Linear‐like properties arise naturally in the adaptive control setting

Abstract

Summary Classical discrete‐time adaptive controllers typically provide asymptotic stabilization and tracking; usually the affect of the noise is at best bounded‐input bounded‐output. Recently we have shown that if you design a discrete‐time adaptive controller in just the right way, then in a variety of situations you not only obtain exponential stability, but also a bounded gain on the noise in every p−norm, as well as a never‐before‐seen linear‐like convolution bound on the input–output behavior. Quite surprisingly, the approach is very natural, and relies on the use of the unmodified, original projection algorithm to carry out parameter estimation; if the set of plant uncertainty is not convex, then a multi‐estimator and switching are used. The goal of this paper is to provide an overview of the approach, discuss the results‐to‐date, and list some of the open problems.

Inherent robustness in the adaptive control of a large class of systems

“…We observe here that the controller (29a,b) and ( 30) fits into the paradigm of Section 2 with z ← ∅. In Reference 29 it is proven that (29a,b) and (30) provides exponential stability and a convolution bound for (23); by Theorems 1 and 2 we immediately see that the same is true in the presence of time-variation and/or unmodelled dynamics.…”

Supervisory Adaptive Control Revisited: Linear-Like Convolution Bounds

“…The objective of this thesis is to design an adaptive controller which provides the same linear-like property of [35,36,38], while relaxing some of the assumptions; namely, we remove the need to know the plant delay and the sign of the high frequency gain. To achieve this, we apply Supervisory Control in discrete-time to both the classical d-stepahead adaptive tracking problem (this is, in itself, new) and to the step tracking problem using an integral pole placement controller.…”

“…Observe that this estimator is the same as the weighted least-squares estimator, with two exceptions. Firstly, the true parameters θ * and estimated parameters θ are confined to a known set S in much the same way as in the Projection Algorithm setup used in [36,38] and other successful adaptive controllers. Secondly, instead of allowing the estimator to change its estimate at every time-step, it is required to 'pause' for the duration of the dwell time.…”

“…In much of the literature, this algorithm is modified to avoid having to treat the case ϕ(t) = 0 separately, but doing so makes it lose the nice property we are interested in[38].…”

Supervisory adaptive control revisited: Linear‐like convolution bounds and tolerance of slow time‐variations

“…The impetus for this research is recent work by the second author in References 18–23, wherein it is shown that highly desirable linear‐like input–output behavior can be achieved in a particular adaptive control setting via a carefully designed recursive parameter estimation algorithm. This is expressed as a convolution bound, which confers exponential stability and a bounded gain on the noise in every $$$ p $$$‐norm; such convolutions bounds are, as far as the authors are aware, a first in adaptive control and it allows one to use a modular approach* to prove robustness to a degree of unmodeled dynamics and tolerance to slow time‐varying parameters, 24,25 which is a highly desirable feature.…”