On Complexity of Lyapunov Functions for Switched Linear Systems ?
نویسندگان
چکیده
We show that for any positive integer d, there are families of switched linear systems—in fixed dimension and defined by two matrices only—that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree ≤ d, or (ii) a polytopic Lyapunov function with ≤ d facets, or (iii) a piecewise quadratic Lyapunov function with ≤ d pieces. This implies that there cannot be an upper bound on the size of the linear and semidefinite programs that search for such stability certificates. Several constructive and non-constructive arguments are presented which connect our problem to known (and rather classical) results in the literature regarding the finiteness conjecture, undecidability, and non-algebraicity of the joint spectral radius. In particular, we show that existence of a sum of squares Lyapunov function implies the finiteness property of the optimal product.
منابع مشابه
Design of Observer-based H∞ Controller for Robust Stabilization of Networked Systems Using Switched Lyapunov Functions
In this paper, H∞ controller is synthesized for networked systems subject to random transmission delays with known upper bound and different occurrence probabilities in the both of feedback (sensor to controller) and forward (controller to actuator) channels. A remote observer is employed to improve the performance of the system by computing non-delayed estimates of the sates. The closed-loop s...
متن کاملOn common linear/quadratic Lyapunov functions for switched linear systems
Using duality and complementarity ideas, and Z-transformations, in this article, we discuss equivalent ways of describing the existence of common linear/quadratic Lyapunov functions for switched linear systems. In particular, we extend a recent result of Mason-Shorten on positive switched system with two constituent linear time-invariant systems to an arbitrary finite system.
متن کاملA new switching strategy for exponential stabilization of uncertain discrete-time switched linear systems in guaranteed cost control problem
Uncertain switched linear systems are known as an important class of control systems. Performance of these systems is affected by uncertainties and its stabilization is a main concern of recent studies. Existing work on stabilization of these systems only provides asymptotical stabilization via designing switching strategy and state-feedback controller. In this paper, a new switching strate...
متن کاملLocal stabilization for a class of nonlinear impulsive switched system with non-vanishing uncertainties under a norm-bounded control input
Stability and stabilization of impulsive switched system have been considered in recent decades, but there are some issues that are not yet fully addressed such as actuator saturation. This paper deals with expo-nential stabilization for a class of nonlinear impulsive switched systems with different types of non-vanishing uncertainties under the norm-bounded control input. Due to the constraine...
متن کاملCharacterization of Stabilizing Switching Sequences in Switched Linear Systems Using Piecewise Linear Lyapunov Functions
In this paper, the stability of switched linear systems is investigated using piecewise linear Lyapunov functions. Given a switched linear system, we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system. We assume that each individual subsystem is stable and admits a piecewise linear Lyapunov function. Based on these Lyapunov...
متن کاملA new approach based on state conversion to stability analysis and control design of switched nonlinear cascade systems
In this paper, the problems of control and stabilization of switched nonlinear cascade systems is investigated. The so called simultaneous domination limitation (SDL) is introduced in previous works to assure the existence of a common quadratic Lyapunov function (CQLF) for switched nonlinear cascade systems. According to this idea, if all subsystems of a switched system satisfy the SDL, a CQLF ...
متن کامل