Convexification of Optimal Decentralized Control Without a Stabilizing Controller

The problem of finding an optimal decentralized controller is considered, where both the plant and the controllers under consideration are rational. It has been shown that a condition called quadratic invariance, which relates the plant and the constraints imposed on the desired controller, allows the optimal decentralized control problem to be cast as a convex optimization problem, provided that a controller is given which is both stable and stabilizing. This paper shows how, even when such a controller is not provided, the optimal decentralized control problem may still be cast as a convex optimization problem, albeit a more complicated one. The solution of the resulting convex problem is then discussed. The result that quadratic invariance gives convexity is thus extended to all finite-dimensional linear problems. In particular, this result may now be used for plants which are not strongly stabilizable, or for which a stabilizing controller is simply difficult to find. The results hold in continuous-time or discrete-time.