Conditions for the existence of a common quadratic Lyapunov function via stability analysis of matrix families

This paper presents a necessary and sufficient condition for the existence of a common quadratic Lyapunov function (CQLF) for two stable second order linear time invariant systems via the route of stability analysis of a convex combination of matrices. In a recent paper, it was noted that a necessary and sufficient condition for the existence of a CQLF for two stable second order linear time invariant systems with plant matrices A₁ and A₂ is that the convex combinations of A₁ and A₂, as well as that of A₁ and A₂^-1 be both Hurwitz stable. However, in a recent paper of Yedavalli (2002), a necessary and sufficient ´extreme point´ solution was presented to test the Hurwitz stability of a convex combination of l arbitrary Hurwitz stable matrices. Thus, extending those results to the CQLF problem results in simple set of necessary and sufficient conditions for the CQLF directly in terms of the two plant matrices under consideration. This paper thus establishes an interesting ´connection´ between the CQLF problem in linear switched systems and stability analysis of matrix families.