The study of discrete singularly perturbed linear-quadratic control systems

This chapter studies the reduced-order parallel algorithms for solving the algebraic Lyapunov and Riccati equations of discrete-time singularly perturbed systems. The algebraic Riccati equation is solved efficiently by using a bilinear transformation. The obtained results are extended to the discrete-time linear regulator problem. An F-8 aircraft model is used to demonstrate the presented technique. The chapter discusses studies the discrete-time Kalman filtering and the corresponding linear optimal stochastic control problem. It examines that the algebraic regulator and filter Riccati equations of singularly perturbed discrete-time control systems are completely and exactly decomposed into reduced-order continuous-time algebraic Riccati equations corresponding to the slow and fast time scales. The exact solution of the global discrete algebraic Riccati equation is obtained in terms of the reduced-order pure-slow and pure-fast non-symmetric continuous-time algebraic Riccati equations. The optimal global Kalman filter is decomposed into pure-slow and pure-fast local optimal filters both driven by the system measurements and the system optimal control input. It is shown that these two filters are implemented independently in the different time scales. As a result, the optimal linear-quadratic Gaussian control problem for singularly perturbed linear discrete systems takes the complete decomposition and parallelism between pure-slow and pure-fast filters and controllers.