Classical d-Step-Ahead Adaptive Control Revisited: Linear-Like Convolution Bounds and Exponential Stability (Extended Version)

TL;DR

This paper extends the analysis of adaptive control to d-step-ahead controllers, demonstrating linear-like bounds, exponential stability, and bounded noise gain, which improve robustness and performance over classical methods.

Contribution

It generalizes previous results to d-step-ahead adaptive controllers, establishing linear-like convolution bounds and exponential stability.

Findings

Proves linear-like bounds for d-step-ahead adaptive control

Establishes exponential stability with the extended approach

Demonstrates robustness to unmodelled dynamics and parameter variations

Abstract

Classical discrete-time adaptive controllers provide asymptotic stabilization and tracking; neither exponential stabilization nor a bounded noise gain is typically proven. In recent work it has been shown, in both the pole placement stability setting and the first-order one-step-ahead tracking setting, that if the original, ideal, Projection Algorithm is used (subject to the common assumption that the plant parameters lie in a convex, compact set and that the parameter estimates are restricted to that set) as part of the adaptive controller, then a linear-like convolution bound on the closed loop behaviour can be proven; this immediately confers exponential stability and a bounded noise gain, and it can be leveraged to provide tolerance to unmodelled dynamics and plant parameter variation. In this paper we extend the approach to the d-step-ahead adaptive controller setting and prove comparable properties.