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Notes on data-driven output-feedback control of linear MIMO systems

Mohammad Alsalti, Victor G. Lopez, Matthias A. Müller

TL;DR

The paper tackles the challenge of data-driven dynamic output-feedback control for MIMO LTI systems by showing that conventional non-minimal states fail to satisfy essential rank conditions when $pℓ≥n$. It introduces a data-driven non-minimal state $z_t$ constructed from past inputs and outputs so that the data matrix $U_0Z_0$ has full row rank under persistent excitation, enabling direct stabilization with a data-based gain via SDP-based formulations. A noisy-data extension uses SVD truncation and robust control tools (e.g., Petersen's lemma) to maintain stabilization in the presence of noise, demonstrated through a numerical example where the proposed approach outperforms the standard $ ext{ξ}$-based method. Overall, the work relaxes the $pℓ=n$ restriction for MIMO systems, broadening the applicability of data-driven output-feedback control and paving the way for robust and optimal extensions as well as nonlinear generalizations.

Abstract

Recent works have approached the data-driven design of dynamic output-feedback controllers for discrete-time LTI systems by constructing non-minimal state vectors composed of past inputs and outputs. Depending on the system's complexity (order $n$, lag $\ell$ and number of outputs $p$), it was observed in several works that such an approach presents significant limitations. In particular, many works require to restrict the class of LTI systems to those satisfying the relation $p\ell=n$. In this note, we show how to address the general MIMO case (for which $p\ell\geq n$ in general) by constructing an alternative non-minimal state vector from data. Different from the existing literature, our method guarantees the satisfaction of certain rank conditions when the system is persistently excited, thereby facilitating the direct data-driven dynamic output-feedback control of MIMO systems by applying methods that were originally developed for the input-state data setting.

Notes on data-driven output-feedback control of linear MIMO systems

TL;DR

The paper tackles the challenge of data-driven dynamic output-feedback control for MIMO LTI systems by showing that conventional non-minimal states fail to satisfy essential rank conditions when . It introduces a data-driven non-minimal state constructed from past inputs and outputs so that the data matrix has full row rank under persistent excitation, enabling direct stabilization with a data-based gain via SDP-based formulations. A noisy-data extension uses SVD truncation and robust control tools (e.g., Petersen's lemma) to maintain stabilization in the presence of noise, demonstrated through a numerical example where the proposed approach outperforms the standard -based method. Overall, the work relaxes the restriction for MIMO systems, broadening the applicability of data-driven output-feedback control and paving the way for robust and optimal extensions as well as nonlinear generalizations.

Abstract

Recent works have approached the data-driven design of dynamic output-feedback controllers for discrete-time LTI systems by constructing non-minimal state vectors composed of past inputs and outputs. Depending on the system's complexity (order , lag and number of outputs ), it was observed in several works that such an approach presents significant limitations. In particular, many works require to restrict the class of LTI systems to those satisfying the relation . In this note, we show how to address the general MIMO case (for which in general) by constructing an alternative non-minimal state vector from data. Different from the existing literature, our method guarantees the satisfaction of certain rank conditions when the system is persistently excited, thereby facilitating the direct data-driven dynamic output-feedback control of MIMO systems by applying methods that were originally developed for the input-state data setting.
Paper Structure (7 sections, 7 theorems, 45 equations, 1 figure)

This paper contains 7 sections, 7 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation & Preliminaries
  3. Limitations of existing non-minimal state vectors
  4. An alternative non-minimal state
  5. Handling noisy data
  6. Example
  7. Discussion and Conclusions

Key Result

Theorem 1

Consider a controllable DT-LTI system as in eqn_LTI. If the input sequence $\{u^d_k\}_{k=0}^{N-1}$ is persistently exciting of order $L+n$ (for $L\in\mathbb{Z}_{>0}$), then the following holds where $\{x^d_k\}_{k=0}^{N-1}$ is the corresponding state sequence that results from applying $u^d$ to eqn_LTI.

Figures (1)

  • Figure 1: An illustration of the closed-loop system, where a feedback controller based on a non-minimal state $z_t$ is used.

Theorems & Definitions (15)

  • Theorem 1: Willems05
  • Theorem 2: Persis20
  • Theorem 3
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • Remark 1
  • Remark 2
  • ...and 5 more