Damping With Varying Regularization in Optimal Decentralized Control

We study a homotopy continuation method for the design of an optimal static or dynamic decentralized controller to minimize a quadratic cost functional. The proposed method involves a combination of the classical local search technique in the space of control policies, a gradual damping of the system dynamics, and a gradual variation of a parameterized cost functional. A series of optimal decentralized control (ODC) problems is generated via a continuous variation of parameters. Unlike the classical homotopy literature, which focuses on tracking a specific trajectory, we study the ensemble of critical controller trajectories and show how the properties of the ensemble can be leveraged to find a globally optimal solution of the ODC problem. After guaranteeing the continuity and asymptotic properties of the proposed method, we prove that with enough damping, there is no spurious locally optimal controller for a block-diagonal control structure. This leads to a sufficient condition under which an iterative algorithm can find a global solution to a class of optimal decentralized control problems. The "damping property" introduced in this analysis is shown to be unique for general system matrices. Empirical observations are presented for instances with an exponential number of locally optimal decentralized controllers, where the developed method could find the global solution even when initialized at a poor local solution.