Towards robust Lie-algebraic stability conditions for switched linear systems

Abstract

This paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching, which involve the commutators (Lie brackets) among the given matrices generating the switched system. The main novel feature of these stability criteria is that, unlike their earlier counterparts, they are robust with respect to small perturbations of the system parameters. Two distinct approaches are investigated. For discrete-time switched linear systems, we formulate a stability condition in terms of an explicit upper bound on the norms of the Lie brackets. For continuous-time switched linear systems, we develop two stability criteria which capture proximity of the associated matrix Lie algebra to a solvable or a "solvable plus compact" Lie algebra, respectively.

Global Uniform Asymptotic Stability of a Class of Switched Linear Systems with an Infinite Number of Subsystems

“…By Definition 1 and Theorem 7, any infinite product of this kind of switching linear system under these conditions converges to 0. By definition of the joint spectral radius mentioned in (5),…”

“…Some new sufficient conditions for exponential stability of switched linear systems under arbitrary switching were proposed in [5]. Hien et al considered the problem of exponential stability and stabilization of switched linear timedelay systems.…”

Topological Entropy Bounds for Switched Linear Systems with Lie Structure

“…Naturally things become more complicated for nonlinear systems, but results tying nonlinear systems to the Lie algebra structure can be found in [21] and [29]. Research into robustness conditions can be found in [2], and the works of [7,8] investigate state feedback which induces simultaneous triangularizability in the closed loop. Other references to switched systems can be found in [16] and [30].…”

On Lie-algebraic-solvability-based feedback stabilization of systems with controller-driven sampling

“…One stability result for switched systems states that if the closed-loop subsystem evolution matrices A cl i are stable (i.e., have spectral radius less than 1) and the Lie algebra generated by {A cl i : i ∈ n} is solvable, then the closed-loop DTSS (6) will be stable under arbitrary switching (Gurvits, 1995;Theys, 2005). For continuous-time switched systems, corresponding and more general stability results are given in Agrachev and Liberzon (2001). Solvability of the Lie algebra generated by {A cl i : i ∈ n} is equivalent to the existence of an invertible matrix T ∈ C n×n such that T −1 A cl i T is upper triangular for i ∈ n. Definition 2.…”

“…A recent paper dealing with such robustification is Agrachev et al (2010), where conditions formulated directly in terms of the Lie brackets are given which guarantee stability under arbitrary switching and are robust to small perturbations of the system parameters. In relation to stability results involving the Lie-algebraic solvability property, Agrachev et al point out that "the indicated lack of robustness is a shortcoming of the existing stability tests and is not an attribute of the [stabilization] problem itself ".…”