Computational Methods for Distributed Control of Heterogeneous Cyclic Interconnection Structures

We address the problem of fast system analysis and controller synthesis for heterogeneous subsystems interconnected in a loop. Such distributed systems have state-space realizations with a special matrix structure, which we show to be a generalization of the Sequentially Semi-Separable (SSS) structure employed in recent research on systems interconnected in a Cartesian array. By extending the O(N) structure-preserving SSS arithmetic to the matrices induced by this circular type of interconnection, we introduce a new cyclic SSS matrix structure and arithmetic which leads to fast and efficient procedures for linear computational complexity optimal distributed controller synthesis for arbitrarily heterogeneous subsystems connected in a loop. In the homogeneous case, where all subsystems are identical, the computational complexity reduces to O(1) and an interesting relationship with the infinite case is demonstrated. CSSS matrices can be non-Toeplitz (as compared to circulant matrices), a great expansion of domain, but when restricted to be Toeplitz (and thus circulant), the arithmetic reduces to O(1), a huge cost-savings. The procedures are demonstrated on two computational examples, using a freely available MATLAB toolbox implementation of these algorithms.