Adaptive control via adaptive observation and asymptotic feedback matrix synthesis

For unknown linear, time invariant systems an adaptive feedback control scheme is established. It is composed of a stable adaptive state and parameter observation part, and a stable asymptotic synthesis of an adaptive feedback matrix part, the latter based on the current parameter estimates. Both adaptive state feedback control and adaptive feedback control, using the adaptively observed state instead of the true state, are shown to result in closed loop control systems, which behave globally asymptotically stable in the sense of Lyapunov with respect to the initial uncertainty, provided only that the input of the unknown system is sufficiently excited by means of an external command signal. Thereby an algebraic separation property for adaptive state feedback control is also established, extending an earlier nonadaptive result. In particular, no assumptions on the parameters of the unknown system nor on the speed of the adaptive observer and the adaptive feedback matrix generator are made.