Valid RBFNN Adaptive Control for Nonlinear Systems With Unmatched Uncertainties.

In this article, an adaptive tracking controller based on radial basis function neural networks (RBFNNs) is proposed for nonlinear plants with unmatched uncertainties and smooth reference signals. The concept of valid RBFNN adaptive control is introduced where all closed-loop arguments of the involved RBFNNs should always remain inside their corresponding compact sets. Considering the local approximation capacity of RBFNNs, validity requirements are necessary for ensuring reliable closed-loop approximation accuracy and stability. To obtain valid RBFNN adaptive controllers, a novel iterative design method is proposed and embedded into the traditional backstepping approach. In the initial iteration, an ideal RBFNN and its online estimated version are introduced in each step where the initial compact set guarantees the validity requirement for only a finite time interval. Then, by carefully investigating the dependence among different signals and introducing some auxiliary variables, the compact sets are redesigned for prolonging the time interval satisfying validity requirements to infinity as the iteration goes on. Consequently, a closed-loop system model can be formulated during the entire control process, which underlies a rigorous proof on closed-loop stability and some guidelines on practical implementation. Meanwhile, rigorous analysis from validity requirements reveals, for the first time, a new feature of RBFNN adaptive controllers in the presence of unmatched uncertainties: excessively large scales of RBFNNs in intermediate steps may impair the closed-loop performance. Finally, simulation results are provided to illustrate the efficiency and feasibility of the obtained results.