Linear quadratic regulation problem for discrete-time systems with multi-channel multiplicative noise

This paper investigates the linear quadratic regulation (LQR) problem for discrete-time systems with multiplicative noise. Multiplicative noise is usually assumed to be a scalar in existing literature works. Motivated by recent applications of networked control systems and MIMO communication technology, we consider multi-channel multiplicative noise represented by a diagonal matrix. We first show that the finite horizon LQR problem can be solved using a generalized Riccati equation. We then prove the convergence of the generalized Riccati equation under the conditions of stabilization and exact observability, and obtain the solution to the infinite horizon LQR problem. Finally, we provide a numerical example to demonstrate the proposed approach.