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The linear quadratic regulator for periodic hybrid systems. (English) Zbl 1440.93089

Summary: 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.

MSC:

93B52 Feedback control
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93D20 Asymptotic stability in control theory
49N10 Linear-quadratic optimal control problems
93C05 Linear systems in control theory
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