Stability of uncertain systems using Lyapunov functions with non-monotonic terms

Márcio J. Lacerda, Peter J Seiler Jr

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

This paper is concerned with the problem of robust stability of uncertain linear time-invariant systems in polytopic domains. The main contribution is to present a systematic procedure to check the stability of the uncertain systems by using an arbitrary number of quadratic functions within higher order derivatives of the vector field in the continuous-time case and higher order differences of the vector field in the discrete-time case. The matrices of the Lyapunov function appear decoupled from the dynamic matrix of the system in the conditions. This fact leads to sufficient conditions that are given in terms of Linear Matrix Inequalities defined at the vertices of the polytope. The proposed method does not impose sign condition constraints in the quadratic functions that compose the Lyapunov function individually. Moreover, some of the quadratic functions do not decrease monotonically along trajectories. However, if the sufficient conditions are satisfied, then a monotonic standard Lyapunov function that depends on the dynamics of the uncertain system can be constructed a posteriori. Numerical examples from the literature are provided to illustrate the proposed approach.

Original languageEnglish (US)
Pages (from-to)187-193
Number of pages7
JournalAutomatica
Volume82
DOIs
StatePublished - Aug 1 2017

Fingerprint

Uncertain systems
Lyapunov functions
Linear matrix inequalities
Trajectories
Derivatives

Keywords

  • Continuous and discrete-time uncertain systems
  • Non-monotonic Lyapunov functions
  • Robust stability
  • Time-invariant uncertainty

Cite this

Stability of uncertain systems using Lyapunov functions with non-monotonic terms. / Lacerda, Márcio J.; Seiler Jr, Peter J.

In: Automatica, Vol. 82, 01.08.2017, p. 187-193.

Research output: Contribution to journalArticle

@article{32e83b2e79aa48eaac6567bb7e36865f,
title = "Stability of uncertain systems using Lyapunov functions with non-monotonic terms",
abstract = "This paper is concerned with the problem of robust stability of uncertain linear time-invariant systems in polytopic domains. The main contribution is to present a systematic procedure to check the stability of the uncertain systems by using an arbitrary number of quadratic functions within higher order derivatives of the vector field in the continuous-time case and higher order differences of the vector field in the discrete-time case. The matrices of the Lyapunov function appear decoupled from the dynamic matrix of the system in the conditions. This fact leads to sufficient conditions that are given in terms of Linear Matrix Inequalities defined at the vertices of the polytope. The proposed method does not impose sign condition constraints in the quadratic functions that compose the Lyapunov function individually. Moreover, some of the quadratic functions do not decrease monotonically along trajectories. However, if the sufficient conditions are satisfied, then a monotonic standard Lyapunov function that depends on the dynamics of the uncertain system can be constructed a posteriori. Numerical examples from the literature are provided to illustrate the proposed approach.",
keywords = "Continuous and discrete-time uncertain systems, Non-monotonic Lyapunov functions, Robust stability, Time-invariant uncertainty",
author = "Lacerda, {M{\'a}rcio J.} and {Seiler Jr}, {Peter J}",
year = "2017",
month = "8",
day = "1",
doi = "10.1016/j.automatica.2017.04.042",
language = "English (US)",
volume = "82",
pages = "187--193",
journal = "Automatica",
issn = "0005-1098",
publisher = "Elsevier Limited",

}

TY - JOUR

T1 - Stability of uncertain systems using Lyapunov functions with non-monotonic terms

AU - Lacerda, Márcio J.

AU - Seiler Jr, Peter J

PY - 2017/8/1

Y1 - 2017/8/1

N2 - This paper is concerned with the problem of robust stability of uncertain linear time-invariant systems in polytopic domains. The main contribution is to present a systematic procedure to check the stability of the uncertain systems by using an arbitrary number of quadratic functions within higher order derivatives of the vector field in the continuous-time case and higher order differences of the vector field in the discrete-time case. The matrices of the Lyapunov function appear decoupled from the dynamic matrix of the system in the conditions. This fact leads to sufficient conditions that are given in terms of Linear Matrix Inequalities defined at the vertices of the polytope. The proposed method does not impose sign condition constraints in the quadratic functions that compose the Lyapunov function individually. Moreover, some of the quadratic functions do not decrease monotonically along trajectories. However, if the sufficient conditions are satisfied, then a monotonic standard Lyapunov function that depends on the dynamics of the uncertain system can be constructed a posteriori. Numerical examples from the literature are provided to illustrate the proposed approach.

AB - This paper is concerned with the problem of robust stability of uncertain linear time-invariant systems in polytopic domains. The main contribution is to present a systematic procedure to check the stability of the uncertain systems by using an arbitrary number of quadratic functions within higher order derivatives of the vector field in the continuous-time case and higher order differences of the vector field in the discrete-time case. The matrices of the Lyapunov function appear decoupled from the dynamic matrix of the system in the conditions. This fact leads to sufficient conditions that are given in terms of Linear Matrix Inequalities defined at the vertices of the polytope. The proposed method does not impose sign condition constraints in the quadratic functions that compose the Lyapunov function individually. Moreover, some of the quadratic functions do not decrease monotonically along trajectories. However, if the sufficient conditions are satisfied, then a monotonic standard Lyapunov function that depends on the dynamics of the uncertain system can be constructed a posteriori. Numerical examples from the literature are provided to illustrate the proposed approach.

KW - Continuous and discrete-time uncertain systems

KW - Non-monotonic Lyapunov functions

KW - Robust stability

KW - Time-invariant uncertainty

UR - http://www.scopus.com/inward/record.url?scp=85019366449&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019366449&partnerID=8YFLogxK

U2 - 10.1016/j.automatica.2017.04.042

DO - 10.1016/j.automatica.2017.04.042

M3 - Article

VL - 82

SP - 187

EP - 193

JO - Automatica

JF - Automatica

SN - 0005-1098

ER -