# Low-complexity Modeling of Partially Available Second-order Statistics via Matrix Completion

Tuesday, October 6, 2015 - 1:25pm - 2:25pm

Lind 305

Mihailo Jovanovic (University of Minnesota, Twin Cities)

We study the problem of completing partially known state statistics of

complex dynamical systems using low-complexity linearized models. State

statistics of linear systems satisfy certain structural constraints that

arise from the underlying dynamics and the directionality of input

disturbances. The dynamical interaction between state variables is known

while the directionality of input excitation is uncertain. Thus, the goal

of the inverse problem that we formulate is to identify the dynamics and

directionality of input excitation so as to explain the observed sample

statistics. In particular, we seek to explain the data with the least

number of possible input disturbance channels. This can be formulated as a

rank minimization problem, and for its solution, we employ a convex

relaxation based on the nuclear norm. The resulting optimization problem

can be cast as a semidefinite program and solved efficiently using

general-purpose solvers for small- and medium-size problems. We develop a

customized alternating minimization algorithm (AMA) to solve the problem

for large-scale systems. We demonstrate that AMA works as a proximal

gradient for the dual problem and provide examples to illustrate that

identified colored-in-time stochastic disturbances represent an effective

means for explaining available second-order state statistics.

complex dynamical systems using low-complexity linearized models. State

statistics of linear systems satisfy certain structural constraints that

arise from the underlying dynamics and the directionality of input

disturbances. The dynamical interaction between state variables is known

while the directionality of input excitation is uncertain. Thus, the goal

of the inverse problem that we formulate is to identify the dynamics and

directionality of input excitation so as to explain the observed sample

statistics. In particular, we seek to explain the data with the least

number of possible input disturbance channels. This can be formulated as a

rank minimization problem, and for its solution, we employ a convex

relaxation based on the nuclear norm. The resulting optimization problem

can be cast as a semidefinite program and solved efficiently using

general-purpose solvers for small- and medium-size problems. We develop a

customized alternating minimization algorithm (AMA) to solve the problem

for large-scale systems. We demonstrate that AMA works as a proximal

gradient for the dual problem and provide examples to illustrate that

identified colored-in-time stochastic disturbances represent an effective

means for explaining available second-order state statistics.

MSC Code:

37Fxx

Keywords: