Distributed and Localized Covariance Control of Coupled Systems: A System Level Approach

TL;DR

This work addresses finite-horizon covariance steering for dynamically coupled, stochastic linear systems under local communication constraints. It recasts the localized CS problem in the System Level Synthesis (SLS) framework, enabling affine, convex locality constraints on system responses $\boldsymbol{\Phi}_x$ and $\boldsymbol{\Phi}_u$ and a convex terminal covariance condition; it further introduces a transformation to handle nonseparable objectives and employs a consensus ADMM to distribute computation. The main contributions are (i) a convex SLS formulation of localized CS with locality, (ii) a transformation that restores separability for general instances, and (iii) a provably convergent distributed algorithm that leverages only local information and neighbor communications. Numerical validation on a 36-subsystem power network demonstrates convergence of the state covariance to the target and feasibility under localized information exchange. This work enables scalable, distributed control of large, interconnected networks subject to Gaussian disturbances, with practical relevance to power grids and other networked systems.

Abstract

This work is concerned with the finite-horizon optimal covariance steering of networked systems governed by discrete-time stochastic linear dynamics. In contrast with existing work that has only considered systems with dynamically decoupled agents, we consider a dynamically coupled system composed of interconnected subsystems subject to local communication constraints. In particular, we propose a distributed algorithm to compute the localized optimal feedback control policy for each individual subsystem, which depends only on the local state histories of its neighboring subsystems. Utilizing the system-level synthesis (SLS) framework, we first recast the localized covariance steering problem as a convex SLS problem with locality constraints. Subsequently, exploiting its partially separable structure, we decompose the latter problem into smaller subproblems, introducing a transformation to deal with nonseparable instances. Finally, we employ a variation of the consensus alternating direction method of multipliers (ADMM) to distribute computation across subsystems on account of their local information and communication constraints. We demonstrate the effectiveness of our proposed algorithm on a power system with 36 interconnected subsystems.

Figures (2)

• Figure 1: Network topology of the $6 \times 6$ power grid.
• Figure 2: Results for the simulation example with $d = 1$.

Theorems & Definitions (13)

• Definition 1: $d$-Hop Neighbors
• Remark 1
• Remark 2
• Theorem 1
• Remark 3
• Definition 2: $d$-Localized System Responses
• Definition 3: $d$-Locality Constraints
• Proposition 1
• proof
• Remark 4