Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributionally robust LMI synthesis for LTI systems

Dennis Gramlich, Shuhao Yan, Carsten W. Scherer, Christian Ebenbauer%

TL;DR

This paper addresses control under distributional uncertainty by formulating distributionally robust control (DRC) for LTI systems with Wasserstein ambiguity sets. It develops exact moment-relaxation-based LMIs that convert infinite-horizon DRC into finite-dimensional convex programs and proves an exact equivalence with robust $H_2$ synthesis. The authors provide convex synthesis frameworks for both correlated and independent disturbances, using Schur complements and Scherer parametrization to enable controller optimization within SDP solvers. Simulation on a wind-turbine model demonstrates substantial robustness gains of DR controllers against worst-case correlated disturbances, with practical out-of-sample performance validated via variance and Bode analyses. The work offers a modular, scalable approach to incorporating disturbance distributional information into robust control design, with potential extensions to hybrid ambiguity sets.

Abstract

This article shows that distributionally robust controller synthesis as investigated in \cite{taskesen2024distributionally} can be formulated as a convex linear matrix inequality (LMI) synthesis problem. To this end, we rely on well-established convexification techniques from robust control. The LMI synthesis problem we propose has the advantage that it can be solved efficiently using off-the-shelf semi-definite programming (SDP) solvers. In addition, our formulation exposes the studied distributionally robust controller synthesis problem as an instance of robust $H_2$ synthesis.

Distributionally robust LMI synthesis for LTI systems

TL;DR

This paper addresses control under distributional uncertainty by formulating distributionally robust control (DRC) for LTI systems with Wasserstein ambiguity sets. It develops exact moment-relaxation-based LMIs that convert infinite-horizon DRC into finite-dimensional convex programs and proves an exact equivalence with robust synthesis. The authors provide convex synthesis frameworks for both correlated and independent disturbances, using Schur complements and Scherer parametrization to enable controller optimization within SDP solvers. Simulation on a wind-turbine model demonstrates substantial robustness gains of DR controllers against worst-case correlated disturbances, with practical out-of-sample performance validated via variance and Bode analyses. The work offers a modular, scalable approach to incorporating disturbance distributional information into robust control design, with potential extensions to hybrid ambiguity sets.

Abstract

This article shows that distributionally robust controller synthesis as investigated in \cite{taskesen2024distributionally} can be formulated as a convex linear matrix inequality (LMI) synthesis problem. To this end, we rely on well-established convexification techniques from robust control. The LMI synthesis problem we propose has the advantage that it can be solved efficiently using off-the-shelf semi-definite programming (SDP) solvers. In addition, our formulation exposes the studied distributionally robust controller synthesis problem as an instance of robust synthesis.

Paper Structure

This paper contains 13 sections, 14 theorems, 64 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and problem statement
  2. Related work
  3. Contributions
  4. Preliminaries
  5. Robust performance analysis
  6. Robust performance analysis for correlated disturbances
  7. Robust performance analysis for independent disturbances
  8. Robust synthesis
  9. Relation to robust $H_2$ synthesis
  10. Simulation Results
  11. Conclusion
  12. Proof of Theorem \ref{['thm:finiteHorizonProblemEquivalent']}
  13. Wasserstein and Autocorrelation Conditions

Key Result

Theorem IV.2

For two distributions $\mathbb{P}_1$, $\mathbb{P}_2$ on $\mathbb{R}^d$ with mean vectors $\mu_1$ and $\mu_2$ and covariance matrices $\Sigma_1$ and $\Sigma_2$, respectively, the following holds:

Figures (2)

  • Figure 1: This figure shows a closed-loop system described by matrices ${\color{blue}\mathcal{A}},{\color{blue}\mathcal{B}},{\color{blue}\mathcal{C}},{\color{blue}\mathcal{D}}$. The system has an uncertain real block $\Delta$ in the performance channel from $\hat{w}$ to $z$. The symbol $Z_-$ denotes the back shift operator.
  • Figure 2: Bode diagrams from the maximum singular value of the disturbance to each performance output, $\omega$, $\phi$, $h$, $u$.

Theorems & Definitions (25)

  • Definition IV.1
  • Theorem IV.2: gelbrich1990formula
  • Theorem V.1
  • Proposition V.2
  • proof
  • Proposition V.3
  • proof
  • Theorem V.4
  • proof
  • Theorem V.5
  • ...and 15 more