Closed-Loop Position Control of PreisachHystereses

R.B. Gorbet∗

Dept. of Electrical EngineeringUniversity of Waterloo

Waterloo, Ontario, Canada

K.A. MorrisDept. of Applied Mathematics

University of WaterlooWaterloo, Ontario, Canada

ABSTRACT

We discuss closed-loop control of hysteretic systems having Preisach rep-resentations. We apply dissipativity methods, the key component of whichis the determination of an appropriate supply rate. The concept of cyclo-dissipation is used to identify a new supply rate for Preisach hystereses anddissipativity with respect to this supply rate is shown. The result is useful inthe design of output feedback controllers, and complements a previous resultusing feedback of the output derivative. We consider the specific example ofa shape memory alloy actuator and identify a class of controllers which pro-vide stability of position feedback with current as the control variable. Thisdemonstrates one way in which system dynamics that couple with hysteresiscan be included in controller design.

Keywords: hysteresis, smart materials, position control, stability, Preisach,shape memory alloy, dissipativity

∗Corresponding author: University of Waterloo, ECE Dept., 200 University AvenueWest, Waterloo, ON N2L 3G1, CANADA. +1-519-885-1211 ×3489 (voice), +1-519-746-3077 (fax), rbgorbet@uwaterloo.ca

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1 Introduction

Hysteresis non-linearities are present in the dynamics of many physical sys-tems, and notably in a class of materials known as “smart materials.” Smartmaterials such as shape memory alloys (SMA), piezoceramics, and magne-tostrictive materials hold high promise for the design of a new generationof actuation systems. Although each has particular advantages, actuatorsmade from these so-called smart materials are generally scalable, lightweight,smaller than more traditional alternatives, and have fewer moving parts. Akey disadvantage of these actuators is the presence, to varying degrees indifferent materials, of a hysteresis non-linearity. This non-linearity can becomplex in shape, introduces memory into the system, and is not generallysector-bounded. This behaviour, combined with uncertain identification ofmodel parameters, can make design of reliable controllers for these systemsdifficult.

It is little surprise that piezoceramic actuators, which have the smallesthysteresis of any of these materials, are those found most often in commer-cial applications. While SMA have enjoyed reasonable success in “on-off”applications, SMA and other smart materials have yet to achieve broad ac-ceptance as alternative options for controlled actuation. This is due at leastin part to the complexity of their behaviour. In order to extract the fullpotential from closed-loop systems incorporating these materials, designersmust take non-linear behaviour into account.

The prime consideration in the current work is proving the stability ofthe closed-loop control system. This stability is of the utmost importancein many proposed applications of smart actuators, especially in the medicaland aerospace fields. A number of different researchers have worked withPreisach and related models, and have been successful in designing controllersfor hysteretic systems with Preisach representations. Because of the non-linearities involved, however, few have considered the theoretical stability ofthe resulting control systems. We list first some of the common approachesto improving performance, then describes past work on closed-loop stabilityfor Preisach models.

Under certain reasonable parameter assumptions, the Preisach model canbe inverted in a relatively straightforward and computationally inexpensivemanner. By implementing a model-based inverse as part of the closed-loopcontroller[14, e.g.] or as a feed-forward compensator[9, 23], the hysteresis canbe (approximately) linearized. A linear feedback controller can be used to

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reduce any residual error due to model uncertainty. While this approach hasshown good results in the literature, no attempt has been made to demon-strate the stability of the resulting closed-loop system.

In [24], the authors introduce an adaptive inverse hysteresis model. Usedto reduce the hysteresis and in conjunction with an adaptive feedback con-troller, the authors show good tracking performance for unknown hystereses.While the hysteresis inverse model appears quite general, stability results areavailable only for hystereses demonstrating a certain amount of symmetry.SMA actuators are one example of a system which would not satisfy thesymmetry constraints.

A novel approach to hysteresis compensation is found in [5], where the au-thors model the hysteretic behaviour as a frequency-dependent delay. Theyintroduce the concept of a “phaser,” a constant-phase, unity-gain linear con-troller. An approximation to this ideal phaser is used to cancel the identifiedsystem phase delay in the frequency band of interest, resulting in remark-able linearization for periodic inputs. If the range of operating frequencies islarge and the delay varies in this region, multiple phasers can be combinedto (mostly) linearize the hysteresis. This method works quite well, especiallyfor piezoceramics, which have a relatively well-behaved hysteresis curve. Un-fortunately, the method is only suited for periodic signals. Stability is notinvestigated.

In this work, we consider hysteresis non-linearities which can be repre-sented by the Preisach hysteresis model. Few researchers have investigatedthe stability of closed-loop control for smart materials using this general hys-teresis model. Closed-loop stability has been shown in the past for feedbackusing the derivative of the output[12, 21]. Other researchers have addressedthe stability question for specific material models, for example, the one-dimensional energy-based model for SMA actuators in [20]. The focus ofthis paper being the Preisach model, the reader is referred to [2, 7, 20] forexamples of non-Preisach work.

Dissipativity theory, which is a generalization of the well-known passivitytheory, is applied. Mathematically, dissipativity is a property of the generalmodel structure and is independent of any parameter errors or assumptionsabout linearity. Thus, controllers designed using these energy-based methodsenjoy a large degree of robustness.

We provide two general results for use in the design of stabilizing control-lers: the first for feedback of the output derivative, and the second for outputposition feedback. We then apply the second result to an SMA actuator, as

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an example of how coupled dynamics (in this case, the heating dynamics)can be accounted for in the controller design. This is the first work to ourknowledge to demonstrate closed-loop stability for position feedback.

The Preisach hysteresis model is adopted for its generality. While themodel was originally developed to represent magnetic hysteresis, experimen-tal evidence indicates that it also describes other hysteretic dynamics, suchas phase transition problems [11, 14, e.g.]. As a result, the main results areuseful for a broad class of systems.

Section 2 of the paper discusses dissipative systems, and dissipation inhysteretic systems in particular. We use the concept of cyclo-dissipation topostulate a new supply rate for hysteretic systems. In Section 3, we give abrief description of the Preisach model of hysteresis. Section 4 demonstratesdissipativity of the Preisach model with respect to the supply rates uy anduy. The two are not equivalent, and the connection is non-trivial. In Section5, the dissipativity result is applied to define a class of stabilizing controllersfor position control of an SMA actuator, using position feedback. Section6 presents experimental results showing robust position control of an SMAactuator.

2 Dissipative Systems

In this section, we recall definitions of passivity and dissipativity. We thengive a formal definition of a dissipative system that is more general thanearlier definitions. We define cyclo-dissipativity and show its connectionto dissipative systems. Cyclo-dissipativity is used to motivate appropriatesupply rates for hysteretic systems.

We first present a formal definition of a dynamical system. If u(t) is afunction defined on the real line,

uT (t) :=

{u(t) t < T,0 t ≥ T.

Definition 2.1 [25] We define a dynamical system Σ through sets U , U , andX and a map φ : IR2 ×X × U 7→ X. The following axioms are satisfied:

1. The input space, U , consists of a class of U-valued functions on thereal line, for which uT (t) ∈ L2(t0,∞;U) for all t0 ∈ IR, T > t0.

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2. Define IR+2 := {(t2, t1) ∈ IR× IR; t2 ≥ t1}. The state transition function

φ : IR+2 × X × U 7→ X and defines the state x ∈ X through the rela-

tion x(t2) = φ(t2, t1, x(t1), u). This function satisfies the usual axiomsfor autonomous dynamical systems: For all t0, t1 ∈ IR, x0 ∈ X andu, u1, u2 ∈ U ,

(a) Consistency of initial condition: φ(t, t, x0, u) = x0,

(b) Semigroup property: φ(t2, t1, φ(t1, t0, x0, u), u) = φ(t2, t0, x0, u),

(c) Causality: u1(t) = u2(t) for t0 < t < t1 implies φ(t, t0, x0, u1) =φ(t, t0, x0, u2) for t0 < t < t1,

(d) Time Invariance: φ(t1 + T, t2 + T, x0, u2) = φ(t1, t2, x0, u1) for allT ≥ 0, t2 ≥ t1, u1 ∈ U , u2(t) = u1(t+ T ).

For a given dynamical system, we will say that (x, u) is a path on [t0, t1] ifthere is x0 ∈ X such that x(t) = φ(t, t0, x0, u), t0 ≤ t ≤ t1. The setX is calledthe state space, and U the input space. An important feature of a dynamicalsystem representation is that the state must account for all memory in thesystem. For most systems, the state cannot be measured. We define anmeasured output y(t) in a set Y through the operator r : X × U → Y :

y(t) = r(x(t), u(t)).

Note the output at time t depends only on the values of the state x and inputu at time t.

Passivity

Passivity theory is a set of energy-based definitions and theorems which canbe useful in analyzing the stability of interconnected systems. Consider thenon-linear time-varying scalar function

y = f(t, u). (2.1)

This system is said to be passive if there exist non-negative constants δ andε such that ∫ T

0

u(t)y(t)dt ≥ ε ‖u‖2 + δ ‖y‖2 (2.2)

for all inputs u defined on [0, T ]. System (2.1) is said to be strictly inputpassive if ε > 0 and strictly output passive if δ > 0.

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A nice physical interpretation of equation (2.2) exists when the units ofthe product u y are power. In this case, u y is the energy supply rate, and theintegral in (2.2) is the total energy supplied over [0, T ]. Equation (2.2) saysthat the energy added to a passive system over [0, T ] must no greater thanthe energy supplied. In other words, a passive system cannot generate energyinternally. System passivity is a very useful concept in feedback systems, asthe passivity characteristics of interconnected systems can be used to showstability of the closed loop system.

Most dynamical systems, such as a capacitor, have the ability to storeenergy for later recovery. Suppose that such a system were allowed to havestored energy at time t = 0, which was then recovered over [0, T ]. Thiswould result in a negative net energy over [0, T ], and a violation of equation(2.2). One way to modify the definitions above is to qualify the definition(and any results) with the assumption that the system starts in a state ofminimum stored energy: a “relaxed” state. This assumption is generallymade implicitly. For instance, for many systems the zero state is a point ofminimum energy, and it is typically assumed that all systems start with zeroinitial state.

An extension of this concept to allow for energy storage is the basis ofdissipativity theory.

Dissipativity

Definition 2.2 [25] A dynamical system is dissipative if there exists a non-negative function V : X 7→ IR+ and a function w : U × Y 7→ IR such that forall t1 ≥ t0,

V (x0) +

∫ t1

t0

w(u, y)dt ≥ V (x1) (2.3)

where u ∈ U , x1 = φ(t1, t0, x0, u) and y = r(x, u).

The function V is generally called the storage function. If V is identifiedwith energy stored in the system, and w identified with the power supplied;then dissipative systems are identified as those for which the stored energyis less than or equal to the initial energy plus the work done on the system.The supply rate and storage function are, in general, not unique. (In fact,the storage function is unique if and only if the system is conservative i.e.equality holds in the dissipation inequality [25].) Some possible abstract

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storage functions are discussed in [25]. One common definition of supplyrate is

w(u, y) = 〈y,Qy〉+ 〈y,Ru〉+ 〈u, Su〉, (2.4)

where Q, R, S are real matrices of appropriate dimensions. This is the basisof the definition of dissipativity in [19].

Given the supply rate in (2.4) and the trivial storage function V (x) = 0,passivity is the special case where Q = S = 0 and R = I. Strict inputpassivity is obtained by defining S = −ε2I and R = I.

The definitions of passivity and dissipativity considered thus far havean intuitive interpretation when the units of u y are power. However, theassociated definitions and results hold, even when the “supply rate” and“storage function” have no actual physical meaning. Still, the determinationof these functions is key to demonstrating dissipativity for a given system,and it is useful to be able to look to energy as a guide.

Many systems which are physically dissipative have more general energysupply rates than that in Defn. 2.2. In many applications the supply rate w isof the form σ(x, u)(x) where σ(x, u) is a linear functional on the state-space.For example, in mechanical systems we are often concerned with controllingthe position y of an object by applying an input force u. The instantaneouspower in this case is given by u y, rather than u y. Similarly, in the classicalexample of linear elasticity, setting x to be strain and σ to be stress, thesystem is generally dissipative with w = σx. In many cases, the measuredquantity is not velocity x and so the system is not dissipative in the classicalsense. It is therefore useful to introduce more general definitions.

Definition 2.3 Let w : X × X × U → R. Define for an arbitrary interval[t0, t1] and state x0 ∈ X the work function W

W[t0,t1](x0, u) =

∫ t1

t0

w(x0, x(t), u(t))dt

where x(t) = φ(t, t0, x0, u). The domain of W is all (x0, u) ∈ X×U such thatw(x0, x(·), u(·)) is well-defined and in L1(t0, t1). We say that u is admissibleon [t0, t1] for a given x0 if (x0, u) is in the domain of the work function.

Note that the output of a system is not defined or used here. The followingdefinition of dissipativity is identical to Definition 2.2 except that we allow

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work functions other than ∫ t1

t0

w(u, y)dt.

Definition 2.4 A dynamical system Σ is dissipative with supply rate w ifthere exists a non-negative functional V : X 7→ IR+, such that for all t0, t1 ∈IR, x0 ∈ X and all admissible u, the dissipation inequality

V (x0) +

∫ t1

t0

w(x0, x(τ), u(τ))dτ ≥ V (x1) (2.5)

is satisfied, where x1 = φ(t1, t0, x0, u).

The following example is a system that clearly dissipates energy. However,it is not passive, nor dissipative in the usual sense. It is dissipative in themore general sense defined above.

Example 2.1 [17, Theorem 4.1] Consider the second-order system

Mz(t) +Dz(t) +Kz(t) = Fu(t)

with position sensorsy(t) = Cdz(t)

where we assume there exists Q such that C ′dQ = F . Let x = (z, z) be thestate and assume that the state-space is reachable from 0. Choose any c soc2M < D and p so that p2C ′dCd < c2K. The system is dissipative with supplyrate

−p2〈y, y〉+ c2〈y,Qu〉+ 〈y, Qu〉.

If two systems which both dissipate energy are connected in feedback,it does not necessarily follow that the closed-loop system is stable. This isillustrated by the following simple example.

Example 2.2

z1(t) + 12z1(t) + 100z1(t) = u1(t) z2(t) + .001z2(t) + .1z2(t) = u2(t)

u1 = r1 − z2 u2 = r2 + z1

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Each system can be considered as a model of damped linear spring, and eachsystem dissipates energy. However, the interconnected system is unstable.The connection of two dissipative systems affects the stability of the closedloop.

Dissipativity theory is important in controller design because dissipationof subsystems can be used to show stability of the closed loop. There are twomain approaches: (1) show that a closed loop storage function composed ofstorage functions for the subsystems is actually a Lyapunov function e.g. [26]or (2) combine the supply rates for the subsystems and show input/outputstability for the closed loop e.g. [19]. The passivity theorem, small gaintheorem and other sector-based stability results are all special cases of thesecond approach. Both techniques are demonstrated in [17].

Cyclo-dissipativity

As mentioned above, the choice of storage function can be guided by theenergy within the system, or one of the more abstract choices may be made.Choice of supply rate can be more difficult. The following concept of cyclo-dissipativity has been found to be useful in that regard.

Many physical systems have the property that positive net work must bedone on the system in order to bring it through a cycle from a given stateback to the same state. This motivates the following definition, similar to[18, Defn. 8].

Definition 2.5 A dynamical system Σ is cyclo-dissipative with supply ratew if ∫ t1

t0

w(x0, x, u)dτ ≥ 0 (2.6)

whenever x(t1) = x(t0) = x0.

Notice that every dissipative system is cyclo-dissipative; the converse is false.However, cyclo-dissipative systems do have several important connectionswith dissipative systems. As in [26], we will say that a system is partiallydissipative if there is a function V : X 7→ IR, not necessarily non-negative,such that the dissipation inequality (2.5) is satisfied.

Many cyclo-dissipative systems are at least partially dissipative [26, Theo-rem IV.7]. Partial dissipativity can be used in closed loop stability/instabilitytheorems in a manner identical to that for connections of dissipative systems.However, one of the most useful aspects of the notion of cyclo-dissipativity

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y+

y− ld(y) la(y) u+u−

y

Figure 1: Hysteresis Loop

is that it provides a guide to an appropriate supply rate for a system. Thisapproach is very useful in the context of hysteretic systems.

Many hysteretic systems have a dynamic behaviour which is characterisedby nested loops and branching in the input-output plane. Figure 1 illustratesone such loop.

Suppose that for any path on [t0, t2] where (u(t0), y(t0)) = (u(t2), y(t2))the loop L is counter-clockwise, as shown in Figure 1. Consider first, withoutloss of generality, the case where y increases from y− at time t0 to y+ at someintermediate time t1, and then decreases to y− again at time t2. Followinga similar analysis in terms of energy in [4], denote the value of the input onthe ascending loop by la(y) and on the descending loop by ld(y). Since theloop is counter-clockwise, la(y) > ld(y), y− < y < y+. The area enclosed byany loop L is

0 ≤∫ y+

y−

la(y)− ld(y)dy (2.7)

=

∫ y+

y−

la(y)dy −∫ y+

y−

ld(y)dy

=

∫ t1

t0

la(y)ydt−∫ t1

t2

ld(y)ydt

=

∫ t1

t0

u(t)ydt−∫ t1

t2

u(t)ydt

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=

∫ t1

t0

u(t)ydt

We obtain an identical inequality if y is first decreased to y− and then in-creased to y+. Comparing equations (2.6) and (2.7), the system appears tobe cyclo-dissipative with supply rate u(t)y(t).

Now, suppose that instead the system output is defined as

e = y+ − y.

A simple sketch shows that the loops are now clockwise. Denote the valueof y as u increases from u− to u+ by la(u), and by ld(u) as u decreases fromu+ to u−. We obtain, in a manner similar to that above, the inequality

0 ≤∫Lu(t)e(t)dt

over any loop L.We have thus identified two potential supply rates for hysteretic systems:

uy and ue. In order to show dissipativity with respect to these supply rates,we need to know more about the internal dynamics of the system. In otherwords, we need a model. We consider here the Preisach model because of itsgenerality and ability to reproduce minor loops.

3 Preisach Model of Hysteresis

In this work, we use the Preisach model, which is appealing for its generalstructure. It was first developed to describe the hysteresis in magnetic mate-rials. However, several experimental results suggest that it describes the be-haviour in many other hysteretic materials, for instance SMA’s e.g. [11, 14].The chief assumption is that the hysteresis is rate-independent. Rate inde-pendence means that the curves (u, y) ∈ IR2 are invariant for changes in theinput rate, such as changes in frequency.

A very brief description of the Preisach model is given below. A morecomplete description can be found in [12]. A detailed study of Preisachmodels can be found in [4, 16]. The basic building block of the Preisachmodel is the hysteresis relay γ. A relay is characterized by its half-widthr > 0 and the input offset s, and is denoted γr,s. The behaviour of therelay is described schematically in Figure 2. The structure of the model is

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+1

−1

r

us

γr,s

Figure 2: Hysteresis Relay

u y

µ(r1, s1)

µ(rn, sn)

Figure 3: Preisach Model Structure

illustrated in Figure 3. The output is computed as the weighted sum of relayoutputs; the value µ(r, s) represents the weighting of the relay γr,s. Therelay output, and hence the Preisach model, is only defined for continuousinputs u. As this input varies with time, each individual relay adjusts itsoutput according to the current input value, and the weighted sum of allrelay outputs provides the overall system output

y(t) =

∫ ∞

0

∫ ∞

−∞µ(r, s)γr,s[u](t)dsdr. (3.1)

The half-plane IR+× IR is often referred to as the Preisach plane, P . Thecollection of weights µ(r, s) forms the Preisach weighting function µ : P 7→ IR.This weighting function is experimentally determined for a given system; see

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[1, 16] for two different approaches to the identification problem. In anyphysical setup, there are limitations which can be interpreted as a restrictionon the support of µ. For instance, control input saturation, say at u, meansthat some relays in P can never be exercised and cannot contribute to achange in output. This effectively restricts the domain of non-zero µ to atriangle Pr = {(r, s) ∈ P| |s| ≤ u − r}. The fact that relays outside Pr donot contribute to output is modelled by setting µ = 0 outside Pr.

For the remainder of this work, we will assume that the weighting functionµ belongs to the set Mp, defined as follows.

Definition 3.1 Mp is the set of all weighting functions µ : P 7→ IR whichare bounded, piece-wise continuous, and non-negative.

The first two properties are true for physical systems having continuous,bounded outputs. The non-negativity requirement will limit the class ofPreisach models for which dissipativity can be shown, but this limit is notoverly restrictive. For example, models satisfying µ ∈Mp have been reportedfor SMA actuators[11] and for piezoceramics [14].

In particular, it is easy to show that µ ∈Mp means the maximum outputvalue y is finite and can be written

y =

∫ ∞

0

∫ ∞

−∞µ(r, s)dsdr, (3.2)

the value of output when all relays are “turned on.”The Preisach plane can be used to track individual relay states by ob-

serving the evolution of the Preisach plane boundary, ψ, in Pr. Figure 4illustrates the Preisach plane concepts which follow.

First, the relays are divided into two time-varying sets P− and P+ definedas follows:

P±(t) = {(r, s) ∈ Pr | output of γr,s at t is ± 1}. (3.3)

Each set is connected; ψ is the line separating P+ from P−, with P+ below P−.When an arbitrary input is applied, monotonically increasing input segmentsgenerate boundary segments of slope -1, while monotonically decreasing inputsegments generate boundary segments of slope +1. Input reversals causecorners in the boundary. The boundary is piecewise linear and Lipschitzcontinuous with Lipschitz constant 1.

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u(t)

u

−u

P−

Pr

P+

s

r

ψ

Figure 4: Preisach Plane with Sample Boundary, ψ

The boundary ψ always intersects the axis r = 0 at the current inputvalue. With ψ written as a map IR × IR+ 7→ IR, then ψ(t, 0) = u(t). Ifthe boundary at time t is ψ(t, r), applying an input for which u(t) 6= ψ(t, 0)amounts to applying an input with a discontinuity at t. Such an input wouldnot be admissible.

Defining the constant

yo =

∫ ∞

0

∫ 0

−∞µ(r, s)dsdr −

∫ ∞

0

∫ ∞

0

µ(r, s)dsdr,

then since P+ is below ψ and P− above, the output of a classical Preisachoperator (3.1) with boundary ψ can be rewritten

y(t) =

∫ ∞

0

2

∫ ψ(t,r)

0

µ(r, s)dsdr + yo. (3.4)

The history of past input reversals—and hence of hysteresis branchingbehaviour—is stored in the corners of the boundary[10]. In other words, thePreisach plane boundary is the memory of the Preisach model. This suggeststhat it is an appropriate choice of state.

Theorem 3.1 [10] The Preisach model described above forms a dynamicalsystem with input space

U = {u ∈ C0(−∞,∞) ‖u‖∞ ≤ u and limt→−∞

u(t) = 0}

and state-space X the subset of Lipschitz continuous functions ψ ∈ C[0, u]with ψ(u) = 0.

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Henceforth, a dynamical system having state space representation de-scribed above and with output behaviour described by (3.1) will be called a“Preisach system”.

4 Dissipation in the Preisach Model

In Section 2, we used the looping behaviour of hysteretic materials to definetwo potential supply rates. Here we show that Preisach systems are dissi-pative with respect to each of these supply rates. The proofs of this requiredefining a storage function for each supply rate, and then showing that in-equality (2.5) holds. The first result of this section has been demonstratedpreviously, and is recalled here for completeness.

Theorem 4.1 [12] Hysteretic systems admitting a Preisach model for whichµ ∈Mp are dissipative with respect to the supply rate uy.

One storage function satisfying the dissipation inequality (2.5) with thissupply rate can be found in [12]. In that work, the storage function is derviedfrom an analysis of energy loss in individual relays. Another, purely mathe-matical storage function for the same supply rate can be found in [4, Prop.2.5.4], in the context of a general analysis of Preisach operators.

Graphical analysis in section 2 suggested that if e is taken as the output,we also have dissipation with respect to the supply rate ue. In order to provethat this is true, we first define a storage function candidate V , and thenshow that the supply rate ue and V satisfy the dissipation inequality (2.5).

Lemma 4.1 Define

V (ψ) =

∫ ∞

0

2Q(r)dr, (4.1)

where

Q(r) =

∫ ψ

−∞

∫ ∞

s

µ(r, ξ)dξds

+

∫ ∞

ψ

µ(r, s)ds [ψ(t, 0)− ψ(t, r) + r]

+r

∫ ψ

−∞µ(r, s)ds.

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The function V : X 7→ IR+, where X is the state space defined in Theo-rem 3.1, is finite and non-negative and is therefore a valid storage functioncandidate.

Proof. Since µ ∈ Mp, µ(r, s) = 0 for all |s| > u or r > u. Therefore, Q(r)is finite for all r ∈ [0, u] and zero for all r > u. As a result, V is finite for allstates ψ.

To show that V is non-negative we will show that Q is non-negative. Sinceµ ≥ 0 and r ≥ 0, the only term in Q(r, s) that is not obviously non-negativeis the second term: ∫ ∞

ψ

µ(r, s)ds [ψ(t, 0)− ψ(t, r) + r] .

However, for all t, ψ is a Lipschitz function of r, so |ψ(t, 0)− ψ(t, r)| ≤ r andQ (and therefore V ) is non-negative.

Lemma 4.2 For each t ≥ 0,

ψ(t, r) (u(t)− ψ(t, r)) ≥ 0

for all r for which ψ(t, r) is defined.

Proof. This result follows from the wiping-out property of the Preisachmodel [16]. Choose any time t. For every r > 0,

|u(t)− ψ(t, r)| ≤ r.

If |u(t) − ψ(t, r)| < r, then by the wiping-out property ψ(t, r) = 0. Ifu(t) − ψ(t, r) = r, then we necessarily have that u is increasing and sou(t) = ψ(t, r) > 0 if this derivative is defined. Similarly, if u(t)−ψ(t, r) = −r,then u(t) = ψ(t, r) < 0.

Theorem 4.2 Hysteretic systems admitting a Preisach model for which µ ∈Mp are dissipative with respect to the supply rate ue.

Proof. In order to show that the dissipation inequality (2.5) holds, we willshow that

u(t)e(t) ≥ d

dtV (ψ(t))

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at all times t where the storage function V is defined in (4.1).Combining equations (3.2) and (3.4) gives

e(t) = y − y(t) =

∫ ∞

0

2

∫ ∞

ψ(t,r)

µ(r, s)dsdr. (4.2)

We rewrite

V (ψ) =

∫ ∞

0

2

[∫ ψ

0

∫ ∞

s

µ(r, ξ)dξds

+

∫ ∞

ψ

µ(r, s)ds [ψ(t, 0)− ψ(t, r)]

]dr + c

where the constant

c =

∫ ∞

0

2

[∫ 0

−∞

∫ ∞

s

µ(r, ξ)dξds+ r

∫ ∞

−∞µ(r, s)ds

]dr.

We define

ε(t, r) = ψ(t, 0)− ψ(t, r) = u(t)− ψ(t, r).

Then,

u(t)e(t)− d

dtV (ψ)

=

∫ ∞

0

2

[u(t)

∫ ∞

ψ(t,r)

µ(r, s)ds− ψ(t, r)

∫ ∞

ψ(t,r)

µ(r, ξ)dξ

+µ(r, ψ)ψ(t, r)ε(t, r)− ε(t, r)

∫ ∞

ψ(t,r)

µ(r, s)ds

]dr

=

∫ ∞

0

2[µ(r, ψ)ψ(t, r)ε(t, r)

]dr.

Since µ is non-negative by assumption and from Lemma 4.2, ψ(t, r)ε(t, r) ≥ 0for all r,

u(t)e(t)− d

dtV (ψ) ≥ 0

and inequality (2.5) holds.

17

Dissipativity of a dynamical system with respect to a particular supplyrate may be used to design controllers for the system. Theorem 4.1 is useful insystems with output-derivative feedback. Such strategies have been appliedin the past for beam vibration control using piezoceramic actuators [8, e.g.].However, for many applications, the output position is the measured variableand that which we desire to control. In addition, it is sometimes easier tocontrol u rather than u itself; for example, it is easier to control thermalrate (proportional to power input) than it is to control temperature. In thenext section we use the dissipativity result of Theorem 4.2 to obtain robustposition control of an SMA actuator.

5 SMA Actuators

In SMA, hysteresis occurs as a result of temperature- and stress-inducedphase transitions. When a shape memory alloy undergoes a temperature-induced phase transformation, its deformation characteristics are altered.This change is significant enough that SMA wire can be used as an actu-ator by cycling the temperature through the transformation range of thealloy. In the low-temperature phase, a mechanically loaded wire will stretchsignificantly; when it is heated to the high-temperature phase, it will con-tract, exerting a force on the load and recovering up to 6% strain with nopermanent deformation. SMA actuators have been employed successfully inrobotics research applications e.g. [13], and are beginning to be used in morecommercial applications such as valves and dampers e.g. [27].

In [11] it was demonstrated that a Preisach model with µ ∈Mp yields agood fit to experimental data. Thus, we may conclude that SMA actuatorsare dissipative with respect to both supply rates discussed in the previoussection. In [12] dissipation with respect to rate feedback was shown. However,it is difficult to control temperature and also to measure velocity of the wire.Furthermore, we are interested in controlling wire position, not velocity.

Theorem 4.2 could be used directly to obtain stability of a control systemwhere rate of temperature change in the wire is controlled and the mea-surement is position. However, this ignores heat losses in the wire and alsoassumes that we control power. In practice, the closed loop control systemis more accurately modelled as shown in Figure 5. A reference position r1is fed into the closed-loop system, and a controller H reacts to the errorbetween measured position y3 and the reference. The controller attempts to

18

Himax

SMAwire

r1

u1 y1 u2 y2

r2

u3 y3

i P TT = f(T,P)

-

Figure 5: Block Diagram of Current-Controlled SMA Actuator

control the contraction of the SMA wire, and thus the actuator position, bymodifying the temperature. This is done by controlling the current in thewire, i. Noise on the control signal is modelled by the exogenous input r2.

The two centre blocks describe the relationship between the control vari-able, current i, and heat flow, T . The first of these two blocks models theconversion from current to power, and includes overcurrent protection at imaxas well as negative current cutoff to model the lack of active wire cooling.The model for this block is

P (t) =

R i2max i > imaxR i2(t) 0 < i ≤ imax0 i ≤ 0

(5.1)

The second block in the mapping i 7→ T is a first-order model for therelationship between heating and power. The temperature of the wire aboveambient, T , is related to the power by

c1T (t) = −c2T (t) + P (t)

y2 = −c2c1T (t) +

1

c1P (t). (5.2)

The constants c1 and c2 are lumped physical parameters (ci > 0).We now apply the dissipativity result of Theorem 4.2 to derive a robust

stability result for the SMA actuator position control system shown in Figure5.

Lemma 5.1 The heating dynamics with input current u2 = i and outputy2 = T has finite gain and so is dissipative with supply rate

w2(y2, u2) = −〈y2, y2〉+ k〈u2, u2〉 (5.3)

where k is the overall L2-gain.

19

Proof. First consider the model of current i to power P given in (5.1).As a result of the saturation, the model has finite gain Rimax. The heatingequation (5.2) with input P , output y2 and state T also has finite gain.

The series connection from u2 = i to y2 = T therefore has finite gain, andthe heating dynamics are dissipative with supply rate (5.3).

The model (5.2) ignores the effect of latent heats of phase transformation.These latent heats affect heating in the following way: heating is slowedduring the phase transformation from martensite to austenite, and coolingis slowed during the inverse transformation[3]. Also, equation (5.1) assumesa constant electrical resistance when in fact, the resistance of an SMA varieshysteretically with temperature. However, the stability proof which followsdepends only on the fact that each of these equations has finite gain, and soincluding these effects would not have any effect on the stability result.

A large class of controllers will now be shown to stabilize the system.As a corollary, we will show that practical PID controllers can be used forposition control of this system.

Theorem 5.2 The current-controlled SMA shown in Figure 5 is stable if

1. the heating dynamics with input current and output rate of temperaturechange have finite gain k2,

w2(y2, u2) = −〈y2, y2〉+ k2〈u2, u2〉

2. the controller is externally stable, strictly passive:

〈y1, u1〉 > ≥ δ1〈u1, u1〉

with L2-gain k1 of the controller satisfying k1 < δ21.

Proof. The system can be considered to consist of three blocks, as shown inFigure 5: the controller, the heating dynamics, and the SMA wire hysteresis.The dissipativity of each of these blocks will be determined, and the resultcombined to show overall closed-loop stability. Refer to the block diagramof Figure 5 for notation.

Controller: The controller H is assumed to be strictly passive with finitegain. This means that it is dissipative with supply rate

w1(y1, u1) = −α1〈y1, y1〉+ (k1α1 − δ1)〈u1, u1〉+ 〈y1, u1〉

20

where k1 is the gain and α1 > 0 is arbitrary.

Heating: From Lemma 5.1, the heating dynamics are dissipative with supplyrate

w2(y2, u2) = −α2〈y2, y2〉+ k2α2〈u2, u2〉where k2 is the gain and α2 > 0 is arbitrary.

SMA Wire: From Theorem 4.2, the SMA wire is dissipative with supplyrate

w3(y3, u3) = α3〈y3, u3〉where u3 = T and α3 > 0 is arbitrary.

The individual dissipativity characteristics of the three blocks can now becombined to show closed-loop stability. Defining

u =

u1

u2

u3

, y =

y1

y2

y3

, r =

r1r20

,then the interconection matrix

H =

0 0 1−1 0 00 −1 0

,satisfies (c.f., Figure 5)

u = r −Hy. (5.4)

Define the diagonal matrices

Q = diag(−α1,−α2, 0)

R = diag(k1α1 − δ1, k2α2, 0)

S = diag(1, 0, α3).

The dissipativity of each block implies that

〈y,Qy〉+ 〈u,Ru〉+ 〈y, Su 〉 ≥ 0.

Using (5.4) to eliminate u and rearranging,

〈y, Qy〉 − 〈y, Sr〉 ≤ 〈r, Rr〉

21

where

Q = −Q−H ′RH +1

2[SH +H ′S ′] , S = S − 2H ′R.

Now, if Q is positive definite, then it will follow as in [19, Thm. 1] that forsome k4 > 0,

‖y‖ ≤ k4‖r‖.

The matrix Q will be positive definite if all three principal minors havepositive determinants e.g. [22, Thm 6B].

We will now show that if the free parameters α1, α2, α3 are chosen care-fully, then Q is positive definite as required. Define α1,α2 in terms of α3 anda new parameter ε:

α1 = 2α3 + ε

α2 = 2α3/k2.

The 3 determinants are

D1 = ε,

D2 =2α3ε

k2

,

D3 =α3F

4k2

whereF = (−8k1)ε

2 + (8δ1 − α3k2 − 16k1α3)ε− 2.

The determinants D1 and D2 will be positive if and only if ε > 0 and α3 > 0.We will choose ε and α3 so that the third determinant D3 is also positive.Choose ε to maximize F :

ε =8δ1 − α3(k2 + 16k1)

16k1

. (5.5)

Provided that α3 > 0 is sufficiently small, ε > 0 as required. Substitutingthis value of ε into F

F =(k2 + 16k1)

2

32k1

α23 −

δ1(k2 + 16k1)

2k1

α3 + 2

(δ21

k1

− 1

).

The function F is a quadratic in α3 and describes a parabola opening up,

with F (0) = 2(δ21k1− 1

). Since k1 < δ2

1, this quantity is positive, so α3 > 0

22

can be chosen sufficiently small that both F (α3) > 0 and the maximizingvalue of ε (5.5) are positive. With these values of the parameters, all threedeterminants are positive. It now follows that Q is positive definite, and theclosed loop is externally stable.

While Theorem 5.2 is motivated by the configuration in Figure 5, thestability result can be generalized. The only property required of the heatingdynamics was finite gain. As a result, Theorem 5.2 applies to any systemwith dynamics coupled in series with a hysteresis, so long as

• the coupled dynamics has finite gain; and

• the hysteresis can be represented by a Preisach model for which theweighting function µ is bounded, piece-wise continuous, and non-negative.

The result allows one to define a class of stabilizing controllers for these non-linear systems, which includes several standard linear controllers, as shownbelow.

Corollary 5.3 If the PID controller with transfer function

H(s) = KP +KI

s+ εI+

KDs

εDs+ 1

where ε,KP , KI , KD > 0 satisfies K2P > k1 where k1 is the L2-gain of the

controller, then it stabilizes a SMA actuator.

Proof: It is straightforward to show that the controller has passivityconstant δ1 = KP . The result then follows from the theorem.

It is difficult to write down a formula for the gain k1 of a general PIDcontroller. However, note that for a pure PI controller, the gain is KP +KI/εI , while for a pure PD controller this quantity is KP + KD/εD. Thecondition thus represents a constraint on the lower bound of the proportionalactionKP . PID controller parameters can be chosen to optimize performanceconsiderations, provided that K2

P > k1 so that the closed-loop is stable.

6 Experimental Results

This section describes the experimental apparatus used to verify the con-troller design results. Closed-loop stability results for non-linear systems are

23

PC-386 currentamplifier

800g

SMA wire

encoderinput

Figure 6: Experimental SMA Actuator Setup

often quite conservative, with the result that controllers which meet the sta-bility requirements result in poor performance. Experimental data is shownwhich demonstrates that the class of stabilizing controllers defined in thiswork contains useful controllers.

The experimental verification is carried out on a one-wire SMA actuatorunder constant-load conditions. Figure 6 shows a diagram of the experimen-tal position control setup.

The actuator consists of a 79cm (loaded, martensite) length of 0.25mmdiameter NiTi “Flexinol” wire from Dynalloy Inc., biased by a load of 800g.This represents a constant stress on the wire of roughly 163 MPa. Theaustenite transformation temperature of the wire is given as 90C; the alloyis martensite at room temperature. The wire is routed over a 3cm diameterpulley, with a contact arc length of approximately 90◦. The shaft of thepulley drives a 2000 count/revolution optical encoder, for a linear resolutionof approximately 47µm. The encoder output is converted to position data,the control variable. To heat the wire, control current is driven through theSMA by a voltage-controlled current amplifier. Wire cooling is not actively

24

controlled. Control calculations are performed in real-time on a 50MHz i386PC with an i387 numeric coprocessor, at a sampling rate of 100Hz.

The control structure is similar to that shown in Figure 5. The wire is notactively cooled, so a negative control output is cut off at zero. In addition,the wire current is saturated at 1 amp in order to protect the wire fromoverheating. These non-linearities were included in the heating model usedin the stability analysis of the previous section.

The following figures show experimental responses using various control-lers of the form

C(s) = Kp +Ki

s+ ε+

Kds

εs+ 1.

This is an approximated PID controller; in all cases, ε = 0.01. These control-lers were discretized and implemented as difference equations on the controlcomputer.

Figure 7 shows the step response of the actuator to various input stepsizes. The PI controller used was tuned manually for a reference step of20mm. Figure 8 shows the expected reduction in the overshoot after theaddition of a derivative term in the controller.

Figure 9 shows the tracking response of the system to a reference which isa combination of sinusoids, and exercises the minor loop branching behaviourof the hysteresis.

The controllers used for these three previous experimental responses sat-isfy the stability condition of Corollary 5.3, K2

p > Kp + Ki/ε. Outside theclass of stabilizing controllers predicted by Corollary 5.3, it is possible toobtain marginal stability (see Figure 10). This behaviour is unexpected,and demonstrates the importance of complete modelling and proper stabilitywork.

As with the majority of non-linear stability results, Theorem 5.2 is asufficient condition. It is possible to see an improvement in performance usingcontrollers which are outside the class of stabilizing controllers predicted byCorollary 5.3 (see Figure 11).

7 Conclusions

A new form of dissipativity, useful for the design of output-feedback control-lers, was shown for Preisach systems with µ ∈ Mp. Using this new dissipa-tivity result, energy-based techniques were applied to demonstrate position

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

Time (seconds)

Wire

con

trac

tion

(mm

)

Figure 7: PI-controlled step response for various target step sizes (Kp =2.6, Ki = 0.04, Kd = 0)

feedback stability for a one-wire SMA actuator under constant load. Heatingdynamics and saturation non-linearities were included in the modelling andstability analysis, in addition to the hysteresis non-linearity inherent in theSMA wire.

A common concern is the conservative nature of some non-linear stabilityresults. Experiments demonstrate here that useful controllers are includedin the class of stabilizing controllers. In addition, it was demonstrated ex-perimentally that marginal stability could be observed using a simple PIcontroller, and the class of stabilizing controllers excludes the controller pro-ducing this behaviour.

26

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

Time (seconds)

Wire

con

trac

tion

(mm

)

Figure 8: PID-controlled step response (Kp = 2.6, Ki = Kd = 0.04)

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

Time(seconds)

Wire

con

trac

tion

(mm

)

exp outRef inp.

Figure 9: P-controlled tracking response (Kp = 1.1)

27

0 20 40 60 80 100 120 140 1600

5

10

15

20

25

30

Time (seconds)

Wire

con

trac

tion

(mm

)

Figure 10: PI-controlled marginally stable step response (Kp = 0.1, Ki =1, Kd = 0)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

Time (seconds)

Wire

con

trac

tion

(mm

)

Figure 11: Improved-performance PI-controlled step response (Kp =0.5, Ki = 0.06, Kd = 0)

28

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