Constructing piecewise-polynomial lyapunov functions for local stability of nonlinear systems using Handelman's theorem

Reza Kamyar, Chaitanya Murti, Matthew Peet

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

In this paper, we propose a new convex approach to stability analysis of nonlinear systems with polynomial vector fields. First, we consider an arbitrary convex polytope that contains the equilibrium in its interior. Then, we decompose the polytope into several convex sub-polytopes with a common vertex at the equilibrium. Then, by using Handelman's theorem, we derive a new set of affine feasibility conditions -solvable by linear programming- on each sub-polytope. Any solution to this feasibility problem yields a piecewise polynomial Lyapunov function on the entire polytope. This is the first result which utilizes Handelman's theorem and decomposition to construct piecewise polynomial Lyapunov functions on arbitrary polytopes. In a computational complexity analysis, we show that for large number of states and large degrees of the Lyapunov function, the complexity of the proposed feasibility problem is less than the complexity of certain semi-definite programs associated with alternative methods based on Sum-of-Squares or Polya's theorem. Using different types of convex polytopes, we assess the accuracy of the algorithm in estimating the region of attraction of the equilibrium point of the reverse-time Van Der Pol oscillator.

Original languageEnglish (US)
Article number7040246
Pages (from-to)5481-5487
Number of pages7
JournalUnknown Journal
Volume2015-February
Issue numberFebruary
DOIs
StatePublished - 2014

Fingerprint

Piecewise Polynomials
Local Stability
Lyapunov functions
Polynomial function
Systems Analysis
Polytope
Lyapunov Function
Nonlinear systems
Nonlinear Systems
Polynomials
Linear Programming
Polytopes
Theorem
Decompose
Polynomial Vector Fields
Convex Polytope
Convex Polytopes
Van Der Pol Oscillator
Semidefinite Program
Complexity Analysis

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

Constructing piecewise-polynomial lyapunov functions for local stability of nonlinear systems using Handelman's theorem. / Kamyar, Reza; Murti, Chaitanya; Peet, Matthew.

In: Unknown Journal, Vol. 2015-February, No. February, 7040246, 2014, p. 5481-5487.

Research output: Contribution to journalArticle

@article{42c80d66f02147828156db08cafa2c38,
title = "Constructing piecewise-polynomial lyapunov functions for local stability of nonlinear systems using Handelman's theorem",
abstract = "In this paper, we propose a new convex approach to stability analysis of nonlinear systems with polynomial vector fields. First, we consider an arbitrary convex polytope that contains the equilibrium in its interior. Then, we decompose the polytope into several convex sub-polytopes with a common vertex at the equilibrium. Then, by using Handelman's theorem, we derive a new set of affine feasibility conditions -solvable by linear programming- on each sub-polytope. Any solution to this feasibility problem yields a piecewise polynomial Lyapunov function on the entire polytope. This is the first result which utilizes Handelman's theorem and decomposition to construct piecewise polynomial Lyapunov functions on arbitrary polytopes. In a computational complexity analysis, we show that for large number of states and large degrees of the Lyapunov function, the complexity of the proposed feasibility problem is less than the complexity of certain semi-definite programs associated with alternative methods based on Sum-of-Squares or Polya's theorem. Using different types of convex polytopes, we assess the accuracy of the algorithm in estimating the region of attraction of the equilibrium point of the reverse-time Van Der Pol oscillator.",
author = "Reza Kamyar and Chaitanya Murti and Matthew Peet",
year = "2014",
doi = "10.1109/CDC.2014.7040246",
language = "English (US)",
volume = "2015-February",
pages = "5481--5487",
journal = "Scanning Electron Microscopy",
issn = "0586-5581",
publisher = "Scanning Microscopy International",
number = "February",

}

TY - JOUR

T1 - Constructing piecewise-polynomial lyapunov functions for local stability of nonlinear systems using Handelman's theorem

AU - Kamyar, Reza

AU - Murti, Chaitanya

AU - Peet, Matthew

PY - 2014

Y1 - 2014

N2 - In this paper, we propose a new convex approach to stability analysis of nonlinear systems with polynomial vector fields. First, we consider an arbitrary convex polytope that contains the equilibrium in its interior. Then, we decompose the polytope into several convex sub-polytopes with a common vertex at the equilibrium. Then, by using Handelman's theorem, we derive a new set of affine feasibility conditions -solvable by linear programming- on each sub-polytope. Any solution to this feasibility problem yields a piecewise polynomial Lyapunov function on the entire polytope. This is the first result which utilizes Handelman's theorem and decomposition to construct piecewise polynomial Lyapunov functions on arbitrary polytopes. In a computational complexity analysis, we show that for large number of states and large degrees of the Lyapunov function, the complexity of the proposed feasibility problem is less than the complexity of certain semi-definite programs associated with alternative methods based on Sum-of-Squares or Polya's theorem. Using different types of convex polytopes, we assess the accuracy of the algorithm in estimating the region of attraction of the equilibrium point of the reverse-time Van Der Pol oscillator.

AB - In this paper, we propose a new convex approach to stability analysis of nonlinear systems with polynomial vector fields. First, we consider an arbitrary convex polytope that contains the equilibrium in its interior. Then, we decompose the polytope into several convex sub-polytopes with a common vertex at the equilibrium. Then, by using Handelman's theorem, we derive a new set of affine feasibility conditions -solvable by linear programming- on each sub-polytope. Any solution to this feasibility problem yields a piecewise polynomial Lyapunov function on the entire polytope. This is the first result which utilizes Handelman's theorem and decomposition to construct piecewise polynomial Lyapunov functions on arbitrary polytopes. In a computational complexity analysis, we show that for large number of states and large degrees of the Lyapunov function, the complexity of the proposed feasibility problem is less than the complexity of certain semi-definite programs associated with alternative methods based on Sum-of-Squares or Polya's theorem. Using different types of convex polytopes, we assess the accuracy of the algorithm in estimating the region of attraction of the equilibrium point of the reverse-time Van Der Pol oscillator.

UR - http://www.scopus.com/inward/record.url?scp=84988041809&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988041809&partnerID=8YFLogxK

U2 - 10.1109/CDC.2014.7040246

DO - 10.1109/CDC.2014.7040246

M3 - Article

VL - 2015-February

SP - 5481

EP - 5487

JO - Scanning Electron Microscopy

JF - Scanning Electron Microscopy

SN - 0586-5581

IS - February

M1 - 7040246

ER -