Bill, Adam

Guiver, Christopher

Logemann, Hartmut

Townley, Stuart

We identify a stability/instability trichotomy for a class of non-negative continuous-time Lur’e systems. Asymptotic as well as input-to-state stability concepts (ISS) are considered. The presented trichotomy rests on Perron-Frobenius theory, absolute stability theory and recent ISS results forLur’e systems

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        Citation for published version:Bill, A, Guiver, C, Logemann, H & Townley, S 2014, A stability/instability  trichotomy  for non-negative Lur’esystems. in Proceedings of the MTNS 2014.Publication date:2014Document VersionPublisher's PDF, also known as Version of recordLink to publicationPublisher RightsCC BYUniversity of BathGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 13. May. 2019A stability/instability trichotomy for non-negative Lur’e systems*Adam Bill 1, Chris Guiver 2, Hartmut Logemann 1 and Stuart Townley 2Abstract— We identify a stability/instability trichotomy fora class of non-negative continuous-time Lur’e systems. Asymp-totic as well as input-to-state stability concepts (ISS) are con-sidered. The presented trichotomy rests on Perron-Frobeniustheory, absolute stability theory and recent ISS results forLur’e systems.I. INTRODUCTIONLet A = (aij) ∈ Rn×n and b, c ∈ Rn and consider thecorresponding single-input single-output non-negative linearsystemx˙ = Ax+ bu, x(0) = ξ ∈ Rn+; y = cTx. (1)We assume that(A1) A is Metzler, b, c ∈ Rn+ and b, c 6= 0 holds.We recall that A = (aij) is Metzler if aij ≥ 0 for i 6= j(all off-diagonal elements are non-negative).System (1) is said to be non-negative if (A1) holds andu ≥ 0. Non-negative systems of the form (1) occur naturallyin biological, ecological and economic contexts.We impose the following assumptions.(A2) A is Hurwitz.(A3) There exist non-negative numbers α and κ such thatαI +A+ κbcT is primitive.Recall that (A3) means that the matrix (αI+A+κbcT )kis a positive matrix for some k ∈ N.In the following, let G denote the transfer function of (1),that is, G(s) := cT (sI −A)−1b.Lemma 1.1: Assume that (A1)-(A3) hold. Then G(0) >0 and ‖G‖H∞ = G(0).A proof of Lemma 1.1 can be found in [1].Applying nonlinear non-negative feedback u = f(y) to(1), where f : R+ → R+ is locally Lipschitz, leads to thefollowing non-negative Lur’e systemx˙ = Ax+ bf(cTx), x(0) = ξ ∈ Rn+. (2)We assume that the following assumption holds.(A4) f : R+ → R+ is locally Lipschitz and f(0) = 0.Whilst absolute stability of Lur’e systems is a classicaltopic in control theory (see, for example, [2], [3], [8]), itseems that non-negative Lur’e systems have not received*This work was supported by EPSRC (Grant EP/I019456/1)1 Department of Mathematical Sciences, University of Bath, Bath BA27AY, UK A.R.Bill@bath.ac.uk, H.Logemann@bath.ac.uk2 Environment and Sustainability Institute, University of Exeter, PenrynCampus, Penryn, Cornwall, TR10 9EZ, UKC.Guiver@exeter.ac.uk, S.B.Townley@ex.ac.ukmuch attention (see however [7] which provides an analysisof the stability properties of a class of non-negative discrete-time Lur’e systems).Assuming that (A1)-(A4) hold, we setp :=1G(0),and consider the following three cases.Case 1. f(z)/z ≤ p for all z > 0.Case 2. infz>0 f(z)/z > p.Case 3. There exists y∗ > 0 such that f(y∗) = py∗ and∣∣∣∣f(z)− f(y∗)z − y∗∣∣∣∣≤ p for all z > 0, z 6= y∗.The condition in Case 3 means that the graph of f is“sandwiched” between the straight lines l1 and l2 given byl1(z) = pz and l2(z) = 2py∗ − pz, see Figure 1.zf(z)l2l12py∗f0 y∗Fig. 1. Case 3: graph of f “sandwiched” between the lines l1 and l2.II. LYAPUNOV STABILITY RESULTSIn this section, we present results which describe thestability/instability properties in each of three cases, where“stability” is interpreted in the sense of Lyapunov.Let x(· ; ξ) denote the unique maximally defined forwardsolution of (2) with maximal interval of existence [0, ωξ),where 0 < ωξ ≤ ∞.The proposition below relates to Case 1. It follows fromwell known results in absolute stability theory, see, forexample, [3].Proposition 2.1: Assume that (A1)-(A4) hold.(a) If f(z)/z ≤ p for all z > 0, then the equilibrium 0 isstable in the large in the sense that there exists Γ ≥ 1 suchthat, for every ξ ∈ Rn+, ωξ =∞ and‖x(t; ξ)‖ ≤ Γ‖ξ‖ ∀ t ≥ 0.21st International Symposium onMathematical Theory of Networks and SystemsJuly 7-11, 2014. Groningen, The Netherlands ISBN: 978-90-367-6321-9 1752(b) If f(z)/z < p for all z > 0, then the equilibrium 0is globally asymptotically stable. In particular, for everyξ ∈ Rn+, ωξ =∞ and x(t; ξ)→ 0 as t→∞.(c) If supz>0 f(z)/z < p, then the equilibrium 0 is globallyexponentially stable, that is, there exist N ≥ 1 and ν > 0such that, for every ξ ∈ Rn+, ωξ =∞ and‖x(t; ξ)‖ ≤ Ne−νt‖ξ‖ ∀ t ≥ 0.In Case 2, the solutions of (2) diverge to ∞ for every non-zero initial condition. More precisely, we have the followingresult.Theorem 2.2: Assume that (A1)-(A4) hold andinfz>0 f(z)/z > p. Let ξ ∈ Rn+, ξ 6= 0, be suchthat the solution x(t; ξ) exists for all t ≥ 0. Thenlimt→ωξxi(t; ξ) =∞ ∀ i ∈ {1, . . . , n},where xi(· ; ξ) denotes the i-th component of x(· ; ξ).We proceed to consider Case 3.Theorem 2.3: Assume that (A1)-(A4) hold.(a) If there exists y∗ > 0 such that f(y∗) = py∗ and∣∣∣∣f(z)− f(y∗)z − y∗∣∣∣∣≤ p ∀ z ≥ 0, z 6= y∗then x∗ = −pA−1by∗ ∈ Rn+ is an equilibrium of (2) andx∗ is stable in the large in the sense that there there existsΓ ≥ 1 such that, for every ξ ∈ Rn+, ωξ =∞ and‖x(t; ξ)− x∗‖ ≤ Γ‖ξ − x∗‖ ∀ t ≥ 0.(b) If there exists y∗ > 0 such that f(y∗) = py∗ and∣∣∣∣f(z)− f(y∗)z − y∗∣∣∣∣< p ∀ z > 0, z 6= y∗then 0 and x∗ = −pA−1by∗ ∈ Rn+ are the only equilibria of(2) and x∗ is globally asymptotically stable in the sense thatx∗ is stable in the large (see statement (a) of this theorem)and, for every ξ ∈ Rn+ such that ξ 6= 0, ωξ = ∞ andx(t; ξ)→ x∗ as t→∞.(c) If there exists y∗ > 0 such that f(y∗) = py∗,∣∣∣∣f(z)− f(y∗)z − y∗∣∣∣∣< p ∀ z > 0, z 6= y∗andlim supy→y∗∣∣∣∣f(z)− f(y∗)y − y∗∣∣∣∣< p,and iflim infz→0f(z)z> p, (3)then 0 and x∗ = −pA−1by∗ ∈ Rn+ are the only equilibriaof (2) and x∗ is “semi-globally” exponentially stable in thesense that, for every compact set K ⊂ Rn+ with 0 /∈ K,there exists N ≥ 1 and ν > 0 such that, for every ξ ∈ K,ωξ =∞ and‖x(t; ξ)− x∗‖ ≤ Ne−νt‖ξ − x∗‖ ∀ t ≥ 0.(d) If (3) holds and there exists y∗ > 0 such that f(y∗) =py∗ and, for every ε > 0,supz≥ε, z 6=y∗∣∣∣∣f(z)− f(y∗)z − y∗∣∣∣∣< p,then 0 and x∗ = −pA−1by∗ ∈ Rn+ are the only equilibriaof (2) and x∗ is “quasi-globally” exponentially stable in thesense that, for every δ > 0 there exist N ≥ 1 and ν > 0such that, for every ξ ∈ Rn+ with ‖ξ‖ ≥ δ, ωξ =∞ and‖x(t; ξ)− x∗‖ ≤ Ne−νt‖ξ − x∗‖ ∀ t ≥ 0. (4)We remark that “global” exponential stability of x∗ (in thesense that there exist N ≥ 1 and ν > 0 such that (4) issatisfied for all ξ ∈ Rn+ with ξ 6= 0) does not hold. Thisis an immediate consequence of the following result whichfollows from continuity properties of the flow generated bythe Lur’e system (2).Proposition 2.4: Assume that (A1)-(A4) hold and thatthere exists y∗ > 0 such that f(y∗) = py∗. Then, for everysequence (tn) in R+ with tn →∞ as n→∞, there existsa sequence (ξn) in Rn+ with ξn 6= 0 and ξn → 0 as n→∞and such thatlimn→∞‖x(tn; ξn)− x∗‖‖ξn − x∗‖= 1,where x∗ = −pA−1by∗.Discrete-time results similar to statement (b) of Proposition2.1, Theorem 2.2 and statement (b) of Theorem 2.3 can befound in [7].Proofs of the results in Section II can be found in [1].III. INPUT-TO-STATE STABILITY RESULTSFinally, we investigate the stability behaviour of (2)subject to non-negative disturbances, that is, we analyzeinput-to-state stability (ISS) properties of the forced Lur’esystemx˙ = Ax+ b(f(cTx) + d), x(0) = ξ ∈ Rn+, (5)where d : R+ → R+ is locally essentially bounded. Theunique maximally defined forward solution of (5) is denotedby x(· ; ξ, d).For an overview of ISS theory, the reader is referred to[6]. We recall some terminology and notation relating tocomparison functions. Let K denote the set of all continuosfunctions ϕ : R+ → R+ such that ϕ(0) = 0 and ϕ isstrictly increasing. Moreover, define K∞ := {ϕ ∈ K :lims→∞ ϕ(s) =∞}. We denote by KL the set of functionsin two variables ψ : R+ × R+ → R+ with the followingproperties: ψ(· , t) ∈ K for all t ≥ 0, and ψ(s, ·) isnonincreasing with limt→∞ ψ(s, t) = 0 for all s ≥ 0.The following proposition is a consequence of recent ISSresults for Lur’e systems, see [4], [5].Proposition 3.1: Assume that (A1)-(A4) hold. If thereexists ρ ∈ K∞ such thatf(z) ≤ pz − ρ(z) ∀ z ≥ 0,MTNS 2014Groningen, The Netherlands1753then the equilibrium 0 of the unforced Lur’e system (2) isISS in the sense that there exist ψ ∈ KL and ϕ ∈ K suchthat for all ξ ∈ Rn+ and all non-negative d ∈ L∞loc(R+),x(· ; ξ, d) is defined on R+ and‖x(t; ξ, d)‖ ≤ ψ(‖ξ‖, t) + ϕ(‖d‖L∞(0,t)) ∀ t ≥ 0.The following theorem shows that, under suitable assump-tions, the equilibrium x∗ has stability properties which aresimilar to ISS.Theorem 3.2: Assume that (A1)-(A4) hold and that thereexists y∗ > 0 such that f(y∗) = py∗ and∣∣∣∣f(z)− f(y∗)z − y∗∣∣∣∣< p ∀ z > 0, z 6= y∗, (6)Furthermore, assume that (3) holds andpz − f(z)→∞ as z →∞. (7)Then 0 and x∗ = −pA−1by∗ ∈ Rn+ are the only equilibriaof the unforced Lur’e system (2) and x∗ is “quasi ISS” inthe sense that, for every δ > 0, there there exist ψ ∈ KLand ϕ ∈ K such that for all ξ ∈ Rn+ with ‖ξ‖ ≥ δ and allnon-negative d ∈ L∞loc(R+), x(· ; ξ, d) is defined on R+ and‖x(t; ξ, d)−x∗‖ ≤ ψ(‖ξ−x∗‖, t)+ϕ(‖d‖L∞(0,t)) ∀ t ≥ 0.(8)To relate the conditions (6) and (7) to those in Proposition3.1, we note that if (6) and (7) hold, then, for every ε > 0,there exists ρ ∈ K∞ such that|f(z)−f(y∗)| ≤ p|z−y∗|−ρ(|z−y∗|) ∀ z ≥ ε, z 6= y∗.The proof of Theorem 3.2 is based on Proposition 3.1 andthe following lemma.Lemma 3.3: Assume that (A1)-(A4) hold. If (3) is satis-fied and there exists y∗ > 0 such that f(y∗) = py∗ and (6)holds, then, for every δ > 0, there exist constants η > 0and τ ≥ 0 such that for all ξ ∈ Rn+ with ‖ξ‖ ≥ δ and allnon-negative d ∈ L∞loc(R+), x(· ; ξ, d) is defined on R+ andcTx(t; ξ, d) ≥ η ∀ t ≥ τ.This lemma also plays a key roll in the proof of statements(b)-(d) of Theorem 2.3 (with disturbance d = 0). Detailedproofs of Proposition 3.1, Theorem 3.2 and Lemma 3.3 canbe found in [1].Finally, it follows from Proposition 2.4 that “global” ISSof x∗ (in the sense that there exist ψ ∈ KL and ϕ ∈ Ksuch that (8) is satisfied for all ξ ∈ Rn+ with ξ 6= 0 and allnon-negative d ∈ L∞loc(R+)) does not hold.REFERENCES[1] A. Bill, C. Guiver, H. Logemann and S. Townley. Stability of non-negative Lur’e systems, in preparation, 2014.[2] W.M. Haddad and V. Chellaboina. Nonlinear Dynamical Systems andControl, Princeton University Press, Princeton, 2008.[3] B. Jayawardhana, H. Logemann and E.P. Ryan. The circle criterionand input-to-state stability: new perspectives on a classical result,IEEE Control Systems Magazine, 31, 32-67, (August 2011).[4] E.Sarkans and H. Logemann. Input-to-state stability for Lur’e sys-tems, in preparation, 2013.[5] E. Sarkans. Absolute stability and input-to-state stability of Lur’esystems, PhD thesis, in preparation, University of Bath, 2014.[6] E. D. Sontag. Input to state stability: basic concepts and results, inNonlinear and Optimal Control Theory, P. Nistri and G. Stefani, Eds.Berlin: Springer-Verlag, 2006, pp. 163-220.[7] S. Townley, R. Rebarber and B.Tenhumberg. Feedback control sys-tems analysis of density dependent population dynamics, Systems &Control Letters, 61, 309-315, 2012.[8] M. Vidyasagar. Nonlinear Systems Analysis, 2nd ed.,Prentice-Hall,Englewood Cliffs, NJ, 1993.MTNS 2014Groningen, The Netherlands1754

A stability/instability trichotomy for non-negative Lur’e systems

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A stability/instability trichotomy for non-negative Lur’e systems

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