Adaptive control of pure feedback systems without inversion of parameter estimates

This paper presents a novel adaptive control technique that avoids the need to invert parameter estimates for stabilising a class of nonlinear systems with uncertain or unknown input maps. The classical backstepping technique requires the inversion of the input maps, making it inapplicable in the case where the estimate of the input map is updated concurrently. Since the estimates may cross zero, the control law becomes ill-conditioned, and closed-loop stability cannot be ensured. The extension of the adaptive backstepping framework presented in this paper constructs an adaptive control law that does not require the inversion of any parameter estimates. A rigorous theoretical analysis based on the Barbashin-Krasovskii-LaSalle invariance principle is used to establish the stability of the resulting closed-loop system. Finally, the proposed technique is demonstrated by designing an adaptive controller for nonlinear systems with input map uncertainty.