Adaptive optimal observer design via approximate dynamic programming

This paper presents an optimal observer design framework using a recently emerging method, approximate dynamic programming (ADP), to minimize a predefined cost function. We first exploit the duality between the linear optimal observer and the linear quadratic tracking (LQT) control. We show that the optimal observer design can be formulated as an optimal control problem subject to a specific cost function, and thus the solution can be obtained by solving an algebraic Riccati equation (ARE). For nonlinear systems, we further introduce an optimal observer design formulation and suggest a modified policy iteration method. Finally, to solve the problem online we propose a framework based on ADP and specifically on an approximator structure. Namely, a critic approximator is used to estimate the optimal value function, and a newly developed tuning law is proposed to find the parameters online. The stability and the performance are guaranteed with rigorous proofs. Numerical simulations are given to validate the theoretical studies.