2020/09/25 by Shuang Gao, Aditya Mahajan
Computer Science · Engineering · Mathematics · #Adjacency matrix #Algebra over a field #Applied mathematics #Arithmetic #Combinatorics #Computer science #Coupling (piping) #Degeneracy (biology) #Discrete mathematics #Distributed Control Multi-Agent Systems #Eigendecomposition of a matrix #Eigenvalues and eigenvectors #Graph #Graph theory #Laplacian matrix #Mathematics #Matrix (chemical analysis) #Matrix decomposition #Neural Networks Stability and Synchronization #Nonlinear Dynamics and Pattern Formation #Notation #Physics #Pure mathematics #Spectral graph theory #Symmetric matrix #Topology (electrical circuits) #Voltage graph #cs.SY #eess.SY
paper · pdf · doi:10.1109/tcns.2021.3124259
published as IEEE Transactions on Control of Network Systems, 2021
openalex publication_date 2020/09/25 · arxiv created 2021/02/15 · arxiv updated 2021/11/05 · openalex created_date 2021/11/08 · openalex updated_date 2026/08/05
In this paper, we investigate optimal control of network-coupled subsystems where the dynamics and the cost couplings depend on an underlying undirected weighted graph. The graph coupling matrix in the dynamics may be the adjacency matrix, the Laplacian matrix, or any other symmetric matrix corresponding to the underlying graph. The cost couplings can be any polynomial function of the underlying coupling matrix. We use the spectral decomposition of the graph coupling matrix to decompose the overall system into (L+1) systems with decoupled dynamics and cost, where L is the rank of the coupling matrix. Furthermore, the optimal control input at each subsystem can be computed by solving (Ldist + 1) decoupled Riccati equations where Ldist (Ldist ≤ L) is the number of distinct non-zero eigenvalues of the coupling matrix. A salient feature of the result is that the solution complexity depends on the number of distinct eigenvalues of the coupling matrix rather than the size of the network. Therefore, the proposed solution framework provides a scalable method for synthesizing and implementing optimal control laws for large-scale network-coupled subsystems.