A note on stability conditions for planar switched systems

Abstract

This paper is concerned with the stability problem for the planar linear switched systemẋ(t) = u(t)A1x(t)+(1−u(t))A2x(t), where the real matrices A1, A2 ∈ R 2×2 are Hurwitz and u(·) : [0, ∞[→ {0, 1} is a measurable function. We give coordinate-invariant necessary and sufficient conditions on A1 and A2 under which the system is asymptotically stable for arbitrary switching functions u(·). The new conditions unify those given in previous papers and are simpler to be verified since we are reduced to study 4 cases instead of 20. Most of the cases are analyzed in terms of the function Γ(A1, A2) = 1 2 (tr(A1)tr(A2) − tr(A1A2)).

On the marginal instability of linear switched systems

“…Indeed, the balls with respect to a Barabanov norm are invariant for (1). On the other hand, for every initial condition x 0 , there exists a trajectory of (1) lying on the sphere v −1 (v(x 0 )), hence not converging to 0.…”

“…In the following, a switched system of the form (1) will be often identified with the corresponding set of matrices A. This paper is concerned with stability issues for (1), where the stability properties are assumed to be uniform with respect to the switching law A(·).…”

“…In particular, for discrete-time linear switched systems, several results have been obtained in the case in which the spectral radius is equal to one under particular assumptions (see for instance [11] and references therein). This case corresponds to the situation ρ(A) = 0 for continuous-time systems of the form (1).…”

“…The Lyapunov exponent is a "measure" of the asymptotic stability of (1). Indeed the system is (uniformly) exponentially stable if and only if ρ(A) < 0.…”

Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems

“…This variational approach for analyzing GUAS was pioneered by E. S. Pyatntisky in the context of the celebrated absolute stability problem (Pyatnitskii (1970(Pyatnitskii ( , 1971), and proved to be quite useful in the analysis of continuous-time switched systems; see the survey papers (Barabanov (2005); Margaliot (2006)) as well as the more recent papers (Sharon and Margaliot (2007); Margaliot and Branicky (2009)), and the related work in (Boscain (2002); Balde and Boscain (2008); Balde et al (2009)). Recently, the variational approach was also used to analyze the stability of discrete-time linear switched systems (Monovich and Margaliot (2011a,b)).…”