Optimal linear estimation for multiplicative noise systems with time delay and its duality

This paper investigates the duality between the estimation and control problems for time-delay systems. We first consider the estimation problem for linear discrete-time systems in the presence of multiplicative noise and time delays, where the delays appear in both state and measurement equations. Based on the innovation analysis approach, the linear minimum mean square error estimators are developed in terms of a forward partial difference Riccati equation and forward Lyapunov equations. The Riccati equation is of the same dimension as the plant, therefore compared with the conventional augmentation approach, the presented approach greatly lessens the computational demand when the delay is large. Then the LQR problem for deterministic time-delay systems is discussed based on the dynamic programming technique, and the controller is given in terms of a backward partial difference Riccati equation and backward Lyapunov equations. Finally, after comparing the estimation and control results, we establish a duality between the estimation problem for time-delay systems with multiplicative noise and the LQR problem for deterministic time-delay systems with constraint conditions.