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Probabilistic Tube-based Control Synthesis of Stochastic Multi-Agent Systems under Signal Temporal Logic

Eleftherios E. Vlahakis, Lars Lindemann, Pantelis Sopasakis, Dimos V. Dimarogonas

TL;DR

This work tackles stochastic discrete-time linear MASs subject to a global STL specification that must be satisfied with probability θ. It decomposes each agent’s dynamics into a deterministic part and an additive error, and constructs a probabilistic reachable tube (PRT) as the Cartesian product of per-agent error reachables driven by disturbances confined to confidence regions; a probabilistic bound links the PRT to the STL satisfaction probability, enabling constraint tightening that yields a tractable deterministic problem. A distributed, recursively feasible control synthesis solves agent-level subproblems under scheduling, exploiting a clique-based STL decomposition to manage joint tasks. The approach demonstrates scalability on a ten-agent example where centralized methods are impractical, and highlights a trade-off between horizon-induced conservatism and computational efficiency, with future work aimed at data-driven tightening to reduce conservatism.

Abstract

We consider the control design of stochastic discrete-time linear multi-agent systems (MASs) under a global signal temporal logic (STL) specification to be satisfied at a predefined probability. By decomposing the dynamics into deterministic and error components, we construct a probabilistic reachable tube (PRT) as the Cartesian product of reachable sets of the individual error systems driven by disturbances lying in confidence regions (CRs) with a fixed probability. By bounding the PRT probability with the specification probability, we tighten all state constraints induced by the STL specification by solving tractable optimization problems over segments of the PRT, and relax the underlying stochastic problem with a deterministic one. This approach reduces conservatism compared to tightening guided by the STL structure. Additionally, we propose a recursively feasible algorithm to attack the resulting problem by decomposing it into agent-level subproblems, which are solved iteratively according to a scheduling policy. We demonstrate our method on a ten-agent system, where existing approaches are impractical.

Probabilistic Tube-based Control Synthesis of Stochastic Multi-Agent Systems under Signal Temporal Logic

TL;DR

This work tackles stochastic discrete-time linear MASs subject to a global STL specification that must be satisfied with probability θ. It decomposes each agent’s dynamics into a deterministic part and an additive error, and constructs a probabilistic reachable tube (PRT) as the Cartesian product of per-agent error reachables driven by disturbances confined to confidence regions; a probabilistic bound links the PRT to the STL satisfaction probability, enabling constraint tightening that yields a tractable deterministic problem. A distributed, recursively feasible control synthesis solves agent-level subproblems under scheduling, exploiting a clique-based STL decomposition to manage joint tasks. The approach demonstrates scalability on a ten-agent example where centralized methods are impractical, and highlights a trade-off between horizon-induced conservatism and computational efficiency, with future work aimed at data-driven tightening to reduce conservatism.

Abstract

We consider the control design of stochastic discrete-time linear multi-agent systems (MASs) under a global signal temporal logic (STL) specification to be satisfied at a predefined probability. By decomposing the dynamics into deterministic and error components, we construct a probabilistic reachable tube (PRT) as the Cartesian product of reachable sets of the individual error systems driven by disturbances lying in confidence regions (CRs) with a fixed probability. By bounding the PRT probability with the specification probability, we tighten all state constraints induced by the STL specification by solving tractable optimization problems over segments of the PRT, and relax the underlying stochastic problem with a deterministic one. This approach reduces conservatism compared to tightening guided by the STL structure. Additionally, we propose a recursively feasible algorithm to attack the resulting problem by decomposing it into agent-level subproblems, which are solved iteratively according to a scheduling policy. We demonstrate our method on a ten-agent system, where existing approaches are impractical.
Paper Structure (16 sections, 7 theorems, 14 equations, 3 figures, 1 algorithm)

This paper contains 16 sections, 7 theorems, 14 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Notation and Preliminaries
  4. Multi-agent system
  5. Dynamics
  6. Disturbance
  7. STL specification
  8. Problem statement
  9. Main results
  10. Error dynamics and construction of probabilistic tubes
  11. Constraint tightening
  12. Distributed control synthesis
  13. Decomposition of STL formula $\psi$
  14. Iterative algorithm
  15. Example
  16. ...and 1 more sections

Key Result

Lemma 1

Let $\mathcal{X}_i,\mathcal{Y}_i$$\subseteq {\rm I\!R}^{n_i}$, $i\in{\rm I\!N}_{[1,M]}$. Then, $(\mathcal{X}_1\times \cdots \times \mathcal{X}_M) \oplus (\mathcal{Y}_1\times \cdots \times \mathcal{Y}_M)=(\mathcal{X}_1\oplus \mathcal{Y}_1)\times \cdots\times (\mathcal{X}_M\oplus \mathcal{Y}_M)$.

Figures (3)

  • Figure 1: The ten agents of the MAS and the cliques $\mathcal{K}_\phi$ in $\phi$.
  • Figure 2: Nominal trajectories (solid lines) by Alg. \ref{['alg:control_syn']}, with initial states marked by crosses. Tubes (transparent covers) around trajectories, at probability levels $\Theta_i = 0.965$.
  • Figure 3: Compute times (log scale) for solving \ref{['eq:multi_agent_problem_tight']} via Alg. \ref{['alg:control_syn']} and a centralized approach for varying agent numbers.

Theorems & Definitions (19)

  • Lemma 1: Distributivity of Minkowski sum
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • ...and 9 more