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Closing the Gap to Quadratic Invariance: a Regret Minimization Approach to Optimal Distributed Control

Daniele Martinelli, Andrea Martin, Giancarlo Ferrari-Trecate, Luca Furieri

TL;DR

This work derives convex relaxations of the resulting regret minimization problem that are compatible with any desired controller sparsity, while revealing a renewed role of the Quadratic Invariance (QI) condition in designing in-formative benchmarks to measure regret.

Abstract

In this work, we focus on the design of optimal controllers that must comply with an information structure. State-of-the-art approaches do so based on the H2 or Hinfty norm to minimize the expected or worst-case cost in the presence of stochastic or adversarial disturbances. Large-scale systems often experience a combination of stochastic and deterministic disruptions (e.g., sensor failures, environmental fluctuations) that spread across the system and are difficult to model precisely, leading to sub-optimal closed-loop behaviors. Hence, we propose improving performance for these scenarios by minimizing the regret with respect to an ideal policy that complies with less stringent sensor-information constraints. This endows our controller with the ability to approach the improved behavior of a more informed policy, which would detect and counteract heterogeneous and localized disturbances more promptly. Specifically, we derive convex relaxations of the resulting regret minimization problem that are compatible with any desired controller sparsity, while we reveal a renewed role of the Quadratic Invariance (QI) condition in designing informative benchmarks to measure regret. Last, we validate our proposed method through numerical simulations on controlling a multi-agent distributed system, comparing its performance with traditional H2 and Hinfty policies.

Closing the Gap to Quadratic Invariance: a Regret Minimization Approach to Optimal Distributed Control

TL;DR

This work derives convex relaxations of the resulting regret minimization problem that are compatible with any desired controller sparsity, while revealing a renewed role of the Quadratic Invariance (QI) condition in designing in-formative benchmarks to measure regret.

Abstract

In this work, we focus on the design of optimal controllers that must comply with an information structure. State-of-the-art approaches do so based on the H2 or Hinfty norm to minimize the expected or worst-case cost in the presence of stochastic or adversarial disturbances. Large-scale systems often experience a combination of stochastic and deterministic disruptions (e.g., sensor failures, environmental fluctuations) that spread across the system and are difficult to model precisely, leading to sub-optimal closed-loop behaviors. Hence, we propose improving performance for these scenarios by minimizing the regret with respect to an ideal policy that complies with less stringent sensor-information constraints. This endows our controller with the ability to approach the improved behavior of a more informed policy, which would detect and counteract heterogeneous and localized disturbances more promptly. Specifically, we derive convex relaxations of the resulting regret minimization problem that are compatible with any desired controller sparsity, while we reveal a renewed role of the Quadratic Invariance (QI) condition in designing informative benchmarks to measure regret. Last, we validate our proposed method through numerical simulations on controlling a multi-agent distributed system, comparing its performance with traditional H2 and Hinfty policies.
Paper Structure (15 sections, 3 theorems, 38 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 15 sections, 3 theorems, 38 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Spatial regret
  4. Review of convex design of distributed controllers
  5. Main results
  6. Synthesis of the oracle KHat
  7. Convex reformulation of the spatial regret problem
  8. Numerical results
  9. Conclusions
  10. Example of oracle not guaranteeing well-posedness
  11. Proof of Proposition \ref{['prop:well_posed_regret']}
  12. Example that using SI to design both K, KHat does not guarantee well-posedness
  13. Proof of Theorem \ref{['th:well_posed_with_QI']}
  14. Proof of Proposition \ref{['prop:dual_regret']}
  15. Implementation details

Key Result

Proposition 1

Assume an oracle $\mathbf{\hat{K}}$ is derived by solving the optimization problem where $f(\mathbf{\hat{K}})$ is $\mathbb{E}_{\boldsymbol{\delta} \sim \mathcal{D}} [{J(\boldsymbol{\delta},\mathbf{\hat{K}})}]$ or $\max_{\lVert \boldsymbol{\delta} \rVert_{2} \leq 1}(J(\boldsymbol{\delta},\mathbf{\hat{K}}))$. Then, $\operatorname{SpRegret} (\mathbf{K}, \mathbf{\hat{K}}) \geq 0$ for

Figures (2)

  • Figure 1: Percentage of times a control policy outperforms the remaining three as a function of the maximum number of masses affected by uniformly distributed disturbances in the interval $[-0.5,1]$ in a system of $10$ masses. Shaded areas around the dotted lines represent the $95\%$ confidence interval around the corresponding mean values.
  • Figure 2: Percentage of times a control policy outperforms the remaining two as a function of the number of masses constituting the large-scale systems when all the masses are affected by uniformly distributed disturbances in the interval $[-0.5,1]$. Shaded areas around the dotted lines represent the $95\%$ confidence interval around the corresponding mean values.

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Definition 1
  • Proposition 1
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Proposition 2