Controller architectures: Tradeoffs between performance and structure

Abstract

This review article describes the design of static controllers that achieve an optimal tradeoff between closed-loop performance and controller structure. Our methodology consists of two steps. First, we identify controller structure by incorporating regularization functions into the optimal control problem and, second, we optimize the controller over the identified structure. For large-scale networks of dynamical systems, the desired structural property is captured by limited information exchange between physical and controller layers and the regularization term penalizes the number of communication links. Although structured optimal control problems are, in general, nonconvex, we identify classes of convex problems that arise in the design of symmetric systems, undirected consensus and synchronization networks, optimal selection of sensors and actuators, and decentralized control of positive systems. Examples of consensus networks, drug therapy design, sensor selection in flexible wing aircrafts, and optimal wide-area control of power systems are provided to demonstrate the effectiveness of the framework.

Structured Decentralized Control of Positive Systems With Applications to Combination Drug Therapy and Leader Selection in Directed Networks

“…The regularization function g in (3) can be any convex function, e.g., a quadratic penalty u T Ru with R 0 to limit the magnitude of u, an 1 penalty to promote sparsity of u, or the indicator function associated with a convex set C to ensure that u ∈ C. We refer the reader to [32], [33] for an overview of recent uses of regularization in control-theoretic problems. We now review some recent results.…”

Multi‐layer distributed protocols for robust cooperative tracking in interconnected nonlinear multiagent systems

“…Let Let  1m be the set of all admissible (linear static state feedback) stabilizing control signals for a completely known dynamical system in the modified LQR problem (16), and u x = K x e x be the control signal that minimizes this quadratic cost function. The matrix  cx = −K x = R −1 x P characterizes the candidate control layer graph topology  cx for the first-order tracking problem (10) if the solution P = P T ∈ ℝ N×N ≻ 0 of the associated N × N nonlinear matrix equation (17) satisfies condition (18).…”

“…We note that, in order to be a valid candidate reduced-order Laplacian matrix,  cx must be an M-matrix, all of its eigenvalues must be in the right-half plane, and at least one of its row sums must be positive, whereas the rest of them are nonnegative scalars. The latest condition on the row sums are ensured by the additional condition (18) in Design Problem 1. In the rest, we establish the closed-form solutions for the candidate control layer topology and comment that the other two requirements will be satisfied when we design the control layer topology using Design Problem 1.…”

“…In particular, Rieger et al proposed the concept of a multi‐layer control framework for large‐scale systems as well as its potential benefits and applications to the US national power systems, and Egerstedt showed that the architectural aspects of cyber‐physical systems can be captured using graph theoretic approaches while reaching a global objective based on an MAS of completely known first‐order agents. While the previous two studies are mainly about the (graph‐ and) control‐oriented formulations, Jovanovic and Dhingra provided a summary of some optimization techniques that could be used to develop the required (sparse) control systems for large‐scale systems. Obtaining the closed‐form solutions was unlikely due to the numerical nature of the solution approaches.…”

Design Of Real-Time Implementable Distributed Suboptimal Control: An LQR Perspective

“…In [31] and [32], sparsity inducing regularization terms were added to the cost function to design sparse controller architectures. Then optimal controller matrices with specified sparsity structures were computed by solving optimization problems.…”