On gain adaptation in adaptive control

The adaptive high-gain output feedback strategy u(t)=-k(t)y(t), (d/dt)k(t)=||y(t)||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)⊂C₊; 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_h(μ), encompassing nonlinear systems modeled by functional differential equations, where the parameter h≥0 quantifies system memory and the continuous function μ:[0,∞)→[0,∞), with μ(0)=0, relates to the allowable system nonlinearities. Next, the linear control law is replaced by u(t)=-k(t)[y(t)+μ(||y(t)||)/||y(t)||]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)=ψ(||y(t)||), where the continuous function ψ satisfies certain growth conditions determined by μ (in particular cases, e.g., linear systems, a bounded function ψ is admissible). The above performance objectives 1) and 2) are shown to persist in the generalized framework.