**5. Model-based control strategies for smart actuators**

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

**Figure 14.** Block scheme of the 2-variables operator (a); Example of reconstruction of the *ζ* function,

In order to show the effectiveness of the described procedure, it is important now to show the performances of a multi-variable model as the one described in eqn. (23). In particular, in Fig. 14-(b) it is shown the reconstructed *ζ* function, while in Fig. 14-(a), the scheme of the model is shown. It undergoes the input variable *x*<sup>1</sup> and *x*2, according to eqn. (23), which in the specific case are the magnetic field and the stress, respectively. The output *y*∗(*t*) is compared to the actual measured strain function and the results are sketched in Fig. 15. In particular, in Fig. 15-(a) it is shown the time variation of the input variables, i.e. *x*1(*t*) = *H*(*t*) and *x*2(*t*) = *σ*(*t*), in the first two frames, while in the third, it is described the output of the *ζ*-function, to which the hysteresis operator W is subject. In Fig. 15-(b) it is shown the system response, compared to the time behavior of the measured strain. Here the performances of the model described so far are compared to those of a model proposed in [10], where the


(b)


 - -


 

**Figure 15.** Comparison of the model's behavior to experiments for a prescribed history. Stress (load), magnetic field (curr) and *ζ* (corr-curr) vs. time (a); output displacement compared to the experimental

In conclusion of this section, it is worth to be further stressed that the procedure described so far is of general breath and can be exploited in modeling and compensation of any material/system with multi-variable hysteresis as most of multifunctional materials behave.

 

 

(b)

 

 

 

 






 

 

> 

 

data (b).

 


(a)

named as "effective current" in the vertical axis of the plot (b).

dependence of memory on the stress is not taken into account.

(a)

 

 

 

After a complete review on the modeling and compensation strategies of actual systems with hysteresis, in general working conditions, has been settled, it is possible to discuss different approaches to their control through the definition of specific application frameworks. As a first step, let us start from the classical *model based* control approach, already sketched in Fig. 8, sect. 3

There, the employment of the compensation algorithm allows a easy design of the controller, which could be, as a first attempt, a classical PI system, with all its limits, but with the great advantage of its implementation neatness. This viewpoint has been widely used in smart actuation tasks where Piezo, SMAs or magnetostrictives were employed, [23, 24, 27, 35, 39]. In Fig. 16-(a) the whole control system is shown. In this figure, the block *D*(*s*) = *H*(*s*)*G*(*s*) merge together the linear dynamic part of the actuator and the linear dynamic plant. So the series connection of W and *D*(*s*) represents the *smart actuator* and the plant (SmA). This is a quite general scheme allowing to discuss the control system of the actuator both when loaded by the plant or *unloaded*. The part (b) of the same Fig. 16 shows the same system actuator-plant when it is affected by two inputs (*x*1, *x*2). The detailed discussion of this scheme will be provided later, at the end of this section.

In the example shown in Fig. 17 a triangular waveform with period *T* = 0.02 s is tracked by the control system sketched in Fig. 16-(a), with and without the compensation algorithm described in the previous sections, [12]. The effects of compensation evidence the increase of the tracking performances of the actuator. In [9] it is also discussed the improvements of the control performances also in term of stability.

(a) Model-based control scheme

(b) Control scheme with two variables feeding the Sma

**Figure 16.** Model-based control scheme (a) and its generalization to a multi-variable, model-based controller (b). In both cases, the smart actuator and the plant (SmA) are represented by the connection of W and *D*(*s*)

The examples considered until now refers to *slow* actuation applications, such as micro positioning tasks (cfr. also [8]). When the assumption of low input variation cannot be assumed as the general working condition, a more accurate discussion should be provided. To this aim, let us again consider the Fig 2, showing how the mechanical response of the material is strongly affected by the applied constant stress (i.e. the *pre-stress*). Normally the magnetostrictive material is subject to a prescribed (and optimal) stress, with the aim to maximize such response. When the actuator is working to actuate a prescribed mechanical load, the total stress experienced by the material has very low fluctuations with respect to the applied stress, whenever slow variations of the applied field are concerned. For high frequencies application tasks this assumption no longer holds and the material experiences

#### 16 Will-be-set-by-IN-TECH 160 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

a simultaneous variation of stress and field. In that case the employment of a compensation algorithm with two variable is necessary. To this aim the *multi-variable* hysteresis modeling described in the section above could be an interesting starting point to define suitable multi-variable compensation and control strategies. A first attempt to tackle such a quite demanding problem was settled in [10], where a simple generalization of the Preisach model, taking into account the presence of a second variable (i.e. the stress) in the distribution function was stated and discussed. Such approach had the drawback that only a correction of the operator's output was concerned while no effects on the state were considered. It should be said, however, that even so the system showed promising characteristics, which yielded to further improvements and to the definition of the model described above (eqn. (21)). In the following we will sketch how the latter model can be employed to define suitable and well behaved multi-variable control strategies, [15–17]. However, a preliminary discussion to settle the existence conditions of the compensator for such a kind of multi-variable hysteresis operator is required. Let us now refer to the model specified in eqn. (21) which is re-arranged

(b) Green: compensated control; Blue: uncompensated control

**Figure 17.** Model-based control scheme (a) and its performance in tracking a reference signal (b)

as follows:

$$z \equiv \mathcal{W}^{-1}[y - q] = \zeta(\mathfrak{x}\_1, \mathfrak{x}\_2), \tag{24}$$

where

$$\mathcal{J}: (\mathfrak{x}\_1, \mathfrak{x}\_2) \in \Omega \longrightarrow \mathbb{R}$$

is a continuous function and Ω the set where it is defined. Moreover, consider now the set where the above eqn. (24) admits at least one solution:

$$\mathcal{S} = \left\{ (\mathfrak{x}\_2, \mathfrak{z}) \in \mathbb{R}^2 : \exists \mathfrak{x}\_1 \text{ with } \mathfrak{z}(\mathfrak{x}\_1, \mathfrak{x}\_2) \equiv \mathfrak{z} \right\} \tag{25}$$

and the function:

$$\Phi: \mathfrak{x}\_1 \in X \subseteq \mathbb{R} \longrightarrow (\mathfrak{x}\_2, z) \in \mathbb{S}. \tag{26}$$

It should be noticed that the latter defines a *single valued* function, being *ζ*(*x*1, *x*2) a *single valued* function. Moreover, recall that *e* = *y* − *q* is the model's output without the pure mechanical response represented by *q*(*x*2).

This function could be either injective or not, depending on the characteristics of the real material, as evidenced hereafter. To this aim, let us consider the material, showing the relation between the output *y* − *q* and the input variables *x*<sup>1</sup> and *x*<sup>2</sup> as shown in Fig. 18-(a). There with *z* it has been indicated the input variable corresponding to the output *y* − *q* when *x*<sup>2</sup> = *x*<sup>∗</sup> 2 . If now *x*<sup>2</sup> = *x*� <sup>2</sup> two distinct values of *x*<sup>1</sup> are obtained. This implies that the function Φ is *not injectve* (Fig 18-(b)). Conversely, if we refer to Fig. 19-(a), the value of *x*<sup>1</sup> corresponding

(a) Non monotone response characteristic of a general smart material.

(b) Example of non-injective Φ function

**Figure 18.** Example of the Φ function reconstructed from input-output available data. It can be stressed that the non-monotonicity of the response functions, as sketched in (a), yields to *∂ζ <sup>∂</sup>x*<sup>1</sup> that can assume any sign for assigned *x*<sup>2</sup>

to a given *z* (which, as before, refers to the assigned value of output *y* − *q* on the curve corresponding to *x*<sup>2</sup> = *x*∗ <sup>2</sup> ), is now unique, when it exists. In fact in the same figure, the possibility that no *x*<sup>1</sup> values corresponding to a given couple (*x*2, *z*) is also considered. This example evidences that not all values (*x*2, *z*) belongs to the domain of definition of the function Φ. In this case the function is *injective* in *S*. Moreover, it would appear quite evident that the properties of the Φ function are strictly related to the monotonicity of the functions (*y* − *q*) ↔ *z* or, in other words to the derivatives *∂ζ*(*x*1,*x*¯2) *<sup>∂</sup>x*<sup>1</sup> , as detailed hereafter.

The function

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

a simultaneous variation of stress and field. In that case the employment of a compensation algorithm with two variable is necessary. To this aim the *multi-variable* hysteresis modeling described in the section above could be an interesting starting point to define suitable multi-variable compensation and control strategies. A first attempt to tackle such a quite demanding problem was settled in [10], where a simple generalization of the Preisach model, taking into account the presence of a second variable (i.e. the stress) in the distribution function was stated and discussed. Such approach had the drawback that only a correction of the operator's output was concerned while no effects on the state were considered. It should be said, however, that even so the system showed promising characteristics, which yielded to further improvements and to the definition of the model described above (eqn. (21)). In the following we will sketch how the latter model can be employed to define suitable and well behaved multi-variable control strategies, [15–17]. However, a preliminary discussion to settle the existence conditions of the compensator for such a kind of multi-variable hysteresis operator is required. Let us now refer to the model specified in eqn. (21) which is re-arranged

> --

 

 

control

**Figure 17.** Model-based control scheme (a) and its performance in tracking a reference signal (b)

*ζ* : (*x*1, *x*2) ∈ Ω −→ **R** is a continuous function and Ω the set where it is defined. Moreover, consider now the set

(*x*¯2, *<sup>z</sup>*¯) <sup>∈</sup> **<sup>R</sup>**<sup>2</sup> : <sup>∃</sup>*x*<sup>1</sup> with *<sup>ζ</sup>*(*x*1, *<sup>x</sup>*¯2) <sup>≡</sup> *<sup>z</sup>*¯

It should be noticed that the latter defines a *single valued* function, being *ζ*(*x*1, *x*2) a *single valued* function. Moreover, recall that *e* = *y* − *q* is the model's output without the pure mechanical

This function could be either injective or not, depending on the characteristics of the real material, as evidenced hereafter. To this aim, let us consider the material, showing the relation between the output *y* − *q* and the input variables *x*<sup>1</sup> and *x*<sup>2</sup> as shown in Fig. 18-(a). There with *z* it has been indicated the input variable corresponding to the output *y* − *q* when *x*<sup>2</sup> = *x*<sup>∗</sup>

*not injectve* (Fig 18-(b)). Conversely, if we refer to Fig. 19-(a), the value of *x*<sup>1</sup> corresponding

<sup>2</sup> two distinct values of *x*<sup>1</sup> are obtained. This implies that the function Φ is


*<sup>z</sup>* ≡ W−1[*<sup>y</sup>* <sup>−</sup> *<sup>q</sup>*] = *<sup>ζ</sup>*(*x*1, *<sup>x</sup>*2), (24)

Φ : *x*<sup>1</sup> ∈ *X* ⊆ **R** −→ (*x*2, *z*) ∈ *S*. (26)


 

(b) Green: compensated control; Blue: uncompensated




(25)

2 .


as follows:

and the function:

If now *x*<sup>2</sup> = *x*�

response represented by *q*(*x*2).

where

 

 

 

 

where the above eqn. (24) admits at least one solution:

*S* = 

(a) Model-based control scheme

$$z = \zeta(x\_1, x\_2)$$

can be rearranged by the aid of the function: *<sup>g</sup>* : **<sup>R</sup>**<sup>2</sup> <sup>→</sup> **<sup>R</sup>**2, having components:

$$\begin{cases} z = \mathfrak{f}(\mathfrak{x}\_1, \mathfrak{x}\_2); \\ \mathfrak{x} = \mathfrak{x}\_{2\prime} \end{cases} \tag{27}$$

and *Jacobian*:

$$J\_{\mathcal{S}}(\mathbf{x}\_1, \mathbf{x}\_2) = \begin{vmatrix} \frac{\partial \zeta}{\partial \mathbf{x}\_1} & \frac{\partial \zeta}{\partial \mathbf{x}\_2} \\ 0 & 1 \end{vmatrix} = \begin{vmatrix} \frac{\partial \zeta}{\partial \mathbf{x}\_1} \end{vmatrix} . \tag{28}$$

(a) Monotone response characteristic of a general smart material.

(b) Example of injective Φ function

**Figure 19.** Example of the Φ function reconstructed from input-output available data. It can be stressed that the strict monotonicity of the response functions, as sketched in (a), yields to *∂ζ <sup>∂</sup>x*<sup>1</sup> > 0 for assigned *<sup>x</sup>*2.

#### 18 Will-be-set-by-IN-TECH 162 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

If now we define the set:

$$A = \left\{ (P \equiv (\mathbf{x}\_1, \mathbf{x}\_2) \in \Omega : f(P) \neq \mathbf{0} \right\},\tag{29}$$

it is possible to conclude that whenever *A* is a non-empty set, for every *P* ∈ *A* the *g* function *g* is locally invertible and so the equation *ζ*(*x*1, *x*2) = *z* has only one *x*<sup>1</sup> solution. The sets

$$\mathcal{S}^\* = \left\{ (\mathfrak{x}\_2, \mathfrak{z}) \in \mathbb{R}^2 : (\mathfrak{x}\_1, \mathfrak{x}\_2) \in A \right\} \tag{30}$$

is a subset of *S*, i.e. *S*<sup>∗</sup> ⊆ *S*. As a consequence, Φ is injective in *S*<sup>∗</sup> and finally it admits the inverse function

$$
\Phi^{-1} : \mathbb{S}^\* \longrightarrow X^\*,
$$

being *X*∗ a subset of *X*. This line of reasoning shows the conditions which guarantee the existence of the Φ−<sup>1</sup> function and hence the possibility to define the compensator of the 2-variables operator with hysteresis. This is a crucial point since only with this condition a generalization of the classical *model-based* control strategy can be settled for multi-variable systems with rate independent memory.

The availability of a procedure enabling the inversion of a hysteresis operator depending on two input variables opens the opportunity to extend *model-based* control strategies to systems having in realty two inputs (for example, desired strain and mechanical stress). This need arises in specific dynamic working conditions of magnetostrictives, (i.e. active vibration suppression of structures), where the device could experience forces of comparable magnitude of the applied stress, but could be extended to similar conditions involving different smart materials.

The simplest control strategy exploiting the results concerning the inversion of multi-variable hysteresis operators, is sketched in Fig. 16-(b). There we can observe the compensation algorithm described so far. In particular the operator Γ−<sup>1</sup> which, in open loop, takes the variable *e* = *y* − *q* and provides the corresponding input function for the reference case (i.e. *x*<sup>2</sup> = *x*∗ <sup>2</sup> ), that is *x*<sup>1</sup> = *z*. The inverse of the Φ function, takes *z* and *x*<sup>2</sup> and provides the control variable *x*1. If a given desired signal is provided, in presence of a time varying *x*<sup>2</sup> = *σ*(*t*), the global behavior of the actuators drastically changes and, without a suitable *stress-dependent* compensation algorithm, the controller performances suddenly worsen. For the sake of example, in Fig. 20 it is shown the relative error in tracking a sin(*t*)/*t* function provided a time variation of the applied force over the full simulation time interval of about 400N. It is evident the improved behavior of the 2-variables control (2VC) system with respect to the case when stress corrections are not provided, [18]. The present approach allows to compensate the effects of nonlinearity and hysteresis in more general working conditions, guaranteeing a close-to-linear behavior of the actuator and making so easier the controller design. Unfortunately, the effects of the compensation algorithm together with the PI controller are beneficial for the whole control system whenever the desired signal is slowly varying, but its performances decrease for faster actuation purposes. Further, multi-variable compensation and control is mostly required exactly when faster signal to track are involved. Thus, different control schemes able to guarantee sufficient performances in a wider frequency range are necessary, [19, 22]. As a first attempt to address the issue, let us recall that closed loop control represents a necessary solution, in presence of disturbances model uncertainties, guaranteeing performances otherwise unachievable by open loop solutions. On the contrary, feedforward controls are easier to implement, do not affect the stability of the overall system

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

it is possible to conclude that whenever *A* is a non-empty set, for every *P* ∈ *A* the *g* function *g* is locally invertible and so the equation *ζ*(*x*1, *x*2) = *z* has only one *x*<sup>1</sup> solution. The sets

is a subset of *S*, i.e. *S*<sup>∗</sup> ⊆ *S*. As a consequence, Φ is injective in *S*<sup>∗</sup> and finally it admits the

<sup>Φ</sup>−<sup>1</sup> : *<sup>S</sup>*<sup>∗</sup> −→ *<sup>X</sup>*∗, being *X*∗ a subset of *X*. This line of reasoning shows the conditions which guarantee the existence of the Φ−<sup>1</sup> function and hence the possibility to define the compensator of the 2-variables operator with hysteresis. This is a crucial point since only with this condition a generalization of the classical *model-based* control strategy can be settled for multi-variable

The availability of a procedure enabling the inversion of a hysteresis operator depending on two input variables opens the opportunity to extend *model-based* control strategies to systems having in realty two inputs (for example, desired strain and mechanical stress). This need arises in specific dynamic working conditions of magnetostrictives, (i.e. active vibration suppression of structures), where the device could experience forces of comparable magnitude of the applied stress, but could be extended to similar conditions involving different smart

The simplest control strategy exploiting the results concerning the inversion of multi-variable hysteresis operators, is sketched in Fig. 16-(b). There we can observe the compensation algorithm described so far. In particular the operator Γ−<sup>1</sup> which, in open loop, takes the variable *e* = *y* − *q* and provides the corresponding input function for the reference case

the control variable *x*1. If a given desired signal is provided, in presence of a time varying *x*<sup>2</sup> = *σ*(*t*), the global behavior of the actuators drastically changes and, without a suitable *stress-dependent* compensation algorithm, the controller performances suddenly worsen. For the sake of example, in Fig. 20 it is shown the relative error in tracking a sin(*t*)/*t* function provided a time variation of the applied force over the full simulation time interval of about 400N. It is evident the improved behavior of the 2-variables control (2VC) system with respect to the case when stress corrections are not provided, [18]. The present approach allows to compensate the effects of nonlinearity and hysteresis in more general working conditions, guaranteeing a close-to-linear behavior of the actuator and making so easier the controller design. Unfortunately, the effects of the compensation algorithm together with the PI controller are beneficial for the whole control system whenever the desired signal is slowly varying, but its performances decrease for faster actuation purposes. Further, multi-variable compensation and control is mostly required exactly when faster signal to track are involved. Thus, different control schemes able to guarantee sufficient performances in a wider frequency range are necessary, [19, 22]. As a first attempt to address the issue, let us recall that closed loop control represents a necessary solution, in presence of disturbances model uncertainties, guaranteeing performances otherwise unachievable by open loop solutions. On the contrary, feedforward controls are easier to implement, do not affect the stability of the overall system

<sup>2</sup> ), that is *x*<sup>1</sup> = *z*. The inverse of the Φ function, takes *z* and *x*<sup>2</sup> and provides

(*x*¯2, *<sup>z</sup>*¯) <sup>∈</sup> **<sup>R</sup>**<sup>2</sup> : (*x*1, *<sup>x</sup>*2) <sup>∈</sup> *<sup>A</sup>*

*S*∗ = 

*A* = {(*P* ≡ (*x*1, *x*2) ∈ Ω : *J*(*P*) �= 0} , (29)

(30)

If now we define the set:

inverse function

materials.

(i.e. *x*<sup>2</sup> = *x*∗

systems with rate independent memory.

(a) Desired and measured output vs. time. Up: PI controller; bottom; PI+2VC (b) Applied stress and relative tracking error vs. time

**Figure 20.** Example of real time control task, by considering the effects of compensation with one or two variables

and show faster behavior with respect to feedback solutions since its action depends only on the feedforward controller and not on process dynamics. Modern control systems often employ both actions with the aim to obtain the advantages of both configuration, avoiding, as far as possible, their drawbacks so yielding to a controller with higher performances. Such a configuration is called *two-degrees-of-freedom*, 2-DoF, control system since in this way it is possible to design a controller that weights independently the reference signal and the measurements coming from the process [36, 38]. Examples of 2-DoF control systems for

**Figure 21.** Block schemes of 2-DoF control systems. Hysteresis compensation on the feedback loop (a); Hysteresis compensation on the feedforward action (b)

actuators with memory, proposed in [19], are shown in Fig. 21. In both of the diagrams it is possible to locate a feedback and a feedforward action. In part -(a) of the figure, the feedforward action trivially consists in adding the desired output to the output of the linear controller *C*(*s*). The compensation, is located on the feedback action. In that case both the compensator and actuator with hysteresis experiences exactly the same input history and so they share the same internal state. Conversely, the scheme reported on the part -(b) of the same figure shows the compensation located on the feedforward action, while the feedback loop is without compensation. In such a scheme it is quite evident that the real actuator and its compensator doesn't experiences the same input history. For this reason we could expect a different behavior with respect to the previous system.

**Figure 22.** Tracking performances of a PI control system with 2VC for low frequency application. In this case the period of the desired output is *T* = 5s

**Figure 23.** Tracking performances of a PI control system with 2VC for higher frequency application. In this case the period of the desired output is *T* = 0.1s

In order to evidence the drawbacks of a classical feedback loop employing a simple PI controller, and the improvements provided by the 2-DoF control system, let us discuss the experiments performed by a *real-time* environment where the compensation algorithm and the PI controller are implemented in a PC with Matlab, and a xPC Target toolbox with a 16 bit acquisition/generation card are also employed. Further details are discussed in [19]. In particular, in Fig. 22 it is shown the performances of the feedback system, sketched in Fig. 16, in tracking a triangular waveform, with period of 5 *s*. Here the relative error lies below 3%. When the frequency is increased to 10Hz, the problems in tracking the same waveform dramatically arise and tracking error goes well beyond 15%, as evidenced in Fig 23. The performances results of the scheme reported in Fig. 21-(b) are shown in Fig. 24, with an increase of tracking performances. In Fig. 25, conversely, the performances of the 2-DoF control system presented in Fig. 21-(a) are shown. It can be observed a further increase of the tracking performances of the system with quite low tracking error, since the system provides the same state evolution of both the actuator and compensator.

**Figure 24.** Tracking performances of a 2-DoF control system with 2VC for higher frequency application. In this case the period of the desired output is *T* = 0.1s and the controller is specified in Fig. 21-(b)

**Figure 25.** Tracking performances of a 2-DoF control system with 2VC for higher frequency application. In this case the period of the desired output is *T* = 0.1s and the controller is specified in Fig. 21-(a)

In conclusion the availability of suitable and well assessed tools for the modeling and compensation of actuators with hysteresis enables to define control algorithms that fully exploits this high modeling capabilities. The result is the availability of a model-based control system which gather good performances and quite low computational effort, guaranteeing its suitability for real time and fast actuation tasks.

#### **6. Conclusion**

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





 

 

 


 

**Figure 23.** Tracking performances of a PI control system with 2VC for higher frequency application. In

In order to evidence the drawbacks of a classical feedback loop employing a simple PI controller, and the improvements provided by the 2-DoF control system, let us discuss the experiments performed by a *real-time* environment where the compensation algorithm and the PI controller are implemented in a PC with Matlab, and a xPC Target toolbox with a 16 bit acquisition/generation card are also employed. Further details are discussed in [19]. In particular, in Fig. 22 it is shown the performances of the feedback system, sketched in Fig. 16, in tracking a triangular waveform, with period of 5 *s*. Here the relative error lies below 3%. When the frequency is increased to 10Hz, the problems in tracking the same waveform dramatically arise and tracking error goes well beyond 15%, as evidenced in Fig 23. The performances results of the scheme reported in Fig. 21-(b) are shown in Fig. 24, with an increase of tracking performances. In Fig. 25, conversely, the performances of the 2-DoF control system presented in Fig. 21-(a) are shown. It can be observed a further increase of the tracking performances of the system with quite low tracking error, since the system provides

 

**Figure 22.** Tracking performances of a PI control system with 2VC for low frequency application. In this



 

 

 (b)

 

(b)




 -

> --

 

 

 

 

 

 -

 

 

 

 


case the period of the desired output is *T* = 5s


 ---

 

the same state evolution of both the actuator and compensator.

 

(a)

this case the period of the desired output is *T* = 0.1s


 

(a)


 !

 

> The chapter aimed to provide a view on the basic problems involved when modeling and control of smart materials, normally affected by memory phenomena, are concerned. So, after a brief review of some of the contributions provided in this field, the attention was focused specifically on materials showing a magnetostrictive behavior, such as Terfenol-D and Galfenol. However, all the ideas and tools introduced and discussed could be widely exploited for other smart materials with non-negligible memory effects.

22 Will-be-set-by-IN-TECH 166 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

In Sect. 2 a brief description of the phenomena involved in magnetostrictive materials is drawn and the ideas of their modeling are also provided. In this respect, the basic operators describing systems with hysteresis is presented, with a specific attention to the classical Preisach model. Although the most known and applied approach is the one in the 'rotated' system of coordinates, according to Mayergoyz's book, [28], the adopted formalism allover the chapter is that preferred in the books [7] and [26], showing some interesting characteristic from the modeling viewpoint. In any case, the relationship between them is explicitly pointed out.

In the next section 3 the attention is focused on the main modeling and compensation issues which are normally adopted for control purposes. Here, a specific attention is devoted to the congruency property of Preisach operator, which forces a specific attention in the inversion of the operator. Finally, a basic point is discussed with some detail, that is, the definition of compensation algorithms which doesn't require further computational effort with respect to the 'direct' operator, i.e. the Preisach operator. This specific point becomes crucial when a *model-based* real time control approach is of concern.

In order to put the issues discussed in the chapter in the current framework of smart materials and devices, in section 5 the need to manage two independent variables in controlling the device is emphasized, by proposing a well-behaved procedure to handle the stress and magnetic field simultaneously. The effectiveness of this approach for a more precise description and control of smart devices is also discussed by comparison to measured data. Such discussion paves the way to the last section of the chapter, where some application in *real time* controlling smart materials are presented. In particular, 'standard' and 2-DoF control strategies are presented, all fulfilling the constraint to keep a low computational complexity of the whole control system, still ensuring good tracking and stability performances.
