On the optimal design of structured feedback gains for interconnected systems

We consider the design of optimal static feedback gains for interconnected systems subject to architectural constraints on the distributed controller. These constraints are in the form of sparsity requirements for the feedback matrix, which means that each controller has access to information from only a limited number of subsystems. We derive necessary conditions for the optimality of structured static feedback gains in the form of coupled matrix equations. In general these equations have multiple solutions, each of which is a stationary point of the objective function. For stable open-loop systems, we show that in the limit of expensive control, the optimal controller can be found analytically using perturbation techniques. We use this feedback gain to initialize homotopy-based gradient and Newton iterations that find an optimal solution to the original (non-expensive) control problem. For unstable open-loop systems, the centralized truncated gain is used as an initial estimate for the iterative schemes aimed at finding the optimal structured feedback gain. We consider both spatially invariant and spatially varying problems and illustrate our developments with several examples.