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 L2=Ly or weighted L2=Ly norm bounds. We analyse a standard Lyapunovbased 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 e¤ort. 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.