**On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System**

Mauricio Zapateiro, Francesc Pozo and Ningsu Luo

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50325

### **1. Introduction**

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 deflection within an acceptable level [11, 21].

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 suspension bottoming.

©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 cited. ©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.

### 2 Will-be-set-by-IN-TECH 138 Advances on Analysis and Control of Vibrations – Theory and Applications On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System <sup>3</sup>

Backstepping is a recursive design for systems with nonlinearities not constrained by linear bounds. The ease with which backstepping incorporated uncertainties and unknown parameters contributed to its instant popularity and rapid acceptance. Applications of this technique have been recently reported ranging from robotics to industry or aerospace [6, 7, 15, 22, 24]. Backstepping control has also been explored in some works about suspension systems. For example, [26] designed a semiactive backstepping control combined with neural network (NN) techniques for a system with MR damper. In that work, the controller was formulated for an experimental platform, whose MR damper was modeled by means of an artificial neural network. The control input was updated with a backstepping controller. On the other hand, [16] studied a hybrid control of active suspension systems for quarter-car models with two-degrees-of-freedom. This hybrid control was implemented by controlling the linear part with *H*∞ techniques and the nonlinear part with an adaptive controller based on backstepping.

variable structure controllers will be complemented with the comparison of a model-based controller which has been successfully applied by the authors in other works: backstepping. As it was mentioned earlier, backstepping is well suited to this kind of problems because it can account for robustness and nonlinearities. It has been used by the authors to analyze this

On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System 139

The chapter is organized as follows. Section 2 presents the mathematical details of the system to be controlled. In Section 3, the three variable structure controllers are developed. In Section 4, the backstepping control formulation details are outlined. Section 5 shows the numerical

The suspension system can be modeled as a quarter car model, as shown in Figure 1. The system can be viewed as a composition of two subsystems: the tyre subsystem and the suspension subsystem. The tyre subsystem is represented by the wheel mass *mu* while the suspension subsystem consists of a sprung mass, *ms*, that resembles the vehicle mass. This way of seeing the system will be useful later on when designing the model-based semi active controller. The compressibility of the wheel pneumatic is *kt*, while *cs* and *ks* are the damping and stiffness of the uncontrolled suspension system. The quarter car model equations are

Taking *x*1, *x*2, *x*<sup>3</sup> and *x*<sup>4</sup> as state variables allows us to formulate the following state-space

*x*˙1 = *x*<sup>2</sup> − *d <sup>x</sup>*˙2 <sup>=</sup> <sup>−</sup> *kt mu*

*x*˙3 = −*x*<sup>2</sup> + *x*<sup>4</sup>

where *ρ* = *ms*/*mu*, *d* is the velocity of the disturbance input and *u* is the acceleration input

*msx*˙4 + *cs*(*x*<sup>4</sup> − *x*2) + *ksx*<sup>3</sup> − *f*mr = 0 (1) *mux*˙2 − *cs*(*x*<sup>4</sup> − *x*2) − *ksx*<sup>3</sup> + *ktx*<sup>1</sup> + *f*mr = 0 (2)

*<sup>x</sup>*<sup>1</sup> <sup>+</sup> *<sup>ρ</sup><sup>u</sup>* (3)

*<sup>x</sup>*˙4 <sup>=</sup> <sup>−</sup>*<sup>u</sup>* (4)

particular problem [28] with interesting results.

results, and in Section 6, the conclusions are drawn.

**2. Suspension system model**

given by:

where:

representation:

• Tyre subsystem:

• Suspension subsystem:

due to the damping subsystem. The input *u* is given by:

• *x*<sup>1</sup> is the tyre deflection

• *x*<sup>2</sup> is the unsprung mass velocity • *x*<sup>3</sup> is the suspension deflection • *x*<sup>4</sup> is the sprung mass velocity.

Some works on Quantitative Feedback Theory (QFT) applied to the control of suspension systems can be found in the literature. For instance, [1] analyzed *H*∞ and QFT controllers designed for an active suspension system in order to account for the structured and unstructured uncertainties of the system. As a result, the vertical body acceleration in QFT-controlled is lower than that of the *H*∞-controlled and its performance is superior. In the presence of a hydraulic actuator, the QFT-controlled system performance degrades but it is still comparable to that of the *H*∞-control. [28] addressed a study leading to compare the performance of backstepping and QFT controllers in active and semiactive control of suspension systems. In this case, the nonlinearities were treated as uncertainties in the model so that the linear QFT could be applied to the control formulation. As a result, similar performances between both classes of controllers were achieved.

In this chapter, we will analyze three model-free variable structure controllers for a class of semiactive vehicle suspension systems equipped with MR dampers. The variable structure control (VSC) is a control scheme which is well suited for nonlinear dynamic systems [12]. VSC was firstly studied in the early 1950's for systems represented by single-input high-order differential equations. A rise of interest became in the 1970's because the robustness of VSC were step by step recognized. This control method can make the system completely insensitive to time-varying parameter uncertainties, multiple delayed state perturbations and external disturbances [17]. Nowadays, research and development continue to apply VSC control to a wide variety of engineering areas, such as aeronautics (guidance law of small bodies [29]), electric and electronic engineering (speed control of an induction motor drive [3]). By using this kind of controllers, it is possible to take the best out of several different systems by switching from one to the other. The first strategy that we propose in this work, *σ*1, is based on the difference between the body angular velocity and the wheel angular velocity. The second strategy, *σ*2, more complex, is based both on the difference between the body angular velocity and the wheel angular velocity, and on the difference between the body angular position and the wheel angular position. In this case, the resulting algorithm can be viewed as the clipped control in [9], but with some differences. Finally, the last strategy presented is based on a time variable depending on the absolute value of the difference between the body angular velocity and the wheel angular velocity, and on the difference between the body angular position and the wheel angular position. The study of the three variable structure controllers will be complemented with the comparison of a model-based controller which has been successfully applied by the authors in other works: backstepping. As it was mentioned earlier, backstepping is well suited to this kind of problems because it can account for robustness and nonlinearities. It has been used by the authors to analyze this particular problem [28] with interesting results.

The chapter is organized as follows. Section 2 presents the mathematical details of the system to be controlled. In Section 3, the three variable structure controllers are developed. In Section 4, the backstepping control formulation details are outlined. Section 5 shows the numerical results, and in Section 6, the conclusions are drawn.

### **2. Suspension system model**

The suspension system can be modeled as a quarter car model, as shown in Figure 1. The system can be viewed as a composition of two subsystems: the tyre subsystem and the suspension subsystem. The tyre subsystem is represented by the wheel mass *mu* while the suspension subsystem consists of a sprung mass, *ms*, that resembles the vehicle mass. This way of seeing the system will be useful later on when designing the model-based semi active controller. The compressibility of the wheel pneumatic is *kt*, while *cs* and *ks* are the damping and stiffness of the uncontrolled suspension system. The quarter car model equations are given by:

$$m\_s \dot{\mathbf{x}}\_4 + c\_s(\mathbf{x}\_4 - \mathbf{x}\_2) + k\_s \mathbf{x}\_3 - f\_{\rm mr} = \mathbf{0} \tag{1}$$

$$m\_u \dot{\mathbf{x}}\_2 - c\_s(\mathbf{x}\_4 - \mathbf{x}\_2) - k\_s \mathbf{x}\_3 + k\_l \mathbf{x}\_1 + f\_{\text{mr}} = \mathbf{0} \tag{2}$$

where:

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

Backstepping is a recursive design for systems with nonlinearities not constrained by linear bounds. The ease with which backstepping incorporated uncertainties and unknown parameters contributed to its instant popularity and rapid acceptance. Applications of this technique have been recently reported ranging from robotics to industry or aerospace [6, 7, 15, 22, 24]. Backstepping control has also been explored in some works about suspension systems. For example, [26] designed a semiactive backstepping control combined with neural network (NN) techniques for a system with MR damper. In that work, the controller was formulated for an experimental platform, whose MR damper was modeled by means of an artificial neural network. The control input was updated with a backstepping controller. On the other hand, [16] studied a hybrid control of active suspension systems for quarter-car models with two-degrees-of-freedom. This hybrid control was implemented by controlling the linear part with *H*∞ techniques and the nonlinear part with an adaptive controller based

Some works on Quantitative Feedback Theory (QFT) applied to the control of suspension systems can be found in the literature. For instance, [1] analyzed *H*∞ and QFT controllers designed for an active suspension system in order to account for the structured and unstructured uncertainties of the system. As a result, the vertical body acceleration in QFT-controlled is lower than that of the *H*∞-controlled and its performance is superior. In the presence of a hydraulic actuator, the QFT-controlled system performance degrades but it is still comparable to that of the *H*∞-control. [28] addressed a study leading to compare the performance of backstepping and QFT controllers in active and semiactive control of suspension systems. In this case, the nonlinearities were treated as uncertainties in the model so that the linear QFT could be applied to the control formulation. As a result, similar

In this chapter, we will analyze three model-free variable structure controllers for a class of semiactive vehicle suspension systems equipped with MR dampers. The variable structure control (VSC) is a control scheme which is well suited for nonlinear dynamic systems [12]. VSC was firstly studied in the early 1950's for systems represented by single-input high-order differential equations. A rise of interest became in the 1970's because the robustness of VSC were step by step recognized. This control method can make the system completely insensitive to time-varying parameter uncertainties, multiple delayed state perturbations and external disturbances [17]. Nowadays, research and development continue to apply VSC control to a wide variety of engineering areas, such as aeronautics (guidance law of small bodies [29]), electric and electronic engineering (speed control of an induction motor drive [3]). By using this kind of controllers, it is possible to take the best out of several different systems by switching from one to the other. The first strategy that we propose in this work, *σ*1, is based on the difference between the body angular velocity and the wheel angular velocity. The second strategy, *σ*2, more complex, is based both on the difference between the body angular velocity and the wheel angular velocity, and on the difference between the body angular position and the wheel angular position. In this case, the resulting algorithm can be viewed as the clipped control in [9], but with some differences. Finally, the last strategy presented is based on a time variable depending on the absolute value of the difference between the body angular velocity and the wheel angular velocity, and on the difference between the body angular position and the wheel angular position. The study of the three

performances between both classes of controllers were achieved.

on backstepping.


Taking *x*1, *x*2, *x*<sup>3</sup> and *x*<sup>4</sup> as state variables allows us to formulate the following state-space representation:

• Tyre subsystem:

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= \mathbf{x}\_2 - d\\ \dot{\mathbf{x}}\_2 &= -\frac{k\_t}{m\_u}\mathbf{x}\_1 + \rho u \end{aligned} \tag{3}$$

• Suspension subsystem:

$$\begin{aligned} \dot{x}\_3 &= -x\_2 + x\_4 \\ \dot{x}\_4 &= -u \end{aligned} \tag{4}$$

where *ρ* = *ms*/*mu*, *d* is the velocity of the disturbance input and *u* is the acceleration input due to the damping subsystem. The input *u* is given by:

**Figure 1.** Quarter car suspension model

$$
\mu = \frac{1}{m\_{\rm s}} (k\_{\rm s} \mathbf{x}\_{\rm 3} + c\_{\rm s} (\mathbf{x}\_{\rm 4} - \mathbf{x}\_{\rm 2}) - f\_{\rm mr}) \tag{5}
$$

where *V*max is the maximum voltage to the current driver associated with saturation of the magnetic field in the MR damper, *H*(·) is the Heaviside step function, *f*<sup>d</sup> is the desired control

> � 1, [(*f*<sup>d</sup> <sup>−</sup> *<sup>f</sup>*mr) *<sup>&</sup>gt;* 0 and *<sup>f</sup>*mr *<sup>&</sup>gt;* <sup>0</sup>] or [(*f*<sup>d</sup> <sup>−</sup> *<sup>f</sup>*mr) *<sup>&</sup>lt;* 0 and *<sup>f</sup>*mr *<sup>&</sup>lt;* <sup>0</sup>] −1, [(*f*<sup>d</sup> − *f*mr) *>* 0 and *f*mr *<* 0] or [(*f*<sup>d</sup> − *f*mr) *<* 0 and *f*mr *>* 0]

On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System 141

*V*max, *f*<sup>d</sup> *> f*mr and *f*mr *>* 0 *V*max, *f*<sup>d</sup> *< f*mr and *f*mr *<* 0 0, *f*<sup>d</sup> *> f*mr and *f*mr *<* 0 0, *f*<sup>d</sup> *< f*mr and *f*mr *>* 0

fMR

v = 0

v = 0

v = Vmax

� 1, [ *f*<sup>d</sup> *> f*mr and *f*mr *>* 0] or [ *f*<sup>d</sup> *< f*mr and *f*mr *<* 0] −1, [ *f*<sup>d</sup> *> f*mr and *f*mr *<* 0] or [ *f*<sup>d</sup> *< f*mr and *f*mr *>* 0]

Finally, the full expression in equation (9) can be rewritten as a piecewise function in the

⎧ ⎪⎪⎨

⎪⎪⎩

This algorithm for selecting the command signal is graphically represented in Figure 2. More precisely, the shadowed area in Figure 2 is the area where *f*<sup>d</sup> *> f*mr and *f*mr *>* 0, or *f*<sup>d</sup> *< f*mr and *f*mr *<* 0. Note that in that particular work, they used the voltage as the control signal

fd

force and *f*mr is the measured force of the MR damper.

=

=

=

*V*max

⎧ ⎪⎪⎨

⎪⎪⎩

<sup>2</sup> [sgn [(*f*<sup>d</sup> <sup>−</sup> *<sup>f</sup>*mr)*f*mr] <sup>+</sup> <sup>1</sup>] <sup>=</sup>

because that is the way that current driver can be controlled.

v = 0

v = 0

v = Vmax

**Figure 2.** Graphical representation of the algorithm in equation (8) for selecting the command signal.

sgn [(*f*<sup>d</sup> − *f*mr) *f*mr] =

following way:

The sign part of equation (9) can be transformed in the following way:

� 1, (*f*<sup>d</sup> <sup>−</sup> *<sup>f</sup>*mr) *<sup>f</sup>*mr *<sup>&</sup>gt;* <sup>0</sup> −1, (*f*<sup>d</sup> − *f*mr) *f*mr *<* 0

1, *f*<sup>d</sup> *> f*mr and *f*mr *>* 0 1, *f*<sup>d</sup> *< f*mr and *f*mr *<* 0 −1, *f*<sup>d</sup> *> f*mr and *f*mr *<* 0 −1, *f*<sup>d</sup> *< f*mr and *f*mr *>* 0

where *f*mr is the damping force generated by the semiactive device. In this study, we assume that the semiactive device is magnetorheological (MR) damper. It is modeled according to the following Bouc-Wen model [19]:

$$f\_{\rm mr} = c\_0(v)z\_4 + k\_0(v)z\_3 + \alpha(v)\zeta \tag{6}$$

$$\dot{\zeta} = -\delta |z\_4| \zeta |\zeta|^{n-1} - \beta z\_4 |\zeta|^n + \kappa z\_4 \tag{7}$$

where *ζ* is an evolutionary variable that describes the hysteretic behavior of the damper, *z*<sup>4</sup> is the piston velocity, *z*<sup>3</sup> is the piston deflection and *v* is a voltage input that controls the current that generates the magnetic field; *δ*, *β*, *κ* and *n* are parameters that are chosen so to adjust the hysteretic dynamics of the damper; *c*0(*v*) = *c*0*<sup>a</sup>* + *c*0*bv* represents the voltage-dependent damping, *k*0(*v*) = *k*0*<sup>a</sup>* + *k*0*bv* represents the voltage-dependent stiffness and *α*(*v*) = *α<sup>a</sup>* + *αbv* is a voltage-dependent scaling factor.

### **3. Variable structure controller formulation**

Feedback control radically alters the dynamics of a system: it affects its natural frequencies, its transient response as well as its stability. The MR damper of the quarter-car model considered in this study is voltage-controlled, so the voltage (*v*) is updated by a feedback control loop.

It is well known that the force generated by the MR damper cannot be commanded; only the voltage *v* applied to the current driver for the MR damper can be directly changed. One of the first control approaches involving an MR damper was proposed by [9] and called it clipped optimal control. In this approach, the command voltage takes one of two possible values: zero or the maximum. This is chosen according to the following algorithm:

$$v = V\_{\text{max}} H \{ (f\_{\text{d}} - f\_{\text{mr}}) f\_{\text{mr}} \} \tag{8}$$

$$\dot{\lambda} = \frac{V\_{\text{max}}}{2} \left[ \text{sgn} \left[ (f\_{\text{d}} - f\_{\text{mr}}) f\_{\text{mr}} \right] + 1 \right] \,\tag{9}$$

where *V*max is the maximum voltage to the current driver associated with saturation of the magnetic field in the MR damper, *H*(·) is the Heaviside step function, *f*<sup>d</sup> is the desired control force and *f*mr is the measured force of the MR damper.

The sign part of equation (9) can be transformed in the following way:

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

where *f*mr is the damping force generated by the semiactive device. In this study, we assume that the semiactive device is magnetorheological (MR) damper. It is modeled according to the

where *ζ* is an evolutionary variable that describes the hysteretic behavior of the damper, *z*<sup>4</sup> is the piston velocity, *z*<sup>3</sup> is the piston deflection and *v* is a voltage input that controls the current that generates the magnetic field; *δ*, *β*, *κ* and *n* are parameters that are chosen so to adjust the hysteretic dynamics of the damper; *c*0(*v*) = *c*0*<sup>a</sup>* + *c*0*bv* represents the voltage-dependent damping, *k*0(*v*) = *k*0*<sup>a</sup>* + *k*0*bv* represents the voltage-dependent stiffness and *α*(*v*) = *α<sup>a</sup>* + *αbv*

Feedback control radically alters the dynamics of a system: it affects its natural frequencies, its transient response as well as its stability. The MR damper of the quarter-car model considered in this study is voltage-controlled, so the voltage (*v*) is updated by a feedback control loop. It is well known that the force generated by the MR damper cannot be commanded; only the voltage *v* applied to the current driver for the MR damper can be directly changed. One of the first control approaches involving an MR damper was proposed by [9] and called it clipped optimal control. In this approach, the command voltage takes one of two possible values: zero

*<sup>n</sup>*−<sup>1</sup> <sup>−</sup> *<sup>β</sup>z*4|*ζ*<sup>|</sup>

(*ksx*<sup>3</sup> + *cs*(*x*<sup>4</sup> − *x*2) − *f*mr) (5)

*<sup>n</sup>* + *κz*<sup>4</sup> (7)

*f*mr = *c*0(*v*)*z*<sup>4</sup> + *k*0(*v*)*z*<sup>3</sup> + *α*(*v*)*ζ* (6)

*v* = *V*max*H*{(*f*<sup>d</sup> − *f*mr)*f*mr} (8)

<sup>2</sup> [sgn [(*f*<sup>d</sup> <sup>−</sup> *<sup>f</sup>*mr)*f*mr] <sup>+</sup> <sup>1</sup>] , (9)

**Figure 1.** Quarter car suspension model

following Bouc-Wen model [19]:

is a voltage-dependent scaling factor.

**3. Variable structure controller formulation**

or the maximum. This is chosen according to the following algorithm:

<sup>=</sup> *<sup>V</sup>*max

*<sup>u</sup>* <sup>=</sup> <sup>1</sup> *ms*

˙

*ζ* = −*δ*|*z*4|*ζ*|*ζ*|

sgn [(*f*<sup>d</sup> − *f*mr) *f*mr] = � 1, (*f*<sup>d</sup> <sup>−</sup> *<sup>f</sup>*mr) *<sup>f</sup>*mr *<sup>&</sup>gt;* <sup>0</sup> −1, (*f*<sup>d</sup> − *f*mr) *f*mr *<* 0 = � 1, [(*f*<sup>d</sup> <sup>−</sup> *<sup>f</sup>*mr) *<sup>&</sup>gt;* 0 and *<sup>f</sup>*mr *<sup>&</sup>gt;* <sup>0</sup>] or [(*f*<sup>d</sup> <sup>−</sup> *<sup>f</sup>*mr) *<sup>&</sup>lt;* 0 and *<sup>f</sup>*mr *<sup>&</sup>lt;* <sup>0</sup>] −1, [(*f*<sup>d</sup> − *f*mr) *>* 0 and *f*mr *<* 0] or [(*f*<sup>d</sup> − *f*mr) *<* 0 and *f*mr *>* 0] = � 1, [ *f*<sup>d</sup> *> f*mr and *f*mr *>* 0] or [ *f*<sup>d</sup> *< f*mr and *f*mr *<* 0] −1, [ *f*<sup>d</sup> *> f*mr and *f*mr *<* 0] or [ *f*<sup>d</sup> *< f*mr and *f*mr *>* 0] = ⎧ ⎪⎪⎨ ⎪⎪⎩ 1, *f*<sup>d</sup> *> f*mr and *f*mr *>* 0 1, *f*<sup>d</sup> *< f*mr and *f*mr *<* 0 −1, *f*<sup>d</sup> *> f*mr and *f*mr *<* 0 −1, *f*<sup>d</sup> *< f*mr and *f*mr *>* 0

Finally, the full expression in equation (9) can be rewritten as a piecewise function in the following way:

$$\frac{V\_{\text{max}}}{2} \left[ \text{sgn} \left[ (f\_{\text{d}} - f\_{\text{mr}}) f\_{\text{mr}} \right] + 1 \right] = \begin{cases} V\_{\text{max}}, f\_{\text{d}} > f\_{\text{mr}} \text{ and } f\_{\text{mr}} > 0 \\ V\_{\text{max}}, f\_{\text{d}} < f\_{\text{mr}} \text{ and } f\_{\text{mr}} < 0 \\ 0, \qquad f\_{\text{d}} > f\_{\text{mr}} \text{ and } f\_{\text{mr}} < 0 \\ 0, \qquad f\_{\text{d}} < f\_{\text{mr}} \text{ and } f\_{\text{mr}} > 0 \end{cases}$$

This algorithm for selecting the command signal is graphically represented in Figure 2. More precisely, the shadowed area in Figure 2 is the area where *f*<sup>d</sup> *> f*mr and *f*mr *>* 0, or *f*<sup>d</sup> *< f*mr and *f*mr *<* 0. Note that in that particular work, they used the voltage as the control signal because that is the way that current driver can be controlled.

**Figure 2.** Graphical representation of the algorithm in equation (8) for selecting the command signal.

In this paper we consider the same idea of changing the voltage. This control signal is computed according to the following control strategies, computed as a function of the sprung mass velocity (*x*4), the unsprung mass velocity (*x*2), and the suspension deflection (*x*3):

$$\sigma\_1: \quad v(\mathbf{x\_2}, \mathbf{x\_4}) = \frac{V\_{\text{max}}}{2} \left[ \text{sgn}(\mathbf{x\_4} - \mathbf{x\_2}) + 1 \right] \tag{10}$$

Δ α

(**e***T***Re** <sup>−</sup> *<sup>γ</sup>*2**w***T***w**)*dt* (19)

(20)

II

v = 0

1

Δω

IV

Semi-active control have two essential characteristics. The first is that the these devices offer the adaptability of active control devices without requiring the associated large power sources. The second is that the device cannot inject energy into the system; hence semi-active control devices do not have the potential to destabilize (in the bounded input–bounded output sense) the system [20]. As a consequence, the stability of the closed-loop system is guaranteed.

In this section we present the formulation of a model-based controller. The objective, as explained in the Introduction, is to make a comparison between this model-based controller and the VSC controllers. We will appeal to the backstepping technique that has been developed in previous works for this kind of systems.The objective is to design an adaptive backstepping controller to regulate the suspension deflection with the aid of an MR damper thus providing safety and comfort while on the road. The adaptive backstepping controller will be designed in such a way that, for a given *γ >* 0, the state-dependent error variables *e*<sup>1</sup>

where **<sup>e</sup>** = (*e*1,*e*2)<sup>T</sup> is a vector of controlled signals, **<sup>R</sup>** <sup>=</sup> diag{*r*1,*r*2} is a positive definite

In order to formulate the backstepping controller, the state space model (3) - (4) must be first written in strict feedback form [14]. Therefore, the following coordinate transformation is

> *ρ* + 1 *x*3

*<sup>x</sup>*<sup>2</sup> <sup>+</sup> *<sup>ρ</sup> ρ* + 1 *x*4

*<sup>z</sup>*<sup>1</sup> <sup>=</sup> *<sup>x</sup>*<sup>1</sup> <sup>+</sup> *<sup>ρ</sup>*

*<sup>z</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> *ρ* + 1

*z*<sup>3</sup> = *x*<sup>3</sup> *z*<sup>4</sup> = −*x*<sup>2</sup> + *x*<sup>4</sup>

The system, represented in the new coordinates, is given by:

– 1

v = 0

III

On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System 143

v = 5

I

v = 5

**Figure 3.** Graphical representation of the strategy *σ*<sup>2</sup> for selecting the command signal.

and *e*<sup>2</sup> (to be defined later) accomplish the following H<sup>∞</sup> performance *J*<sup>∞</sup> *<* 0:

 ∞ 0

*J*<sup>∞</sup> =

**4. Backstepping controller formulation**

matrix and **w** is an energy-bounded disturbance.

performed [13]:

$$\sigma\_2: \quad v(\mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) = \frac{V\_{\text{max}}}{2} \left[ \text{sgn} \left( \text{sgn} (\mathbf{x}\_4 - \mathbf{x}\_2) + \mathbf{x}\_3 \right) + 1 \right] \tag{11}$$

$$\sigma\_3: \quad v(\mathbf{x\_2}, \mathbf{x\_4}) = \frac{V\_{\text{max}}}{2} [\text{sgn}(r) + 1], \\ \frac{dr}{dt} = -100r|\mathbf{x\_4} - \mathbf{x\_2}| - 10(\mathbf{x\_4} - \mathbf{x\_2}) \tag{12}$$

Variable structure controllers (VSC) are a very large class of robust controllers [10]. The distinctive feature of VSC is that the structure of the system is intentionally changed according to an assigned law. This can be obtained by switching on or cutting off feedback loops, scheduling gains and so forth. By using VSC, it is possible to take the best out of several different systems (more precisely structures), by switching from one to the other. The control law defines various regions in the phase space and the controller switches between a structure and another at the boundary between two different regions according to the control law.

The three strategies presented in this section can be viewed as variable structure controllers, since the value of the control signal is set to be zero or one, as can be seen in the following transformations:

$$v\_1 \cdot v\_1 \cdot \quad v(\mathbf{x\_2}, \mathbf{x\_4}) = \frac{V\_{\text{max}}}{2} \left[ \text{sgn}(\mathbf{x\_4} - \mathbf{x\_2}) + 1 \right] \tag{13}$$

$$= \begin{cases} 0, & \text{if } \Delta \omega < 0, \\\\ V\_{\text{max}\omega} & \text{if } \Delta \omega \ge 0 \end{cases} \tag{14}$$

$$\sigma\_2 \colon \quad v(\mathbf{x\_2}, \mathbf{x\_3}, \mathbf{x\_4}) = \frac{V\_{\text{max}}}{2} \left[ \text{sgn} \left( \text{sgn} (\mathbf{x\_4} - \mathbf{x\_2}) + \mathbf{x\_3} \right) + 1 \right] \tag{15}$$

$$= \begin{cases} 0, & \text{if } \Delta \omega < 0, \, \mathbf{x\_3} < 1 \text{ (region IV)},\\ 0, & \text{if } \Delta \omega > 0, \, \mathbf{x\_3} < -1 \text{ (region II)},\\ V\_{\text{max}}, \, \text{if } \Delta \omega < 0, \, \mathbf{x\_3} \ge 1 \text{ (region II)},\\ V\_{\text{max}}, \, \text{if } \Delta \omega \ge 0, \, \mathbf{x\_3} \ge -1 \text{ (region III)} \end{cases} \tag{16}$$

$$v\_3: \quad v(\mathbf{x\_2}, \mathbf{x\_4}) = \frac{V\_{\text{max}}}{2} [\text{sgn}(r) + 1] \tag{17}$$

$$= \begin{cases} 0, & \text{if } r < 0, \\ V\_{\text{max}\nu} & \text{if } r \ge 0 \end{cases}, \frac{dr}{dt} = -100r|\mathbf{x}\_4 - \mathbf{x}\_2| - 10(\mathbf{x}\_4 - \mathbf{x}\_2) \tag{18}$$

where Δ*ω* = *x*<sup>4</sup> − *x*2. In Figure 3 we hace depicted the graphical representation of the strategy *σ*<sup>2</sup> for selecting the command signal.

**Figure 3.** Graphical representation of the strategy *σ*<sup>2</sup> for selecting the command signal.

Semi-active control have two essential characteristics. The first is that the these devices offer the adaptability of active control devices without requiring the associated large power sources. The second is that the device cannot inject energy into the system; hence semi-active control devices do not have the potential to destabilize (in the bounded input–bounded output sense) the system [20]. As a consequence, the stability of the closed-loop system is guaranteed.

### **4. Backstepping controller formulation**

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

In this paper we consider the same idea of changing the voltage. This control signal is computed according to the following control strategies, computed as a function of the sprung mass velocity (*x*4), the unsprung mass velocity (*x*2), and the suspension deflection (*x*3):

<sup>2</sup> [sgn(*r*) + <sup>1</sup>], *dr*

Variable structure controllers (VSC) are a very large class of robust controllers [10]. The distinctive feature of VSC is that the structure of the system is intentionally changed according to an assigned law. This can be obtained by switching on or cutting off feedback loops, scheduling gains and so forth. By using VSC, it is possible to take the best out of several different systems (more precisely structures), by switching from one to the other. The control law defines various regions in the phase space and the controller switches between a structure and another at the boundary between two different regions according to the control law.

The three strategies presented in this section can be viewed as variable structure controllers, since the value of the control signal is set to be zero or one, as can be seen in the following

> 0, if Δ*ω <* 0, *x*<sup>3</sup> *<* 1 (region IV), 0, if Δ*ω >* 0, *x*<sup>3</sup> *<* −1 (region I), *V*max, if Δ*ω <* 0, *x*<sup>3</sup> ≥ 1 (region II), *V*max, if Δ*ω* ≥ 0, *x*<sup>3</sup> ≥ −1 (region III)

� 0, if Δ*ω <* 0,

*V*max, if Δ*ω* ≥ 0

� 0, if *r <* 0, *<sup>V</sup>*max, if *<sup>r</sup>* <sup>≥</sup> <sup>0</sup> , *dr*

where Δ*ω* = *x*<sup>4</sup> − *x*2. In Figure 3 we hace depicted the graphical representation of the strategy

<sup>2</sup> [sgn(*x*<sup>4</sup> <sup>−</sup> *<sup>x</sup>*2) + <sup>1</sup>] (10)

<sup>2</sup> [sgn (sgn(*x*<sup>4</sup> <sup>−</sup> *<sup>x</sup>*2) + *<sup>x</sup>*3) <sup>+</sup> <sup>1</sup>] (11)

<sup>2</sup> [sgn(*x*<sup>4</sup> <sup>−</sup> *<sup>x</sup>*2) + <sup>1</sup>] (13)

<sup>2</sup> [sgn (sgn(*x*<sup>4</sup> <sup>−</sup> *<sup>x</sup>*2) + *<sup>x</sup>*3) <sup>+</sup> <sup>1</sup>] (15)

<sup>2</sup> [sgn(*r*) + <sup>1</sup>] (17)

*dt* <sup>=</sup> <sup>−</sup>100*r*|*x*<sup>4</sup> <sup>−</sup> *<sup>x</sup>*2| − <sup>10</sup>(*x*<sup>4</sup> <sup>−</sup> *<sup>x</sup>*2) (18)

*dt* <sup>=</sup> <sup>−</sup>100*r*|*x*<sup>4</sup> <sup>−</sup> *<sup>x</sup>*2| − <sup>10</sup>(*x*<sup>4</sup> <sup>−</sup> *<sup>x</sup>*2) (12)

(14)

(16)

*<sup>σ</sup>*<sup>1</sup> : *<sup>v</sup>*(*x*2, *<sup>x</sup>*4) = *<sup>V</sup>*max

*<sup>σ</sup>*<sup>3</sup> : *<sup>v</sup>*(*x*2, *<sup>x</sup>*4) = *<sup>V</sup>*max

*<sup>σ</sup>*<sup>1</sup> : *<sup>v</sup>*(*x*2, *<sup>x</sup>*4) = *<sup>V</sup>*max

*<sup>σ</sup>*<sup>2</sup> : *<sup>v</sup>*(*x*2, *<sup>x</sup>*3, *<sup>x</sup>*4) = *<sup>V</sup>*max

*<sup>σ</sup>*<sup>3</sup> : *<sup>v</sup>*(*x*2, *<sup>x</sup>*4) = *<sup>V</sup>*max

*σ*<sup>2</sup> for selecting the command signal.

=

=

=

⎧ ⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

*<sup>σ</sup>*<sup>2</sup> : *<sup>v</sup>*(*x*2, *<sup>x</sup>*3, *<sup>x</sup>*4) = *<sup>V</sup>*max

transformations:

In this section we present the formulation of a model-based controller. The objective, as explained in the Introduction, is to make a comparison between this model-based controller and the VSC controllers. We will appeal to the backstepping technique that has been developed in previous works for this kind of systems.The objective is to design an adaptive backstepping controller to regulate the suspension deflection with the aid of an MR damper thus providing safety and comfort while on the road. The adaptive backstepping controller will be designed in such a way that, for a given *γ >* 0, the state-dependent error variables *e*<sup>1</sup> and *e*<sup>2</sup> (to be defined later) accomplish the following H<sup>∞</sup> performance *J*<sup>∞</sup> *<* 0:

$$J\_{\infty} = \int\_{0}^{\infty} (\mathbf{e}^{T} \mathbf{R} \mathbf{e} - \gamma^{2} \mathbf{w}^{T} \mathbf{w}) dt \tag{19}$$

where **<sup>e</sup>** = (*e*1,*e*2)<sup>T</sup> is a vector of controlled signals, **<sup>R</sup>** <sup>=</sup> diag{*r*1,*r*2} is a positive definite matrix and **w** is an energy-bounded disturbance.

In order to formulate the backstepping controller, the state space model (3) - (4) must be first written in strict feedback form [14]. Therefore, the following coordinate transformation is performed [13]:

$$\begin{aligned} z\_1 &= x\_1 + \frac{\rho}{\rho + 1} x\_3\\ z\_2 &= \frac{1}{\rho + 1} x\_2 + \frac{\rho}{\rho + 1} x\_4\\ z\_3 &= x\_3\\ z\_4 &= -x\_2 + x\_4 \end{aligned} \tag{20}$$

The system, represented in the new coordinates, is given by:

	- Tyre subsystem:

$$\begin{aligned} \dot{z}\_1 &= z\_2 - d\\ \dot{z}\_2 &= -k\_l[m\_\iota(\rho+1)]^{-1}z\_1 + \rho k\_l[m\_\iota(\rho+1)^2]^{-1}z\_3 \end{aligned} \tag{21}$$

In order to begin with the adaptive backstepping design, we firstly define the following error

*<sup>V</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> 2 *e*2

<sup>1</sup> = *e*2*z*<sup>4</sup> = *e*1(*e*<sup>2</sup> − *r*1*e*1) = *e*1*e*<sup>2</sup> − *r*1*e*

On the other hand, the derivatives of the errors of the uncertain parameter estimations are

Thus, by using (35) - (39) and the fact that *ak* = *a*˜*<sup>k</sup>* + *a*ˆ*<sup>k</sup>* and *ac* = *a*˜*<sup>c</sup>* + *a*ˆ*c*, the derivative of *V*

*<sup>k</sup> <sup>a</sup>*˜*<sup>k</sup>* ˙

*<sup>a</sup>*ˆ*k*) <sup>−</sup> *<sup>a</sup>*ˆ*kz*3*e*<sup>2</sup> <sup>−</sup> *<sup>a</sup>*˜*<sup>c</sup>* (*z*4*e*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*−<sup>1</sup> *<sup>c</sup>* ˙

˙ *<sup>a</sup>*˜*<sup>k</sup>* <sup>=</sup> <sup>−</sup> ˙

˙ *<sup>a</sup>*˜*<sup>c</sup>* <sup>=</sup> <sup>−</sup> ˙

1 2 *e* 2 <sup>2</sup> + 1 2*rk a*˜ 2 *<sup>k</sup>* + 1 2*rc a*˜ 2

<sup>1</sup> <sup>+</sup> *<sup>e</sup>*2*di* <sup>−</sup> *akz*3*e*<sup>2</sup> <sup>−</sup> *acz*4*e*<sup>2</sup> <sup>+</sup> *<sup>a</sup> <sup>f</sup> <sup>f</sup>*mr*e*<sup>2</sup> <sup>−</sup> *<sup>r</sup>*1*z*4*e*<sup>2</sup> <sup>−</sup> *<sup>r</sup>*−<sup>1</sup>

*k* ˙

*V*˙

Equation (30) can be stabilized with the following virtual control input:

where *r*<sup>1</sup> *>* 0. Now define a second error variable and its derivative:

*V*˙

Now, an augmented Lyapunov function candidate is chosen:

*V* = *V*<sup>1</sup> +

*e*<sup>1</sup> = *z*<sup>3</sup> (29) *e*˙1 = *z*˙3 = *z*<sup>4</sup> (30)

On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System 145

<sup>1</sup> = *e*1*e*˙1 = *e*1*z*<sup>4</sup> (32)

*z*4*<sup>d</sup>* = −*r*1*e*<sup>1</sup> (33) *z*˙4*<sup>d</sup>* = −*r*1*e*˙1 = −*r*1*z*<sup>4</sup> (34)

*e*<sup>2</sup> = *z*<sup>4</sup> − *z*4*<sup>d</sup>* (35)

*e*˙2 = *z*˙4 − *z*˙4*<sup>d</sup>* (36)

*a*ˆ*<sup>k</sup>* (38)

*a*ˆ*c* (39)

<sup>1</sup> (37)

*<sup>c</sup>* (40)

2

*<sup>k</sup> <sup>a</sup>*˜*<sup>k</sup>* ˙

*<sup>a</sup>*ˆ*<sup>k</sup>* <sup>−</sup> *<sup>r</sup>*−<sup>1</sup>

*<sup>a</sup>*ˆ*<sup>k</sup>* <sup>−</sup> (*a*˜*<sup>k</sup>* <sup>+</sup> *<sup>a</sup>*ˆ*k*)*z*3*e*<sup>2</sup> <sup>−</sup> (*a*˜*<sup>c</sup>* <sup>+</sup> *<sup>a</sup>*ˆ*<sup>c</sup>* )*z*4*e*<sup>2</sup> <sup>−</sup> *<sup>r</sup>*−<sup>1</sup>

*<sup>c</sup> a*˜*ca*˙ *<sup>c</sup>*

*a*ˆ*c*) − *a*ˆ*cz*4*e*<sup>2</sup> + *a <sup>f</sup>* +

*<sup>c</sup> a*˜*<sup>c</sup>* ˙ *a*ˆ*c*

(41)

<sup>1</sup> (31)

variable and its derivative:

whose first-order derivative is:

Therefore,

given by:

yields:

*V*˙ =*e*1*e*˙1 + *e*2*e*˙2 + *r*−<sup>1</sup>

2

2

2

*f*mr*e*<sup>2</sup> − *r*1*z*4*e*<sup>2</sup>

=*e*1*e*<sup>2</sup> − *r*1*e*

=*e*1*e*<sup>2</sup> − *r*1*e*

=*e*1*e*<sup>2</sup> − *r*1*e*

*<sup>k</sup> <sup>a</sup>*˜*<sup>k</sup>* ˙

*a*˜*<sup>k</sup>* + *r*−<sup>1</sup> *<sup>c</sup> a*˜*<sup>c</sup>* ˙ *a*˜*c*

<sup>1</sup> <sup>+</sup> *<sup>e</sup>*2*di* <sup>−</sup> *<sup>a</sup>*˜*k*(*z*3*e*<sup>3</sup> <sup>+</sup> *<sup>r</sup>*−<sup>1</sup>

<sup>1</sup> <sup>+</sup> *<sup>e</sup>*2*di* <sup>+</sup> *<sup>a</sup> <sup>f</sup> <sup>f</sup>*mr*e*<sup>2</sup> <sup>−</sup> *<sup>r</sup>*1*z*4*e*<sup>2</sup> <sup>−</sup> *<sup>r</sup>*−<sup>1</sup>

Now, the following Lyapunov function candidate is chosen:

• Suspension subsystem:

$$\begin{aligned} \dot{z}\_3 &= z\_4\\ \dot{z}\_4 &= k\_l m\_u^{-1} z\_1 - k\_l \rho [m\_u(\rho+1)]^{-1} z\_3 - (\rho+1)u \end{aligned} \tag{22}$$

Substitution of the expression for *u* (5) into (22) yields:

$$\begin{aligned} \dot{z}\_3 &= z\_4\\ \dot{z}\_4 &= k\_l m\_u^{-1} z\_1 - k\_t \rho [m\_u(\rho + 1)]^{-1} z\_3 - \\ &\quad (\rho + 1) m\_s^{-1} [k\_s x\_3 + c\_s (x\_4 - x\_2) - f\_{\rm{mr}}] \\ &= - \left[ k\_l m\_s \rho (\rho + 1)^{-1} + (\rho + 1) k\_s m\_u \right] (m\_u m\_s)^{-1} z\_3 + \\ &\quad k\_l m\_u^{-1} z\_1 - (\rho + 1) m\_s^{-1} c\_s z\_4 + (\rho + 1) m\_s^{-1} f\_{\rm{mr}} \\ &= d\_l - a\_k z\_3 - a\_c z\_4 + a\_f f\_{\rm{mr}} \end{aligned} \tag{23}$$

where *ak* = [*ktmsρ*(*ρ* + 1)−<sup>1</sup> + (*ρ* + 1)*ksmu*](*mums*)−1, *ac* = (*ρ* + 1)*m*−<sup>1</sup> *<sup>s</sup> cs* and *a <sup>f</sup>* = (*ρ* + 1)*m*−<sup>1</sup> *<sup>s</sup>* ; *di* = *ktm*−<sup>1</sup> *<sup>s</sup> z*<sup>1</sup> reflects the fact that the disturbance enters to the suspension subsystem through the tyre subsystem.

Assume that *ak* and *ac* in (23) are uncertain constant parameters whose estimated values are *a*ˆ*<sup>k</sup>* and *a*ˆ*c*, respectively. Thus, the errors between the estimates and the actual values are given by:

$$
\mathfrak{a}\_k = a\_k - \mathfrak{a}\_k \tag{24}
$$

$$
\mathfrak{A}\_{\mathfrak{c}} = \mathfrak{a}\_{\mathfrak{c}} - \mathfrak{A}\_{\mathfrak{c}} \tag{25}
$$

Let *ad* = *kt*[*mu*(*ρ* + 1)]−1, *an* = *ρkt*[*mu*(*ρ* + 1)2] <sup>−</sup><sup>1</sup> and *am* = *ktm*−<sup>1</sup> *<sup>u</sup>* . From (21) - (22), it can be shown that the transfer functions from *d*(*t*) and *f*mr(*t*) to *z*1(*t*) are:

$$\frac{Z\_1(s)}{D(s)} = \frac{-s(s^2 + a\_c s + a\_k)}{s^4 + a\_c s^3 + (a\_d + a\_k)s^2 + a\_d a\_c s + a\_d a\_k - a\_m a\_n} \tag{26}$$

$$\frac{Z\_1(s)}{F\_{\rm{mr}}(s)} = \frac{a\_{\rm{n}}a\_f}{s^4 + a\_c s^3 + (a\_d + a\_k)s^2 + a\_d a\_c s + a\_d a\_k - a\_m a\_n} \tag{27}$$

If the poles of the transfer functions (26) and (27) are in the left side of the *s* plane, then we can guarantee the bounded input - bounded output (BIBO) stability of *Z*1(*s*) for any bounded input *D*(*s*) and *F*mr(*s*). Thus, the disturbance input *di*(*t*) in (23) is also bounded. This boundedness condition will be necessary later in the controller stability condition.

Finally, since *di*(*t*) is the only disturbance input to the suspension subsystem, the vector **w** of the H∞ performance objective as given in (19) becomes:

$$J\_{\infty} = \int\_{0}^{\infty} (\mathbf{e}^{T} \mathbf{R} \mathbf{e} - \gamma^{2} d\_{i}^{2}) dt \tag{28}$$

In order to begin with the adaptive backstepping design, we firstly define the following error variable and its derivative:

$$\mathbf{z\_1} = \mathbf{z\_3} \tag{29}$$

$$
\dot{z}\_1 = \dot{z}\_3 = z\_4 \tag{30}
$$

Now, the following Lyapunov function candidate is chosen:

$$V\_1 = \frac{1}{2}e\_1^2\tag{31}$$

whose first-order derivative is:

$$
\dot{V}\_1 = e\_1 \dot{e}\_1 = e\_1 z\_4 \tag{32}
$$

Equation (30) can be stabilized with the following virtual control input:

$$z\_{4d} = -r\_1 e\_1 \tag{33}$$

$$
\dot{z}\_{4d} = -r\_1 \dot{e}\_1 = -r\_1 z\_4 \tag{34}
$$

where *r*<sup>1</sup> *>* 0. Now define a second error variable and its derivative:

$$e\_2 = z\_4 - z\_{4d} \tag{35}$$

$$
\dot{\varepsilon}\_2 = \dot{z}\_4 - \dot{z}\_{4d} \tag{36}
$$

Therefore,

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

*<sup>z</sup>*˙2 <sup>=</sup> <sup>−</sup>*kt*[*mu*(*<sup>ρ</sup>* <sup>+</sup> <sup>1</sup>)]−1*z*<sup>1</sup> <sup>+</sup> *<sup>ρ</sup>kt*[*mu*(*<sup>ρ</sup>* <sup>+</sup> <sup>1</sup>)2]

*<sup>u</sup> <sup>z</sup>*<sup>1</sup> <sup>−</sup> *ktρ*[*mu*(*<sup>ρ</sup>* <sup>+</sup> <sup>1</sup>)]−1*z*3<sup>−</sup>

*<sup>u</sup> <sup>z</sup>*<sup>1</sup> <sup>−</sup> (*<sup>ρ</sup>* <sup>+</sup> <sup>1</sup>)*m*−<sup>1</sup>

=*di* − *akz*<sup>3</sup> − *acz*<sup>4</sup> + *a <sup>f</sup> f*mr

*<sup>s</sup>* [*ksx*<sup>3</sup> + *cs*(*x*<sup>4</sup> − *x*2) − *f*mr] <sup>=</sup> <sup>−</sup> [*ktmsρ*(*<sup>ρ</sup>* <sup>+</sup> <sup>1</sup>)−<sup>1</sup> + (*<sup>ρ</sup>* <sup>+</sup> <sup>1</sup>)*ksmu*](*mums*)−1*z*3<sup>+</sup>

where *ak* = [*ktmsρ*(*ρ* + 1)−<sup>1</sup> + (*ρ* + 1)*ksmu*](*mums*)−1, *ac* = (*ρ* + 1)*m*−<sup>1</sup> *<sup>s</sup> cs* and *a <sup>f</sup>* = (*ρ* + 1)*m*−<sup>1</sup> *<sup>s</sup>* ; *di* = *ktm*−<sup>1</sup> *<sup>s</sup> z*<sup>1</sup> reflects the fact that the disturbance enters to the suspension subsystem

Assume that *ak* and *ac* in (23) are uncertain constant parameters whose estimated values are *a*ˆ*<sup>k</sup>* and *a*ˆ*c*, respectively. Thus, the errors between the estimates and the actual values are given

*<sup>s</sup>*<sup>4</sup> + *acs*<sup>3</sup> + (*ad* + *ak*)*s*<sup>2</sup> + *adacs* + *adak* − *aman*

*<sup>s</sup>*<sup>4</sup> + *acs*<sup>3</sup> + (*ad* + *ak*)*s*<sup>2</sup> + *adacs* + *adak* − *aman*

If the poles of the transfer functions (26) and (27) are in the left side of the *s* plane, then we can guarantee the bounded input - bounded output (BIBO) stability of *Z*1(*s*) for any bounded input *D*(*s*) and *F*mr(*s*). Thus, the disturbance input *di*(*t*) in (23) is also bounded. This boundedness condition will be necessary later in the controller stability condition.

Finally, since *di*(*t*) is the only disturbance input to the suspension subsystem, the vector **w** of

(**e***T***Re** <sup>−</sup> *<sup>γ</sup>*2*d*<sup>2</sup>

*<sup>s</sup> csz*<sup>4</sup> + (*<sup>ρ</sup>* + <sup>1</sup>)*m*−<sup>1</sup>

*<sup>s</sup> f*mr

*a*˜*<sup>k</sup>* = *ak* − *a*ˆ*<sup>k</sup>* (24) *a*˜*<sup>c</sup>* = *ac* − *a*ˆ*<sup>c</sup>* (25)

<sup>−</sup><sup>1</sup> and *am* = *ktm*−<sup>1</sup> *<sup>u</sup>* . From (21) - (22), it can be

*<sup>i</sup>* )*dt* (28)

<sup>−</sup>1*z*<sup>3</sup>

*<sup>z</sup>*˙4 <sup>=</sup> *ktm*−<sup>1</sup> *<sup>u</sup> <sup>z</sup>*<sup>1</sup> <sup>−</sup> *ktρ*[*mu*(*<sup>ρ</sup>* <sup>+</sup> <sup>1</sup>)]−1*z*<sup>3</sup> <sup>−</sup> (*<sup>ρ</sup>* <sup>+</sup> <sup>1</sup>)*<sup>u</sup>* (22)

(21)

(23)

(26)

(27)

*z*˙1 = *z*<sup>2</sup> − *d*

*z*˙3 = *z*<sup>4</sup>

Substitution of the expression for *u* (5) into (22) yields:

(*ρ* + 1)*m*−<sup>1</sup>

*ktm*−<sup>1</sup>

*z*˙3 =*z*<sup>4</sup> *z*˙4 =*ktm*−<sup>1</sup>

Let *ad* = *kt*[*mu*(*ρ* + 1)]−1, *an* = *ρkt*[*mu*(*ρ* + 1)2]

*Z*1(*s*)

*Z*1(*s*)

the H∞ performance objective as given in (19) becomes:

shown that the transfer functions from *d*(*t*) and *f*mr(*t*) to *z*1(*t*) are:

*<sup>D</sup>*(*s*) <sup>=</sup> <sup>−</sup>*s*(*s*<sup>2</sup> <sup>+</sup> *acs* <sup>+</sup> *ak*)

*<sup>F</sup>*mr(*s*) <sup>=</sup> *ana <sup>f</sup>*

*J*<sup>∞</sup> =

 ∞ 0

• Tyre subsystem:

• Suspension subsystem:

through the tyre subsystem.

by:

$$
\dot{V}\_1 = e\_2 z\_4 = e\_1 (e\_2 - r\_1 e\_1) = e\_1 e\_2 - r\_1 e\_1^2 \tag{37}
$$

On the other hand, the derivatives of the errors of the uncertain parameter estimations are given by:

$$
\dot{\vec{a}}\_k = -\dot{\vec{a}}\_k \tag{38}
$$

$$
\dot{\vec{a}}\_{\mathcal{L}} = -\dot{\vec{a}}\_{\mathcal{L}} \tag{39}
$$

Now, an augmented Lyapunov function candidate is chosen:

$$V = V\_1 + \frac{1}{2}c\_2^2 + \frac{1}{2r\_k}\vec{a}\_k^2 + \frac{1}{2r\_c}\vec{a}\_c^2\tag{40}$$

Thus, by using (35) - (39) and the fact that *ak* = *a*˜*<sup>k</sup>* + *a*ˆ*<sup>k</sup>* and *ac* = *a*˜*<sup>c</sup>* + *a*ˆ*c*, the derivative of *V* yields:

$$\begin{aligned} \dot{V} &= e\_1 \dot{e}\_1 + e\_2 \dot{e}\_2 + r\_k^{-1} \ddot{a}\_k \dot{\bar{a}}\_k + r\_c^{-1} \ddot{a}\_c \dot{\bar{a}}\_c \\ &= e\_1 e\_2 - r\_1 e\_1^2 + e\_2 d\_i - a\_k z\_3 e\_2 - a\_c z\_4 e\_2 + a\_f f\_{\text{mr}} e\_2 - r\_1 z\_4 e\_2 - r\_k^{-1} \ddot{a}\_k \dot{\bar{a}}\_k - r\_c^{-1} \ddot{a}\_c \dot{\bar{a}}\_c \\ &= e\_1 e\_2 - r\_1 e\_1^2 + e\_2 d\_i + a\_f f\_{\text{mr}} e\_2 - r\_1 z\_4 e\_2 - r\_k^{-1} \ddot{a}\_k \dot{\bar{a}}\_k - (\ddot{a}\_k + \ddot{a}\_k) z\_3 e\_2 - (\ddot{a}\_c + \ddot{a}\_c) z\_4 e\_2 - r\_c^{-1} \ddot{a}\_c \dot{\bar{a}}\_c \\ &= e\_1 e\_2 - r\_1 e\_1^2 + e\_2 d\_i - \ddot{a}\_k (z\_3 e\_3 + r\_k^{-1} \ddot{\bar{a}}\_k) - \ddot{a}\_k z\_3 e\_2 - \ddot{a}\_c (z\_4 e\_2 + r\_c^{-1} \ddot{\bar{a}}\_c) - \ddot{a}\_c z\_4 e\_2 + a\_f + \\ &\qquad f\_{\text{mr}} e\_2 - r\_1 z\_4 e\_2 \end{aligned} \tag{41}$$

### 10 Will-be-set-by-IN-TECH 146 Advances on Analysis and Control of Vibrations – Theory and Applications On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System <sup>11</sup>

Now consider the following adaptation laws:

$$z\_3 e\_1 + r\_k^{-1} \dot{a}\_k = 0 \tag{42}$$

The control force given by (45) can be used to drive an actively controlled damper. However, the fact that semiactive devices cannot inject energy into a system, makes necessary the modification of this control law in order to implement it with a semiactive damper; that is, semiactive dampers cannot apply force to the system, only absorb it. There are different ways to perform this [2, 27]. In this work, we will calculate the MR damper voltage making use of

*<sup>v</sup>* <sup>=</sup> <sup>−</sup>*e*<sup>1</sup> <sup>−</sup> *<sup>a</sup>*ˆ*zz*<sup>3</sup> <sup>+</sup> *<sup>a</sup>*ˆ*cz*<sup>4</sup> <sup>+</sup> *<sup>r</sup>*1*z*<sup>4</sup> <sup>−</sup> *<sup>r</sup>*2*e*<sup>2</sup> <sup>−</sup> *<sup>e</sup>*2(2*γ*)−<sup>2</sup> <sup>+</sup> *<sup>a</sup> <sup>f</sup>*(*c*0*az*<sup>4</sup> <sup>+</sup> *<sup>k</sup>*0*az*<sup>3</sup> <sup>+</sup> *<sup>α</sup>aζ*)

The same process followed to obtain the control law (45) can be used to demonstrate that the control law (4) does stabilize the system. Begin by replacing (6) into (44) in order to obtain:

<sup>1</sup> + *e*2*di* + *e*2[*e*<sup>1</sup> − *a*ˆ*kz*<sup>3</sup> − *a*ˆ*cz*<sup>4</sup> + *a <sup>f</sup>*(*c*0*az*<sup>4</sup> + *k*0*az*<sup>3</sup> + *αaζ*)+

*<sup>a</sup> <sup>f</sup>*(*c*0*bz*<sup>4</sup> <sup>+</sup> *<sup>k</sup>*0*bz*<sup>3</sup> <sup>+</sup> *<sup>α</sup>bζ*)*<sup>v</sup>* <sup>−</sup> *<sup>r</sup>*1*z*4] (51)

<sup>−</sup>*<sup>a</sup> <sup>f</sup> <sup>c</sup>*0*bx*<sup>2</sup> <sup>+</sup> *<sup>a</sup> <sup>f</sup> <sup>k</sup>*0*bx*<sup>3</sup> <sup>+</sup> *<sup>a</sup> <sup>f</sup> <sup>c</sup>*0*bx*<sup>4</sup> <sup>+</sup> *<sup>a</sup> <sup>f</sup> <sup>α</sup>b<sup>ζ</sup>* <sup>+</sup>

*<sup>a</sup> <sup>f</sup>*(*c*0*bz*<sup>4</sup> <sup>+</sup> *<sup>k</sup>*0*bz*<sup>3</sup> <sup>+</sup> *<sup>α</sup>bζ*) (50)

On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System 147

<sup>−</sup><sup>1</sup> <sup>−</sup> *<sup>a</sup>*ˆ*<sup>z</sup>* <sup>−</sup> *<sup>r</sup>*1*r*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*1(2*γ*)−<sup>2</sup> <sup>+</sup> *<sup>a</sup> <sup>f</sup> <sup>k</sup>*0*<sup>a</sup>*

<sup>1</sup> <sup>−</sup> *<sup>r</sup>*2*e*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2*d*<sup>2</sup>

 *x*3

*<sup>i</sup>* and, as

(52)

its mathematical model. Thus, the following control law is proposed:

Thus, by replacing the control law of (4) into (51) we also get *<sup>V</sup>*˙ ≤ −*r*1*e*<sup>2</sup>

Finally, we can write the control law in terms of the state variables as follows:

 *x*<sup>2</sup> + 

*x*<sup>4</sup> + *a <sup>f</sup> αaζ*

In this section we will analyze the performance results obtained form simulations performed in Matlab/SImulink. The numerical values of the model that we used in this study. Thus: *α<sup>a</sup>* = 332.7 N/m, *α<sup>b</sup>* = 1862.5 N·V/m, *c*0*<sup>a</sup>* = 7544.1 N·s/m, *c*0*<sup>b</sup>* = 7127.3 N·s·V/m, *<sup>k</sup>*0*<sup>a</sup>* <sup>=</sup> 11375.7 N/m, *<sup>k</sup>*0*<sup>b</sup>* <sup>=</sup> 14435.0 N·V/m, *<sup>δ</sup>* <sup>=</sup> 4209.8 m−2, *<sup>κ</sup>* <sup>=</sup> 10246 and *<sup>n</sup>* <sup>=</sup> 2. This is a scaled version of the MR damper found in [5]. The parameter values of the suspension system are [13]: *ms*=11739 kg, *mu*=300 kg, *ks*=252000 N/m, *cs*=10000 N·s/m and *kt*=300000 N/m. In order to facilitate the analysis, we will quantify the performance results by means of the indices shown in Table 1. Indices *J*<sup>1</sup> - *J*<sup>3</sup> show the ratio between the peak response of the controlled suspension system (displacement, velocity and acceleration) and that of the uncontrolled system. Indices *J*<sup>4</sup> - *J*<sup>6</sup> are the normalized ITSE (integral of the time squared error) signals that indicate how much the displacement, velocity and acceleration are attenuated compared to the uncontrolled case. Index *J*<sup>7</sup> is the relative maximum control effort with respect to the weight of the suspension system. Small indices indicate good control performance. Two scenarios are considered: an uneven road, simulated by random vibrations

provided that *a <sup>f</sup>*(*c*0*bz*<sup>4</sup> + *k*0*bz*<sup>3</sup> + *αbζ*) �= 0; otherwise, *v* = 0.

previously stated, the stability of the system is guaranteed.

*<sup>V</sup>*˙ <sup>=</sup> <sup>−</sup> *<sup>r</sup>*1*<sup>e</sup>*

*v* = 

2

<sup>−</sup>*a*ˆ*<sup>c</sup>* <sup>−</sup> *<sup>r</sup>*<sup>1</sup> <sup>+</sup> *<sup>r</sup>*<sup>2</sup> + (2*γ*)−<sup>2</sup> <sup>+</sup> *<sup>a</sup> <sup>f</sup> <sup>c</sup>*0*<sup>a</sup>*

*<sup>z</sup>*ˆ*<sup>c</sup>* <sup>+</sup> *<sup>r</sup>*<sup>1</sup> <sup>−</sup> *<sup>r</sup>*<sup>2</sup> <sup>−</sup> (2*γ*)−<sup>2</sup> <sup>+</sup> *<sup>a</sup> <sup>f</sup> <sup>c</sup>*0*<sup>a</sup>*

and the presence of a bump on the road.

**5. Numerical simulations**

−*a <sup>f</sup> c*0*bx*<sup>2</sup> + *a <sup>f</sup> k*0*bx*<sup>3</sup> + *a <sup>f</sup> c*0*bx*4 + *a <sup>f</sup> αbζ*

$$z\_4 e\_2 + r\_c^{-1} \dot{\mathfrak{a}}\_c = 0 \tag{43}$$

Substitution of (42) and (43) into (41) yields:

$$\dot{V} = -r\_1 \mathbf{e}\_1^2 + e\_2 d\_i + e\_2 (e\_1 - \mathbf{\hat{a}}\_k z\_3 - \mathbf{\hat{a}}\_c z\_4 + a\_f f\_{\text{mr}} - r\_1 z\_4) \tag{44}$$

By choosing the following control law:

$$f\_{\rm mr} = -\frac{e\_1 - \hbar\_k z\_3 - \hbar\_c z\_4 - r\_1 z\_4 + r\_2 e\_2 + e\_2 (2\gamma)^{-2}}{a\_f} \tag{45}$$

with *γ >* 0 and *r*<sup>2</sup> *>* 0, we get:

$$\begin{aligned} \dot{V} &= -r\_1 e\_1^2 + e\_2 d\_i - r\_2 e\_2^2 - e\_2^2 (2\gamma)^{-2} \\ &= -r\_1 e\_1^2 + e\_2 d\_i - r\_2 e\_2^2 - e\_2^2 (2\gamma)^{-2} + \gamma^2 d\_i^2 - \gamma^2 d\_i^2 \\ &= -r\_1 e\_1^2 - r\_2 e\_2^2 + \gamma^2 d\_i^2 - (\gamma d\_i - e\_2 (2\gamma)^{-2})^2 \\ \dot{V} &\le -r\_1 e\_1^2 - r\_2 e\_2^2 + \gamma^2 d\_i^2 \end{aligned} \tag{46}$$

The objective of guaranteeing global boundedness of trajectories is equivalently expressed as rendering *V*˙ negative outside a compact region. As stated earlier, the disturbance input *di* is bounded as long as the poles of the transfer functions (26) and (27) are in the left side of the *s* plane. When this is the case, the boundedness of the input disturbance *di* guarantees the existence of a small compact region *<sup>D</sup>* <sup>⊂</sup> **<sup>R</sup>**<sup>2</sup> (depending on *<sup>γ</sup>* and *di* itself) such that *<sup>V</sup>*˙ is negative outside this set. More precisely, when *r*1*e*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>r</sup>*2*e*<sup>2</sup> <sup>2</sup> *<sup>&</sup>lt; <sup>γ</sup>*2*d*<sup>2</sup> *<sup>i</sup>* , *<sup>V</sup>*˙ is positive and then the error variables are increasing values. Finally, when the expression *r*1*e*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>r</sup>*2*e*<sup>2</sup> <sup>2</sup> is greater than *γ*2*d*<sup>2</sup> *<sup>i</sup>* , *<sup>V</sup>*˙ is then negative. This implies that all the closed-loop trajectories have to remain bounded, as we wanted to show. Now, under zero initial conditions, from 46 we can write:

$$\int\_0^\infty \dot{V} \, dt \le -\int\_0^\infty r\_1 e\_1^2 \, dt - \int\_0^\infty r\_2 e\_2^2 \, dt + \int\_0^\infty \gamma^2 d\_i^2 \, dt \tag{47}$$

or, equivanlently,

$$|V|\_{t=\infty} - V|\_{t=0} \le -\int\_0^\infty \mathbf{e}^T \mathbf{R} \mathbf{e} \, dt + \gamma^2 \int\_0^\infty d\_l^2 \, dt\tag{48}$$

Then, it can be shown that

$$J\_{\infty} = \int\_{0}^{\infty} \left( \mathbf{e}^{T} \mathbf{R} \mathbf{e} - \gamma^{2} d\_{i}^{2} \right) dt \le -V|\_{t=\infty} \le 0 \tag{49}$$

Thus, the adaptive backstepping controller satisfies both the H∞ performance and the asymptotic stability of the system.

The control force given by (45) can be used to drive an actively controlled damper. However, the fact that semiactive devices cannot inject energy into a system, makes necessary the modification of this control law in order to implement it with a semiactive damper; that is, semiactive dampers cannot apply force to the system, only absorb it. There are different ways to perform this [2, 27]. In this work, we will calculate the MR damper voltage making use of its mathematical model. Thus, the following control law is proposed:

$$v = \frac{-e\_1 - \hbar z\_2 z\_3 + \hbar c z\_4 + r\_1 z\_4 - r\_2 e\_2 - e\_2 (2\gamma)^{-2} + a\_f (c\_{0a} z\_4 + k\_{0a} z\_3 + a\_d \zeta)}{a\_f (c\_{0b} z\_4 + k\_{0b} z\_3 + a\_b \zeta)} \tag{50}$$

provided that *a <sup>f</sup>*(*c*0*bz*<sup>4</sup> + *k*0*bz*<sup>3</sup> + *αbζ*) �= 0; otherwise, *v* = 0.

The same process followed to obtain the control law (45) can be used to demonstrate that the control law (4) does stabilize the system. Begin by replacing (6) into (44) in order to obtain:

$$\begin{split} \dot{V} &= -r\_1 e\_1^2 + e\_2 d\_i + e\_2 [e\_1 - \mathfrak{a}\_k z\_3 - \mathfrak{a}\_\iota z\_4 + a\_f (c\_{0a} z\_4 + k\_{0a} z\_3 + a\_d \zeta) + \\ a\_f (c\_{0b} z\_4 + k\_{0b} z\_3 + a\_b \zeta) v - r\_1 z\_4] \end{split} \tag{51}$$

Thus, by replacing the control law of (4) into (51) we also get *<sup>V</sup>*˙ ≤ −*r*1*e*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>r</sup>*2*e*<sup>2</sup> <sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2*d*<sup>2</sup> *<sup>i</sup>* and, as previously stated, the stability of the system is guaranteed.

Finally, we can write the control law in terms of the state variables as follows:

$$\begin{aligned} v &= \frac{\left(-\mathfrak{d}\_c - r\_1 + r\_2 + (2\gamma)^{-2} + a\_f c\_{0a}\right) \mathfrak{x}\_2 + \left(-1 - \mathfrak{d}\_z - r\_1 r\_2 + r\_1 (2\gamma)^{-2} + a\_f k\_{0a}\right) \mathfrak{x}\_3}{-a\_f c\_{0b} \mathfrak{x}\_2 + a\_f k\_{0b} \mathfrak{x}\_3 + a\_f c\_{0b} \mathfrak{x}\_4 + a\_f a\_b \mathfrak{f}} + \\ &\frac{\left(\mathfrak{z}\_c + r\_1 - r\_2 - (2\gamma)^{-2} + a\_f c\_{0a}\right) \mathfrak{x}\_4 + a\_f a\_d \mathfrak{f}}{-a\_f c\_{0b} \mathfrak{x}\_2 + a\_f k\_{0b} \mathfrak{x}\_3 + a\_f c\_{0b} \mathfrak{x}\_4 + a\_f a\_b \mathfrak{f}}\end{aligned} \tag{52}$$

### **5. Numerical simulations**

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

*a*ˆ*<sup>k</sup>* = 0 (42)

*a*ˆ*<sup>c</sup>* = 0 (43)

(45)

(46)

<sup>1</sup> + *e*2*di* + *e*2(*e*<sup>1</sup> − *a*ˆ*kz*<sup>3</sup> − *a*ˆ*cz*<sup>4</sup> + *a <sup>f</sup> f*mr − *r*1*z*4) (44)

*z*3*e*<sup>1</sup> + *r*−<sup>1</sup> *k* ˙

*z*4*e*<sup>2</sup> + *r*−<sup>1</sup> *<sup>c</sup>* ˙

*<sup>f</sup>*mr <sup>=</sup> <sup>−</sup> *<sup>e</sup>*<sup>1</sup> <sup>−</sup> *<sup>a</sup>*ˆ*kz*<sup>3</sup> <sup>−</sup> *<sup>a</sup>*ˆ*cz*<sup>4</sup> <sup>−</sup> *<sup>r</sup>*1*z*<sup>4</sup> <sup>+</sup> *<sup>r</sup>*2*e*<sup>2</sup> <sup>+</sup> *<sup>e</sup>*2(2*γ*)−<sup>2</sup>

2 <sup>2</sup> − *e* 2 2(2*γ*)−<sup>2</sup>

2 <sup>2</sup> − *e* 2

The objective of guaranteeing global boundedness of trajectories is equivalently expressed as rendering *V*˙ negative outside a compact region. As stated earlier, the disturbance input *di* is bounded as long as the poles of the transfer functions (26) and (27) are in the left side of the *s* plane. When this is the case, the boundedness of the input disturbance *di* guarantees the existence of a small compact region *<sup>D</sup>* <sup>⊂</sup> **<sup>R</sup>**<sup>2</sup> (depending on *<sup>γ</sup>* and *di* itself) such that *<sup>V</sup>*˙

bounded, as we wanted to show. Now, under zero initial conditions, from 46 we can write:

 ∞ 0

Thus, the adaptive backstepping controller satisfies both the H∞ performance and the

(**e***T***Re** <sup>−</sup> *<sup>γ</sup>*2*d*<sup>2</sup>

*a f*

<sup>2</sup>(2*γ*)−<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2*d*<sup>2</sup>

*<sup>i</sup>* <sup>−</sup> (*γdi* <sup>−</sup> *<sup>e</sup>*2(2*γ*)−2)<sup>2</sup>

<sup>1</sup> <sup>+</sup> *<sup>r</sup>*2*e*<sup>2</sup>

*<sup>i</sup>* , *<sup>V</sup>*˙ is then negative. This implies that all the closed-loop trajectories have to remain

 ∞ 0 *r*2*e* 2 <sup>2</sup> *dt* +

**e***T***Re** *dt* + *γ*<sup>2</sup>

<sup>2</sup> *<sup>&</sup>lt; <sup>γ</sup>*2*d*<sup>2</sup>

 ∞ 0

 ∞ 0 *d*2

*γ*2*d*<sup>2</sup>

*<sup>i</sup>* ) *dt* ≤ −*V*|*t*=<sup>∞</sup> ≤ 0 (49)

*<sup>i</sup>* , *<sup>V</sup>*˙ is positive and then

*<sup>i</sup> dt* (47)

*<sup>i</sup> dt* (48)

<sup>2</sup> is greater

<sup>1</sup> <sup>+</sup> *<sup>r</sup>*2*e*<sup>2</sup>

*<sup>i</sup>* <sup>−</sup> *<sup>γ</sup>*2*d*<sup>2</sup> *i*

Now consider the following adaptation laws:

Substitution of (42) and (43) into (41) yields:

By choosing the following control law:

with *γ >* 0 and *r*<sup>2</sup> *>* 0, we get:

than *γ*2*d*<sup>2</sup>

or, equivanlently,

Then, it can be shown that

asymptotic stability of the system.

*<sup>V</sup>*˙ <sup>=</sup> <sup>−</sup>*r*1*<sup>e</sup>*

*<sup>V</sup>*˙ <sup>=</sup> <sup>−</sup> *<sup>r</sup>*1*<sup>e</sup>*

= − *r*1*e* 2

= − *r*1*e* 2 <sup>1</sup> − *r*2*e* 2 <sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2*d*<sup>2</sup>

*<sup>V</sup>*˙ ≤ − *<sup>r</sup>*1*<sup>e</sup>*

is negative outside this set. More precisely, when *r*1*e*<sup>2</sup>

*V dt* ˙ ≤ −

*J*<sup>∞</sup> =

*V*|*t*=<sup>∞</sup> − *V*|*t*=<sup>0</sup> ≤ −

 ∞ 0

 ∞ 0

2

2 <sup>1</sup> − *r*2*e* 2 <sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2*d*<sup>2</sup> *i*

<sup>1</sup> + *e*2*di* − *r*2*e*

<sup>1</sup> + *e*2*di* − *r*2*e*

the error variables are increasing values. Finally, when the expression *r*1*e*<sup>2</sup>

 ∞ 0 *r*1*e* 2 <sup>1</sup> *dt* −

2

In this section we will analyze the performance results obtained form simulations performed in Matlab/SImulink. The numerical values of the model that we used in this study. Thus: *α<sup>a</sup>* = 332.7 N/m, *α<sup>b</sup>* = 1862.5 N·V/m, *c*0*<sup>a</sup>* = 7544.1 N·s/m, *c*0*<sup>b</sup>* = 7127.3 N·s·V/m, *<sup>k</sup>*0*<sup>a</sup>* <sup>=</sup> 11375.7 N/m, *<sup>k</sup>*0*<sup>b</sup>* <sup>=</sup> 14435.0 N·V/m, *<sup>δ</sup>* <sup>=</sup> 4209.8 m−2, *<sup>κ</sup>* <sup>=</sup> 10246 and *<sup>n</sup>* <sup>=</sup> 2. This is a scaled version of the MR damper found in [5]. The parameter values of the suspension system are [13]: *ms*=11739 kg, *mu*=300 kg, *ks*=252000 N/m, *cs*=10000 N·s/m and *kt*=300000 N/m. In order to facilitate the analysis, we will quantify the performance results by means of the indices shown in Table 1. Indices *J*<sup>1</sup> - *J*<sup>3</sup> show the ratio between the peak response of the controlled suspension system (displacement, velocity and acceleration) and that of the uncontrolled system. Indices *J*<sup>4</sup> - *J*<sup>6</sup> are the normalized ITSE (integral of the time squared error) signals that indicate how much the displacement, velocity and acceleration are attenuated compared to the uncontrolled case. Index *J*<sup>7</sup> is the relative maximum control effort with respect to the weight of the suspension system. Small indices indicate good control performance. Two scenarios are considered: an uneven road, simulated by random vibrations and the presence of a bump on the road.

We assume that the car has laser sensors that allow us to read the position of the sprung and unsprung masses. Since the velocities are needed for control implementation, these are obtained by first low-pass filtering the displacement readings and then applying a filter of the form *<sup>s</sup>* (*λ<sup>s</sup>* <sup>+</sup> <sup>1</sup>)*<sup>q</sup>* that approximates the derivative of the signal. In this filter, *<sup>λ</sup>* is a sufficiently small constant that can be obtained from the ratio between the two-norm of the second derivative of the signal and the noise amplitude; *q* is the order of the filter which should be at lest equal to 2. Choosing parameters this way, allows for minimizing the error between the real and the estimated signal derivatives [4].

**<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>10</sup> −2**

On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System 149

**<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>10</sup> −0.5**

**<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>10</sup> −2**

**<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>10</sup> −0.4**

**<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>10</sup> −2**

**<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>10</sup> −10**

time [s]

**Figure 4.** Uneven road disturbance and tyre subsystem response.

**Figure 5.** Uneven road disturbance and tyre subsystem response.

time [s]

**uncontrolled sigma1 sigma2 sigma3**

**uncontrolled sigma1 sigma2 sigma3**

**0**

**0**

**0**

**−0.2 0 0.2** susp. defl. [m]

**0**

**0**

spr. mass accel. [m/s2

]

**10**

spr. mass vel. [m/s]

**2**

unsp. mass vel. [m/s]

**2**

tyre defl. [m]

**0.5**

rand dist. [m/s]

**2**


**Table 1.** Performance indices.

In the first scenario, the unevenness of the road was simulated by random vibration, as shown in Figure 4. This figure also compares the performance of the three *σ* controllers. What we can see for this figure, is that the three VSC controllers perform in a similar way and satisfactorily control the deflection of the tyre subsystem. In Figure 5, we see the performance of the same controllers at regulating the suspension deflection. Once again, the three controllers accomplish the objective in a similar way. This visual observations can be confirmed by analyzing the performance indices of Table 2. In Figures 6 and 7, we can see a comparison of the *σ*<sup>3</sup> controller and the backstepping controller. A notable superiority of the VSC controller is observed over the backstepping controller. It can be due to the fact that this kind of controllers are more sensitive to the fast-changing dynamics of a signal, in this case, the velocity, which can make it react faster. The performance indices of Table 2 also show that it is harder for the backstepping controller to keep the peak acceleration, velocity and displacement under acceptable limits, despite its control effort is much higher than that of the VSC controllers.

**Figure 4.** Uneven road disturbance and tyre subsystem response.

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

We assume that the car has laser sensors that allow us to read the position of the sprung and unsprung masses. Since the velocities are needed for control implementation, these are obtained by first low-pass filtering the displacement readings and then applying a filter of the

**Index Definition**

(*λ<sup>s</sup>* <sup>+</sup> <sup>1</sup>)*<sup>q</sup>* that approximates the derivative of the signal. In this filter, *<sup>λ</sup>* is a sufficiently small constant that can be obtained from the ratio between the two-norm of the second derivative of the signal and the noise amplitude; *q* is the order of the filter which should be at lest equal to 2. Choosing parameters this way, allows for minimizing the error between

Norm. peak suspension deflection.

Norm. peak sprung mass velocity.

Norm. peak sprung mass acceleration.

<sup>3</sup>*unc*(*t*) *dt* Norm. suspension deflection ITSE.

<sup>4</sup>*unc*(*t*) *dt* Norm. sprung mass velocity ITSE.

<sup>4</sup>*unc*(*t*) *dt* Norm. sprung mass acceleration ITSE.

In the first scenario, the unevenness of the road was simulated by random vibration, as shown in Figure 4. This figure also compares the performance of the three *σ* controllers. What we can see for this figure, is that the three VSC controllers perform in a similar way and satisfactorily control the deflection of the tyre subsystem. In Figure 5, we see the performance of the same controllers at regulating the suspension deflection. Once again, the three controllers accomplish the objective in a similar way. This visual observations can be confirmed by analyzing the performance indices of Table 2. In Figures 6 and 7, we can see a comparison of the *σ*<sup>3</sup> controller and the backstepping controller. A notable superiority of the VSC controller is observed over the backstepping controller. It can be due to the fact that this kind of controllers are more sensitive to the fast-changing dynamics of a signal, in this case, the velocity, which can make it react faster. The performance indices of Table 2 also show that it is harder for the backstepping controller to keep the peak acceleration, velocity and displacement under acceptable limits, despite its control effort is much higher than that of the VSC controllers.

Maximum control effort.

form *<sup>s</sup>*

the real and the estimated signal derivatives [4].

*<sup>J</sup>*<sup>1</sup> <sup>=</sup> *max*|*x*3(*t*)|*cont max*|*x*3(*t*)|*unc*

*<sup>J</sup>*<sup>2</sup> <sup>=</sup> *max*|*x*4(*t*)|*cont max*|*x*4(*t*)|*unc*

*<sup>J</sup>*<sup>3</sup> <sup>=</sup> *max*|*x*˙4(*t*)|*cont max*|*x*˙4(*t*)|*unc*

<sup>3</sup>*cont*(*t*) *dt*

<sup>4</sup>*cont*(*t*) *dt*

 *T* <sup>0</sup> *tx*<sup>2</sup>

 *T* <sup>0</sup> *tx*<sup>2</sup>

 *<sup>T</sup>* <sup>0</sup> *tx*<sup>2</sup>

 *<sup>T</sup>* <sup>0</sup> *tx*<sup>2</sup>

 *T* <sup>0</sup> *tx*˙ 2 <sup>4</sup>*cont*(*t*) *dt*

 *T* <sup>0</sup> *tx*˙ 2

*<sup>J</sup>*<sup>7</sup> <sup>=</sup> *max*<sup>|</sup> *fmr*(*t*)<sup>|</sup> *ws*

*J*<sup>4</sup> =

*J*<sup>5</sup> =

*J*<sup>6</sup> =

**Table 1.** Performance indices.

**Figure 5.** Uneven road disturbance and tyre subsystem response.

**Index** *σ*<sup>1</sup> *σ*<sup>2</sup> *σ*<sup>3</sup> **Backstepping** *J*<sup>1</sup> 0.1288 0.1280 0.0993 0.5144

On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System 151

*J*<sup>2</sup> 0.1485 0.1488 0.2157 0.9800

*J*<sup>3</sup> 0.2205 0.2803 0.2803 1.2586

*J*<sup>4</sup> 0.0059 0.0058 0.0053 0.2317

*J*<sup>5</sup> 0.0090 0.0089 0.0181 0.5852

*J*<sup>6</sup> 0.0189 0.0187 0.0310 0.9615

*J*<sup>7</sup> 0.1268 0.1279 0.1471 0.4538

In the second scenario a bump on the road is simulated as seen in Figure 8. In this case, the VSC controllers have a similar performance and it happened in the previous scenario. The performance indices of Table 3 confirm this fact. In comparison, the *σ*<sup>3</sup> controller seems to perform slightly better, specially at reusing the peak response of the suspicion and tyre deflections as can be seen in Figure 10 and 11 where a comparison between then *σ*<sup>3</sup> and backstepping controllers is illustrated. These results are in the line than those of the first scenario. It is worth noting the fact that the VSC controllers perform better with less control

**<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>10</sup> −4**

**<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>10</sup> −0.4**

**<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>10</sup> −4**

time [s]

**uncontrolled sigma1 sigma2 sigma3**

**Table 2.** Performance indices of the random unevenness disturbance case.

**Figure 8.** Bump on the road disturbance and tyre subsystem response.

effort.

**−0.2 0 0.2** tyre defl. [m]

unsp. mass vel. [m/s]

bump dist. [m/s]

**Figure 6.** Uneven road disturbance and tyre subsystem response.

**Figure 7.** Uneven road disturbance and car subsystem response.


**Table 2.** Performance indices of the random unevenness disturbance case.

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

**<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>10</sup> −2**

**<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>10</sup> −0.4**

**<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>10</sup> −2**

**<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>10</sup> −0.4**

**<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>10</sup> −2**

**<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>10</sup> −10**

time [s]

**Figure 6.** Uneven road disturbance and tyre subsystem response.

**Figure 7.** Uneven road disturbance and car subsystem response.

time [s]

**uncontrolled backstepping sigma3**

**uncontrolled backstepping sigma3**

**0**

**−0.2 0 0.2** tyre defl. [m]

**0**

**−0.2 0 0.2** susp. defl. [m]

**0**

**0**

spr. mass accel. [m/s2

]

**10**

spr. mass vel. [m/s]

**2**

unsp. mass vel. [m/s]

**2**

rand dist. [m/s]

**2**

In the second scenario a bump on the road is simulated as seen in Figure 8. In this case, the VSC controllers have a similar performance and it happened in the previous scenario. The performance indices of Table 3 confirm this fact. In comparison, the *σ*<sup>3</sup> controller seems to perform slightly better, specially at reusing the peak response of the suspicion and tyre deflections as can be seen in Figure 10 and 11 where a comparison between then *σ*<sup>3</sup> and backstepping controllers is illustrated. These results are in the line than those of the first scenario. It is worth noting the fact that the VSC controllers perform better with less control effort.

**Figure 8.** Bump on the road disturbance and tyre subsystem response.

**<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>10</sup> −4**

On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System 153

**<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>10</sup> −0.4**

**<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>10</sup> −4**

**<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>10</sup> −0.2**

**<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>10</sup> −0.5**

**<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>10</sup> −5**

time [s]

**Figure 10.** Bump on the road disturbance and tyre subsystem response.

**Figure 11.** Bump on the road disturbance and car subsystem response.

time [s]

**uncontrolled backstepping sigma3**

**uncontrolled backstepping sigma3**

**−0.2 0 0.2** tyre defl. [m]

> > **0**

**0 0.5 1**

> **0 5 10**

spr. mass accel. [m/s2

]

susp. defl. [m]

spr. mass vel. [m/s]

**0.2**

unsp. mass vel. [m/s]

bump dist. [m/s]

**Figure 9.** Bump on the road disturbance and tyre subsystem response.


**Table 3.** Performance indices of the road bump disturbance case.

**Figure 10.** Bump on the road disturbance and tyre subsystem response.

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

**<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>10</sup> −0.2**

**<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>10</sup> −0.5**

**<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>10</sup> −5**

**Index** *σ*<sup>1</sup> *σ*<sup>2</sup> *σ*<sup>3</sup> **Backstepping** *J*<sup>1</sup> 0.8317 0.8325 0.4584 0.4271 *J*<sup>2</sup> 1.1507 1.1505 1.3430 1.3892 *J*<sup>3</sup> 1.1157 1.2623 1.2623 1.3007 *J*<sup>4</sup> 0.1605 0.1625 0.2241 0.0703 *J*<sup>5</sup> 0.1827 0.1797 0.2681 0.4702 *J*<sup>6</sup> 0.4168 0.4113 0.5884 1.0308 *J*<sup>7</sup> 0.3613 0.3614 0.4100 0.4431

**Figure 9.** Bump on the road disturbance and tyre subsystem response.

**Table 3.** Performance indices of the road bump disturbance case.

time [s]

**uncontrolled sigma1 sigma2 sigma3**

**0**

**0 0.5 1**

> **0 5 10**

spr. mass accel. [m/s2

]

susp. defl. [m]

spr. mass vel. [m/s]

**0.2**

**Figure 11.** Bump on the road disturbance and car subsystem response.

### **6. Conclusions**

In this chapter we presented the problem of the vibration control in vehicles. One model-based and three variable structure controllers were analyzed and compared in order to study their performance during typical road disturbances. The performance of the controller were also analyzed for the particular situation in which the suspension system is made up of a magnetorheological damper, which is well-known to be a nonlinear device. All of the controllers performed satisfactorily at regulating the suspension deflection while keeping the acceleration, velocity and displacement variables within acceptable limits. One important result obtained in this work was that despite the simplicity of these controllers, they performed significantly better than the model-based controller. It is to be noted that further studies -theoretical and experimental- should be performed in order to get a better insight of the performance of such controllers and the possibilities of being used in real systems.

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### **Acknowledgements**

M. Zapateiro has been supported by the 'Juan de la Cierva' fellowship from the Government of Spain. This study has also been partially supported by Secretaría de Estado de Investigación, Desarrollo e Innovación (formerly, Ministry of Science and Innovation) through DPI2011-27567-C02-02, DPI2011-28033-C03-01 and DPI 2011-27567-C02-01.

## **Author details**

Mauricio Zapateiro and Francesc Pozo *Department of Applied Mathematics III, Universitat Politècnica de Catalunya, Barcelona, Spain*

Ningsu Luo *Institute of Informatics and Applications, University of Girona, Girona, Spain*

### **7. References**


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

In this chapter we presented the problem of the vibration control in vehicles. One model-based and three variable structure controllers were analyzed and compared in order to study their performance during typical road disturbances. The performance of the controller were also analyzed for the particular situation in which the suspension system is made up of a magnetorheological damper, which is well-known to be a nonlinear device. All of the controllers performed satisfactorily at regulating the suspension deflection while keeping the acceleration, velocity and displacement variables within acceptable limits. One important result obtained in this work was that despite the simplicity of these controllers, they performed significantly better than the model-based controller. It is to be noted that further studies -theoretical and experimental- should be performed in order to get a better insight of the performance of such controllers and the possibilities of being used in real systems.

M. Zapateiro has been supported by the 'Juan de la Cierva' fellowship from the Government of Spain. This study has also been partially supported by Secretaría de Estado de Investigación, Desarrollo e Innovación (formerly, Ministry of Science and Innovation) through

*Department of Applied Mathematics III, Universitat Politècnica de Catalunya, Barcelona, Spain*

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**Acknowledgements**

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Ningsu Luo

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© 2012 Karimi, 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 Karimi, 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.

**A Computational Approach to Vibration** 

**Control of Vehicle Engine-Body Systems** 

In recent years, the noise and vibration of cars have become increasingly important [20, 23, 29, 30, 35]. A major comfort aspect is the transmission of engine-induced vibrations through powertrain mounts into the chassis (see Figure 1). Engine and powertrain mounts are usually designed according to criteria that incorporate a trade-off between the isolation of the engine from the chassis and the restriction of engine movements. The engine mount is an efficient passive means to isolate the car chassis structure from the engine vibration. However, the passive means for isolation is efficient only in the high frequency range. However the vibration disturbance generated by the engine occurs mainly in the low frequency range [8, 19, 23, 30]. These vibrations are result of the fuel explosion in the cylinder and the rotation of the different parts of the engine (see Figure 2). In order to attenuate the low frequency disturbances of the engine vibration while keeping the space

A variety of control techniques, such as Proportional-Integral-Derivative (PID) or Lead-Lag

control have been used in active vibration systems [1, 3, 4, 10, 11, 15, 24, 26, 31, 32, 34, 35]. The main characteristic of feedforward control is that information about the disturbance source is available and is usually realised with the Filtered-X Least-Mean-Squares (Fx-LMS) algorithms. However, the disturbance source is assumed to be unknown in feedback control, then different strategies of feedback control for vibration attenuation of unknown disturbance exist ranging from classical methods to a more advanced methods. Recently, the performance result obtained by *H* feedback controller with the result obtained by feedforward controller using

Fx-LMS algorithms for vehicle engine-body vibration system was compared in [30, 35].

On the other hand, wavelet theory is a relatively new and an emerging area in mathematical research [2]. It has been applied in a wide range of engineering disciplines such as signal

Hamid Reza Karimi

http://dx.doi.org/10.5772/50295

**1. Introduction** 

Additional information is available at the end of the chapter

and price constant, active vibration means are necessary.

compensation, Linear Quadratic Gaussian (LQG), *H*<sup>2</sup> , *H* ,

## **A Computational Approach to Vibration Control of Vehicle Engine-Body Systems**

Hamid Reza Karimi

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

[23] E.R. Wang, X.Q. Ma, S. Rakheja, C.Y. Su. Semi-active control of vehicle vibration with MR-dampers, *Proc. of the 42nd IEEE Conference on Decision and Control*, Maui, Hawaii

[24] T. Wang, S. Tong, Y. Li. Robust adaptive fuzzy control for nonlinear system with dynamic uncertainties based on backstepping, *International Journal of Innovative*

[25] G.Z. Yao, F.F: Yap, G. Chen, W.H. Li, S.H. Yeo. MR damper and its application for semi-active control of vehicle suspension system, *Mechatronics*, vol. 12(7), pp. 963-973. [26] M. Zapateiro, N. Luo, H.R. Karimi, J. Vehí. Vibration control of a class of semiactive suspension system using neural network and backstepping techniques, *Mechanical*

[27] M. Zapateiro, H.R. Karimi, N. Luo, B.F. Spencer, Jr. Real-time hybrid testing of semiactive control strategies for vibration reduction in a structure with MR damper,

[28] M. Zapateiro, F. Pozo, H.R. Karimi, N. Luo, Semiactive control methodologies for suspension control with magnetorheological dampers, *IEEE/ASME Transactions on*

[29] Z. Zexu, W. Weidong, L. Litao, H. Xiangyu, C. Hutao, L. Shuang, and C. Pingyuan. Robust sliding mode guidance and control for soft landing on small bodies, *Journal of*

*Computing, Information and Control*, Vol. 5, no. 9, pp. 2675-2688, 2009.

*Structural Control and Health Monitoring*, Vol. 17(4), pp.427-451, 2010.

*Systems and Signal Processing*, Vol. 23, 1946-1953, 2009.

*The Franklin Institute*, vol. 349, no. 2, pp. 493–509, 2012.

*Mechatronics*, vol. 17(2), pp. 370-380, 2012.

USA, 2003.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50295

## **1. Introduction**

In recent years, the noise and vibration of cars have become increasingly important [20, 23, 29, 30, 35]. A major comfort aspect is the transmission of engine-induced vibrations through powertrain mounts into the chassis (see Figure 1). Engine and powertrain mounts are usually designed according to criteria that incorporate a trade-off between the isolation of the engine from the chassis and the restriction of engine movements. The engine mount is an efficient passive means to isolate the car chassis structure from the engine vibration. However, the passive means for isolation is efficient only in the high frequency range. However the vibration disturbance generated by the engine occurs mainly in the low frequency range [8, 19, 23, 30]. These vibrations are result of the fuel explosion in the cylinder and the rotation of the different parts of the engine (see Figure 2). In order to attenuate the low frequency disturbances of the engine vibration while keeping the space and price constant, active vibration means are necessary.

A variety of control techniques, such as Proportional-Integral-Derivative (PID) or Lead-Lag compensation, Linear Quadratic Gaussian (LQG), *H*<sup>2</sup> , *H* , -synthesis and feedforward control have been used in active vibration systems [1, 3, 4, 10, 11, 15, 24, 26, 31, 32, 34, 35]. The main characteristic of feedforward control is that information about the disturbance source is available and is usually realised with the Filtered-X Least-Mean-Squares (Fx-LMS) algorithms. However, the disturbance source is assumed to be unknown in feedback control, then different strategies of feedback control for vibration attenuation of unknown disturbance exist ranging from classical methods to a more advanced methods. Recently, the performance result obtained by *H* feedback controller with the result obtained by feedforward controller using Fx-LMS algorithms for vehicle engine-body vibration system was compared in [30, 35].

On the other hand, wavelet theory is a relatively new and an emerging area in mathematical research [2]. It has been applied in a wide range of engineering disciplines such as signal

© 2012 Karimi, 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 Karimi, 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.

processing, pattern recognition and computational graphics. Recently, some of the attempts are made in solving surface integral equations, improving the finite difference time domain method, solving linear differential equations and nonlinear partial differential equations and modelling nonlinear semiconductor devices [5, 6, 7, 13, 16, 17, 18, 21, 27].

A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 159

Orthogonal functions like Haar wavelets (HWs) [13, 16], Walsh functions [7], block pulse functions [27], Laguerre polynomials [14], Legendre polynomials [5], Chebyshev functions [12] and Fourier series [28], often used to represent an arbitrary time functions, have received considerable attention in dealing with various problems of dynamic systems. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations for the solution of problems described by differential equations, such as analysis of linear time-invariant, time-varying systems, model reduction, optimal control and system identification. Thus, the solution, identification and optimisation procedure are either greatly reduced or much simplified accordingly. The available sets of orthogonal functions can be divided into three classes such as piecewise constant basis functions (PCBFs) like HWs, Walsh functions and block pulse functions; orthogonal polynomials like Laguerre, Legendre and Chebyshev as well as sine-cosine functions in

In the present paper, we, for the first time, introduce a computational solution to the finitetime robust optimal control problem of the vehicle engine-body vibration system based on HWs. To this aim, mathematical model of the engine-body vibration structure is presented such the actuators and sensors used to investigate the robust optimal control are selected to be collocated. Moreover, the properties of HWs, Haar wavelet integral operational matrix and Haar wavelet product operational matrix are given and are utilized to provide a systematic computational framework to find the approximated robust optimal trajectory and finite-time *H* control of the vehicle engine-body vibration system with respect to a *H* performance by solving only the linear algebraic equations instead of solving the differential equations. One of the main advantages is solving linear algebraic equations instead of solving nonlinear differential Riccati equation to optimize the control problem of the vehicle engine-body vibration system. We demonstrate the applicability of the technique

The rest of this paper is organized as fallows. Section 2 introduces properties of the HWs. Mathematical model of the engine-body vibration structure is stated in Section 3. Algebraic solution of the engine-body system is given in Section 4 and Haar wavelet-based optimal trajectories and robust optimal control are presented in Sections 5 and 6, respectively. Simulation results of the robust optimal control of the vehicle engine-body vibration system

The notations used throughout the paper are fairly standard. The matrices *rI* , 0*r* and 0*r s* are the identity matrix with dimension *r r* and the zero matrices with dimensions *r r* and *r s* , respectively. The symbol and *tr A*( ) denote Kronecker product and trace of the matrix *A* , respectively. Also, operator *vec X*( ) denotes the vector obtained by putting

.

2 2 0 () () () *<sup>T</sup> x t x t x t dt* 

*x t*( ) denotes the 2 *L* norm of *x t*( ) ;

are shown in Section 7 and finally the conclusion is discussed.

matrix *X* into one column. Finally, given a signal *x t*( ) , 2

Fourier series [21].

by the simulation results.

i.e.,

engine mount point

.

**Figure 1.** Front axis of AUDI A 8 from [22, 30] (Werkbild Audi AG).

**Figure 2.** Chassis excited by the engine vibration.

Orthogonal functions like Haar wavelets (HWs) [13, 16], Walsh functions [7], block pulse functions [27], Laguerre polynomials [14], Legendre polynomials [5], Chebyshev functions [12] and Fourier series [28], often used to represent an arbitrary time functions, have received considerable attention in dealing with various problems of dynamic systems. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations for the solution of problems described by differential equations, such as analysis of linear time-invariant, time-varying systems, model reduction, optimal control and system identification. Thus, the solution, identification and optimisation procedure are either greatly reduced or much simplified accordingly. The available sets of orthogonal functions can be divided into three classes such as piecewise constant basis functions (PCBFs) like HWs, Walsh functions and block pulse functions; orthogonal polynomials like Laguerre, Legendre and Chebyshev as well as sine-cosine functions in Fourier series [21].

158 Advances on Analysis and Control of Vibrations – Theory and Applications

**Figure 1.** Front axis of AUDI A 8 from [22, 30] (Werkbild Audi AG).

chassis subframe

X

Y

Crank

Pitch

Z: Bounce

O

**Figure 2.** Chassis excited by the engine vibration.

processing, pattern recognition and computational graphics. Recently, some of the attempts are made in solving surface integral equations, improving the finite difference time domain method, solving linear differential equations and nonlinear partial differential equations

> . engine mount point

> > Engine Block

Chassis

Piston

Engine Mount

and modelling nonlinear semiconductor devices [5, 6, 7, 13, 16, 17, 18, 21, 27].

In the present paper, we, for the first time, introduce a computational solution to the finitetime robust optimal control problem of the vehicle engine-body vibration system based on HWs. To this aim, mathematical model of the engine-body vibration structure is presented such the actuators and sensors used to investigate the robust optimal control are selected to be collocated. Moreover, the properties of HWs, Haar wavelet integral operational matrix and Haar wavelet product operational matrix are given and are utilized to provide a systematic computational framework to find the approximated robust optimal trajectory and finite-time *H* control of the vehicle engine-body vibration system with respect to a *H* performance by solving only the linear algebraic equations instead of solving the differential equations. One of the main advantages is solving linear algebraic equations instead of solving nonlinear differential Riccati equation to optimize the control problem of the vehicle engine-body vibration system. We demonstrate the applicability of the technique by the simulation results.

The rest of this paper is organized as fallows. Section 2 introduces properties of the HWs. Mathematical model of the engine-body vibration structure is stated in Section 3. Algebraic solution of the engine-body system is given in Section 4 and Haar wavelet-based optimal trajectories and robust optimal control are presented in Sections 5 and 6, respectively. Simulation results of the robust optimal control of the vehicle engine-body vibration system are shown in Section 7 and finally the conclusion is discussed.

The notations used throughout the paper are fairly standard. The matrices *rI* , 0*r* and 0*r s* are the identity matrix with dimension *r r* and the zero matrices with dimensions *r r* and *r s* , respectively. The symbol and *tr A*( ) denote Kronecker product and trace of the matrix *A* , respectively. Also, operator *vec X*( ) denotes the vector obtained by putting matrix *X* into one column. Finally, given a signal *x t*( ) , 2 *x t*( ) denotes the 2 *L* norm of *x t*( ) ; i.e.,

$$\left\|\mathbf{x}(t)\right\|\_{2}^{2} = \int\_{0}^{\infty} \mathbf{x}(t)^{T} \mathbf{x}(t) \, dt \,\,\,.$$

### **2. Properties of Haar Wavelets**

Properties of HWs, which will be used in the next sections, are introduced in this section.

### **2.1. Haar Wavelets (HWs)**

The oldest and most basic of the wavelet systems is named Haar wavelet that is a group of square waves with magnitude of . <sup>1</sup> . in the interval 0,1 [6]. In other words, the HWs are defined on the interval 0,1 as

$$\begin{aligned} \left. \nu\_0(t) = 1 \right. & \qquad t \in \left[0, 1\right) . \\ \left. \nu\_1(t) = \begin{cases} 1 & \text{for } \quad t \in \left[0, \frac{1}{2}\right) . \\ -1 & \text{for } \quad t \in \left[\frac{1}{2}, 1\right) . \end{cases} \end{aligned} \tag{1}$$

A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 161

*m m y t yt yt a H* . (5)

*m mm t dt P t* (6)

*T*

*T*

10 0 0 0

00 0 0 1 1 1 1 , 1 10 0 0 0 0 0

00 1 10 0 0 0 00 0 0 1 10 0 00 0 0 0 0 1 1

at the resolution *<sup>m</sup>* .

(7)

01 1 ˆˆ ˆ ()() ( ) *<sup>T</sup>*

() ()

1

( ) , () () ()

2 2

<sup>2</sup> ( )

*m*

where the matrix *Hm* defined in (4) and also the

*mm m*

*elements*

*m m*

2 2 1

*m m*

*mP H*

2 0

: (1,1,2,2,4,4,4,4, ,( ),( ), ,( )) 22 2

for 2 *m* . For example, at resolution scale *j* 3 , the matrices *H*8 and 8*P* are represented

( ) () () 11 1 1 1 1 1 1 () () () 11 1 1 1 1 1 1

*r*

*d t r dr t dt*

where *i i* <sup>1</sup> *m m it* and using (2), we get

where the matrix

with <sup>1</sup>

as

and

vector *r* is represented by

8

*H*

The integration of the vector ( ) *<sup>m</sup> t* can be approximated by

0

 

represents the integral operator matrix for PCBFs on the interval 0,1

*m*

*P*

00 01 07 1 0 11 17 20 21 27 3 0 31 37

*t t t tt t tt t t t t*

() () () 11 1

 

 

 

 

 

 

 

 

40 41 47 5 0 51 57 6 0 61 67 70 71 77

*tt t t t t tt t t t t*

( ) () () () () () ( ) () () () () () ( ) () ()

<sup>1</sup> <sup>2</sup> *<sup>P</sup>* and <sup>1</sup> <sup>1</sup> ( ) *<sup>T</sup> H H diag r m m <sup>m</sup>*

0 0 0

For HWs, the square matrix *mP* satisfies the following recursive formula [13]:

*mm m m m P*

<sup>2</sup> <sup>1</sup>

*m H* 

*t t*

*t*

and 1 ( ) (2 ) *<sup>j</sup> i t tk* for 1 *i* and we write 2*<sup>j</sup> i k* for *j* 0 and 0 2*<sup>j</sup> k* . We can easily see that the 0 ( )*t* and 1 ( )*t* are compactly supported, they give a local description, at different scales *j* , of the considered function.

### **2.2. Function approximation**

The finite series representation of any square integrable function *y*( )*t* in terms of an orthogonal basis in the interval 0,1 , namely *y*ˆ( )*<sup>t</sup>* , is given by

$$\hat{\boldsymbol{y}}(t) = \sum\_{i=0}^{m-1} a\_i \boldsymbol{\nu}\_i(t) \; := \boldsymbol{a}^T \, \Psi\_m(t) \tag{2}$$

where 01 1 : *<sup>T</sup> <sup>m</sup> a aa a* and 01 1 ( ): ( ) ( ) ( ) *<sup>T</sup> m m t tt t* for 2*<sup>j</sup> <sup>m</sup>* and the Haar coefficients *<sup>i</sup> a* are determined to minimize the mean integral square error 1 2 0 ( ( ) ( )) *<sup>T</sup> m <sup>y</sup> t a t dt* and are given by

$$a\_i = 2^j \int\_0^1 y(t) \, \nu\_i(t) \, dt \tag{3}$$

*Remark 1*. The approximation error, ( ): ( ) ( ) ˆ *<sup>y</sup> m yt yt* , is depending on the resolution *m* and is approaching zero by increasing parameter of the resolution.

The matrix *Hm* can be defined as

$$\boldsymbol{H}\_m = \begin{bmatrix} \boldsymbol{\Psi}\_m(t\_0)\_\prime \, \boldsymbol{\Psi}\_m(t\_1)\_\prime \cdots \, \boldsymbol{\Psi}\_m(t\_{m-1})\_\cdot \end{bmatrix} \tag{4}$$

where *i i* <sup>1</sup> *m m it* and using (2), we get

$$\left[\left(\hat{\mathcal{y}}(t\_0)\,\hat{\mathcal{y}}(t\_1)\,\ldots\,\hat{\mathcal{y}}(t\_{m-1})\right)\right] = a^T \, H\_m \,. \tag{5}$$

The integration of the vector ( ) *<sup>m</sup> t* can be approximated by

$$\int\_{0}^{t} \Psi\_{m}(t) \, dt = P\_{m} \Psi\_{m}(t) \tag{6}$$

where the matrix

160 Advances on Analysis and Control of Vibrations – Theory and Applications

0

( )

*t*

Properties of HWs, which will be used in the next sections, are introduced in this section.

The oldest and most basic of the wavelet systems is named Haar wavelet that is a group of square waves with magnitude of . <sup>1</sup> . in the interval 0,1 [6]. In other words, the HWs are

<sup>1</sup> <sup>1</sup>

 

The finite series representation of any square integrable function *y*( )*t* in terms of an

ˆ() (): () *<sup>m</sup> <sup>T</sup> i i m*

coefficients *<sup>i</sup> a* are determined to minimize the mean integral square error

1

0 2 () () *<sup>j</sup> i i a y t t dt* 

*Remark 1*. The approximation error, ( ): ( ) ( ) ˆ *<sup>y</sup> m yt yt* , is depending on the resolution *m*

*y t at a t* 

1

0

*i*

and 01 1 ( ): ( ) ( ) ( ) *<sup>T</sup>*

and is approaching zero by increasing parameter of the resolution.

, namely *y*ˆ( )*<sup>t</sup>* , is given by

*m m t tt t* 

 

( ) 1, 0,1 ,

*t t*

1, 0, ,

*for t*

*for t*

1, ,1 ,

*t tk* for 1 *i* and we write 2*<sup>j</sup> i k* for *j* 0 and 0 2*<sup>j</sup> k* . We can

 

( )*t* are compactly supported, they give a local description, at

(2)

(3)

for 2*<sup>j</sup> <sup>m</sup>* and the Haar

01 1 ( ), ( ), , ( ) *H tt t m m m mm* (4)

(1)

1 2

2

**2. Properties of Haar Wavelets** 

**2.1. Haar Wavelets (HWs)** 

defined on the interval 0,1 as

and 1 ( ) (2 ) *<sup>j</sup> i* 

easily see that the 0

 

**2.2. Function approximation** 

where 01 1 : *<sup>T</sup> <sup>m</sup> a aa a*

> ( ( ) ( )) *<sup>T</sup> m*

The matrix *Hm* can be defined as

1

0

orthogonal basis in the interval 0,1

2

*<sup>y</sup> t a t dt* and are given by

( )*t* and 1

different scales *j* , of the considered function.

$$P\_m = \left\langle \stackrel{t}{\Psi}\_m(\tau) \, d\tau, \, \Psi\_m(t) > \stackrel{1}{=} \int\limits\_{0}^{t} \Psi\_m(r) \, dr \, \Psi\_m^T(t) \, dt \right\rangle$$

represents the integral operator matrix for PCBFs on the interval 0,1 at the resolution *<sup>m</sup>* . For HWs, the square matrix *mP* satisfies the following recursive formula [13]:

$$P\_m = \frac{1}{2m} \begin{bmatrix} 2mP\_{\frac{m}{2}} & -H\_{\frac{m}{2}} \\ H\_{\frac{m}{2}}^{-1} & 0\_{\frac{m}{2}} \end{bmatrix} \tag{7}$$

with <sup>1</sup> <sup>1</sup> <sup>2</sup> *<sup>P</sup>* and <sup>1</sup> <sup>1</sup> ( ) *<sup>T</sup> H H diag r m m <sup>m</sup>* where the matrix *Hm* defined in (4) and also the vector *r* is represented by

$$r \coloneqq (1, 1, 2, 2, 4, 4, 4, 4, 4, \dots, \underbrace{(\frac{m}{2}), (\frac{m}{2}), \dots, (\frac{m}{2})}\_{\text{(\frac{m}{2}) elements}}^T)^T$$

for 2 *m* . For example, at resolution scale *j* 3 , the matrices *H*8 and 8*P* are represented as

00 01 07 1 0 11 17 20 21 27 3 0 31 37 8 40 41 47 5 0 51 57 6 0 61 67 70 71 77 ( ) () () 11 1 1 1 1 1 1 () () () 11 1 1 1 1 1 1 () () () 11 1 ( ) () () () () () ( ) () () () () () ( ) () () *t t t tt t tt t t t t H tt t t t t tt t t t t* 10 0 0 0 00 0 0 1 1 1 1 , 1 10 0 0 0 0 0 00 1 10 0 0 0 00 0 0 1 10 0 00 0 0 0 0 1 1 

and

$$P\_8 = \frac{1}{16} \begin{bmatrix} 8 & -4H\_1 & -2H\_2 \\ -H\_1^{-1} & 0 & -2H\_2 \\ \hline \\ - & 4H\_2^{-1} & 0 \\ \end{bmatrix} = \frac{1}{16} \begin{bmatrix} 16P\_4 & -H\_4 \\ H\_4^{-1} & 0 \end{bmatrix}$$

$$H\_4^{-1} = \begin{bmatrix} 32 & -16 & -8 & -8 & -4 & -4 & -4 & -4 \\ 16 & 0 & -8 & 8 & -4 & -4 & 4 & 4 \\ 4 & 4 & 0 & 0 & -4 & 4 & 0 & 0 \\ 4 & -4 & 0 & 0 & 0 & 0 & -4 & 4 \\ 1 & 1 & 2 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & -2 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 2 & 0 & 0 & 0 & 0 \\ \end{bmatrix},$$

$$\begin{bmatrix} 1 & -1 & 0 & 2 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & -2 & 0 & 0 & 0 & 0 \\ \end{bmatrix}$$

for further information see [13, 25].

### **2.3. The product operational matrix**

In the study of time-varying state-delayed systems, it is usually necessary to evaluate the product of two Haar function vectors [13]. Let us define

$$\boldsymbol{R}\_m(t) \coloneqq \; \Psi\_m(t)\Psi\_{\;\;m}^T(t) \tag{8}$$

A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 163

( )

*b*

(12)

*T*

(13)

2 2

*m m*

1

*m T*

0 1 1

*a aa a at*

*a a ta t a t*

A schematic of the vehicle engine-body vibration structure is shown in Figure 3. The actuator and sensor used to this control framework are selected to be collocated, since this arrangement is ideal to ensure the stability of the closed loop system for a slightly damped structure [26]. In our study, only the bounce and pitch vibrations in the engine and body are considered [35]. The engine with mass *Me* and inertia moment *eI* is mounted in the body by the engine mounts *<sup>e</sup> k* and *<sup>e</sup> c* . The front mount is the active mount, the output force of

2L

*b m*

: , , , ()

: ( ), ( ), , ( ) .

2 2

*m m*

*a*

2l

**Figure 3.** The sketch of engine-body vibration system

**3. Mathematical model description** 

*<sup>a</sup> diag a H diag a H* 

<sup>2</sup> <sup>2</sup>

2 2

*m m*

*T*

1 1

*<sup>m</sup> <sup>m</sup>*

() ( )

*a H diag a*

*b a*

where 1 0 *a a* and

with

where ( ) *R t <sup>m</sup>* satisfies the following recursive formula

$$R\_m(t) = \frac{1}{2m} \begin{bmatrix} R\_{\frac{m}{2}}(t) & H\_{\frac{m}{2}} \operatorname{diag} \{ \Psi\_b(t) \} \\\\ \left( H\_{\frac{m}{2}} \operatorname{diag} \{ \Psi\_b(t) \} \right)^T & \operatorname{diag} \{ H\_{\frac{m}{2}}^{-1} \Psi\_a(t) \} \end{bmatrix} \tag{9}$$

with 1 00 () () () *<sup>T</sup> Rt t t* and

$$\begin{cases} \boldsymbol{\Psi}\_{\boldsymbol{a}}(t) \coloneqq \left[ \boldsymbol{\nu}\_{0}(t), \boldsymbol{\nu}\_{1}(t), \cdots, \boldsymbol{\nu}\_{\frac{m}{2}-1}(t) \right]^{T} = \boldsymbol{\Psi}\_{\frac{m}{2}}(t) \\\\ \boldsymbol{\Psi}\_{\boldsymbol{b}}(t) \coloneqq \left[ \boldsymbol{\nu}\_{\frac{m}{2}}(t), \boldsymbol{\nu}\_{\frac{m}{2}+1}(t), \cdots, \boldsymbol{\nu}\_{m-1}(t) \right]^{T} . \end{cases} \tag{10}$$

Moreover, the following relation is important for solving optimal control problem of timevarying state-delayed system:

$$R\_m(t)a\_m = \tilde{a}\_m \Psi\_m(t) \tag{11}$$

where 1 0 *a a* and

$$\tilde{a}\_{\underline{m}} = \begin{bmatrix} \tilde{a}\_{\frac{\underline{m}}{2}} & H\_{\frac{\underline{m}}{2}} \operatorname{diag}(a\_b) \\\\ \operatorname{diag}(a\_b) H\_{\frac{\underline{m}}{2}}^{-1} & \operatorname{diag}(a\_a^T H\_{\frac{\underline{m}}{2}}) \end{bmatrix} \tag{12}$$

with

162 Advances on Analysis and Control of Vibrations – Theory and Applications

1

*H*

*P*

for further information see [13, 25].

**2.3. The product operational matrix** 

product of two Haar function vectors [13]. Let us define

where ( ) *R t <sup>m</sup>* satisfies the following recursive formula

*R t*

and

*a*

with 1 00 () () () *<sup>T</sup> Rt t t* 

varying state-delayed system:

8 4

4 0

1 1 2

*H*

1 4

*H*

8 1 1

2

*H*

1 1 16

4

*H*

2 4

32 16 8 8 4 4 4 4 16 0 8 8 4 4 4 4 4 4 0 0 44 0 0

<sup>1</sup> 4 4 0 0 0 0 44 , <sup>64</sup> 1 1 200000

In the study of time-varying state-delayed systems, it is usually necessary to evaluate the

( ): () ( ) *<sup>m</sup>*

2 2

( ) ( ( )) <sup>1</sup> ( ) <sup>2</sup> ( ( ( ))) ( ( )) *m m*

0 1 1

( ) : ( ), ( ), , ( ) .

Moreover, the following relation is important for solving optimal control problem of time-

() () *R ta a t m m mm* (11)

*t tt t*

( ) : ( ), ( ), , ( ) ( )

 

*t tt t t*

*m T*

2 2

*m m*

*b m*

2 2

1 1

 

*m m*

*m H diag t diag H t* 

*b a*

2 2

*T*

*m m*

*T*

*R t H diag t*

1 1 20 0 0 0 0 1 1020000 1 1 0 20 0 0 0

0

16 4 0 0 16

*H H*

4 4

*P H*

*<sup>T</sup> Rt t t m m* (8)

*b*

(9)

(10)

1

$$\begin{cases} a\_a := \left[ a\_0, a\_1, \dots, a\_{\frac{m}{2}-1} \right]^T = a\_{\frac{m}{2}}(t) \\\\ a\_b := \left[ a\_{\frac{m}{2}}(t), a\_{\frac{m}{2}+1}(t), \dots, a\_{m-1}(t) \right]^T \end{cases} \tag{13}$$

**Figure 3.** The sketch of engine-body vibration system

### **3. Mathematical model description**

A schematic of the vehicle engine-body vibration structure is shown in Figure 3. The actuator and sensor used to this control framework are selected to be collocated, since this arrangement is ideal to ensure the stability of the closed loop system for a slightly damped structure [26]. In our study, only the bounce and pitch vibrations in the engine and body are considered [35]. The engine with mass *Me* and inertia moment *eI* is mounted in the body by the engine mounts *<sup>e</sup> k* and *<sup>e</sup> c* . The front mount is the active mount, the output force of

which can be controlled by an electric signal. The active mount consists of a main chamber where an oscillating mass (inertia mass) is moving up and down. The inertia mass is driven by an electro-magnetic force generated by a magnetic coil which is controlled by the input current.

A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 165

2 2

*l k l k*

*e e*

22 2 2

.

; normalizing the system Eq. (15) with

 

(16)

 

(17)

 

( (0) (0)) .

*Mx Cx d*

*f*

, (18)

, (19)

1 1

> *l L*

220 2( ) 2 2( ) 0 2( ) 0 02 2

*e e e e eb e*

*k k L lk k kk L lk*

2 2( ) 2 ( ( 2 ) ) 2

Taking displacement of the chassis 2 ( ( )) *x t* as an output then a comparison of the displacement response respect to the input force *f*( )*t* and the external disturbance ( ) *<sup>e</sup> d t* in the frequency range up to 1 KHz is depicted in Figure 4a) and 4b). Three relevant modes occur around the frequencies 1, 5 and 9 Hz, respectively, which represent the dynamics of

In this section, we study the problem of solving the second-order differential equations of the engine-body system (14) in terms of the input control and exogenous disturbance using

Based on HWs definition on the interval time 0, 1 , we need to rescale the finite time

() () () () () *Mx Cx Kx B f B d f de*

 

( ( ) (0)) ( ) ( ) ( ) ( )

 

*M x x C x d K x dd B f dd B d dd*

22 2 0 0 0 0 0 0 0

*f f f f fd e*

By using the Haar wavelet expansion (2), we express the solution of Eq. (15), input force

in terms of HWs in the forms

 

  0

, we obtain

 

*e ee e b*

*lk L l k l k L L l k L k*

*d B* ,

 ,

*f B*

*K*

the main degrees of freedom (DOFs) of the system.

**4. Algebraic solution of system equations** 

HWs and develop appropriate algebraic equations.

interval 0, *<sup>f</sup> <sup>T</sup>* into 0, 1 by considering *<sup>f</sup> <sup>t</sup>*

Now by integrating the system above in an interval 0,

() () *<sup>m</sup> x X*

() () *<sup>m</sup> f F*

 

and engine disturbance ( ) *<sup>e</sup> d*

the time scale would be as follows

*f*( ) 

The vehicle body with mass *Mb* and inertia moment *bI* is supported by front and rear tires, each of which is modeled as a system consisting of a spring *<sup>b</sup> k* and a damping device *<sup>b</sup> c* . Therefore, a four degree-of-freedom vibration suspension model shown in Figure 3 can be described by the following equations

1 112 2 4 4 2 2 21 1 4 4 2222 3 3 34 4 2 4 2 2 2 2 2( ) 2( ) ( ) ( ) 2( ) 2( ) 2 2 2( ) 2( ) ( ) 2 2 2 2 () (( ( 2 *ee ee e e e e b eb eb e e e e e e ee e b Mx c x kx cx k x L lcx L lk x f t d t M x c c x k k x cx k x L lcx L lkx f t I x l cx l kx l cx l kx lf t Ix L L l* 22 2 2 2 2 4 4 3 2 31 1 2 2 ) ) 2 ) (( ( 2 ) ) 2 ) 2 2 2 2 2( ) 2( ) ( ) *e b eb e ee e e e c Lc x L L l k Lk x l cx l k x lc x lk x L l c x L l k x L f t* (14)

where the states 123 *xtxtxt* ( ), ( ), ( ) and 4 *x t*( ) are the bounces and pitches of the engine and body, respectively, where displacement of the chassis 2 ( ( )) *x t* is usually taken as an output. Input force, *f*( )*t* , is used as the active force to compensate the vibration transmitted to vehicle body. Moreover, engine disturbance ( ) *<sup>e</sup> d t* can be the excitation, generated by the motion up/down of the different parts inside the engine;

The system Eq. (14) can be represented in the following state-space form

$$\begin{cases} M\ddot{\boldsymbol{x}}(t) + \mathbf{C}\dot{\boldsymbol{x}}(t) + K\mathbf{x}(t) = B\_f \, f(t) + B\_d \, d\_e(t), & t \in \left[0, T\_f\right] \\\\ \mathbf{z}(t) = \begin{bmatrix} \mathbf{C}\_1 \mathbf{x}(t) \\ \mathbf{C}\_2 \dot{\mathbf{x}}(t) \\ \mathbf{C}\_3 \, f(t) \end{bmatrix} \end{cases} \tag{15}$$

where <sup>4</sup> *x t*( ) is the state; *f*( )*<sup>t</sup>* is the control input; ( ) *<sup>e</sup> d t* is the disturbance input which belongs to 2 *<sup>L</sup>* [0, ) ; and <sup>3</sup> *z t*( ) is the controlled output with 1 4 *<sup>C</sup>*<sup>1</sup> , 1 4 *<sup>C</sup>*<sup>2</sup> and *C*3 is a positive scalar. The state-space matrices are also defined as

$$M = \begin{bmatrix} M\_{\varepsilon} & 0 & 0 & 0 \\ 0 & M\_{b} & 0 & 0 \\ 0 & 0 & I\_{\varepsilon} & 0 \\ 0 & 0 & 0 & I\_{b} \end{bmatrix}, \mathbf{C} = \begin{vmatrix} 2c\_{\varepsilon} & -2c\_{\varepsilon} & 0 & -2(L-l)c\_{\varepsilon} \\ -2c\_{\varepsilon} & 2(c\_{\varepsilon}+c\_{b}) & 0 & 2(L-l)c\_{\varepsilon} \\ 0 & 0 & 2l^{2}c\_{\varepsilon} & -2l^{2}c\_{\varepsilon} \\ -2lc\_{\varepsilon} & 2(L-l)c\_{\varepsilon} & -2l^{2}c\_{\varepsilon} & 0 \end{vmatrix}, \mathbf{f}$$

.

$$\begin{aligned} \mathcal{B}\_f = \begin{bmatrix} 1 \\ -1 \\ l \\ -L \end{bmatrix}' \end{aligned} \quad \mathcal{B}\_d = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}' $$

$$K = \begin{bmatrix} 2\,k\_{\varepsilon} & -2\,k\_{\varepsilon} & 0 & -2(L-l)k\_{\varepsilon} \\ -2\,k\_{\varepsilon} & 2(k\_{\varepsilon}+k\_{b}) & 0 & 2(L-l)k\_{\varepsilon} \\ 0 & 0 & 2l^{2}k\_{\varepsilon} & -2l^{2}k\_{\varepsilon} \\ -2\,lk\_{\varepsilon} & 2(L-l)k\_{\varepsilon} & -2l^{2}k\_{\varepsilon} & (L^{2}+(L-2l)^{2})k\_{\varepsilon}+2L^{2}k\_{b} \end{bmatrix}$$

Taking displacement of the chassis 2 ( ( )) *x t* as an output then a comparison of the displacement response respect to the input force *f*( )*t* and the external disturbance ( ) *<sup>e</sup> d t* in the frequency range up to 1 KHz is depicted in Figure 4a) and 4b). Three relevant modes occur around the frequencies 1, 5 and 9 Hz, respectively, which represent the dynamics of the main degrees of freedom (DOFs) of the system.

### **4. Algebraic solution of system equations**

164 Advances on Analysis and Control of Vibrations – Theory and Applications

2222 3 3 34 4

motion up/down of the different parts inside the engine;

1 2 3

*C xt*

*C ft*

0 00 0 00 00 0 0 00

*e*

*I*

*b*

*I*

*b*

*M*

*e*

*M*

*M*

() ()

*zt C xt*

( )

( )

*I x l cx l kx l cx l kx lf t*

*e e ee e*

2 2 2 2 ()

described by the following equations

2

*Ix L L l*

(( ( 2

4

2

*b*

current.

 

which can be controlled by an electric signal. The active mount consists of a main chamber where an oscillating mass (inertia mass) is moving up and down. The inertia mass is driven by an electro-magnetic force generated by a magnetic coil which is controlled by the input

The vehicle body with mass *Mb* and inertia moment *bI* is supported by front and rear tires, each of which is modeled as a system consisting of a spring *<sup>b</sup> k* and a damping device *<sup>b</sup> c* . Therefore, a four degree-of-freedom vibration suspension model shown in Figure 3 can be

2 2 21 1 4 4

*M x c c x k k x cx k x L lcx L lkx f t*

) ) 2 ) (( ( 2 ) ) 2 ) 2

where the states 123 *xtxtxt* ( ), ( ), ( ) and 4 *x t*( ) are the bounces and pitches of the engine and body, respectively, where displacement of the chassis 2 ( ( )) *x t* is usually taken as an output. Input force, *f*( )*t* , is used as the active force to compensate the vibration transmitted to vehicle body. Moreover, engine disturbance ( ) *<sup>e</sup> d t* can be the excitation, generated by the

( ) ( ) ( ) ( ) ( ), 0,

*Mx t Cx t Kx t B f t B d t t T <sup>f</sup> d e <sup>f</sup>*

where <sup>4</sup> *x t*( ) is the state; *f*( )*<sup>t</sup>* is the control input; ( ) *<sup>e</sup> d t* is the disturbance input

, 2 2

*lc L l c l c*

*e ee*

2 2 0 2( ) 2 2( ) 0 2( ) 0 02 2 2 2( ) 2 0

*e e e e eb e*

*c c L lc c cc L lc*

which belongs to 2 *<sup>L</sup>* [0, ) ; and <sup>3</sup> *z t*( ) is the controlled output with 1 4 *<sup>C</sup>*<sup>1</sup>

and *C*3 is a positive scalar. The state-space matrices are also defined as

*C*

*e b eb e*

*c Lc x L L l k Lk x l cx*

*ee ee e e e e*

*Mx c x kx cx k x L lcx L lk x f t d t*

*b eb eb e e e e*

22 2 2 2 2

2 2 2 2 2( ) 2( ) ( ) ( )

2( ) 2( ) 2 2 2( ) 2( ) ( )

4 4 3

(15)

2

*e e*

*lc lc*

(14)

, 1 4 *<sup>C</sup>*<sup>2</sup>

,

1 112 2 4 4

31 1 2 2

*ee e e e*

The system Eq. (14) can be represented in the following state-space form

2 2 2 2( ) 2( ) ( )

*l k x lc x lk x L l c x L l k x L f t*

In this section, we study the problem of solving the second-order differential equations of the engine-body system (14) in terms of the input control and exogenous disturbance using HWs and develop appropriate algebraic equations.

Based on HWs definition on the interval time 0, 1 , we need to rescale the finite time interval 0, *<sup>f</sup> <sup>T</sup>* into 0, 1 by considering *<sup>f</sup> <sup>t</sup>* ; normalizing the system Eq. (15) with the time scale would be as follows

$$M\ddot{\boldsymbol{x}}(\sigma) + \mathbb{C}\dot{\boldsymbol{x}}(\sigma) + K\boldsymbol{x}(\sigma) = B\_{\boldsymbol{f}}\,\boldsymbol{f}(\sigma) + B\_{\boldsymbol{d}}\,\boldsymbol{d}\_{\boldsymbol{\varepsilon}}(\sigma) \tag{16}$$

Now by integrating the system above in an interval 0, , we obtain

22 2 0 0 0 0 0 0 0 0 ( ( ) (0)) ( ) ( ) ( ) ( ) ( (0) (0)) . *f f f f fd e f M x x C x d K x dd B f dd B d dd Mx Cx d* (17)

By using the Haar wavelet expansion (2), we express the solution of Eq. (15), input force *f*( ) and engine disturbance ( ) *<sup>e</sup> d* in terms of HWs in the forms

$$\boldsymbol{x}(\sigma) = \boldsymbol{X} \,\, \boldsymbol{\Psi}\_{\boldsymbol{m}}(\sigma) \,, \tag{18}$$

$$f(\sigma) = F\left(\Psi\_m(\sigma)\right),\tag{19}$$

 () () *e em d D* , (20)

A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 167

*fm fm m P C P KI M* with dimension 4 4 *m m* only

and it is also clear that to find the approximated solution of the system, we have to calculate

(a)

Frequency (Hz) 10-1 100 101 102 103

(b)

Frequency (Hz) 10-1 100 101 102 103

the inverse of the matrix 2 2 ( )( ) *T T*







Displacement (dB)

0

10

20

30





Displacement (dB)

0

10

20

30

**Figure 4.** Displacement of the chassis respect to *f*( )*t* (a) and ( ) *<sup>e</sup> d t* (b).

once.

where <sup>4</sup> *<sup>m</sup> <sup>X</sup>* , <sup>1</sup> *<sup>m</sup> <sup>F</sup>* and <sup>1</sup> *<sup>m</sup> De* denote the wavelet coefficients of *x*( ) , *f* ( ) and ( ) *<sup>e</sup> d* , respectively. The initial conditions of *x*(0) and *x*(0) are also represented by <sup>0</sup> (0) ( ) *<sup>m</sup> x X* and 0 (0) ( ) *<sup>m</sup> x X* , where the matrices <sup>4</sup> 0 0 {, } *<sup>m</sup> X X* are defined, respectively, as

$$X\_0 := \left[ \mathbf{x}(0) \underbrace{\mathbf{0}\_{4 \times 1} \dots \mathbf{0}\_{4 \times 1}}\_{\text{(m-1)}} \right] \tag{21}$$

$$\overline{X}\_0 := \left[ \dot{\mathbf{x}}(0) \underbrace{\mathbf{0}\_{4 \times 1} \dots \mathbf{0}\_{4 \times 1}}\_{\text{(m-1)}} \right] \tag{22}$$

$$\text{Therefore, using the wavelet expansions (18)-(20), the relation (17) becomes}$$

$$\text{C}\_{f}\text{M}(\text{X}-\text{X}\_{0}) + \text{T}\_{f}\text{C}\text{X}\text{P}\_{m} + \text{T}\_{f}^{2}\text{K}\text{X}\text{P}\_{m}^{2} = \text{T}\_{f}^{2}\text{B}\_{f}\text{F}\text{P}\_{m}^{2} + \text{T}\_{f}^{2}\text{B}\_{d}\text{D}\_{e}\text{P}\_{m}^{2} + \left(\text{M}\overline{\text{X}}\_{0} + \text{T}\_{f}\text{C}\text{X}\_{0}\right)\text{P}\_{m} \tag{23}$$

( 1)

*m*

For calculating the matrix *X* , we apply the operator *vec*(.) to Eq. (23) and according to the property of the Kronecker product, i.e. ( ) ( ) () *<sup>T</sup> vec ABC C A vec B* , we have:

0 4 1 4 1

$$\begin{split} (\mathbf{I}\_m \otimes \mathbf{M})(\text{vec}(\mathbf{X}) - \text{vec}(\mathbf{X}\_0)) + \mathbf{T}\_f \left( \mathbf{P}\_m^T \otimes \mathbf{C} \right) \text{vec}(\mathbf{X}) + \mathbf{T}\_f^2 \left( \mathbf{P}\_m^{2T} \otimes \mathbf{K} \right) \text{vec}(\mathbf{X}) \\ = \mathbf{T}\_f^2 \left( \mathbf{P}\_m^{2T} \otimes \mathbf{B}\_f \right) \text{vec}(\mathbf{F}) + \mathbf{T}\_f^2 \left( \mathbf{P}\_m^{2T} \otimes \mathbf{B}\_d \right) \text{vec}(\mathbf{D}\_\epsilon) \\ + \mathbf{T}\_f \left( \mathbf{P}\_m^T \otimes \mathbf{C} \right) \text{vec}(\mathbf{X}\_0) + \left( \mathbf{P}\_m^T \otimes \mathbf{M} \right) \text{vec}(\mathbf{\overline{X}}\_0). \end{split} \tag{24}$$

Solving Eq. (24) for *vec X*( ) leads to

$$\text{vec}(X) = \Delta\_1 \text{vec}(F) + \Delta\_2 \text{vec}(D\_e) + \Delta\_3 \text{vec}(X\_0) + \Delta\_4 \text{vec}(\bar{X}\_0) \tag{25}$$

where the matrices <sup>4</sup> 1 2 {, } *m m* and 4 4 3 4 {, } *m m* are defined as

$$\begin{cases} \Lambda\_{1} = \mathrm{T}\_{f}^{2} \left( \mathrm{T}\_{f} \left( \mathrm{P}\_{m}^{T} \otimes \mathrm{C} \right) + \mathrm{T}\_{f}^{2} \left( \mathrm{P}\_{m}^{2T} \otimes \mathrm{K} \right) + I\_{m} \otimes \mathrm{M} \right)^{-1} \left( \mathrm{P}\_{m}^{2T} \otimes \mathrm{B}\_{f} \right) \\ \Lambda\_{2} = \mathrm{T}\_{f}^{2} \left( \mathrm{T}\_{f} \left( \mathrm{P}\_{m}^{T} \otimes \mathrm{C} \right) + \mathrm{T}\_{f}^{2} \left( \mathrm{P}\_{m}^{2T} \otimes \mathrm{K} \right) + I\_{m} \otimes \mathrm{M} \right)^{-1} \left( \mathrm{P}\_{m}^{2T} \otimes \mathrm{B}\_{d} \right) \\ \Lambda\_{3} = \mathrm{T}\_{f} \left( \mathrm{P}\_{m}^{T} \otimes \mathrm{C} \right) + \mathrm{T}\_{f}^{2} \left( \mathrm{P}\_{m}^{2T} \otimes \mathrm{K} \right) + I\_{m} \otimes \mathrm{M} \right)^{-1} \left( I\_{m} \otimes \mathrm{M} + \mathrm{T}\_{f} \left( \mathrm{P}\_{m}^{T} \otimes \mathrm{C} \right) \right) \\ \Lambda\_{4} = \left( \mathrm{T}\_{f} \left( \mathrm{P}\_{m}^{T} \otimes \mathrm{C} \right) + \mathrm{T}\_{f}^{2} \left( \mathrm{P}\_{m}^{2T} \otimes \mathrm{K} \right) + I\_{m} \otimes \mathrm{M} \right)^{-1} \left( \mathrm{P}\_{m}^{T} \otimes \mathrm{M} \right) . \end{cases} \tag{26}$$

Consequently, using (25) and (26) and the properties of the Kronecker product, the solution of system (15) is

$$\mathbf{x}(\sigma) = (\Psi\_m^T(\sigma) \otimes I\_4) \text{ vere}(X) \tag{27}$$

and it is also clear that to find the approximated solution of the system, we have to calculate the inverse of the matrix 2 2 ( )( ) *T T fm fm m P C P KI M* with dimension 4 4 *m m* only once.

166 Advances on Analysis and Control of Vibrations – Theory and Applications

() () *e em d D*

and 0 (0) ( ) *<sup>m</sup> x X*

where <sup>4</sup> *<sup>m</sup> <sup>X</sup>* , <sup>1</sup> *<sup>m</sup> <sup>F</sup>* and <sup>1</sup> *<sup>m</sup> De*

and ( ) *<sup>e</sup> d* 

<sup>0</sup> (0) ( ) *<sup>m</sup> x X*

respectively, as

Solving Eq. (24) for *vec X*( ) leads to

where the matrices <sup>4</sup>

1

 

of system (15) is

2

3

4

Therefore, using the wavelet expansions (18)-(20), the relation (17) becomes

property of the Kronecker product, i.e. ( ) ( ) () *<sup>T</sup> vec ABC C A vec B* , we have:

*m f m f m*

1 2 {, } *m m* and 4 4

2 2 2 1 2

*<sup>m</sup> x* 

2 2 2 1 2

2 2 1

( ( ) ( ) )( )

*P C P KI M I M P C*

( ( ) ( ) )( ) ( ( ) ( ) )( )

*P C P KI M P B P C P KI M P B*

*TT T f fm fm m m f TT T f fm fm m m d T T T f m f m m m fm*

2 2 1

Consequently, using (25) and (26) and the properties of the Kronecker product, the solution

<sup>4</sup> () ( () ) () *<sup>T</sup>*

 

*P C P KI M P M*

( ( ) ( ) ) ( ).

*TT T fm fm m m*

0

, respectively. The initial conditions of *x*(0) and *x*(0) are also represented by

( 1)

( 1)

2 2 2 2

*T T f m m*

*T T f m f fm d e*

1 2 3 04 0 ( ) () ( ) ( ) ( ) *<sup>e</sup> vec X vec F vec D vec X vec X* (25)

3 4 {, } *m m* are defined as

   

*T T*

*m*

*m*

0 41 41

0 4 1 4 1

2 2 2 22 2 <sup>0</sup> 0 0 ( ) ( ) *M X X CXP KXP B FP B D P MX CX P <sup>f</sup> m f m f f m f d em f m* (23)

For calculating the matrix *X* , we apply the operator *vec*(.) to Eq. (23) and according to the

( )( ( ) ( )) ( ) ( ) ( ) ( )

*I M vec X vec X P C vec X P K vec X*

: (0) 0 0

*X x*

: (0) 0 0

*X x*

denote the wavelet coefficients of *x*( )

, where the matrices <sup>4</sup>

, (20)

(21)

(22)

2 2

( ) () ( ) ( )

*P B vec F P B vec D*

( ) ( ) ( ) ( ).

*P C vec X P M vec X*

0 0

*I vec X* (27)

 , *f* ( ) 

(24)

(26)

0 0 {, } *<sup>m</sup> X X* are defined,

**Figure 4.** Displacement of the chassis respect to *f*( )*t* (a) and ( ) *<sup>e</sup> d t* (b).

### **5. Optimal control design**

The control objective is to find the optimal control *f*( )*t* with respect to a quadratic cost functional approximately such acts as the active force to compensate the vibration transmitted to vehicle body. The quadratic cost functional weights the states and their derivatives with respect to time in the cost function as follows:

$$J = \frac{1}{2} \mathbf{x}^T(\mathbf{T}\_f) \mathbf{S}\_1 \mathbf{x}(\mathbf{T}\_f) + \frac{1}{2} \dot{\mathbf{x}}^T(\mathbf{T}\_f) \mathbf{S}\_2 \dot{\mathbf{x}}(\mathbf{T}\_f) + \frac{1}{2} \int\_0^{\mathbf{T}\_f} \left( \mathbf{x}^T(t) \mathbf{Q}\_1 \mathbf{x}(t) + \dot{\mathbf{x}}^T(t) \mathbf{Q}\_2 \dot{\mathbf{x}}(t) + \mathbf{R} \, f(t)^2 \right) dt \tag{28}$$

where 1 *S* :4 4 , 2 *S* :4 4 , 1 *Q* :4 4 and 2 *Q* :4 4 are positive-definite matrices and *R* is a positive scalar. We can rewrite the cost function (28) as follows:

$$J = \frac{1}{2} \mathbf{\dot{x}}^T \begin{pmatrix} \mathbf{1} \end{pmatrix} \quad \mathbf{T}\_{\boldsymbol{\dot{\gamma}}}^{-1} \dot{\mathbf{x}}^T \begin{pmatrix} \mathbf{1} \end{pmatrix} \tilde{\mathbf{S}} \begin{bmatrix} \mathbf{x}(1) \\ \mathbf{T}\_{\boldsymbol{\dot{\gamma}}}^{-1} \dot{\mathbf{x}}(1) \end{bmatrix} + \frac{\mathbf{T}\_{\boldsymbol{\dot{\gamma}}}}{2} \begin{bmatrix} \mathbf{I} \end{bmatrix} \mathbf{\dot{x}}^T \begin{pmatrix} \mathbf{1} \end{pmatrix} \mathbf{\dot{x}}^T \begin{pmatrix} \mathbf{1} \end{pmatrix} \tilde{\mathbf{Q}} \begin{bmatrix} \mathbf{x}(\sigma) \\ \mathbf{T}\_{\boldsymbol{\dot{\gamma}}}^{-1} \dot{\mathbf{x}}(\sigma) \end{bmatrix} + \mathbf{R} \begin{bmatrix} \mathbf{f}(\sigma)^2 \end{bmatrix} \, d\sigma \,. \tag{29}$$

where 1 2 *S diag S S* (,) and 1 2 *Q diag Q Q* (,) with the time scale *<sup>f</sup> <sup>t</sup>* .

From (15) and the relation () () *<sup>m</sup> x X* , where *X m* :4 denotes the wavelet coefficients of *x*( ) after its expansion in terms of HFs, we read

$$\mathbb{E}\left[\begin{array}{c}\mathfrak{x}(\sigma)\\\mathrm{T}^{-1}\_{\gamma}\acute{\mathsf{x}}(\sigma)\end{array}\right] = \left[\begin{array}{c}X\\\mathrm{T}^{-1}\_{\gamma}\overline{X}\end{array}\right]\mathbb{1}\_{m}(\sigma) \coloneqq X\_{\mathrm{aug}}\,\,\Psi\_{m}(\sigma) \tag{30}$$

A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 169

*i i i m m* 

(35)

, then for finding the

(38)

By substituting the definition (31) in (33) and using the properties of the operator *tr*(.) in

<sup>1</sup> ( ) *T T mf f <sup>M</sup> S MQ* and *m fm* <sup>2</sup> *R M* , respectively,

optimal control law, which minimizes the cost functional *J*(.) , the following necessary

( ) *J vec F*

By considering ( ) *aug vec X* , which is a function of *vec F*( ) , and using the properties of

1 1 41 <sup>2</sup> [ ( )] ( ) ( ) ( ) *<sup>f</sup>*

*<sup>J</sup> P I vec X vec F*

21 1 4 1 ( ) [ ( )] ( ) *<sup>f</sup>*

4 1 21 1 4 1 1 1 2 1 2

*mm m m <sup>T</sup> e m <sup>m</sup>*

*P I vec X vec X P I*

1 1 41 1 1 3 04 0 4

0 [ ( )] ) ( ) ( )), ( )

*m*

Consequently, the optimal vectors of *vec X*( ) and *vec F*( ) are found, respectively, in the

*m m aug m*

4 1 11 1 1

*I*

*f*

4

(36)

*<sup>m</sup> <sup>m</sup> m aug vec F P I vec X* (37)

*<sup>d</sup>* and : (1) (1) *<sup>T</sup> Mmf m m* , respectively.

0

*T T*

1 2 <sup>1</sup> ( ( ) ( ) ( ) ( ))

*aug m aug <sup>m</sup> J vec X vec X vec F vec F* (34)

Appendix A1, the cost function (28) is given by

2

1

0 : () () *<sup>T</sup> Mm mm* 

*vec F*

*f*

condition should be satisfied

following forms

and

where the matrices 1 :8 8 *<sup>m</sup> m m* and 2 : *<sup>m</sup> m m* are defined as

 

It is clear that the cost function of *J*(.) is a function of <sup>1</sup>

derivatives of inner product of Kronecker product in Appendix A2, we find

1 1

Then the wavelet coefficients of the optimal control law will be in vector form as

1 11

( ) ( ( [ ( )] )( ( )( ( )

*T T m*

*vec X I P I vec D P I*

*f*

*T T*

*T T*

4 1 1

*m m T*

*f*

*T T m*

and the matrices : *Mm m m* and : *Mmf m m* are defined as

where 1 *f aug X <sup>X</sup> <sup>X</sup>* and

$$\operatorname{vec}(X\_{\text{aug}}) = \left[ \operatorname{vec}^T \begin{pmatrix} X \end{pmatrix} \quad \operatorname{T}^{-1}\_{\text{/}} \operatorname{vec}^T \begin{pmatrix} \overline{X} \end{pmatrix} \right]^T \tag{31}$$

*Remark* 2. By substituting () () *<sup>m</sup> x X* into 0 *x x x t dt* ( ) (0) ( ) , we have:

$$X\,\,\Psi\_m(\sigma) - X\_0\,\,\Psi\_m(\sigma) = \int\_0^{\sigma} \overline{X}\,\,\Psi\_m(\sigma)\,d\sigma \,\,\,\,\tag{32}$$

and using (4), we read *X X XP* <sup>0</sup> *<sup>m</sup>* . Then, by applying the operator of *vec*(.) and according to the properties of Kronecker product in Appendix A1, we obtain

$$\vec{\text{vec}}(X) - \vec{\text{vec}}(X\_0) = (P\_m^T \otimes I\_n) \text{ } \vec{\text{vec}}(\overline{X}) \tag{33}$$

By substituting the definition (31) in (33) and using the properties of the operator *tr*(.) in Appendix A1, the cost function (28) is given by

$$J = \frac{1}{2} (vec^T (X\_{\text{ang}}) \Pi\_{m1} \text{vec}(X\_{\text{ang}}) + \text{vec}^T(F) \Pi\_{m2} \text{vec}(F)) \tag{34}$$

where the matrices 1 :8 8 *<sup>m</sup> m m* and 2 : *<sup>m</sup> m m* are defined as

$$
\Pi\_{m1} = M\_f^T \otimes \tilde{\mathbf{S}} + \mathbf{T}\_f (M^T \otimes \tilde{\mathbf{Q}}) \text{ and } \Pi\_{m2} = \mathbf{R} \mathbf{T}\_f M\_{m'} \text{ respectively.}
$$

and the matrices : *Mm m m* and : *Mmf m m* are defined as

$$M\_m \coloneqq \mathop{\!}^1\_0 \Psi\_m(\sigma) \Psi\_m^T(\sigma) \, d\sigma \text{ and } \; M\_{mf} \coloneqq \Psi\_m(1) \Psi\_m^T(1) \text{, respectively.} $$

It is clear that the cost function of *J*(.) is a function of <sup>1</sup> *i i i m m* , then for finding the optimal control law, which minimizes the cost functional *J*(.) , the following necessary condition should be satisfied

$$\frac{\partial \mathcal{J}}{\partial \text{vec}(F)} = 0 \tag{35}$$

By considering ( ) *aug vec X* , which is a function of *vec F*( ) , and using the properties of derivatives of inner product of Kronecker product in Appendix A2, we find

$$\frac{\partial \hat{\mathbb{J}}}{\partial \text{vec}(F)} = \left[\boldsymbol{\Delta}\_1^T \quad \boldsymbol{\Gamma}\_f^{-1} \boldsymbol{\Delta}\_1^T (\boldsymbol{P}\_m^{-1} \otimes \boldsymbol{I}\_4) \right] \boldsymbol{\Pi}\_{m1} \text{vec}(\mathbf{X}\_{aug}) + \boldsymbol{\Pi}\_{m2} \text{vec}(F) \tag{36}$$

Then the wavelet coefficients of the optimal control law will be in vector form as

$$\text{vec}\{F\} = -\boldsymbol{\Pi}\_{m2}^{-1}\{\boldsymbol{\Delta}\_1^T \quad \mathbf{T}\_{\boldsymbol{\gamma}}^{-1}\boldsymbol{\Delta}\_1^T \{\mathbf{P}\_m^{-1}\otimes I\_4\}\} \text{ } \boldsymbol{\Pi}\_{m1} \text{ } \text{vec}\{\mathbf{X}\_{\text{aug}}\} \tag{37}$$

Consequently, the optimal vectors of *vec X*( ) and *vec F*( ) are found, respectively, in the following forms

$$\begin{split} \text{vec}(\mathbf{X}) &= \left( I\_{4m} + \Delta\_1 \boldsymbol{\Pi}\_{m2}^{-1} \boldsymbol{\Delta}\_1^T \boldsymbol{\Delta}\_1^T (\mathbf{P}\_m^{-1} \otimes \boldsymbol{I}\_4) \right) \boldsymbol{\Pi}\_{m1} \begin{bmatrix} I\_{4m} \\ \boldsymbol{\Gamma}\_{\boldsymbol{\gamma}}^{-1} (\boldsymbol{P}\_m^T \otimes \boldsymbol{I}\_4)^{-1} \end{bmatrix}^{-1} \{ \boldsymbol{\Delta}\_2 \text{vec}(\boldsymbol{D}\_\epsilon) + \left( \boldsymbol{\Delta}\_1 \boldsymbol{\Pi}\_{m2}^{-1} \right)^{-1} \} \\ &\times \left[ \boldsymbol{\Delta}\_1^T \quad \boldsymbol{\Gamma}\_{\boldsymbol{\gamma}}^{-1} \boldsymbol{\Delta}\_1^T (\mathbf{P}\_m^{-1} \otimes \boldsymbol{I}\_4) \right] \boldsymbol{\Pi}\_{m1} \begin{bmatrix} \mathbf{0}\_{4m} \\ \boldsymbol{\Gamma}\_{\boldsymbol{\gamma}}^{-1} (\boldsymbol{P}\_m^T \otimes \boldsymbol{I}\_4)^{-1} \end{bmatrix} + \boldsymbol{\Delta}\_3 \right) \text{vec}(\mathbf{X}\_0) + \boldsymbol{\Delta}\_4 \text{vec}(\mathbf{\overline{X}}\_0) \rangle, \end{split} \tag{38}$$

and

168 Advances on Analysis and Control of Vibrations – Theory and Applications

derivatives with respect to time in the cost function as follows:

positive scalar. We can rewrite the cost function (28) as follows:

*T T f T T*

where 1 2 *S diag S S* (,) and 1 2 *Q diag Q Q* (,) with the time scale *<sup>f</sup> <sup>t</sup>*

1 1

*x X*

after its expansion in terms of HFs, we read

( )

 

From (15) and the relation () () *<sup>m</sup> x X*

of *x*( ) 

where 1

*aug*

*<sup>X</sup> <sup>X</sup>*

*f*

*Remark* 2. By substituting () () *<sup>m</sup> x X*

*X*

 

and

The control objective is to find the optimal control *f*( )*t* with respect to a quadratic cost functional approximately such acts as the active force to compensate the vibration transmitted to vehicle body. The quadratic cost functional weights the states and their

1 2 12 0 <sup>111</sup> ( ) ( ) ( ) ( ) ( () () () () () ) <sup>222</sup> *f*

1

0 (1) ( ) <sup>1</sup> [ (1) (1)] ([ ( ) ( )] ( )) 2 2 *f f* (1) ( ) *f f*

*J x xS x x Q Rf d x x*

( ): ( ) ( ) *f f*

<sup>1</sup> ( ) () () *<sup>f</sup>*

into

0

according to the properties of Kronecker product in Appendix A1, we obtain

( ) ( ) () *X X Xd mm m*

and using (4), we read *X X XP* <sup>0</sup> *<sup>m</sup>* . Then, by applying the operator of *vec*(.) and

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

*T T*

0

*<sup>X</sup> x X*

*m aug m*

*ff ff J x S x x S x x t Q x t x t Q x t R f t dt* (28)

where 1 *S* :4 4 , 2 *S* :4 4 , 1 *Q* :4 4 and 2 *Q* :4 4 are positive-definite matrices and *R* is a

1 1 2 1 1

. (29)

*x x*

*T T T T*

2

 

.

, where *X m* :4 denotes the wavelet coefficients

*T*

, we have:

, (32)

*aug vec X vec X vec X* (31)

0 *x x x t dt* ( ) (0) ( ) 

*m n vec X vec X P I vec X* (33)

 

(30)

**5. Optimal control design** 

4 1 11 21 1 4 1 1 1 4 4 1 11 1 4 1 21 1 4 1 1 1 4 ( ) [ ( )] { ( ) ( [ ( )] ) ( ) *f f f f T T m m mm T m T T m mm m m T m I vec F P I P I I <sup>I</sup> P I P I* 0 

A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 171

*T*

*aug vec X vec X vec X* (44)

*<sup>m</sup> vec X vec X P I vec X* (45)

*m m*

  are defined as

*<sup>p</sup> tr ABC vec A I B vec C* ,

(47)

  (42)

1 2 2 22 1 3

*f e*

*m aug m*

(43)

( ) (( ( ) ( )) ( ) ( )) <sup>2</sup> ( )

*f*

( ): ( ) ( ) *f f*

*T T*

Moreover, according to Remark 2 in [18], the following relation is already satisfied between

0 4 () ( ) ( ) () *<sup>T</sup>*

<sup>2</sup> <sup>3</sup> ( ( )) ( ( ) ( ) ( )) <sup>2</sup> *T f T TT*

(46)

2 2 1 32 2 <sup>1</sup> ( ( ) ( ) ( ) ( ) ( ) ( ))

*m m <sup>M</sup>* , respectively.

*P I vec X vec X* (48)

*m*

*mf aug aug m aug aug <sup>m</sup> me e J tr M X SX tr M X C X tr C M F F tr M D D*

Using the property of the Kronecker product, i.e. ( ) ( ) ( ) ( ) *T T*

2 2 1

( )( ) *A C D B AD CB* and ( ) ( ) () *<sup>T</sup> vec ABC C A vec B* , we can write (42) as

*T TT aug m aug <sup>m</sup> em e J vec X vec X C vec F vec F vec D vec D*

*m m*

, 2

*f*

21 1 1 2 22 2 4 1 ( ) ( ) ( ): ( ) *<sup>f</sup>*

*<sup>e</sup> <sup>m</sup> <sup>m</sup> m aug md aug vec D*

1

It is easy to show that the worst-case disturbance in Eq. (47) occurs when

*T T*

*m*

*<sup>X</sup> x X*

*x x x C Cf d d <sup>x</sup>* 

1

Using the relation () () *<sup>m</sup> x X* 

*f*

*X*

and

By using the definition (44) in Eq. (45), we have

where the matrices 8 8

*m mf <sup>M</sup> <sup>m</sup> S MC* and 2 <sup>2</sup>

<sup>1</sup> <sup>2</sup> ( ) *<sup>f</sup>*

 

where *<sup>f</sup> t*

where 1

*aug*

*vec X*( ) and *vec X*( )

2

*<sup>X</sup> <sup>X</sup>*

1

0

2 1

( )

<sup>1</sup> ( ) () () *<sup>f</sup>*

*J x xS <sup>x</sup>*

*f T T*

*T T f*

(1) <sup>1</sup> ( (1) (1)) (1)

*f*

, 1 2 *S diag S S* (,) and 11 22 (,) *T T C diag C C C C* .

1 1

*x X*

, we read

*x*

$$\begin{aligned} &\times \{\Delta\_2 \operatorname{vec}(\boldsymbol{D}\_\varepsilon) + \{\Delta\_1 \boldsymbol{\Pi}\_{m2}^{-1} \boldsymbol{\Delta}\_1^T \boldsymbol{\Delta}\_1^{-1} \boldsymbol{\Delta}\_1^T (\boldsymbol{P}\_m^{-1} \otimes \boldsymbol{I}\_4) \} \boldsymbol{\Pi}\_{m1} \left[ \begin{matrix} \boldsymbol{\sigma}\_{4m} \\ \mathbf{T}\_m^{-1} (\boldsymbol{P}\_m^T \otimes \boldsymbol{I}\_4)^{-1} \end{matrix} + \boldsymbol{\Delta}\_3 \right] \operatorname{vec}(\boldsymbol{X}\_0) \\ &+ \boldsymbol{\Delta}\_4 \operatorname{vec}(\overline{\boldsymbol{X}}\_0) \big{]} - \begin{bmatrix} \boldsymbol{\mathcal{O}}\_{4m} \\ \mathbf{T}\_m^{-1} (\boldsymbol{P}\_m^T \otimes \boldsymbol{I}\_4)^{-1} \end{bmatrix} \operatorname{vec}(\boldsymbol{X}\_0) \big{]}. \end{aligned} \tag{39}$$

Finally, the Haar function-based optimal trajectories and optimal control are obtained approximately from Eq. (27) and () () ( ) *<sup>T</sup> <sup>m</sup> f t t vec F* .

### **6. Robust optimal control design**

In this section, an optimal state feedback controller is to be determined computationally such that the following requirements are satisfied:


The control objective is to find the approximated robust optimal control *f*( )*t* with *H* performance such *f*( )*t* acts as the active force to compensate the vibration transmitted to vehicle body, i.e. guarantees desired 2 *L* gain performance. Next, we shall establish the *H* performance of the system (15) under zero initial condition. To this end, we introduce

$$\mathbf{J} = \left\{ \mathbf{J}^{\prime} \mathbf{x}^{T} (\mathbf{T}\_{f}) \mathbf{S}\_{1} \mathbf{x} (\mathbf{T}\_{f}) + \frac{1}{2} \dot{\mathbf{x}}^{T} (\mathbf{T}\_{f}) \mathbf{S}\_{2} \dot{\mathbf{x}} (\mathbf{T}\_{f}) + \bigvee\_{0}^{T\_{f}} \left( \mathbf{z}^{T} (\mathbf{t}) \mathbf{z}(\mathbf{t}) - \boldsymbol{\gamma}^{2} \mathbf{d}\_{\boldsymbol{\gamma}}^{2} (\mathbf{t}) \right) \, \mathrm{d}t. \tag{40}$$

It is well known that a sufficient condition for achieving robust disturbance attenuation is that the inequality *J* 0 for every 2 ( ) [0, ) *<sup>e</sup> dt L* [33, 36]. Therefore, we will establish conditions under which

$$\inf\_{\text{vec}(F)} \sup\_{\text{vec}(D\_\varepsilon)} \text{J}(\text{vec}(F), \text{vec}(D\_\varepsilon)) \le 0 \tag{41}$$

From (15), the Eq. (40) can be represented as

### A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 171

$$\begin{split} \mathbf{J} &= \mathbf{J}\_{2}^{\prime}(\mathbf{x}^{T}(\mathbf{l}) \quad \frac{\mathbf{1}}{\mathbf{T}\_{f}} \dot{\mathbf{x}}^{T}(\mathbf{l})) \tilde{\mathbf{S}} \begin{pmatrix} \mathbf{x}(\mathbf{l}) \\ \mathbf{T}\_{\gamma}^{-1} \dot{\mathbf{x}}(\mathbf{l}) \end{pmatrix} \\ &+ \frac{\mathbf{T}\_{f}}{2} \int\_{0}^{1} (\mathbf{l}(\mathbf{x}^{T}(\sigma) \quad \mathbf{T}\_{\gamma}^{-1} \dot{\mathbf{x}}^{T}(\sigma)) \tilde{\mathbf{C}} \begin{pmatrix} \mathbf{x}(\sigma) \\ \mathbf{T}\_{\gamma}^{-1} \dot{\mathbf{x}}(\sigma) \end{pmatrix} + \mathbf{C}\_{3}^{2} f^{2}(\sigma) - \gamma^{2} d\_{\gamma}^{2}(\sigma)) \, d\sigma \end{split} \tag{42}$$

where *<sup>f</sup> t* , 1 2 *S diag S S* (,) and 11 22 (,) *T T C diag C C C C* .

Using the relation () () *<sup>m</sup> x X* , we read

$$\begin{bmatrix} \mathbf{x}(\sigma) \\ \mathbf{T}^{-1}\_{\prime} \dot{\mathbf{x}}(\sigma) \end{bmatrix} = \begin{bmatrix} X \\ \mathbf{T}^{-1}\_{\prime} \overline{X} \end{bmatrix} \Psi\_{m}(\sigma) \mathbf{:=} X\_{\text{aug}} \Psi\_{m}(\sigma) \tag{43}$$

where 1 *f aug X <sup>X</sup> <sup>X</sup>* and

170 Advances on Analysis and Control of Vibrations – Theory and Applications

4 4 0 1 1 0

 

*m T m*

0 ( )) ( )}. ( )

*vec X vec X P I*

*f*

approximately from Eq. (27) and () () ( ) *<sup>T</sup>*

such that the following requirements are satisfied:

i. the closed-loop system is asymptotically stable;

**6. Robust optimal control design** 

non-zero ( ) [0, ) *<sup>e</sup> d t* where

we introduce

conditions under which

From (15), the Eq. (40) can be represented as

4

*f*

*em m m T*

4 1 11 21 1 4 1 1 1

*T T m m mm T*

*f*

4 1 11 2 1 21 1 4 1 1 1 3 0

( ( ) ( [ ( )] ) () ( )

Finally, the Haar function-based optimal trajectories and optimal control are obtained

In this section, an optimal state feedback controller is to be determined computationally

0 is a prescribed scalar.

The control objective is to find the approximated robust optimal control *f*( )*t* with *H* performance such *f*( )*t* acts as the active force to compensate the vibration transmitted to vehicle body, i.e. guarantees desired 2 *L* gain performance. Next, we shall establish the *H* performance of the system (15) under zero initial condition. To this end,

2 2 1 1

*ff ff J x S x x S x z t z t d t dt*

It is well known that a sufficient condition for achieving robust disturbance attenuation is that the inequality *J* 0 for every 2 ( ) [0, ) *<sup>e</sup> dt L* [33, 36]. Therefore, we will establish

<sup>1</sup> ( ) ( ) ( ) ( ) ( ( ) ( ) ( )) . <sup>2</sup>

( ( ), ( )) 0

0

*Inf Sup J vec F vec D* (41)

*f*

*T*

*<sup>m</sup> f t t vec F* .

ii. under zero initial condition, the closed-loop system satisfies 2 2 () () *<sup>e</sup> zt d t*

2 2 1 2

() ( )

*e <sup>e</sup> vec F vec D*

*TT T*

*T T m*

*vec D P I vec X P I*

( ) [ ( )] { ( )

*vec F P I P I*

*f*

4

*f*

4

*m*

*I*

4

*e*

(40)

for any

(39)

4 1 11 1

*f*

*T T m*

*f*

*m*

*I*

 

0

*m*

4 1 21 1 4 1 1 1

*mm m m T*

*<sup>I</sup> P I P I*

( [ ( )] ) ( )

$$\operatorname{vec}(\mathbf{X}\_{\text{aug}}) = \left[ \operatorname{vec}^T(\mathbf{X}) \quad \mathbf{T}\_f^{-1} \operatorname{vec}^T(\overline{\mathbf{X}}) \right]^T \tag{44}$$

Moreover, according to Remark 2 in [18], the following relation is already satisfied between *vec X*( ) and *vec X*( )

$$\text{vec}(X) - \text{vec}(X\_0) = (P\_m^T \otimes I\_4) \text{ } \text{vec}(\overline{X}) \tag{45}$$

By using the definition (44) in Eq. (45), we have

$$J = \left\{ \right\} \left( tr\left( M\_{m\text{f}} X\_{\text{aug}}^T \tilde{S} X\_{\text{aug}} \right) \right\} + \frac{T\_f}{2} \left( tr\left( M\_m X\_{\text{aug}}^T \tilde{C} X\_{\text{aug}} \right) + tr\left( \mathbb{C}\_3^2 M\_m F^T F \right) - \gamma^2 tr\left( M\_m D\_\varepsilon^T D\_\varepsilon \right) \right) \tag{46}$$

Using the property of the Kronecker product, i.e. ( ) ( ) ( ) ( ) *T T <sup>p</sup> tr ABC vec A I B vec C* , ( )( ) *A C D B AD CB* and ( ) ( ) () *<sup>T</sup> vec ABC C A vec B* , we can write (42) as

$$J = \frac{1}{2} (\text{vec}^T \{ \mathbf{X}\_{\text{aug}} \} \Pi\_{m1} \text{vec} \{ \mathbf{X}\_{\text{aug}} \} + \mathbf{C}\_3^2 \{ \text{vec}^T \{ \mathbf{F} \} \Pi\_{m2} \text{vec} \{ \mathbf{F} \} - \boldsymbol{\gamma}^2 \text{vec}^T \{ \mathbf{D}\_\epsilon \} \Pi\_{m2} \text{vec} \{ \mathbf{D}\_\epsilon \}) \tag{47}$$

where the matrices 8 8 1 *m m m* , 2 *m m m* are defined as <sup>1</sup> <sup>2</sup> ( ) *<sup>f</sup> m mf <sup>M</sup> <sup>m</sup> S MC* and 2 <sup>2</sup> *f m m <sup>M</sup>* , respectively.

It is easy to show that the worst-case disturbance in Eq. (47) occurs when

$$\text{vec}^\*(\boldsymbol{D}\_{\boldsymbol{\epsilon}}) = \boldsymbol{\gamma}^{-2} \boldsymbol{\Pi}\_{m2}^{-1} \begin{bmatrix} \boldsymbol{\Lambda}\_2^T & \boldsymbol{\Gamma}\_{\boldsymbol{\epsilon}}^{-1} \boldsymbol{\Lambda}\_2^T (\boldsymbol{P}\_m^{-1} \otimes \boldsymbol{I}\_4) \end{bmatrix} \boldsymbol{\Pi}\_{m1} \text{vec} \{\boldsymbol{X}\_{\text{aug}}\} \coloneqq \boldsymbol{\gamma}^{-2} \boldsymbol{\Pi}\_{m1} \text{vec} \{\boldsymbol{X}\_{\text{aug}}\} \tag{48}$$

By substituting Eq. (48) into Eq. (47) we obtain

$$\begin{array}{cc} \text{Inf} & \text{Sup } f(\text{vec}(F), \text{vec}(D\_e)) = \text{Inf} & f(\text{vec}(F), \text{vec}^\*(D\_e)) \text{} \end{array} \tag{49}$$

A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 173

1 1 {, } *S S* and the vectors*C*<sup>1</sup> , *C*2 and the

In this section the proposed computational methodology is applied to the vehicle enginebody vibration system (15) such the exogenous disturbance ( ) *<sup>e</sup> d t* is assumed to be a *Sin* (.) function at the frequency of 10 *Hz* . The system parameters, used for the design and simulation are given in Tables 1 and 2 in the *Appendix B*. Table 3 in the *Appendix* gives the pole-zero locations of 8th –order model of the vehicle engine-body vibration system. It is clear that the vehicle engine-body vibration system is unstable and has the nonminimum phase property. The objective is to find the approximated robust optimal displacement of the chassis and robust optimal input force with *H* performance using HWs at the finite

scalar *C*3 in the controlled output *z t*( ) in Eq. (15) are chosen as 124 *S S* 0 , <sup>1</sup> *C* [0, 1, 1, 2],

**Figure 5.** Comparison of displacement of the chassis found by HWs at resolution level *j* 5 (solid)

the resolution level equal to 3.15 and 5 , respectively, i.e. 3.15

To compare the approximate solutions <sup>2</sup> *x t*( ) and *f*( )*t* , found by HWs, to the analytical solution found by Theorem 1 in the Appendix C, we choose the performance bound and

curves found are plotted in Figures 5 and 6. It is clear that the effect of the engine

and *j* 5 . The time

time interval 0, 1 . Moreover, the matrices 4 4

**7. Numerical results** 

<sup>2</sup> *C* [3, 1, 0,1] and 3 *C* 1 .

and by analytical solution (dashed).

Minimizing the right-hand side of Eq. (49) results in the algebraic relation between wavelet coefficients of the robust optimal control and of the optimal state trajectories in the following closed form

$$\begin{split} \text{vec}(F) &= -\mathbb{C}\_{3}^{-2} \Pi\_{m2}^{-1} \begin{bmatrix} \boldsymbol{\Lambda}\_{1}^{T} & \boldsymbol{\Gamma}\_{\cdot}^{-1} \boldsymbol{\Lambda}\_{1}^{T} (\boldsymbol{P}\_{m}^{-1} \otimes \boldsymbol{I}\_{4}) \end{bmatrix} (\boldsymbol{\Pi}\_{m1} - \boldsymbol{\gamma}^{-2} \boldsymbol{\Pi}\_{\cdot m}^{T} \boldsymbol{\Pi}\_{m2} \boldsymbol{\Pi}\_{m4}) \text{vec}(\mathbf{X}\_{\text{aug}}) \\ &:= \boldsymbol{\Pi}\_{mf} \text{vec}(\mathbf{X}\_{\text{aug}}). \end{split} \tag{50}$$

As a result we have

$$\begin{array}{cccc}\text{Inf} & \text{Sup } & \text{J} \text{(vec}(F), \text{vec}(D\_{\varepsilon})) \leq \text{vec}^{T} \langle \mathbf{X}\_{\text{aug}} \rangle \langle \Pi\_{m1} + R\Pi\_{m}^{T} \Pi\_{m2} \Pi\_{m\not\leq} - \gamma^{2} \Pi\_{m}^{T} \Pi\_{m2} \Pi\_{m\not\leq} \text{(vec} \langle \mathbf{X}\_{\text{aug}} \rangle \text{ (51)}\\ \text{vec}(F) & \text{vec}(D\_{\varepsilon}) & \end{array}$$

Consequently, if there exists positive scalar to the matrix inequality

$$
\boldsymbol{\Pi}\_{m1} + \boldsymbol{\mathsf{C}}\_{3}^{2} \boldsymbol{\Pi}\_{m\!\!\!/ }^{T} \boldsymbol{\Pi}\_{m2} \boldsymbol{\Pi}\_{m\!\!\!/ } - \boldsymbol{\chi}^{2} \boldsymbol{\Pi}\_{m\!\!\!/ }^{T} \boldsymbol{\Pi}\_{m2} \boldsymbol{\Pi}\_{m\!\!\!/ } \leq 0 \tag{52}
$$

then inequality (41) is concluded.

From the relations above we obtain the robust optimal vectors of *vec X*( ) and *vec F*( ) after some matrix calculations, respectively, in the following forms

$$\begin{split} \text{vec}\{\mathbf{X}\} &= \left(I\_{4m} - \left(\boldsymbol{\Delta}\_{1}\boldsymbol{\Pi}\_{mf} + \boldsymbol{\gamma}^{-2}\boldsymbol{\Delta}\_{2}\boldsymbol{\Pi}\_{md}\right)\right] \begin{bmatrix} I\_{4m} \\ \mathbf{T}\_{\boldsymbol{\gamma}}^{-1} \left(\boldsymbol{P}\_{m}^{T} \otimes I\_{4}\right)^{-1} \end{bmatrix} \text{-} \begin{bmatrix} \left(\boldsymbol{\Delta}\_{3} - \left(\boldsymbol{\Delta}\_{1}\boldsymbol{\Pi}\_{mf} + \boldsymbol{\gamma}^{-2}\boldsymbol{\Delta}\_{2}\boldsymbol{\Pi}\_{md}\right) \right) \\\\ \boldsymbol{\gamma} \begin{bmatrix} \boldsymbol{0}\_{4m} \\ \mathbf{T}\_{\boldsymbol{\gamma}}^{-1} \left(\boldsymbol{P}\_{m}^{T} \otimes I\_{4}\right)^{-1} \end{bmatrix} \text{vec}\left(\mathbf{X}\_{0}\right) + \boldsymbol{\Delta}\_{4} \text{vec}\left(\boldsymbol{\overline{X}}\_{0}\right) \end{bmatrix} , \end{split} \tag{53}$$

and

4 4 2 1 1 1 41 2 1 1 4 4 4 4 2 31 2 1 11 1 0 4 4 4 1 1 4 4 ( ) {( (( ( ) ) ( ) ( ) 0 0 ( ( ) ) ) () () () ( ( ( ) *f f f f f m m mf T T m mf md m m m m mf md T T m m m T m m I I vec F <sup>I</sup> P I P I vec X PI PI I <sup>I</sup> P I* 4 2 1 1 2 1 1 4 0 4 ) ) ( ))} ( ) *<sup>f</sup> m mf md T m I vec X P I* (54)

Finally, the Haar wavelet-based robust optimal trajectories and robust optimal control are obtained approximately from Eq. (27) and () () ( ) *<sup>T</sup> <sup>m</sup> f t t vec F* , respectively.

### **7. Numerical results**

172 Advances on Analysis and Control of Vibrations – Theory and Applications

*e*

() ( ) ( )

21 1 1 2

( ( ), ( )) ( ( ), ( ))

*Inf Sup J vec F vec D Inf J vec F vec D* (49)

*e e vec F vec D vec F*

Minimizing the right-hand side of Eq. (49) results in the algebraic relation between wavelet coefficients of the robust optimal control and of the optimal state trajectories in the

3 21 1 4 1 <sup>2</sup> ( ) ( )( )( )

*vec F C P I vec X*

*<sup>e</sup> aug m m mf m md aug vec F vec D*

13 2 <sup>2</sup> 0 *mf md T T <sup>m</sup> C m mf m md* 

From the relations above we obtain the robust optimal vectors of *vec X*( ) and *vec F*( ) after

41 2 1 1 31 2

*I*

*m m mf md T mf md m*

 

() ( ( ) ) (( ( ) ( )

4 4 2 31 2 1 11 1 0

( ( ) ) ) () () ()

*f f*

*mf md T*

*m m*

*I I*

( ) {( (( ( ) ) ( ) ( )

*mf T T m mf md*

*vec F <sup>I</sup> P I P I*

*mf md T T*

Finally, the Haar wavelet-based robust optimal trajectories and robust optimal control are

*f f*

2 2

1 1 04 0

*vec X vec X P I*

) ( ) ( )), ( )

*f*

*Inf Sup J vec F vec D vec X R*

some matrix calculations, respectively, in the following forms

4

obtained approximately from Eq. (27) and () () ( ) *<sup>T</sup>*

*vec X I P I*

*m T m*

0

*f*

4 1 1 4 4

 

*I*

*m*

*m*

*f*

( ( ( )

*T m*

*<sup>I</sup> P I*

4

*f md T T T*

( ( ), ( )) ( ) ( )( ) *mf md*

*T TT*

*m m m m md aug*

12 2

to the matrix inequality

4 2 12

4 4 2 1 1 1 41 2 1 1 4 4

*m m*

0 0

*m m*

*m m*

4 4

4 2 1 1 2 1 1 4 0

*<sup>m</sup> f t t vec F* , respectively.

 

*vec X PI PI*

*m*

*m*

*I*

4 ) ) ( ))} ( ) *<sup>f</sup>*

*vec X P I*

4

2

(50)

(53)

(54)

*vec X* (51)

(52)

 

By substituting Eq. (48) into Eq. (47) we obtain

: ( ).

Consequently, if there exists positive scalar

then inequality (41) is concluded.

*mf aug*

*vec X*

following closed form

As a result we have

*e*

() ( )

and

In this section the proposed computational methodology is applied to the vehicle enginebody vibration system (15) such the exogenous disturbance ( ) *<sup>e</sup> d t* is assumed to be a *Sin* (.) function at the frequency of 10 *Hz* . The system parameters, used for the design and simulation are given in Tables 1 and 2 in the *Appendix B*. Table 3 in the *Appendix* gives the pole-zero locations of 8th –order model of the vehicle engine-body vibration system. It is clear that the vehicle engine-body vibration system is unstable and has the nonminimum phase property. The objective is to find the approximated robust optimal displacement of the chassis and robust optimal input force with *H* performance using HWs at the finite time interval 0, 1 . Moreover, the matrices 4 4 1 1 {, } *S S* and the vectors*C*<sup>1</sup> , *C*2 and the scalar *C*3 in the controlled output *z t*( ) in Eq. (15) are chosen as 124 *S S* 0 , <sup>1</sup> *C* [0, 1, 1, 2], <sup>2</sup> *C* [3, 1, 0,1] and 3 *C* 1 .

**Figure 5.** Comparison of displacement of the chassis found by HWs at resolution level *j* 5 (solid) and by analytical solution (dashed).

To compare the approximate solutions <sup>2</sup> *x t*( ) and *f*( )*t* , found by HWs, to the analytical solution found by Theorem 1 in the Appendix C, we choose the performance bound and the resolution level equal to 3.15 and 5 , respectively, i.e. 3.15 and *j* 5 . The time curves found are plotted in Figures 5 and 6. It is clear that the effect of the engine

disturbance is attenuated onto the displacement of the chassis as the output as well. In other words, *f*( )*t* compensates the vibration transmitted to the chassis. Compare the Haar wavelet based solutions to the continuous solutions using the differential Riccati equation, the approximate solutions (53) and (54) deliver both, robust control *f*( )*t* and state trajectory *x t*( ) in one step by solving linear algebraic equations instead of solving nonlinear differential Riccati equation, while accuracy can easily be improved by increasing the resolution level *j* .

A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 175

**9. Appendix** 

**9.1. Appendix A** 

results can be established:

**9.2. Appendix B** 

**Table 1.** The vehicle body parameter.

A1. *Some properties of Kronecker product*

A2. *Derivatives of inner products of Kronecker product*

Let *A* :*p q* , *Bq r* : ,*Cr s* : and *Dq t* : be fixed matrices, then we have:

( ) ( ) ( ),

*T T*

*tr ABC vec A I B vec C tr ABC vec A I B vec C A C D B AD CB*

 

*T T T*

*vec ABC C A vec B*

( )( ) .

( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ),

Let *A* :*n n* be fixed constants and : 1 *x n* be a vector of variables. Then, the following

*x Ax Ax Ax*

Let be a *p q* matrix whose entries are a matrix function of the elements of *Yst* : , where *Y* is a function of matrix : *m n* . That is, 1 ( ) *Y* , where 2 *Y X* ( ) . The matrix

*p n*

( ) ( ),

*Ax vec A*

( ) ,

*Ax <sup>A</sup> x*

( )

A3. *Chain rule for matrix derivatives using Kronecker product*

of derivatives of with respect to is given by

*x*

( )

*vec Y I I*

Parameters Values *Mb* 1000 [*kg*] *bI* 810 [ <sup>2</sup> *kg m* ] *<sup>b</sup> k* 20000 [*N/m*] *<sup>b</sup> c* 300 [*N/m/s*] *<sup>b</sup> L* 2.5 [*m*]

 

*T*

*T T*

*x*

*p*

*p*

*T*

( )

.

*vec Y*

**Figure 6.** Comparison of input force found by HWs at resolution level *j* 5 (solid) and by analytical solution (dashed).

### **8. Conclusion**

This paper presented the modelling of engine-body vibration structure to control of bounce and pitch vibrations using HWs. To this aim, the Haar wavelet-based optimal control for vibration reduction of the engine-body system was developed computationally. The Haar wavelet properties were introduced and utilized to find the approximate solutions of optimal trajectories and robust optimal control by solving only algebraic equations instead of solving the Riccati differential equation. Numerical results were presented to illustrate the advantage of the approach.

### **9. Appendix**

174 Advances on Analysis and Control of Vibrations – Theory and Applications

increasing the resolution level *j* .

solution (dashed).

**8. Conclusion** 

advantage of the approach.

disturbance is attenuated onto the displacement of the chassis as the output as well. In other words, *f*( )*t* compensates the vibration transmitted to the chassis. Compare the Haar wavelet based solutions to the continuous solutions using the differential Riccati equation, the approximate solutions (53) and (54) deliver both, robust control *f*( )*t* and state trajectory *x t*( ) in one step by solving linear algebraic equations instead of solving nonlinear differential Riccati equation, while accuracy can easily be improved by

**Figure 6.** Comparison of input force found by HWs at resolution level *j* 5 (solid) and by analytical

This paper presented the modelling of engine-body vibration structure to control of bounce and pitch vibrations using HWs. To this aim, the Haar wavelet-based optimal control for vibration reduction of the engine-body system was developed computationally. The Haar wavelet properties were introduced and utilized to find the approximate solutions of optimal trajectories and robust optimal control by solving only algebraic equations instead of solving the Riccati differential equation. Numerical results were presented to illustrate the

### **9.1. Appendix A**

### A1. *Some properties of Kronecker product*

Let *A* :*p q* , *Bq r* : ,*Cr s* : and *Dq t* : be fixed matrices, then we have:

$$\begin{aligned} \text{vec}(ABC) &= \text{(C}^{T} \otimes A) \text{ } \text{vec}(B), \\ \text{tr}(ABC) &= \text{vec}^{T} \text{(A}^{T} \text{) (I}\_{p} \otimes B) \text{ } \text{vec}(C), \\ \text{tr}(ABC) &= \text{vec}^{T} \text{(A}^{T} \text{) (I}\_{p} \otimes B) \text{ } \text{vec}(C), \\ \text{(A} \otimes \text{C}) \text{ (D} \otimes B) &= A \, D \otimes \text{C} \, B. \end{aligned}$$

### A2. *Derivatives of inner products of Kronecker product*

Let *A* :*n n* be fixed constants and : 1 *x n* be a vector of variables. Then, the following results can be established:

$$\begin{aligned} \frac{\hat{\boldsymbol{\mathcal{O}}}(A\boldsymbol{\mathbf{x}})}{\hat{\boldsymbol{\mathcal{O}}}\boldsymbol{\boldsymbol{x}}} &= \text{vec}\left(\boldsymbol{A}\right)\boldsymbol{\prime} \\ \frac{\hat{\boldsymbol{\mathcal{O}}}(A\boldsymbol{\boldsymbol{x}})}{\hat{\boldsymbol{\mathcal{O}}}\boldsymbol{x}^{T}} &= \boldsymbol{A}\boldsymbol{\prime} \\ \frac{\hat{\boldsymbol{\mathcal{O}}}(\boldsymbol{x}^{T}A\boldsymbol{\boldsymbol{x}})}{\hat{\boldsymbol{\mathcal{O}}}\boldsymbol{x}} &= A\,\boldsymbol{x} + \boldsymbol{A}^{T}\boldsymbol{x} \end{aligned}$$

### A3. *Chain rule for matrix derivatives using Kronecker product*

Let be a *p q* matrix whose entries are a matrix function of the elements of *Yst* : , where *Y* is a function of matrix : *m n* . That is, 1 ( ) *Y* , where 2 *Y X* ( ) . The matrix of derivatives of with respect to is given by

$$\frac{\partial Z}{\partial \mathbf{X}} = \left\{ \frac{\partial \operatorname{vec}^T(Y)}{\partial \mathbf{X}} \otimes I\_p \right\} \left\{ I\_n \otimes \frac{\partial Z}{\partial \operatorname{vec}(Y)} \right\} \dots$$

### **9.2. Appendix B**


**Table 1.** The vehicle body parameter.


A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 177

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[4] Cavallo A., Maria G., and Setola R., 'A Sliding Manifold approach for Vibration

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[10] Hino M., Iwai Z., Mizumoto I., and Kohzawa R., 'Active Vibration Control of a Multi-Degree-of-Freedom Structure by the Use of Robust Decentralized Simple Adaptive

[11] Hong J., and Bernstein D. S., 'Bode Integral Constraints, Collocation and Spill Over in Active Noise and Vibration Control' *IEEE Trans. on Control Systems Technology*, 6(1), 1998. [12] Horng I.R., and Chou J.H., 'Analysis, Parameter Estimation and Optimal Control of Time-Delay Systems via Chebyshev series' *Int. J. Control*, 41, 1221-1234, 1985. [13] Hsiao C.H. and Wang W.J., 'State Analysis and Parameter Estimation of Bilinear Systems via Haar Wavelets' *IEEE Trans. Circuits and Systems I: Fundamental Theory and* 

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[16] Karimi H.R., Lohmann B., Jabehdar Maralani P. and Moshiri B. 'A Computational Method for Solving Optimal Control and Parameter Estimation of Linear Systems Using

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215-232, 1984.

35, October, 1993.

176 Advances on Analysis and Control of Vibrations – Theory and Applications

**Table 2.** The engine parameters.


**Table 3.** Pole-zero locations of the 8th -order model.

### **9.3. Appendix C**

*Theorem 1* (State Feedback) [9]. Consider dynamical system

$$\begin{cases}
\dot{\mathfrak{x}}(t) = A\,\mathfrak{x}(t) + B\_1\,\mathfrak{u}(t) + B\_2\,\mathfrak{w}(t) \\
\boldsymbol{z}(t) = C\,\mathfrak{x}(t) + D\,\mathfrak{u}(t)
\end{cases}$$

under assumption 1 (, ,) *A B C* is stabilizable. For a given 0 , the differential Riccati equation

$$\dot{X} = A^T \ X + X \ A + X(\gamma^{-2} B\_2 \ B\_2^T - B\_1 \ B\_1^T) X + \mathcal{C}^T \mathcal{C}^T$$

has a positive semi-definite solution *X t*( ) such that <sup>2</sup> 11 22 ( ) () *T T <sup>A</sup> BB B B Xt* is asymptotically stable. Then the control law 1 ( ) ( ) ( ): ( ) ( ) *<sup>T</sup> ut B Xt xt Kt Xt* is stabilizing and satisfies 2 2 *zt wt* () () .

### **Author details**

Hamid Reza Karimi

*Department of Engineering, Faculty of Engineering and Science, University of Agder, Grimstad, Norway* 

### **10. References**

176 Advances on Analysis and Control of Vibrations – Theory and Applications

**Table 2.** The engine parameters.

**9.3. Appendix C** 

equation

satisfies

*Norway* 

**Author details** 

Hamid Reza Karimi

**Table 3.** Pole-zero locations of the 8th -order model.

2 2 *zt wt* () () .

*Theorem 1* (State Feedback) [9]. Consider dynamical system

under assumption 1 (, ,) *A B C* is stabilizable. For a given 0

Parameters Values

*Me* 250 [*kg*]

*eI* 8.10 [ <sup>2</sup> *kg m* ]

*<sup>e</sup> k* 200000 [*N/m*]

*<sup>e</sup> c* 200 [*N/m/s*]

*<sup>e</sup> L* 0.5 [*m*]

Poles Zeros -6.2313 *i* 111.62 -6.23 *i* 111.69


0.14 *i* 29.48 0.03 *i* 26.86

1 2 () () () ()

, the differential Riccati

11 22 ( ) () *T T <sup>A</sup> BB B B Xt* is

*x t Ax t B u t B w t*

2 22 11 ( ) *<sup>T</sup> TT T X A X XA X B B B B X C C* 

asymptotically stable. Then the control law 1 ( ) ( ) ( ): ( ) ( ) *<sup>T</sup> ut B Xt xt Kt Xt* is stabilizing and

*Department of Engineering, Faculty of Engineering and Science, University of Agder, Grimstad,* 

() () ()

*z t Cx t Du t* 

has a positive semi-definite solution *X t*( ) such that <sup>2</sup>


	- [19] Karkosch H.J., Svaricek F., Shoureshi R. and Vance, J.L., 'Automotive Applications of Active Vibration Control' *Proc. ECC*, 2000.

**Chapter 0**

**Chapter 8**

**Multi-Objective Control Design with Pole**

Tore Bakka and Hamid Reza Karimi

http://dx.doi.org/10.5772/46403

**1. Introduction**

Additional information is available at the end of the chapter

from the wind is proportional to the cube of the wind speed.

work is properly cited.

**Placement Constraints for Wind Turbine System**

The demand for energy world wide is increasing every day. And in these "green times" renewable energy is a hot topic all over the world. Wind energy is currently one of the most popular energy sectors. The growth in the wind power industry has been tremendous over the last decade, its been increasing every year and it is nowadays one of the most promising sources for renewable energy. Since the early 1990s wind power has enjoyed a renewed interest, particularly in the European Union where the annual growth rate is about 20%. It is also a growing interest in offshore wind turbines, either bottom fixed or floating. Offshore wind is higher and less turbulent than the conditions we find onshore. In order to sustain this growth in interest and industry, wind turbine performance must continue to be improved. The wind turbines are getting bigger and bigger which in turn leads to larger torques and loads on critical parts of the structure. This calls for a multi-objective control approach, which means we want to achieve several control objectives at the same time. E.g. maximize the power output while mitigating any unwanted oscillations in critical parts of the wind turbine structure. One of the major reasons the wind turbine is a challenging task to control is due to the nonlinearity in the relationship between turning wind into power. The power extracted

A wind turbines power production capability is often presented in relation to wind speed, as shown in Fig 1. From the figure we see that the power capability vs. wind speed is divided into four regions of operation. Region I is the start up phase. As the wind accelerates beyond the cut-in speed, we enter region II. A common control strategy in this region is to keep the pitch angle constant while controlling the generator torque. At the point where the wind speed is higher than the rated wind speed of the turbine (rated speed), we enter region III. In this region the torque is kept constant and the controlling parameter is the pitching angle. This is the region we are concerned with in this paper, i.e. the above rated wind speed conditions. The last region is the shutting-down phase (cut-out). The expression for power produced by

> ©2012 Reza Karimi and Bakka, 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

©2012 Reza Karimi and Bakka, 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.

