A Practical Excitation Condition of Simultaneous Parameter Estimation and Adaptive Tracking

In this study, a practical excitation condition for the simultaneous implementation of estimation and tracking is provided. This condition is easy to verify for realizing tracking and estimating under adaptation law without filters. For the general parameterized system and backstepping-based adaptive control law, a framework is proposed to analyze the parameter convergence. First, the time axis is divided, and in each segment, the estimation error corresponds to the solution of linear equations with perturbation. Then, by estimating the solutions of a family of perturbed linear equations, a sufficient convergence condition is given. The concept of a determinant of excitation (DE) is introduced for the first time. Compared with previous convergence analysis methods, the exponential convergence is no longer required. Therefore, the parameter convergence condition derived by the study analysis method (i.e., the DE condition) is weaker and easier to verify than the long-standing persistence of the excitation condition (PE). Furthermore, the theoretical results are verified by numerical simulation.