On gain adaptation in adaptive control

The adaptive high-gain output feedback strategy ( ) = ( ) ( ), ( ) ( ) = ( ) is well established in the context of linear, minimum-phase, -input -output systems ( , , ) with the property that spec( ) ; 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 ( ), encompassing nonlinear systems modeled by functional differential equations, where the parameter 0 quantifies system memory and the continuous function : [0 ) [0 ), with (0) = 0, relates to the allowable system nonlinearities. Second, the linear control law is replaced by ( ) = ( )[ ( ) + ( ( ) ) ( ) ] ( ), wherein the additional nonlinear term counteracts the system nonlinearities. Third, the quadratic adaptation law is replaced by the law ( ) ( ) = ( ( ) ), where the continuous function satisfies certain growth conditions determined by (in particular cases, e.g., linear systems, a bounded function is admissible). Performance objectives 1) and 2) above are shown to persist in the generalized framework.