Convergence Properties of Adaptive Systems and the Definition of Exponential Stability
For researchers in adaptive control, this paper resolves confusion about stability definitions and excitation conditions, though it is an incremental theoretical clarification.
This paper clarifies that persistent excitation of the reference model in adaptive control only guarantees weak persistent excitation, leading to uniform asymptotic stability but not exponential stability in the large. It demonstrates the existence of an infinite region with bounded state rate in both open-loop and closed-loop reference model adaptive systems.
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.