A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We present a framework for stability analysis of systems of coupled linear Partial-Differential Equations (PDEs). The class of PDE systems considered in this paper includes parabolic, elliptic and hyperbolic systems with Dirichelet, Neuman and mixed boundary conditions. The results in this paper apply to systems with a single spatial variable and assume existence and continuity of solutions except in such cases when existence and continuity can be inferred from existence of a Lyapunov function. Our approach is based on a new concept of state for PDE systems which allows us to express the derivative of the Lyapunov function as a Linear Operator Inequality directly on L2 and allows for any type of suitably well-posed boundary conditions. This approach obviates the need for integration by parts, spacing functions or similar mathematical encumbrances. The resulting algorithms are implemented in Matlab, tested on several motivating examples, and the codes have been posted online. Numerical testing indicates the approach has little or no conservatism for a large class of systems and can analyze systems of up to 20 coupled PDEs.

Original languageEnglish (US)
Title of host publication2018 IEEE Conference on Decision and Control, CDC 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages545-550
Number of pages6
ISBN (Electronic)9781538613955
DOIs
StatePublished - Jan 18 2019
Event57th IEEE Conference on Decision and Control, CDC 2018 - Miami, United States
Duration: Dec 17 2018Dec 19 2018

Publication series

NameProceedings of the IEEE Conference on Decision and Control
Volume2018-December
ISSN (Print)0743-1546

Conference

Conference57th IEEE Conference on Decision and Control, CDC 2018
CountryUnited States
CityMiami
Period12/17/1812/19/18

Fingerprint

State-space Representation
Lyapunov Stability
Partial differential equations
Stability Analysis
Partial differential equation
Lyapunov functions
Lyapunov Function
Boundary conditions
Parabolic-elliptic System
Integration by parts
Operator Inequality
Mixed Boundary Conditions
Linear partial differential equation
Mathematical operators
Systems of Partial Differential Equations
Hyperbolic Systems
Linear Operator
Spacing
MATLAB
Linear Inequalities

ASJC Scopus subject areas

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

Cite this

Peet, M. (2019). A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework. In 2018 IEEE Conference on Decision and Control, CDC 2018 (pp. 545-550). [8619025] (Proceedings of the IEEE Conference on Decision and Control; Vol. 2018-December). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2018.8619025

A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework. / Peet, Matthew.

2018 IEEE Conference on Decision and Control, CDC 2018. Institute of Electrical and Electronics Engineers Inc., 2019. p. 545-550 8619025 (Proceedings of the IEEE Conference on Decision and Control; Vol. 2018-December).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Peet, M 2019, A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework. in 2018 IEEE Conference on Decision and Control, CDC 2018., 8619025, Proceedings of the IEEE Conference on Decision and Control, vol. 2018-December, Institute of Electrical and Electronics Engineers Inc., pp. 545-550, 57th IEEE Conference on Decision and Control, CDC 2018, Miami, United States, 12/17/18. https://doi.org/10.1109/CDC.2018.8619025
Peet M. A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework. In 2018 IEEE Conference on Decision and Control, CDC 2018. Institute of Electrical and Electronics Engineers Inc. 2019. p. 545-550. 8619025. (Proceedings of the IEEE Conference on Decision and Control). https://doi.org/10.1109/CDC.2018.8619025
Peet, Matthew. / A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework. 2018 IEEE Conference on Decision and Control, CDC 2018. Institute of Electrical and Electronics Engineers Inc., 2019. pp. 545-550 (Proceedings of the IEEE Conference on Decision and Control).
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