A Dual to Lyapunov's Second Method for Linear Systems With Multiple Delays and Implementation Using SOS

Abstract

We present a dual form of Lyapunov–Krasovskii functional which allows the problem of controller synthesis for multidelay systems to be formulated and solved in a convex manner. First, we give a generalized version of the dual stability condition formulated in terms of Lyapunov operators which are positive, self-adjoint, and preserve the structure of the state space. Second, we provide a class of such operators and express the stability conditions as positivity and negativity of quadratic Lyapunov–Krasovskii functional forms. Next, we adapt the Sum of Squares (SOS) methodology to express positivity and negativity of these forms as Linear Matrix Inequalities (LMIs), describing a new set of polynomial manipulation tools designed for this purpose. We apply the resulting LMIs to a battery of numerical examples and demonstrate that the stability conditions are not significantly conservative. Finally, we formulate a test for controller synthesis for systems with multiple delays, apply the test to a numerical example, and simulate the resulting closed-loop system.