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Data-Driven Stabilization Using Prior Knowledge on Stabilizability and Controllability

Amir Shakouri, Henk J. van Waarde, Tren M. J. T. Baltussen, W. P. M. H. Heemels

TL;DR

This work extends data-driven stabilization of LTI systems by incorporating prior knowledge on stabilizability and controllability. It proves that knowing controllability does not relax data-informativity requirements, whereas stabilizability can yield weaker necessary and sufficient conditions, especially when state-data are rank-deficient, enabling stabilization from data that would be non-informative otherwise. The authors derive tractable LMIs and a Θ-based design to compute stabilizing gains that work for all data-consistent systems under the prior, and validate the approach with a three-tank numerical example showing substantial gains in informativity when adopting stabilizability as prior. Overall, the paper provides a principled framework and practical LMIs for data-driven stabilization that leverages structure in the true system, with clear implications for reducing data requirements and conservatism in controller design.

Abstract

In this work, we study data-driven stabilization of linear time-invariant systems using prior knowledge of system-theoretic properties, specifically stabilizability and controllability. To formalize this, we extend the concept of data informativity by requiring the existence of a controller that stabilizes all systems consistent with the data and the prior knowledge. We show that if the system is controllable, then incorporating this as prior knowledge does not relax the conditions required for data-driven stabilization. Remarkably, however, we show that if the system is stabilizable, then using this as prior knowledge leads to necessary and sufficient conditions that are weaker than those for data-driven stabilization without prior knowledge. In other words, data-driven stabilization is easier if one knows that the underlying system is stabilizable. We also provide new data-driven control design methods in terms of linear matrix inequalities that complement the conditions for informativity.

Data-Driven Stabilization Using Prior Knowledge on Stabilizability and Controllability

TL;DR

This work extends data-driven stabilization of LTI systems by incorporating prior knowledge on stabilizability and controllability. It proves that knowing controllability does not relax data-informativity requirements, whereas stabilizability can yield weaker necessary and sufficient conditions, especially when state-data are rank-deficient, enabling stabilization from data that would be non-informative otherwise. The authors derive tractable LMIs and a Θ-based design to compute stabilizing gains that work for all data-consistent systems under the prior, and validate the approach with a three-tank numerical example showing substantial gains in informativity when adopting stabilizability as prior. Overall, the paper provides a principled framework and practical LMIs for data-driven stabilization that leverages structure in the true system, with clear implications for reducing data requirements and conservatism in controller design.

Abstract

In this work, we study data-driven stabilization of linear time-invariant systems using prior knowledge of system-theoretic properties, specifically stabilizability and controllability. To formalize this, we extend the concept of data informativity by requiring the existence of a controller that stabilizes all systems consistent with the data and the prior knowledge. We show that if the system is controllable, then incorporating this as prior knowledge does not relax the conditions required for data-driven stabilization. Remarkably, however, we show that if the system is stabilizable, then using this as prior knowledge leads to necessary and sufficient conditions that are weaker than those for data-driven stabilization without prior knowledge. In other words, data-driven stabilization is easier if one knows that the underlying system is stabilizable. We also provide new data-driven control design methods in terms of linear matrix inequalities that complement the conditions for informativity.

Paper Structure

This paper contains 11 sections, 12 theorems, 66 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Recap of Data-driven stabilization
  4. Data-driven stabilization using prior knowledge
  5. Controllability as Prior Knowledge
  6. Stabilizability as Prior Knowledge
  7. Necessary conditions
  8. Necessary and sufficient conditions with full rank state data
  9. Necessary and sufficient conditions with rank-deficient state data
  10. Numerical Example
  11. Conclusions

Key Result

Proposition 2

The data $\mathcal{D}$ are informative for stabilization if and only if there exists $\Theta\in\mathbb{R}^{T\times n}$ such that Moreover, $A+BK$ is Schur for all $(A,B)\in\Sigma_\mathcal{D}$if and only if$K=U_-\Theta(X_-\Theta)^{-1}$ for some $\Theta$ satisfying eq:dd_lmi.

Figures (2)

  • Figure 1: Set of data-consistent systems, set of uncontrollable systems, and their intersection for Example \ref{['ex:2']}.
  • Figure 2: Schematic representation of the three-tank system.

Theorems & Definitions (25)

  • Definition 1: van2020data
  • Proposition 2: van2020data
  • Definition 3
  • Example 4
  • Theorem 5
  • Example 6
  • Lemma 7
  • proof
  • Lemma 8: horn2012matrix
  • Lemma 9
  • ...and 15 more