Decentralized Control Of Constrained Linear Systems Via Convex Optimization Methods

Decentralized control problems naturally arise in the control of large-scale networked systems. Such systems are regulated by a collection of local controllers in a decentralized manner, in the sense that each local controller is required to specify its control input based on its locally accessible sensor measurements. In this dissertation, we consider the decentralized control of discrete-time, linear systems subject to exogenous disturbances and polyhedral constraints on the state and input trajectories. The underlying system is composed of a finite collection of dynamically coupled subsystems, each of which is assumed to have a dedicated local controller. The decentralization of information is expressed according to sparsity constraints on the sensor measurements that each local controller has access to. In its most general form, the decentralized control problem amounts to an infinite-dimensional nonconvex program that is, in general, computationally intractable. The primary difficulty of the decentralized control problem stems from the potential informational coupling between the controllers. Specifically, in problems with nonclassical information structures, the actions taken by one controller can affect the information acquired by other controllers acting on the system. This gives rise to an incentive for controllers to communicate with each other via the actions that they undertake--the so-called signaling incentive. To complicate matters further, there may be hard constraints coupling the actions and local states being regulated by different controllers that must be jointly enforced with limited communication between the local controllers. In this dissertation, we abandon the search for the optimal decentralized control policy, and resort to approximation methods that enable the tractable calculation of feasible decentralized control policies. We first provide methods for the tractable calculation of decentralized control policies that are affinely parameterized in their measurement history. For problems with partially nested information structures, we show that the optimization over such a policy space admits an equivalent reformulation as a semi-infinite convex program. The optimal solution to these semi-inifinite programs can be calculated through the solution of a finite-dimensional conic program. For problems with nonclassical information structures, however, the optimization over such a policy space amounts to a semi-infinite nonconvex program. With the objective of alleviating the nonconvexity in such problems, we propose an approach to decentralized control design in which the information-coupling states are effectively treated as disturbances whose trajectories are constrained to take values in ellipsoidal "contract" sets whose location, scale, and orientation are jointly optimized with the affine decentralized control policy being used to control the system. The resulting problem is a semidefinite program, whose feasible solutions are guaranteed to be feasible for the original decentralized control design problem. Decentralized control policies that are computed according to such convex optimization methods are, in general, suboptimal. We, therefore, provide a method of bounding the suboptimality of feasible decentralized control policies through an information-based convex relaxation. Specifically, we characterize an expansion of the given information structure, which maximizes the optimal value of the decentralized control design problem associated with the expanded information structure, while guaranteeing that the expanded information structure be partially nested. The resulting decentralized control design problem admits an equivalent reformulation as an infinite-dimensional convex program. We construct a further constraint relaxation of this problem via its partial dualization and a restriction to affine dual control policies, which yields a finite-dimensional conic program whose optimal value is a provable lower bound on the minimum cost of the original decentralized control design problem. Finally, we apply our convexd programming approach to control design to the decentralized control of distributed energy resources in radial power distribution systems. We investigate the problem of designing a fully decentralized disturbance-feedback controller that minimizes the expected cost of serving demand, while guaranteeing the satisfaction of individual resource and distribution system voltage constraints. A direct application of our aforementioned control design methods enables both the calculation of affine controllers and the bounding of their suboptimality through the solution of finite-dimensional conic programs. A case study demonstrates that the decentralized affine controller we compute can perform close to optimal.