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Stability and stabilization of polynomial dynamical systems using Bernstein polynomials

Authors:

Mohamed Amin Ben Sassi,

Sriram SankaranarayananAuthors Info & Claims

HSCC '15: Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control

Pages 291 - 292

https://doi.org/10.1145/2728606.2728639

Published: 14 April 2015 Publication History

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Abstract

In this work, we examine relaxations for the stability analysis and synthesis of stabilizing controllers for polynomial dynamical systems. It is well-known that such problems can be naturally solved using a reduction to polynomial optimization problems. The Sum of Squares (SOS) programming relaxation further relaxes these polynomial optimization problems to convex Semi-Definite Programming (SDP) problems. However, their application, in practice, to formal verification and correct-by-construction synthesis has been made harder due to numerical stability issues encountered while solving SDPs. Our work proposes a new approach to relaxations based on Bernstein polynomials to yield linear programming (LP) relaxations for polynomial optimization problems. This allows us to find polynomial Lyapunov functions that certify the (asymptotic) stability of a system. The approach is also extended to synthesizing a stabilizing feedback control law for a controlled system that will stabilize the closed-loop dynamics to a specified equilibrium.

References

[1]

M. A. Ben Sassi and S. Sankaranarayanan and X. Chen and E. Abraham. Linear Relaxations of Polynomial Positivity for Polynomial Lyapunov Function Synthesis. IMA Journal of Mathematical Control and Information, In Press, 2015.

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[2]

M. A. Ben Sassi and S. Sankaranarayanan. Stabilization of polynomial dynamical systems using linear programming based on Bernstein polynomials.Arxiv:1501.04578, 2015 (Draft: under revision).

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[3]

R. T. Farouki. The Bernstein polynomial basis: A centennial retrospective. Comput. Aided Geom. Des., 29(6): 379--419, August 2012.

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[4]

S. Gaubert, E. Goubault, A. Taly, and S. Zennou. Static analysis by policy iteration on relational domains. In R. De Nicola, editor, Programming Languages and Systems, volume 4421 of Lecture Notes in Computer Science, pages 237--252. Springer Berlin Heidelberg, 2007.

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Wang QChen MXue BZhan NKatoen J(2022)Encoding inductive invariants as barrier certificates: synthesis via difference-of-convex programmingInformation and Computation10.1016/j.ic.2022.104965(104965)Online publication date: Sep-2022
https://doi.org/10.1016/j.ic.2022.104965
Izadbakhsh AGholizade‐Narm HDeylami A(2022)Observer‐based adaptive controller design for chaos synchronization using Bernstein‐type operatorsInternational Journal of Robust and Nonlinear Control10.1002/rnc.602632:7(4318-4335)Online publication date: 24-Jan-2022
https://doi.org/10.1002/rnc.6026
Chen XPeng CLin WYang ZZhang YLi X(2020)A Novel Approach for Solving the BMI Problem in Barrier Certificates GenerationComputer Aided Verification10.1007/978-3-030-53288-8_29(582-603)Online publication date: 14-Jul-2020
https://doi.org/10.1007/978-3-030-53288-8_29
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Index Terms

Stability and stabilization of polynomial dynamical systems using Bernstein polynomials
1. Information systems
  1. Information systems applications
    1. Decision support systems
      1. Expert systems
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials

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Published In

HSCC '15: Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control

April 2015

321 pages

ISBN:9781450334334

DOI:10.1145/2728606

Program Chairs:
Antoine Girard
Université Joseph Fourier, Grenoble, France
,
Sriram Sankaranarayanan
University of Colorado at Boulder

Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author.

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Association for Computing Machinery

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Published: 14 April 2015

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HSCC '15: 18th International Conference on Hybrid Systems: Computation and Control

April 14 - 16, 2015

Washington, Seattle

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Wang QChen MXue BZhan NKatoen J(2022)Encoding inductive invariants as barrier certificates: synthesis via difference-of-convex programmingInformation and Computation10.1016/j.ic.2022.104965(104965)Online publication date: Sep-2022
https://doi.org/10.1016/j.ic.2022.104965
Izadbakhsh AGholizade‐Narm HDeylami A(2022)Observer‐based adaptive controller design for chaos synchronization using Bernstein‐type operatorsInternational Journal of Robust and Nonlinear Control10.1002/rnc.602632:7(4318-4335)Online publication date: 24-Jan-2022
https://doi.org/10.1002/rnc.6026
Chen XPeng CLin WYang ZZhang YLi X(2020)A Novel Approach for Solving the BMI Problem in Barrier Certificates GenerationComputer Aided Verification10.1007/978-3-030-53288-8_29(582-603)Online publication date: 14-Jul-2020
https://doi.org/10.1007/978-3-030-53288-8_29
Izadbakhsh AKhorashadizadeh S(2020)Robust adaptive control of robot manipulators using Bernstein polynomials as universal approximatorInternational Journal of Robust and Nonlinear Control10.1002/rnc.491330:7(2719-2735)Online publication date: 26-Feb-2020
https://doi.org/10.1002/rnc.4913

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