LANS Informal Seminar: Reza Kamyar

Seminar Title: Distributed parallel algorithms for problems in polynomial optimization and robust control
Speaker: Reza Kamyar, Ph.D. student at Illinois Institute of Technology

Date/Time: 2012-01-11 15:00
Location: Building 240, 1404-1405

Description:
Numerous physical phenomena and the devices which involve such phenomena are aptly modeled by Partial Differential Equations. For this reason, the stability and control of the systems modeled by PDEs is of significant importance. Unfortunately, current analytical methods for stability analysis and control of PDEs in infinite-dimensional spaces are only applicable to simple classes of PDEs. An alternative approach is to discretize the PDE at specific points in the space and represent it as a set of ODEs in finite-dimensional spaces. Although the resulting state-space system is much easier to analyze and control by using a wide range of available tools in finite-dimensional linear analysis, the large dimension of the state-space and the presence of uncertain parameters in the system makes the stability and control problem computationally intractable.

In this study, we propose a distributed computing approach for large-scale robust stability and control problems. First, we design a decentralized algorithm which uses Polya’s algorithm to convert the robust stability and control conditions into a set of highly structured Linear Matrix Inequalities (LMIs). Then we show that a common implementation of a primal-dual interior-point method for solving these LMIs has a block-diagonal structure which is preserved at each iteration. By exploiting this property, we create a highly scalable cluster-computing implementation of our algorithm for stability analysis and control of large systems. The theoretical and experimental results for speed-up verify the scalability of the algorithms. Numerical examples demonstrate the ability to perform robust analysis of systems with more than one-hundred states and several uncertain parameters.