A solution to the common Lyapunov function problem for continuous-time systems

Stability of switched systems: a Lie-algebraic condition

“…Regarding perturbations of upper-triangular matrices, one can obtain explicit bounds that have to be satisfied by the elements below the diagonal so that the quadratic common Lyapunov function for the unperturbed systems remains a common Lyapunov function for the perturbed ones [8]. Unfortunately, the condition of Theorem 2 is not robust.…”

“…The existence of a quadratic common Lyapunov function for a family of linear systems whose matrices can be simultaneously put into the upper-triangular form has been pointed out before (see, e.g., [8,15] and related earlier work in [1]). It is important to recognize, however, that while it is a nontrivial matter to find a basis in which all matrices take the triangular form or even decide whether such a basis exists, the Lie-algebraic condition given by Theorem 2 is formulated in terms of the original data and can always be checked in a finite number of steps if P is a finite set.…”

“…The problem of finding conditions that guarantee asymptotic stability of (1) for an arbitrary switching signal σ has recently attracted a considerable amount of attention-see the work reported in [2,3,8,9,10,11,14,15] and the references therein. Some of the aforementioned results suggest that certain properties of the Lie algebra {A p : p ∈ P} LA generated by the matrices A p may be of relevance to the question of stability of (1).…”

“…The authors wish to thank Leonid Gurvits and George Seligman for useful discussions that have contributed to this work, and Kumpati Narendra for calling their attention to the reference [8].…”

A new methodology for the stability analysis of pairwise triangularizable and related switching systems

“…gular (Mori, Mori & Kuroe 1996, Mori, Mori & Kuroe 1997, Liberzon, Hespanha & Morse 1998, Shorten & Narendra 1998. The existence of such a function, referred to as a c ommon quadratic Lyapunov function (CQLF), is su cient to guarantee the exponential stability of the switching system _ x = A(t)x A(t) 2 A .…”