PhD thesis

Structure identification and optimal design of large-scale networks of dynamical systems
Fu Lin
University of Minnesota, Minneapolis, 2012

Abstract

This dissertation is about structure identification and optimal control of large-scale networks of dynamical systems. It contains four parts. In Part I, we focus on identification of controller architectures that strike a balance between the performance of the system and the sparsity of the controller. This is achieved by solving a parameterized family of sparsity-promoting optimal control problems whose solution traces the trade-off curve that starts at the centralized controller and ends at the sparse controller of interest.

Part II is devoted to the design of sparse communication graphs for consensus networks. This class of problems is commonly seen in distributed estimation and control. We show that the sparsity-promoting control problem can be formulated as a semidefinite program whose globally optimal solution can be computed efficiently.

In Part III, we consider optimal localized control of vehicular formations with nearest neighbor interactions. We identify a class of convex problems by restricting the controller to symmetric feedback gains. For the design of non-symmetric gains, we solve a parameterized family of problems whose solution gradually changes from the spatially uniform gain to the optimal non-symmetric gain. We investigate the performance of localized controllers in large formations and show that the best performance is achieved with non-symmetric and spatially-varying controllers.

Finally, in Part IV, we consider the leader selection problem in consensus networks, in which leaders have access to their own state in addition to relative information exchange with neighbors. We are interested in selecting an a priori specified number of nodes as leaders such that the steady-state variance of the deviation from consensus is minimized. For this combinatorial optimization problem, we develop efficient algorithms to compute lower and upper bounds on the global optimal value.