Existence of harmonic Riccati equations and their solutions in continuous-time periodic systems

Abstract

By means of the harmonic Lyapunov equation and the contraction mapping theorem, the existence of Toeplitz-operator-valued Riccati equations in finite-dimensional linear continuous-time periodic (FDLCP) systems is verified for the first time in this paper. Properties of solutions of Toeplitz-operator-valued Riccati equations are also derived. Through this novel method different from the frequently adopted Hamiltonian matrix analysis approach, it is revealed that periodic solutions of periodic matrix Riccati equations exist under appropriate assumptions on FDLCP systems matrices and cost function matrices, and analytic properties of the periodic solutions are also investigated. Based on the contraction mapping theorem, an iterative algorithm is suggested for asymptotic computation of the periodic solution of periodic matrix Riccati equations. The iterative algorithm only involves solutions of algebraic Lyapunov equations and Fourier coefficients calculations of periodically time-varying matrices.