Distributed and Constrained $ \mathcal{H}_2 $ Control Design via System Level Synthesis and Dual Consensus ADMM

Abstract

Design of optimal distributed linear feedback controllers to achieve a desired aggregate behavior, while simultaneously satisfying state and input constraints, is a challenging but important problem in many applications. System level synthesis is a recent technique which has been used to reparametrize the optimal control problem as a convex program. Prior work on system level synthesis with state and input constraints has included closed-loop finite impulse response and locality constraints or, in the case where these constraints were lifted using a simple pole approximation, only a centralized design was considered. However, closed-loop finite impulse response and locality constraints cannot be satisfied in many applications. Furthermore, the centralized design using the simple pole approximation lacks robustness to communication failures and disturbances, has high computational cost and does not preserve data privacy of local controllers. The main contribution of this work is to develop a distributed solution to system level synthesis with the simple pole approximation in order to incorporate state and input constraints without closed-loop finite impulse response or locality constraints, and in a distributed implementation. To achieve this, it is first shown that the dual of this problem is a distributed consensus problem. Then, an algorithm is developed based on the alternating direction method of multipliers to solve the dual while recovering a primal solution, and a convergence certificate is provided. Finally, the method's performance is demonstrated on a test case of control design for distributed energy resources that collectively provide stability services to the power grid.