Direct adaptive regulation of discrete time nonlinear systems with arbitrary nonlinearities by backstepping

This paper presents a direct adaptive regulation result for a class of strict feedback discrete time nonlinear systems with arbitrary nonlinearities. This results is an extension of the current state of the art in discrete time adaptive backstepping, where until now results are restricted to the case where the plant's nonlinearities can be expressed as a linear combination of known functions and restrictive linear growth conditions are imposed. Here, a more general approach is followed by using function approximation and online update of the approximators. The regulation theorem guarantees convergence of the state norm to a neighborhood of the origin, and the size of the neighborhood depends exclusively on the chosen approximator structure and ideal approximation errors. The stability is semi-global, and no high-gain bounding terms are required, since explicit bounds for the state are found from the stability analysis. The proposed method is a direct adaptive scheme where a stabilizing control law, rather than the plant nonlinearities, is directly approximated. An example is provided to illustrate the method.