State statistics of linear systems satisfy certain structural constraints that arise from the underlying dynamics and the directionality of input disturbances. In this talk, we study the problem of completing partially known state statistics. Our aim is to develop tools that can be used in the context of control-oriented modeling of large-scale dynamical systems. For the type of applications we have in mind, the dynamical interaction between state variables is known while the directionality and dynamics of input excitation is often uncertain. Thus, the goal of the mathematical problem that we formulate is to identify the dynamics and directionality of input excitation in order to explain and complete observed sample statistics. More specifically, we seek to explain correlation data with the least number of possible input disturbance channels. We formulate this inverse problem as rank minimization, and for its solution, we employ a convex relaxation based on the nuclear norm. The resulting optimization problem is cast as a semidefinite program and can be solved efficiently using general-purpose solvers for small- and medium-size problems. For large-scale systems, we develop a customized alternating minimization algorithm (AMA). We interpret AMA as a proximal gradient for the dual problem and prove sub-linear convergence for the algorithm with fixed step-size. We conclude with an example that illustrates the utility of our modeling and optimization framework.
Mihailo Jovanovic received the PhD degree from UC Santa Barbara, in 2004, under the direction of Bassam Bamieh. He is a professor of Electrical and Computer Engineering at the University of Minnesota and has held visiting positions with Stanford University and the Institute for Mathematics and its Applications. His current research focuses on dynamics and control of fluid flows, design of controller architectures, and fundamental limitations in the control of large networks of dynamical systems. He serves as an Associate Editor of the SIAM Journal on Control and Optimization and had served as an Associate Editor of the IEEE Control Systems Society Conference Editorial Board from July 2006 until December 2010. Prof. Jovanovic received a CAREER Award from the National Science Foundation in 2007, the George S. Axelby Outstanding Paper Award from the IEEE Control Systems Society in 2013, and the Distinguished Alumni Award from UC Santa Barbara in 2014.