On gain adaptation in adaptive control

The adaptive high-gain output feedback strategy u(t)=-k(t)y(t), (d/dt)k(t)=/spl par/y(t)/spl par//sup 2/ is well established in the context of linear, minimum-phase, m-input m-output systems (A, B, C) with the property that spec(CB)/spl sub//spl Copf//sub +/; the strategy applied to any such linear system achieves the performance objectives of: 1) global attractivity of the zero state; and 2) convergence of the adapting gain to a finite limit. Here, these results are generalized in three aspects. First, the class of systems is enlarged to a class N/sub h/(/spl mu/), encompassing nonlinear systems modeled by functional differential equations, where the parameter h/spl ges/0 quantifies system memory and the continuous function /spl mu/:[0,/spl infin/)/spl rarr/[0,/spl infin/), with /spl mu/(0)=0, relates to the allowable system nonlinearities. Next, the linear control law is replaced by u(t)=-k(t)[y(t)+/spl mu/(/spl par/y(t)/spl par/)//spl par/y(t)/spl par/]y(t), wherein the additional nonlinear term counteracts the system nonlinearities. Then, the quadratic adaptation law is replaced by the law (d/dt)k(t)=/spl psi/(/spl par/y(t)/spl par/), where the continuous function /spl psi/ satisfies certain growth conditions determined by /spl mu/ (in particular cases, e.g., linear systems, a bounded function /spl psi/ is admissible). The above performance objectives 1) and 2) are shown to persist in the generalized framework.