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Robust H-infinity control under stochastic requirements: minimizing conditional value-at-risk instead of worst-case performance

Ervan Kassarian, Francesco Sanfedino, Daniel Alazard, Andrea Marrazza

TL;DR

The paper addresses excessive conservatism in traditional $\mathcal{H}_\infty$ robustness by replacing worst-case minimization with a stochastic CVAR criterion, modeling uncertainty as random via density $p(\delta)$ and optimizing tail risk rather than absolute worst performance. It develops a convex reformulation using $F^{zw}_{\beta}(k,\alpha)$ to compute $\mathrm{CVAR}_{\beta}^{zw}(k)$ and $\mathrm{VAR}_{\beta}^{zw}(k)$, and applies a sample-average approximation to enable tractable optimization, with an interval-of-confidence framework to quantify estimation accuracy. The optimization procedure combines active-configuration stability with SAA-based nonsmooth optimization, enabling iterative refinement and stability enforcement across the uncertainty set. Applied to a realistic spacecraft attitude-control benchmark, the stochastic CVAR approach yields large gains in nominal and typical performance while tolerating rare worst-case degradations, demonstrating reduced conservatism and practical scalability for stochastic robust control problems.

Abstract

Conventional robust $\mathcal H_2/\mathcal H_\infty$ control minimizes the worst-case performance, often leading to a conservative design driven by very rare parametric configurations. To reduce this conservatism while taking advantage of the stochastic properties of Monte-Carlo sampling and its compatibility with parallel computing, we introduce an alternative paradigm that optimizes the controller with respect to a stochastic criterion, namely the conditional value at risk. We illustrate the potential of this approach on a realistic satellite benchmark, showing that it can significantly improve overall performance by tolerating some degradation in very rare worst-case scenarios.

Robust H-infinity control under stochastic requirements: minimizing conditional value-at-risk instead of worst-case performance

TL;DR

The paper addresses excessive conservatism in traditional robustness by replacing worst-case minimization with a stochastic CVAR criterion, modeling uncertainty as random via density and optimizing tail risk rather than absolute worst performance. It develops a convex reformulation using to compute and , and applies a sample-average approximation to enable tractable optimization, with an interval-of-confidence framework to quantify estimation accuracy. The optimization procedure combines active-configuration stability with SAA-based nonsmooth optimization, enabling iterative refinement and stability enforcement across the uncertainty set. Applied to a realistic spacecraft attitude-control benchmark, the stochastic CVAR approach yields large gains in nominal and typical performance while tolerating rare worst-case degradations, demonstrating reduced conservatism and practical scalability for stochastic robust control problems.

Abstract

Conventional robust control minimizes the worst-case performance, often leading to a conservative design driven by very rare parametric configurations. To reduce this conservatism while taking advantage of the stochastic properties of Monte-Carlo sampling and its compatibility with parallel computing, we introduce an alternative paradigm that optimizes the controller with respect to a stochastic criterion, namely the conditional value at risk. We illustrate the potential of this approach on a realistic satellite benchmark, showing that it can significantly improve overall performance by tolerating some degradation in very rare worst-case scenarios.

Paper Structure

This paper contains 14 sections, 2 theorems, 20 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Computation of VAR and CVAR
  4. Formulation as an optimization problem
  5. Sample average approximation (SAA)
  6. Interval of confidence
  7. Optimization
  8. Optimization procedure
  9. Open challenges
  10. Application
  11. Benchmark
  12. Formulation as a deterministic control problem
  13. Formulation as a stochastic control problem
  14. Conclusion

Key Result

Proposition 1

Under Assumptions assumption_bounded, assumption_stability and assumption, let us define the function where $[x]_+ = \text{max}(x, 0)$. The $\beta$-CVAR of the loss function $L_{zw}(k,\delta)$ is the minimum of $F_\beta(k, \alpha)$ as a function of $\alpha$: and the $\beta$-VAR is the left-end value of the set of all $\alpha$ that attain this minimum (possibly reducing to a single point): Moreo

Figures (5)

  • Figure 1: LFR representation of $T_{zw}(\mathrm s, k,\delta)$
  • Figure 2: Probability density $p^{zw}_k(\alpha)$ of the loss function $|| T_{zw}(\mathrm s, k,\delta) ||$ with $\beta$-VAR, $\beta$-CVAR and deterministic worst case (only defined under Assumption \ref{['assumption_bounded']}).
  • Figure 3: Benchmark: closed-loop system
  • Figure 4: Empirical probability densities with solution $\hat{k}_{det}$
  • Figure 5: Empirical probability densities with solution $\hat{k}_{sto}$

Theorems & Definitions (3)

  • Definition 1: $\beta$-VAR, $\beta$-CVAR
  • Proposition 1: Expression of $\beta$-VAR and $\beta$-CVAR as solutions of a convex minimization problem Rockafellar2000
  • Corollary 1: Optimal controller for $\beta$-CVAR minimization