Jayawardhana, B.

Logemann, H.

Ryan, E.P.

English

University of Groningen

   University of GroningenResults of ISS type for hysteretic Lur'e systemsJayawardhana, B.; Logemann, H.; Ryan, E.P.Published in:18th Symposium on Mathematical Theory of Networks and SystemsIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2008Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Jayawardhana, B., Logemann, H., & Ryan, E. P. (2008). Results of ISS type for hysteretic Lur'e systems: adifferential inclusions approach. In 18th Symposium on Mathematical Theory of Networks and SystemsSIAM.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 25-04-2023Results of ISS Type forHysteretic Lur’eSystems: a DifferentialInclusions Approach∗B. Jayawardhana†, H. Logemann‡, and E.P. Ryan§1 IntroductionThe paper comprises a study of absolute stability, input-to-state stability, andboundedness properties of a feedback interconnection of a finite-dimensional, linear,m-input, m-output system (A,B,C) and a set-valued nonlinearity Φ. With refer-ence to Figure 1, we assume that D is a set-valued map in which input or distur-bance signals are embedded. The analytical framework is of sufficient generalityD (A,B,C)Φy+−Figure 1. Feedback interconnection of linear system (A,B,C) and nonlinearity Φto encompass feedback systems with hysteresis operators (that is, a causal rate-independent operator) in the feedback loop. To illustrate this, let F be a causal op-erator from dom(F ) ⊂ L1loc(R+, Rm) to L1loc(R+, Rm), where R+ := [0,∞), and con-sider the feedback system (structurally of Lur’e type), with input d ∈ L∞loc(R+, Rm),∗Based on research supported by the UK Engineering & Physical Sciences Research Council(Grant Ref: GR/S94582/01).†Department of Discrete Technology and Production Automation, University of Groningen,9747 AG Groningen, The Netherlands. Email: B.Jayawardhana@rug.nl‡Department of Mathematical Sciences, University of Bath, Bath BA2 7AY,United Kingdom. Email: hl@maths.bath.ac.uk§Department of Mathematical Sciences, University of Bath, Bath BA2 7AY,United Kingdom. Email: epr@maths.bath.ac.ukgiven by the functional differential equationẋ(t) = Ax(t) + B(d(t) − (F (Cx))(t)). (1)Assume that F can be embedded in a set-valued map Φ in the sense thaty ∈ dom(F ) =⇒ (F (y))(t) ∈ Φ(y(t)) for a.a. t ∈ R+ .If the input d is such that d(t) ∈ D(t) for almost all t, then any solution of (1) is afortiori a solution of the feedback interconnection in Figure 1. In this sense, prop-erties of solutions of the feedback interconnection are inherited by solutions of (1).Under particular regularity assumptions on D and Φ, generalized sector conditionson Φ, and positive-real conditions related to the linear component (A,B,C), weestablish input-to-state stability (in the sense of [10], but extended to differentialinclusions) and boundedness properties of solutions of the system in Figure 1.2 Set-valued nonlinearities and differential inclusionsA set-valued map y 7→ Φ(y) ⊂ Rm, with non-empty values and defined on Rm, issaid to be upper semicontinuous at y ∈ Rm if, for every open set U containing Φ(y),there exists an open neighbourhood Y of y such that Φ(Y ) := ∪z∈Y Φ(z) ⊂ U ; themap Φ is said to be upper semicontinuous if it is upper semicontinuous at everyy ∈ Rm. The set of upper semicontinuous compact-convex-valued mapsΦ : Rm → {S ⊂ Rm | S non-empty, compact and convex}is denoted by U . Let D : R+ → {S ⊂ Rm |S 6= ∅} be a set-valued map. The mapD is said to be measurable if the preimage D−1(U) := {t ∈ R+ |D(t) ∩ U 6= ∅} ofevery open set U ⊂ Rm is Lebesgue measurable; D is said to be locally essentiallybounded if D is measurable and the function t 7→ |D(t)| is in L∞loc(R+). The set ofall locally essentially bounded set-valued maps R+ → {S ⊂ Rm |S 6= ∅} is denotedby B. For D ∈ B, I ⊂ R+ an interval and 1 ≤ p ≤ ∞, the Lp-norm of the restrictionof the function t 7→ |D(t)| to the interval I is denoted by ‖D‖Lp(I).The feedback system shown in Figure 1 corresponds to the initial-value problemẋ(t) − Ax(t) ∈ B (D(t) − Φ(Cx(t))) , x(0) = x0 ∈ Rn, D ∈ B , (2)where A ∈ Rn×n, B ∈ Rn×m, C ∈ Rm×n and Φ ∈ U . By a solution of (2) we meanan absolutely continuous function x : [0, ω) → Rn, 0 < ω ≤ ∞, such that x(0) = x0and the differential inclusion in (2) is satisfied almost everywhere on [0, ω); a solutionis maximal if it has no proper right extension that is also a solution; a solution isglobal if it exists on [0,∞). We record the following existence result (a consequenceof, for example, [3, Corollary 5.2]).Lemma 1. Let Φ ∈ U . For each x0 ∈ Rn and each D ∈ B, the initial-value problem(2) has a solution. Moreover, every solution can be extended to a maximal solutionx : [0, ω) → Rn and, if x is bounded, then x is global.3 Input-to-state stability: the main resultsIn the context of the differential inclusion (2), the transfer-function matrix of thelinear system given by (A,B,C) is denoted by G, i.e., G(s) = C(sI − A)−1B.We assemble the following hypotheses which will be variously invoked in the theorydeveloped below. Recall that K∞ is the set of all functions ϕ : R+ → R+ that arecontinuous, strictly-increasing and unbounded with ϕ(0) = 0; KL is the set of allfunctions β : R+ × R+ → R+ such that β(·, t) ∈ K∞ for each t ∈ R+ and, for eachr ∈ R+, β(r, t) ↓ 0 as t → ∞.(H1) There exist numbers a < b and δ > 0 such that〈ay − v, by − v〉 ≤ 0 ∀ v ∈ Φ(y), ∀ y ∈ Rm , (3)G(I + aG)−1 ∈ H∞ and (I + bG)(I + aG)−1 − δI is positive real.(H2) Φ(0) = {0} and there exist numbers a > 0, δ ∈ [0, 1) and θ ≥ 0 such thata‖y‖2 ≤ 〈y, v〉 ∀ v ∈ Φ(y), ∀ y ∈ Rm , (4)‖v − aδy‖ ≤ 〈y, v − aδy〉 ∀ v ∈ Φ(y), ∀ y ∈ Rm with ‖y‖ ≥ θ (5)and G(I + δaG)−1 is positive real.(H3) There exist ϕ ∈ K∞ and numbers b > 0 and δ ∈ [0, 1) such thatmax{ϕ(‖y‖)‖y‖, ‖v‖2/b}≤ 〈y, v〉 ∀ v ∈ Φ(y), ∀ y ∈ Rm (6)and (δ/b)I + G is positive real.(H4) Φ(0) = {0} and there exist ϕ ∈ K∞ and a number θ ≥ 0 such thatϕ(‖y‖)‖y‖ ≤ 〈y, v〉 ∀ v ∈ Φ(y), ∀ y ∈ Rm , (7)‖v‖ ≤ 〈y, v〉 ∀ v ∈ Φ(y), ∀ y ∈ Rm with ‖y‖ ≥ θ (8)and G is positive real.Remark 2. (a) (H1) is a set-valued version of the familiar multivariable sectorcondition.(b) If m = 1 (the single-input, single-output case), then the combined frequency-domain assumptions in (H1) admit a graphical characterization in terms of theNyquist diagram of G (see, e.g., [5, pp. 268]).(c) Conditions (4) and (7) can be viewed as the limits of (3) and (6), respectively,as b → ∞.(d) A sufficient condition for (6) to hold is the “nonlinear” sector condition〈ϕ(y)‖y‖−1y − v , by − v〉≤ 0 ∀ v ∈ Φ(y), ∀ y ∈ Rm, (9)(e) If m = 1 and (4) holds, then (5) is trivially satisfied for any θ ≥ 1 and anyδ ∈ [0, 1). Similarly, if m = 1 and (7) holds, then (8) is satisfied for every θ ≥ 1.(f) If (6) holds for some ϕ ∈ K∞ and for some b > 0, then Φ(0) = {0} and,furthermore, (8) is satisfied for any θ > 0 satisfying ϕ(θ) ≥ b.Definition 3. System (2) is said to be input-to-state stable with bias c ≥ 0 ifevery maximal solution of (2) is global, and there exist β1 ∈ KL and β2 ∈ K∞ suchthat, for all x0 ∈ Rn and all D ∈ B, every global solution x satisfies‖x(t)‖ ≤ max{β1(‖x0‖, t) , β2(‖D‖L∞[0,t] + c)}∀ t ∈ R+ . (10)System (2) is input-to-state stable if it is input-to-state stable with bias 0.System (2) has the converging-input-converging-state property if, for all x0 ∈ Rn andall D ∈ B with ‖D‖L∞[t,∞) → 0 as t → ∞, every maximal solution x of (2) is globaland satisfies x(t) → 0 as t → ∞. The following lemma shows in particular that ifsystem (2) is input-to-state stable, then it has the converging-input-converging-stateproperty.Lemma 4. Assume that system (2) is input-to-state stable with bias c ≥ 0 and letβ1 and β2 be as in Definition 3. Then, for all x0 ∈ Rn and all D ∈ B, every globalsolution x of (2) satisfieslim supt→∞‖x(t)‖ ≤ lim supt→∞β2(‖D‖L∞[t,2t] + c) .We now arrive at the main results on input-to-state stability (proofs of whichcan be found in [4]).Theorem 5. Let the linear system (A,B,C) be stabilizable and detectable. Assumethat (H1) holds. Then, every maximal solution of (2) is global and there existpositive constants c1, c2 and ε such that, for all x0 ∈ Rn and D ∈ B, every globalsolution x satisfies‖x(t)‖ ≤ c1e−εt‖x0‖ + c2‖D‖L∞[0,t] ∀ t ∈ R+ .In particular, system (2) is input-to-state stable.Theorem 6. Let the linear system (A,B,C) be minimal. Assume that at least oneof hypotheses (H2), (H3) or (H4) holds. Then system (2) is input-to-state stable.In [1] it is has been proved, for single-valued Φ and D, that, if (A3) holds,then (2) is input-to-state stable. Therefore, Theorem 6 can be considered as ageneralization of the main result in [1].In the following corollaries (to Theorems 5 and 6, respectively), we will consider notonly nonlinearities satisfying at least one of the conditions (3), (4), (6) and (7) forall arguments y ∈ Rm, but also nonlinearities Φ ∈ U with the property that thereexist a set-valued map Φ̃ ∈ U satisfying at least one of the conditions (3), (4), (6)and (7) and a compact set K ⊂ Rm such thaty ∈ Rm\K =⇒ Φ(y) ⊂ Φ̃(y) . (11)In particular, single-input, single-output hysteretic elements can be subsumed bythis set-valued formulation provided that the “characteristic diagram” of the hys-teresis is contained in the graph of some Φ ∈ U .Corollary 7. Let the linear system (A,B,C) be stabilizable and detectable. LetΦ ∈ U be such that there exist a set-valued map Φ̃ ∈ U and a compact set K ⊂ Rmsuch that (11) holds. Assume that (H1) holds with Φ replaced by Φ̃. Then, everymaximal solution of (2) is global and there exist positive constants c1, c2 and ε suchthat, for all x0 ∈ Rn and D ∈ B, every global solution x satisfies‖x(t)‖ ≤ c1e−εt‖x0‖ + c2(‖D‖L∞[0,t] + E) ∀ t ∈ R+ ,whereE := supy∈K supv∈Φ(y) inf ṽ∈Φ̃(y)‖v − ṽ‖ . (12)Corollary 8. Let the linear system (A,B,C) be minimal and let Φ ∈ U be suchthat there exist a set-valued map Φ̃ ∈ U and a compact set K ⊂ Rm such that (11)holds. Assume that at least one of the hypotheses (A1), (A2) or (A3) holds with Φreplaced by Φ̃. Then system (2) is input-to-state stable with bias E given by (12).4 Hysteretic feedback systemsWe return to the feedback interconnection of Figure 1, but now in a single-input,single-output setting and with a hysteresis operator F in the feedback path. Anoperator F : C(R+) → C(R+) is a hysteresis operator if it is causal and rateindependent. Here rate independence means that F (y ◦ ζ) = (Fy) ◦ ζ for everyy ∈ C(R+) and every time transformation ζ, where ζ : R+ → R+ is said to be atime transformation if it is continuous, non-decreasing and surjective. Conditionson F which ensure well-posedness of the feedback interconnection (existence anduniqueness of solutions of the associated initial-value problem) are expounded in, forexample, [8] and [9]. The so-called Preisach operators are among the most generaland most important hysteresis operators: in particular, they can model complexhysteresis effects such as nested loops in input-output characteristics. Therefore,and for clarity of presentation, we focus on the class of Preisach operators.A basic building block for these operators is the backlash operator. A discussionof the backlash operator (also called play operator) can be found in a number ofreferences, see for example [2], [6] and [7]. Let σ ∈ R+ and introduce the functionbσ : R2 → R given bybσ(v1, v2) := max{v1 − σ , min{v1 + σ, v2}}=v1 − σ, if v2 < v1 − σv2, if v2 ∈ [v1 − σ, v1 + σ]v1 + σ, if v2 > v1 + σ .Let Cpm(R+) denote the space of continuous piecewise monotone functions definedon R+. For all σ ∈ R+ and ξ ∈ R, define the operator Bσ, ξ : Cpm(R+) → C(R+) byBσ, ξ(y)(t) ={bσ(y(0), ξ) for t = 0 ,bσ(y(t), (Bσ, ξ(u))(ti)) for ti < t ≤ ti+1, i = 0, 1, 2, . . . ,where 0 = t0 < t1 < t2 < . . ., limn→∞ tn = ∞ and u is monotone on eachinterval [ti, ti+1]. We remark that ξ plays the role of an “initial state”. It is notdifficult to show that the definition is independent of the choice of the partition(ti). Figure 2 illustrates how Bσ, ξ acts. It is well-known that Bσ, ξ extends to ayBσ,ξ(y)−σσFigure 2. Backlash hysteresisLipschitz continuous operator on C(R+) (with Lipschitz constant L = 1), the so-called backlash operator, which we shall denote by the same symbol Bσ, ξ. It iswell-known that Bσ, ξ is a hysteresis operator.Let ξ : R+ → R be a compactly supported and globally Lipschitz function withLipschitz constant 1. Let µ be a signed Borel measure on R+ such that |µ|(K) < ∞for all compact sets K ⊂ R+, where |µ| denotes the total variation of µ. DenotingLebesgue measure on R by µL, let w : R×R+ → R be a locally (µL ⊗µ)-integrablefunction and let w0 ∈ R. The operator Pξ : C(R+) → C(R+) defined by(Pξ(y))(t) =∫ ∞0∫ (Bσ, ξ(σ)(y))(t)0w(s, σ)µL(ds)µ(dσ) + w0∀u ∈ C(R+) , ∀ t ∈ R+ ,(13)is called a Preisach operator, cf. [2, p. 55]. It is well-known that Pξ is a hysteresisoperator (this follows from the fact that Bσ, ξ(σ) is a hysteresis operator for every σ ≥0). Under the assumption that the measure µ is finite and w is essentially bounded,the operator Pξ is Lipschitz continuous with Lipschitz constant L = |µ|(R+)‖w‖∞(see [7]) in the sense thatsupt∈R+|Pξ(y1)(t) − Pξ(y2)(t)| ≤ L supt∈R+|y1(t) − y2(t)| ∀ y1, y2 ∈ C(R+).This property ensures the well-posedness of the feedback interconnection.Setting w(·, ·) = 1 and w0 = 0 in (13), we obtain the Prandtl operator Pξ : C(R+) →C(R+) defined byPξ(y)(t) =∫ ∞0(Bσ, ξ(σ)(y))(t)µ(dσ) ∀u ∈ C(R+) , ∀ t ∈ R+ . (14)For ξ ≡ 0 and µ given by µ(E) =∫Eχ[0,5](σ)dσ (where χ[0,5] denotes the indicatorfunction of the interval [0, 5]), the Prandtl operator is illustrated in Figure 3. The10−200040tP0(y)y−5 10−20040P0(y)yFigure 3. Example of Prandtl hysteresisnext proposition identifies conditions under which the Preisach operator (13) satis-fies a generalized sector bound. For simplicity, we assume that the measure µ andthe function w are non-negative (an important case in applications), although theproposition can be extended to signed measures µ and sign-indefinite functions w.Proposition 9. Let Pξ be the Preisach operator defined in (13). Assume thatthe measure µ is non-negative, a1 := µ(R+) < ∞ and a2 :=∫ ∞0σµ(dσ) < ∞.Furthermore, assume thatb1 := ess inf(s,σ)∈R×R+w(s, σ) ≥ 0 , b2 := ess sup(s,σ)∈R×R+w(s, σ) < ∞and setaP := a1b1, bP := a1b2 , cP := a2b2 + |w0| . (15)Then∀ y ∈ C(R+) ∀ t ∈ R+ , y(t) ≥ 0 =⇒ aPy(t) − cP ≤ (Pξ(y))(t) ≤ bPy(t) + cP ,(16)∀ y ∈ C(R+) ∀ t ∈ R+ , y(t) ≤ 0 =⇒ bPy(t) − cP ≤ (Pξ(y))(t) ≤ aPy(t) + cP ,(17)and, for every η > 0,∀ y ∈ C(R+) ∀ t ∈ R+ , |y(t)| ≥ cP/η =⇒ (aP−η)y2(t) ≤ (Pξ(y))(t)y(t) ≤ (bP+η)y2(t) .(18)Let Pξ be a Preisach operator satisfying the hypotheses of Proposition 9. LetaP , bP and cP be given by (15) and define Φ, Φ̃ ∈ U byΦ(y) :={{v ∈ R | aPy − cP ≤ v ≤ bPy + cP}, y ≥ 0{v ∈ R | bPy − cP ≤ v ≤ aPy + cP}, y < 0.Φ̃(y) := {v ∈ R | (aP − η)y2 ≤ vy ≤ (bP + η)y2},where η > 0. In view of (16) and (17),y ∈ C(R+) =⇒ (Pξ(y))(t) ∈ Φ(y(t)) ∀ t ∈ R+.Moreover, writing K := [−cP/η , cP/η], we haveΦ(y) ⊂ Φ̃(y) ∀ y ∈ R\K and E := supy∈K supv∈Φ(y) inf ṽ∈Φ̃(y)|v − ṽ| = cP .Let the linear system (A,B,C) (with transfer function G) be stabilizable and de-tectable. Write a := aP − η, b := bP + η and assume that G/(1 + aG) ∈ H∞and, for some δ > 0, (1 + bG)/(1 + aG) − δ is positive real. Then hypothesis (H1)holds with m = 1 and Φ̃ replacing Φ. We are now in a position to invoke Corollary7 to conclude properties of solutions of the single-input, single-output, functionaldifferential equationẋ(t) = Ax(t) + B[d(t) − (Pξ(Cx))(t)], x(0) = x0. (19)We reiterate that, for each x0 ∈ Rn and d ∈ L∞loc(R+), (19) has unique globalsolution. An application of Corollary 7 (with D(t) = {d(t)} for all t ∈ R+) yieldsthe existence of constants ε, c1, c2 > 0 such that, for every global solution x,‖x(t)‖ ≤ c1e−εt‖x0‖ + c2(‖d‖L∞[0,t] + cP)∀ t ∈ R+, (20)showing in particular that (19) is input-to-state stable with bias cP . Furthermore,by Lemma 4,limt→∞d(t) = 0 =⇒ lim supt→∞‖x(t)‖ ≤ c2cP . (21)Bibliography[1] M. Arcak & A. Teel, Input-to-state stability for a class of Lurie systems,Automatica, 38 (2002), pp. 1945–1949.[2] M. Brokate & J. Sprekels, Hysteresis and Phase Transitions, Springer Ver-lag, New York, 1996.[3] K. Deimling, Multivalued Differential Equations, de Gruyter, Berlin, 1992.[4] B. Jayawardhana, H. Logemann & E.P. Ryan, Input-to-state stabil-ity of differential inclusions with applications to hysteretic and quantizedfeedback systems, submitted for publication, 2007: preprint available at:http://www.maths.bath.ac.uk/∼epr/iss.pdf[5] H.K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Upper SaddleRiver, NJ, 2002.[6] M.A. Krasnosel’skii & A.V. Pokrovskii, Systems with Hysteresis. SpringerVerlag, Berlin, 1989.[7] H. Logemann & A.D. Mawby, Low-gain integral control of infinite-dimensional regular linear systems subject to input hysteresis, in Advances inMathematical Systems Theory, F. Colonius et al. eds., Birkhäuser, Boston, 2001,pp. 255–293.[8] H. Logemann & E.P. Ryan, Systems with hysteresis in the feedback loop:existence, regularity and asymptotic behaviour of solutions, ESAIM Control,Optimiz. & Calc. of Variations, 9 (2003), pp. 169–196.[9] H. Logemann, E.P. Ryan & I. Shvartsman, A class of differential-delaysystems with hysteresis: asymptotic behaviour of solutions, Nonlinear Analysis(2007), doi: 10.1016/j.na.2007.05.025.[10] E.D. Sontag, Input to state stability: basic concepts and results, in Nonlinearand Optimal Control Theory, P. Nistri and G. Stefani eds., Springer Verlag,Berlin, 2006, pp. 163-220.

Results of ISS type for hysteretic Lur'e systems:a differential inclusions approach

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Results of ISS type for hysteretic Lur'e systems:a differential inclusions approach

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