LQR and Stabilization for Discrete-Time Systems With Multiplicative Noises and Input Delays

This paper is concerned with linear quadratic regulation (LQR) and stabilization problems for discrete-time stochastic systems with multiple input channels and input delays. In comparison with single-input-channel problems, a key difficulty, induced by the different delays in different input channels, is that the information sets for multiple controllers are required to be asymmetric and coupled. A novel technique, referred to as orthogonal decomposition-reorganization, is proposed to conquer the coupling between channels. After the technique is used, a new method based on some refined decompositions is provided to solve the delayed forward-backward stochastic difference equations(D-FBSDEs), which overcomes the asymmetry and amounts to finding the solvability condition of the underlying LQR problems. The complete solutions of the finite- and infinite-horizon LQR problems are given by newly developing Riccati-type equations. Also, a necessary and sufficient stabilizing condition is obtained for the system restricted by a performance index. Finally, the effectiveness of the results is evaluated by a numerical example.