A Convex Approach to Analysis, State and Output Feedback Control Parabolic PDEs Using Sum-of-Squares

In this paper we use optimization-based methods to analyze the stability and design state and output-feedback controllers for a class of one-dimensional parabolic partial differential equations. The output may be the complete state measurement or the boundary measurement of the state. The input considered is Neumann boundary actuation. We use Lyapunov operators, duality, and the Luenberger observer framework to reformulate the synthesis problem as a convex optimization problem expressed as a set of Linear-Operator-Inequalities (LOIs). We then show how feasibility of these LOIs may be tested using Semidefinite Programming (SDP) and the Sum-of-Squares methodology. Moreover, we provide numerical results which prove that the method can be generalized for application to systems with other types of boundary conditions.

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