We consider the adaptive tracking problem for a chain of integrators, where the uncertainty is static and functional. The uncertainty is specified by L or weighted L norm bounds. We analyse a standard Lyapunov based adaptive design which utilizes a function approximator to induce a parametric uncertainty, on which the adaptive design is completed. Performance is measured by a modified LQ cost functional, penalising both the tracking error transient and the control effort. With such a cost functional, it is shown that a standard control design has divergent performance when the resolution of a `mono-resolution' approximator is increased. The class of `mono-resolution' approximators includes models popular in applications. A general construction of a class of approximators and their associated controllers which have a uniformly bounded performance independent of the resolution of the approximator is given.

We consider a tracking problem for an uncertain strict feedback system, where the uncertainties are memoryless nonlinearities specified by weighted L norms about a nominal system. These nonparametric uncertainties are converted to a parametric form by the use of function approximators, and adaptive backstepping designs are considered. It is shown that when multi-resolution function approximator structures are utilized, it is possible to achieve a transient LQ performance bound independant of the resolution of the approximator. In contrast, mono-resolution approximators which can cause an LQ performance to diverge with approximator resolution [5].