Sparsity-promoting optimal control of spatially-invariant systems

We study the optimal design of sparse and block sparse feedback gains for spatially-invariant systems on a circle. For this class of systems, the state-space matrices are jointly diagonalizable via the discrete Fourier transform. We exploit this structure to develop an ADMM-based algorithm that significantly reduces the computational complexity relative to standard approaches. Specifically, the complexity of the developed algorithm scales linearly with the number of subsystems. This is in contrast to a cubic scaling when circulant structure is not exploited. Two examples are provided to illustrate the effectiveness of the developed approach.