On infinity norms as Lyapunov functions for piecewise affine systems

Abstract

This paper considers off-line synthesis of stabilizing static feedback control laws for discrete-time piecewise affine (PWA) systems. Two of the problems of interest within this framework are: (i) incorporation of the S -procedure in synthesis of a stabilizing state feedback control law and (ii) synthesis of a stabilizing output feedback control law. Tackling these problems via (piecewise) quadratic Lyapunov function candidates yields a bilinear matrix inequality at best. A new solution to these problems is proposed in this work, which uses infinity norms as Lyapunov function candidates and, under certain conditions, requires solving a single linear program. This solution also facilitates the computation of piecewise polyhedral positively invariant (or contractive) sets for discrete-time PWA systems.

Finite bisimulations for switched linear systems

“…Under certain conditions, a Lyapunov function with piecewise polytopic sublevel sets for system (14) exists [28]. System (14) with a piecewise polyhedral Lyapunov function satisfies the properties given above.…”

Finite Bisimulations for Switched Linear Systems

“…In addition, using a Lyapunov function with a minimal contraction rate can decrease the computation time. Computation of such Lyapunov functions are out of the scope of this paper, but for example, minimization of the contraction rate is possible via the algorithm presented in [28].…”

“…The system is adapted from [28], where stabilizing static feedback control laws for discrete-time piecewise affine (PWA) systems are synthesized. The synthesis framework involves computation of piecewise linear Lyapunov functions that admit piecewise polytopic sublevel sets.…”

“…We define the operating regions, {X σ } σ∈Σ , and the working set, X = ∪ σ∈Σ X σ , with respect to the conic partition of R 2 , {Ω σ } σ∈Σ , used in [28] and the piecewise linear Lyapunov function of system:…”

“…The matrices, {L σ } σ∈Σ , of the Lyapunov function (25) are omitted due to space reasons, but can be found in [28]. We set Γ X = 19.75 (P Γ X = {x ∈ R n | V (x) ≤ Γ X } as before) Γ D = 10, and X σ = X ∩ Ω σ for all σ ∈ Σ.…”