Stability of a certain class of hybrid dynamical systems

Abstract

In this work we introduce a type of hybrid equation closely related to those that appear when a digital controller or a hybrid controller of a certain class is interconnected with a full non-linear continuous-time system. For these equations we obtain results concerning existence, uniqueness and noncontinuability of solutions. This results are used to study the behaviour of the hybrid dynamical systems associated with those equations. We establish relationships between the stability of these hybrid dynamical systems and the ones corresponding to their associated discretized systems on one hand, and to their associated linearized systems on the other. We also characterize their stability, uniform asymptotic stability and exponential stability. These results are applied to the exponential stabilization of a non-linear control system by digital or hybrid controllers.

Stabilization by Means of Time-varying Hybrid Feedback

“…The main result of the present section states that non-uniform in time GAOC for the control system (18) implies the existence of a hybrid feedback such that the closed-loop system is a non-uniform in time RGAOS hybrid system for appropriate partitions. (18) is non-uniformly in time GAOC.…”

“…The proof is divided into two parts: in the first part we construct the feedback function and a function V : + × n → + , which is going to be used as a Lyapunov function. As remarked already in the introduction V : + × n → + has certain properties such that this function may be characterized as a CLF for (18) in the sense of [32]. In the second part we assume a partition π = {τ i } ∞ i=0 of + with finite diameter, inf {τ i+1 − τ i − δ(τ i ); i = 0, 1, .…”

“…, for all t ≥ t 0 ; (S3) for every ξ ∈ n and t 1 ≥ t 0 the solution φ(t, t 1 , ξ; u( · , t 0 , x 0 )) of (18) exists for all t ≥ t 1 and satisfies |φ(t, t 1 , ξ; u( · , t 0 , x 0 ))| ≤ γ (t) a (|x 0 | + |ξ |) for all t ≥ t 1 .…”

“…We denote by x(t) = φ(t, t 0 , x 0 ; u) with t ≥ t 0 ≥ 0 the solution of (18) which corresponds to some measurable and locally essentially bounded input u : + → m , initiated from x(t 0 ) = x 0 ∈ n . (18) is non-uniformly in time globally asymptotically output controllable (GAOC), if there exist functions µ, β, γ ∈ K + , σ ∈ K L and a ∈ K ∞ , such that for each (t 0 , x 0 ) ∈ + × n there exists a measurable and locally bounded input u( · , t 0 , x 0 ) : + → m with the following properties: (18) exists for all t ≥ t 0 and satisfies…”

“…In these works the concepts of sampled solutions or π-solutions of control systems as well as the fundamental concept of the Control Lyapunov Function (CLF, see [12,26,29,31]) are utilized. In [5,6,18] sufficient conditions for the local asymptotic stability of hybrid systems that arise from sampled-data feedback laws are provided. Recently, attempts for extending results to the time-varying case are given in [1,14,39,40] and the problem of "adding an integrator" for this type of feedback is examined in [38].…”