Systems & Control Letters 61 (2012) 347–353 Contents lists available at SciVerse ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle On robust Lie-algebraic stability conditions for switched linear systems Andrei A. Agrachev a , Yuliy Baryshnikov b , Daniel Liberzon b,∗ a International School for Advanced Studies, S.I.S.S.A., via Beirut 4, 34014 Trieste, Italy b Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61821, USA article info Article history: Received 6 June 2011 Received in revised form 23 November 2011 Accepted 28 November 2011 Available online 5 January 2012 Keywords: Switched linear system Asymptotic stability Commutator Lie algebra Robustness 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. © 2011 Elsevier B.V. All rights reserved. 1. Introduction A switched system is described by a family of systems and a rule that orchestrates the switching between them (see [1] for an overview). In the large body of literature devoted to stability analysis of switched systems, a specific research direction that has received a lot of attention is to develop stability criteria that take into account commutation relations among the constituent systems. In the simplest case when these systems pairwise commute, stability is preserved under arbitrary switching; this can be shown either by directly studying the solutions (which is straightforward for linear systems but takes more effort for nonlinear systems [2]) or by constructing a common Lyapunov function (which was done for linear systems in [3], for nonlinear exponentially stable systems in [4], and for general nonlinear asymptotically stable systems in [5]). To build on this observation, one can consider the Lie algebra generated by the constituent systems (a matrix Lie algebra in the linear case or a Lie algebra of vector fields in general) and ask whether the structure of this Lie algebra can be used to verify stability of the switched system. Provided that the constituent systems are linear and stable, it was shown that the switched system remains stable under arbitrary switching if the Lie algebra is nilpotent [6], solvable [7,8], or has a compact semisimple part [9,10]; each of these classes of Lie algebras strictly contains the ∗ Corresponding author. E-mail addresses: agrachev@sissa.it (A.A. Agrachev), ymb@illinois.edu (Y. Baryshnikov), liberzon@illinois.edu (D. Liberzon). previous one, and the existence of a quadratic common Lyapunov function is guaranteed for all of them. Moreover, it was shown in [10] that no further generalization is possible based solely on the properties of the Lie algebra. For nonlinear systems the story is much less complete, but recently some results connecting Lie brackets and stability of switched nonlinear systems (beyond the already mentioned commuting case) were established in [11,12]. For other methods dealing with stability of switched systems, see, e.g., the references in [1,13]. While mathematically quite elegant, the available stability conditions based on commutation relations suffer from one serious drawback: they are not robust with respect to small perturbations of the system data. For example, if we take two matrices that commute with each other and perturb one of them slightly, they will cease to commute. If we take a family of matrices generating a nilpotent or solvable matrix Lie algebra and introduce arbitrarily small errors in their entries, the new Lie algebra will no longer possess any helpful structure (see [10, Section A.6] for a precise result along these lines). For this reason, the results mentioned in the previous paragraph have very limited applicability and serve primarily academic interests. It is important to note that stability itself, as well as the existence of a (quadratic) common Lyapunov function, are properties that do have inherent robustness to small perturbations; see [1, Section 2.2.4] for a detailed discussion of this issue. Thus the indicated lack of robustness is a shortcoming of the existing stability tests and is not an attribute of the problem itself. To get a handle on robustness and obtain more satisfactory re- sults, we must characterize ‘‘closeness’’ of a given collection of systems to one with ‘‘nice’’ commutation relations. Rather than 0167-6911/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2011.11.016