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Distributionally Robust Control for Chance-Constrained Signal Temporal Logic Specifications

Arash Bahari Kordabad, Eleftherios E. Vlahakis, Lars Lindemann, Dimos V. Dimarogonas, Sadegh Soudjani

TL;DR

This work addresses probabilistic STL satisfaction for discrete-time stochastic linear systems with unknown disturbance distributions. It leverages Lipschitz STL predicates and a concentration-of-measure property to convert a chance-constrained problem into an expectation-based surrogate, then applies Wasserstein DRO to handle distributional ambiguity with finite-sample guarantees. A data-driven reformulation eliminates decision variables in the probability space while preserving feasibility and providing out-of-sample performance bounds. A numerical case study demonstrates robust STL satisfaction under unknown disturbances and highlights improved robustness and efficiency relative to sample-average approaches. The framework offers a practical path for enforcing temporal logic specifications in uncertain environments.

Abstract

We consider distributionally robust optimal control of stochastic linear systems under signal temporal logic (STL) chance constraints when the disturbance distribution is unknown. By assuming that the underlying predicate functions are Lipschitz continuous and the noise realizations are drawn from a distribution having a concentration of measure property, we first formulate the underlying chance-constrained control problem as stochastic programming with constraints on expectations and propose a solution using a distributionally robust approach based on the Wasserstein metric. We show that by choosing a proper Wasserstein radius, the original chance-constrained optimization can be satisfied with a user-defined confidence level. A numerical example illustrates the efficacy of the method.

Distributionally Robust Control for Chance-Constrained Signal Temporal Logic Specifications

TL;DR

This work addresses probabilistic STL satisfaction for discrete-time stochastic linear systems with unknown disturbance distributions. It leverages Lipschitz STL predicates and a concentration-of-measure property to convert a chance-constrained problem into an expectation-based surrogate, then applies Wasserstein DRO to handle distributional ambiguity with finite-sample guarantees. A data-driven reformulation eliminates decision variables in the probability space while preserving feasibility and providing out-of-sample performance bounds. A numerical case study demonstrates robust STL satisfaction under unknown disturbances and highlights improved robustness and efficiency relative to sample-average approaches. The framework offers a practical path for enforcing temporal logic specifications in uncertain environments.

Abstract

We consider distributionally robust optimal control of stochastic linear systems under signal temporal logic (STL) chance constraints when the disturbance distribution is unknown. By assuming that the underlying predicate functions are Lipschitz continuous and the noise realizations are drawn from a distribution having a concentration of measure property, we first formulate the underlying chance-constrained control problem as stochastic programming with constraints on expectations and propose a solution using a distributionally robust approach based on the Wasserstein metric. We show that by choosing a proper Wasserstein radius, the original chance-constrained optimization can be satisfied with a user-defined confidence level. A numerical example illustrates the efficacy of the method.
Paper Structure (13 sections, 5 theorems, 39 equations, 1 figure)

This paper contains 13 sections, 5 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Discrete-Time Stochastic Linear Systems
  4. STL specifications
  5. Chance-constrained optimization
  6. Concentration of measure property
  7. Solution Approach
  8. Case study
  9. Conclusions
  10. Proof of Lemma \ref{['lem:lip:minmax']}
  11. Proof of Theorem \ref{['the:lip']}
  12. Proof of Theorem \ref{['the:EP']}
  13. Proof of Theorem \ref{['the:data:DRO']}

Key Result

Lemma 1

For any two Lipschitz functions $f_1:X\rightarrow\mathbb R$ and $f_2:X\rightarrow\mathbb R$, $\max(f_1, f_2)$ and $\min(f_1, f_2)$ are Lipschitz functions with $L := \max(L_1, L_2)$, where $L_i$ is the Lipschitz constant of $f_i$, $i\in\{1,2\}$.

Figures (1)

  • Figure 1: System trajectories for different realizations for deterministic system (black), ECP solution using the sample average approximation (green), and the proposed DRP solution (blue).

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 1
  • Lemma 2
  • ...and 5 more