An Asymptotic Scaling Analysis of LQ Performance for an Approximate Adaptive Control Design

We consider the adaptive tracking problem for a chain of integrators, where the uncertainty is static and functional. The uncertainty is specified by L ²/L ^∞ or weighted L ²/L ^∞ norm bounds. We analyse a standard Lyapunov-based adaptive design which utilises 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. Date received: 20 April 1999. Date revised: 29 October 2001.