Data-Driven Distributionally Robust Control for Interacting Agents under Logical Constraints

TL;DR

The paper tackles robust control synthesis for interacting stochastic agents under signal temporal logic (STL) constraints, formulating the problem as a chance-constrained program (CCP) that must hold under uncertainty tubes from uncontrollable agents. It develops two CCP-to-ECP reformulations using concentration of measure (CoM) and conditional value at risk (CVaR), and augments them with a distributionally robust optimization (DRO) layer based on Wasserstein ambiguity to provide finite-sample guarantees with unknown disturbance distributions. To solve the resulting non-smooth, data-driven problems, it employs smooth min/max approximations and a sequential quadratic programming (SQP) framework, along with data-driven reformulations that ensure probabilistic satisfaction guarantees. Case studies compare CoM and CVaR approaches and demonstrate robustness benefits of the DRO extension in a two-agent scenario, highlighting trade-offs between conservativeness, feasibility, and performance under limited data.

Abstract

In this paper, we propose a distributionally robust control synthesis for an agent with stochastic dynamics that interacts with other agents under uncertainties and constraints expressed by signal temporal logic (STL). We formulate the control synthesis as a chance-constrained program (CCP) with STL specifications that must be satisfied with high probability under all uncertainty tubes induced by the other agents. To tackle the CCP, we propose two methods based on concentration of measure (CoM) theory and conditional value at risk (CVaR) and compare the required assumptions and resulting optimizations. These approaches convert the CCP into an expectation-constrained program (ECP), which is simpler to solve than the original CCP. To estimate the expectation using a finite set of observed data, we adopt a distributionally robust optimization (DRO) approach. The underlying DRO can be approximated as a robust data-driven optimization that provides a probabilistic under-approximation to the original ECP, where the probability depends on the number of samples. Therefore, under feasibility, the original STL constraints are satisfied with two layers of designed confidence: the confidence of the chance constraint and the confidence of the approximated data-driven optimization, which depends on the number of samples. We then provide details on solving the resulting robust data-driven optimization numerically. Finally, we compare the two proposed approaches through case studies.

Figures (12)

• Figure 1: The controlled agent in blue, aims to reach the location indicated by a green circle while avoiding collisions with the other agent. The other agent in black is modeled as an uncontrollable agent with an uncertainty tube from the perspective of the controlled agent. The red color indicates collisions along the path of the controlled agent indicated by blue circles.
• Figure 2: Illustration of the different risk measures $\mathbb E[R]$, $\mathrm{VaR}_{1-\varepsilon}$ and $\mathrm{CVaR}_{1-\varepsilon}$ for random variable $R$ with probability density function $\mathrm{f}_R$.
• Figure 3: Feasible domains for the original CCP (green), the CVaR-based approximation (blue), and the CoM-based approximation (red).
• Figure 4: Feasible domain for the DRP-CoM (in red) and DRP-CVaR (in blue) optimizations for varying probabilistic thresholds $\varepsilon$ and fixed Wasserstein radius $r=1$ (top), and for varying Wasserstein radii $r$ and fixed probabilistic threshold $\varepsilon=0.6$ (bottom). It can be observed that, depending on the problem parameters such as $\varepsilon$ and $r$, either the CVaR or the CoM approach may be employed effectively.
• Figure 5: Comparison of the objective function distribution for CoM (top) and CVaR (bottom) with their respective DRP solutions.

Theorems & Definitions (17)

• Remark 1
• Remark 2
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1
• Proposition 1
• proof
• Theorem 1