Synthesis of Nonsmooth Systems

Abstract

In this chapter, the state-space approach is extended to the nonsmooth setting. Both the full information case with perfect state measurements and the incomplete information case with output disturbance-corrupted measurements are studied side by side. Sufficient conditions for the existence of a global solution of the problem are carried out in terms of an appropriate solvability of two Hamilton–Jacobi–Isaacs partial differential inequalities, which arise in the state-feedback and output-injection design, respectively, and which may not admit continuously differentiable solutions. The present $\mathcal{L}_{2}$-gain analysis follows the line of reasoning where the corresponding Hamilton–Jacobi–Isaacs expressions are viewed in the sense of Clarke proximal superdifferentials and are required to be negative definite. The resulting controller is associated with specific proximal solutions of the Hamilton–Jacobi–Isaacs partial differential inequalities. Developed in the general time-varying setting, the nonsmooth synthesis is then specified for periodic and autonomous systems with focus on the periodic and time-invariant controller designs, respectively. Local output-feedback synthesis is additionally presented over sampled-data measurements. A linear matrix inequality-based extension of the state-space approach to a class of nonsmooth distributed parameter systems finalizes the present chapter.