Optimal Parameter Estimation Under Finite Excitation

Although parameter estimation of continuous-time systems has been studied for decades, the gradient-descent algorithm and its advancements (e.g., least-squares) were all derived to minimize the error between the measured system output and the predictor/observer output, rather than the estimation error (difference between the unknown parameters and their estimates). Hence, the transient convergence response of the estimation error that depends on the manually set learning gains is difficult to analyze. The main contribution of this article is to propose an optimal parameter estimation method, which can directly minimize a cost function of the estimation error to retain the optimal parameter estimation. For this purpose, filter operations and auxiliary variables are used to derive a constructive formulation of estimation error. Then, a cost function of the extracted estimation error is established and minimized to derive a new parameter update law by using the optimality principle. In this framework, a time-varying gain is obtained to handle the effect of regressor and guarantee the exponential convergence under the classical persistent excitation (PE) condition. Moreover, a further tailored parameter update law including a switching term with an excitation increasing mechanism is studied to adapt a weak finite excitation (FE) condition, where both the transient optimal and steady-state exponential convergence can be still retained. Finally, the efficacy of the proposed estimators is verified via numerical simulations and practical experiments on a proton exchange membrane fuel cell system.