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

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.