Conditions for uniform solvability of parameter-dependent Lyapunov equations with applications

We consider the problem of finding a common quadratic Lyapunov function to demonstrate stability of a family of matrices which incorporate design freedoms. Generically, this can be viewed as picking a family of controller (or observer) gains so that the family of closed-loop system matrices admits a common Lyapunov function. We provide several conditions, necessary and sufficient, for various structures of matrix families. Families of matrices containing a subset of diagonal matrices invariant under the design freedoms is also considered since this is a case that occurs in many applications. Conditions for uniform solvability of the Lyapunov equations are explicitly given and involve inequalities regarding relative magnitudes of terms in the matrices. Motivating applications of the obtained results to observer and controller designs for time-varying, switched, and nonlinear systems are highlighted.