Command Filtered Adaptive Backstepping

Abstract

Implementation of adaptive backstepping controllers requires analytic calculation of the partial derivatives of certain stabilizing functions. It is well documented that, as the order of a nonlinear system increases, analytic calculation of these derivatives becomes prohibitive. Therefore, in practice, either alternative control approaches are used or the derivatives are neglected in the implementation. Neglecting the derivatives results in the loss of all guarantees proven by Lyapunov methods for the adaptive backstepping approach and may result in instability. This paper presents a new implementation approach for adaptive backstepping control. The main objectives are to facilitate the derivation and implementation of the adaptive backstepping approach, with performance guarantees proven by Lyapunov methods, for applications that were prohibitively difficult using the standard analytic implementation approach. The new approach uses filtering methods to produce certain command signals and their derivatives which eliminates the requirement of analytic differentiation. The approach also introduces filters to generate certain compensating signals necessary to compute compensated tracking errors suitable for adaptive parameter estimation. We present a set of Lemmas and Theorems to analyze the performance both during the initialization and the operating phases. We show that the initialization phase is of finite duration that can be controlled by selection of a design parameter. We also show that all signals within the system are bounded during this short initialization phase. During the operating phase, we show that the command filtered implementation approach has theoretical properties identical to those of the conventional approach. The general approach is presented and analyzed for systems in generalized parameter strict feedback form. Extensions of the approach are presented to demonstrate the application of the method to a land vehicle trajectory following application. Application and effectiveness of the proposed method is shown by simulation results.