Bridging Model Reference Adaptive Control and Data Informativity

TL;DR

The paper addresses the data requirements of model reference adaptive control (MRAC) by introducing a data informativity framework and deriving necessary and sufficient online conditions for convergence to a solution of the MRAC matching equations $A_{ m s}+B_{ m s}K=A_{ m m}$ and $B_{ m s}L=B_{ m m}$. It bridges offline data informativity with online MRAC by designing an adaptive scheme that online checks informativity (through an informative time $T^{*}$) and switches inputs to ensure data become informative after a finite time. The main result shows that if $T^{*}$ is finite, the gains $( ilde{K}(t), ilde{L}(t))$ converge to a solution of the matching equations and the tracking error $e(t)$ tends to zero, with no requirement for unique system identification or knowledge of the input matrix. Simulations on a numerical system and a highly maneuverable aircraft illustrate that informative data can be achieved with modest data excitation and that convergence occurs even when the data do not satisfy the full-rank condition, underscoring the practical impact of the proposed online MRAC framework.

Abstract

The goal of model reference adaptive control (MRAC) is to ensure that the trajectories of an unknown dynamical system track those of a given reference model. This is done by means of a feedback controller that adaptively changes its gains using data collected online from the closed-loop system. One of the approaches to solve the MRAC problem is to impose conditions on the data that guarantee convergence of the gains to a solution of the so-called matching equations. In the literature, various extensions of the concept of persistent excitation have been proposed in an effort to weaken the conditions on the data required for this convergence.Despite these efforts, it is not well-understood what are the weakest possible data requirements ensuring convergence of MRAC. In this paper, we propose a new framework to study the MRAC problem, using the concept of data informativity. Our main contribution is to provide \textit{necessary and sufficient} conditions for the asymptotic convergence of the adaptive gains to a solution of the matching equations. These necessary and sufficient conditions can be readily checked online as new data are generated by the closed-loop system. Our results reveal that existing excitation conditions impose stronger requirements on the collected data than required. Notably, the necessary and sufficient conditions provided in this paper are weaker than those for unique system identification.

Figures (4)

• Figure 1: State tracking errors, norms of the matching error and input signals in $\mathcal{S}_1$ and $\mathcal{S}_2$.
• Figure 2: State tracking errors, norms of the matching error and input signals in $\mathcal{S}_3$ and $\mathcal{S}_4$.
• Figure 3: State tracking errors, norms of the matching error and input signals in $\mathcal{S}_5$ and $\mathcal{S}_6$.
• Figure 4: State tracking errors, norms of the matching error and input signals in $\mathcal{S}_7$ and $\mathcal{S}_8$.

Theorems & Definitions (24)

• Definition 1
• Lemma 1
• Definition 2
• Lemma 2
• Definition 3
• Lemma 3
• Lemma 3
• proof
• Proposition 1
• proof