Control of Distributed Systems with Interconnection Constraints

As components shrink in size and cost, analysis and synthesis of distributed coordination and control algorithms for networked control systems with communication constraints has become an active area of control theory research. From quantization systems for networked control systems (NCS) to fundamental limitations of control systems under information constraints, and from consensus problems to formation control and sensing and coverage problems, researchers have been interested in distributed control algorithms that achieve global objectives with minimum communication requirements. The main objective of this dissertation is to report novel results and techniques for controlling interconnected systems under communication constraints. We study the dynamics and classical control approaches (such as LQR, H̢蠁_) in complex interconnected systems under various communication and interconnection constraints. Of particular interest are theoretical questions regarding the influence of a complex system's structure and limited communication on its stability and robustness, and the design of distributed algorithms and protocols to access and modify the global behavior of the system. The first problem considered deals in this dissertation with the classical linear systems with bit-rate constraints in the feedback channel. Based on the bit-rate constraints, an optimization problem is formulated to maximize the convergence rate of quantization error for these systems. The quantization error goes to zero if and only if the bit-rate is above some threshold determined by the unstable modes of the plants. This known result is shown here via an alternative proof which leads to the minimum bit-rate required to stabilize linear systems. We further show that a dynamic quantization algorithm will approach the analytic results of our optimization problem. Furthermore, we consider the performance of standard LQR problems under bit-rate constraint. Our results show that the performance of standard LQR problems under bit-rate constrain consists of two parts: the perfect information part and the cost introduced by stepwise quantization error. In this way, we can quantify the penalty of limited communication on LQR problems. In the remainder of this dissertation, several novel results toward a distributed control theory for spatially invariant and distributed systems are introduced. For the spatially invariant systems, a multidimensional KYP-like lemma exists, such that, the H̢蠁_ performance can be characterized by a set of LMIs. These results are later utilized in distributed controller synthesis. For distributed systems, a distributed stability condition to guarantee global quadratic performance under various communication constraints is presented. The idea is based on a theory of separation of graphs from an operator's point of view, where the plant and the interconnections as modelled as two interconnected operators in proper signal space. The essential tools we are using here are two versions of S-procedures which are similar to Lagrange relaxation techniques, initially developed by researchers in the former Soviet Union. It is later shown that these distributed stability conditions can be utilized in distributed controller synthesis in a manner similar to those gain-scheduling controllers in Linear Parameter Varying (LPV) systems. The synthesis is based on the elimination lemma, and convex conditions are derived for the existence of distributed controllers. Comparisons of distributed controllers, centralized and decentralized controllers are also briefly discussed in the introduction and summary.