SOS Methods for Multi-Delay Systems: A Dual Form of Lyapanov-Krasovskii Functional

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We present a dual form of Lyapunov-Krasovskii functional which allows the problem of controller synthesis of multi-delay systems to be formulated and solved in a convex manner. First, we give a general form of 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 SOS methodology to express positivity and negativity of these forms as LMIs, describing a new set of polynomial manipulation tools designed for this purpose. Finally, we apply the resulting LMIs to a battery of numerical examples and demonstrate that the stability conditions are not conservative. The results of this paper are significant in that they open the way for dynamic output H-infinity optimal control of systems with multiple time-delays.