**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]