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Scenario Reduction with Guarantees for Stochastic Optimal Control of Linear Systems

Francesco Cordiano, Bart De Schutter

TL;DR

This work addresses the computational burden of scenario-based stochastic MPC for chance-constrained linear systems with additive, discrete uncertainty by introducing a problem-dependent clustering-based scenario reduction. It minimizes a loss that directly relates to the control objective and augments the reduced problem with constraint tightening and cost over-approximation to guarantee out-of-sample feasibility and performance against the original distribution. The authors prove that the tightened/over-approximated formulation overbounds the original problem and yields OOS guarantees with reduced computational load; the approach is validated through a numerical example showing favorable trade-offs between conservatism, performance, and computation. The proposed method offers a practical route to scalable stochastic control with rigorous guarantees, and it suggests extensions to nonlinear systems and more general uncertainty models.

Abstract

Scenario reduction algorithms can be an effective means to provide a tractable description of the uncertainty in optimal control problems. However, they might significantly compromise the performance of the controlled system. In this paper, we propose a method to compensate for the effect of scenario reduction on stochastic optimal control problems for chance-constrained linear systems with additive uncertainty. We consider a setting in which the uncertainty has a discrete distribution, where the number of possible realizations is large. We then propose a reduction algorithm with a problem-dependent loss function, and we define sufficient conditions on the stochastic optimal control problem to ensure out-of-sample guarantees (i.e., against the original distribution of the uncertainty) for the controlled system in terms of performance and chance constraint satisfaction. Finally, we demonstrate the effectiveness of the approach on a numerical example.

Scenario Reduction with Guarantees for Stochastic Optimal Control of Linear Systems

TL;DR

This work addresses the computational burden of scenario-based stochastic MPC for chance-constrained linear systems with additive, discrete uncertainty by introducing a problem-dependent clustering-based scenario reduction. It minimizes a loss that directly relates to the control objective and augments the reduced problem with constraint tightening and cost over-approximation to guarantee out-of-sample feasibility and performance against the original distribution. The authors prove that the tightened/over-approximated formulation overbounds the original problem and yields OOS guarantees with reduced computational load; the approach is validated through a numerical example showing favorable trade-offs between conservatism, performance, and computation. The proposed method offers a practical route to scalable stochastic control with rigorous guarantees, and it suggests extensions to nonlinear systems and more general uncertainty models.

Abstract

Scenario reduction algorithms can be an effective means to provide a tractable description of the uncertainty in optimal control problems. However, they might significantly compromise the performance of the controlled system. In this paper, we propose a method to compensate for the effect of scenario reduction on stochastic optimal control problems for chance-constrained linear systems with additive uncertainty. We consider a setting in which the uncertainty has a discrete distribution, where the number of possible realizations is large. We then propose a reduction algorithm with a problem-dependent loss function, and we define sufficient conditions on the stochastic optimal control problem to ensure out-of-sample guarantees (i.e., against the original distribution of the uncertainty) for the controlled system in terms of performance and chance constraint satisfaction. Finally, we demonstrate the effectiveness of the approach on a numerical example.
Paper Structure (10 sections, 3 theorems, 25 equations, 1 figure, 1 algorithm)

This paper contains 10 sections, 3 theorems, 25 equations, 1 figure, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Notation
  4. Problem statement
  5. Scenario reduction
  6. Enforcing out-of-sample guarantees
  7. Adaption of the constraint set
  8. Adaption of the cost function
  9. Numerical example
  10. CONCLUSIONS

Key Result

Lemma 1

Algorithm alg:red converges to a (suboptimal) solution of clus in a finite number of iterations. In particular, if $p^{(h)}=\frac{1}{M}, h=1,...,M$, the update for $\Tilde{\boldsymbol{\eta}}^{(j)}$ in step 3) in Algorithm alg:red corresponds to:

Figures (1)

  • Figure 1: Performance analysis for the real system. In particular, we notice that P2 over-approximates the original problem \ref{['sbocp']} both in terms of constraint satisfaction and performance, and the associated computational cost is comparable to P1.

Theorems & Definitions (9)

  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 3