Actuator Motion

Transcription

1 Control of Hysteretic Systems: R.B. Gorbet Systems Control Group University of Toronto Toronto, Ontario M5S 3G4 A State-Space Approach K.A. Morris Dept. of Applied Mathematics University of Waterloo Waterloo, Ontario N2L 3G1 D.W.L. Wang Dept. of Electrical & Computer Engineering University of Waterloo Waterloo, Ontario N2L 3G1 1 Introduction Hysteresis occurs in a number of applications, and has many causes. A very important class of hysteretic systems are \smart materials" such as shape memory alloys (SMA), magnetostrictive materials and piezoceramics. When a shape memory alloy undergoes a temperature-induced phase transformation, its deformation characteristics are altered. This change is signicant enough that an SMA wire can be used as an actuator by cycling the temperature through the transformation range of the alloy. SMA actuators have been employed successfully in robotics research applications for many years[8, e.g.], and are beginning to be used in more commercial applications such as valves and dampers[14]. The hysteresis in a typical SMA actuator is illustrated in Figure 1. The highly non-linear behaviour and poor identication of hysteretic systems can make design of reliable controllers for these systems dicult. We are interested in control of SMA actuators, and hysteretic systems in general. Several experimental results suggest that a model rst developed to describe the dynamics in magnetic materials, the Preisach model, also describes the behaviour of SMAs and other phase transformation problems[6, 9,e.g.].In this paper we describe the Preisach model and show that it can be formulated as a classical dynamical system. Using a purely abstract analysis of the Preisach operator, it is shown that these systems are dissipative, in a generalization of the sense rst described in [13]. An initial diculty in studying hysteresis is to arrive at a precise denition. The word comes from the Greek word meaning \to lag behind". It is tempting to characterize hysteresis in terms of this lag, or perhaps by the fact that hysteretic systems 1

2 y 0.6 y + Actuator Motion u( ) = sin(4 ) u y ; u ; u + Temp [C] u( ) = sin( ) -0.6 Figure 1: SMA Actuator Hysteresis Figure 2: Output of LTI system: y(t) = R t 0 e;(t; ) u()d t =4::12 have \memory", or that they generate \loops"[10]. While these are all characteristics of hysteretic systems, many linear systems also have delays or memory. As well, for certain inputs, some linear systems can display looping behaviour (cf. Figure 2). The most useful denition for our purposes is obtained by restricting ourselves to rate independent or static hysteresis. Rate independence means that the curves described in R 2 by the couple (u y) areinvariant forchanges in the input rate, such as changes of the frequency. Linear systems are not rate-independent, as evidenced by the dierent outputs obtained in Figure 2. While this restriction excludes some types of hysteresis, it does include many commonly encountered, such as those occurring in phase transitions[3]. Many static hysteretic systems of practical interest, including piezoceramics and shape memory alloys, can be modelled by a Preisach operator. Essentially, the behaviour of the system is described by a weighted sum over a number of relays, each of which has output +1 or ;1. In magnetic materials these relays haveaninterpretation as representing individual magnetic dipoles. A physical description is more elusive for other systems however, the validity of these models has been demonstrated by a number of experiments[6, 9, e.g.]. An important consequence of the rate independence of these systems is that the output is dependent only on the past input extrema, not on all input values. This property will be signicant in obtaining a state-space description of Preisach systems. 2 Preisach Operators The basic building block of the Preisach model is the hysteresis relay. A relay is characterized by its half-width r>0 and the input oset s, and is denoted by r s. The behaviour of the relay is described schematically in Figure 3. The structure of the model is illustrated in Figure 4. The output is computed as the weighted sum of relay outputs the value (r s) represents the weighting of the relay r s. The relay 2

3 r s (r1 s1) +1 u y s r u (rn sn) ;1 Figure 3: Hysteresis Relay Figure 4: Preisach Model Structure output, and hence the Preisach model, is only dened for continuous inputs u. As this input varies with time, each individual relay adjusts its output according to the current input value, and the weighted sum of all relay outputs provides the overall system output (cf. Figure 4) y(t) = Z 1 Z 1 (r s) r s [u](t)dsdr: (1) 0 ;1 The notation ; will be used to denote the operator which maps a continuous input function to some output function according to (1): y =;u. The identication of a relay r s with the point (r s) allows each relay to be uniquely represented as a point inir + IR. This half-plane plays an important role in understanding the Preisach model, and is often referred to as the Preisach plane, P. The collection of weights (r s)formsthepreisach weighting function : P 7! IR. This weighting function is experimentally determined for a given system see [2, 10] for two dierent approaches to the identication problem. In any physical setup, there are limitations which can be interpreted as a restriction on the support of. For instance, control input saturation, say at ^u, means that some relays in P can never be exercised and cannot contribute to a change in output. This eectively restricts the domain of to a triangle in P dened by P r = f(r s) 2Pjjsj ^u ; rg, illustrated in Figure 5. In this case, eectively has compact support P r. We will henceforth assume that is only non-zero in some region P r. We will also assume that is bounded, piece-wise continuous, and non-negative inside P r the set of such will be called M p. These are common assumptions when dealing with Preisach models for physical systems[3, 10, e.g.], and a model of an SMA actuator identied in [6] satised 2M p. In [5], it was shown that if is bounded and piece-wise continuous, then ;:C 0 7! C 0. If furthermore is non-negative, then 1 ;:W1 2 7! W1 2. R 1 W 2 1 is the Sobolev space: the space of real-valued functions satisfying 1;1 (_u 2 + u 2 )dt < 1. 3

4 s ^u s ^u s ^u 3 P ; P ; P ; P r 3 P r 3 P r P + r P + r -1 P + r ;^u a ;^u b ;^u c Figure 5: Preisach Boundary Behaviour One ambiguity remains in this denition of the model, and that is the question of the initial state of the relays. Obviously, the output y depends not only on u but on the initial conguration of the relays of P r. It is common to assume an initial relay output of ;1ifs>0 and +1 otherwise[3, e.g.]. There is some physical justication for this choice. Since magnetic materials haveweighting functions which are symmetric about s = 0, the model output corresponding to this assumed initial state is zero. It represents the de-magnetized state of a magnetic material: the state in which no remnant magnetization is present[12]. The Preisach plane can be used to track individual relay states by observing the evolution of the Preisach plane boundary,, inp r. First, the relays are divided into two time-varying sets, represented by the regions P ; and P + dened as follows: P (t) =f(r s) 2P r j output of r s at t is 1g: (2) It will become clear that each set is connected is the line separating P + from P ;. The boundary corresponding to the assumed initial relay conguration (the line s = 0) will be denoted. As an example, suppose the input u starts at u = 0 and increases monotonically to u =3. Initially, is the line s =0inP r. As u increases, relay outputsswitch from ;1 to +1 when u = s + r. Hence, P + grows at the expense of P ;, and the moving boundary dening this growth is the line s = u ; r. In Figure 5a, the thick line represents for some input value 0 <u<3 the arrow indicates that the sloped segment moves upwards as u increases. The line in Figure 5b shows the state of P r when u = 3. Similarly, if the input reverses direction at u = 3 and decreases monotonically to u = ;1, relays in P + switch over to P ; when u = s ; r. A new segment is generated on, corresponding to the line s = u + r (cf. Figure 5c). Subsequent input reversals generate further segments of. Note that the boundary always intersects the axis r = 0 at the current input value. With written as a map IR IR + 7! IR, then (t 0) = u(t). If the boundary at time t is (t r), applying an input for which u(t) 6= (t 0) amounts to applying 4

5 s ^u 3 P ; s s ^u ^u 5 P ; P ; P r 3 P r 3 P r -1 P + r -1 P + r -1 P + r ;^u a ;^u b ;^u c Figure 6: Wiping Out Property an input with a discontinuity at t. In this case, the output is not dened. This observation leads to the following denition. Denition 2.1 (Admissible Inputs) An input u 2 U is said to be admissible to state at time t if u(t) = (t 0). The Preisach plane boundary is the memory of the Preisach model. When an arbitrary input is applied, monotonically increasing input segments generate boundary segments of slope -1, while monotonically decreasing input segments generate boundary segments of slope +1. Input reversals cause corners in the boundary. The history of past input reversals and hence of hysteresis branching behaviour is stored in the corners of the boundary. We have seen that input extrema generate the corners of the Preisach boundary, and that this boundary represents the memory of the Preisach model. The wiping out property states that some input extrema can remove the eects of previous extrema, essentially \wiping out" the memory of the model. That a system display this behaviour has been shown to be one of two necessary and sucient conditions for existence of a Preisach model. For more on representation conditions, see [10]. The wiping out behaviour is sketched in Figure 6, and explained as follows. Consider once more the input of Figure 5. Continuing from Figure 5c, suppose that u reverses and increases monotonically until it reaches u = 5. A segment of slope -1 sweeps upward through P r, switching relays from P ; to P + (cf. Figure 6a). As the input reaches u = 3 and continues to increase, the two corners which had been generated by previous reversals at u =3andu = ;1 are eliminated (cf. Figure 6b). For u>3, the inuence of those two previous input extrema has been completely removed (cf. Figure 6c). So input extrema which exceed previous extrema in magnitude can \wipe out" part of the memory. Those extrema which remain in memory at any time form a reduced memory sequence, which will be discussed further in Section

6 3 State Space Representation The previous section described the traditional input-output representation of the Preisach model. Sector conditions, such as passivity, form a very important part of non-linear input-output stability theory. Unfortunately some hystereses, such as those found in magnetic materials, do not satisfy a sector condition. This section is concerned with developing a state-space representation for Preisach models for which 2 M p. By placing the model in a state-space framework, more general stability techniques such aslyapunov and dissipativity theory may be applied. The dynamical system framework used is that of [13], where a complete denition can be found. The system is dened through the input, output and state spaces U Y and X,aswell as the state transition operator and the read-out operator r. :IR 2 X U 7!X must satisfy the standard axioms: consistency: (t o t o x o u)=x o for all t o 2 IR x o 2X u 2U determinism: (t 1 t o x o u 1 )=(t 1 t o x o u 2 ) for all t o t 1 2 IR t 1 t o x o 2X and all u 1 u 2 2U satisfying u 1 (t) =u 2 (t) for all t o t t 1 semi-group: (t 2 t o x o u)=(t 2 t 1 (t 1 t o x o u) u) for all t o t 1 t 2 x o 2X u 2U stationarity: (t 1 + T t o + T x o T u) = (t 1 t o x o u) for all t o 2 IR t 1 t o T 2 IR x o 2X and u 2U. T is the shift operator: T u(t) =u(t + T ). The input space U is dened, for some system-dependent ^u >0, as U = fu 2 C 0 (;1 1)jkuk 1 ^u and lim u(t) =0g: t!;1 For anyinterval [t 0 t 1 ]inir, the notation u [t0 t 1] denotes the restriction of u to [t 0 t 1 ]. The notation U[t 0 t 1 ] denotes the set obtained when every element ofu is restricted to [t 0 t 1 ]. The input restriction kuk 1 ^u allows the Preisach plane to be bounded, and arises naturally in systems where input signals are subject to saturation. In Section 2, it was seen that in order to be admissible to state at time t, an input must satisfy u(t) = (t 0) (cf. Denition 2.1). Since the initial boundary is assumed to be the line s = 0, this implies that lim t!;1 u(t) =lim t!;1 (t 0) = 0. Hence the second limitation on the input space. The output space Y is the set of real-valued functions C 0 (;1 1). Since the boundary embodies the memory of the model, it is a natural choice for the state. The following denition captures the salient features of the boundaries, and ts Willems' denition of a state space. 6

7 Denition 3.1 (The State Space) The state space X is dened to be the set of continuous functions :[0 ^u] 7! IR which satisfy the following properties: (BP1) Lipschitz condition: j (r 1 ) ; (r 2 )jjr 1 ; r 2 j 8r 1 r 2 2 [0 ^u] (BP2) initial condition: (^u) =0. The Lipschitz property is more general than required, since boundaries may only be composed of segments of slope 1. However, including all functions with Lipschitz constant 1 leads to a complete state space, as will be seen shortly. The second property, along with (BP1), ensures that elements of X are within the triangle dened by P r. Before proceeding to dene the operators and r, we rst examine some properties of the state space: boundedness, completeness and reachability. The distance between two states 1 2 2X is dened as the area between the boundaries, d( 1 2 )= Z ^u 0 j 1 (r) ; 2 (r)j dr: It is clear that d( ) is a metric. From the denition of X, d( 1 2 ) ^u 2 for all 1 2 2X, and the state space is bounded. Denition 3.2 (Reachability) We say that o 2X is reachable if there exists nite T and u 2U(;1 T] so that (T )= o. As the input to the Preisachmodelevolves, it generates boundaries composed only of alternating segments of slope 1, plus a segment on the line s = 0 if any memory of the initial condition remains. However, boundaries in X may be continuous curves. Inputs which decrease in amplitude and increase in frequency as they approach T may generate a countably innite number of boundary segments, accumulating at the r = 0 axis in P r. For nite time T,however, there is no input in U(;1 T] which can generate a smooth curve over all of [0 ^u], so the entire state space X is not reachable. Theorem 3.1 (Reachable Boundary) Aboundary 2X is reachable from if there exists c 2 [0 ^u] such that is composed ofaniteorcountably innite number of segments of slope exactly 1 over [0 c], and =0over [c ^u]. Proof. The proof is clear from the boundary evolution rules described in the previous section. The set of all reachable boundaries will be denoted X r. The following concept of approximate reachability is similar to that of approximate controllability for innite-dimensional systems, proposed in [4]. Denition 3.3 (Approximate Reachability) The state space (X d) of a dynamical system is said to be \approximately reachable from x o " if, for any x 2 X and ">0, there exists a state x r 2X,reachable from x o, such that d(x r x) <". 7

8 Theorem 3.2 (Approximate Reachability) The state space X is approximately reachable from the initial state. Proof. We need to show that X r = X. Consider any 2 X and " > 0. For each n =1 2 ::: let fr i g i=0:::n be a uniform partition of [0 ^u], and dene a second partition fr i g i=1:::n by R i = (r i) ; (r i;1 ) 2 + r i + r i;1 : 2 Because has Lipschitz constant 1,wehave r i;1 R i r i. For every n (and every corresponding partition), dene the function n by (ri;1 )+(r ; r n(r) = i;1 ) r i;1 r R i (r i ) ; (r ; r i ): R i <r<r i By construction, every n is composed of alternating segments of slope 1, and satises n (^u) = (^u) =0,sof n gx r. Also, n (r i )= (r i )fori =0:::n. Since and all n are uniformly continuous then for any ">0 there exists a nite N such thatd( n ) <"for all n>n,and X r = X. Theorem 3.3 (Completeness) The metric space (X d) is complete. Proof. First, we show that X forms an equicontinuous family of curves. Choose any ">0. For any 2X and any r o 2 [0 ^u], if jr ; r o j <"then j (r) ; (r o )j jr ; r o j < ": Since " is arbitrary, the family of curves X is equicontinuous. Choose any Cauchy sequence f n g in X. It is required to show that there exists d 2X with n ;!. For every n, n (^u) =0and is Lipschitz continuous, so for every point r 2 [0 ^u], j n (r)j = j n (r) ; n (^u)j jr ; ^uj and the sequence f n g is point-wise bounded over X. By the Arzela-Ascoli theorem, f n g has a subsequence f nk g which converges both uniformly and point-wise to a continuous function. This implies that and so nk d ;!. Z ^u 0 j nk (r) ; (r)j dr ;! 0 8

9 Also, d is simply the L 1 norm on C 0 [0 ^u], and L 1 is the completion of C 0. Since f n gc 0 [0 ^u] is Cauchy, f n g converges in d to some limit o 2 L 1 [0 ^u]. Since the limit of a sequence which converges in a metric space is unique, o =, and d ;!. n d We nowhave convergence of the Cauchy sequence n ;!. To prove completeness, it remains to show that this limit is in X. We know already that is continuous. Choose any r 1 r 2 2 [0 ^u]. Recall that f n (r)g is a bounded sequence of real numbers. Then j (r 2 ) ; (r 1 )j = lim n!1 n (r 2 ) ; lim n!1 n (r 1 ) = lim ( n(r 2 ) ; n (r 1 )) n!1 = lim j( n(r 2 ) ; n (r 1 ))j n!1 lim n!1 jr 2 ; r 1 j = jr 2 ; r 1 j so is Lipschitz continuous with constant 1. The point-wise convergence nk ;! implies that (^u) =0,since nk (^u) =0 for every n k. So also satises (BP2), and 2X. Hence, (X d) is complete. 3.1 Reduced Memory Sequences In this section, we introduce an intermediate space S, which will be used in the construction of the state transition operator. The wiping out property of the Preisach model was described in Section 2. In essence, any input maximum which exceeds previous maxima will wipe out the memory of those maxima minima can be similarly \wiped out". At agiven time t, only certain past extrema are retained and aect the output. They form an alternating set of input maxima and minima, in which each maximum is smaller in amplitude than the previous one, and each minimum is larger than the previous one. The two series converge to u(t). This alternating sequence is known as the reduced memory sequence, and the following mathematical construction is based on that of [12]. A dierent but equivalent approach to memory sequences in hysteresis operators can be found in [3]. For any input u 2U(;1 T] and any T,sets 0 =0and =max t2(;1 ] ju(t)j. This is well-dened, since lim t!;1 u(t) =0. Let t 1 = maxft 2 (;1 ]jju(t)j = g, and dene the elements s i j i=1 2 ::: of the reduced memory sequence s(u ) as follows: i =1: s 1 = u(t 1 ) s i;1 <s i;2 : s i = max u(t) t2(t i;1 ] and t i =maxft 2 (t i;1 ]ju(t) =s i g s i;1 >s i;2 : s i = min u(t) t2(t i;1 ] and t i =maxft 2 (t i;1 ]ju(t) =s i g (3) 9

10 U F S F r G ;1 H G H ` r X r Y Figure 7: Relationship Between System Spaces terminating the sequence if t i =. Note that the values s i are well-dened: by denition of s i;1 in (3), u(t) >s i;1 (or u(t) < s i;1 ) over (t i;1 ]. Since u is continuous, the required maximum (or minimum) is well-dened. The times t i are similarly well-dened, since the maximum is being taken over a non-empty set and is nite. The sequence ft i g is merely used to construct fs i g and then discarded: the time at which extrema occur is of no signicance in the Preisach model due to its static nature. If the input u has a nite number of extrema in (;1 ], the above sequence has nite length N, t N = and u(t N ) = u(). In this case, the tail of the sequence is formed by setting s i = s N for i > N. If the sequence is innite, then setting t = supft i g, the input u must be constant over [t ]. Note that in both cases, lim i!1 s i = u(). This section will make use of the notation N(s) = supfijs i;1 6= s i g. For any reduced memory sequence s(u ), this is the index beyond which s i is constant and equal to u(). If u has a nite number of extrema in (;1 ], then N(s(u )) is nite otherwise, N(s) may be innite. Also, for any sequence s = fs i g,lets e and s o denote the even and odd subsequences s e = fs i g i=2 4 ::: = fs e ig and s o = fs i g i=1 3 ::: = fs o i g. Denition 3.4 (Space of Reduced Memory Sequences) The space of reduced memory sequences, S l 1, is composed of all sequences s with ksk 1 ^u, and for which the even subsequence s e and odd subsequence s o satisfy 1. s e is strictly decreasing (strictly increasing) up to N(s) and, if N(s) < 1, constant thereafter 2. s o is strictly increasing (strictly decreasing) up to N(s) and, if N(s) < 1, constant thereafter 3. lim i!1 s e i =lim i!1 s o i. Figure 7 shows the spaces dened thus far. We now dene mappings between these spaces, and note some of their properties. F : U 7! S For any time < 1 and any input u 2 U(;1 ], the reduced memory sequence F u = s(u ) is dened as in (3). 10

11 F r : S 7! U The reduced memory sequence s(u ) captures only information regarding dominant extrema of u (;1 ]. There are therefore an innite number of inputs u i 6= u which are equivalent, in the sense that they have the same reduced memory sequence: F u i = F u. Hence, no inverse of F exists. Aright-inverse F r : S 7! U(;1 ]is dened below. For any s( ) 2S, it is required to construct an input u 2U(;1 ] with extrema equal to the elements of s and satisfying u() = lim i!1 s i. Choose any t 0 <,and let ft i g be a partition of [t 0 ] dened for all i 1by t i = t o + ; t 0 2 Note that lim i!1 t i =. Set s 0 = 0 and dene u(t) on (;1 ] by straight-line interpolation between the points (t i s i ): u(t) = 8 < : Xi;1 k=0 1 2 k : 0 t t 0 s i t = t i (t ; t i;1 ) s i;s i;1 t s i;t i;1 i;1 t i;1 <t<t i : The resulting output u 2U(;1 ] has extrema corresponding to elements of s( ), and u() =u(lim i!1 t i)= lim i!1 s i: (4) We now give some properties of these operators. 3 : C 0 C 0 7! C 0 is dened by The concatenation operator u (to t 1]3v (t1 t 2] = u to <t t 1 v t 1 <t t 2 : Lemma 3.1 The operators F F r satisfy the following properties, for any < F r is a right-inverse of F : for any s 2S, F F r s( )=s( ). 2. F is deterministic that is, for any < 1 and u 1 u 2 2 U(;1 ] satisfying u 1 (t) =u 2 (t) over (;1 ], then F u 1 = F u For any u 2 U(;1 T] the composition F r F input for every <T thatis, ; F r F u (;1 ] 3u( T] 2U(;1 T]: preserves the continuity of the 4. For any T > and u 2U(;1 T], theoperators satisfy the identity F T ; F r F u (;1 ] 3u( T] = FT u: (5) 11

12 Proof. The rst two properties follow from the denitions of F and F r. To prove the other two, choose any u 2U(;1 T] <T. Let the reduced memory sequence at be s(u ) =F u, and the input which is reconstructed from s(u ) be~u = F r s. From (4), ~u() = lim i!1 s i. But from the denition of F, lim i!1 s i = u(). So ~u() =u() and F r F u3u ( T] 2U(;1 T], proving Property 3. By construction, ~u above contains all of the same extremum information as u (;1 ]. Therefore, ~u3u ( T] has extrema identical to those of u (;1 T ]. Since ~u3u ( T] 2U(;1 T], F T (~u3u ( T] ) is dened and the identity (5) holds. G : S 7! X r Any reduced memory sequence s(u ) 2 S, denes a corresponding boundary = G(s). The elements of s correspond to the corners of the boundary curve, as dened below. For all i<1, dene the set of points p i 2 IR 2 p 0 = (^u 0) p 1 = (js 1 j 0) p i = 8 < : si;1;s i s i;1+s i 2 2 si;s i;1 s i+s i;1 2 2 s i <s i;1 s i >s i;1 and G(s) to be the linear interpolate between the points p i. Note that if N(s) < 1 then for all i>n(s), s i = s i;1 = s N(s) and p i =(0 s N(s) )=(0 u()). If N(s) is innite, then the boundary G(s) has an innite number of corners p i. In this case, since lim i!1 s e i = lim i!1 s o i = u() then lim i!1 p i =(0 u()). In both cases, the result is as expected for the Preisach model: the boundary at time intersects the axis r = 0 at the point (0 u()): ( 0) = u(). Note that the range of G, Ra(G), is the set of all curves 2 X which have a nite or countably innite number of alternating segments of slope 1. This set is not quite X, since X also contains continuous curves. However, Ra(G) =X r. G ;1 : X r 7!S For every sequence s 2 S, the boundary G(s) is unique, by denition of G. Since Ra(G) = X r, the inverse mapping G ;1 : X r 7! S exists. The construction of a sequence s 2Sfrom any boundary in 2X r is done by extracting the coordinates of the corners and calculating the corresponding extrema from (6). If the number of corners N is nite, the reduced memory sequence is completed by setting the tail to s i = s N for all i>n. (6) H ` : U 7! X r H : X r 7!U The mappings H ` : U 7! X r and H : X r 7! U are dened as the compositions H = F rg;1 H ` = GF. Lemma 3.2 The operators H H ` satisfy the following properties, for any < H ` is a left-inverse of H : for any 2X r, H ` H =. 12

13 2. H ` is deterministic that is, for any < 1 and u 1 u 2 2U(;1 ] satisfying u 1 (t) =u 2 (t) over (;1 ], then H ` u 1 u For any T > and u 2U(;1 T], theoperators satisfy the identity H ` T ; H H ` u (;1 ] 3u( T] T u: (7) Proof. The proof follows from the denitions of H and H `, and Lemma State Transition and Read-Out Operators The state transition operator determines the state = (t 1 t o o u) which results at time t 1 from applying an input u 2 U[t o t 1 ] to a system starting in state o at time t o. For this operation to be well-posed, the state o must be reachable and u must be admissible to o at t o that is, u(t o )= (t o 0) (cf. Denition 2.1). The state transition operator is dened using the mappings introduced in the previous section. Given some interval [t 0 t 1 ], some initial state o 2X r, and some input u 2U[t 0 t 1 ] admissible to o at t o, (t 1 t o o u) is dened as follows: 1. determine the memory sequence for the initial state: s( t 0 )=G ;1 o, 2. construct an input u o 2U(;1 t 0 ]which generates s( t 0 ): u o = F r t 0 s( t 0 ), 3. concatenate the inputs u o and u to form ~u = u o 3u 2U(;1 t 1 ], 4. determine the corresponding boundary 1 at time t 1 : 1 = GF t1 ~u. Recalling that H = F r G;1 and H ` = GF, the state transition function is given by (t 1 t o o u)=h ` t 1 Ht0 o 3u (t0 t 1] : (8) We now prove that the state transition operator as dened in (8) satises the four required axioms. consistency: Choose any t o 2 IR, o 2X r, and admissible u 2U. Then (t o t o o u) t o Hto o 3u (to t o] = o : t o Hto o determinism: Choose any t 1 t o, o 2X r, and admissible u 1 u 2 2Usuch that u 1 (t) =u 2 (t) over [t o t 1 ]. Let u o = H to o. Then (t 1 t o o u 1 ) t 1 ; uo 3u 1(to t 1] and (t 1 t o o u 2 ) t 1 ; uo 3u 2(to t 1] : 13

14 But u 1(to t 1] = u 2(to t 1], and from Lemma 3.2 H ` is deterministic, so we have (t 1 t o o u 1 )=(t 1 t o o u 2 ). semi-group: Choose any t o 2 IR t 2 t 1 t o, o 2 X r and admissible u 2 U. Let u o = H to o, ~u = u o 3u (to t 2], and note that u o 3u (to t 1] = ~u (;1 t1] and u (t1 t 2] =~u (t1 t 2]. Then (t 2 t 1 (t 1 t o o u) u) t 2 Ht1 f(t 1 t o o u)g3u (t1 t 2] by Property 3 of Lemma 3.2. But t Ht1 2 H ` t Hto 1 o 3u (to t 1] 3u(t1 t 2] t Ht1 2 H ` t uo 1 3u (to t 1] 3u(t1 t 2] t 2 Ht1 H ` t 1 ~u (;1 t1]3~u (t1 t 2] t 2 ~u (t 2 t 0 o u) t Hto 2 o 3u (to t 2] and so satises the semi-group property. t 2 uo 3u (to t 2] t 2 ~u stationarity: Choose any t o 2 IR t 1 t o o 2X r, admissible u 2U and T < 1. It is required to show that Recall that T Then H ` t 1 Hto o 3u (to t 1] is the shift operator. Set and similarly, s(u 2 t o + T )=G ;1 t 1+T u 1 = H to o 2U(;1 t o ] u 2 = H to+t o 2U(;1 t o + T ]: s(u 1 t o )=F to u 1 = F to F r t o G ;1 o = G ;1 o o. Now dene ~u 1 = u 1 3u (to t 1] ~u 2 = u 2 3 T u (to t 1]: Hto+T o3 T u (to t 1] : (9) Using these denitions in (9), it must be shown that H `t 1 ~u 1 t ~u 1+T 2. Since s(u 1 t o )=s(u 2 t o + T ), then s(~u 1 t 1 )=F t1 ~u 1 is the same reduced memory sequence as s(~u 2 t 1 + T )=F ~u t1+t 2. Thus and is stationary. H ` t 1 ~u 1 = GF t1 ~u 1 = GF t1+t ~u 2 t 1+T ~u 2 14

15 The read-out function r gives the system output which corresponds to a particular state. Recall that the Preisach model output is (1): y(t) = Z 1 Z 1 (r s) r s [u](t)dsdr: 0 ;1 Since relays below have output +1 and relays above, -1, r can be dened as a function of : y(t) =r( (t)) = Z 1 Z (t) Z 1 Z 1 (r s)dsdr ; (r s)dsdr: 0 ;1 0 (t) 4 Dissipativity of the Preisach Model In his pioneering work on dissipative dynamical systems[13], Willems shows that the major input-output stability results can all be cast as special cases of dissipativity theory. Dissipativity is dened in terms of the relationship between two functions known as the supply rate and the storage function. Denition 4.1 (Dissipativity [13]) A dynamical system is said to be dissipative with respect to the (locally integrable) supply rate w : U Y 7! IR if there exists a non-negative function S : X 7! IR +, called the storage function, such that for all t 1 t o x o 2X and u 2U, S(x o )+ Z t1 t o w(u(t) y(t))dt S((t 1 t o x o u)): (10) In this section, it is shown that the Preisach model is \dissipative" in a more general sense, since the supply rate will include the derivative of the output. This type of supply rate has been investigated in [11]. Essentially, for a system to be dissipative, the sum of the storage in the initial state and the supply generated by the input must not be less than the storage in the nal state. In other words, there is no internal generation of storage. The word \energy" is conspicuously absent from this description: while dissipativity theory is based on energy concepts, the supply rate and storage function are generalizations of the physical concepts of \rate of energy supply" and \amount of stored energy". There need not be any physical energy interpretation in order for the denition or related results to hold. 4.1 Energy Storage in the Preisach Model In general, storage functions for physical systems are not unique. However, it is often the case that the actual energy stored in a system is a storage function with some related supply rate. 15

16 In [5], a formula was derived for Q, the energy stored in the Preisach model, by looking at energy transfer and storage in individual relays. It was assumed that the input and output variables were related such that the units of the product u _y are power. This is often the case in actuators, where u is some form of electrical or mechanical force, and y is displacement. It was shown that the energy transfer to a relay in a switch from -1 to +1 was q + = 2(r s)(s + r) the energy transfer in switching from +1 to -1 is q ; = ;2(r s)(s ; r). Note that the energy transferred to any relay during a complete cycle is q + + q ; =4r(r s), which is the area of the weighted relay. This agrees with the well-known result that the hysteretic energy loss in each cycle of a magnetic circuit is equal to the area of the hysteresis loop[1, e.g.]. Dening the regions Q 1 = f(r s) 2P r j s>rg Q 2 = f(r s) 2P r jjsjrg Q 3 = f(r s) 2P r j s<;rg we observe that if (r s) 0, q + 0 for relays in Q 3 and q ; 0 for relays in Q 1. Energy transfer is positive for all other switches. Negative energy transfer represents energy being recovered from the system: relays whose next switch will result in negative energy transfer are storing energy. The formula for total stored energy is Q( (t)) = 2 Z Q 1\P +(t) (r s)(s ; r)dsdr ; 2 Z Q 3\P;(t) (r s)(s + r)dsdr: (11) Recall from their denition in equation (2) that the regions P + (t) and P ; (t) are entirely dened by the boundary (t). If 2M p then Q( ) 0, since s>rin Q 1 and s<;r in Q 3. Proposition 4.1 (Minimum Energy) If 2 M p, then whenever u(t) = 0, the Preisach model is in a state of minimum stored energy. Proof. If u(t) = 0, then (t 0) = 0. Since boundaries have Lipschitz constant 1, must be entirely contained in the region Q 2. Thus Q 1 \P + = and similarly, Q 3 \P ; =. So from (11), Q( ) = 0 since the areas of integration are empty. Itwillbeshown that the Preisach model satises the dissipation inequality (10) with the generalized supply rate w = u _y. We do this by demonstrating that the recoverable stored energy Q is a storage function for this supply rate. For a more abstract approach to the topic of dissipation in hysteresis operators, see the work on hysteresis potentials in [3]. Theorem 4.1 If 2 M p, the Preisach model satises the generalized dissipation inequality Q( o )+ Z t1 t o u _ydt Q( 1 ) (12) for any o 2X, t 1 t o and u 2U[t o t 1 ] such that (t 1 t o o u)= 1. 16

17 Proof. Recall that if 2 M p then Q( ) 0. Also, and P r are bounded, so Q( ) < 1 for all, andq : X 7! IR +. It remains to show that for any initial state o and u 2U[t o t 1 ] such that 1 = (t 1 t o o u), the inequality (12) is satised. The remainder of the proof is outlined below for a detailed proof, see [5]. The idea is to consider an arbitrary input u and write out the expression for the energy transferred by u to the system on a relay by relay basis. This gives the expression Z t1 t o u _ydt = Z P r n + (r s)q + dsdr + Z P r n ; (r s)q ; dsdr where n + (r s) is the number of times the relay r s is switched from -1 to +1, and n ; (r s) is the number of times it is switchedfrom+1to-1. We then remove from the right hand side quantities which are positive: energy transferred when a relay is fully cycled positive energy transfer during partial cycling, depending on the region of P r. After a non-trivial step in regrouping terms on the right-hand side, this results in the inequality Z t1 u _ydt 2 t o Z Q 1\P +(t 1) ;2 Z Q 1\P +(t o) = Q( (t 1 )) ; Q( (t o )) Z (r s)(s ; r)dsdr ; 2 (r s)(s ; r)dsdr +2 which shows that the dissipation inequality (12) is satised. (r s)(s + r)dsdr Q 3\P;(t 1) Z (r s)(s + r)dsdr Q 3\P;(t o) The Passivity Theorem allows the determination of classes of feedback stabilizing controllers for systems which are passive. Denition 4.2 (Passivity) An operator G is said to be passive if there exists 0 such that, for all T < 1 and u 2 Do(G), Z T Z T ugu dt ;1 ;1 G is said to be strictly passive if equation (13) holds for >0. u 2 dt: (13) If the product ugu has units of power, this says that the net energy input over time is positive for every possible input. In other words, the system does not generate any power internally. The underlying assumption in the above denition is that the system starts in a state of minimum stored energy. Otherwise, an input could be generated which recovers any stored energy, causing R ugu dt to be negative. The denition of passivity (with = 0) is a special case of dissipativity, in which the supply rate is w(u y) =uy and the storage function is zero. Like dissipativity, the theory is motivated by the study of energy storage, where the input-output pair represents instantaneous power. Again, the theory and related results continue to hold when such an interpretation is not available. 17

18 Corollary 4.1 (Preisach Model Passivity) If 2 M p, the composite operator d ;:W 2 dt 1 7! L 2 is passive. Proof. The assumption on guarantees that the Preisach model satises the inequality (12). Suppose the system starts in a state of minimum energy storage, i.e. u(t o )=0. Then Q( o )=0(cf. Proposition 4.1) and Z t1 u _ydt Q((t 1 t o o u)) 0: t o But since lim t!;1 u(t) = 0, then for any nite T and u 2 Do(;), we have which completes the proof. Z T u _ydt 0 ;1 This passivity result has been reported in [7], where a Passivity Theorem was applied to obtain a stability result for rate feedback of Preisach models for which 2M p. The derivation presented here takes a more general approach, emphasizing that passivity is simply a special case of dissipativity. 5 Conclusion We have shown that the Preisach hysteresis operator can be placed in the standard dynamical system framework. Although the system has memory, it is entirely contained in the state, allowing treatment of these non-linearities in the more familiar state-space setting. What distinguishes these static hysteretic systems from more common dynamical systems is that the time scale is irrelevant. This framework is used to show a generalized form of dissipativity, and the passivity of the relationship from input to output derivative. Current research goals involve the extension of the rate feedback result of [7] to position feedback. Research directions specic to SMA actuators include the incorporation of time-varying actuator stresses in the dissipativity and stability results. References [1] J.C. Anderson, Magnetism and Magnetic Materials, Chapman and Hall, [2] H.T. Banks, A.J. Kurdilla and G. Webb, \Identication of Hysteretic Control In- uence Operators Representing Smart Actuators: Convergent Approximations", CRSC-TR97-7, Center for Research in Scientic Computation, NCSU, [3] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer,

19 [4] R.F. Curtain and A.J. Pritchard, Innite Dimensional Linear Systems Theory, Springer-Verlag, [5] R.B. Gorbet, Control of Hysteretic Systems with Preisach Representations, Ph.D. Thesis, University ofwaterloo, Waterloo, Canada, [6] R.B. Gorbet, D.W.L. Wang and K.A. Morris, \Preisach Model Identication of a Two-Wire SMA Actuator", IEEE ICRA 1998, submitted, October, [7] R.B. Gorbet, K.A. Morris and D.W.L. Wang, \Stability of Control Systems for the Preisach Hysteresis Model", Journal of Engineering Design and Automation, to appear. [8] M. Hashimoto, M. Takeda, H. Sagawa, I. Chiba and K. Sato, \Application of shape memory alloy to robotic actuators", Journal of Robotic Systems, Vol. 2, No. 1, pp. 3-25, [9] D. Hughes and J.T. Wen, \Preisach modeling of piezoceramic and shape memory alloy hysteresis", 1995 IEEE Control Conference on Applications, Albany, New York, September [10] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, [11] K.A. Morris and J.N. Juang, \Dissipative Controller Designs for Second-Order Dynamic Systems", IEEE Transactions on Automatic Control, Vol. 39, No. 5, pp , [12] A. Visintin, Dierential Models of Hysteresis, Springer-Verlag, [13] J.C. Willems, \Dissipative Dynamical Systems, Part I: General Theory", Archives for Rational Mechanics and Analysis, Vol. 45, pp , [14] S. Yokota, K. Yoshida, K. Bandoh and M. Suhara, \Response of proportional valve using shape-memory-alloy array actuators", 13th IFAC World Congress, pp , IFAC,