Distributed Design for Decentralized Control Using Chordal Decomposition and ADMM

Abstract

We propose a distributed design method for decentralized control by exploiting the underlying sparsity properties of the problem. Our method is based on chordal decomposition of sparse block matrices and the alternating direction method of multipliers (ADMM). We first apply a classical parameterization technique to restrict the optimal decentralized control into a convex problem that inherits the sparsity pattern of the original problem. The parameterization relies on a notion of strongly decentralized stabilization, and sufficient conditions are discussed to guarantee this notion. Then, chordal decomposition allows us to decompose the convex restriction into a problem with partially coupled constraints, and the framework of ADMM enables us to solve the decomposed problem in a distributed fashion. Consequently, the subsystems only need to share their model data with their direct neighbours, not needing a central computation. Numerical experiments demonstrate the effectiveness of the proposed method.

Controllability Analysis and Optimal Control of Mixed Traffic Flow with Human-driven and Autonomous Vehicles

“…Formulation (24) is known as the structured optimal control problem [34], [42]. This problem is in general non-convex and computationally hard to find a globally optimal solution.…”

“…The structured optimal control problem (24) (or its variants) has attracted some attention in the literature. A few methods have been proposed to find an approximation solution, such as using convex approximations [42], or directly employing nonconvex optimization techniques [34]. However, many existing methods require that the system is completely controllable, and therefore they are not applicable to our problem since the mixed traffic system is not completely controllable, as proved in Theorem 1.…”

“…One key step in ( 31) is to design two sparsity patterns T and S that satisfy (30) for a given K. A typical method, e.g., [42], is to assume X to be block diagonal and Z ∈ K; see Fig. 5a for illustration.…”

Distributed Optimization for Control and Estimation

“…These systems comprise a significant number of interconnected sub-systems which cooperate with each other to achieve a desired objective. Therefore, using a centralized optimization approach will require a considerable number of communications due to the distributed nature of them which is not desired, (Yang et al, 2019). The third limitation is the incapability of the centralized optimization approaches in handling applications with structural constraints.…”

“…The applications of distributed optimization in control and estimation are very broad, e.g., wireless sensor networks, (Biswas et al, 2006;Simonetto and Leus, 2014;Wang et al, 2008), power systems, (Kargarian et al, 2016;Molzahn et al, 2017), autonomous 1 Introduction vehicles, (Bian et al, 2019;Cao et al, 2012;Khayatian et al, 2020;Malikopoulos et al, 2018;Tajalli and Hajbabaie, 2018), machine learning, (Boyd et al, 2011;Nedic, 2020), robotic networks, (Bullo et al, 2008), communication networks, (Johansson, 2008Sayed, 2014), and multi-agent systems, (Fioretto et al, 2018;Ren and Cao, 2011;Zhu and Martínez, 2015), to name a few. For a thorough and recent review of distributed optimization algorithms, refer to Boyd et al (2011);Molzahn et al (2017); Nedić and Liu (2018); Nedich et al (2015); Yang and Johansson (2010); Yang et al (2019).…”