Salar Fattahi

Transformation of optimal centralized controllers into near-global static distributed controllers

This paper is concerned with the optimal decentralized control problem for linear discrete-time deterministic and stochastic systems. The objective is to design a stabilizing static distributed controller with a given structure, whose performance is close to that of the optimal centralized controller. To this end, we derive a necessary and sufficient condition under which there exists a distributed controller that generates the same input and state trajectories as the optimal centralized one. This condition is then translated into a convex optimization problem. Subsequently, a regularization term is incorporated into the objective of the proposed optimization problem to indirectly account for the stability of the distributed control system. The designed optimization has a closed-form solution (explicit formula), which depends on the optimal centralized controller as well as the prescribed controller structure. If the optimal objective value of the proposed optimization problem is small enough at the explicit solution, the resulting controller is stabilizing and has a high performance. The derived formula may help partially answer some open problems, such as finding the minimum number of free elements required in the distributed controller under design to achieve a performance close to the optimal centralized one. The proposed approach is tested on a power network and several random systems.

On the convexity of optimal decentralized control problem and sparsity path

This paper is concerned with an important special case of the stochastic optimal decentralized control (SODC) problem, where the objective is to design a static structurally constrained controller for a stable stochastic system. This problem is non-convex and hard to solve in general. We show that if either the measurement noise covariance or the input weighting matrix is not too small, the problem is locally convex. Under such circumstances, the design of a decentralized controller with a bounded norm subject to an arbitrary sparsity pattern is naturally a convex problem. We also study the problem of designing a sparse controller using a regularization technique, where the control structure is not pre-specified but penalized in the objective function. Under some genericity assumptions, we prove that this method is able to design a decentralized controller with any arbitrary sparsity level. Although this paper is focused on stable systems, the results can be generalized to unstable systems as long as an initial stabilizing controller with a desirable structure is known a priori.

Theoretical guarantees for the design of near globally optimal static distributed controllers

This paper is concerned with the optimal distributed control problem for linear discrete-time deterministic and stochastic systems. The objective is to design a stabilizing static distributed controller whose performance is close to that of the optimal centralized controller (if such controller exists). In our previous work, we have developed a computational framework to transform centralized controllers into distributed ones for deterministic systems. By building on this result, we derive strong theoretical lower bounds on the optimality guarantee of the designed distributed controllers and show that the proposed mathematical framework indirectly maximizes the derived lower bound while striving to achieve a closed-loop stability. Furthermore, we extend the proposed design method to stochastic systems that are subject to input disturbance and measurement noise. The developed optimization problem has a closed-form solution (explicit formula) and can be easily deployed for large-scale systems that require low computational efforts. The proposed approach is tested on a power network and random systems to demonstrate its efficacy.

A strong semidefinite programming relaxation of the unit commitment problem

The unit commitment (UC) problem aims to find an optimal schedule of generating units subject to the demand and operating constraints for an electricity grid. The majority of existing algorithms for the UC problem rely on solving a series of convex relaxations by means of branch-and-bound or cutting-planning methods. In this paper, we develop a strengthened semidefinite program (SDP) for the UC problem by first deriving certain valid quadratic constraints and then relaxing them to linear matrix inequalities. These valid inequalities are obtained by the multiplication of the linear constraints of the UC problem such as the flow constraints of two different lines. The performance of the proposed convex relaxation is evaluated on several instances of the UC problem. For most of the instances, globally optimal integer solutions are obtained by solving a single convex problem. Since the proposed technique leads to a large number of valid quadratic inequalities, an iterative procedure is devised to impose a small number of such valid inequalities. For the cases where the strengthened SDP does give a global integer solution, we incorporate other valid inequalities, including a set of Boolean quadric polytope constraints. The proposed relaxations are extensively tested on various IEEE power systems in simulations.

Convexification of generalized network flow problem

This paper is concerned with the minimum-cost flow problem over an arbitrary flow network. In this problem, each node is associated with some possibly unknown injection and each line has two unknown flows at its ends that are related to each other via a nonlinear function. Moreover, all injections and flows must satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which is shown to always obtain globally optimal injections. This relaxation may fail to find optimal flows because the mapping from injections to flows is not unique in general. We show that the proposed relaxation, named convexified GNF (CGNF), obtains a globally optimal flow vector if the optimal injection vector is a Pareto point. More generally, the network can be decomposed into two subgraphs such that the lines between the subgraphs are congested at optimality and that CGNF finds correct optimal flows over all lines of one of these subgraphs. We also fully characterize the set of all globally optimal flow vectors, based on the optimal injection vector found via CGNF. In particular, we show that this solution set is a subset of the boundary of a convex set, and may include an exponential number of disconnected components. A primary application of this work is in optimization over electrical power networks.

Convex analysis of generalized flow networks

This paper is concerned with the generalized network flow (GNF) problem, which aims to find a minimum-cost solution for a generalized flow network. The objective is to determine the optimal injections at the nodes as well as optimal flows over the lines of the network. In this problem, each line is associated with two flows in opposite directions that are related to each other via a given nonlinear function. Under some monotonicity and convexity assumptions, we have shown in our recent work that a convexified generalized network flow (CGNF) problem always finds optimal injections for GNF, but may fail to find optimal flows. In this paper, we develop three results to explore the possibility of obtaining optimal flows. First, we show that CGNF yields optimal flows for GNF if the optimal injection vector is a Pareto point. Second, we show that if CGNF fails to find an optimal flow vector, then the graph can be decomposed into two subgraphs, where the lines between the subgraphs are congested at optimality and CGNF finds correct optimal flows over the lines of one of these subgraphs. Third, we fully characterize the set of all optimal flow vectors. In particular, we show that this non-convex set is a subset of the boundary of a convex set, and may include an exponential number of disconnected components.