A Dual to Lyapanov's Second Method for Linear Systems with Multiple Delays and Implementation using SOS

Research output: Contribution to journalArticle

Abstract

We present a dual form of Lyapunov-Krasovskii functional which allows the problem of controller synthesis for multi-delay systems to be formulated and solved in a convex manner. First, we give a generalized version of the dual stability condition formulated in terms of Lyapunov operators which are positive, self-adjoint and preserve the structure of the state-space. Second, we provide a class of such operators and express the stability conditions as positivity and negativity of quadratic Lyapunov-Krasovskii functional forms. Next, we adapt the SOS methodology to express positivity and negativity of these forms as LMIs, describing a new set of polynomial manipulation tools designed for this purpose. We apply the resulting LMIs to a battery of numerical examples and demonstrate that the stability conditions are not significantly conservative. Finally, we formulate a test for controller synthesis for systems with multiple delays, apply the test to a numerical example, and simulate the resulting closed-loop system.

Original languageEnglish (US)
JournalIEEE Transactions on Automatic Control
DOIs
StateAccepted/In press - May 1 2018

Fingerprint

Linear systems
Controllers
Closed loop systems
Polynomials

Keywords

  • Asymptotic stability
  • Controller Synthesis
  • Delay Systems
  • Delays
  • LMIs
  • Lyapunov methods
  • Numerical stability
  • Optimization
  • Stability criteria

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering

Cite this

@article{f8eb24a3a7284cb9a965196d1e326630,
title = "A Dual to Lyapanov's Second Method for Linear Systems with Multiple Delays and Implementation using SOS",
abstract = "We present a dual form of Lyapunov-Krasovskii functional which allows the problem of controller synthesis for multi-delay systems to be formulated and solved in a convex manner. First, we give a generalized version of the dual stability condition formulated in terms of Lyapunov operators which are positive, self-adjoint and preserve the structure of the state-space. Second, we provide a class of such operators and express the stability conditions as positivity and negativity of quadratic Lyapunov-Krasovskii functional forms. Next, we adapt the SOS methodology to express positivity and negativity of these forms as LMIs, describing a new set of polynomial manipulation tools designed for this purpose. We apply the resulting LMIs to a battery of numerical examples and demonstrate that the stability conditions are not significantly conservative. Finally, we formulate a test for controller synthesis for systems with multiple delays, apply the test to a numerical example, and simulate the resulting closed-loop system.",
keywords = "Asymptotic stability, Controller Synthesis, Delay Systems, Delays, LMIs, Lyapunov methods, Numerical stability, Optimization, Stability criteria",
author = "Matthew Peet",
year = "2018",
month = "5",
day = "1",
doi = "10.1109/TAC.2018.2832470",
language = "English (US)",
journal = "IEEE Transactions on Automatic Control",
issn = "0018-9286",
publisher = "Institute of Electrical and Electronics Engineers Inc.",

}

TY - JOUR

T1 - A Dual to Lyapanov's Second Method for Linear Systems with Multiple Delays and Implementation using SOS

AU - Peet, Matthew

PY - 2018/5/1

Y1 - 2018/5/1

N2 - We present a dual form of Lyapunov-Krasovskii functional which allows the problem of controller synthesis for multi-delay systems to be formulated and solved in a convex manner. First, we give a generalized version of the dual stability condition formulated in terms of Lyapunov operators which are positive, self-adjoint and preserve the structure of the state-space. Second, we provide a class of such operators and express the stability conditions as positivity and negativity of quadratic Lyapunov-Krasovskii functional forms. Next, we adapt the SOS methodology to express positivity and negativity of these forms as LMIs, describing a new set of polynomial manipulation tools designed for this purpose. We apply the resulting LMIs to a battery of numerical examples and demonstrate that the stability conditions are not significantly conservative. Finally, we formulate a test for controller synthesis for systems with multiple delays, apply the test to a numerical example, and simulate the resulting closed-loop system.

AB - We present a dual form of Lyapunov-Krasovskii functional which allows the problem of controller synthesis for multi-delay systems to be formulated and solved in a convex manner. First, we give a generalized version of the dual stability condition formulated in terms of Lyapunov operators which are positive, self-adjoint and preserve the structure of the state-space. Second, we provide a class of such operators and express the stability conditions as positivity and negativity of quadratic Lyapunov-Krasovskii functional forms. Next, we adapt the SOS methodology to express positivity and negativity of these forms as LMIs, describing a new set of polynomial manipulation tools designed for this purpose. We apply the resulting LMIs to a battery of numerical examples and demonstrate that the stability conditions are not significantly conservative. Finally, we formulate a test for controller synthesis for systems with multiple delays, apply the test to a numerical example, and simulate the resulting closed-loop system.

KW - Asymptotic stability

KW - Controller Synthesis

KW - Delay Systems

KW - Delays

KW - LMIs

KW - Lyapunov methods

KW - Numerical stability

KW - Optimization

KW - Stability criteria

UR - http://www.scopus.com/inward/record.url?scp=85046364974&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046364974&partnerID=8YFLogxK

U2 - 10.1109/TAC.2018.2832470

DO - 10.1109/TAC.2018.2832470

M3 - Article

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

ER -