Adaptive Control with Multiresolution Bases

This paper presents new results on adaptive nonlinear control using wavelet basis functions. First, it considers how to deal effectively with a potentially infinite number of unknown parameters, rather than using a priori truncations of wavelet expansions. This is done by constructing formally an ideal “infinite” controller, able to manage infinitely many unknown parameters in a convergent fashion, and only then designing a way to approximate its behavior with a finite controller. Besides being theoretically satisfying, the existence of such a consistent underlying infinite controller is easy to guarantee in practice and considerably improves convergence properties in the high frequency range of the unknown function. The paper also shows the advantages of specific multiresolution analysis wavelets over the “Mexican hat”–type wavelet frames now commonly used in control and learning applications. Finally, it discusses several possible constructions in the multidimensional case, and their properties.