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 of piecewise affine systems through discontinuous piecewise quadratic Lyapunov functions

State-dependent switched systems characterized by piecewise affine (PWA) dynamics in a polyhedral partition of the state space are considered. Sufficient conditions on the vectors fields such that the solution crosses the common boundaries of the polyhedra are expressed in terms of quadratic inequalities constrained to the polyhedra intersections. A piece-wise quadratic (PWQ) function, not necessarily continuous, is proposed as a candidate Lyapunov function (LF). The sign conditions and the negative jumps at the boundaries are expressed in terms of linear matrix inequalities (LMIs) via cone-copositivity. A sufficient condition for the asymptotic stability of the PWA system is then obtained by finding a PWQ-LF through the solution of a set LMIs. Numerical results with a conewise linear system and an opinion dynamics model show the effectiveness of the proposed approach

Asymptotic stability of piecewise affine systems with Filippov solutions via discontinuous piecewise Lyapunov functions

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 the first part of the paper 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

Piecewise semi-ellipsoidal control invariant sets

Computing control invariant sets is paramount in many applications. The families of sets commonly used for computations are ellipsoids and polyhedra. However, searching for a control invariant set over the family of ellipsoids is conservative for systems more complex than unconstrained linear time invariant systems. Moreover, even if the control invariant set may be approximated arbitrarily closely by polyhedra, the complexity of the polyhedra may grow rapidly in certain directions. An attractive generalization of these two families are piecewise semi-ellipsoids. We provide in this paper a convex programming approach for computing control invariant sets of this family.Comment: 7 pages, 3 figures, to be published in IEEE Control Systems Letter

Automated Stability Analysis of Piecewise Affine Dynamics Using Vertices

This paper presents an automated algorithm to analyze the stability of piecewise affine (PWA) dynamical systems due to their broad applications. We parametrize the Lyapunov function as a PWA function, with polytopic regions defined by the PWA dynamics. Using this parametrization, Stability conditions can be expressed as linear constraints restricted to polytopes so that the search for a Lyapunov function involves solving a linear program. However, a valid Lyapunov function might not be found given these polytopic regions. A natural response is to increase the size of the parametrization of the Lyapunov function by dividing regions and solving the new linear program. This paper proposes two new methods to divide each polytope into smaller ones. The first approach divides a polytope based on the sign of the derivative of the candidate Lyapunov function, while the second divides it based on the change in the vector field of the PWA dynamical system. In addition, we propose using Delaunay triangulation to achieve automated division of regions and preserve the continuity of the PWA Lyapunov function. Examples involving learned models and explicit MPC controllers demonstrate that the proposed method of dividing regions leads to valid Lyapunov functions with fewer regions than existing methods, reducing the computational time taken for stability analysisComment: 11 pages, 11 figure

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