Stability and approximator convergence in nonparametric nonlinear adaptive control

This paper investigates nonparametric nonlinear adaptive control under passive learning conditions. Passive learning refers to the normal situation in control applications in which the system inputs cannot be selected freely by the learning system. This article also analyzes the stability of both the system state and approximator parameter estimates. Stability results are presented for both parametric (known model structure with unknown parameters) and nonparametric (unknown model structure resulting in epsilon-approximation error) adaptive control applications. Upper bounds on the tracking error are developed. The article also analyzes the persistence (PE) of excitation conditions required for parameter convergence. In addition, to a general PE analysis, the article presents a specific analysis pertinent to approximators that are composed of basis elements with local support. In particular, the analysis shows that as long as a reduced dimension subvector of the regressor vector is PE, then a specialized form of exponential convergence will be achieved. This condition is critical, since the general PE conditions are not practical in most control applications. In addition to the PE results, this article explicitly defines the regions over which the approximator converges when locally supported basis elements are used. The results are demonstrated throughout via examples.