An asymptotic scaling analysis of LQ performance for an approximate adaptive control design (Q1611002)

The paper studies the adaptive tracking problem for a chain of integrators, where the uncertainty is static and functional. The uncertainty is specified by \(L^2/L^\infty\) or weighted \(L^2/L^\infty\) norm bounds. A standard Lyapunov-based adaptive design is analyzed, 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, penalizing 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. A geneal 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.