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Online Data-Driven Adaptive Control for Unknown Linear Time-Varying Systems

Shenyu Liu, Kaiwen Chen, Jaap Eising

TL;DR

The stability of the closed-loop system is analyzed in detail, which shows that under some mild assumptions, the proposed online data-driven adaptive control scheme can guarantee practical global exponential stability.

Abstract

This paper proposes a novel online data-driven adaptive control for unknown linear time-varying systems. Initialized with an empirical feedback gain, the algorithm periodically updates this gain based on the data collected over a short time window before each update. Meanwhile, the stability of the closed-loop system is analyzed in detail, which shows that under some mild assumptions, the proposed online data-driven adaptive control scheme can guarantee practical global exponential stability. Finally, the proposed algorithm is demonstrated by numerical simulations and its performance is compared with other control algorithms for unknown linear time-varying systems.

Online Data-Driven Adaptive Control for Unknown Linear Time-Varying Systems

TL;DR

The stability of the closed-loop system is analyzed in detail, which shows that under some mild assumptions, the proposed online data-driven adaptive control scheme can guarantee practical global exponential stability.

Abstract

This paper proposes a novel online data-driven adaptive control for unknown linear time-varying systems. Initialized with an empirical feedback gain, the algorithm periodically updates this gain based on the data collected over a short time window before each update. Meanwhile, the stability of the closed-loop system is analyzed in detail, which shows that under some mild assumptions, the proposed online data-driven adaptive control scheme can guarantee practical global exponential stability. Finally, the proposed algorithm is demonstrated by numerical simulations and its performance is compared with other control algorithms for unknown linear time-varying systems.
Paper Structure (10 sections, 3 theorems, 49 equations, 3 figures, 1 algorithm)

This paper contains 10 sections, 3 theorems, 49 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Data-driven and adaptation mechanism
  4. Periodically updated control law
  5. A data-driven model of the LTV system
  6. Stability analysis and algorithmic realization
  7. Stability analysis
  8. The control algorithm
  9. Simulation
  10. Conclusion

Key Result

Proposition III.1

Given a scalar $\lambda\in(0,1)$ and the collected data $X_i$, $X_i^+$, $U_i$ over $\mathcal{T}^W_i$. If there exist scalars $\alpha_1\geqslant 0$, $\alpha_2\geqslant 0$, a positive definite matrix $Q_i\in\mathbb R^{n\times n}$, and a matrix $L_i\in\mathbb R^{m\times n}$ such that where $\bar{M}$, $\bar{N}_1$, and $\bar{N}_2$ are as in def:bar_M and $\Pi$ is as defined in def:Pi. Then, the matric

Figures (3)

  • Figure 1: Illustration of the timing of data collection and control gain update. The purple curve represents a "trajectory" of $[A(t), B(t)]\in\mathbb R^{n\times n}\times \mathbb R^{n\times m}$. At each time instant $t_i^S$, the controller estimates a set to which $[A(t),B(t)]$, $t\in\mathcal{T}_i$, belongs, represented by the green frustum-shaped region. Such estimation is made based on the data collected over the time window $\mathcal{T}_i^W$.
  • Figure 2: Semi-logarithmic time histories of $|x(t)|$ of the LTV system in the four control schemes.
  • Figure 3: Semi-logarithmic time histories of $|x(t)|$ of the LTI system in the four control schemes.

Theorems & Definitions (5)

  • Remark 1: Choice of the time window length
  • Proposition III.1: An LMI for the feedback gain
  • Remark 2: Proposition \ref{['thm:1']} is only sufficient
  • Theorem IV.1: Stability of the closed-loop system
  • Corollary IV.2: Functionality of the ODDAC algorithm