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Incremental stability of Lur’e systems through piecewise-affine approximations
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress The International Federation of Automatic Control Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International Federation of Control The International Federation of Automatic Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 1673–1679
Incremental stability of Lur’e systems Incremental Incremental stability stability of of Lur’e Lur’e systems systems through piecewise-affine approximations through piecewise-affine approximations through piecewise-affine approximations
S´ ergio Waitman ∗∗ Laurent Bako ∗∗ Paolo Massioni ∗∗ ∗∗ ∗ Laurent ∗ Paolo Massioni ∗∗ S´ e rgio G´ Waitman Bako ∗ ∗∗∗ ∗ ∗∗ S´ e Waitman Laurent Bako Paolo erard Scorletti S´ ergio rgio G´ Waitman Laurent Bako ∗ Fromion Paolo Massioni Massioni ∗ Vincent ∗∗∗ ∗ ∗∗∗ e rard Scorletti Vincent Fromion ∗ Vincent Fromion ∗∗∗ G´ e rard Scorletti G´ erard Scorletti Vincent Fromion ∗ ere, UMR CNRS 5005, Ecole Centrale de Lyon, ∗ Laboratoire Amp` ∗ Amp` eere, UMR CNRS 5005, Ecole Centrale de Lyon, ∗ Laboratoire Laboratoire Amp` UMR CNRS Ecole Centrale de Universit´ e de Ecully, France (e-mail: Laboratoire Amp` ere, re,Lyon, UMRF-69134 CNRS 5005, 5005, Ecole Centrale de Lyon, Lyon, Universit´ e de Lyon, F-69134 Ecully, France (e-mail: Universit´ de Lyon, Lyon, F-69134 F-69134 Ecully, Ecully, France France (e-mail: [email protected], {laurent.bako, Universit´ ee de (e-mail: [email protected], {laurent.bako, [email protected], {laurent.bako, gerard.scorletti}@ec-lyon.fr) [email protected], {laurent.bako, gerard.scorletti}@ec-lyon.fr) ∗∗ gerard.scorletti}@ec-lyon.fr) Laboratoire Amp` e re, UMR CNRS 5005, INSA de Lyon, Universit´e gerard.scorletti}@ec-lyon.fr) ∗∗ ∗∗ Amp` eere, UMR CNRS 5005, INSA de Lyon, Universit´ee ∗∗ Laboratoire Laboratoire Amp` re, UMR CNRS 5005, INSA de Lyon, de Lyon, F-69621 Villeurbanne, France Laboratoire Amp`ere, UMR Villeurbanne, CNRS 5005, INSA de(e-mail: Lyon, Universit´ Universit´e de Lyon, F-69621 France (e-mail: de Lyon, Lyon, [email protected]) F-69621 Villeurbanne, Villeurbanne, France France (e-mail: (e-mail: de F-69621 [email protected]) ∗∗∗ [email protected]) Universit´e Paris-Saclay, F-78350 Jouy-en-Josas, ∗∗∗ MaIAGE, INRA,[email protected]) ∗∗∗ INRA, Universit´ ee Paris-Saclay, F-78350 Jouy-en-Josas, ∗∗∗ MaIAGE, MaIAGE, INRA, Universit´ France (e-mail: [email protected]) MaIAGE,France INRA,(e-mail: Universit´ e Paris-Saclay, Paris-Saclay, F-78350 F-78350 Jouy-en-Josas, Jouy-en-Josas, [email protected]) France (e-mail: [email protected]) France (e-mail: [email protected]) Abstract: Lur’e-type nonlinear systems are virtually ubiquitous in applied control theory, which Abstract: nonlinear systems are ubiquitous in control Abstract: Lur’e-type nonlinear systems are virtually virtually ubiquitous in applied applied control theory, theory, which explains theLur’e-type great interest they have attracted throughout the years. The purpose of this which paper Abstract: Lur’e-type nonlinear systems are virtually ubiquitous in applied control theory, which explains the great interest they have attracted throughout the years. The purpose of this paper explains the great interest they have attracted throughout the years. The purpose of this paper is to propose conditions tothey assess incremental asymptoticthe stability of Lur’e systems that are explains the great interest have attracted throughout years. The purpose of this paper is to propose conditions to assess incremental asymptotic stability of Lur’e systems that are is to propose conditions to assess incremental asymptotic stability of Lur’e systems that are less conservative than those obtained with the incremental circle criterion. The method is based is to propose conditions to assess incremental asymptotic stability of Lur’e systems that are less conservative than those obtained with the incremental circle criterion. The method is based less conservative than those obtained with the incremental circle criterion. The method is based on the approximation of the nonlinearity by a piecewise-affine function. The Lur’e system can less conservative than those obtained with the incremental circle criterion. The method is based on the by function. The system on the the approximation of the nonlinearity nonlinearity by aaa piecewise-affine piecewise-affine function. The Lur’e Lur’e system can can then beapproximation rewritten as a of so-called piecewise-affine Lur’e system, for which sufficient conditions for on the approximation of the nonlinearity by piecewise-affine function. The Lur’e system can then be rewritten as a so-called piecewise-affine Lur’e system, for which sufficient conditions for then be rewritten as a so-called piecewise-affine Lur’e system, for which sufficient conditions for asymptotic incremental stabilitypiecewise-affine are provided. These conditions are expressed as conditions linear matrix then be rewritten as a so-called Lur’e system, for which sufficient for asymptotic incremental stability provided. These conditions are expressed as linear matrix asymptotic incremental stability are provided. These conditions are matrix inequalities (LMIs) allowing the are construction a continuous piecewise-quadratic incremental asymptotic incremental stability are provided. of These conditionspiecewise-quadratic are expressed expressed as as linear linear matrix inequalities (LMIs) allowing the construction of a continuous incremental inequalities (LMIs) allowing the construction of a continuous piecewise-quadratic incremental Lyapunov function, which can be efficiently solved numerically. The results are illustrated with inequalities (LMIs) allowing the construction of a continuous piecewise-quadratic incremental Lyapunov function, Lyapunov examples. function, which which can can be be efficiently efficiently solved solved numerically. numerically. The The results results are are illustrated illustrated with with numerical Lyapunov function, which can be efficiently solved numerically. The results are illustrated with numerical examples. numerical numerical examples. examples. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: incremental stability, Lur’e systems, incremental circle criterion, piecewise-affine Keywords:piecewise-affine incremental stability, stability, Lur’e systems, systems, incremental circle criterion, criterion, piecewise-affine Keywords: incremental Lur’e incremental piecewise-affine systems, approximation, Lyapunov methods.circle Keywords: incremental stability, Lur’e systems, incremental systems, piecewise-affine piecewise-affine approximation, Lyapunov methods.circle criterion, piecewise-affine systems, approximation, Lyapunov methods. systems, piecewise-affine approximation, Lyapunov methods. 1. INTRODUCTION criterion. These conditions are established for nonlineari1. INTRODUCTION INTRODUCTION criterion. These conditions areinestablished established for nonlineari1. criterion. These are ties belonging to conditions a sector and, this sense, for thenonlinearinonlinear1. INTRODUCTION criterion. These conditions areinestablished for nonlinearities belonging to a sector and, this sense, the nonlinearties belonging to a sector and, in this sense, the nonlinearity can be seen as a bounded perturbation on the linear ties belonging to a sector and, in this sense, the The so-called Lur’e-type nonlinear systems, given by the ity can be seen as a bounded perturbation on nonlinearthebounds linear can be as aa bounded perturbation on the linear The so-called so-called Lur’e-type nonlinear nonlinear systems, given by by the ity dynamics ofseen the system. The description via sector ity can be seen as bounded perturbation on the linear The Lur’e-type systems, given the feedback interconnection of a linear time-invariant (LTI) The so-called Lur’e-type nonlinear systems, given by the dynamics of the system. The description via sector bounds dynamics of the system. The description via sector bounds feedbackand interconnection of nonlinearity a linear linear time-invariant time-invariant (LTI) yields stability results that tend to be quite conservative, dynamics of the system. The description via sector bounds feedback interconnection of a (LTI) system a memoryless ϕ, represent an feedback interconnection of nonlinearity a linear time-invariant (LTI) yields stability resultsgives thatatend tend tocrude be quite quite conservative, results that be conservative, system and and memoryless ϕ,application represent an as the stability sector bound veryto representation of yields stability resultsgives thatatend tocrude be quite conservative, system aa represent important of systems nonlinearity with practicalϕ, in yields system andclass a memoryless memoryless nonlinearity ϕ,application represent an an as the sector bound very representation of as the sector bound gives a very crude representation of important class of systems with practical in the nonlinear operator. For stability analysis, an attempt as the sector bound gives a very crude representation of important class of systems with practical application in virtually any domain of system theory. The study in of the nonlinear operator. For stability analysis, an attempt important class of systems with practical application nonlinear For stability analysis, an attempt virtually any isdomain domain of system systemwith theory. The study study of of the to reduce theoperator. conservatism was made by transforming the nonlinear operator. For stability analysis, an attempt virtually any of theory. The these systems closely connected the development virtually any isdomain of systemwith theory. The study of to to reduce reduce theloop conservatism was made made by the conservatism was by transforming transforming theseabsolute systems closelyproblem connected the development the feedback via the addition of so-called Popovto reduce theloop conservatism was made by transforming these systems is closely connected with development of the stability (see e.g.the Liberzon (2006)), these systems is closely connected with the development of the feedback via the addition of so-called Popovthe feedback loop via the addition of so-called Popovthe absolute stability problem (see e.g. Liberzon (2006)), Zames-Falb frequency-dependent multipliers (Zames, 1966; the feedback loop via the addition of so-called the absolute problem (see which consistsstability in establishing conditions to ensure(2006)), asymp- Zames-Falb frequency-dependent multipliers (Zames,Popovthe absolute stability problemconditions (see e.g. e.g. Liberzon Liberzon (2006)), 1966; Zames-Falb frequency-dependent multipliers (Zames, 1966; which consists in establishing to ensure asympZames and Falb, 1968). However, it turns out that this Zames-Falb frequency-dependent multipliers (Zames, 1966; which consists in establishing conditions to ensure asymptotic stability of the origin forconditions a set of nonlinear functions which consists in establishing to ensure asympZames and Falb, 1968). However, it turns out that this Zames and 1968). However, it turns out that this totic stability of of the the origin origin for for aa set set of of nonlinear nonlinear functions approach is Falb, not applicable when incremental stability is Zames and Falb, 1968). However, it turns out that this totic stability functions in a sector. totic stability of the origin for a set of nonlinear functions approach is not applicable when incremental stability is approach is not applicable when incremental stability is in a sector. considered (Kulkarni and Safonov, 2002). Fromion and approach is not applicable when incremental stability is in sector. in aathis sector. considered (Kulkarni and Safonov, 2002). Fromion and In paper, we are interested in assessing incremental considered (Kulkarni Safonov, 2002). Fromion and Safonov (2004) showedand that there exist Lur’e nonlinear considered (Kulkarni and Safonov, 2002). Fromion and In this paper, wesystems, are interested interested in assessing incremental Safonov (2004) showed that there exist Lur’e nonlinear In this paper, we are in assessing incremental stability of Lur’e i.e. the stability of every system Safonov (2004) showed that there exist Lur’e nonlinear systems for which multiplier-based analysis ensures finite In this paper, we are interested in assessing incremental Safonov (2004) showed that there exist Lur’e nonlinear stability of of with Lur’e respect systems,toi.e. i.e. the the stability stability of every every system system systems systems for which which multiplier-based analysis ensuresstable. finite stability Lur’e systems, of trajectory other. Several for ensures finite stability, butmultiplier-based which are not analysis incrementally stability of with Lur’e respect systems,toi.e.each the stability of everydifferent system gain systems for which multiplier-based analysis ensuresstable. finite trajectory each other. Several different gain stability, but which are not incrementally trajectory with respect to each other. Several different notions of incremental e.g. Fromion, stability, but which are not stable. On the other hand, necessary and incrementally sufficient conditions trajectory with respectstability to eachcoexist other.(see Several different gain gain stability, but which are not incrementally stable. notions of incremental stability coexist (see e.g. Fromion, On incremental the other other hand, necessary and sufficient conditions notions of stability coexist (see Fromion, 1997; Lohmiller and Slotine, Angeli, 2002; Pavlov On the necessary sufficient for stability of Lur’eand systems wereconditions proposed notions of incremental incremental stability1998; coexist (see e.g. e.g. Fromion, On the other hand, hand, necessary and sufficient 1997; Lohmiller and Slotine, 1998; Angeli, 2002; Pavlov for Fromion incremental stability of Lur’e systems wereconditions proposed 1997; Lohmiller and Slotine, 1998; Angeli, 2002; Pavlov et al., 2004), but all have in common the fact that they enfor incremental stability of Lur’e systems were proposed by et al. (2003), but with the drawback of being 1997; Lohmiller and Slotine, 1998; Angeli, 2002; Pavlov for incremental stability of Lur’e systems were proposed et al., al.,strong 2004),qualitative but all all have have in common common the fact thatbehavior, they enen- by by Fromion Fromion et al. al.is (2003), (2003), but with with the drawbackapproach of being being et 2004), but in that they sure properties on the the fact system et but the drawback of NP-hard. There then a need for an alternative et al., 2004), but all have in common the fact that they enby Fromion et al.is (2003), but with the drawbackapproach of being sure strong qualitative properties on the system behavior, NP-hard. There then a need for an alternative sure strong qualitative properties on the system behavior, such as asymptotic independence of initial conditions and NP-hard. There then aa need an to the assessment incremental stability of Lur’eapproach systems, sure strong qualitative properties on the system behavior, NP-hard. There is isof need for for an alternative alternative approach suchunicity as asymptotic asymptotic independence of initial conditions and to to the the assessment assessment ofthen incremental stability of Lur’e Lur’e systems, such as independence initial conditions and the of the steady state. For of this reason, incremental of incremental stability of systems, which is less conservative than the celebrated incremental such as asymptotic independence of initial conditions and to the assessment of incremental stability of Lur’e systems, the unicity unicity ofoften the steady steady state. Forwith this problems reason, incremental incremental which criterion is less less conservative conservative than the celebrated celebrated incremental the of the state. For this reason, stability is used to cope involving which is than the incremental circle while being efficiently solvable. For this the unicity of the steady state. For this reason, incremental which is less conservative than the celebrated incremental stability is is often often used used to toand cope with problems problems involving circle criterion criterion whilethe being efficiently solvable. For For this stability cope with tracking/synchronization anti-windup controlinvolving (see e.g. circle while being efficiently solvable. reason, we consider analysis via piecewise-affine apstability is often used to cope with problems involving circle criterion whilethe being efficiently solvable. For this this tracking/synchronization and anti-windup control (see e.g. reason, we consider analysis via piecewise-affine aptracking/synchronization and anti-windup control (see e.g. Rantzer, 2000; Kim and de Persis, 2015). control (see e.g. reason, we consider the analysis via piecewise-affine approximations. tracking/synchronization and anti-windup reason, we consider the analysis via piecewise-affine apRantzer, 2000; 2000; Kim Kim and and de de Persis, Persis, 2015). 2015). proximations. Rantzer, proximations. Rantzer, 2000; Kim and de Persis, 2015). proximations. In the framework of input-output stability, Zames (1966) Piecewise-affine (PWA) systems are nonlinear systems deIn the the framework framework ofconditions input-output stability, Zames (1966) (1966) Piecewise-affine (PWA) systems systems are nonlinear nonlinear systems deIn input-output stability, Zames proposed graphicalof to ensure (incremental) sta- Piecewise-affine (PWA) are systems described by piecewise-affine differential equations. They can In the framework ofconditions input-output stability, Zames (1966) Piecewise-affine (PWA) systems are nonlinear systems deproposed graphical to ensure (incremental) sta- scribed scribed by by piecewise-affine piecewise-affine differential equations. They can can proposed graphical conditions to ensure (incremental) stability of Lur’e systems, known as the (incremental) circle differential equations. They proposed graphical conditions to ensure (incremental) stascribed by piecewise-affine differential equations. They can bility of of Lur’e systems, bility systems, known known as as the the (incremental) (incremental) circle circle bility of Lur’e Lur’e systems, known as the (incremental) circle Copyright 2017 IFAC 1709Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright 2017 IFAC 1709 Copyright © 2017 IFAC 1709 Peer review© of International Federation of Automatic Copyright ©under 2017 responsibility IFAC 1709Control. 10.1016/j.ifacol.2017.08.491
Proceedings of the 20th IFAC World Congress 1674 Sérgio Waitman et al. / IFAC PapersOnLine 50-1 (2017) 1673–1679 Toulouse, France, July 9-14, 2017
(resp. v 0) is equivalent to the componentwise inequality vi > 0 (resp. vi ≥ 0), ∀i ∈ {1, . . . , n}. For a matrix A ∈ Rn×n , A 0 (resp. A 0) denotes that A is positive definite (resp. semi-definite). The column concatenation of two matrices A and B of compatible dimensions, denoted A by col, is such that col(A, B) = B .
ϕ
ϕ
+
−1 p
+
⇐⇒ LTI
q
ϕPWA ⇐⇒ −1
−1 p
p LTI
PWA
q
q PWA
Fig. 1. Block diagram illustrating the approach taken in this paper. be used to naturally describe systems containing piecewiseaffine nonlinearities (such as saturations, relays and dead zones), or as an approximation of more general nonlinear systems. The interest in this class of systems lies in the fact that their description is quite close to that of LTI systems, allowing transposition of classic results on stability and performance analysis while being able to present quite complex nonlinear dynamics. For example, Johansson and Rantzer (1998) introduced piecewise-quadratic Lyapunov functions to the analysis of PWA systems through the use of the S-procedure. The approach was extended to consider the analysis of incremental properties of PWA systems by Waitman et al. (2016). In this paper, we propose a method to assess incremental stability of Lur’e systems through piecewise-affine approximations. The nonlinearity ϕ is replaced by a piecewiseaffine function ϕPWA plus an approximation error , which is characterized by its Lipschitz constant. This allows us to rewrite the Lur’e system as the interconnection of a PWA system with the approximation error, in what we may call a PWA Lur’e system (see Fig. 1). Through the refinement of ϕPWA , we are able to control the approximation error, and hence expect to obtain less conservative results. In this sense, we address two technical questions: obtain sufficient conditions to assess incremental stability of PWA Lur’e systems; and propose a method allowing the construction of ϕPWA . The proposed conditions are based on linear matrix inequalities (LMI) that can be efficiently solved. Although techniques to construct piecewise-affine approximations exist in the literature (see e.g. Zavieh and Rodrigues, 2013; Azuma et al., 2010), we introduce an approximation method ensuring a given upper bound on the Lipschitz constant of the approximation error. The paper is organized as follows. Section 2 states the problem of ensuring incremental asymptotic stability of Lur’e systems. The proposed approach is presented in Section 3. In Section 4, sufficient conditions for incremental asymptotic stability of PWA Lur’e systems are presented. Section 5 proposes a method to construct ϕPWA that ensures an upper bound on the Lipschitz constant of the approximation. Finally, Section 6 contains numerical examples illustrating the results obtained with the proposed approach. Notation We denote by · the Euclidean norm. The real half line [0, +∞) is denoted by R+ . The interior of a set A is denoted int (A). For a vector v = [v1 , . . . , vn ] ∈ Rn , v 0
The function φ : R+ × R+ × X → X is called the state transition map and is such that x = φ(t, t0 , x0 ) is the state x ∈ X attained at instant t when the system evolves from x0 ∈ X at the instant t0 .
The derivative of the function ϕ(q) with respect to q is denoted ϕ (q). A function ρ : R+ → R+ is said to be positive definite if it is such that ρ(0) = 0 and ρ(r) > 0, ∀r = 0. We denote by K the class of continuous and strictly increasing functions α : R+ → R+ for which α(0) = 0. A function α is of class K∞ if it is of class K and unbounded. A continuous function β : R+ × R+ → R+ is of class KL if for any fixed t ≥ 0, β(·, t) ∈ K and, for fixed s, β(s, ·) is decreasing and satisfies limt→∞ β(s, t) = 0. 2. PROBLEM FORMULATION In this paper, we are interested in establishing conditions to assess the incremental asymptotic stability of nonlinear Lur’e systems given by x(t) ˙ = Ax(t) + Bp(t) q(t) = Cx(t) (1) p(t) = −ϕ(q(t)) x(0) = x0 where x(t) ∈ X ⊆ Rn is the state, p(t), q(t) ∈ R are internal signals and ϕ is a given memoryless Lipschitz nonlinearity with ϕ(0) = 0. Let us recall the following definition, adapted from Angeli (2002). Definition 1. (Incremental asymptotic stability). We say that system (1) is incrementally asymptotically stable if there exists a function β of class KL so that for all ˜0 ∈ X and all t ≥ 0 the following holds x0 , x ˜0 , t) (2) x(t) − x ˜(t) ≤ β( x0 − x ˜(t) = φ(t, 0, x ˜0 ). If X = Rn , with x(t) = φ(t, 0, x0 ) and x the system is said to be incrementally globally asymptotically stable.
Parallel to standard stability conditions, incremental asymptotic stability may be shown to be equivalent to a Lyapunov-like condition. In view of the adapted definition adopted in this paper, let us recall the following theorem, adapted from Angeli (2002). Theorem 2. System (1) is incrementally asymptotically stable as in Definition 1 if there exist a continuous function V : X × X → R+ , called an incremental Lyapunov function, and K∞ functions α1 and α2 such that ˜ ≤ V (x, x ˜) ≤ α2 x − x ˜ (3) α1 x − x for every x, x ˜ ∈ X, and along any two trajectories ˜0 ∈ X, V satisfies x(t), x ˜(t), starting respectively from x0 , x for any t ≥ 0 t ˜0 ) ≤ − ρ x(τ ) − x ˜(τ ) dτ (4) V (x(t), x ˜(t)) − V (x0 , x 0
˜(t) = φ(t, 0, x ˜0 ) and ρ a positive with x(t) = φ(t, 0, x0 ), x definite function.
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3. PROPOSED APPROACH
ϕ(q)
Proof. The proof follows after straightforward manipulations. Indeed, it suffices to replace ϕ(q) by the sum ϕPWA (q)+(q). Then, using the fact that ϕPWA (q) = ri q + si = ri Cx + si , the nonlinear system (1) may be rewritten as x˙ = Ax − B(ϕPWA (q) + (q)) = Ax − B(ri Cx + si + (q)) = (A − ri BC)x − si B − B(q) (6) =: Ai x + ai + Bp By performing analysis on (5), we replace the test for every ϕ ∈ {ϕ | ϕ ∈ [−L, L]} by the test for every ϕ ∈ {ϕ | ϕ = ϕPWA + , with ∈ [−η, η]}. As we are able to control the approximation error through the refinement of ϕPWA (and
ϕ(q)
ϕ(q)
q
The traditional approach to assess incremental stability of Lur’e systems (1) is to use the incremental circle criterion (see e.g. Zames (1966); Fromion et al. (1999)). This involves embedding ϕ in a so-called incremental sector. Definition 3. (Incremental sector). The nonlinearity ϕ is said to belong to the incremental sector [κ1 , κ2 ] if κ1 ≤ (ϕ(q) − ϕ(˜ q ))/(q − q˜)) ≤ κ2 , for all q, q˜ ∈ R, with q = q˜. From Definition 3, it is clear that a Lipschitz nonlinearity ϕ, with Lipschitz constant L, belongs to the sector [−L, L]. The incremental circle criterion gives conditions to assess incremental stability of every nonlinearity inside an incremental sector. By doing so, we obtain tractable conditions to perform the analysis, but at the price of some conservatism. This is due to the fact that, in general, incremental sector conditions provide a very crude description of ϕ. To cope with this problem, we propose computing a piecewise-affine approximation ϕPWA of the nonlinearity ϕ, so that (1) is transformed into the interconnection of a PWA system with the approximation error: x(t) ˙ = Ai x(t) + ai + Bp (t) q(t) = C x(t) + c + Dp (t) for x(t) ∈ Xi i i (5) p (t) = −(q(t)) x(0) = x0 We shall refer to (5) as a PWA Lur’e system. We make the assumption that the approximation error is Lipschitz with Lipschitz constant η. The regions Xi , for i ∈ I := {1, . . . , N }, are closed convex polyhedral sets Xi = {x ∈ X | Gi x + gi 0} with non-empty and pairwise disjoint interiors such that i∈I Xi = X. Then, {Xi }i∈I constitutes a finite partition of X. From the geometry of Xi , the intersection Xi ∩ Xj between two different regions is always contained in a hyperplane, i.e. Xi ∩ Xj ⊆ {x ∈ X | Eij x + eij = 0}. The approach is illustrated in Fig. 1, and formalized in the next proposition. Proposition 4. Let Ri ⊂ R, i ∈ I = {1, . . . , N }, be nonempty intervals with pairwise disjoint interiors, such that {Ri }i∈I forms a partition of R. Let the scalar nonlinearity ϕ in (1) be decomposed as ϕ(q) = ϕPWA (q) + (q), with ϕPWA a piecewise-affine function given by ϕPWA (q) = ri q+ si , for q ∈ Ri . Then, the Lur’e system (1) is equivalent to the PWA Lur’e system (5), with (q) := ϕ(q) − ϕPWA (q), Ai := A − ri BC, ai := −si B, Ci = C, ci = 0, D = 0 and Xi = {x ∈ X | Cx ∈ Ri }.
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q
⇒
ϕ (q)
ϕ(q)
ϕ (q) ϕ (q)
q
ϕ (q)
⇒
q
Fig. 2. Comparison between the sectors describing the nonlinearity ϕ for the incremental circle criterion (left) and the piecewise-affine approach (right). thus to control η), this allows us to obtain a PWA Lur’e system whose nonlinearity is described by much tighter sector bounds (see Fig. 2). Hence, the analysis provides potentially less conservative results for the incremental analysis of Lur’e systems. The approach is presented in the next algorithm. Algorithm 1. Given a Lur’e system (1) with a memoryless Lipschitz nonlinearity ϕ: 1. Compute a piecewise-affine approximation ϕPWA so that = ϕ − ϕPWA is Lipschitz, with a Lipschitz constant η smaller than a given upper bound ηref . 2. Use Proposition 4 to construct an equivalent PWA Lur’e system (5) from (1). 3. Assess incremental asymptotic stability of (5), and, if positive, conclude on the incremental asymptotic stability of (1). In the next section, we present how to assess incremental asymptotic stability of PWA Lur’e systems through the analysis of an augmented system. Then, in Section 5, we address the problem on how to construct piecewiseaffine approximations of static nonlinearities. The goal is to ensure an upper bound on the Lipschitz constant of the approximation error, so that it is contained in an incremental sector. For the sake of clarity, the method to construct the piecewise-affine approximation is presented for SISO nonlinearities. Nevertheless, this method can be developed in the MIMO case as well. 4. INCREMENTAL STABILITY OF PIECEWISE-AFFINE LUR’E SYSTEMS In this section we propose conditions to assess incremental asymptotic stability of PWA Lur’e systems given by (5). The results are based on the construction of a piecewisequadratic incremental Lyapunov function via the application of Theorem 5. When studying incremental properties, it is standard to consider a fictitious augmented system composed of two copies of the original system (see e.g. Angeli (2002); Fromion (1997)). Incremental stability can then be studied via the analysis of the difference between the trajectories x, q˜, p˜ε ) denote the of both systems. Let (x, q, pε ) and (˜ state, the input and the output of each copy of the system.
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Considering the PWA structure of (5), we can define the augmented system given by ˙ x(t) = Aij x(t) + Bp(t) for x(t) ∈ Xij (7) q(t) = C ij x(t) + Dp(t) x(0) = x0 where x = col(x, x ˜, 1), p = col(p , p˜ ), q = q − q˜ and Ai 0 ai B 0 Aij = 0 Aj aj B= 0 B (8) 0 0 0 0 0 C ij = [Ci −Cj ci − cj ] D = [D −D] . The space X is defined as X = X × X × {1}, and regions Xij are defined as Xij = {x ∈ X | x ∈ Xi and x ˜ ∈ Xj }. Each region Xij is described by Xij = {x ∈ X | Gij x 0} where Gi 0 gi . (9) Gij = 0 Gj gj Analogously to the state partition {Xi }i∈I of system (5), the intersection between any two regions Xij and Xkl of (7) is either empty or contained in the hyperplane given by Xij ∩ Xkl ⊆ x ∈ X | E ijkl x = 0 . (10)
We shall propose conditions to compute an incremental Lyapunov function possessing the following piecewisequadratic structure: ˜) for x ∈ Xii (x − x ˜)T Pi (x − x (11) V (x, x ˜) = T x P ij x for x ∈ Xij , i = j
As presented in Waitman et al. (2016), the choice of a quadratic function on (x − x ˜) on regions Xii does not lead to any loss of generality. Indeed, it is a consequence of the fact that V (x, x) = 0, for every x ∈ X, due to (3).
Let In be the n × n identity matrix, and let I n ∈ R and J n ∈ R(2n+1)×(2n+1) be the matrices defined by In −In I 0 n . (12) In = Jn = −In In 0 0 We are then able to state the following theorem. Theorem 5. Let (5) be a PWA Lur’e system, and let be Lipschitz continuous with Lipschitz constant η > 0. If there exist symmetric matrices Pi ∈ Rn×n and P ij ∈ R(2n+1)×(2n+1) ; Uij , Rij , Wij ∈ Rpij ×pij with nonnegative coefficients and zero diagonal; Lijkl ∈ R(2n+1)×1 and positive scalars σ1 , σ2 , σ3 such that Pi − σ1 In 0 Pi − σ 2 I n 0 T Ai Pi + Pi Ai + CiT Ci + σ3 In Pi B + CiT D 0 B T Pi + D T C i DT D − η −2 Ip (13) for i ∈ I, T P ij − σ1 J n − Gij Uij Gij 0 T P ij − σ2 J n + Gij Rij Gij 0 T T Aij P ij + P ij Aij + C ij C ij + T P B + C D ij T ij σ3 J n + Gij Wij Gij 0 T T T −2 B P ij + D C ij D D − η Ip (14) 2n×2n
for (i, j) ∈ I × I, i = j, and
T
P ij = P kl + Lijkl E ijkl + E ijkl LTijkl
(15)
for (i, j), (k, l) such that Xij ∩ Xkl = ∅ are satisfied, then the PWA Lur’e system (5) is incrementally asymptotically stable. Proof. According to Theorem 2, (5) is incrementally asymptotically stable if there exists a continuous incremental Lyapunov function V , which is lower and upper bounded by class K∞ functions, and respects the integral constraint (4). We shall prove the theorem by showing that feasibility of (13)–(15) implies the existence of such a function possessing the structure (11). Continuity - We first show that V is a continuous function of x. This is clearly the case inside every cell, so we just need to show continuity on the boundaries. From (10), E ijkl x = 0 for all x ∈ Xij ∩ Xkl , then (15) implies that xT P ij x = xT P kl x for x ∈ Xij ∩ Xkl and hence that V is continuous. Norm bounds - The first inequality in (14), post and pre multiplied respectively by x and xT , implies that xT P ij x− 2
T
˜ ≥ xT Gij Uij Gij x. Since Uij is composed of σ1 x − x nonnegative coefficients, the right-hand side of the previous inequality is nonnegative whenever x ∈ Xij . This implies that xT P ij x ≥ σ1 x − x ˜
2
for x ∈ Xij .
(16)
The first inequality in (13) implies that V (x, x ˜) ≥ 2 ˜ for all x ∈ Xii . With (16), this guarantees σ1 x − x that 2 ˜ , ∀x, x ˜ ∈ X. (17) V (x, x ˜) ≥ σ1 x − x Proceeding exactly as before, the second inequalities in (13) and (14) imply that 2
˜ , ∀x, x ˜ ∈ X. (18) V (x, x ˜) ≤ σ2 x − x Inequalities (17) and (18) imply that the continuous piecewise quadratic function V given by (11) is such that 2
2
˜ ≤ V (x, x ˜) ≤ σ2 x − x ˜ . σ1 x − x
(19)
Integral constraint - We now show that the incremental Lyapunov function respects the integral constraint (4). Using the same arguments as before, the last inequality in (14), post and pre multiplied by col(x, p)T and col(x, p), implies that xT P ij (Aij x + Bp) + (Aij x + Bp)T P ij x + 2
˜ (C ij x + Dp)T (C ij x + Dp) − η −2 pT I p p ≤ −σ3 x − x (20) for all p ∈ R2 and all x ∈ Xij . Let ta and tb be two time instants such that the state trajectory of system (7) remains in Xij on the interval [ta , tb ]. By noticing that x˙ = Aij x + Bp, and integrating from ta to tb along trajectories of (7), we have
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x(tb )T P ij x(tb ) − x(ta )T P ij x(ta ) + tb 2 2
∆q(τ ) − η −2 ∆p (τ ) dτ tb ta 2 σ3 ∆x(τ ) dτ ≤− ta
(21)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Sérgio Waitman et al. / IFAC PapersOnLine 50-1 (2017) 1673–1679
with ∆x := x − x ˜, and ∆q and ∆p similarly defined. The same reasoning can be applied to the last inequality in (13), post and pre multiplying by col(x − x ˜, p − p˜ )T and col(x − x ˜, p − p˜ ), which yields ∆x(tb )T Pi ∆x(tb ) − ∆x(ta )T Pi ∆x(ta ) + tb 2 2 ∆q(τ ) − η −2 ∆p (τ ) dτ tb ta 2 σ3 ∆x(τ ) dτ. ≤−
(22)
ta
We note that the first terms in (21) and (22) represent the incremental Lyapunov function (11). Let us consider a trajectory x(τ ), ∀τ ∈ [0, t]. The time t1 can be decomposed n−1 as t = t − tin,n + k=0 (tout,k − tin,k ), with tout,k = tin,k+1 and tin,0 = 0, so that during each time interval [tin,k , tout,k ] the trajectory stays in a given region. Then, replacing ta by tin,k and tb by tout,k in (21) and (22), adding up to n for every region Xij crossed, and using the continuity of V yields ˜0 ) + V (x(t), x ˜(t)) − V (x0 , x t 2 2 ∆q(τ ) − η −2 ∆p (τ ) dτ 0 t 2 σ3 ∆x(τ ) dτ. ≤−
there is (ri , si ) ∈ R2 such that ϕPWA (q) = ri q + si , for q ∈ Ri , where i ∈ I = {1, . . . , N }. Since ϕ is continuous and is Lipschitz continuous, ϕPWA must be continuous. This implies that ∀q ∈ Ri ∩Rj , ri q +si = rj q +sj . We also fix ϕPWA (0) = 0, so that for any i such that q = 0 ∈ Ri , we have si = 0. We shall make the following assumption on the nonlinearity ϕ. Assumption 6. The memoryless nonlinearity ϕ is continuously differentiable, i.e. ϕ ∈ C 1 (R), and asymptotically affine, i.e. there exist k1 , k2 ∈ R such that limq→−∞ |ϕ (q) − k1 | = 0 and limq→∞ |ϕ (q) − k2 | = 0. Assumption 6 ensures that we are able to construct an approximation ϕPWA with a finite partition, i.e. with N < ∞. We are interested in finding ϕPWA that best approximates ϕ. We shall measure the approximation error by its Lipschitz constant, i.e., by its incremental gain. This may be formalized as minimize
η
subject to
(q) − (˜ q ) ≤ η q − q˜ q, q˜ ∈ R
ϕPWA ∈Φ(N )
(23)
0
Since is Lipschitz with a Lipschitz constant equal to η, 2 2 the quantity ∆q − η −2 ∆p is always positive, and we obtain t1 2 V (x(t), x ˜(t)) − V (x0 , x ˜0 ) ≤ − σ3 ∆q(τ ) dτ. (24) 0
Then V satisfies the conditions in Theorem 2 with αi (r) := 2 2 σi r , for i ∈ {1, 2} and ρ(r) := σ3 r . The function V is then an incremental Lyapunov function and system (5) is incrementally asymptotically stable, which concludes the proof. Testing the conditions in Theorem 5 is a feasibility problem involving LMIs, and thus can be efficiently solved. This result is of independent interest, as it extends the incremental circle criterion to the framework of PWA Lur’e systems. Indeed, by taking N = 1, we recover the LMI conditions of the classic incremental circle criterion (see e.g. Fromion et al. (1999)). In the proof of Theorem 5, we construct an incremental Lyapunov function that ensures incremental asymptotic stability. Another interpretation can be given in view of the framework of dissipative systems (Willems, 1972). Indeed, Theorem 5 can be seen as an incremental small gain theorem between the PWA system and the Lipschitz nonlinearity, where V would play the role of the storage 2 function, with supply rate w(q, p, x) = η −2 p − p˜ − 2 2 ˜ . q − q˜ − σ3 x − x 5. PIECEWISE-AFFINE APPROXIMATION OF SCALAR NONLINEARITIES Let us define Φ(N ) as the set of piecewise-affine functions ϕPWA : R → R defined on a partition of size N . That is, Φ(N ) is the set of piecewise-affine functions for which there exists a partition {Ri }i∈I of R, with |I| = N . Then,
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(P1)
As we refine the partition {Ri }i∈I , by choosing a larger N , the approximation error decreases, while the complexity of ϕPWA increases. This indicates a trade-off between the accuracy of the description and the complexity of the analysis. We shall search for a value of N ensuring a given upper bound ηref on the Lipschitz constant of the approximation error. This allows us to apply Theorem 5 to assess the incremental asymptotic stability of (1). The next proposition gives a method to obtain ϕPWA respecting the desired upper bound on the approximation. Proposition 7. Let ϕ be a function satisfying Assumption 6. Let ηref > 0, and let {Ri }i∈I , with I = {1, . . . , N }, be a partition of R obtained by a uniform division of the image of ϕ under R, i.e. l(ϕ (Ri )) = l(ϕ (Rj )), for all i, j ∈ I, where l(·) denotes the length of an interval. Also, let ri = (supq∈int(Ri ) ϕ (q) + inf q∈int(Ri ) ϕ (q))/2 and si be chosen to ensure continuity of ϕPWA . Then, by choosing N such that l(ϕ (Ri )) ≤ 2ηref , the obtained approximation ϕPWA ensures that is Lipschitz with a Lipschitz constant η ≤ ηref . Proof. We first use the fact that Lipschitz continuity is equivalent to boundedness of the derivative, for almost every q ∈ R. Then, we show that the proposed partition method ensures the desired upper bound on the Lipschitz constant. We begin by recalling a known fact about Lipschitz functions. Let η > 0. For an arbitrary partition {Ri }i∈I , the following two statements are equivalent: (i) (q) − (˜ q ) ≤ η q − q˜ , for all q, q˜ ∈ R. (ii) (q) ≤ η, for almost all q ∈ R.
We recall that (q) = ϕ (q) − ri , for all q ∈ int (Ri ). Let ηi > 0 be such that supq∈int(Ri ) (q) ≤ ηi . By choosing ri = (supq∈int(Ri ) ϕ (q) + inf q∈int(Ri ) ϕ (q))/2, we ensure that ηi = l(ϕ (Ri ))/2. Since ϕ is Lipschitz continuous, its derivative is bounded on R. Then, we can use the proposed partition so that the image of ϕ under R is
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uniformly divided, and we have ηi = l(ϕ (Ri ))/2 ≤ ηref , ∀i ∈ I. Then, by defining η = ηi and using the equivalent statements in the beginning of the proof, we have that (q) − (˜ q ) ≤ η q − q˜, for all q, q˜ ∈ R, with η ≤ ηref , which concludes the proof.
ϕ (q)
Proof. Due to the oddness of ϕ, we can focus on R+ and obtain the remaining by symmetry. Let {Ri }i∈I be an arbitrary partition of R+ , with I = {0, . . . , m}. Also, let ηi > 0 be as in Proposition 7. Then, by taking η := maxi∈I ηi , we have that (q) ≤ η, for almost all q ∈ R+ . It is clear that, for each region, the choice of ri that minimizes ηi is given by ri = (supq∈int(Ri ) ϕ (q) + inf q∈int(Ri ) ϕ (q))/2. In this case, we have ηi = (supq∈int(Ri ) ϕ (q) − inf q∈int(Ri ) ϕ (q))/2. As ϕ is Lipschitz, ϕ is bounded on R+ . Since the derivative ϕ is continuous and nondecreasing on R+ , we have m m supq∈int(Ri ) ϕ (q) − inf q∈int(Ri ) ϕ (q) ηi = 2 i=0 i=0
(ϕ (R+ )) . (25) 2 From this, we are interested minimizing η = maxi∈I ηi , in m subject to ηi ≥ 0 and i=0 ηi = (ϕ (R+ ))/2. The minimum is obtained when all ηi have the same value, which is obtained by taking a partition such that the image of ϕ under R+ is uniformly divided. This yields η = (ϕ (R+ ))/(2(m + 1)). Then, proceeding as in Proposition 7, we conclude that ϕPWA obtained by this method ensures that (q) − (˜ q ) ≤ η q − q˜, for all q, q˜ ∈ R, with η minimal. =
Despite the fact that Problem (P1) is non-convex due to the need to define the partition {Ri }i∈I , Proposition 9 shows that, in the case where ϕ satisfies Assumption 8, the optimal solution is known and quite easy to compute. The partitioning strategy is illustrated in Fig. 3. In this case, we may explicitly compute N such that the error bound is guaranteed to be inferior to ηref , as stated in the next proposition. Proposition 10. Let ϕ be a nonlinearity satisfying Assumptions 6 and 8. Let ηref > 0 be the desired upper bound on the Lipschitz constant of the approximation error. Let · denote the ceiling function. Then, if (ϕ (R+ )) − 1 > 0, (26) m= 2ηref
r4
ϕ min + 4·(2η) r3 ϕ min + 3·(2η)
The regions Ri = [qi , qi+1 ] can be defined by solving scalar nonlinear equations, which can be done by standard techniques such as the bisection method. We remark that, since ϕ is asymptotically linear, the leftmost and rightmost regions Ri may be unbounded.
One could wonder whether the partition method in Proposition 7 gives the optimal solution to (P1). It turns out that this is true, provided that ϕ satisfies some new assumptions, as stated in the following. Assumption 8. The memoryless nonlinearity ϕ is odd, monotone, and so that ϕ is nondecreasing on R+ . Proposition 9. Let ϕ be a nonlinear function respecting Assumptions 6 and 8. Then, the partition method described in Proposition 7 yields ϕPWA that is the optimal solution to (P1).
ϕ (q)
ϕ min + 5·(2η)
r2 ϕ min + 2·(2η) r1 ϕ min + 2η r0 ϕ min
q0
q
q1 q2 q3 q4 R0
R1 R2 R3
R4
···
Fig. 3. Partitioning strategy presented in Proposition 9, based on the uniform division of the image of ϕ under R. with N := 2m+1, and ϕPWA is obtained by the method in Proposition 9, then the approximation error is Lipschitz with a Lipschitz constant η ≤ ηref . Proof. This is a simple consequence of the fact that the partitioning strategy presented in Proposition 9 ensures that η = (ϕ (R+ ))/(2(m + 1)). With the techniques presented in this section and Section 4, we have all the tools to apply Algorithm 1 to the study of the incremental asymptotic stability of Lur’e systems (1). This shall be illustrated in the next section through some numerical examples. 6. NUMERICAL EXAMPLES Example 11. Consider the nonlinear system given by (1) with −1 0 1 A= B= C = [0 1] (27) 3 −2 0
and ϕ(q) = 2q 3 , for |q| ≤ 1, and ϕ(q) = 6q − 4 sign(q), for |q| > 1. ϕ satisfies Assumption 6, and belongs to the incremental sector [0, 6]. Analysis via the incremental circle criterion does not lead to a conclusion on the incremental stability of the system. We aim to obtain a piecewise-affine approximation ϕPWA over R, so that we can apply Theorem 5. Let us fix the desired maximal Lipschitz constant as ηref = 0.8. Using the approach proposed in Section 5, we obtain the approximation illustrated in Fig. 4, with N = 7 and η = 0.75. Using Proposition 4, the system is transformed in the interconnection of a PWA system and a Lipschitz nonlinearity. We then successfully apply Theorem 5 to construct a piecewise-affine incremental Lyapunov function, and conclude that this system is globally incrementally asymptotically stable. Example 12. Let us consider the nonlinear missile benchmark presented in Reichert (1992). The incremental behavior of the closed-loop system with a PI controller has been previously studied in Fromion et al. (1999). In this reference, the closed-loop system is written as an LTI system fedback through a nonlinearity ϕ(α) = −(an α3 + bn |α| α), with α being the angle of attack (see Fromion et al. (1999) for complete model and details). This model is assumed to be valid for |α| less than 20◦ (or 0.34 rad).
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7
5
ϕ(q) ϕPWA(q)
4 3
ϕ (q) ϕ PWA(q)
6
5
2 1
4
0 3
−1 −2
2
−3
1
−4 −5 −1.5
−1
−0.5
0
0.5
q
1
0
1.5
−1
−0.5
0
q
0.5
1
Fig. 4. Nonlinear function ϕ(q) = 2q 3 in Example 11 and the piecewise-affine approximation ϕPWA . The dotted lines represent the partition {Ri }i∈I . 3
ϕ(α) ϕPWA(α)
2
1
0
−1 −2 −3
−0.3
−0.2
−0.1
0
α
0.1
0.2
0.3
Fig. 5. Nonlinear function ϕ(α) = −(an α3 + bn |α| α) in Example 12 and the piecewise-affine approximation ϕPWA . Using again the techniques in the previous section with ηref = 3.5, we obtain the approximation ϕPWA presented in Fig. 5, with N = 5 and η = 2.4293. Application of Theorem 5 allows us to assess the incremental asymptotic stability of the closed-loop system, which concurs with the observations on Fromion et al. (1999) about the good behavior provided by the PI controller. 7. CONCLUSION In this paper we have proposed a new method to assess incremental asymptotic stability of Lur’e systems, based on piecewise-affine approximations. As a byproduct, we extended the celebrated incremental circle criterion to the analysis of PWA Lur’e systems, with conditions that can be solved very efficiently by interior point solvers. Perspectives for future work include the extension of the approach in Section 5 to the case of multivariable nonlinearities, and the establishment of local results, e.g. in the case when the nonlinearity ϕ is not asymptotically linear and a global approximation ϕPWA with a finite partition is not possible. Finally, the results in Section 5 may be coupled with robustness analysis to ensure robust incremental stability of Lur’e systems. REFERENCES Angeli, D. (2002). A Lyapunov approach to incremental stability properties. IEEE Transactions on Automatic Control, 47(3), 410–421. Azuma, S., Imura, J., and Sugie, T. (2010). Lebesgue piecewise affine approximation of nonlinear systems. Nonlinear Analysis: Hybrid Systems, 4(1), 92 – 102.
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Fromion, V. (1997). Some results on the behavior of Lipschitz continuous systems. In European Control Conference (ECC), 2011–2016. Brussels, Belgium. Fromion, V. and Safonov, M.G. (2004). Popov-Zames-Falb multipliers and continuity of the input/output map. In IFAC Symposium on Nonlinear Control Systems (NOLCOS). Stuttgart, Germany. Fromion, V., Safonov, M.G., and Scorletti, G. (2003). Necessary and sufficient conditions for Lur’e system incremental stability. In European Control Conference (ECC), 71–76. Cambridge, United Kingdom. Fromion, V., Scorletti, G., and Ferreres, G. (1999). Nonlinear performance of a PI controlled missile: an explanation. International Journal of Robust and Nonlinear Control, 9(8), 485–518. Johansson, M. and Rantzer, A. (1998). Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Transactions on Automatic Control, 43(4), 555–559. Kim, H. and de Persis, C. (2015). Output synchronization of Lur’e-type nonlinear systems in the presence of input disturbances. In IEEE Conference on Decision and Control (CDC), 4145–4150. Osaka, Japan. Kulkarni, V.V. and Safonov, M.G. (2002). Incremental positivity nonpreservation by stability multipliers. IEEE Transactions on Automatic Control, 47(1), 173–177. Liberzon, M.R. (2006). Essays on the absolute stability theory. Automation and Remote Control, 67(10), 1610– 1644. Lohmiller, W. and Slotine, J.J.E. (1998). On contraction analysis for non-linear systems. Automatica, 34(6), 683– 696. Pavlov, A., Pogromsky, A., van de Wouw, N., and Nijmeijer, H. (2004). Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Systems & Control Letters, 52(3–4), 257 – 261. Rantzer, A. (2000). A performance criterion for antiwindup compensators. European Journal of Control, 6(5), 449–452. Reichert, R.T. (1992). Dynamic scheduling of modernrobust-control autopilot designs for missiles. IEEE Control Systems, 12(5), 35–42. Waitman, S., Massioni, P., Bako, L., Scorletti, G., and Fromion, V. (2016). Incremental L2 -gain analysis of piecewise-affine systems using piecewise quadratic storage functions. In IEEE Conference on Decision and Control. Las Vegas, USA. Willems, J.C. (1972). Dissipative dynamical systems parts I and II. Archive for Rational Mechanics and Analysis, 45(5), 321–393. Zames, G. and Falb, P.L. (1968). Stability conditions for systems with monotone and slope-restricted nonlinearities. SIAM Journal on Control, 6(1), 89–108. Zames, G. (1966). On the input-output stability of timevarying nonlinear feedback systems—parts I and II. IEEE Transactions on Automatic Control, 11(2), 228– 238, 465–476. Zavieh, A. and Rodrigues, L. (2013). Intersection-based piecewise affine approximation of nonlinear systems. In Mediterranean Conference on Control Automation (MED), 640–645. Platanias-Chania, Greece.
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IFAC PapersOnLine 50-1 (2017) 1673–1679
Incremental stability of Lur’e systems Incremental Incremental stability stability of of Lur’e Lur’e systems systems through piecewise-affine approximations through piecewise-affine approximations through piecewise-affine approximations
S´ ergio Waitman ∗∗ Laurent Bako ∗∗ Paolo Massioni ∗∗ ∗∗ ∗ Laurent ∗ Paolo Massioni ∗∗ S´ e rgio G´ Waitman Bako ∗ ∗∗∗ ∗ ∗∗ S´ e Waitman Laurent Bako Paolo erard Scorletti S´ ergio rgio G´ Waitman Laurent Bako ∗ Fromion Paolo Massioni Massioni ∗ Vincent ∗∗∗ ∗ ∗∗∗ e rard Scorletti Vincent Fromion ∗ Vincent Fromion ∗∗∗ G´ e rard Scorletti G´ erard Scorletti Vincent Fromion ∗ ere, UMR CNRS 5005, Ecole Centrale de Lyon, ∗ Laboratoire Amp` ∗ Amp` eere, UMR CNRS 5005, Ecole Centrale de Lyon, ∗ Laboratoire Laboratoire Amp` UMR CNRS Ecole Centrale de Universit´ e de Ecully, France (e-mail: Laboratoire Amp` ere, re,Lyon, UMRF-69134 CNRS 5005, 5005, Ecole Centrale de Lyon, Lyon, Universit´ e de Lyon, F-69134 Ecully, France (e-mail: Universit´ de Lyon, Lyon, F-69134 F-69134 Ecully, Ecully, France France (e-mail: [email protected], {laurent.bako, Universit´ ee de (e-mail: [email protected], {laurent.bako, [email protected], {laurent.bako, gerard.scorletti}@ec-lyon.fr) [email protected], {laurent.bako, gerard.scorletti}@ec-lyon.fr) ∗∗ gerard.scorletti}@ec-lyon.fr) Laboratoire Amp` e re, UMR CNRS 5005, INSA de Lyon, Universit´e gerard.scorletti}@ec-lyon.fr) ∗∗ ∗∗ Amp` eere, UMR CNRS 5005, INSA de Lyon, Universit´ee ∗∗ Laboratoire Laboratoire Amp` re, UMR CNRS 5005, INSA de Lyon, de Lyon, F-69621 Villeurbanne, France Laboratoire Amp`ere, UMR Villeurbanne, CNRS 5005, INSA de(e-mail: Lyon, Universit´ Universit´e de Lyon, F-69621 France (e-mail: de Lyon, Lyon, [email protected]) F-69621 Villeurbanne, Villeurbanne, France France (e-mail: (e-mail: de F-69621 [email protected]) ∗∗∗ [email protected]) Universit´e Paris-Saclay, F-78350 Jouy-en-Josas, ∗∗∗ MaIAGE, INRA,[email protected]) ∗∗∗ INRA, Universit´ ee Paris-Saclay, F-78350 Jouy-en-Josas, ∗∗∗ MaIAGE, MaIAGE, INRA, Universit´ France (e-mail: [email protected]) MaIAGE,France INRA,(e-mail: Universit´ e Paris-Saclay, Paris-Saclay, F-78350 F-78350 Jouy-en-Josas, Jouy-en-Josas, [email protected]) France (e-mail: [email protected]) France (e-mail: [email protected]) Abstract: Lur’e-type nonlinear systems are virtually ubiquitous in applied control theory, which Abstract: nonlinear systems are ubiquitous in control Abstract: Lur’e-type nonlinear systems are virtually virtually ubiquitous in applied applied control theory, theory, which explains theLur’e-type great interest they have attracted throughout the years. The purpose of this which paper Abstract: Lur’e-type nonlinear systems are virtually ubiquitous in applied control theory, which explains the great interest they have attracted throughout the years. The purpose of this paper explains the great interest they have attracted throughout the years. The purpose of this paper is to propose conditions tothey assess incremental asymptoticthe stability of Lur’e systems that are explains the great interest have attracted throughout years. The purpose of this paper is to propose conditions to assess incremental asymptotic stability of Lur’e systems that are is to propose conditions to assess incremental asymptotic stability of Lur’e systems that are less conservative than those obtained with the incremental circle criterion. The method is based is to propose conditions to assess incremental asymptotic stability of Lur’e systems that are less conservative than those obtained with the incremental circle criterion. The method is based less conservative than those obtained with the incremental circle criterion. The method is based on the approximation of the nonlinearity by a piecewise-affine function. The Lur’e system can less conservative than those obtained with the incremental circle criterion. The method is based on the by function. The system on the the approximation of the nonlinearity nonlinearity by aaa piecewise-affine piecewise-affine function. The Lur’e Lur’e system can can then beapproximation rewritten as a of so-called piecewise-affine Lur’e system, for which sufficient conditions for on the approximation of the nonlinearity by piecewise-affine function. The Lur’e system can then be rewritten as a so-called piecewise-affine Lur’e system, for which sufficient conditions for then be rewritten as a so-called piecewise-affine Lur’e system, for which sufficient conditions for asymptotic incremental stabilitypiecewise-affine are provided. These conditions are expressed as conditions linear matrix then be rewritten as a so-called Lur’e system, for which sufficient for asymptotic incremental stability provided. These conditions are expressed as linear matrix asymptotic incremental stability are provided. These conditions are matrix inequalities (LMIs) allowing the are construction a continuous piecewise-quadratic incremental asymptotic incremental stability are provided. of These conditionspiecewise-quadratic are expressed expressed as as linear linear matrix inequalities (LMIs) allowing the construction of a continuous incremental inequalities (LMIs) allowing the construction of a continuous piecewise-quadratic incremental Lyapunov function, which can be efficiently solved numerically. The results are illustrated with inequalities (LMIs) allowing the construction of a continuous piecewise-quadratic incremental Lyapunov function, Lyapunov examples. function, which which can can be be efficiently efficiently solved solved numerically. numerically. The The results results are are illustrated illustrated with with numerical Lyapunov function, which can be efficiently solved numerically. The results are illustrated with numerical examples. numerical numerical examples. examples. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: incremental stability, Lur’e systems, incremental circle criterion, piecewise-affine Keywords:piecewise-affine incremental stability, stability, Lur’e systems, systems, incremental circle criterion, criterion, piecewise-affine Keywords: incremental Lur’e incremental piecewise-affine systems, approximation, Lyapunov methods.circle Keywords: incremental stability, Lur’e systems, incremental systems, piecewise-affine piecewise-affine approximation, Lyapunov methods.circle criterion, piecewise-affine systems, approximation, Lyapunov methods. systems, piecewise-affine approximation, Lyapunov methods. 1. INTRODUCTION criterion. These conditions are established for nonlineari1. INTRODUCTION INTRODUCTION criterion. These conditions areinestablished established for nonlineari1. criterion. These are ties belonging to conditions a sector and, this sense, for thenonlinearinonlinear1. INTRODUCTION criterion. These conditions areinestablished for nonlinearities belonging to a sector and, this sense, the nonlinearties belonging to a sector and, in this sense, the nonlinearity can be seen as a bounded perturbation on the linear ties belonging to a sector and, in this sense, the The so-called Lur’e-type nonlinear systems, given by the ity can be seen as a bounded perturbation on nonlinearthebounds linear can be as aa bounded perturbation on the linear The so-called so-called Lur’e-type nonlinear nonlinear systems, given by by the ity dynamics ofseen the system. The description via sector ity can be seen as bounded perturbation on the linear The Lur’e-type systems, given the feedback interconnection of a linear time-invariant (LTI) The so-called Lur’e-type nonlinear systems, given by the dynamics of the system. The description via sector bounds dynamics of the system. The description via sector bounds feedbackand interconnection of nonlinearity a linear linear time-invariant time-invariant (LTI) yields stability results that tend to be quite conservative, dynamics of the system. The description via sector bounds feedback interconnection of a (LTI) system a memoryless ϕ, represent an feedback interconnection of nonlinearity a linear time-invariant (LTI) yields stability resultsgives thatatend tend tocrude be quite quite conservative, results that be conservative, system and and memoryless ϕ,application represent an as the stability sector bound veryto representation of yields stability resultsgives thatatend tocrude be quite conservative, system aa represent important of systems nonlinearity with practicalϕ, in yields system andclass a memoryless memoryless nonlinearity ϕ,application represent an an as the sector bound very representation of as the sector bound gives a very crude representation of important class of systems with practical in the nonlinear operator. For stability analysis, an attempt as the sector bound gives a very crude representation of important class of systems with practical application in virtually any domain of system theory. The study in of the nonlinear operator. For stability analysis, an attempt important class of systems with practical application nonlinear For stability analysis, an attempt virtually any isdomain domain of system systemwith theory. The study study of of the to reduce theoperator. conservatism was made by transforming the nonlinear operator. For stability analysis, an attempt virtually any of theory. The these systems closely connected the development virtually any isdomain of systemwith theory. The study of to to reduce reduce theloop conservatism was made made by the conservatism was by transforming transforming theseabsolute systems closelyproblem connected the development the feedback via the addition of so-called Popovto reduce theloop conservatism was made by transforming these systems is closely connected with development of the stability (see e.g.the Liberzon (2006)), these systems is closely connected with the development of the feedback via the addition of so-called Popovthe feedback loop via the addition of so-called Popovthe absolute stability problem (see e.g. Liberzon (2006)), Zames-Falb frequency-dependent multipliers (Zames, 1966; the feedback loop via the addition of so-called the absolute problem (see which consistsstability in establishing conditions to ensure(2006)), asymp- Zames-Falb frequency-dependent multipliers (Zames,Popovthe absolute stability problemconditions (see e.g. e.g. Liberzon Liberzon (2006)), 1966; Zames-Falb frequency-dependent multipliers (Zames, 1966; which consists in establishing to ensure asympZames and Falb, 1968). However, it turns out that this Zames-Falb frequency-dependent multipliers (Zames, 1966; which consists in establishing conditions to ensure asymptotic stability of the origin forconditions a set of nonlinear functions which consists in establishing to ensure asympZames and Falb, 1968). However, it turns out that this Zames and 1968). However, it turns out that this totic stability of of the the origin origin for for aa set set of of nonlinear nonlinear functions approach is Falb, not applicable when incremental stability is Zames and Falb, 1968). However, it turns out that this totic stability functions in a sector. totic stability of the origin for a set of nonlinear functions approach is not applicable when incremental stability is approach is not applicable when incremental stability is in a sector. considered (Kulkarni and Safonov, 2002). Fromion and approach is not applicable when incremental stability is in sector. in aathis sector. considered (Kulkarni and Safonov, 2002). Fromion and In paper, we are interested in assessing incremental considered (Kulkarni Safonov, 2002). Fromion and Safonov (2004) showedand that there exist Lur’e nonlinear considered (Kulkarni and Safonov, 2002). Fromion and In this paper, wesystems, are interested interested in assessing incremental Safonov (2004) showed that there exist Lur’e nonlinear In this paper, we are in assessing incremental stability of Lur’e i.e. the stability of every system Safonov (2004) showed that there exist Lur’e nonlinear systems for which multiplier-based analysis ensures finite In this paper, we are interested in assessing incremental Safonov (2004) showed that there exist Lur’e nonlinear stability of of with Lur’e respect systems,toi.e. i.e. the the stability stability of every every system system systems systems for which which multiplier-based analysis ensuresstable. finite stability Lur’e systems, of trajectory other. Several for ensures finite stability, butmultiplier-based which are not analysis incrementally stability of with Lur’e respect systems,toi.e.each the stability of everydifferent system gain systems for which multiplier-based analysis ensuresstable. finite trajectory each other. Several different gain stability, but which are not incrementally trajectory with respect to each other. Several different notions of incremental e.g. Fromion, stability, but which are not stable. On the other hand, necessary and incrementally sufficient conditions trajectory with respectstability to eachcoexist other.(see Several different gain gain stability, but which are not incrementally stable. notions of incremental stability coexist (see e.g. Fromion, On incremental the other other hand, necessary and sufficient conditions notions of stability coexist (see Fromion, 1997; Lohmiller and Slotine, Angeli, 2002; Pavlov On the necessary sufficient for stability of Lur’eand systems wereconditions proposed notions of incremental incremental stability1998; coexist (see e.g. e.g. Fromion, On the other hand, hand, necessary and sufficient 1997; Lohmiller and Slotine, 1998; Angeli, 2002; Pavlov for Fromion incremental stability of Lur’e systems wereconditions proposed 1997; Lohmiller and Slotine, 1998; Angeli, 2002; Pavlov et al., 2004), but all have in common the fact that they enfor incremental stability of Lur’e systems were proposed by et al. (2003), but with the drawback of being 1997; Lohmiller and Slotine, 1998; Angeli, 2002; Pavlov for incremental stability of Lur’e systems were proposed et al., al.,strong 2004),qualitative but all all have have in common common the fact thatbehavior, they enen- by by Fromion Fromion et al. al.is (2003), (2003), but with with the drawbackapproach of being being et 2004), but in that they sure properties on the the fact system et but the drawback of NP-hard. There then a need for an alternative et al., 2004), but all have in common the fact that they enby Fromion et al.is (2003), but with the drawbackapproach of being sure strong qualitative properties on the system behavior, NP-hard. There then a need for an alternative sure strong qualitative properties on the system behavior, such as asymptotic independence of initial conditions and NP-hard. There then aa need an to the assessment incremental stability of Lur’eapproach systems, sure strong qualitative properties on the system behavior, NP-hard. There is isof need for for an alternative alternative approach suchunicity as asymptotic asymptotic independence of initial conditions and to to the the assessment assessment ofthen incremental stability of Lur’e Lur’e systems, such as independence initial conditions and the of the steady state. For of this reason, incremental of incremental stability of systems, which is less conservative than the celebrated incremental such as asymptotic independence of initial conditions and to the assessment of incremental stability of Lur’e systems, the unicity unicity ofoften the steady steady state. Forwith this problems reason, incremental incremental which criterion is less less conservative conservative than the celebrated celebrated incremental the of the state. For this reason, stability is used to cope involving which is than the incremental circle while being efficiently solvable. For this the unicity of the steady state. For this reason, incremental which is less conservative than the celebrated incremental stability is is often often used used to toand cope with problems problems involving circle criterion criterion whilethe being efficiently solvable. For For this stability cope with tracking/synchronization anti-windup controlinvolving (see e.g. circle while being efficiently solvable. reason, we consider analysis via piecewise-affine apstability is often used to cope with problems involving circle criterion whilethe being efficiently solvable. For this this tracking/synchronization and anti-windup control (see e.g. reason, we consider analysis via piecewise-affine aptracking/synchronization and anti-windup control (see e.g. Rantzer, 2000; Kim and de Persis, 2015). control (see e.g. reason, we consider the analysis via piecewise-affine approximations. tracking/synchronization and anti-windup reason, we consider the analysis via piecewise-affine apRantzer, 2000; 2000; Kim Kim and and de de Persis, Persis, 2015). 2015). proximations. Rantzer, proximations. Rantzer, 2000; Kim and de Persis, 2015). proximations. In the framework of input-output stability, Zames (1966) Piecewise-affine (PWA) systems are nonlinear systems deIn the the framework framework ofconditions input-output stability, Zames (1966) (1966) Piecewise-affine (PWA) systems systems are nonlinear nonlinear systems deIn input-output stability, Zames proposed graphicalof to ensure (incremental) sta- Piecewise-affine (PWA) are systems described by piecewise-affine differential equations. They can In the framework ofconditions input-output stability, Zames (1966) Piecewise-affine (PWA) systems are nonlinear systems deproposed graphical to ensure (incremental) sta- scribed scribed by by piecewise-affine piecewise-affine differential equations. They can can proposed graphical conditions to ensure (incremental) stability of Lur’e systems, known as the (incremental) circle differential equations. They proposed graphical conditions to ensure (incremental) stascribed by piecewise-affine differential equations. They can bility of of Lur’e systems, bility systems, known known as as the the (incremental) (incremental) circle circle bility of Lur’e Lur’e systems, known as the (incremental) circle Copyright 2017 IFAC 1709Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright 2017 IFAC 1709 Copyright © 2017 IFAC 1709 Peer review© of International Federation of Automatic Copyright ©under 2017 responsibility IFAC 1709Control. 10.1016/j.ifacol.2017.08.491
Proceedings of the 20th IFAC World Congress 1674 Sérgio Waitman et al. / IFAC PapersOnLine 50-1 (2017) 1673–1679 Toulouse, France, July 9-14, 2017
(resp. v 0) is equivalent to the componentwise inequality vi > 0 (resp. vi ≥ 0), ∀i ∈ {1, . . . , n}. For a matrix A ∈ Rn×n , A 0 (resp. A 0) denotes that A is positive definite (resp. semi-definite). The column concatenation of two matrices A and B of compatible dimensions, denoted A by col, is such that col(A, B) = B .
ϕ
ϕ
+
−1 p
+
⇐⇒ LTI
q
ϕPWA ⇐⇒ −1
−1 p
p LTI
PWA
q
q PWA
Fig. 1. Block diagram illustrating the approach taken in this paper. be used to naturally describe systems containing piecewiseaffine nonlinearities (such as saturations, relays and dead zones), or as an approximation of more general nonlinear systems. The interest in this class of systems lies in the fact that their description is quite close to that of LTI systems, allowing transposition of classic results on stability and performance analysis while being able to present quite complex nonlinear dynamics. For example, Johansson and Rantzer (1998) introduced piecewise-quadratic Lyapunov functions to the analysis of PWA systems through the use of the S-procedure. The approach was extended to consider the analysis of incremental properties of PWA systems by Waitman et al. (2016). In this paper, we propose a method to assess incremental stability of Lur’e systems through piecewise-affine approximations. The nonlinearity ϕ is replaced by a piecewiseaffine function ϕPWA plus an approximation error , which is characterized by its Lipschitz constant. This allows us to rewrite the Lur’e system as the interconnection of a PWA system with the approximation error, in what we may call a PWA Lur’e system (see Fig. 1). Through the refinement of ϕPWA , we are able to control the approximation error, and hence expect to obtain less conservative results. In this sense, we address two technical questions: obtain sufficient conditions to assess incremental stability of PWA Lur’e systems; and propose a method allowing the construction of ϕPWA . The proposed conditions are based on linear matrix inequalities (LMI) that can be efficiently solved. Although techniques to construct piecewise-affine approximations exist in the literature (see e.g. Zavieh and Rodrigues, 2013; Azuma et al., 2010), we introduce an approximation method ensuring a given upper bound on the Lipschitz constant of the approximation error. The paper is organized as follows. Section 2 states the problem of ensuring incremental asymptotic stability of Lur’e systems. The proposed approach is presented in Section 3. In Section 4, sufficient conditions for incremental asymptotic stability of PWA Lur’e systems are presented. Section 5 proposes a method to construct ϕPWA that ensures an upper bound on the Lipschitz constant of the approximation. Finally, Section 6 contains numerical examples illustrating the results obtained with the proposed approach. Notation We denote by · the Euclidean norm. The real half line [0, +∞) is denoted by R+ . The interior of a set A is denoted int (A). For a vector v = [v1 , . . . , vn ] ∈ Rn , v 0
The function φ : R+ × R+ × X → X is called the state transition map and is such that x = φ(t, t0 , x0 ) is the state x ∈ X attained at instant t when the system evolves from x0 ∈ X at the instant t0 .
The derivative of the function ϕ(q) with respect to q is denoted ϕ (q). A function ρ : R+ → R+ is said to be positive definite if it is such that ρ(0) = 0 and ρ(r) > 0, ∀r = 0. We denote by K the class of continuous and strictly increasing functions α : R+ → R+ for which α(0) = 0. A function α is of class K∞ if it is of class K and unbounded. A continuous function β : R+ × R+ → R+ is of class KL if for any fixed t ≥ 0, β(·, t) ∈ K and, for fixed s, β(s, ·) is decreasing and satisfies limt→∞ β(s, t) = 0. 2. PROBLEM FORMULATION In this paper, we are interested in establishing conditions to assess the incremental asymptotic stability of nonlinear Lur’e systems given by x(t) ˙ = Ax(t) + Bp(t) q(t) = Cx(t) (1) p(t) = −ϕ(q(t)) x(0) = x0 where x(t) ∈ X ⊆ Rn is the state, p(t), q(t) ∈ R are internal signals and ϕ is a given memoryless Lipschitz nonlinearity with ϕ(0) = 0. Let us recall the following definition, adapted from Angeli (2002). Definition 1. (Incremental asymptotic stability). We say that system (1) is incrementally asymptotically stable if there exists a function β of class KL so that for all ˜0 ∈ X and all t ≥ 0 the following holds x0 , x ˜0 , t) (2) x(t) − x ˜(t) ≤ β( x0 − x ˜(t) = φ(t, 0, x ˜0 ). If X = Rn , with x(t) = φ(t, 0, x0 ) and x the system is said to be incrementally globally asymptotically stable.
Parallel to standard stability conditions, incremental asymptotic stability may be shown to be equivalent to a Lyapunov-like condition. In view of the adapted definition adopted in this paper, let us recall the following theorem, adapted from Angeli (2002). Theorem 2. System (1) is incrementally asymptotically stable as in Definition 1 if there exist a continuous function V : X × X → R+ , called an incremental Lyapunov function, and K∞ functions α1 and α2 such that ˜ ≤ V (x, x ˜) ≤ α2 x − x ˜ (3) α1 x − x for every x, x ˜ ∈ X, and along any two trajectories ˜0 ∈ X, V satisfies x(t), x ˜(t), starting respectively from x0 , x for any t ≥ 0 t ˜0 ) ≤ − ρ x(τ ) − x ˜(τ ) dτ (4) V (x(t), x ˜(t)) − V (x0 , x 0
˜(t) = φ(t, 0, x ˜0 ) and ρ a positive with x(t) = φ(t, 0, x0 ), x definite function.
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3. PROPOSED APPROACH
ϕ(q)
Proof. The proof follows after straightforward manipulations. Indeed, it suffices to replace ϕ(q) by the sum ϕPWA (q)+(q). Then, using the fact that ϕPWA (q) = ri q + si = ri Cx + si , the nonlinear system (1) may be rewritten as x˙ = Ax − B(ϕPWA (q) + (q)) = Ax − B(ri Cx + si + (q)) = (A − ri BC)x − si B − B(q) (6) =: Ai x + ai + Bp By performing analysis on (5), we replace the test for every ϕ ∈ {ϕ | ϕ ∈ [−L, L]} by the test for every ϕ ∈ {ϕ | ϕ = ϕPWA + , with ∈ [−η, η]}. As we are able to control the approximation error through the refinement of ϕPWA (and
ϕ(q)
ϕ(q)
q
The traditional approach to assess incremental stability of Lur’e systems (1) is to use the incremental circle criterion (see e.g. Zames (1966); Fromion et al. (1999)). This involves embedding ϕ in a so-called incremental sector. Definition 3. (Incremental sector). The nonlinearity ϕ is said to belong to the incremental sector [κ1 , κ2 ] if κ1 ≤ (ϕ(q) − ϕ(˜ q ))/(q − q˜)) ≤ κ2 , for all q, q˜ ∈ R, with q = q˜. From Definition 3, it is clear that a Lipschitz nonlinearity ϕ, with Lipschitz constant L, belongs to the sector [−L, L]. The incremental circle criterion gives conditions to assess incremental stability of every nonlinearity inside an incremental sector. By doing so, we obtain tractable conditions to perform the analysis, but at the price of some conservatism. This is due to the fact that, in general, incremental sector conditions provide a very crude description of ϕ. To cope with this problem, we propose computing a piecewise-affine approximation ϕPWA of the nonlinearity ϕ, so that (1) is transformed into the interconnection of a PWA system with the approximation error: x(t) ˙ = Ai x(t) + ai + Bp (t) q(t) = C x(t) + c + Dp (t) for x(t) ∈ Xi i i (5) p (t) = −(q(t)) x(0) = x0 We shall refer to (5) as a PWA Lur’e system. We make the assumption that the approximation error is Lipschitz with Lipschitz constant η. The regions Xi , for i ∈ I := {1, . . . , N }, are closed convex polyhedral sets Xi = {x ∈ X | Gi x + gi 0} with non-empty and pairwise disjoint interiors such that i∈I Xi = X. Then, {Xi }i∈I constitutes a finite partition of X. From the geometry of Xi , the intersection Xi ∩ Xj between two different regions is always contained in a hyperplane, i.e. Xi ∩ Xj ⊆ {x ∈ X | Eij x + eij = 0}. The approach is illustrated in Fig. 1, and formalized in the next proposition. Proposition 4. Let Ri ⊂ R, i ∈ I = {1, . . . , N }, be nonempty intervals with pairwise disjoint interiors, such that {Ri }i∈I forms a partition of R. Let the scalar nonlinearity ϕ in (1) be decomposed as ϕ(q) = ϕPWA (q) + (q), with ϕPWA a piecewise-affine function given by ϕPWA (q) = ri q+ si , for q ∈ Ri . Then, the Lur’e system (1) is equivalent to the PWA Lur’e system (5), with (q) := ϕ(q) − ϕPWA (q), Ai := A − ri BC, ai := −si B, Ci = C, ci = 0, D = 0 and Xi = {x ∈ X | Cx ∈ Ri }.
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q
⇒
ϕ (q)
ϕ(q)
ϕ (q) ϕ (q)
q
ϕ (q)
⇒
q
Fig. 2. Comparison between the sectors describing the nonlinearity ϕ for the incremental circle criterion (left) and the piecewise-affine approach (right). thus to control η), this allows us to obtain a PWA Lur’e system whose nonlinearity is described by much tighter sector bounds (see Fig. 2). Hence, the analysis provides potentially less conservative results for the incremental analysis of Lur’e systems. The approach is presented in the next algorithm. Algorithm 1. Given a Lur’e system (1) with a memoryless Lipschitz nonlinearity ϕ: 1. Compute a piecewise-affine approximation ϕPWA so that = ϕ − ϕPWA is Lipschitz, with a Lipschitz constant η smaller than a given upper bound ηref . 2. Use Proposition 4 to construct an equivalent PWA Lur’e system (5) from (1). 3. Assess incremental asymptotic stability of (5), and, if positive, conclude on the incremental asymptotic stability of (1). In the next section, we present how to assess incremental asymptotic stability of PWA Lur’e systems through the analysis of an augmented system. Then, in Section 5, we address the problem on how to construct piecewiseaffine approximations of static nonlinearities. The goal is to ensure an upper bound on the Lipschitz constant of the approximation error, so that it is contained in an incremental sector. For the sake of clarity, the method to construct the piecewise-affine approximation is presented for SISO nonlinearities. Nevertheless, this method can be developed in the MIMO case as well. 4. INCREMENTAL STABILITY OF PIECEWISE-AFFINE LUR’E SYSTEMS In this section we propose conditions to assess incremental asymptotic stability of PWA Lur’e systems given by (5). The results are based on the construction of a piecewisequadratic incremental Lyapunov function via the application of Theorem 5. When studying incremental properties, it is standard to consider a fictitious augmented system composed of two copies of the original system (see e.g. Angeli (2002); Fromion (1997)). Incremental stability can then be studied via the analysis of the difference between the trajectories x, q˜, p˜ε ) denote the of both systems. Let (x, q, pε ) and (˜ state, the input and the output of each copy of the system.
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Considering the PWA structure of (5), we can define the augmented system given by ˙ x(t) = Aij x(t) + Bp(t) for x(t) ∈ Xij (7) q(t) = C ij x(t) + Dp(t) x(0) = x0 where x = col(x, x ˜, 1), p = col(p , p˜ ), q = q − q˜ and Ai 0 ai B 0 Aij = 0 Aj aj B= 0 B (8) 0 0 0 0 0 C ij = [Ci −Cj ci − cj ] D = [D −D] . The space X is defined as X = X × X × {1}, and regions Xij are defined as Xij = {x ∈ X | x ∈ Xi and x ˜ ∈ Xj }. Each region Xij is described by Xij = {x ∈ X | Gij x 0} where Gi 0 gi . (9) Gij = 0 Gj gj Analogously to the state partition {Xi }i∈I of system (5), the intersection between any two regions Xij and Xkl of (7) is either empty or contained in the hyperplane given by Xij ∩ Xkl ⊆ x ∈ X | E ijkl x = 0 . (10)
We shall propose conditions to compute an incremental Lyapunov function possessing the following piecewisequadratic structure: ˜) for x ∈ Xii (x − x ˜)T Pi (x − x (11) V (x, x ˜) = T x P ij x for x ∈ Xij , i = j
As presented in Waitman et al. (2016), the choice of a quadratic function on (x − x ˜) on regions Xii does not lead to any loss of generality. Indeed, it is a consequence of the fact that V (x, x) = 0, for every x ∈ X, due to (3).
Let In be the n × n identity matrix, and let I n ∈ R and J n ∈ R(2n+1)×(2n+1) be the matrices defined by In −In I 0 n . (12) In = Jn = −In In 0 0 We are then able to state the following theorem. Theorem 5. Let (5) be a PWA Lur’e system, and let be Lipschitz continuous with Lipschitz constant η > 0. If there exist symmetric matrices Pi ∈ Rn×n and P ij ∈ R(2n+1)×(2n+1) ; Uij , Rij , Wij ∈ Rpij ×pij with nonnegative coefficients and zero diagonal; Lijkl ∈ R(2n+1)×1 and positive scalars σ1 , σ2 , σ3 such that Pi − σ1 In 0 Pi − σ 2 I n 0 T Ai Pi + Pi Ai + CiT Ci + σ3 In Pi B + CiT D 0 B T Pi + D T C i DT D − η −2 Ip (13) for i ∈ I, T P ij − σ1 J n − Gij Uij Gij 0 T P ij − σ2 J n + Gij Rij Gij 0 T T Aij P ij + P ij Aij + C ij C ij + T P B + C D ij T ij σ3 J n + Gij Wij Gij 0 T T T −2 B P ij + D C ij D D − η Ip (14) 2n×2n
for (i, j) ∈ I × I, i = j, and
T
P ij = P kl + Lijkl E ijkl + E ijkl LTijkl
(15)
for (i, j), (k, l) such that Xij ∩ Xkl = ∅ are satisfied, then the PWA Lur’e system (5) is incrementally asymptotically stable. Proof. According to Theorem 2, (5) is incrementally asymptotically stable if there exists a continuous incremental Lyapunov function V , which is lower and upper bounded by class K∞ functions, and respects the integral constraint (4). We shall prove the theorem by showing that feasibility of (13)–(15) implies the existence of such a function possessing the structure (11). Continuity - We first show that V is a continuous function of x. This is clearly the case inside every cell, so we just need to show continuity on the boundaries. From (10), E ijkl x = 0 for all x ∈ Xij ∩ Xkl , then (15) implies that xT P ij x = xT P kl x for x ∈ Xij ∩ Xkl and hence that V is continuous. Norm bounds - The first inequality in (14), post and pre multiplied respectively by x and xT , implies that xT P ij x− 2
T
˜ ≥ xT Gij Uij Gij x. Since Uij is composed of σ1 x − x nonnegative coefficients, the right-hand side of the previous inequality is nonnegative whenever x ∈ Xij . This implies that xT P ij x ≥ σ1 x − x ˜
2
for x ∈ Xij .
(16)
The first inequality in (13) implies that V (x, x ˜) ≥ 2 ˜ for all x ∈ Xii . With (16), this guarantees σ1 x − x that 2 ˜ , ∀x, x ˜ ∈ X. (17) V (x, x ˜) ≥ σ1 x − x Proceeding exactly as before, the second inequalities in (13) and (14) imply that 2
˜ , ∀x, x ˜ ∈ X. (18) V (x, x ˜) ≤ σ2 x − x Inequalities (17) and (18) imply that the continuous piecewise quadratic function V given by (11) is such that 2
2
˜ ≤ V (x, x ˜) ≤ σ2 x − x ˜ . σ1 x − x
(19)
Integral constraint - We now show that the incremental Lyapunov function respects the integral constraint (4). Using the same arguments as before, the last inequality in (14), post and pre multiplied by col(x, p)T and col(x, p), implies that xT P ij (Aij x + Bp) + (Aij x + Bp)T P ij x + 2
˜ (C ij x + Dp)T (C ij x + Dp) − η −2 pT I p p ≤ −σ3 x − x (20) for all p ∈ R2 and all x ∈ Xij . Let ta and tb be two time instants such that the state trajectory of system (7) remains in Xij on the interval [ta , tb ]. By noticing that x˙ = Aij x + Bp, and integrating from ta to tb along trajectories of (7), we have
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x(tb )T P ij x(tb ) − x(ta )T P ij x(ta ) + tb 2 2
∆q(τ ) − η −2 ∆p (τ ) dτ tb ta 2 σ3 ∆x(τ ) dτ ≤− ta
(21)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Sérgio Waitman et al. / IFAC PapersOnLine 50-1 (2017) 1673–1679
with ∆x := x − x ˜, and ∆q and ∆p similarly defined. The same reasoning can be applied to the last inequality in (13), post and pre multiplying by col(x − x ˜, p − p˜ )T and col(x − x ˜, p − p˜ ), which yields ∆x(tb )T Pi ∆x(tb ) − ∆x(ta )T Pi ∆x(ta ) + tb 2 2 ∆q(τ ) − η −2 ∆p (τ ) dτ tb ta 2 σ3 ∆x(τ ) dτ. ≤−
(22)
ta
We note that the first terms in (21) and (22) represent the incremental Lyapunov function (11). Let us consider a trajectory x(τ ), ∀τ ∈ [0, t]. The time t1 can be decomposed n−1 as t = t − tin,n + k=0 (tout,k − tin,k ), with tout,k = tin,k+1 and tin,0 = 0, so that during each time interval [tin,k , tout,k ] the trajectory stays in a given region. Then, replacing ta by tin,k and tb by tout,k in (21) and (22), adding up to n for every region Xij crossed, and using the continuity of V yields ˜0 ) + V (x(t), x ˜(t)) − V (x0 , x t 2 2 ∆q(τ ) − η −2 ∆p (τ ) dτ 0 t 2 σ3 ∆x(τ ) dτ. ≤−
there is (ri , si ) ∈ R2 such that ϕPWA (q) = ri q + si , for q ∈ Ri , where i ∈ I = {1, . . . , N }. Since ϕ is continuous and is Lipschitz continuous, ϕPWA must be continuous. This implies that ∀q ∈ Ri ∩Rj , ri q +si = rj q +sj . We also fix ϕPWA (0) = 0, so that for any i such that q = 0 ∈ Ri , we have si = 0. We shall make the following assumption on the nonlinearity ϕ. Assumption 6. The memoryless nonlinearity ϕ is continuously differentiable, i.e. ϕ ∈ C 1 (R), and asymptotically affine, i.e. there exist k1 , k2 ∈ R such that limq→−∞ |ϕ (q) − k1 | = 0 and limq→∞ |ϕ (q) − k2 | = 0. Assumption 6 ensures that we are able to construct an approximation ϕPWA with a finite partition, i.e. with N < ∞. We are interested in finding ϕPWA that best approximates ϕ. We shall measure the approximation error by its Lipschitz constant, i.e., by its incremental gain. This may be formalized as minimize
η
subject to
(q) − (˜ q ) ≤ η q − q˜ q, q˜ ∈ R
ϕPWA ∈Φ(N )
(23)
0
Since is Lipschitz with a Lipschitz constant equal to η, 2 2 the quantity ∆q − η −2 ∆p is always positive, and we obtain t1 2 V (x(t), x ˜(t)) − V (x0 , x ˜0 ) ≤ − σ3 ∆q(τ ) dτ. (24) 0
Then V satisfies the conditions in Theorem 2 with αi (r) := 2 2 σi r , for i ∈ {1, 2} and ρ(r) := σ3 r . The function V is then an incremental Lyapunov function and system (5) is incrementally asymptotically stable, which concludes the proof. Testing the conditions in Theorem 5 is a feasibility problem involving LMIs, and thus can be efficiently solved. This result is of independent interest, as it extends the incremental circle criterion to the framework of PWA Lur’e systems. Indeed, by taking N = 1, we recover the LMI conditions of the classic incremental circle criterion (see e.g. Fromion et al. (1999)). In the proof of Theorem 5, we construct an incremental Lyapunov function that ensures incremental asymptotic stability. Another interpretation can be given in view of the framework of dissipative systems (Willems, 1972). Indeed, Theorem 5 can be seen as an incremental small gain theorem between the PWA system and the Lipschitz nonlinearity, where V would play the role of the storage 2 function, with supply rate w(q, p, x) = η −2 p − p˜ − 2 2 ˜ . q − q˜ − σ3 x − x 5. PIECEWISE-AFFINE APPROXIMATION OF SCALAR NONLINEARITIES Let us define Φ(N ) as the set of piecewise-affine functions ϕPWA : R → R defined on a partition of size N . That is, Φ(N ) is the set of piecewise-affine functions for which there exists a partition {Ri }i∈I of R, with |I| = N . Then,
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(P1)
As we refine the partition {Ri }i∈I , by choosing a larger N , the approximation error decreases, while the complexity of ϕPWA increases. This indicates a trade-off between the accuracy of the description and the complexity of the analysis. We shall search for a value of N ensuring a given upper bound ηref on the Lipschitz constant of the approximation error. This allows us to apply Theorem 5 to assess the incremental asymptotic stability of (1). The next proposition gives a method to obtain ϕPWA respecting the desired upper bound on the approximation. Proposition 7. Let ϕ be a function satisfying Assumption 6. Let ηref > 0, and let {Ri }i∈I , with I = {1, . . . , N }, be a partition of R obtained by a uniform division of the image of ϕ under R, i.e. l(ϕ (Ri )) = l(ϕ (Rj )), for all i, j ∈ I, where l(·) denotes the length of an interval. Also, let ri = (supq∈int(Ri ) ϕ (q) + inf q∈int(Ri ) ϕ (q))/2 and si be chosen to ensure continuity of ϕPWA . Then, by choosing N such that l(ϕ (Ri )) ≤ 2ηref , the obtained approximation ϕPWA ensures that is Lipschitz with a Lipschitz constant η ≤ ηref . Proof. We first use the fact that Lipschitz continuity is equivalent to boundedness of the derivative, for almost every q ∈ R. Then, we show that the proposed partition method ensures the desired upper bound on the Lipschitz constant. We begin by recalling a known fact about Lipschitz functions. Let η > 0. For an arbitrary partition {Ri }i∈I , the following two statements are equivalent: (i) (q) − (˜ q ) ≤ η q − q˜ , for all q, q˜ ∈ R. (ii) (q) ≤ η, for almost all q ∈ R.
We recall that (q) = ϕ (q) − ri , for all q ∈ int (Ri ). Let ηi > 0 be such that supq∈int(Ri ) (q) ≤ ηi . By choosing ri = (supq∈int(Ri ) ϕ (q) + inf q∈int(Ri ) ϕ (q))/2, we ensure that ηi = l(ϕ (Ri ))/2. Since ϕ is Lipschitz continuous, its derivative is bounded on R. Then, we can use the proposed partition so that the image of ϕ under R is
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uniformly divided, and we have ηi = l(ϕ (Ri ))/2 ≤ ηref , ∀i ∈ I. Then, by defining η = ηi and using the equivalent statements in the beginning of the proof, we have that (q) − (˜ q ) ≤ η q − q˜, for all q, q˜ ∈ R, with η ≤ ηref , which concludes the proof.
ϕ (q)
Proof. Due to the oddness of ϕ, we can focus on R+ and obtain the remaining by symmetry. Let {Ri }i∈I be an arbitrary partition of R+ , with I = {0, . . . , m}. Also, let ηi > 0 be as in Proposition 7. Then, by taking η := maxi∈I ηi , we have that (q) ≤ η, for almost all q ∈ R+ . It is clear that, for each region, the choice of ri that minimizes ηi is given by ri = (supq∈int(Ri ) ϕ (q) + inf q∈int(Ri ) ϕ (q))/2. In this case, we have ηi = (supq∈int(Ri ) ϕ (q) − inf q∈int(Ri ) ϕ (q))/2. As ϕ is Lipschitz, ϕ is bounded on R+ . Since the derivative ϕ is continuous and nondecreasing on R+ , we have m m supq∈int(Ri ) ϕ (q) − inf q∈int(Ri ) ϕ (q) ηi = 2 i=0 i=0
(ϕ (R+ )) . (25) 2 From this, we are interested minimizing η = maxi∈I ηi , in m subject to ηi ≥ 0 and i=0 ηi = (ϕ (R+ ))/2. The minimum is obtained when all ηi have the same value, which is obtained by taking a partition such that the image of ϕ under R+ is uniformly divided. This yields η = (ϕ (R+ ))/(2(m + 1)). Then, proceeding as in Proposition 7, we conclude that ϕPWA obtained by this method ensures that (q) − (˜ q ) ≤ η q − q˜, for all q, q˜ ∈ R, with η minimal. =
Despite the fact that Problem (P1) is non-convex due to the need to define the partition {Ri }i∈I , Proposition 9 shows that, in the case where ϕ satisfies Assumption 8, the optimal solution is known and quite easy to compute. The partitioning strategy is illustrated in Fig. 3. In this case, we may explicitly compute N such that the error bound is guaranteed to be inferior to ηref , as stated in the next proposition. Proposition 10. Let ϕ be a nonlinearity satisfying Assumptions 6 and 8. Let ηref > 0 be the desired upper bound on the Lipschitz constant of the approximation error. Let · denote the ceiling function. Then, if (ϕ (R+ )) − 1 > 0, (26) m= 2ηref
r4
ϕ min + 4·(2η) r3 ϕ min + 3·(2η)
The regions Ri = [qi , qi+1 ] can be defined by solving scalar nonlinear equations, which can be done by standard techniques such as the bisection method. We remark that, since ϕ is asymptotically linear, the leftmost and rightmost regions Ri may be unbounded.
One could wonder whether the partition method in Proposition 7 gives the optimal solution to (P1). It turns out that this is true, provided that ϕ satisfies some new assumptions, as stated in the following. Assumption 8. The memoryless nonlinearity ϕ is odd, monotone, and so that ϕ is nondecreasing on R+ . Proposition 9. Let ϕ be a nonlinear function respecting Assumptions 6 and 8. Then, the partition method described in Proposition 7 yields ϕPWA that is the optimal solution to (P1).
ϕ (q)
ϕ min + 5·(2η)
r2 ϕ min + 2·(2η) r1 ϕ min + 2η r0 ϕ min
q0
q
q1 q2 q3 q4 R0
R1 R2 R3
R4
···
Fig. 3. Partitioning strategy presented in Proposition 9, based on the uniform division of the image of ϕ under R. with N := 2m+1, and ϕPWA is obtained by the method in Proposition 9, then the approximation error is Lipschitz with a Lipschitz constant η ≤ ηref . Proof. This is a simple consequence of the fact that the partitioning strategy presented in Proposition 9 ensures that η = (ϕ (R+ ))/(2(m + 1)). With the techniques presented in this section and Section 4, we have all the tools to apply Algorithm 1 to the study of the incremental asymptotic stability of Lur’e systems (1). This shall be illustrated in the next section through some numerical examples. 6. NUMERICAL EXAMPLES Example 11. Consider the nonlinear system given by (1) with −1 0 1 A= B= C = [0 1] (27) 3 −2 0
and ϕ(q) = 2q 3 , for |q| ≤ 1, and ϕ(q) = 6q − 4 sign(q), for |q| > 1. ϕ satisfies Assumption 6, and belongs to the incremental sector [0, 6]. Analysis via the incremental circle criterion does not lead to a conclusion on the incremental stability of the system. We aim to obtain a piecewise-affine approximation ϕPWA over R, so that we can apply Theorem 5. Let us fix the desired maximal Lipschitz constant as ηref = 0.8. Using the approach proposed in Section 5, we obtain the approximation illustrated in Fig. 4, with N = 7 and η = 0.75. Using Proposition 4, the system is transformed in the interconnection of a PWA system and a Lipschitz nonlinearity. We then successfully apply Theorem 5 to construct a piecewise-affine incremental Lyapunov function, and conclude that this system is globally incrementally asymptotically stable. Example 12. Let us consider the nonlinear missile benchmark presented in Reichert (1992). The incremental behavior of the closed-loop system with a PI controller has been previously studied in Fromion et al. (1999). In this reference, the closed-loop system is written as an LTI system fedback through a nonlinearity ϕ(α) = −(an α3 + bn |α| α), with α being the angle of attack (see Fromion et al. (1999) for complete model and details). This model is assumed to be valid for |α| less than 20◦ (or 0.34 rad).
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7
5
ϕ(q) ϕPWA(q)
4 3
ϕ (q) ϕ PWA(q)
6
5
2 1
4
0 3
−1 −2
2
−3
1
−4 −5 −1.5
−1
−0.5
0
0.5
q
1
0
1.5
−1
−0.5
0
q
0.5
1
Fig. 4. Nonlinear function ϕ(q) = 2q 3 in Example 11 and the piecewise-affine approximation ϕPWA . The dotted lines represent the partition {Ri }i∈I . 3
ϕ(α) ϕPWA(α)
2
1
0
−1 −2 −3
−0.3
−0.2
−0.1
0
α
0.1
0.2
0.3
Fig. 5. Nonlinear function ϕ(α) = −(an α3 + bn |α| α) in Example 12 and the piecewise-affine approximation ϕPWA . Using again the techniques in the previous section with ηref = 3.5, we obtain the approximation ϕPWA presented in Fig. 5, with N = 5 and η = 2.4293. Application of Theorem 5 allows us to assess the incremental asymptotic stability of the closed-loop system, which concurs with the observations on Fromion et al. (1999) about the good behavior provided by the PI controller. 7. CONCLUSION In this paper we have proposed a new method to assess incremental asymptotic stability of Lur’e systems, based on piecewise-affine approximations. As a byproduct, we extended the celebrated incremental circle criterion to the analysis of PWA Lur’e systems, with conditions that can be solved very efficiently by interior point solvers. Perspectives for future work include the extension of the approach in Section 5 to the case of multivariable nonlinearities, and the establishment of local results, e.g. in the case when the nonlinearity ϕ is not asymptotically linear and a global approximation ϕPWA with a finite partition is not possible. Finally, the results in Section 5 may be coupled with robustness analysis to ensure robust incremental stability of Lur’e systems. REFERENCES Angeli, D. (2002). A Lyapunov approach to incremental stability properties. IEEE Transactions on Automatic Control, 47(3), 410–421. Azuma, S., Imura, J., and Sugie, T. (2010). Lebesgue piecewise affine approximation of nonlinear systems. Nonlinear Analysis: Hybrid Systems, 4(1), 92 – 102.
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