Filterless Least-Squares Based Adaptive Stochastic Continuous-Time Nonlinear Control

Abstract

Abstract In continuous-time system identification and adaptive control, the least-squares parameter estimation algorithm has always been used with regressor filtering, in order to avoid using time-derivatives of the measured state. Filtering adds to the dynamic order of the identifier and affects its performance. We solve the problem of filterless least-squares-based adaptive control for stochastic strict-feedback nonlinear systems with an unknown parameter in the drift term. The novel ingredient in our least-squares identification is that the update law for the parameter estimate is not a simple integrator but it also incorporates a feedthrough effect, namely, the parameter estimator is of relative degree zero (rather than one) relative to the update function. The feedthrough in the update law is a carefully designed nonlinear function, which incorporates the integration with respect to state (and not time) of the regressor function, the purpose of which is to eliminate the need for time-filtering of the regressor. Our backstepping design of the control law compensates the adverse effect of the noise (the Hessian nonlinear term, involving the diffusion nonlinearity, in the Lyapunov analysis) on the least-squares estimator. Such a controller also enables a construction of an single overall Lyapunov function, quadratic in the parameter error and quartic in the transformed state, to guarantee that the equilibrium at the origin of the closed-loop system is globally stable in probability and the states are regulated to zero almost surely.