Adaptive Control with Global Exponential Stability for Parameter-Varying Nonlinear Systems under Unknown Control Gains

TL;DR

This paper presents a novel adaptive control method that guarantees global exponential stability for parameter-varying nonlinear systems with unknown control gains, without requiring persistent excitation, using enhanced Nussbaum functions and nonlinear damping.

Contribution

It introduces a new adaptive control scheme that ensures stability without PE condition and extends to systems with unknown sign and magnitude of control gains using enhanced Nussbaum functions.

Findings

Achieves global exponential stability for parameter-varying systems.

Ensures bounded control input and parameter estimates.

Validated effectiveness through numerical simulations.

Abstract

It is nontrivial to achieve exponential stability even for time-invariant nonlinear systems with matched uncertainties and persistent excitation (PE) condition. In this paper, without the need for PE condition, we address the problem of global exponential stabilization of strict-feedback systems with mismatched uncertainties and unknown yet time-varying control gains. The resultant control, embedded with time-varying feedback gains, is capable of ensuring global exponential stability of parametric-strict-feedback systems in the absence of persistence of excitation. By using the enhanced Nussbaum function, the previous results are extended to more general nonlinear systems where the sign and magnitude of the time-varying control gain are unknown. In particular, the argument of the Nussbaum function is guaranteed to be always positive with the aid of nonlinear damping design, which is critical to perform a straightforward technical analysis of the boundedness of the Nussbaum function. Finally, the global exponential stability of parameter-varying strict-feedback systems, the boundedness of the control input and the update rate, and the asymptotic constancy of the parameter estimate are established. Numerical simulations are carried out to verify the effectiveness and benefits of the proposed methods.