Optimal Adaptive Parameter Estimation with Online Varying Learning Gain

It has been well known that the learning gain plays a crucial role in the adaptive parameter estimation (APE) for guaranteeing fast convergence and robustness. However, the tuning of learning gains in the existing methods is generally empirical and time consuming. To address this issue, this paper presents a novel APE approach to explore the optimality principle in the design of adaptive laws to obtain an online updated optimal learning gain, which is derived to minimize a cost function of estimation error with two weighting matrices. For this purpose, the estimation error is first reconstructed by using filter operations on the system and introducing several auxiliary variables. Then, two constructive APE algorithms driven by the extracted estimation error are developed. Finally, inspired by the duality of control and estimation, the estimation error dynamics are reformulated as a closed-loop system with an analogous control action to minimize the cost function. By using the Hamiltonian function, a Differential Riccati Equation (DRE) is derived to find an optimal solution of learning gain to construct an optimal adaptive law. The robustness of the proposed adaptive laws subjected to disturbances is also analyzed. Comparative numerical simulations are given to illustrate the superiority of the proposed method.