Lyapunov stability for piecewise affine systems via cone-copositivity

Cone-copositive piecewise quadratic Lyapunov functions (PWQ-LFs) for the stability analysis of continuous-time piecewise affine (PWA) systems are proposed. The state space is assumed to be partitioned into a finite number of convex, possibly unbounded, polyhedra. Preliminary conditions on PWQ functions for their sign in the polyhedra and continuity over the common boundaries are provided. The sign of each quadratic function is studied by means of cone-constrained matrix inequalities which are translated into linear matrix inequalities (LMIs) via cone-copositivity. The continuity is guaranteed by adding equality constraints over the polyhedra intersections. An asymptotic stability result for PWA systems is then obtained by finding a continuous PWQ-LF through the solution of a set of constrained LMIs. The effectiveness of the proposed approach is shown by analyzing an opinion dynamics model and two saturating control systems

Stability analysis and stabilization of discrete-time piecewise affine systems

This work addresses the problems of global stabilization and local stability analysis of discrete-time piecewise affine (PWA) systems. To tackle the global stabilization problem, this work considers a PWA state feedback control law, a recently proposed implicit PWA representation and piecewise quadratic (PWQ) Lyapunov candidate functions. Through Finsler’s Lemma, congruence transformations and some structural assumptions, quasi-LMI sufficient conditions to ensure the global exponential stability of the origin of the closed-loop PWA system are derived from the stability conditions. An algorithm is proposed to solve the quasi-LMI conditions and compute the stabilizing gains. Regarding the problem of local stability analysis, this work proposes a method to test the local nonnegativity of PWQ functions using the implicit representation. This method is used to assess the local stability of the origin of PWA systems by considering PWQ Lyapunov candidate functions. Estimates of the Region of Attraction of the Origin (RAO) are obtained as level sets of the Lyapunov function. Approaches to obtain maximized estimates of the RAO are therefore discussed.Este trabalho trata dos problemas de estabilização global e análise de estabilidade local de sistemas afim por partes (PWA, do inglês, Piecewise Affine) de tempo discreto. Para tratar o problema de estabilização global, considera-se uma lei de controle do tipo realimentação de estados afim por partes, uma representação implícita de sistemas PWA e funções de Lyapunov quadraticas por partes (PWQ, do inglês, Piecewise Quadratic). Através do Lema de Finsler, transformações de congruência e algumas suposições de estrutura, condições suficientes na forma de quasi-LMIs para assegurar a estabilidade exponencial global da origem do sistema PWA em malha fechada são derivadas das condições de estabilidade. Um algoritmo para resolver as condições quasi-LMIs e computar os ganhos estabilizantes é proposto. Quanto ao problema de análise local de estabilidade, um método para testar a não negatividade local de funções PWQ usando a representação implícita é proposto. Este método é então utilizado para verificar a estabilidade local da origem de sistemas PWA através de funções de Lyapunov PWQ. Estimativas da região de atração da origem (RAO, do inglês, Region of Attraction of the Origin) são obtidas como curvas de nível da função de Lyapunov. Abordagens para maximizar a estimativa da RAO são então discutida

Stability analysis of conewise linear systems with sliding modes

In this paper cone-copositive piecewise quadratic Lyapunov functions (PWQ-LFs) for the stability analysis of conewise linear systems with the possible presence of sliding modes are proposed. The existence of a PWQ-LF is formulated as the feasibility of a cone-copositive programming problem which is represented in terms of linear matrix inequalities with equality constraints. An algorithm for the construction of a PWQ-LF is provided. Examples show the effectiveness of the approach in the presence of stable and unstable sliding modes