Full-State Feedback of Delayed Systems using SOS: A New Theory of Duality

In this paper, we show that the controller synthesis of delayed systems can be formulated and solved in a convex manner through the use of a duality transformation, a structured class of operators, and the Sum-of-Squares (SOS) methodology. The contributions of this paper are as follows. We show that a dual stability condition can be 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 which can be parameterized using Sum-of-Squares. Next, we show how any operator in this class can be inverted using simple operations on the SOS variables which can be performed in Matlab. Next we use SOS and semidefinite programming to formulate a dual stability test for time-delay systems. Next, we use the dual stability results to formulate a convex test for stabilizability and show how SOS can be used to solve this test and recover the controller. Finally, we give a numerical example. The results of this paper are significant in that they open the way for dynamic output H∞ optimal control of infinite-dimensional systems by giving the first truly convex, numerically realizable full-state feedback controller synthesis criterion.