Sparsity-promoting optimal control of systems with symmetries, consensus and synchronization networks

Optimal control problems in systems with symmetries and consensus/synchronization networks are characterized by structural constraints that arise either from the underlying group structure or the lack of absolute measurements for part of the state vector. Our objective is to design controller structures and resulting control strategies that utilize limited information exchange between subsystems in large-scale networks. To obtain controllers with low communication requirements, we seek solutions to regularized versions of the H2 optimal control problem. Non-smooth regularization terms are introduced to tradeoff network performance with sparsity of the feedback-gain matrix. In contrast to earlier results, our framework allows the state-space representations that are used to quantify the system's performance and sparsity of the static output-feedback controller to be expressed in different sets of coordinates. We show how alternating direction method of multipliers can be leveraged to exploit the underlying structure and compute sparsity-promoting controllers. In particular, for spatially-invariant systems, the computational complexity of our algorithm scales linearly with the number of subsystems. We also identify a class of optimal control problems that can be cast as semidefinite programs and provide an example to illustrate our developments.