Adaptive backstepping with exponential regulation in the absence of persistent excitation

With set point regulation being the most common goal in control engineering, persistence of excitation (PE) is generically absent in adaptive control applications. In the absence of PE, not only is the parameter estimate not guaranteed to converge to the true value, but the state is regulated to the set point at a rate that is not necessarily exponential. In this paper we focus on strict feedback nonlinear systems and propose a strategy that employs time-varying adaptation gains (as well as time-varying control gains, when appropriate) and achieves exponential regulation of the plant state, with an exponential rate that is uniform in the initial condition. This idea fundamentally differs from exponential stability results achieved in the presence of PE because we make the gains (rather than reference signals) time-varying, i.e., we use time-varying tools in a multiplicative (rather than in an additive) fashion. We provide full state feedback results for strict-feedback nonlinear systems. These results establish global uniform stability, exponential regulation of the plant state, boundedness of the control input and the update rate, and the asymptotic constancy (but not convergence to the true value) of the parameter estimate.