The linear quadratic regulator for periodic hybrid systems

The main objective of this paper is to characterize feedback control laws that are optimal with respect to a quadratic cost functional in the framework of linear hybrid systems undergoing time-driven periodic jumps, namely the so-called hybrid Linear-Quadratic Regulator (LQR) problem. The optimal solution to the hybrid LQR problem is determined both in the case of finite-horizon and infinite-horizon optimal control problems by introducing a hybrid (periodic) extension of the classic Differential and Difference Riccati Equations, thus leading to the notion of Monodromy Riccati Equation. Interestingly, due to the periodic nature of the discrete-time events, the computation of the optimal feedback hinges upon the solution of a differential, rather than algebraic, Riccati equation also in the infinite-horizon case, hence yielding a time-varying, periodic control law. Necessary and sufficient conditions that ensure asymptotic stability of the closed-loop system are provided and discussed in detail in the case of infinite-horizon optimal control problems.