Stability of planar switched systems: The nondiagonalizable case

Abstract

Consider the planar linear switched systemẋ(t) = u(t)Ax(t) + (1 − u(t))Bx(t), where A and B are two 2×2 real matrices, x ∈ R 2 , and u(.) : [0, ∞[→ {0, 1} is a measurable function. In this paper we consider the problem of finding a (coordinate-invariant) necessary and sufficient condition on A and B under which the system is asymptotically stable for arbitrary switching functions u(.).This problem was solved in previous works under the assumption that both A and B are diagonalizable. In this paper we conclude this study, by providing a necessary and sufficient condition for asymptotic stability in the case in which A and/or B are not diagonalizable.To this purpose we build suitable normal forms for A and B containing coordinate invariant parameters. A necessary and sufficient condition is then found without looking for a common Lyapunov function but using "worst-trajectory" type arguments.

Some simple but challenging Markov processes

“…The system is said to be unstable if there exists a starting point x 0 and a measurable function u : [0, ∞) → {0, 1} such that the solution of (10) goes to infinity. In [13,4,5], the authors provide necessary and sufficient conditions for the solution of (10) to be unbounded for two matrices A 0 and A 1 in M 2 (R). In the particular case (9), this result reads as follows.…”

On the Stabilization of Persistently Excited Linear Systems

“…It seems to us that one must rather perform a trajectory analysis, on a time interval of length at least equal to T , in order to achieve any information which is uniform with respect to α ∈ G(T, µ). This viewpoint is more similar to the geometric approach to switched systems behind the results in [4,5,6]. As a second consideration, notice that point (i) described above, which is systematically used in the paper, presents formal similarities with the technique of averaging but is rather different from it, since no periodicity nor constant-average assumption is made here.…”

Periodic asymptotic controllability of switched systems

“…For d = N = 2, a complete classification of the families of matrices which are asymptotically stable under arbitrary switching has been given in the papers [11,9,10,18].…”