NSF 1538374: Stability Analysis of Large-Scale Nonlinear Systems using Parallel Computation

Project Description:

In this project, we extend the results of NSF Project 1301660 by using the theory and software developed for that project to study analyze, optimize and control nonlinear systems. In particular, we use variations on Polya's theorem and Handelman's lemma to study both global and local stability of nonlinear differential equations. In particular, we seek algorithms with parallel structure which can be exploited by high-performance computing platforms such as cluster computers, supercomputers and GPU computers. We also are exploring the implications of our algorithms for parallel computing solutions in nonlinear hybrid systems and global optimization.

Project Duration:

09/01/2015-08/31/2018

Project Personnel:

Matthew M. Peet (PI), Arizona State University
Hesameddin Mohammadi (PhD), Arizona State University
Reza Kamyar (PhD), Arizona State University

International Collaborators:

Pedro P. L. D. Peres, The University of Campinas. Campinas, Brasil

Software Products:

Handelman Tools (Matlab Code)

Based on the Paper:
Constructing Piecewise Polynomial Lyapunov Functions Over Arbitrary Convex Polytopes Using Handelman's Basis
53rd IEEE Conference on Decision and Control. Los Angeles, CA. December 15-17, 2014.

Introduction:A Matlab code for local stability analysis of nonlinear ODEs defined by polynomial vector fields using Handelman's theorem for positivity over polytopes.

This toolbox:
1. Requires Matlab 2011a or later.
2. Requires that a working version of SeDuMi be installed.
3. Requires the entire folder be placed in the path along with sub-folders.
4. Is executed using the file handelman_arbitrary_triangles.m

[Handelman_Tool.zip]- Stability Analysis Package

Polya Parallel Tools (MPI)

Based on the Paper:
Solving Large-Scale Robust Control Problems by Exploiting the Parallel Structure of Polya's Theorem
IEEE Transactions on Automatic Control, Vol. 58, No. 8, Aug. 2013, pp. 1931-1947.

Introduction: A Linux Implementation of our parallel computing algorithm for robust stability of linear uncertain systems with uncertain parameters in the unit simplex.

This toolbox:
1. This relatively undocumented code requires MPI, LAPACK, and BLAS, the latter of which are included in the directory SDP3.
2. To install, run the file Makefile file located in the SDP3 directory.
3. For detailed instructions, contact rkamyar@asu.edu.

[Polya_Parallel.zip]- Parallel Computing Framework

Associated Publications at ASU (includes 1301660):

Student Theses:

Reza Kamyar (PhD), Arizona State University
Parallel Optimization of Polynomials for Large-Scale Problems in Stability and Control
Defended January, 2016.
Thesis - [arXiv:math] [.pdf] [.ps]
Defense Talk - [.pdf] [.ps]

Journal Publications:

R. Kamyar and M. M. Peet
Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum-of-Squares
Discrete and Continuous Dynamical Systems - Series B. Special Issue on "Constructive and computational methods in Lyapunov and stability theory". Vol. 20, No. 9, pp2383--2417. (Survey Paper). Oct., 2015.
[arXiv] [.pdf] [.ps]

R. Kamyar, M. M. Peet and Y. Peet
Solving Large-Scale Robust Control Problems by Exploiting the Parallel Structure of Polya's Theorem
IEEE Transactions on Automatic Control, Vol. 58, No. 8, Aug. 2013, pp. 1931-1947.
[arXiv] [.pdf] [.ps]

CDC/ACC Conference Publications:

H. Mohammadi and M. Peet
Combining SOS and Moment Relaxations with Branch and Bound to Extract Solutions to Global Polynomial Optimization Problems
Submitted to the 55th IEEE Conference on Decision and Control. Las Vegas, NV. December 12-14, 2016.
[arXiv] [.pdf] [.ps] [slides]

R. Kamyar and M. Peet
Constructing Piecewise Polynomial Lyapunov Functions Over Arbitrary Convex Polytopes Using Handelman's Basis
53rd IEEE Conference on Decision and Control. Los Angeles, CA. December 15-17, 2014.
[arXiv] [.pdf] [.ps] [slides]

R. Kamyar and M. Peet
Decentralized Polya’s Algorithm for Stability Analysis of Large-scale Nonlinear Systems
52nd IEEE Conference on Decision and Control. Florence, IT. December 10-13, 2013.
[arXiv] [.pdf] [.ps] [slides]

R. Kamyar and M. Peet
Decentralized Computation for Robust Stability of Large-scale Systems with Parameters on the Hypercube
51st IEEE Conference on Decision and Control, Maui, HI. December 15-17, 2012. pp. 6529-6264..
[arXiv] [.pdf] [.ps]

R. Kamyar and M. M. Peet
Solving Large-Scale Robust Control Problems by Exploiting the Parallel Structure of Polya's Theorem
American Control Conference, Montreal, CA. June 27-29, 2012. pp. 5948-5954.
[arXiv] [.pdf] [.ps]

M. M. Peet and Y. V. Peet
A Parallel-Computing Solution for Optimization of Polynomials
Proceedings of the American Control Conference, Baltimore, MD. June 30-July 2, 2010.
[arXiv] [.pdf] [.ps]