Adaptive Control of Discrete-Time Nonlinear bining Nonparametric and Parametric Estimators.pdf

COMMUNICATIONS IN INFORMATION AND SYSTEMS °c 2002 International Press
Vol. 2, No. 1, pp. 69-90, June 2002 004
ADAPTIVE CONTROL OF DISCRETE-TIME NONLINEAR
BINING NONPARAMETRIC AND PARAMETRIC
ESTIMATORS∗
BRUNO PORTIER†
Abstract. In this paper, a new adaptive control bining nonparametric and parametric
estimators is proposed to control stochastic d-dimensional discrete-time nonlinear models of the form
Xn+1 = f(Xn) + Un + εn+1. The unknown function f is assumed to be parametric outside a given
domain of Rd and fully nonparametric inside. The nonparametric part of f is estimated using a
kernel-based method and the parametric one is estimated using the weighted least squares estimator.
The asymptotic optimality of the tracking is established together with some convergence results for
the estimators of f.
Keywords. Adaptive tracking control; Kernel-based estimation; Nonlinear model; Stochastic
systems; Weighted least squared estimator.
1. Introduction. In a recent paper, [Portier and Oulidi 2000] consider the
problem of adaptive control of stochastic d-dimensional discrete-time nonlinear sys-
tems of the form (d ∈ N):
(1) Xn+1 = f(Xn) + Un + εn+1
where Xn, Un and εn are the output, input and noise of the system, respectively. The
state Xn is observed, the function f is unknown, εn is an unobservable noise and the
control Un is to be chosen in order to track a given deterministic reference trajectory
∗
denoted by (Xn)n≥1. To satisfy the control objective, Portier and Oulidi introduce an
adaptive control law using a kernel-based nonparametric estimator (NPE for short)
b
of the function f denoted by fn. Following the certainty-equivalence principle, the
desired control is given by
b ∗
(2) Un = −fn(Xn) + Xn+1
However, pensate for the possible lack of observations which disruptes the
NPE, some a priori knowledge about the function f is required. In a recent paper,
[Xie and Guo 2000] study scalar models of the form (1)