Dynamical State Feedback Control for Linear Input Delay Systems, Part I: Dissipative Stabilization via Semidefinite Programming

Abstract: We propose an SDP-based framework to address the stabilization of input delay systems while taking into account dissipative constraints. A key to our approach is the introduction of the concept of parameterized linear dynamical state feedbacks (LDSFs), which draws inspiration from recent advancements in the analyses of distributed delays. Importantly, the parameterized LDSFs generalize conventional predictor controllers, where the interpretation of state prediction is concealed and their degree of parameterization can be increased by adjusting the integral kernels. A sufficient condition for the existence of dissipative LDSFs is formulated as matrix inequalities by constructing a complete type Krasovski\u{\i} functional. To solve the bilinear matrix inequality in the synthesis condition, we employ an off-line inner convex approximation algorithm that can be initialized using the gains of predictor controllers obtained via explicit construction. Finally, the unknowns of our dissipative LTDS can be directly obtained by solving convex semidefinite programs. Numerical examples and simulations were experimented to demonstrate the validity and effectiveness of our methodology.