Lyapunov stability for piecewise affine systems via cone-copositivity

Iervolino, Raffaele; Tangredi, Domenico; Vasca, Francesco

doi:10.1016/j.automatica.2017.03.011

Cited by 42 publications

(17 citation statements)

References 25 publications

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“…Copositivity is also used in [158,157] for relative degre zero LCS (essentially, with D a P-matrix), using Lyapunov functions of both the state and the multiplier λ in (2.22). Later copositivity has been used for switched positive systems in [288], and for conewise linear systems in [324], for LCS in [321] (the conditions stated in [324,323,321], basing on a characterization of copositive matrices taken from [239,Corollary 2.21] and [514,Theorem 2.1], being essentially the same as those in [280]).…”

Section: Sufficient Lyapunov Conditions Let Us Now Provide Brief Insmentioning

confidence: 99%

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

Brogliato¹,

Tanwani²

2020

SIAM Rev.

119

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This survey article addresses the class of continuous-time systems where a system modeled by ordinary differential equations (ODEs) is coupled with a static and time-varying setvalued operator in the feedback. Interconnections of this form model certain classes of nonsmooth systems including sweeping processes, differential inclusions with maximal monotone right-hand side, complementarity systems, differential and evolution variational inequalities, projected dynamical systems, some piecewise linear switching systems. Such mathematical models have seen applications in electrical circuits, mechanical systems, hysteresis effects, and many more. When we impose a passivity assumption on the open-loop system, and regard the set-valued operator in the feedback as maximally monotone, we obtain a set-valued Lur'e dynamical system. In this article we review the mathematical formalisms, their relationships, main application fields, well-posedness (existence, uniqueness, continuous dependence of solutions), and stability of equilibria. An exhaustive bibliography is provided.

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Section: Sufficient Lyapunov Conditions Let Us Now Provide Brief Insmentioning

confidence: 99%

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

Brogliato¹,

Tanwani²

2020

SIAM Rev.

119

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“…where v i = col(v i , 1) for all i and r j = col(r j , 0) for all j. A sufficient condition for the sign of a quadratic function on a polyhedron can be written in terms of LMIs, so as shown by the following lemmas whose proofs can be easily derived from [12]. Lemma 27: Consider ( 22), x ∈ X, ω = 0 if 0 ∈ X, R the ray matrix of the cone C X , the symmetric matrix P ∈ R (n+1)×(n+1) defined as…”

Section: A Positivity Test For Quadratic Functions On Polyhedral Setsmentioning

confidence: 99%

Asymptotic Stability of Piecewise Affine Systems With Filippov Solutions via Discontinuous Piecewise Lyapunov Functions

Iervolino

Trenn²,

Vasca

2021

IEEE Trans. Automat. Contr.

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Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. In this article, the feasible Filippov solution concept is introduced by characterizing single-mode Caratheodory, sliding mode, and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable, and Caratheodory boundaries. It is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. Several examples illustrate the theoretical arguments and the effectiveness of the stability result.

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“…, D i q } is the set of regulatory domains adjacent to D S . Because any H-representation of a polyhedron P can be converted to its V -representation [12], [7],and given the boundedness of P γ,D s , we get:…”

Section: Problem Formulationmentioning

confidence: 99%

“…for the extremal system Σ k . If, together with the continuity of V in D S , the following set of LMIs is satisfied [7], [6]:…”

Section: Common Lyapunov Function Approachmentioning

confidence: 99%

On piecewise quadratic Lyapunov functions for piecewise affine models of gene regulatory networks

Pasquini

Angeli

2018

2018 IEEE Conference on Decision and Control (CDC)

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In this work we addressed the problem of stability analysis for an uncertain piecewise affine model of a genetic regulatory network. In particular we considered polytopic parameter uncertainties on the proteins production rate functions, giving conditions for the existence of a piecewise quadratic Lyapunov function for any possible realization of the system. In the spirit of other works in literature, the resulting conditions will be given on the vertices of the parameter polytope, while still taking into consideration the piecewise nature of the Lyapunov function and the presence, in general, of sliding modes solutions. An example is shown to prove the validity and applicability of the theoretical results.

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Lyapunov stability for piecewise affine systems via cone-copositivity

Cited by 42 publications

References 25 publications

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

Asymptotic Stability of Piecewise Affine Systems With Filippov Solutions via Discontinuous Piecewise Lyapunov Functions

On piecewise quadratic Lyapunov functions for piecewise affine models of gene regulatory networks

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