[Paper Review] Stochastic Assume-Guarantee Contracts for Cyber-Physical System Design Under Probabilistic Requirements

Jiwei Li, Pierluigi Nuzzo+3 more|arXiv (Cornell University)|May 25, 2017

Formal Methods in Verification18 references3 citations

TL;DR

This paper proposes a stochastic assume-guarantee contract framework for cyber-physical systems using Stochastic Signal Temporal Logic (StSTL) to specify probabilistic requirements. It formulates contract compatibility, consistency, and refinement as mixed-integer programs with conservative approximations of probabilistic constraints, enabling efficient verification and stochastic model predictive control (MPC) synthesis for linear stochastic systems, demonstrated on aircraft power distribution networks with 95% satisfaction rates for battery level constraints.

ABSTRACT

We develop an assume-guarantee contract framework for the design of cyber-physical systems, modeled as closed-loop control systems, under probabilistic requirements. We use a variant of signal temporal logic, namely, Stochastic Signal Temporal Logic (StSTL) to specify system behaviors as well as contract assumptions and guarantees, thus enabling automatic reasoning about requirements of stochastic systems. Given a stochastic linear system representation and a set of requirements captured by bounded StSTL contracts, we propose algorithms that can check contract compatibility, consistency, and refinement, and generate a controller to guarantee that a contract is satisfied, following a stochastic model predictive control approach. Our algorithms leverage encodings of the verification and control synthesis tasks into mixed integer optimization problems, and conservative approximations of probabilistic constraints that produce both sound and tractable problem formulations. We illustrate the effectiveness of our approach on a few examples, including the design of embedded controllers for aircraft power distribution networks.

Motivation & Objective

• To address the lack of formal contract frameworks for stochastic cyber-physical systems with probabilistic requirements.
• To enable early detection of integration errors in complex CPS designs through compositional verification.
• To support robust, non-worst-case designs by expressing requirements as probabilistic constraints rather than deterministic worst-case bounds.
• To develop scalable, tractable algorithms for contract operations and controller synthesis in the presence of uncertainty.
• To demonstrate the framework on real-world applications such as aircraft electric power distribution systems.

Proposed method

• The paper uses Stochastic Signal Temporal Logic (StSTL) to formally specify system behaviors, contract assumptions, and guarantees with probabilistic predicates.
• It formulates contract verification tasks—compatibility, consistency, and refinement—as mixed-integer optimization problems (MIPs) via conservative approximations of probabilistic constraints.
• The approach leverages a mixed-integer linear under-approximation of StSTL formulas to handle complex constraints, such as transient behaviors and battery level thresholds.
• A stochastic Model Predictive Control (MPC) scheme is synthesized at 10-second intervals, using the MIP formulation to generate control actions that satisfy the contracts.
• The framework integrates hierarchical control: a high-level LMS (HL-LMS) computes optimal configurations every 10 seconds, while a low-level LMS (LL-LMS) handles immediate responses to failures.
• The system uses atomic proposition encodings such as $(0.3 - B_j)^{[0.95]}$ to represent probabilistic constraints on battery charge levels with 95% confidence.

Experimental results

Research questions

• RQ1 Can a formal assume-guarantee contract framework be developed for stochastic linear systems under probabilistic requirements?
• RQ2 How can probabilistic constraints in system contracts be encoded efficiently and soundly for verification and control synthesis?
• RQ3 Can contract operations like compatibility, consistency, and refinement be reduced to tractable optimization problems?
• RQ4 To what extent can the framework support real-time control synthesis in safety-critical CPS applications?
• RQ5 How does the framework compare to prior approaches in handling complex transient behaviors and dynamic failures?

Key findings

• The framework successfully verified the consistency of a system-level contract $C_S$ for an aircraft power distribution system using Theorem 5.
• The stochastic MPC scheme achieved a battery level satisfaction rate of over 95% across 500 simulation runs, satisfying the probabilistic constraint $(B_j \geq 0.3)^{[0.95]}$.
• The battery charge level remained above 0.3 for most of the simulation after time 0, confirming the effectiveness of constraint (36).
• In 50 simulation runs, the battery level climbed above 0.4 before time 5 when starting from an initial charge of 0.225, consistent with requirement (37).
• The LL-LMS responded immediately to generator failure at time 34 by connecting GEN3 to bus 2 and shedding loads, while the HL-LMS detected the event at time 40 and recomputed an optimal configuration.
• The optimized configuration reduced load shedding and improved resource utilization by enabling battery charging and power sharing between GEN1 and GEN3, with switching frequency kept within requirements.