CONVERGENCE PROPERTIES OF ADAPTIVE SYSTEMS AND THE DEFINITION OF EXPONENTIAL STABILITY.

The convergence properties of adaptive systems in terms of excitation conditions on the regressor vector are well known. With persistent excitation of the regressor vector in model reference adaptive control the state error and the adaptation error are globally exponentially stable or, equivalently, exponentially stable in the large. When the excitation condition, however, is imposed on the reference input or the reference model state, it is often incorrectly concluded that the persistent excitation in those signals also implies exponential stability in the large. The definition of persistent excitation is revisited so as to address some possible confusion in the adaptive control literature. It is then shown that persistent excitation of the reference model only implies local persistent excitation (weak persistent excitation). Weak persistent excitation of the regressor is still sufficient for uniform asymptotic stability in the large, but not exponential stability in the large. We show that there exists an infinite region in the state-space of adaptive systems where the state rate is bounded. This infinite region with finite rate of convergence is shown to exist not only in classic open-loop reference model adaptive systems but also in a new class of closed-loop reference model adaptive systems.

Adaptive dynamic inversion via time-scale separation.

This paper presents a full state feedback adaptive dynamic inversion method for uncertain systems that depend nonlinearly upon the control input. Using a specialized set of basis functions that respect the monotonic property of the system nonlinearities with respect to control input, a state predictor is defined for derivation of the adaptive laws. The adaptive dynamic inversion controller is defined as a solution of a fast dynamical equation, which achieves time-scale separation between the state predictor and the controller dynamics. Lyapunov-based adaptive laws ensure that the predictor tracks the state of the nonlinear system with bounded errors. As a result, the system state tracks the desired reference model with bounded errors. Benefits of the proposed design method are demonstrated using Van der Pol dynamics with nonlinear control input.

Coordinated decentralized adaptive output feedback control of interconnected systems.

A decentralized adaptive output feedback control design is proposed for large-scale interconnected systems. It is assumed that all the controllers share prior information about the system reference models. Based on that information, a linearly parameterized neural network is introduced for each subsystem to partially cancel the effect of the interconnections on tracking performance. Boundedness of error signals is shown through Lyapunov's direct method.