On the Dynamic Instability of a Class of Switching System

A sufficient condition for the existence of a destabilising switching sequence for the system x˙=A(t)x, A(t)∈{A1,A2,…,AM}, Ai∈IRN×N, where the Ai are Hurwitz matrices, is that there exists non-negative real constants α1,α2, ...,αм, αi≥0, ∑i=1Mαi>0, such that the matrix pencil ∑i=1MαiAi has at east one eigenvalue with a positive real part. An informal proof of this result based upon Floquet theory was presented in (Shorten, 1996; Shorten and Narendra, 1997). In this paper we present a rigourous basis for the proof of this result. Further, we use this result to identify several classes of linear switching systems, which admit the existence of a destabilising switching sequence. These systems provide insights into the relationship between the existence of a common quadratic Lyapunov function and the existence of a destabilising switching sequence for low order systems, as well as the robustness of a class of switching system that is known to be exponentially stable.