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Differentially-Private Distributed Model Predictive Control of Linear Discrete-Time Systems with Global Constraints

Kaixiang Zhang, Yongqiang Wang, Ziyou Song, Zhaojian Li

TL;DR

This work addresses DMPC for linear discrete-time systems with coupled global constraints while preserving subsystem privacy. It integrates a differential-privacy noise mechanism into a distributed dual-gradient DMPC framework, and employs diminishing weakening and step-size sequences to mitigate DP-noise effects, achieving almost-sure convergence to the global optimum and provable $ε$-differential privacy with a finite privacy budget. An implementation strategy ensures recursive feasibility and stability of the closed-loop system, including a privacy-preserving consensus step to verify global constraints. Numerical results on a four-subsystem example demonstrate reduced variance and constraint satisfaction under privacy preservation compared to non-private alternatives, validating practicality. The approach offers a scalable, privacy-aware DMPC solution with theoretical guarantees and practical feasibility for large-scale, networked control systems.

Abstract

Distributed model predictive control (DMPC) has attracted extensive attention as it can explicitly handle system constraints and achieve optimal control in a decentralized manner. However, the deployment of DMPC strategies generally requires the sharing of sensitive data among subsystems, which may violate the privacy of participating systems. In this paper, we propose a differentially-private DMPC algorithm for linear discrete-time systems subject to coupled global constraints. Specifically, we first show that a conventional distributed dual gradient algorithm can be used to address the considered DMPC problem but cannot provide strong privacy preservation. Then, to protect privacy against the eavesdropper, we incorporate a differential-privacy noise injection mechanism into the DMPC framework and prove that the resulting distributed optimization algorithm can ensure both provable convergence to a global optimal solution and rigorous $ε$-differential privacy. In addition, an implementation strategy of the DMPC is designed such that the recursive feasibility and stability of the closed-loop system are guaranteed. Simulation results are provided to demonstrate the effectiveness of the developed approach.

Differentially-Private Distributed Model Predictive Control of Linear Discrete-Time Systems with Global Constraints

TL;DR

This work addresses DMPC for linear discrete-time systems with coupled global constraints while preserving subsystem privacy. It integrates a differential-privacy noise mechanism into a distributed dual-gradient DMPC framework, and employs diminishing weakening and step-size sequences to mitigate DP-noise effects, achieving almost-sure convergence to the global optimum and provable -differential privacy with a finite privacy budget. An implementation strategy ensures recursive feasibility and stability of the closed-loop system, including a privacy-preserving consensus step to verify global constraints. Numerical results on a four-subsystem example demonstrate reduced variance and constraint satisfaction under privacy preservation compared to non-private alternatives, validating practicality. The approach offers a scalable, privacy-aware DMPC solution with theoretical guarantees and practical feasibility for large-scale, networked control systems.

Abstract

Distributed model predictive control (DMPC) has attracted extensive attention as it can explicitly handle system constraints and achieve optimal control in a decentralized manner. However, the deployment of DMPC strategies generally requires the sharing of sensitive data among subsystems, which may violate the privacy of participating systems. In this paper, we propose a differentially-private DMPC algorithm for linear discrete-time systems subject to coupled global constraints. Specifically, we first show that a conventional distributed dual gradient algorithm can be used to address the considered DMPC problem but cannot provide strong privacy preservation. Then, to protect privacy against the eavesdropper, we incorporate a differential-privacy noise injection mechanism into the DMPC framework and prove that the resulting distributed optimization algorithm can ensure both provable convergence to a global optimal solution and rigorous -differential privacy. In addition, an implementation strategy of the DMPC is designed such that the recursive feasibility and stability of the closed-loop system are guaranteed. Simulation results are provided to demonstrate the effectiveness of the developed approach.
Paper Structure (14 sections, 7 theorems, 36 equations, 1 figure, 4 algorithms)

This paper contains 14 sections, 7 theorems, 36 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Problem Description
  4. Distributed Dual-Gradient Method
  5. On Differential Privacy
  6. Differentially-Private Distributed Dual-Gradient Algorithm
  7. Algorithm Description
  8. Convergence Analysis
  9. Privacy Analysis
  10. Implementation of Privacy-Preserving DMPC
  11. Algorithm Implementation
  12. Feasibility and Stability
  13. Numerical Simulations
  14. Conclusion

Key Result

Theorem 1

Suppose Assumptions assump:system, assump:L, and assump:dp-noise hold. If the non-negative weakening factor sequence $\{\chi^k\}$ and the step-size sequence $\{\gamma^k\}$ in Algorithm algo:dual_privacy satisfy $\sum_{k=0}^{\infty}\chi^k=\infty$, $\sum_{k=0}^{\infty}(\chi^k)^2<\infty$, and $\sum_{k=

Figures (1)

  • Figure 1: Evolution of (a) subsystem 1 and (b) global constraint.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2: $\epsilon$-differential privacy, Huang2015
  • Remark 1
  • Theorem 1
  • proof
  • Lemma 1: Lemma 11, Polyak1987
  • Theorem 2
  • proof
  • Remark 2
  • Definition 3
  • ...and 8 more