Nonparametric estimation and adaptive control of functional autoregressive models

This paper deals with nonparametric estimation and adaptive control of nonlinear systems of the form Xn+1 = f (X-n) + U-n + xi(n+1) (n is an element of N) where the state X-n is observed, f is an unknown function, and the control U-n is chosen in order to track a given reference trajectory. We estimate the function f using a nonparametric estimator and study two adaptive control laws built from this nonparametric estimator and derived from the self-tuning control. The rst one can be used for open-loop stable systems and requires an additional exciting noise. The second one needs some a priori knowledge on function f but allows us to control open-loop unstable systems. We establish some general results on the nonparametric estimator of f like the uniform almost sure convergence over dilating sets and then prove that both adaptive control laws are asymptotically optimal in quadratic mean. In addition, we give a strongly consistent estimator of the covariance matrix of the unobservable white noise xi(n).