**2. Hysteresis operators and their basic properties**

The general approach to materials showing nonlinear coupling and hysteresis between the involved variables, can be written as

$$y\_1 = \mathcal{F}\_1(\mathbf{x}\_1, \mathbf{x}\_2),\tag{1}$$

146 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges Modeling, Compensation and Control of Smart Devices with Hysteresis <sup>3</sup> Modeling, Compensation and Control of Smart Devices with Hysteresis 147

$$y\_2 = \mathcal{F}\_2(x\_1, x\_2),\tag{2}$$

being (*x*1, *x*2) the couple of input variables, and (*y*1, *y*2) the couple of output variables. Of course, the choice of input and output variables is, in principle, arbitrary and is performed according to the specific application needs. Moreover, F<sup>1</sup> and F<sup>2</sup> represents arbitrary and general operators with memory, which "link" the input to the output functions. This is a quite generic picture, by which we would just acquaint the reader on the complex phenomena lying below the behavior of such materials. In particular, we would evidence that, generally speaking, equations above define a *multi-variate operator* with memory which could be handled by means of *vector operators* with hysteresis, [28], [25]. Unfortunately, they require huge efforts in identification procedures and high computational weight, so resulting unsuitable for *sensing* or *actuation* tasks. As a consequence, other approaches have been considered with the aim to gather good generality and modeling capabilities to a relative simple handling and computational effort, [28], [1], [42]. There, the basic concern was to fully exploit the "machinery" of the classical hysteresis operators, well suited to link together *one input* to *one output*. They are usually referred to as SISO (Single Input-Single Output) systems. As a starting point, any functional material can be modeled as a SISO system whenever only one of the output variables is of concern and, moreover, one of the inputs is kept constant. For the sake of example, Fig.1-(a) and -(b) sketches the typical elastic response driven by magnetic field or the flux density induced by applied stress, respectively. In each of these pictures, a specific characteristic with memory is described and can easily be modeled through a classical SISO hysteresis operator. In a more general framework, as shown in Fig.2-(a) and -(b), a SISO modeling is still applicable whenever the interest is only on one of the output variables (i.e. the *strain* in picture -(a)), and when *stress* can be assumed as stationary (same picture). Of course, the same reasoning can be applied when the *flux density* as output variable is of concern (picture -(b)). A quite unifying approach to define a wide class of hysteresis operator

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

focused on these materials and their applications, that everyone can experience, navigating on the web, an impressive "blooming" of ideas and proposals. The key element is the consciousness that smart materials achieve new functionalities going well beyond the "sum" of the individual properties and allowing to develop new and really innovative devices.

Among them, those materials coupling mechanical to other physical quantities (thermal, electric or magnetic) received a special attention since their suitability for sensing or actuation goals. The latter involved many researchers in a multi-disciplinary frame in designing and producing really smart actuators, able to provide high forces, or high precision

In particular, as we can find in the scientific literature, the employ of Piezo-electric materials, [23], or magnetostrictives, [21], which are able to cover complementary sets of applications, play a primary role and many devices suited for micropositioning, active vibration control,

Similar conclusions can be drawn for Shape Memory Alloys (SMA) which show huge deformations driven by temperature variations and therefore are generally quite slow [45]. The Ni-Mn-Ga materials, also referred to as Ferromagnetic Shape Memory Alloys (FSMA),

However, all of them, share *rate independent* memory properties which strongly affect the global behavior of the device and its performances. Rate independence means that the observed memory behavior doesn't arise from "dynamics" and therefore is still kept also for "quasi-static" input variations, as happens in the well-known behavior of ferromagnetic

In the sequel, a strategy to "handle" devices employing smart materials with hysteresis in general working conditions will be outlined and several applications employing

magnetostrictives will be discussed to check the validity of the proposed techniques.

**Figure 1.** Elastic and magnetic response of a Terfenol-D sample to a magnetic (a) or mechanical (b) input

The general approach to materials showing nonlinear coupling and hysteresis between the


 

 

 

*y*<sup>1</sup> = F1(*x*1, *x*2), (1)

(b) Inverse Magnetostriction at constant field

*micropositioning*, or high speed responses, in dependence of the selected application.

[40] overcome this limitation still preserving high deformations as SMAs.

smart actuation, ultrasonic generators, are available.

materials. The common way to refer to it is *hysteresis*.


 

(a) Magnetostriction at constant stress

involved variables, can be written as

**2. Hysteresis operators and their basic properties**



 

**Figure 2.** Elastic and magnetic response of a Terfenol-D sample to a magnetic (a) or mechanical (b) input at various magnetic fields and stresses

field

is based on the *Preisach memory* updating rules, [28], [25] and we refer to them in the sequel. For the sake of simplicity and assuming some basic knowledge on the Preisach operator, we can introduce the Preisach hysteresis operator as the superposition of a continuous set of ideal relay operators, *<sup>γ</sup>*ˆ*r s* defined by the parameters *<sup>r</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> and *<sup>s</sup>* <sup>∈</sup> **<sup>R</sup>** as specified in the Fig. 3. Here it should be noticed that the *relay* can be represented by means of the *r* and *s* parameters, or by its switching fields *a* and *b* (*a* ≥ *b*), equivalently. In the latter case the formalism is the one exploited by I.D. Mayergoyz in [28], while the former yields to the formalism used in the books [7, 26] and one can be derived from the other by a simple 45◦ axis rotation, [44]. In the

#### 4 Will-be-set-by-IN-TECH 148 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

sequel we decided to use the second approach which allows the reader to refer to those books so exploiting several results there addressed. On the contrary, this choice has the drawback to discuss some tool of the Preisach formalism (i.e. identification, or inversion algorithms) in a less natural way. For this reason we would further evidence the aim of this chapter, focused in providing a sufficiently complete discussion concerning the modeling and compensation of systems with hysteresis and referring for details to the cited references.

Each *relay*, referred to *Rr*,*s*[*x*], has *r* ≥ 0 and *s* ∈ **R**. We assume now that they are distributed into the **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>** half plane with a prescribed distribution function *<sup>μ</sup>*(*r*,*s*) and the output of this set is the linear weighted superposition of the response of each *relay*. In symbols, [28], [25], [26]:

$$y(t) = \int\_{\mathbb{R}^+} \mu(r, s) R\_{rs}[\mathbf{x}](t) dr ds \tag{3}$$

Of course the past input history of each relay of the distribution results in a non trivial way to compute the integral. However, this problem can be overcome by giving a geometric interpretation to it. First of all, let us associate to each point of the half plane (*r*,*s*) the relay having the same parameters. Moreover, recall that each relay of the distribution switches from the state −1 to +1 if the input field exceeds the upper switching point, that is *x*(*t*) ≥ *s* + *r*, while the opposite transition from positive to negative output is performed if the field is below the lower switching point or, in symbols: *x*(*t*) ≤ *s* − *r*. This yields to the following conclusion: the relay operators switched to the +1 state are those below the straight line of equation *s* + *r* = *x*; conversely, those switched from the state +1 to −1 lie above the line *s* − *r* = *x*. If all the relays start from the state −1 (usually referred as the state of *negative saturation*) and applying the input field *x*(*t*), the line separating the +1 from the −1 elements is sketched in Fig. 3-(b) and represents the state of the system. When, from such initial state,

**Figure 3.** (a) The ideal relay with parameters *r* and *s*; (b) geometrical interpretation of the relay switching

the input is decreased, the relays above the line *s* − *r* = *x*(*t*) switch down to −1. The result is sketched Fig. 4-(a), while in Fig. 4-(b) it is represented the state corresponding to an arbitrary *non-monotone* input variation. The *demagnetized* or *virgin state* is represented in the **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>** half plane as the horizontal axis (*s* = 0). We will refer to it as the *h*<sup>0</sup> state.

After this brief resume, it should be quite easy to realize that the above mentioned rules define a specific relation between the set of continuous input functions, *x*(*t*) and the piece-wise linear function *h*(*r*, *t*), as follows:

$$\Theta : \mathfrak{x} \in \mathcal{C}^0[0, T] \to h(r, t) \in \Psi\_{0\prime} \tag{4}$$

with, *h* the staircase of the classical Preisach model (cfr.[28]), and Ψ<sup>0</sup> the set of admissible Preisach states. The properties of this operator, referred to as Preisach Hysteresis Generator (PHG), can be easily sketched. To this aim, let us consider the functions *x*(*t*) and *z*(*t*), shown in Fig. 5-(a), sharing the same sequence of input extrema and linked by the equation *x*(*t*) = *z* ◦ *ρ*(*t*). The function *ρ* is a time rescaling and ◦ the composition operator. In other words the two functions differ only by the rate of variation. It is easy to verify that the staircases corresponding to *x* and *z* coincide (Fig. 5-(b)). The memory operator, therefore, is not affected by the input rate, but only by the sequence of the input extrema.

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

sequel we decided to use the second approach which allows the reader to refer to those books so exploiting several results there addressed. On the contrary, this choice has the drawback to discuss some tool of the Preisach formalism (i.e. identification, or inversion algorithms) in a less natural way. For this reason we would further evidence the aim of this chapter, focused in providing a sufficiently complete discussion concerning the modeling and compensation

Each *relay*, referred to *Rr*,*s*[*x*], has *r* ≥ 0 and *s* ∈ **R**. We assume now that they are distributed into the **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>** half plane with a prescribed distribution function *<sup>μ</sup>*(*r*,*s*) and the output of this set is the linear weighted superposition of the response of each *relay*. In symbols, [28],

Of course the past input history of each relay of the distribution results in a non trivial way to compute the integral. However, this problem can be overcome by giving a geometric interpretation to it. First of all, let us associate to each point of the half plane (*r*,*s*) the relay having the same parameters. Moreover, recall that each relay of the distribution switches from the state −1 to +1 if the input field exceeds the upper switching point, that is *x*(*t*) ≥ *s* + *r*, while the opposite transition from positive to negative output is performed if the field is below the lower switching point or, in symbols: *x*(*t*) ≤ *s* − *r*. This yields to the following conclusion: the relay operators switched to the +1 state are those below the straight line of equation *s* + *r* = *x*; conversely, those switched from the state +1 to −1 lie above the line *s* − *r* = *x*. If all the relays start from the state −1 (usually referred as the state of *negative saturation*) and applying the input field *x*(*t*), the line separating the +1 from the −1 elements is sketched in Fig. 3-(b) and represents the state of the system. When, from such initial state,

**<sup>R</sup>**<sup>+</sup> *<sup>μ</sup>*(*r*,*s*)*Rr s*[*x*](*t*)*drds* (3)

 


(b)

<sup>Θ</sup> : *<sup>x</sup>* ∈ C0[0, *<sup>T</sup>*] <sup>→</sup> *<sup>h</sup>*(*r*, *<sup>t</sup>*) <sup>∈</sup> <sup>Ψ</sup>0, (4)

 

**Figure 3.** (a) The ideal relay with parameters *r* and *s*; (b) geometrical interpretation of the relay switching the input is decreased, the relays above the line *s* − *r* = *x*(*t*) switch down to −1. The result is sketched Fig. 4-(a), while in Fig. 4-(b) it is represented the state corresponding to an arbitrary *non-monotone* input variation. The *demagnetized* or *virgin state* is represented in the **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>**

After this brief resume, it should be quite easy to realize that the above mentioned rules define a specific relation between the set of continuous input functions, *x*(*t*) and the piece-wise linear

of systems with hysteresis and referring for details to the cited references.

*y*(*t*) =

half plane as the horizontal axis (*s* = 0). We will refer to it as the *h*<sup>0</sup> state.



(a)


function *h*(*r*, *t*), as follows:

[25], [26]:

Let us now consider the memory operator with initial state equal to the *h*−<sup>1</sup> *virgin state*, i.e. *<sup>h</sup>*<sup>0</sup> = *<sup>h</sup>*−<sup>1</sup> undergoing the input *<sup>x</sup>*(*t*), shown in Fig. 5-(a). The vertex of the staircase (*x*3, *x*4) are wiped out as soon as the input exceeds *x*<sup>3</sup> as they were never existed, as shown in Fig. 5-(b). The final state generated by the sequence of input extrema (*x*1, *x*2, *x*3, *x*4, *x*5) coincides with the one generated by the *reduced* input sequence (*x*1, *x*2, *x*5), [41]. The latter is the basic rule for memory updating which is universally referred to as *Preisach deletion rule*. In conclusion, the Preisach-memory operator is *rate independent* and fulfills the *wiping-out* property. What we have just introduced is the machinery enabling the classical Preisach operator, defined in eqn. (3), to take into account *rate-independent* memory phenomena. To

**Figure 4.** (a) staircase in the case of non-monotone input variation (b) Staircase corresponding to a non-monotone input *x*(*t*)

**Figure 5.** (a) staircase in the case of non-monotone input variation (b) Staircase corresponding to a non-monotone input *x*(*t*)

fully exploit such operator, it is possible to define a larger class of operators with hysteresis, based on the Preisach memory updating rules, as proposed in [7], where any operator with Preisach memory can be represented as follows:

$$\mathcal{W} = \mathcal{F} \circ \Theta,\tag{5}$$

where Θ is the PHG, while

$$\mathcal{F}: h(r, x) \in \Psi\_0 \to y(t) \in \mathbb{R}. \tag{6}$$

represents a memoryless functional by which a wide class of hysteresis operators, based on the Preisach memory, can be defined. It should be noted that this formalism, adopted in [7, 26], is slightly different with respect to [28], representing the Preisach plane by means of a rotated coordinate system, [44]. For the sake of example, the following functional

(a) Definition of first order reversal curve (b) Triangular domain over which the Everett integral is defined

**Figure 6.** Identification procedure of a Preisach model, based on the exploitation of first order reversal curves

(a) Characteristic of a floppy disk

(b) Everett function of a floppy disk

**Figure 7.** Example of the identification of the Everett function from experimental data coming from a floppy disk

$$\mathcal{F}: h \in \Psi\_0 \to \int\_0^{+\infty} dr \left( \int\_{-\infty}^h \mu(r, s) ds - \int\_h^{+\infty} \mu(r, s) ds \right), \tag{7}$$

defines the Classical Preisach operator, and *μ*ˆ(*r*,*s*) is a density function to be determined. The functional

$$\mathcal{F}: h \in \Psi\_0 \to \int\_0^{+\infty} \mu(r, s) h(r, t) dr,\tag{8}$$

conversely, defines the well known Prandtl-Ishilinskii operator [7]. Such models have been widely exploited in last years by several authors to compensate the memory of real devices, so allowing a model-based control approach. A non exhaustive review can be found in [44]. It should be quite evident that these models are phenomenological in nature and so require well-defined identification procedures to link their parameters (i.e. the mapping functional, ) to the real material, through a suitable set of measured data. It should be said that such a procedure is not addressed for any arbitrary functional, while, in the case of the Preisach operator, a simple and well-behaved identification procedure is available and, for convenience it is reported with some detail hereafter.

In particular, it could be observed that in order to compute the output of a classical Preisach model, it is not necessary to know the Preisach Distribution Function, *μ*(*r*,*s*) [28], but it is sufficient the knowledge of the integrals of *<sup>μ</sup>*(*r*,*s*) over any triangle of the **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>** half plane and vertex (*u*, *v*), according to the procedure described below. Preliminary, let us define the following function known as *Everett function*, as follows:

$$\mathcal{E}(\mu, v) = \int\_0^u dr \left( \int\_{v-u+r}^v \mu(r, s) ds + \int\_v^{v+u-r} \mu(r, s) ds \right). \tag{9}$$

The link between the actual measured data and the Everett function can be outlined as follows. Let us start from negative saturation and increase the input until the value *x*(*t*) = *a* and the corresponding output is *ya* are attained. Then decrease the input until the value *x*(*t*) = *b* is reached; the corresponding output is referred to as *yab*. The hysteresis branch (Fig. 6-(a)) traced from the reversal point *ya* until, again, the negative saturation is known as *first order reversal curve*, or *for*-curve. It is easy to realize (cfr. also [28]) that the following relation linking the output measured data and the Everett integral holds:

$$\mathcal{E}(u,v) = \frac{y\_a - y\_{ab}}{2} \tag{10}$$

with

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

fully exploit such operator, it is possible to define a larger class of operators with hysteresis, based on the Preisach memory updating rules, as proposed in [7], where any operator with

represents a memoryless functional by which a wide class of hysteresis operators, based on the Preisach memory, can be defined. It should be noted that this formalism, adopted in [7, 26], is slightly different with respect to [28], representing the Preisach plane by means of a rotated

 

 

**Figure 6.** Identification procedure of a Preisach model, based on the exploitation of first order reversal

**Figure 7.** Example of the identification of the Everett function from experimental data coming from a

*μ*(*r*,*s*)*ds* −

coordinate system, [44]. For the sake of example, the following functional

(a) Characteristic of a floppy disk

 +∞ 0

*dr h* −∞

: *h* ∈ Ψ<sup>0</sup> →

W = ◦ Θ, (5)

 

(b) Triangular domain over which the Everett integral is defined

 


(b) Everett function of a floppy disk

*μ*(*r*,*s*)*ds*

, (7)

 +∞ *h*



: *h*(*r*, *x*) ∈ Ψ<sup>0</sup> → *y*(*t*) ∈ **R**. (6)

 

 

Preisach memory can be represented as follows:

where Θ is the PHG, while




(a) Definition of first order reversal curve

curves

floppy disk

$$
\mu = \frac{a - b}{2} \tag{11}
$$

$$v = \frac{a+b}{2}.\tag{12}$$

Let us subdivide the input interval into *N* equal intervals of width Δ*x* = *xmax* − *xmin* and assume *x*<sup>1</sup> = *xmax* = *N*Δ*x*, *xk* = (*N* − *k* + 1)Δ*x*. The corresponding reversal curve (Fig. 6-(b)) so has *N* − *k* + 1 measured points, *yakbn* . Hence the total number of measured points to be used for the identification procedure, are *N* × (*N* − 1)/2, which correspond to the number of nodes in the grid in the **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>** half plane. Now, if we associate the value of the Everett integral to each point of the grid, a discrete version of the Everett function is easily available. The details on its construction and use can be found in [28]. An example of the reversal curves for a real material, and the corresponding Everett function is shown in Fig. 7.
