbf{c}\_{2}^{o} = \mathbf{a}\_{4}^{o} \frac{c\_{2}^{2} - s\_{4}^{2} + a\_{4}^{2}}{2a\_{4}c\_{2}} + (\mathbf{a}\_{4}^{o} \times \mathbf{x}\_{r}^{o}) \sqrt{1 - \left(\frac{c\_{2}^{2} - s\_{4}^{2} + a\_{4}^{2}}{2a\_{4}c\_{2}}\right)^{2}} \tag{6}$$

where: : 4 44 *s AC* , 2 4 *<sup>r</sup> c OC* , 4 4 *<sup>r</sup> a OA* .

The versor 4 *<sup>o</sup>* **<sup>s</sup>** is obtained basing on 4 *<sup>o</sup>* **<sup>a</sup>** and 2 *<sup>o</sup>* **c** :

$$\mathbf{s}\_4^o = -\frac{a\_4}{s\_4}\mathbf{a}\_4^o + \frac{c\_2}{s\_4}\mathbf{c}\_2^o \tag{7}$$

The versor 2 *<sup>o</sup>* **d** is obtained basing on *<sup>o</sup> <sup>r</sup>* **x** and 2 *<sup>o</sup>* **c** :

$$\mathbf{d}\_{2}^{o} = c\_{22}\mathbf{c}\_{2}^{o} + \sqrt{1 - c\_{22}^{2}} \left( \mathbf{x}\_{r}^{o} \times \mathbf{c}\_{2}^{o} \right) \tag{8}$$

where: 22 2 2 2 2 cos ( , ) *o o oo <sup>c</sup>* **c d cd** , 2 2 (, ) *o o* **c d** - a known fixed angle. The versor 6 *<sup>o</sup>* **<sup>s</sup>** is obtained basing on 2 *<sup>o</sup>* **d** and [0,1,0] *o T <sup>r</sup>* **y** :

$$\mathbf{s}\_{\boldsymbol{\theta}}^{o} = -\frac{d\_2}{s\_{\boldsymbol{\theta}}} \mathbf{d}\_2^{o} + \frac{a\_3}{s\_{\boldsymbol{\theta}}} \mathbf{y}\_r^{o} \tag{9}$$

where: 2 2 6 2 3 23 2 3 2( ) *o o s d a da* **d a** , 2 2 *<sup>r</sup> d OB* , 3 3 *<sup>r</sup> a OA* .

The versor 3 *<sup>o</sup>* **<sup>d</sup>** is obtained basing on 6 *<sup>o</sup>* **s** i and *<sup>o</sup> <sup>r</sup>* **x** :

$$\mathbf{d}\_3^o = \mathbf{s}\_6^o \frac{d\_3^2 - b\_{23}^2 + s\_6^2}{2s\_6 d\_3} - \mathbf{s}\_6^o \times \mathbf{x}\_r^o \sqrt{1 - \left(\frac{d\_3^2 - b\_{23}^2 + s\_6^2}{2s\_6 d\_3}\right)^2} \tag{10}$$

where: 3 33 *d AB* , 23 2 3 *b BB* .

The versor *<sup>o</sup> <sup>p</sup>* **y** is obtained basing on 6 *<sup>o</sup>* **<sup>s</sup>** and 3 *<sup>o</sup>* **d** :

$$\mathbf{y}\_p^o = \frac{\mathbf{s}\_6}{b\_{23}} \mathbf{s}\_6^o + \frac{d\_3}{b\_{23}} \mathbf{d}\_3^o \tag{11}$$

The Active Suspension of a Cab in a Heavy Machine 115

(17)

*oo oo oo p r pr pr r oo oo oo p pr pr pr oo oo oo p r pr pr*

**xx yx zx**

The solution to the simple problem involving the link position is complete when the positions of points *Ok*. and *Of* are found in relation to *s*1, *s*4, *s*5. The radius vector from the origin of the reference system associated with the frame *Or* to the point *Of* becomes (Fig 5):

where: () () () () [ ] *p pp p p <sup>T</sup> O Op f pf pf x pf y pf z* **<sup>r</sup>** *rrr* - radius vector of the point *Or* in the system

*pf x r* , ( ) *p*

The radius vector from the origin of the reference system associated with the frame *Or* to the

where: () () () () [ ] *p pp p p <sup>T</sup> OO r r r p k pk pk x pk y pk z* **<sup>r</sup>** - radius vector of the point *Ok* in the system

The inverse problem handled in the coordinate system associated with the machine frame involves the orientation of the platform *p*. The platform should be stabilised in the vertical whilst the active suspension system is in use. The platform position is related to the gravity

In order to solve the inverse problem it is required that the lengths of the actuators *s*1*<sup>o</sup>*, *s*4*<sup>o</sup>* should be established, corresponding to the predetermined and expected platform position with respect to the system associated with the frame and expressed by versor coordinates:

anticipated values (indicated with a subscript "o") obtained from solving the inverse problem will be used to derive the error signal required for the control process. The direction of the cab's vertical axis versor should be opposite to that of the gravity force

*<sup>r</sup>* **x** :

*o oo*

*po* **<sup>z</sup>** and *<sup>o</sup>*

*o or p*

*pk x pk y pk z rrr* - fixed coordinates

*<sup>g</sup>* **z** (Fig 4, 5, 6), which can be expressed in the coordinate system associated with

*po* **z** . Actually, the cab will reach the position nearing the expected one. The

*po po r* **y zx** (20)

22 2

22 2

*p*

associated with the platform, () 5

associated with the platform. () () () , , *ppp*

the frame according to the formula (1).

**4.2. Inverse kinematics problem of link position** 

*po* **<sup>y</sup>** is obtained basing on *<sup>o</sup>*

controlled by the seat elevating drive 5. ( )

point *Ok* becomes:

force versor *<sup>o</sup>*

*po* **<sup>y</sup>** and *<sup>o</sup>*

versor *o o po g* **z z** .

The versor *<sup>o</sup>*

*o po* **<sup>x</sup>** , *<sup>o</sup>* *p*

**xz yz zz**

*o or p*

*rf p p pf* **r d y Rr** *d b* (18)

*pf z z r df s* - variable vertical coordinate of the seat

*pf y r* , *<sup>z</sup> df* - fixed coordinates.

*rk p p pk* **r d y Rr** *d b* (19)

**R xy yy zy**

The versor 7 *<sup>o</sup>* **<sup>s</sup>** is obtained basing on 2 *<sup>o</sup>* **<sup>d</sup>** and 1 111 [,,] *o o o oT xyz* **a** *aaa* :

$$\mathbf{s}\_{\mathcal{T}}^{o} = -\frac{d\_{2}}{s\_{\mathcal{T}}} \mathbf{d}\_{2}^{o} + \frac{a\_{1}}{s\_{\mathcal{T}}} \mathbf{a}\_{1}^{o} \tag{12}$$

where: 2 2 7 2 1 21 2 1 2( ) *o o s d a da* **d a** , 1 1 *<sup>r</sup> a OA* .

The versor 8 *<sup>o</sup>* **<sup>s</sup>** is obtained basing on 7 *<sup>o</sup>* **s** and *<sup>o</sup> <sup>p</sup>* **y** :

$$\mathbf{s}\_8^o = \frac{\mathbf{s}\_7}{\mathbf{s}\_8} \mathbf{s}\_7^o + \frac{\mathbf{b}\_2}{\mathbf{s}\_8} \mathbf{y}\_p^o \tag{13}$$

where: 2 2 8 7 2 72 7 2( ) *o o <sup>p</sup> s s b sb* **s y** , 2 2 *<sup>p</sup> b OB* .

The versor 1 *<sup>o</sup>* **<sup>s</sup>** is obtained basing on 7 *<sup>o</sup>* **<sup>s</sup>** and 8 *<sup>o</sup>* **s** :

$$\mathbf{s}\_{1}^{o} = \frac{c\_{71} - c\_{78}c\_{81}}{1 - c\_{78}^{2}} \mathbf{s}\_{7}^{o} + \frac{c\_{81} - c\_{78}c\_{71}}{1 - c\_{78}^{2}} \mathbf{s}\_{8}^{o} - \frac{\sqrt{1 - c\_{78}^{2} - c\_{71}^{2} - c\_{81}^{2} + 2c\_{78}c\_{71}c\_{81}}}{1 - c\_{78}^{2}} (\mathbf{s}\_{7}^{o} \times \mathbf{s}\_{8}^{o}) \tag{14}$$

where: 222 7 1 12 71 1 7 2 *ssb c s s* , 222 782 78 7 8 2 *ssb c s s* , 222 811 81 1 8 2 *ssb c s s* , 1 1 *<sup>p</sup> b OB* , 12 1 2 *b BB* .

The versor *<sup>o</sup> <sup>p</sup>* **x** is obtained basing on 1 *<sup>o</sup>* **<sup>s</sup>** and 8 *<sup>o</sup>* **s** :

$$\mathbf{x}\_p^o = -\frac{s\_1}{b\_1}\mathbf{s}\_1^o + \frac{s\_8}{b\_1}\mathbf{s}\_8^o \tag{15}$$

where: 1 1 *<sup>p</sup> b OB* .

The versor *<sup>o</sup> <sup>p</sup>* **<sup>z</sup>** is obtained basing on *<sup>o</sup> <sup>p</sup>* **<sup>x</sup>** and *<sup>o</sup> <sup>p</sup>* **y** :

$$\mathbf{z}\_p^o = \mathbf{x}\_p^o \times \mathbf{y}\_p^o \tag{16}$$

Eq (11), (15) and (16) yield the versors of the platform p, and the matrix *<sup>r</sup> <sup>p</sup>***R** - the direction matrix of the reference system associated with the platform with respect to the system associated with the frame:

The Active Suspension of a Cab in a Heavy Machine 115

$$\mathbf{^r}\_p \mathbf{R} = \begin{bmatrix} \mathbf{x}\_p^o \cdot \mathbf{x}\_r^o & \mathbf{y}\_p^o \cdot \mathbf{x}\_r^o & \mathbf{z}\_p^o \cdot \mathbf{x}\_r^o \\ \mathbf{x}\_p^o \cdot \mathbf{y}\_r^o & \mathbf{y}\_p^o \cdot \mathbf{y}\_r^o & \mathbf{z}\_p^o \cdot \mathbf{y}\_r^o \\ \mathbf{x}\_p^o \cdot \mathbf{z}\_r^o & \mathbf{y}\_p^o \cdot \mathbf{z}\_r^o & \mathbf{z}\_p^o \cdot \mathbf{z}\_r^o \end{bmatrix} \tag{17}$$

The solution to the simple problem involving the link position is complete when the positions of points *Ok*. and *Of* are found in relation to *s*1, *s*4, *s*5. The radius vector from the origin of the reference system associated with the frame *Or* to the point *Of* becomes (Fig 5):

$$\mathbf{r}\_{r\!\!\!/} = d\_2 \mathbf{d}\_2^o + b\_2 \mathbf{y}\_{\!\!/}^o + {}\_{p}^r \mathbf{R} \ {}^p \mathbf{r}\_{pf} \tag{18}$$

where: () () () () [ ] *p pp p p <sup>T</sup> O Op f pf pf x pf y pf z* **<sup>r</sup>** *rrr* - radius vector of the point *Or* in the system associated with the platform, () 5 *p pf z z r df s* - variable vertical coordinate of the seat controlled by the seat elevating drive 5. ( ) *p pf x r* , ( ) *p pf y r* , *<sup>z</sup> df* - fixed coordinates.

The radius vector from the origin of the reference system associated with the frame *Or* to the point *Ok* becomes:

$$\mathbf{r}\_{rk} = d\_2 \mathbf{d}\_2^o + b\_2 \mathbf{y}\_p^o + {}\_p^r \mathbf{R}^p \mathbf{r}\_{pk} \tag{19}$$

where: () () () () [ ] *p pp p p <sup>T</sup> OO r r r p k pk pk x pk y pk z* **<sup>r</sup>** - radius vector of the point *Ok* in the system associated with the platform. () () () , , *ppp pk x pk y pk z rrr* - fixed coordinates

### **4.2. Inverse kinematics problem of link position**

114 Advances on Analysis and Control of Vibrations – Theory and Applications

*<sup>o</sup>* **<sup>s</sup>** and 3 *<sup>o</sup>* **d** :

*<sup>o</sup>* **s** and *<sup>o</sup>*

*<sup>o</sup>* **<sup>s</sup>** and 8 *<sup>o</sup>* **s** :

78 78 78

*<sup>o</sup>* **<sup>s</sup>** and 8 *<sup>o</sup>* **s** :

*<sup>p</sup>* **<sup>x</sup>** and *<sup>o</sup>*

*p*

Eq (11), (15) and (16) yield the versors of the platform p, and the matrix *<sup>r</sup>*

*cc c* 

> 222 782

7 8 2 *ssb*

*s s* ,

71 78 81 81 78 71 78 71 81 78 71 81 1 2 2 7 8 2 7 8

81

*c*

( ) 11 1 *<sup>o</sup> <sup>o</sup> <sup>o</sup> o o c cc c cc c c c ccc*

> 1 8 1 8 1 1 *o oo*

*s s b b*

*<sup>p</sup>* **y** :

*o oo*

matrix of the reference system associated with the platform with respect to the system

**sss s s** (14)

*<sup>p</sup>* **y** :

7 2 8 7 8 8 *oo o*

*s b s s*

*p*

6 3 6 3 23 23 *ooo*

*<sup>o</sup>* **<sup>d</sup>** and 1 111 [,,] *o o o oT*

2 1 7 21 7 7 *o oo d a s s*

*xyz* **a** *aaa* :

*p*

222

1 2

222 811

1 8 2 *ssb*

*s s*

*b b* **y sd** (11)

**s da** (12)

**ssy** (13)

, 1 1 *<sup>p</sup> b OB* , 12 1 2 *b BB* .

**x ss** (15)

*<sup>p</sup> p p* **z xy** (16)

*<sup>p</sup>***R** - the direction

*s d*

*<sup>p</sup>* **y** is obtained basing on 6

*<sup>o</sup>* **<sup>s</sup>** is obtained basing on 2

7 2 1 21 2 1 2( ) *o o s d a da* **d a** , 1 1 *<sup>r</sup> a OA* .

*<sup>p</sup> s s b sb* **s y** , 2 2 *<sup>p</sup> b OB* .

78

*c*

*<sup>p</sup>* **x** is obtained basing on 1

*<sup>p</sup>* **<sup>z</sup>** is obtained basing on *<sup>o</sup>*

*<sup>o</sup>* **<sup>s</sup>** is obtained basing on 7

*<sup>o</sup>* **<sup>s</sup>** is obtained basing on 7

8 7 2 72 7 2( ) *o o*

222 7 1 12

1 7 2 *ssb*

*s s* ,

where: 3 33 *d AB* , 23 2 3 *b BB* .

The versor *<sup>o</sup>*

The versor 7

The versor 8

where: 2 2

where: 2 2

71

where: 1 1 *<sup>p</sup> b OB* .

associated with the frame:

The versor *<sup>o</sup>*

*c*

The versor *<sup>o</sup>*

The versor 1

where:

The inverse problem handled in the coordinate system associated with the machine frame involves the orientation of the platform *p*. The platform should be stabilised in the vertical whilst the active suspension system is in use. The platform position is related to the gravity force versor *<sup>o</sup> <sup>g</sup>* **z** (Fig 4, 5, 6), which can be expressed in the coordinate system associated with the frame according to the formula (1).

In order to solve the inverse problem it is required that the lengths of the actuators *s*1*<sup>o</sup>*, *s*4*<sup>o</sup>* should be established, corresponding to the predetermined and expected platform position with respect to the system associated with the frame and expressed by versor coordinates: *o po* **<sup>x</sup>** , *<sup>o</sup> po* **<sup>y</sup>** and *<sup>o</sup> po* **z** . Actually, the cab will reach the position nearing the expected one. The anticipated values (indicated with a subscript "o") obtained from solving the inverse problem will be used to derive the error signal required for the control process. The direction of the cab's vertical axis versor should be opposite to that of the gravity force versor *o o po g* **z z** .

The versor *<sup>o</sup> po* **<sup>y</sup>** is obtained basing on *<sup>o</sup> po* **<sup>z</sup>** and *<sup>o</sup> <sup>r</sup>* **x** :

$$\mathbf{y}\_{p\boldsymbol{\wp}}^{\boldsymbol{o}} = \mathbf{z}\_{p\boldsymbol{\wp}}^{\boldsymbol{o}} \times \mathbf{x}\_{r}^{\boldsymbol{o}} \tag{20}$$

**Figure 6.** Solving the inverse problem- schematic diagram

The versor *<sup>o</sup> po* **<sup>x</sup>** is obtained basing on *<sup>o</sup> po* **<sup>y</sup>** and *<sup>o</sup> po* **z** :

$$\mathbf{x}^{o}\_{po} = \mathbf{y}^{o}\_{po} \times \mathbf{z}^{o}\_{po} \tag{21}$$

The Active Suspension of a Cab in a Heavy Machine 117

**s** (27)

1 *o o o o s* **s**

1

*ooo asc* **asc** (28)

**x** (30)

(31)

*p r db a d* **d y yd** (32)

*<sup>r</sup> pr p <sup>r</sup> db d* **ω d ω y ω d** (33)

*r r ss c* **s ω s ω c** (29)

2 22

*o or o r or s* **sx sy sz** , <sup>1</sup>

In order to solve the simple problem to derive velocity of the active suspension system in the coordinate system associated with the frame it is required that the following vectors

*<sup>p</sup>*,*r kr f r* , , **ωωω** - identical angular velocity of the platform *p*, of the cab *k* and the operator

**v** - linear velocity of the c.o.g. in the seat-operator system *Of* as functions of cylinders'

44 44 22

4 4 4 4, 4 2 2, 2 ( )( ) *oo o*

2, 2, *<sup>o</sup>* **<sup>ω</sup>** *r rr* 

2,

The kinematic chain determined by points *A*2*B*2*B*3*A*3 (Fig 5.) satisfies the closing condition:

2 2 23 23 3 3 *o o oo*

2 2, 2 23 , 3 3, 3 ( ) ( )( ) *o oo*

*<sup>r</sup>* **x** determines the direction of the angular velocity vector of the rocker arm 3:

*<sup>o</sup>* **s** yields:

*<sup>r</sup>* **x** determines the direction of the angular velocity vector of the rocker arm 2:

4

2 *r s r*

The kinematic chain determined by points *A*4*C*4*A*2 (Fig 5) satisfies the closing condition:

11 1 1 ( )( )( ) *ooo*

*o o po po* **s d y ax** *d b ab* - vector of the actuator 1.

where: 1 2 2 2 11 1

**v** - linear velocity of the cab's c.o.g *Qk*,

Differentiating Eq (28) over time yields:

Scalar-multiplying Eq (29) by 4

Differentiating Eq (32) over time yields:

where: 2 2 24 [( ) ] *oo o <sup>r</sup> r c* **xcs** .

have to be determined:

seat *f*,

, *O r <sup>k</sup>*

, *O r <sup>f</sup>*

velocity.

The versor *<sup>o</sup>*

The versor *<sup>o</sup>*

*o o oo*

**4.3. Direct kinematics problem of links velocity** 

The modulus of the vector between points *Or* and *C* and its versor are computed using the triangle *OrA*3*C* (Fig 6):

$$s\_9 = \sqrt{a\_{23}^2 + b\_{23}^2 - 2a\_{23}b\_{23}(\mathbf{y}\_{pv}^o \cdot \mathbf{y}\_r^o)}\tag{22}$$

$$\mathbf{s}\_{g}^{o} = \frac{a\_{23}\mathbf{y}\_{r}^{o} - b\_{23}\mathbf{y}\_{po}^{o}}{s\_{9}} \tag{23}$$

The versor 2 *o <sup>o</sup>* **d** is obtained basing on 9 *<sup>o</sup>* **s** , *s*9, *d*2 and *d*3:

$$\mathbf{d}\_{2o}^o = -\sqrt{1 - \left(\frac{d\_2^2 - d\_3^2 + s\_9^2}{2s\_9d\_2}\right)^2} \mathbf{x}\_r^o \times \mathbf{s}\_9^o + \frac{d\_2^2 - d\_3^2 + s\_9^2}{2s\_9d\_2} \mathbf{s}\_9^o \tag{24}$$

The versor 2 *o <sup>o</sup>* **<sup>c</sup>** is obtained basing on *<sup>o</sup> <sup>r</sup>* **x** and 2 *o <sup>o</sup>* **d** :

$$\mathbf{c}\_{2o}^o = c\_{22}\mathbf{d}\_{2o}^o - \sqrt{1 - c\_{22}^2} (\mathbf{x}\_r^o \times \mathbf{d}\_{2o}^o) \tag{25}$$

The expected length of the drive 4 and its versor are obtained from the triangle *A*4*A*2*C*4:

$$\mathbf{s}\_{4o} = \sqrt{(\mathbf{s}\_{4o} \cdot \mathbf{x}\_r^o)^2 + (\mathbf{s}\_{4o} \cdot \mathbf{y}\_r^o)^2 + (\mathbf{s}\_{4o} \cdot \mathbf{z}\_r^o)^2} \, \, \, \qquad \mathbf{s}\_{4o}^o = \frac{\mathbf{s}\_{4o}}{\mathbf{s}\_{4o}} \tag{26}$$

where: 4 22 44 *o o o o* **sca** *c a* - vector of the drive 4.

The expected length of the actuator and its axis versor are obtained on the basis of a polygon *A*1*A*2*B*2*OpB*1:

The Active Suspension of a Cab in a Heavy Machine 117

$$\mathbf{s}\_{1o} = \sqrt{(\mathbf{s}\_{1o} \cdot \mathbf{x}\_r^o)^2 + (\mathbf{s}\_{1o} \cdot \mathbf{y}\_r^o)^2 + (\mathbf{s}\_{1o} \cdot \mathbf{z}\_r^o)^2}, \qquad \mathbf{s}\_{1o}^o = \frac{\mathbf{s}\_{1o}}{\mathbf{s}\_{1o}} \tag{27}$$

where: 1 2 2 2 11 1 *o o oo o o po po* **s d y ax** *d b ab* - vector of the actuator 1.

### **4.3. Direct kinematics problem of links velocity**

In order to solve the simple problem to derive velocity of the active suspension system in the coordinate system associated with the frame it is required that the following vectors have to be determined:

*<sup>p</sup>*,*r kr f r* , , **ωωω** - identical angular velocity of the platform *p*, of the cab *k* and the operator seat *f*,

, *O r <sup>k</sup>* **v** - linear velocity of the cab's c.o.g *Qk*,

116 Advances on Analysis and Control of Vibrations – Theory and Applications

*o r z*

*Or*

*r z*

*o r y*

9 *o s*

*<sup>o</sup> d* <sup>3</sup>

*o*

*C*

<sup>9</sup> *<sup>s</sup>*

**Figure 6.** Solving the inverse problem- schematic diagram

2 *o*

*po* **<sup>x</sup>** is obtained basing on *<sup>o</sup>*

*<sup>o</sup>* **d** is obtained basing on 9

*<sup>o</sup>* **<sup>c</sup>** is obtained basing on *<sup>o</sup>*

*o o* **sca** *c a* - vector of the drive 4.

1

*po* **<sup>y</sup>** and *<sup>o</sup>*

2 2

9

*o r*

44 4 4 ( )( )( ) *ooo*

*<sup>r</sup>* **x** and 2

The expected length of the drive 4 and its versor are obtained from the triangle *A*4*A*2*C*4:

2 22

*o or o r or s* **sx sy sz** , <sup>4</sup>

The expected length of the actuator and its axis versor are obtained on the basis of a polygon

*po* **z** :

*<sup>o</sup> d* <sup>3</sup>

*o po y*

*o po y*

*po z*

*p z*

*<sup>o</sup> d <sup>o</sup>*

*po po po* **x yz** (21)

*o*

*A*3

*p y*

*r y*

*po r s a b ab* **y y** (22)

**s** (23)

*o oo*

*B*<sup>2</sup> *B*<sup>3</sup>

*Op*

*o g z*

The modulus of the vector between points *Or* and *C* and its versor are computed using the

9 23 23 23 23 2( ) *o o*

23 23

*<sup>o</sup>* **s** , *s*9, *d*2 and *d*3:

2 9 9

*o <sup>o</sup>* **d** :

2 2 *o o o o*

9

*o o <sup>o</sup> r po a b s* **y y**

<sup>2</sup> 2 22 <sup>222</sup> 239 239

*dds dds s d s d*

2 2 22 2 22 2 1( ) *o o o o*

9 2 9 2

**d x s s** (24)

*o o ro* **c d xd** *c c* (25)

4

4 *o o o o s* **s**

**s** (26)

The versor *<sup>o</sup>*

The versor 2

The versor 2

triangle *OrA*3*C* (Fig 6):

*o*

*o*

where: 4 22 44

*A*1*A*2*B*2*OpB*1:

*o o*

, *O r <sup>f</sup>* **v** - linear velocity of the c.o.g. in the seat-operator system *Of* as functions of cylinders' velocity.

The kinematic chain determined by points *A*4*C*4*A*2 (Fig 5) satisfies the closing condition:

$$\mathbf{a\_4}\mathbf{a\_4^o} + \mathbf{s\_4}\mathbf{s\_4^o} = \mathbf{c\_2}\mathbf{c\_2^o} \tag{28}$$

Differentiating Eq (28) over time yields:

$$
\dot{\mathbf{s}}\_4 \mathbf{s}\_4^\vartheta + \mathbf{s}\_4 (\boldsymbol{\omega}\_{4,r} \times \mathbf{s}\_4^\vartheta) = \mathbf{c}\_2 (\boldsymbol{\omega}\_{2,r} \times \mathbf{c}\_2^\vartheta) \tag{29}
$$

The versor *<sup>o</sup> <sup>r</sup>* **x** determines the direction of the angular velocity vector of the rocker arm 2:

$$
\boldsymbol{\omega}\_{2,r} = \boldsymbol{\alpha}\_{2,r} \mathbf{x}\_r^\boldsymbol{\boldsymbol{\alpha}} \tag{30}
$$

Scalar-multiplying Eq (29) by 4 *<sup>o</sup>* **s** yields:

$$
\alpha\_{2,r} = \frac{\dot{s}\_4}{r\_2} \tag{31}
$$

where: 2 2 24 [( ) ] *oo o <sup>r</sup> r c* **xcs** .

The kinematic chain determined by points *A*2*B*2*B*3*A*3 (Fig 5.) satisfies the closing condition:

$$d\_2 \mathbf{d}\_2^o + b\_{23} \mathbf{y}\_p^o = a\_{23} \mathbf{y}\_r^o + d\_3 \mathbf{d}\_3^o \tag{32}$$

Differentiating Eq (32) over time yields:

$$d\_2(\boldsymbol{\omega}\_{2,r} \times \mathbf{d}\_2^o) + b\_{23}(\boldsymbol{\omega}\_{p,r} \times \mathbf{y}\_p^o) = d\_3(\boldsymbol{\omega}\_{3,r} \times \mathbf{d}\_3^o) \tag{33}$$

The versor *<sup>o</sup> <sup>r</sup>* **x** determines the direction of the angular velocity vector of the rocker arm 3:

$$
\boldsymbol{\omega}\_{3,r} = \boldsymbol{\alpha}\_{3,r} \mathbf{x}\_r^\circ \tag{34}
$$

The Active Suspension of a Cab in a Heavy Machine 119

 

*<sup>o</sup>* **s** yields the coordinate of the angular

.

*<sup>p</sup>* **y** :

Differentiating Eq (40) over time yields:

accordingly:

where: 11 1 ( ) *<sup>p</sup>*

velocity of actuators *s*1 and *s*4:

1*p*

frame (Fig 5), is expressed as:

Eq (45) can be written in the matrix format:

1 4 **J hh0** , 145

*o p py <sup>r</sup>* **<sup>y</sup> <sup>h</sup>** , 4

where: 1

where:

*o o*

1 1 1 1, 1 1 , 2 2, 2 2 , ( )( ) ( )( ) *oo o o o*

Recalling the relationships expressing the angular velocity vector of the platform (Eq 36) and angular velocity vector of the rocker arm 2 (Eq 30), Eq (41) can be rearranged

1 1 1 1, 1 1 ( , ) ( , ) 2 2 2 ( ,) ( ,)

( ,)

*pr y*

2

The angular velocity vector of the platform, based on (36), (39), (43), (38), will become:

**s ω s y z xd z x**

*<sup>s</sup> ss b d b*

 

velocity of the platform in the direction determined by the versor *<sup>o</sup>*

*py <sup>p</sup> r b* **z s** , 1 1 4

,

4 *pp p*

*<sup>T</sup> sss* **<sup>s</sup>** .

**z**

*p r*,

The radius vector of the point *Ok* - the origin of the reference system associated with the

22 2 *o o*

*ooo p p p px py p rr r* 

**xyz <sup>h</sup>** .

*p*

*py*

*r*

Scalar multiplying vectors present in Eq (42) by 1

4

2 ( )( )( )( ) *p p p p o o o o oo o o r prz p pr y p r prx p prz p*

1 4

*o o p*

**z s**

*o oo o oo pr p p pz px pz*

**y xd z x s**

1 4 *<sup>p</sup> p p*

*b*

 

*py py s s r r*

12 2 2

*bd b b r r rr*

4 14 4

*p r p p p px py py pz s ss s r rr r*

Finally, the angular velocity vector of the platform, cab and the seat is linearly related to the

1 4 ( ) *p pp p*

*o o o*

*p p p*

( )

*r*

(42)

*<sup>r</sup> pr p <sup>r</sup> pr p ss b d b* **s ω s ω x ω d ω y** (41)

(43)

2 1

**<sup>ω</sup> x yz** (44)

*pr kr f r* , , , 11 44 **ωωω** *s s* **h h** (45)

**ω J s** (46)

*rk p pk* **r d yr** *d b* (47)

Substituting (30), (31) and (34) into Eq (33) and scalar-multiplying all vectors by the versor *o <sup>p</sup>* **y** , yields the modulus of angular velocity vector **ω**3,*<sup>r</sup>* :

$$
\alpha\_{3,r} = \frac{\dot{s}\_4}{r\_3} \tag{35}
$$

where: <sup>3</sup> 2 4 2 3 3 2 2 [( ) ][( ) ] ( ) *o o o oo o r pr ooo r p c d r d* **xdy xcs xdy** .

The angular velocity vector of the platform p can be expressed as the sum of three components whose axis directions are determined by versors *<sup>o</sup> <sup>p</sup>* **<sup>x</sup>** , *<sup>o</sup> <sup>p</sup>* **<sup>y</sup>** and *<sup>o</sup> <sup>p</sup>* **z** :

$$\boldsymbol{\alpha}\_{p,r} = \boldsymbol{\alpha}\_{\{p,r\} \boldsymbol{x}\_p} \mathbf{x}\_p^o + \boldsymbol{\alpha}\_{\{p,r\} \boldsymbol{y}\_p} \mathbf{y}\_p^o + \boldsymbol{\alpha}\_{\{p,r\} \boldsymbol{z}\_p} \mathbf{z}\_p^o \tag{36}$$

Recalling Eq (30), (31), (34), (35), (36), Eq (33) can be rewritten as:

.

$$d\_2(\frac{\dot{\mathbf{s}}\_4}{r\_2}\mathbf{x}\_r^o \times \mathbf{d}\_2^o) + b\_{23}a\_{\{p,r\}\mathbf{x}\_p}\mathbf{z}\_p^o - b\_{23}a\_{\{p,r\}\mathbf{z}\_p}\mathbf{x}\_p^o = d\_3(\frac{\dot{\mathbf{s}}\_4}{r\_3}\mathbf{x}\_r^o \times \mathbf{d}\_3^o) \tag{37}$$

Scalar-multiplying Eq (37) by *<sup>o</sup> <sup>p</sup>* **x** yields:

$$
\alpha\_{\{p,r\}z\_p} = \frac{\dot{s}\_4}{r\_{p z\_p}} \tag{38}
$$

where: <sup>23</sup> 2 3 2 3 2 3 ( ) *<sup>p</sup> pz oo o o p r b r d d r r* **xx d d**

Scalar-multiplying Eq (37) by *<sup>o</sup> <sup>p</sup>* **z** yields:

$$
\rho\_{\{p,r\} \times\_p} = \frac{\dot{s}\_4}{r\_{p\infty\_p}} \tag{39}
$$

where: <sup>23</sup> 3 2 3 2 3 2 ( ) *<sup>p</sup> px oo o o p r b r d d r r* **zx d d** .

The closed kinematic chain comprising a actuator *s*1 and represented as a pentagon *A*2*A*1*B*1*OpB*<sup>2</sup> in Fig 5 satisfies the closing condition:

$$a\_1 \mathbf{a}\_1^o + s\_1 \mathbf{s}\_1^o + b\_1 \mathbf{x}\_p^o = d\_2 \mathbf{d}\_2^o + b\_2 \mathbf{y}\_p^o \tag{40}$$

Differentiating Eq (40) over time yields:

118 Advances on Analysis and Control of Vibrations – Theory and Applications

*<sup>p</sup>* **y** , yields the modulus of angular velocity vector **ω**3,*<sup>r</sup>* :

[( ) ][( ) ] ( )

**xdy xcs**

*o o o oo o r pr ooo r p*

**xdy** .

components whose axis directions are determined by versors *<sup>o</sup>*

Recalling Eq (30), (31), (34), (35), (36), Eq (33) can be rewritten as:

where: <sup>3</sup> 2 4 2 3

2 2

*o*

3

*r*

*c d*

*d*

Scalar-multiplying Eq (37) by *<sup>o</sup>*

where: <sup>23</sup>

where: <sup>23</sup>

*<sup>p</sup> px*

*r*

( )

*p r*

*<sup>p</sup> pz*

*r*

( )

Scalar-multiplying Eq (37) by *<sup>o</sup>*

*p r*

3, 3, *<sup>o</sup>* **<sup>ω</sup>** *r rr* 

Substituting (30), (31) and (34) into Eq (33) and scalar-multiplying all vectors by the versor

The angular velocity vector of the platform p can be expressed as the sum of three

, ( ,) ( ,) ( ,) *ppp o oo* **<sup>ω</sup>***<sup>p</sup> r pr x p pr y p pr z p* 

4 4 2 2 23 ( , ) 23 ( , ) 3 3 2 3 ( ) ( ) *p p o o o o oo r prx p prz p r*

( ,) *<sup>p</sup>*

( ,) *<sup>p</sup>*

.

*prx*

The closed kinematic chain comprising a actuator *s*1 and represented as a pentagon

11 11 1 2 2 2 *oo o o o*

*prz*

.

*<sup>p</sup>* **z** yields:

**xd z x xd**

 

4

*pz s r*

*p*

4

*px s r*

*p*

(37)

*s s d bbd r r*

*<sup>p</sup>* **x** yields:

2 3 2 3 2 3

3 2 3 2 3 2

*d d r r*

*oo o o*

*b*

**zx d d**

*A*2*A*1*B*1*OpB*<sup>2</sup> in Fig 5 satisfies the closing condition:

*d d r r*

*oo o o*

*b*

**xx d d**

3,

4

3 *r s r*

**x** (34)

(35)

*<sup>p</sup>* **<sup>x</sup>** , *<sup>o</sup>*

**xyz** (36)

(38)

(39)

*<sup>p</sup> <sup>p</sup> asb d b* **asx d y** (40)

*<sup>p</sup>* **<sup>y</sup>** and *<sup>o</sup>*

*<sup>p</sup>* **z** :

$$\dot{\mathbf{s}}\_1 \mathbf{s}\_1^\vartheta + \mathbf{s}\_1 (\boldsymbol{\omega}\_{1,r} \times \mathbf{s}\_1^\vartheta) + b\_1 (\boldsymbol{\omega}\_{p,r} \times \mathbf{x}\_p^\vartheta) = d\_2 (\boldsymbol{\omega}\_{2,r} \times \mathbf{d}\_2^\vartheta) + b\_2 (\boldsymbol{\omega}\_{p,r} \times \mathbf{y}\_p^\vartheta) \tag{41}$$

Recalling the relationships expressing the angular velocity vector of the platform (Eq 36) and angular velocity vector of the rocker arm 2 (Eq 30), Eq (41) can be rearranged accordingly:

$$\dot{s}\_1 \mathbf{s}\_1^o + s\_1 (\boldsymbol{\omega}\_{1,r} \times \mathbf{s}\_1^o) + b\_1 (\boldsymbol{\alpha}\_{\{p,r\}z\_p} \mathbf{y}\_p^o - \boldsymbol{\alpha}\_{\{p,r\}y\_p} \mathbf{z}\_p^o) = d\_2 (\frac{\dot{S}\_4}{r\_2} \mathbf{x}\_r^o \times \mathbf{d}\_2^o) + b\_2 (\boldsymbol{\alpha}\_{\{p,r\}x\_p} \mathbf{z}\_p^o - \boldsymbol{\alpha}\_{\{p,r\}z\_p} \mathbf{x}\_p^o) \tag{42}$$

Scalar multiplying vectors present in Eq (42) by 1 *<sup>o</sup>* **s** yields the coordinate of the angular velocity of the platform in the direction determined by the versor *<sup>o</sup> <sup>p</sup>* **y** :

$$
\alpha\_{(p,r)y\_p} = \frac{\dot{s}\_1}{r\_{py\_p \cdot 1}} + \frac{\dot{s}\_4}{r\_{py\_p \cdot 4}} \tag{43}
$$

$$\text{where: } r\_{py\_p1} = b\_1 (\mathbf{z}\_p^o \cdot \mathbf{s}\_1^o) \text{ , } r\_{py\_p4} = \frac{b\_1 \mathbf{z}\_p^o \cdot \mathbf{s}\_1^o}{\left[\frac{b\_1}{r\_{pz\_p}} \mathbf{y}\_p^o - \frac{d\_2}{r\_2} (\mathbf{x}\_r^o \times \mathbf{d}\_2^o) - \frac{b\_2}{r\_{px\_p}} \mathbf{z}\_p^o + \frac{b\_2}{r\_{pz\_p}} \mathbf{x}\_p^o \right] \cdot \mathbf{s}\_1^o} \text{ , }$$

The angular velocity vector of the platform, based on (36), (39), (43), (38), will become:

$$\mathbf{Cov}\_{p,r} = \frac{\dot{\mathbf{s}}\_4}{r\_{px\_p}} \mathbf{x}\_p^o + (\frac{\dot{\mathbf{s}}\_1}{r\_{py\_p 1}} + \frac{\dot{\mathbf{s}}\_4}{r\_{py\_p 4}}) \mathbf{y}\_p^o + \frac{\dot{\mathbf{s}}\_4}{r\_{pz\_p}} \mathbf{z}\_p^o \tag{44}$$

Finally, the angular velocity vector of the platform, cab and the seat is linearly related to the velocity of actuators *s*1 and *s*4:

$$
\boldsymbol{\omega}\_{p,r} = \boldsymbol{\omega}\_{k,r} = \boldsymbol{\omega}\_{f,r} = \dot{\mathbf{s}}\_1 \mathbf{h}\_1 + \dot{\mathbf{s}}\_4 \mathbf{h}\_4 \tag{45}
$$

where: 1 1*p o p py <sup>r</sup>* **<sup>y</sup> <sup>h</sup>** , 4 4 *pp p ooo p p p px py p rr r* **z xyz <sup>h</sup>** .

Eq (45) can be written in the matrix format:

$$
\boldsymbol{\omega}\_{p,r} = \mathbf{J}\_{co} \cdot \tag{46}
$$

$$
\mathbf{0} \parallel \cdot \dot{\mathbf{s}} = \begin{bmatrix} \dot{s}\_1 & \dot{s}\_4 & \dot{s}\_5 \end{bmatrix}^T .
$$

where: 1 4 **J hh0** , 145

The radius vector of the point *Ok* - the origin of the reference system associated with the frame (Fig 5), is expressed as:

$$\mathbf{r}\_{rk} = d\_2 \mathbf{d}\_2^o + b\_2 \mathbf{y}\_p^o + \mathbf{r}\_{pk} \tag{47}$$

$$\text{where: } \mathbf{r}\_{pk} = \mathbf{x}\_p^{\ o} \, ^p r\_{(pk)x} + \mathbf{y}\_p^{\ o} \, ^p r\_{(pk)y} + \mathbf{z}\_p^{\ o} \, ^p r\_{(pk)z} \dots$$

Differentiating Eq (47) over time yields the linear velocity vector of the point *Ok*:

$$\mathbf{v}\_{O\_k,r} = d\_2(\boldsymbol{\omega}\_{2,r} \times \mathbf{d}\_2^o) + \boldsymbol{\omega}\_{p,r} \times (\mathbf{y}\_p^o b\_2 + \mathbf{r}\_{pk}) \tag{48}$$

Recalling Eq (30), (31), (45), Eq (48) can be rewritten as:

$$\mathbf{v}\_{O\_k,r} = \dot{\mathbf{s}}\_1 \mathbf{k}\_{1O\_k} + \dot{\mathbf{s}}\_4 \mathbf{k}\_{4O\_k} \tag{49}$$

The Active Suspension of a Cab in a Heavy Machine 121

*po* **x**

where: **ω***r g*, - angular velocity of the frame with respect to road, based on measurement data. Absolute linear velocities of points *Ok* and *Of* expressed in the reference system associated

, , , , ( ) *O g O g r g mr rk O r k m <sup>k</sup>*

, , , , ( ) *O g O g r g mr rf O r f m <sup>f</sup>*

The inverse problem involves finding the drive velocities for the predetermined cab velocity with respect to the road. As the active suspension mechanism displays three degrees of freedom (DOFs), three constraints can be imposed upon the cab velocity. The function of the active suspension system is to stabilise the cab in the vertical direction, hence the condition is adopted prohibiting the absolute rotating motion of the platform around its two axes *<sup>o</sup>*

*po* **y** . The third condition implicates that the absolute value of linear velocity of the point

(,) <sup>0</sup> *<sup>o</sup>*

(,) <sup>0</sup> *<sup>o</sup>*

( ,) 0 *f*

Recalling Eq (55) and conditions (58), (59), we get the formulas expressing the expected

*o oo o o po o o po r g po o oo o o po o o po r g po*

When the actuators 1 and 4 should move at velocities governed by Eq (61), the cab will

1 1 4 4 5 5 , , ( ) ( )( )( ) [ ( )] 0 *f f fm o o oo o o O g o O g o O g O g g r g mr rf o g ss s* **k z k z k zv z ω rr z** (62)

The solution to linear system of equations (61) and (62) can be written as a matrix equation:

11 44 , 11 44 ,

**h x hx ω x hy hy ω y**

*s s s s*

perform a slight rotating motion around in the direction *<sup>o</sup>*

The third condition (60) in relation to Eq (54), (57) gives:

*o*

**v** - measured linear velocity of the control point *Om* associated with the frame

**v v ω rrv** (56)

**v v ω rrv** (57)

*mr mr x mr y mr z* **r** *rrr* - vector between the points *Om* and *Or*

*p g o po* **ω x** (58)

*p g o po* **ω y** (59)

*O go g* **v z** (60)

0 0

(61)

*po* **z** only.

 

*<sup>o</sup>* **s Jv** (63)

with the frame are:

where: , *O g <sup>m</sup>*

and *<sup>o</sup>*

with respect to the road; () () () [,,]*<sup>T</sup>*

expressed in the reference system associated with the frame.

**4.4. Inverse kinematics problem of links velocity** 

*Of* in the direction of the gravity force should be zero:

velocities of actuators 1 and 4:

where: 1 12 ( ) *<sup>k</sup> o <sup>O</sup> p pk* **k h yr** *b* , <sup>2</sup> 4 2 42 2 ()( ) *<sup>k</sup> o o o O r p pk <sup>d</sup> <sup>b</sup> r* **k xd h y r** .

Eq (49) can be expressed in the matrix format:

$$\mathbf{v}\_{O\_k,r} = \mathbf{J}\_{vk}\mathbf{\dot{s}}\tag{50}$$

where: 1 4 *k k vk O O* **Jkk0** .

The radius vector of the c.o.g of the seat-operator system *Of* becomes:

$$\mathbf{r}\_{r\!\!\!/} = d\_2 \mathbf{d}\_2^o + b\_2 \mathbf{y}\_{\!\!/}^o + \mathbf{r}\_{\!\!\!/} \tag{51}$$

where: () () 5 ( ) *ooo p p pf p pf x p pf y p z* **rx y z** *r r s df* .

Differentiating Eq (51) over time yields the linear velocity of the point *Of*:

$$\mathbf{v}\_{O\_{f},r} = d\_{2}\boldsymbol{\omega}\_{2,r} \times \mathbf{d}\_{2}^{\diamond} + \boldsymbol{\omega}\_{p,r} \times (\mathbf{y}\_{p}^{\diamond}\boldsymbol{b}\_{2} + \mathbf{r}\_{p\circ}) + \dot{\mathbf{s}}\_{5}\mathbf{z}\_{p}^{\diamond} \tag{52}$$

Recalling Eq (30), (31), (45), Eq (52) can be rewritten as:

$$\mathbf{v}\_{O\_{f'},r} = \dot{\mathbf{s}}\_1 \mathbf{k}\_{1O\_f} + \dot{\mathbf{s}}\_4 \mathbf{k}\_{4O\_f} + \dot{\mathbf{s}}\_5 \mathbf{k}\_{5O\_f} \tag{53}$$

where: : 1 12 ( ) *<sup>f</sup> o <sup>O</sup> p pf* **k h yr** *<sup>b</sup>* , <sup>2</sup> 4 24 2 2 ()( ) *<sup>f</sup> oo o O r p pf <sup>d</sup> <sup>b</sup> r* **k xd h y r** , 5 *<sup>f</sup> o O p* **k z** .

Eq (53) can be expressed in the matrix format:

$$\mathbf{v}\_{O\_{f'},r} = \mathbf{J}\_{v\not\!f} \mathbf{\dot{s}} \tag{54}$$

where: 145 *fff vf O O O* **Jkk k** .

To define the operating conditions of the drives in the active suspension mechanism, velocity vectors related to the road system are of key importance. The absolute angular velocity of the cab in the reference system associated with the frame becomes:

$$
\boldsymbol{\omega}\_{p,\boldsymbol{\mathcal{g}}} = \boldsymbol{\omega}\_{p,r} + \boldsymbol{\omega}\_{r,\boldsymbol{\mathcal{g}}} = \mathbf{J}\_{\boldsymbol{\alpha}} \dot{\mathbf{s}} + \boldsymbol{\omega}\_{r,\boldsymbol{\mathcal{g}}} \tag{55}
$$

where: **ω***r g*, - angular velocity of the frame with respect to road, based on measurement data. Absolute linear velocities of points *Ok* and *Of* expressed in the reference system associated with the frame are:

$$\mathbf{v}\_{O\_k, \mathcal{g}} = \mathbf{v}\_{O\_{m'}, \mathcal{g}} + \boldsymbol{\omega}\_{r, \mathcal{g}} \times (\mathbf{r}\_{mr} + \mathbf{r}\_{rk}) + \mathbf{v}\_{O\_k, r} \tag{56}$$

$$\mathbf{v}\_{O\_{f},\mathcal{G}} = \mathbf{v}\_{O\_{m},\mathcal{G}} + \boldsymbol{\omega}\_{r,\mathcal{G}} \times (\mathbf{r}\_{mr} + \mathbf{r}\_{rf}) + \mathbf{v}\_{O\_{f},r} \tag{57}$$

where: , *O g <sup>m</sup>* **v** - measured linear velocity of the control point *Om* associated with the frame with respect to the road; () () () [,,]*<sup>T</sup> mr mr x mr y mr z* **r** *rrr* - vector between the points *Om* and *Or* expressed in the reference system associated with the frame.

### **4.4. Inverse kinematics problem of links velocity**

120 Advances on Analysis and Control of Vibrations – Theory and Applications

Recalling Eq (30), (31), (45), Eq (48) can be rewritten as:

*o <sup>O</sup> p pk* **k h yr** *b* , <sup>2</sup>

Eq (49) can be expressed in the matrix format:

Differentiating Eq (47) over time yields the linear velocity vector of the point *Ok*:

2

*r*

The radius vector of the c.o.g of the seat-operator system *Of* becomes:

Differentiating Eq (51) over time yields the linear velocity of the point *Of*:

, 2 2, 2 , 2 ( )( ) *<sup>k</sup> o o*

, 11 44 *Or O O k kk*

()( ) *<sup>k</sup> o o o O r p pk <sup>d</sup> <sup>b</sup>*

**k xd h y r** .

4 2 42

, *O r vk <sup>k</sup>*

22 2 *o o*

, 2 2, 2 , 2 <sup>5</sup> ( ) *<sup>f</sup>*

, 11 44 55 *Or O O O <sup>f</sup> fff*

4 24 2

, *O r vf <sup>f</sup>*

To define the operating conditions of the drives in the active suspension mechanism, velocity vectors related to the road system are of key importance. The absolute angular

*p*,*g pr rg* ,, ,

()( ) *<sup>f</sup> oo o O r p pf <sup>d</sup> <sup>b</sup>*

**k xd h y r** , 5 *<sup>f</sup>*

2

*r*

velocity of the cab in the reference system associated with the frame becomes:

*oo o*

*Or r p r p pf p* **v** *d* **ω d ω yr z** *b s* (52)

**v kkk** *ss s* (53)

**v Js** (54)

*r g* **ω ωω J s ω** (55)

*o O p* **k z** .

*O r <sup>r</sup> p r p pk* **v** *d b* **ω d ω y r** (48)

**v kk** *s s* (49)

**v Js** (50)

*rf p pf* **r d yr** *d b* (51)

*ooo ppp pk p pk x p pk y p pk z* **rx y z** *rrr* .

where: () () ()

where: 1 12 ( ) *<sup>k</sup>*

where: 1 4 *k k vk O O*

where: : 1 12 ( ) *<sup>f</sup>*

where: 145 *fff vf O O O*

**Jkk0** .

where: () () 5 ( ) *ooo p p pf p pf x p pf y p z* **rx y z** *r r s df* .

Recalling Eq (30), (31), (45), Eq (52) can be rewritten as:

*o <sup>O</sup> p pf* **k h yr** *<sup>b</sup>* , <sup>2</sup>

Eq (53) can be expressed in the matrix format:

**Jkk k** .

The inverse problem involves finding the drive velocities for the predetermined cab velocity with respect to the road. As the active suspension mechanism displays three degrees of freedom (DOFs), three constraints can be imposed upon the cab velocity. The function of the active suspension system is to stabilise the cab in the vertical direction, hence the condition is adopted prohibiting the absolute rotating motion of the platform around its two axes *<sup>o</sup> po* **x** and *<sup>o</sup> po* **y** . The third condition implicates that the absolute value of linear velocity of the point *Of* in the direction of the gravity force should be zero:

$$
\boldsymbol{\omega}\_{(p,\emptyset)o} \cdot \mathbf{x}\_{po}^{o} = \mathbf{0} \tag{58}
$$

$$\mathbf{u}\underbrace{\boldsymbol{\omega}}\_{\left(\boldsymbol{p},\boldsymbol{\chi}\right)\circ\cdots\circ\boldsymbol{\mathcal{Y}}\_{\left(\boldsymbol{p}\right)}}\cdot\mathbf{y}^{\boldsymbol{\theta}}\_{\left(\boldsymbol{p}\right)}=\mathbf{0}\tag{59}$$

$$\mathbf{v}\_{\{O\_{f},g\}o} \cdot \mathbf{z}\_{g}^{o} = \mathbf{0} \tag{60}$$

Recalling Eq (55) and conditions (58), (59), we get the formulas expressing the expected velocities of actuators 1 and 4:

$$\begin{vmatrix} \dot{\mathbf{s}}\_{1o} \mathbf{h}\_{1o} \cdot \mathbf{x}\_{po}^{o} + \dot{\mathbf{s}}\_{4o} \mathbf{h}\_{4o} \cdot \mathbf{x}\_{po}^{o} + \boldsymbol{\omega}\_{r,g} \cdot \mathbf{x}\_{po}^{o} = \mathbf{0} \\\\ \dot{\mathbf{s}}\_{1o} \mathbf{h}\_{1o} \cdot \mathbf{y}\_{po}^{o} + \dot{\mathbf{s}}\_{4o} \mathbf{h}\_{4o} \cdot \mathbf{y}\_{po}^{o} + \boldsymbol{\omega}\_{r,g} \cdot \mathbf{y}\_{po}^{o} = \mathbf{0} \end{vmatrix} \tag{61}$$

When the actuators 1 and 4 should move at velocities governed by Eq (61), the cab will perform a slight rotating motion around in the direction *<sup>o</sup> po* **z** only.

The third condition (60) in relation to Eq (54), (57) gives:

$$\dot{\mathbf{s}}\_{1o}(\mathbf{k}\_{1O\_f} \cdot \mathbf{z}\_g^o) + \dot{\mathbf{s}}\_{4o}(\mathbf{k}\_{4O\_f} \cdot \mathbf{z}\_g^o) + \dot{\mathbf{s}}\_{5o}(\mathbf{k}\_{5O\_f} \cdot \mathbf{z}\_g^o) + \mathbf{v}\_{O\_m,g} \cdot \mathbf{z}\_g^o + [\mathbf{u}\_{r,g} \times (\mathbf{r}\_{mr} + \mathbf{r}\_{(rf)a})] \cdot \mathbf{z}\_g^o = 0 \tag{62}$$

The solution to linear system of equations (61) and (62) can be written as a matrix equation:

$$
\dot{\mathbf{s}}\_o = \mathbf{J} \mathbf{v} \tag{63}
$$

$$\begin{aligned} \text{where: } \dot{\mathbf{s}}\_{o} = \begin{bmatrix} \dot{\mathbf{s}}\_{1o} \\ \dot{\mathbf{s}}\_{4o} \\ \dot{\mathbf{s}}\_{5o} \end{bmatrix}, \quad \mathbf{J} = -\begin{bmatrix} \mathbf{h}\_{1o} \cdot \mathbf{x}\_{po}^{o} & \mathbf{h}\_{4o} \cdot \mathbf{x}\_{po}^{o} & 0 \\ \mathbf{h}\_{1o} \cdot \mathbf{y}\_{po}^{o} & \mathbf{h}\_{4o} \cdot \mathbf{y}\_{po}^{o} & 0 \\ \mathbf{k}\_{1O} \cdot \mathbf{z}\_{g}^{o} & \mathbf{k}\_{4O\_{f}} \cdot \mathbf{z}\_{g}^{o} & \mathbf{k}\_{5O\_{f}} \cdot \mathbf{z}\_{g}^{o} \end{bmatrix}^{-1} \begin{bmatrix} \mathbf{0}\_{1\times3} & \mathbf{x}\_{po}^{o} \\ \mathbf{0}\_{1\times3} & \mathbf{y}\_{po}^{o} \\ \mathbf{0}\_{1\times3} & \mathbf{y}\_{po}^{o} \\ \mathbf{z}\_{g}^{o} & \{\mathbf{z}\_{mr} + \mathbf{r}\_{fl}\} \times \mathbf{z}\_{g}^{o} \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} \mathbf{v}\_{O\_{m}\cdot\boldsymbol{\mathcal{S}}} \\ \mathbf{w}\_{r,\cdot\boldsymbol{\mathcal{S}}} \end{bmatrix}. \end{aligned}$$

### **4.5. Constraining the motion of the drive responsible for the seat movement in the vertical direction**

The machine, when in service or during the ride, may change its position in the vertical direction such that in order to stabilise the seat position in this direction the operating range of the actuator 5 should be exceeded. To solve this problem, it is suggested that a penalty function should be introduced, its argument being the instantaneous length of the actuator *s*5:

$$\dot{\mathbf{s}}\_{5k} = \mathbf{K}\_5 \dot{\mathbf{s}}\_{5k \,\mathrm{max}} \left[ \frac{\mathbf{s}\_{5k} - \mathbf{s}\_5}{0.5(\mathbf{s}\_{5 \,\mathrm{max}} - \mathbf{s}\_{5 \,\mathrm{min}})} \right]^{2n - 1} \tag{64}$$

where: 5 5max 5min 0.5( ) *<sup>ś</sup><sup>r</sup> s ss* , *K*5 - amplification factor penalty function 5 max *<sup>k</sup> s* - maximal velocity of the actuator, *n N* .

The final expected velocity of the actuator 5 should involve a term responsible for the seat "drifting" towards the middle position:

$$
\dot{\mathbf{s}}\_{\mathfrak{S}o}^{\*} = \dot{\mathbf{s}}\_{\mathfrak{S}o} + \dot{\mathbf{s}}\_{\mathfrak{S}k} \tag{65}
$$

The Active Suspension of a Cab in a Heavy Machine 123

**a J s Js** (68)

5*s*

, *O r vf vf <sup>f</sup>*

actuator 5

5 max *<sup>k</sup> s*

5*k s*

5 max *<sup>k</sup> s*

5

The inertia loads are determined basing on absolute acceleration values related to the inertial reference system {*Ogxgygzg*}. In accordance with Eq (55), the absolute angular accelerations of the platform, cab and seat expressed in the mobile reference system

<sup>5</sup>*ś<sup>r</sup> s*

5min *s* 5max *s*

*o p* **z**

The range of major impact of the penalty function on velocity in the

Recalling Eq (56), (57), the absolute linear acceleration of the points *Or*, *Ok* and *Of* in the

, ,, , , ( ) *O g O g r g mr r g r g mr r m*

, ,, , , , , , 2 () *O g O g r g rk r g O r r g r g rk O r k r <sup>k</sup> <sup>k</sup>*

, ,, , , , , , 2 () *O g O g k g kf k g O k k g k g kf O k f k <sup>f</sup> <sup>f</sup>*

inertial reference system {*Ogxgygzg*}, **ω***r g*, , *r g*, **ε** - measured angular velocity and acceleration of the machine frame with respect to the inertial reference system {*Ogxgygzg*},

*rk r k O O* 

**a** - measured linear acceleration of a point on the frame *Qm* with respect to the

*<sup>p</sup>*,*g kg f g rg pr rg pr* , ,,, , , **ε ε ε ε εωω** (69)

**a a ε r ω ω r** (70)

**a a ε r ω v ω ω r a** (71)

**a a ε r ω v ω ω r a** (72)

, **r***mr m r O O* .

where: 145

**Figure 7.** Penalty function

where: , *O g <sup>m</sup>*

associated with the frame become:

*kg pg rg pr* , ,,, **ωω ωω** , **r***kf k f O O*

mobile reference system associated with the frame become:

, **r**

*<sup>T</sup>* **s sss** .

where: 5*<sup>o</sup> s* - expected velocity in the actuator 5, derived from formula (63).

The second term in (65) represents the seat movement towards the middle position being superimposed on its relative movement. The assumed penalty function (64) guarantees that the relative velocity during the seat's return movement (Fig 7) to the middle position should be significant at extreme points of the actuator's displacement range.

### **4.6. Cab and seat acceleration**

In order to solve the simple problem involving acceleration of the active suspension mechanism in the reference system associated with the frame, it is required that certain quantities should be determined in the function of length, velocity and acceleration of actuators 1,4,5. These include: *<sup>p</sup>*,*<sup>r</sup>* **<sup>ε</sup>** - angular acceleration of the platform, cab and seat, , *O r <sup>k</sup>* **a** linear acceleration of the cab's c.o.g, , *O r <sup>f</sup>* **a** - linear acceleration of the c.o.g of the seatoperator system. Differentiating Eq (46), (50), (54) with respect to time yields the angular acceleration of the platform, cab and seat and linear acceleration at points *Ok* and *Of*:

$$\mathbf{e}\_{p,r} = \dot{\mathbf{J}}\_{\alpha} \cdot \dot{\mathbf{s}} + \mathbf{J}\_{\alpha} \cdot \ddot{\mathbf{s}} \tag{66}$$

$$\mathbf{a}\_{O\_k,r} = \dot{\mathbf{J}}\_{vk} \cdot \dot{\mathbf{s}} + \mathbf{J}\_{vk} \ddot{\mathbf{s}} \tag{67}$$

$$\mathbf{a}\_{O\_f, r} = \dot{\mathbf{J}}\_{vf} \, \dot{\mathbf{s}} + \mathbf{J}\_{vf} \ddot{\mathbf{s}} \tag{68}$$

where: 145 *<sup>T</sup>* **s sss** .

122 Advances on Analysis and Control of Vibrations – Theory and Applications

1

 

2 1

<sup>555</sup> *ook sss* (65)

**a** - linear acceleration of the c.o.g of the seat-

**a J s Js** (67)

(66)

*n*

(64)

, ,

**v**

, *O g <sup>m</sup> r g*

**<sup>ω</sup>** .

**a** -

 

**v**

1 4 1 3 1 4 1 3

**J hy hy 0 y**

**4.5. Constraining the motion of the drive responsible for the seat movement in** 

The machine, when in service or during the ride, may change its position in the vertical direction such that in order to stabilise the seat position in this direction the operating range of the actuator 5 should be exceeded. To solve this problem, it is suggested that a penalty function should be introduced, its argument being the instantaneous length of the actuator *s*5:

where: 5 5max 5min 0.5( ) *<sup>ś</sup><sup>r</sup> s ss* , *K*5 - amplification factor penalty function 5 max *<sup>k</sup> s* - maximal

The final expected velocity of the actuator 5 should involve a term responsible for the seat

The second term in (65) represents the seat movement towards the middle position being superimposed on its relative movement. The assumed penalty function (64) guarantees that the relative velocity during the seat's return movement (Fig 7) to the middle position should

In order to solve the simple problem involving acceleration of the active suspension mechanism in the reference system associated with the frame, it is required that certain quantities should be determined in the function of length, velocity and acceleration of actuators 1,4,5. These include: *<sup>p</sup>*,*<sup>r</sup>* **<sup>ε</sup>** - angular acceleration of the platform, cab and seat, , *O r <sup>k</sup>*

operator system. Differentiating Eq (46), (50), (54) with respect to time yields the angular

**ε Js Js**

, *O r vk vk <sup>k</sup>*

 

acceleration of the platform, cab and seat and linear acceleration at points *Ok* and *Of*:

*p r*,

0 0

**hx hx 0 x**

*o o o o po o po po o o o o po o po po*

( ) *fff*

5 5

*śr*

5max 5min 0.5( )

*s s* 

**k zk zk z z r r z**

*o o oo o O g O g O g g mr rf g*

145

5 5 5 max

*s s s Ks*

\*

where: 5*<sup>o</sup> s* - expected velocity in the actuator 5, derived from formula (63).

be significant at extreme points of the actuator's displacement range.

*k k*

where:

**s**

1 4 5

 

*s s s*

**the vertical direction** 

velocity of the actuator, *n N* .

**4.6. Cab and seat acceleration** 

linear acceleration of the cab's c.o.g, , *O r <sup>f</sup>*

"drifting" towards the middle position:

 

*o o o o*

,

**Figure 7.** Penalty function

The inertia loads are determined basing on absolute acceleration values related to the inertial reference system {*Ogxgygzg*}. In accordance with Eq (55), the absolute angular accelerations of the platform, cab and seat expressed in the mobile reference system associated with the frame become:

$$\mathfrak{e}\_{p,\mathcal{g}} = \mathfrak{e}\_{k,\mathcal{g}} = \mathfrak{e}\_{f,\mathcal{g}} = \mathfrak{e}\_{r,\mathcal{g}} + \mathfrak{e}\_{p,r} + \mathfrak{w}\_{r,\mathcal{g}} \times \mathfrak{w}\_{p,r} \tag{69}$$

Recalling Eq (56), (57), the absolute linear acceleration of the points *Or*, *Ok* and *Of* in the mobile reference system associated with the frame become:

$$\mathbf{a}\_{O\_r,\mathcal{g}} = \mathbf{a}\_{O\_m,\mathcal{g}} + \mathbf{e}\_{r,\mathcal{g}} \times \mathbf{r}\_{mr} + \mathbf{a}\_{r,\mathcal{g}} \times (\mathbf{a}\_{r,\mathcal{g}} \times \mathbf{r}\_{mr}) \tag{70}$$

$$\mathbf{a}\_{O\_k, \mathcal{g}} = \mathbf{a}\_{O\_r, \mathcal{g}} + \mathbf{e}\_{r, \mathcal{g}} \times \mathbf{r}\_{rk} + 2\mathbf{a}\boldsymbol{\omega}\_{r, \mathcal{g}} \times \mathbf{v}\_{O\_k, r} + \boldsymbol{\omega}\_{r, \mathcal{g}} \times (\boldsymbol{\omega}\_{r, \mathcal{g}} \times \mathbf{r}\_{rk}) + \mathbf{a}\_{O\_k, r} \tag{71}$$

$$\mathbf{a}\_{\mathcal{O}\_{f},\mathcal{g}} = \mathbf{a}\_{\mathcal{O}\_{k},\mathcal{g}} + \mathbf{e}\_{k,\mathcal{g}} \times \mathbf{r}\_{\mathbb{k}\mathcal{f}} + \mathbf{2}\boldsymbol{\omega}\_{k,\mathcal{g}} \times \mathbf{v}\_{\mathcal{O}\_{f},k} + \boldsymbol{\omega}\_{k,\mathcal{g}} \times (\boldsymbol{\omega}\_{k,\mathcal{g}} \times \mathbf{r}\_{\mathbb{k}\mathcal{f}}) + \mathbf{a}\_{\mathcal{O}\_{f},k} \tag{72}$$

where: , *O g <sup>m</sup>* **a** - measured linear acceleration of a point on the frame *Qm* with respect to the inertial reference system {*Ogxgygzg*}, **ω***r g*, , *r g*, **ε** - measured angular velocity and acceleration of the machine frame with respect to the inertial reference system {*Ogxgygzg*}, *kg pg rg pr* , ,,, **ωω ωω** , **r***kf k f O O* , **r** *rk r k O O* , **r***mr m r O O* .

### **5. Inverse problem of dynamics**

The external loads acting on the active suspension mechanism involve the gravity forces, inertia forces and moments of inertial force of the platform together with the cab, the seat and the operator. These are governed by the Newton-Euler equations, referenced in [18]:

$$\mathbf{P}\_{bk} = -m\_k \mathbf{a}\_{O\_k, \mathcal{g}} \tag{73}$$

The Active Suspension of a Cab in a Heavy Machine 125

inducing the machine motion, of the machine suspension, the active suspension mechanism

**Figure 8.** Model of the road input, machine suspension mechanism, active suspension of the cab and

The programme enables the measurements of the actuator length, the angle of frame tilting

cab's angular motion. These are shown in the block diagram "Measurements of the machine frame movements". During the simulation procedure, these quantities are sent to be further

The proposed control strategy to be applied to the active suspension of the cab uses the feedback control system with compensation for the measured disturbances in the form of the machine frame movements. The expected states of the cab motion, determined in the block "Preset cab motion" involve the requirement whereby the cab is to be stabilised in the vertical direction and the seat must not be displaced along the cab's vertical axis, at the same time the operating range of the actuator 5 should be duly taken into account. Once frame movements are known from measurements and assumptions as to the anticipated cab movements being taken into account, an unambiguous procedure is applied to compute drive movements in the active suspension mechanism. On the output from the block "Inverse problem of kinematics of the active suspension mechanism" we get the expected velocities and accelerations of three drives, represented by vectors

measuring their actual lengths 145 [ ]*<sup>T</sup>* **<sup>s</sup>** *sss* in order to determine the control error

, velocity and acceleration *Om* and of velocity and acceleration of the

*o ooo* **s** *sss* . Actuators should be equipped with sensors for

seat developed in MSC visualNASTRAN 4D

handled by Matlab/Simulink (Fig 9).

*<sup>x</sup>* and pitching *<sup>y</sup>*

<sup>145</sup> [ ]*<sup>T</sup>*

*o ooo* **<sup>s</sup>** *sss* , 145 [ ]*<sup>T</sup>*

for the cab and the seat. All these modelled elements are simplified (Fig 8).

$$\mathbf{M}\_{hk} = -\mathbf{e}\_{k,g}\mathbf{I}\_k - \tilde{\mathbf{a}}\mathbf{o}\_{k,g}\mathbf{I}\_k\mathbf{o}\_{k,g} \tag{74}$$

$$\mathbf{P}\_{b\not\!f} = -m\_{f}\mathbf{a}\_{O\_{f\not\!f}\mathcal{S}}\tag{75}$$

$$\mathbf{M}\_{bf} = -\mathbf{e}\_{f,g}\mathbf{I}\_f - \tilde{\mathbf{w}}\_{f,g}\mathbf{I}\_f\boldsymbol{\omega}\_{f,g} \tag{76}$$

where: 0 0 0 *z y z x y x* **ω** , *r k rT k p kp* **I RI R** , *r rT <sup>f</sup> f p fp* **I RI R** ,

*k <sup>k</sup>* **<sup>I</sup>** , *<sup>f</sup> <sup>f</sup>* **I** - mass moments of inertia of the cab and the seat with operator in their own reference systems.

The sum total of instantaneous power applied by the active suspension mechanism and power of the gravity and inertia forces are brought down to zero:

$$\mathbf{s}^{T}\mathbf{F}\_{s} + m\_{k}\mathbf{v}\_{O\_{k},r}^{T}(\mathbf{g} - \mathbf{a}\_{O\_{k},\mathcal{g}}) + \boldsymbol{\mathfrak{w}}\_{k,r}^{T}(-\mathbf{I}\_{k}\boldsymbol{\varepsilon}\_{k,\mathcal{g}} - \boldsymbol{\tilde{\omega}}\_{k,\mathcal{g}}\mathbf{I}\_{k}\boldsymbol{\omega}\_{k,\mathcal{g}}) + \\\\ \mathbf{s}^{T} + m\_{f,r}(-\mathbf{I}\_{f}\boldsymbol{\varepsilon}\_{f,\mathcal{g}} - \boldsymbol{\tilde{\omega}}\_{f,\mathcal{g}}\mathbf{I}\_{f}\boldsymbol{\omega}\_{f,\mathcal{g}}) = \mathbf{0} \tag{77}$$

where: 145 [, ,]*<sup>T</sup> <sup>s</sup>* **F** *FFF* - forces developed by the drives.

Recalling Jacobean matrices (46), (50), (54), Eq (77) can be rewritten as:

$$\mathbf{s}^{T}\mathbf{I}\mathbf{F}\_{s} + m\_{k}\mathbf{J}\_{vk}^{T}(\mathbf{g} - \mathbf{a}\_{O\_{k},g}) + \mathbf{J}\_{\alpha}^{T}(-\mathbf{I}\_{k}\boldsymbol{\varepsilon}\_{k,g} - \tilde{\boldsymbol{\omega}}\_{k,g}\mathbf{I}\_{k}\boldsymbol{\omega}\_{k,g}) + \\\\ \mathbf{J}\_{\alpha}^{T}\mathbf{I} - \mathbf{J}\_{f}\boldsymbol{\varepsilon}\_{f,g} - \tilde{\boldsymbol{\omega}}\_{f,g}\mathbf{I}\_{f}\boldsymbol{\omega}\_{f,g})] = \mathbf{0} \tag{78}$$
 
$$+ m\_{f}\mathbf{J}\_{vf}^{T}(\mathbf{g} - \mathbf{a}\_{O\_{f},g}) + \mathbf{J}\_{\alpha}^{T}(-\mathbf{I}\_{f}\boldsymbol{\varepsilon}\_{f,g} - \tilde{\boldsymbol{\omega}}\_{f,g}\mathbf{I}\_{f}\boldsymbol{\omega}\_{f,g})] = \mathbf{0} \tag{79}$$

Knowing the loads due to gravity and inertia, Eq (78) yields the forces acting in the drives:

$$\mathbf{F}\_{s} = m\_{k}\mathbf{J}\_{vk}^{T}(\mathbf{a}\_{O\_{k},\mathcal{g}} - \mathbf{g}) + m\_{f}\mathbf{J}\_{vf}^{T}(\mathbf{a}\_{O\_{f},\mathcal{g}} - \mathbf{g}) + \mathbf{J}\_{o}^{T}\{(\mathbf{I}\_{k} + \mathbf{I}\_{f})\mathbf{e}\_{k,\mathcal{g}} + [\tilde{\mathbf{w}}\_{k,\mathcal{g}}(\mathbf{I}\_{f} + \mathbf{I}\_{k})\mathbf{w}\_{k,\mathcal{g}}]\} \tag{79}$$

### **6. Simulation of the active suspension system**

The operation of the active suspension system is investigated using two mutually supportive programmes. MSC visualNastran 4D is used to develop the model of the input inducing the machine motion, of the machine suspension, the active suspension mechanism for the cab and the seat. All these modelled elements are simplified (Fig 8).

124 Advances on Analysis and Control of Vibrations – Theory and Applications

The external loads acting on the active suspension mechanism involve the gravity forces, inertia forces and moments of inertial force of the platform together with the cab, the seat and the operator. These are governed by the Newton-Euler equations, referenced in [18]:

, *k*

, *f*

*k p kp* **I RI R** , *r rT <sup>f</sup>*

*<sup>f</sup>* **I** - mass moments of inertia of the cab and the seat with operator in their own

, , , , ,, ( )( ) *k k T T <sup>T</sup>* **sF v g a** *s k O r O g kr k kg kg k kg <sup>m</sup>* **<sup>ω</sup> <sup>I</sup> ε ω <sup>I</sup> <sup>ω</sup>**

, , , , ,, ( )( ) 0 *f f*

, , ,, [ ( )( ) *<sup>k</sup>*

Knowing the loads due to gravity and inertia, Eq (78) yields the forces acting in the drives:

The operation of the active suspension system is investigated using two mutually supportive programmes. MSC visualNastran 4D is used to develop the model of the input

*s k vk O g k k g k g k k g m* **sF J ga J I ε ω I ω**

, , ,, ( )( )] 0 *<sup>f</sup>*

, , ,, , ( ) ( ) {( ) [ ( ) ]} *k f*

*k f kg kg f k kg* **F Ja g Ja g J I I ε ω I I ω** (79)

The sum total of instantaneous power applied by the active suspension mechanism and

**P a** *bk k O g m* (73)

**P a** *bf f O g m* (75)

*bk k g k k g k k g* , , , **M ε I ω I ω** (74)

*bf f g f f g f f g* , , , **M ε I ω I ω** (76)

*f p fp* **I RI R** ,

*T T mf O r O g fr f fg fg f fg* **v ga <sup>ω</sup> <sup>I</sup> ε ω <sup>I</sup> <sup>ω</sup>** (77)

*f fg fg f fg* **J ga J I ε ω I ω** (78)

**5. Inverse problem of dynamics** 

where:

*k <sup>k</sup>* **<sup>I</sup>** , *<sup>f</sup>* 0

> 

reference systems.

where: 145 [, ,]*<sup>T</sup>*

0

 

*z x y x*

0

power of the gravity and inertia forces are brought down to zero:

*<sup>s</sup>* **F** *FFF* - forces developed by the drives.

Recalling Jacobean matrices (46), (50), (54), Eq (77) can be rewritten as:

*TT T*

*T T mf vf O g*

*T TT*

*s k vk O g m mf vf O g*

**6. Simulation of the active suspension system** 

**ω** , *r k rT*

*z y*

**Figure 8.** Model of the road input, machine suspension mechanism, active suspension of the cab and seat developed in MSC visualNASTRAN 4D

The programme enables the measurements of the actuator length, the angle of frame tilting *<sup>x</sup>* and pitching *<sup>y</sup>* , velocity and acceleration *Om* and of velocity and acceleration of the cab's angular motion. These are shown in the block diagram "Measurements of the machine frame movements". During the simulation procedure, these quantities are sent to be further handled by Matlab/Simulink (Fig 9).

The proposed control strategy to be applied to the active suspension of the cab uses the feedback control system with compensation for the measured disturbances in the form of the machine frame movements. The expected states of the cab motion, determined in the block "Preset cab motion" involve the requirement whereby the cab is to be stabilised in the vertical direction and the seat must not be displaced along the cab's vertical axis, at the same time the operating range of the actuator 5 should be duly taken into account. Once frame movements are known from measurements and assumptions as to the anticipated cab movements being taken into account, an unambiguous procedure is applied to compute drive movements in the active suspension mechanism. On the output from the block "Inverse problem of kinematics of the active suspension mechanism" we get the expected velocities and accelerations of three drives, represented by vectors <sup>145</sup> [ ]*<sup>T</sup> o ooo* **<sup>s</sup>** *sss* , 145 [ ]*<sup>T</sup> o ooo* **s** *sss* . Actuators should be equipped with sensors for measuring their actual lengths 145 [ ]*<sup>T</sup>* **<sup>s</sup>** *sss* in order to determine the control error

1 14 45 5 *T ooo s ss ss s* **e** . The control error should tend to zero if the velocities implemented in actuators are in accordance with the formula:

$$
\dot{\mathbf{s}}\_w = \dot{\mathbf{s}}\_o + \mathbf{K}\_P \mathbf{e} \tag{80}
$$

The Active Suspension of a Cab in a Heavy Machine 127

 *<sup>x</sup>* , *<sup>y</sup>* 

. Basing on

anticipated movement of the active suspension mechanism of the cab and of the cab itself. Basing on anticipated cab movements, inertia interactions are found which, alongside the gravity forces, become the major loads applied to the cab. The inverse problem of the active suspension dynamics involves the calculation of the driving forces in the form of a vector 145 [ ]*<sup>T</sup>* **<sup>F</sup>** *FFF* , counterbalancing the external loads. The contribution of gravity

computed loads *F* and the required instantaneous velocities *w***s** , the block " Model of active suspension drives" generates the realisable instantaneous velocities of actuators

*n nnn sss* **s** . Velocity values *n***s** are sent to be further processed by MSCvisualNASTRAN 4D. This work does not include the analysis of the drive model. It is

The control of the active suspension system gives rise to certain errors *e*, and in consequence the constraints imposed on the angular velocity of the cab and linear velocity of the cab and the seat cannot be accurately reproduced. These errors are attributable to inaccurate measurements of the frame movements, the time delay involved in implementation of the drive velocity or the drives' failure to implement the required velocity (moving beyond the

It is demonstrated in [19] that dimensions of key parts of the mechanism *A*2*A*3*B*3*B*2 can be chosen such that the instantaneous centre of the platform rotation with respect to the machine frame *Cpr* should be included in the road unevenness path when the active suspension system is on. When this condition is satisfied, the actuator 4, controlled in accordance with the cab vertical stabilisation requirement, will at the same time reduce the

Assuming the central position of the cab on the machine frame, the points *A*2 and *A*3 should be arranged symmetrically with respect to the frame's longitudinal axis and the lengths of

The road unevenness range and a typical crescent - shaped field of instantaneous centres of the platform rotation with respect to the frame *Cpr* are shown in Fig 10. It is assumed that when the machine frame is in a horizontal position, *Cpr* is found on the line of wheelground contact. The distance of joints in the rocker arm connections *A*2 and *A*3 from the

,

where: *h* - distance between the platform and the points of joints *A*2 and *A*3 for the frame in

3 23 *m m h hh a b*

(81)

2 2 2 3 23 2 *a b h d* 

**7. Dependings of link dimensions of the active suspension cab** 

absolute movement of the platform in the direction transverse to the ride.

the rocker arms 2 and 3 should be identical 2 3 *ddd* .

ground is *hm*. The relevant dimensions are related as follows:

forces to the load of particular drives depends on the frame tilt angles:

145

**mechanism** 

the horizontal position.

*T*

limits of their typical operating range).

assumed in simulations *n w* **s s** .

**Figure 9.** Model of drives control in the active suspension system - schematic diagram

Computed accelerations **s** and measured velocities **s** and displacements **s** of the drives become the inputs to the block "Direct problem of kinematics of the active suspension mechanism problem of kinematics of the active suspension", which calculates the anticipated movement of the active suspension mechanism of the cab and of the cab itself. Basing on anticipated cab movements, inertia interactions are found which, alongside the gravity forces, become the major loads applied to the cab. The inverse problem of the active suspension dynamics involves the calculation of the driving forces in the form of a vector 145 [ ]*<sup>T</sup>* **<sup>F</sup>** *FFF* , counterbalancing the external loads. The contribution of gravity forces to the load of particular drives depends on the frame tilt angles: *<sup>x</sup>* , *<sup>y</sup>* . Basing on computed loads *F* and the required instantaneous velocities *w***s** , the block " Model of active suspension drives" generates the realisable instantaneous velocities of actuators 145 *T n nnn sss* **s** . Velocity values *n***s** are sent to be further processed by MSCvisualNASTRAN 4D. This work does not include the analysis of the drive model. It is assumed in simulations *n w* **s s** .

126 Advances on Analysis and Control of Vibrations – Theory and Applications

*T*

implemented in actuators are in accordance with the formula:

**K** - gain matrix in the position path.

**MSC.visualNastran 4D Matlab Simulink**

+

*s*

**.**

*s*

, *x y*

**Figure 9.** Model of drives control in the active suspension system - schematic diagram

become the inputs to the block "Direct problem of kinematics of the active suspension mechanism problem of kinematics of the active suspension", which calculates the

*K<sup>P</sup>*

*<sup>o</sup> s <sup>o</sup> s*

Inverse problem of kinematics of the active suspension mechanism

+ \_

*e*

+

Direct problem of kinematics of the active suspension mechanism

Inverse problem of dynamics of the active suspension mechanism

*F*

*<sup>w</sup>s <sup>n</sup> s*

Model of drives in the active suspension system

**s** and measured velocities **s** and displacements **s** of the drives

*ooo s ss ss s* **e** . The control error should tend to zero if the velocities

*wo P* **s s Ke** (80)

Preset motion of the cab

**.**

*ws*

*d dt*

 *s*

1 14 45 5

1

*p P p*

*k*

Computed accelerations

0 0

Displacement of the machine frame

AZK

4

 

*k*

0 0 0 0

5

*p*

*k*

where:

The control of the active suspension system gives rise to certain errors *e*, and in consequence the constraints imposed on the angular velocity of the cab and linear velocity of the cab and the seat cannot be accurately reproduced. These errors are attributable to inaccurate measurements of the frame movements, the time delay involved in implementation of the drive velocity or the drives' failure to implement the required velocity (moving beyond the limits of their typical operating range).

### **7. Dependings of link dimensions of the active suspension cab mechanism**

It is demonstrated in [19] that dimensions of key parts of the mechanism *A*2*A*3*B*3*B*2 can be chosen such that the instantaneous centre of the platform rotation with respect to the machine frame *Cpr* should be included in the road unevenness path when the active suspension system is on. When this condition is satisfied, the actuator 4, controlled in accordance with the cab vertical stabilisation requirement, will at the same time reduce the absolute movement of the platform in the direction transverse to the ride.

Assuming the central position of the cab on the machine frame, the points *A*2 and *A*3 should be arranged symmetrically with respect to the frame's longitudinal axis and the lengths of the rocker arms 2 and 3 should be identical 2 3 *ddd* .

The road unevenness range and a typical crescent - shaped field of instantaneous centres of the platform rotation with respect to the frame *Cpr* are shown in Fig 10. It is assumed that when the machine frame is in a horizontal position, *Cpr* is found on the line of wheelground contact. The distance of joints in the rocker arm connections *A*2 and *A*3 from the ground is *hm*. The relevant dimensions are related as follows:

$$h^2 + \left(\frac{a\_3 - b\_{23}}{2}\right)^2 = d^2 \quad , \qquad \frac{h\_m}{a\_3} = \frac{h\_m - h}{b\_{23}} \tag{81}$$

where: *h* - distance between the platform and the points of joints *A*2 and *A*3 for the frame in the horizontal position.

**Figure 10.** Distribution of the field made of points *pr C*

Eliminating *h* from Eq (81) yields the relationship between the dimension of the four bar linkage *A*2*A*3*B*3*B*2 and *hm*. When this condition is satisfied, *Cpr* is found in the road unevenness path:

$$h\_m \left( 1 - \frac{b\_{23}}{a\_3} \right) = \sqrt{d^2 - \left( \frac{a\_3 - b\_{23}}{2} \right)^2} \tag{82}$$

The Active Suspension of a Cab in a Heavy Machine 129

*yr* 

*r* 

**r***PO m* -

*pk x r m* ,

(84)


*k* 

23

2 2 *k x w b w a d*

*b*<sup>23</sup>

*d*

*B*3

*A*<sup>3</sup>

*a*<sup>3</sup> *d*

*A*<sup>2</sup>

*B*<sup>2</sup>

*zr* 

 

3 max cos

where: *wk* - cab width, *w b <sup>k</sup>* <sup>23</sup> , *<sup>w</sup>*

the joint *A*2 or *A*3.

g

*2hg* 

**Figure 11.** Frame tilt angle in relation to the singular position of the four bar linkage

momentarily deviate from the vertical direction.

Machine specification data used in simulations:

*pk z r m*

*pk y r m* , ( ) 0.685 [ ] *<sup>p</sup>*

2.810 [ ] *<sup>m</sup>*

( ) 0.000 [ ] *<sup>p</sup>*

**8. Simulation data of the active suspension system** 

Satisfying the inequality (84) quarantines a fail-safe operation of the four bar linkage *A*2*A*3*B*3*B*2 and of the cab. Conditions (82), (83), (84) yield the dimensions: *a*3, *b*2, *d*. When the machine is operated in uneven terrain where 2*hg* exceeds the predetermined value, the active suspension mechanism can reach the limits of its working field and the cab will

*l m* - distance between the front and rear axle of the machine frame, 1.980 [ ] *w m <sup>m</sup>* - wheel spacing, *wr* =1.4 [m] - frame width, [ 2.1, 0.818,1.6][ ] *<sup>r</sup>*

position vector of the point *Or*, 1.200 [ ] *w m <sup>k</sup>* - cab width, ( ) 0.000 [ ] *<sup>p</sup>*

Displacements of the four bar linkage in the active suspension mechanism are constrained by the occurrence of singular positions. The mechanism should not come near the singular position, when controllability of the system deteriorates and the loads acting upon the drives and mobile connections tend to increase. For the predetermined maximal height of the road unevenness range 2*hg* and for the machine wheel spacing *wm*, the maximal angle max a of the machine tilting with respect to the axis *yg* should be such that the four bar linkage should not assume a singular position (Fig 9):

$$a\_{x\max} < \phi \quad \rightarrow \arcsin\frac{2h\_{\frac{\phi}{\phi}}}{w\_m} < \arccos\frac{a\_3^2 + (b\_{23} + d)^2 - d^2}{2a\_3(b\_{23} + d)}\tag{83}$$

When the dimensions of links in the four bar linkage *A*2*A*3*B*3*B*2 as well as *hg* satisfy the condition (83), the mechanism is able to operate in a single configuration.

Another geometric condition stems from the assumption that the cab can move freely without colliding with the joints *A*2, *A*3, when the machine assumes its extreme position due to tilting by the angle *<sup>x</sup>*max (Fig 9):

The Active Suspension of a Cab in a Heavy Machine 129

$$a\_3 \cos \alpha\_{x\max} > \frac{b\_{23}}{2} + \frac{w\_k}{2} + \delta\_w + d \tag{84}$$

where: *wk* - cab width, *w b <sup>k</sup>* <sup>23</sup> , *<sup>w</sup>* - allowable distance between cab walls and the point of the joint *A*2 or *A*3.

**Figure 11.** Frame tilt angle in relation to the singular position of the four bar linkage

Satisfying the inequality (84) quarantines a fail-safe operation of the four bar linkage *A*2*A*3*B*3*B*2 and of the cab. Conditions (82), (83), (84) yield the dimensions: *a*3, *b*2, *d*. When the machine is operated in uneven terrain where 2*hg* exceeds the predetermined value, the active suspension mechanism can reach the limits of its working field and the cab will momentarily deviate from the vertical direction.

### **8. Simulation data of the active suspension system**

Machine specification data used in simulations:

128 Advances on Analysis and Control of Vibrations – Theory and Applications

*C2* 

*A*<sup>2</sup>

*h d*

*b23 d*

*zr* 

*A4* 

**Figure 10.** Distribution of the field made of points *pr C*

*Cpr* 

linkage should not assume a singular position (Fig 9):

 *<sup>x</sup>*max

*<sup>x</sup>*max (Fig 9):

unevenness path:

*hm* 

to tilting by the angle

Eliminating *h* from Eq (81) yields the relationship between the dimension of the four bar linkage *A*2*A*3*B*3*B*2 and *hm*. When this condition is satisfied, *Cpr* is found in the road

**.**

*L P*

*B*<sup>2</sup> *B*<sup>3</sup>

*a3* 

23 2 3 23

Displacements of the four bar linkage in the active suspension mechanism are constrained by the occurrence of singular positions. The mechanism should not come near the singular position, when controllability of the system deteriorates and the loads acting upon the drives and mobile connections tend to increase. For the predetermined maximal height of the road unevenness range 2*hg* and for the machine wheel spacing *wm*, the maximal angle

max a of the machine tilting with respect to the axis *yg* should be such that the four bar

*g m*

When the dimensions of links in the four bar linkage *A*2*A*3*B*3*B*2 as well as *hg* satisfy the

Another geometric condition stems from the assumption that the cab can move freely without colliding with the joints *A*2, *A*3, when the machine assumes its extreme position due

<sup>2</sup> ( ) arcsin arccos

3

*a*

2 *<sup>m</sup> b ab h d*

1

condition (83), the mechanism is able to operate in a single configuration.

2

*A3 yr* 

2 22

3 23

2( )

3 23

*h abdd w ab d*

(82)

Road unevenness range

(83)

2.810 [ ] *<sup>m</sup> l m* - distance between the front and rear axle of the machine frame, 1.980 [ ] *w m <sup>m</sup>* - wheel spacing, *wr* =1.4 [m] - frame width, [ 2.1, 0.818,1.6][ ] *<sup>r</sup>* **r***PO m* position vector of the point *Or*, 1.200 [ ] *w m <sup>k</sup>* - cab width, ( ) 0.000 [ ] *<sup>p</sup> pk x r m* , ( ) 0.000 [ ] *<sup>p</sup> pk y r m* , ( ) 0.685 [ ] *<sup>p</sup> pk z r m*

$$\begin{aligned} \,^p r\_{(pf)x} &= 0,000 \, [m] \,, \,^p r\_{(pf)y} = 0,000 \, [m] \,, \, \, df\_z = 0.335 \, [m] \,, \, m = 480 \, [kg] \, - \, \text{cab mass}, \, m \neq 160 \, [kg] \, - \, \text{c} \\ &\quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \,\_p \, \Gamma \, \left[ \begin{array}{ccccc} 180 & 0 & 0 \\ & 1 & \ldots & 1 \end{array} \,, \, \right. \\ \left. \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \quad \ldots \right] \end{aligned}$$

The Active Suspension of a Cab in a Heavy Machine 131

**Figure 12.** Coordinate *xr* of angular velocity of the frame and platform in the function of time,

(,) *p g y*

**Figure 13.** Coordinate *yr* of angular velocity of the frame and platform in the function of time,

*med j*

*N*

The mean power expended by the drives (shown in Fig 15) is derived from the formula:

0 ,

 

*j*

*N dt*

(86)

01234

(, ) *rg y*

01234

(, ) *rg x*

time [s]

time [s]

*s*

*T*


*s T*

expressed in the system associated with the frame.

expressed in the system associated with the frame.



0

*y* [rad/s]

0.4

0.8



0

*x* [rad/s]

0.4

0.8

(,) *p g x*

where: *j* = (1, 4, 5), *Nj*

mass of the seat with an operator, <sup>2</sup> 0 208 0 [ ] 0 0 133 *k kJ kg m* - inertia matrix of the cab in

the reference system associated with the cab, <sup>2</sup> 23.8 0 0 0 24.7 0 [ ] 0 0 13.2 *f <sup>f</sup> J kg m* - inertia matrix

of the seat and operator in the system associated with the seat,

Road profile:

2*hg*= 0.250 [*m*] - height of the unevenness range, *Lg*= 2[*m*] - wave length of the road unevenness, / 2[ ] *rad* - the phase shift angle between the left-and right-hand side of the

machine, ,max 2.82 [ / ] <sup>2</sup> *h g Px g <sup>L</sup> <sup>g</sup> <sup>v</sup> m s <sup>h</sup>* maximal speed of the machine ride computed for

the free wheel in contact with the road surface.

Active suspension mechanism for the cab:

2.420 [ ] *mh m* - distance of joints in the rocker arm connections *A*2 and *A*3 from the ground, 0.05 [ ] *<sup>w</sup> m* - admissible distance between the cab's side wall from the joint axis *A*2 or *A*3, <sup>3</sup> *a m* 1.636 [ ], 3 *ddd m* 0.227 [ ], 23 *b m* 1.490 [ ], 2*c m* 0.099 [ ], 2 2 ( , ) 4.5606 [ ] *o o* **c d** *rad* <sup>4</sup> *a m* 0.541[ ], 4 [0.0000,0.2181, 0.9759][ ] *<sup>o</sup>* **<sup>a</sup>** *<sup>m</sup>* , 1*a m* 1.182 [ ] , <sup>1</sup> [ 0.7218,0.6921,0.0000][ ] *<sup>o</sup>* **<sup>a</sup>** *<sup>m</sup>* , 2 3 23 *bb b* 0.5 , 1*b m* 0.850 [ ] , 5min *s m* 0.415 [ ] , 5max *s m* 0.715 [ ] , 5 max 1[ / ] *<sup>k</sup> s ms* , 5 *K* 50 [ ] .

$$\mathbf{K}\_p = \begin{bmatrix} 20 & 0 & 0 \\ 0 & 20 & 0 \\ 0 & 0 & 0 \end{bmatrix} \text{[1/s]} \text{- matrix of gain in the position path. The distance covered during the } p$$

simulation - 10[*m*]. The time step in the simulation procedure - 0.005 [*s*].

Simulation data relating to the cab's and machine frame angular motion are given in Fig 12 and 13, each showing two plots of one angular velocity component in the function of time.

Simulation data relevant to the linear movement of the point *Of* on the seat are given in Fig 14, showing the plots of vibration reduction factors for the three components of the rms acceleration derived from the formula:

$$\mu\_l = \sqrt{\int\_0^T a\_{f,off,l}^2 dt \, \bigg/ \int\_0^T a\_{f,on,l}^2 dt} \tag{85}$$

where: *l* = (*xg, yg, zg*), *f off l* , , *a* - linear acceleration in the direction *l*, the active suspension system being off, *f on l* , , *a* - linear acceleration in the direction *l*, the active suspension system being on; *Ts* - simulation time associated with the ride velocity.

mass of the seat with an operator, <sup>2</sup>

of the seat and operator in the system associated with the seat,

*k*

the reference system associated with the cab, <sup>2</sup>

*pf y r m* , 0.335 [ ] *<sup>z</sup> df m* , *mk*=480 [*kg*] - cab mass, *mf*=160 [*kg*] -

23.8 0 0

0 0 13.2

*<sup>f</sup> J kg m* 

0 24.7 0 [ ]


(85)


0 208 0 [ ]

/ 2[ ] *rad* - the phase shift angle between the left-and right-hand side of the

*<sup>h</sup>* maximal speed of the machine ride computed for

180 0 0

0 0 133

*f*

2*hg*= 0.250 [*m*] - height of the unevenness range, *Lg*= 2[*m*] - wave length of the road

2.420 [ ] *mh m* - distance of joints in the rocker arm connections *A*2 and *A*3 from the ground,

 *<sup>w</sup> m* - admissible distance between the cab's side wall from the joint axis *A*2 or *A*3, *a m* 1.636 [ ], 3 *ddd m* 0.227 [ ], 23 *b m* 1.490 [ ], 2*c m* 0.099 [ ], 2 2 ( , ) 4.5606 [ ] *o o* **c d** *rad a m* 0.541[ ], 4 [0.0000,0.2181, 0.9759][ ] *<sup>o</sup>* **<sup>a</sup>** *<sup>m</sup>* , 1*a m* 1.182 [ ] , [ 0.7218,0.6921,0.0000][ ] *<sup>o</sup>* **<sup>a</sup>** *<sup>m</sup>* , 2 3 23 *bb b* 0.5 , 1*b m* 0.850 [ ] , 5min *s m* 0.415 [ ] ,

**K** - matrix of gain in the position path. The distance covered during the

Simulation data relating to the cab's and machine frame angular motion are given in Fig 12 and 13, each showing two plots of one angular velocity component in the function of time.

Simulation data relevant to the linear movement of the point *Of* on the seat are given in Fig 14, showing the plots of vibration reduction factors for the three components of the rms

> 2 2 , , ,,

0 0

where: *l* = (*xg, yg, zg*), *f off l* , , *a* - linear acceleration in the direction *l*, the active suspension system being off, *f on l* , , *a* - linear acceleration in the direction *l*, the active suspension system

*s s T T l f off l f on l a dt a dt*

simulation - 10[*m*]. The time step in the simulation procedure - 0.005 [*s*].

being on; *Ts* - simulation time associated with the ride velocity.

*kJ kg m* 

( ) 0,000 [ ] *<sup>p</sup>*

Road profile:

unevenness,

0.05 [ ]

20 0 0

0 00 *<sup>P</sup> s* 

0 20 0 [1 / ]

acceleration derived from the formula:

 

machine, ,max 2.82 [ / ] <sup>2</sup> *h g Px*

Active suspension mechanism for the cab:

*g <sup>L</sup> <sup>g</sup> <sup>v</sup> m s* 

the free wheel in contact with the road surface.

5max *s m* 0.715 [ ] , 5 max 1[ / ] *<sup>k</sup> s ms* , 5 *K* 50 [ ] .

*pf x r m* , ( ) 0,000 [ ] *<sup>p</sup>*

**Figure 12.** Coordinate *xr* of angular velocity of the frame and platform in the function of time, expressed in the system associated with the frame.

**Figure 13.** Coordinate *yr* of angular velocity of the frame and platform in the function of time, expressed in the system associated with the frame.

The mean power expended by the drives (shown in Fig 15) is derived from the formula:

$$\begin{aligned} \stackrel{T\_s}{\int N\_j^+ dt} & \tag{86} \\ \mathcal{N}\_{med,j} &= \frac{0}{T\_s} \end{aligned} \tag{86}$$

where: *j* = (1, 4, 5), *Nj* - instantaneous positive power expended by the drive *j*.

The Active Suspension of a Cab in a Heavy Machine 133

suspension mechanism links leads to significant reduction of the cab and seat vibration in the direction *yr*. Reduction of angular cab vibration around *yr* leads to reduction of linear

The operation of the active suspension system involves the real-time measurements of mechanical quantities which can be accurately measured with state-of-the-art sensors: angular velocity and acceleration of the frame, linear acceleration of a selected point on the machine frame, two angles of the frame tilting from the direction of the gravity forces and

Underlying the simulation procedure is that assumption that each computed drive velocity will be implemented without any time delay (provided that is allowed by collaborating programmes). Results therefore can be utilised when selecting drives which, when in extreme conditions, may not be able to perform the required movements. Besides, the overall time constant, taking into account the response time of the measurement system, the controls and

The actuator 4 handles two DOFs (i.e. the cab rotation around the *xr* axis and its translation along the *yr*-axis) and induces slight movements of the cab in the direction of the *zr*-axis,

*Institute of Machine Design, Faculty of Mechanical Engineering, Cracow University of Technology,* 

I am particularly indebted to the Institute of Machine Design at Cracow Polytechnic for financial support needed to prepare this chapter and have it published in the book Vibration Control.

[1] Achen A, Toscano J, Marjoram R, StClair K, McMahon B, Goelz A, Shutto S (2008) Semiactive vehicle cab suspension using magnetorheological (mr) technology. Proc. of

[2] Jonasson M, Roos F (2008) Design and evaluation of an active electromechanical wheel

[3] Al Sayed B, Chatelet E, Baguet S, Jacquet-Richardet G (2011) Dissipated energy and boundary condition effects associated to dry friction on the dynamics of vibrating

[4] Sampaio J V R (2009) Design of a Low Power Active Truck Cab Suspension, Eindhoven

the 7th JFPS Int. Symp. on Fluid Power, TOYAMA, 561-564.

structures, Mechanism and Machine Theory 46: 479–491.

suspension system, Mechatronics, 18: 218–230.

University of Technology, DCT № 119.

drives becomes another limiting factor, particularly at higher frequencies of road input.

seat vibrations in the direction *xr*.

the length and velocities implemented by actuators.

hence its power demand is higher than in drive 1.

**Author details** 

Grzegorz Tora

*Cracow, Poland* 

**Acknowledgement** 

**10. References** 

**Figure 14.** Vibration reduction of the point *Of* on the operator seat, in the directions *xr*, *yr*, *zr*

Plots in Fig 14 and 15 show the relevant parameters in the function of the coefficient linearly related to the machine ride velocity, whilst for ,max *h h Px Px v v* the value of becomes 1.

**Figure 15.** Mean power ratings of the drives 1, 4, 5

### **9. Conclusions**

Simulations of the active suspension system performance have proved its adequacy in vibration reduction of angular vibrations of the cab around the longitudinal axis of the machine *xr* and around the transverse axis *yr*. Seat vibrations along the vertical axis *zr* are successfully controlled, too. The applied procedure of dimension synthesis of the active suspension mechanism links leads to significant reduction of the cab and seat vibration in the direction *yr*. Reduction of angular cab vibration around *yr* leads to reduction of linear seat vibrations in the direction *xr*.

The operation of the active suspension system involves the real-time measurements of mechanical quantities which can be accurately measured with state-of-the-art sensors: angular velocity and acceleration of the frame, linear acceleration of a selected point on the machine frame, two angles of the frame tilting from the direction of the gravity forces and the length and velocities implemented by actuators.

Underlying the simulation procedure is that assumption that each computed drive velocity will be implemented without any time delay (provided that is allowed by collaborating programmes). Results therefore can be utilised when selecting drives which, when in extreme conditions, may not be able to perform the required movements. Besides, the overall time constant, taking into account the response time of the measurement system, the controls and drives becomes another limiting factor, particularly at higher frequencies of road input.

The actuator 4 handles two DOFs (i.e. the cab rotation around the *xr* axis and its translation along the *yr*-axis) and induces slight movements of the cab in the direction of the *zr*-axis, hence its power demand is higher than in drive 1.

### **Author details**

132 Advances on Analysis and Control of Vibrations – Theory and Applications

**Figure 14.** Vibration reduction of the point *Of* on the operator seat, in the directions *xr*, *yr*, *zr*

Plots in Fig 14 and 15 show the relevant parameters in the function of the coefficient

*Nmed* ,1

*Nmed* ,5

*Nmed* ,4

*h h*

[ ] 0 0.2 0.4 0.6 0.8 <sup>1</sup>

Simulations of the active suspension system performance have proved its adequacy in vibration reduction of angular vibrations of the cab around the longitudinal axis of the machine *xr* and around the transverse axis *yr*. Seat vibrations along the vertical axis *zr* are successfully controlled, too. The applied procedure of dimension synthesis of the active

0 0.2 0.4 0.6 0.8 1

[ ]

*Px Px v v* the value of

*x*

*y*

*z*

related to the machine ride velocity, whilst for ,max

**Figure 15.** Mean power ratings of the drives 1, 4, 5

0

1

2

*Nmed,j* [kW]

3

4

1.2

1.6

2

*l* [rad/s]

2.4

2.8

**9. Conclusions** 

becomes 1.

linearly

Grzegorz Tora *Institute of Machine Design, Faculty of Mechanical Engineering, Cracow University of Technology, Cracow, Poland* 

## **Acknowledgement**

I am particularly indebted to the Institute of Machine Design at Cracow Polytechnic for financial support needed to prepare this chapter and have it published in the book Vibration Control.

### **10. References**

	- [5] Du H, Lam J, Sze K Y (2003) Non-fragile output feedback H∞ vehicle suspension control using genetic algorithm, Engineering Applications of Artificial Intelligence 16: 667–680.

**Section 2** 

**Vibration Control Case Studies** 


**Section 2** 

**Vibration Control Case Studies** 

134 Advances on Analysis and Control of Vibrations – Theory and Applications

for Off-Road Vehicles, IEEE, 3: 38-43.

Machine Theory 34: 801-824.

Vienna, July.

[5] Du H, Lam J, Sze K Y (2003) Non-fragile output feedback H∞ vehicle suspension control using genetic algorithm, Engineering Applications of Artificial Intelligence 16: 667–680. [6] Duke M, Goss G, (2007) Investigation of Tractor Driver Seat Performance with Nonlinear Stiffness and On–off Damper, Biosystems Engineering 96 (4): 477–486.

[7] Jarviluoma M, Nevala K (1997) An Active Vibration Damping System of a Driver's Seat

[8] Ruotsalainen P,Nevala K, Marjanen Y (2006) Design of an adjustable hydropneumatic damper for cab suspension, The XIII International Congress on Sound and Vibration,

[9] Savkoor A, Manders S, Riva P (2001) Design of actively controlled aerodynamic devices

[10] He Y, McPhee J (2005) Multidisciplinary design optimization of mechatronic vehicles

[11] Yildirim S (2004) Vibration control of suspension systems using a proposed neural

[12] Liu H, Nonami K, Hagiwara T (2008) Active following fuzzy output feedback sliding mode control of real-vehicle semi-active suspensions, Journal of Sound and Vibration 314: 39–52. [13] Graf Ch, Maas J, Pflug H-Ch (2009) Concept for an Active Cabin Suspension, Proc. of the 2009 IEEE International Conference on Mechatronics. Malaga, Spain, April. [14] Akcay H, Turkay S (2009) Influence of tire damping on mixed 2 *H H*/ synthesis of

[15] Koch G, Fritsch O, Lohmann B (2010) Potential of low bandwidth active suspension control with continuously variable damper, Control Engineering Practice 18: 1251–1262. [16] Du H, Zhang N, Lam J (2008) Parameter-dependent input-delayed control of uncertain

[17] Marzbanrad J, Ahmadi G, Zohoor H, Hojjat Y (2004) Stochastic optimal previewcontrol

[18] Dasgupta B, Choudhury P (1999) A general strategy based on the Newton-Euler approach for the dynamic formulation of parallel manipulators, Mechanism and

[19] Tora G (2008) Kinematyka mechanizmu platformowego w układzie aktywnej redukcji drgań, XXI Ogólnopolska Konferencja Naukowo-Dydaktyczna Teorii Maszyn i Mechanizmów, Wydawnictwo ATH w Bielsku Białej ISBN 978-83-60714-57-7: 357-366.

for reducing pitch and heave of truck cabins, JSAE Review 22: 421–434.

with active suspensions, Journal of Sound and Vibration 283: 217–241.

half-car active suspensions, Journal of Sound and Vibration 322: 15–28.

vehicle suspension, Journal of Sound and Vibration 317: 537–556.

of a vehicle suspension, Journal of Sound and Vibration 275, 973–990.

Network, Journal of Sound and Vibration 277: 1059–1069.

**Chapter 0**

**Chapter 6**

**On Variable Structure Control Approaches to**

Vehicle suspension systems are one of the most critical components of a vehicle and it have been a hot research topic due to their importance in vehicle performance. These systems are designed to provide comfort to the passengers to protect the chassis and the freigt [28]. However, ride comfort, road holding and suspension deflection are often conflicting and a compromise of the requirements must be considered. Among the proposed solutions, active suspension is an approach to improve ride comfort while keeping suspension stroke and tire

In semiactive suspension, the value of the damper coefficient can be controlled and can show reasonable performance as compared to that of an active suspension control. Besides, it does not require external energy. For instance, in the work by [18] a semiactive suspension control of a quarter-car model using a hybrid-fuzzy-logic-based controlled is developed and implemented. [23] formulated a force-tracking PI controller for an MR-damper controlled quarter-car system. The preliminary results showed that the proposed semiactive force tracking PI control scheme could provide effective control of the sprung mass resonance as well as the wheel-hop control. Furthermore, the proposed control yields lower magnitudes of mass acceleration in the ride zone. [25] designed a semi active suspension system using a magnetorhelogical damper. The control law was formulated following the sky-hook technique in which the direction of the relative velocity between the sprung and unsprung masses is compared to that of the velocity of the unsprung mass. Depending on this result, an on-off action is performed. [8] designed a semiactive static output *H*∞ controller for a quarter car system equipped with a magnetorheological damper. In this case, the control law was formulated in order to regulate the vertical acceleration as a measure to keep passengers' comfort within acceptable limits. They also added a constraint in order to keep the transfer function form road disturbance to suspension deflection small enough to prevent excessive

> ©2012 Zapateiro et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2012 Zapateiro et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Semiactive Control of a Quarter Car System**

Mauricio Zapateiro, Francesc Pozo and Ningsu Luo

Additional information is available at the end of the chapter

deflection within an acceptable level [11, 21].

http://dx.doi.org/10.5772/50325

**1. Introduction**

suspension bottoming.

cited.
