TY - JOUR

T1 - Negative Imaginary Theory for a Class of Linear Time-Varying Systems

AU - Kurawa, Suleiman

AU - Bhowmick, Parijat

AU - Lanzon, Alexander

N1 - Funding Information:
Manuscript received March 17, 2020; revised May 28, 2020; accepted June 19, 2020. Date of publication July 7, 2020; date of current version July 24, 2020. This work was supported in part by the Engineering and Physical Sciences Research Council under Grant EP/R008876/1, and in part by the Petroleum Technology Development Fund Overseas Scholarship Scheme. All research data supporting this publication are directly available within this publication. Recommended by Senior Editor V. Ugrinovskii. (Corresponding author: Parijat Bhowmick.) The authors are with the Control Systems Centre, Department of Electrical and Electronic Engineering, School of Engineering, University of Manchester, Manchester M13 9PL, U.K. (e-mail: suleiman.kurawa@postgrad.manchester.ac.uk; parijat.bhowmick@ manchester.ac.uk; alexander.lanzon@manchester.ac.uk). Digital Object Identifier 10.1109/LCSYS.2020.3007403
Publisher Copyright:
© 2017 IEEE.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2021/7/1

Y1 - 2021/7/1

N2 - This letter introduces the notion of linear time-varying (LTV) negative imaginary systems. LTV negative imaginary systems are defined using a time-domain dissipative supply rate $w$ ( $u,\dot {y}$ ) that depends on input to the system ( $u$ ), time-derivative of the system's output ( $\dot {y}$ ) and an index $\delta \geq 0$. For $\delta > 0$ , it gives rise to a strict subclass within the LTV negative imaginary systems, termed as LTV output strictly negative imaginary systems. For characterizing the proposed class of systems, a set of linear differential matrix inequality conditions is derived based on the given state-space realization. Subsequently, LTV negative imaginary theory is specialized to linear parameter-varying (LPV) cases for which, the differential matrix inequality conditions can easily be avoided by considering the rate of variation of the uncertain parameters as independent LMI variables. Finally, a set of sufficient conditions is derived which ensures that the origin is a globally asymptotically stable equilibrium point of an unforced positive feedback interconnection of two uniformly asymptotically stable LTV negative imaginary systems.

AB - This letter introduces the notion of linear time-varying (LTV) negative imaginary systems. LTV negative imaginary systems are defined using a time-domain dissipative supply rate $w$ ( $u,\dot {y}$ ) that depends on input to the system ( $u$ ), time-derivative of the system's output ( $\dot {y}$ ) and an index $\delta \geq 0$. For $\delta > 0$ , it gives rise to a strict subclass within the LTV negative imaginary systems, termed as LTV output strictly negative imaginary systems. For characterizing the proposed class of systems, a set of linear differential matrix inequality conditions is derived based on the given state-space realization. Subsequently, LTV negative imaginary theory is specialized to linear parameter-varying (LPV) cases for which, the differential matrix inequality conditions can easily be avoided by considering the rate of variation of the uncertain parameters as independent LMI variables. Finally, a set of sufficient conditions is derived which ensures that the origin is a globally asymptotically stable equilibrium point of an unforced positive feedback interconnection of two uniformly asymptotically stable LTV negative imaginary systems.

KW - global uniform asymptotic stability

KW - LPV negative imaginary systems

KW - LTV negative imaginary systems

KW - non autonomous systems

U2 - 10.1109/LCSYS.2020.3007403

DO - 10.1109/LCSYS.2020.3007403

M3 - Article

AN - SCOPUS:85089513740

VL - 5

SP - 1001

EP - 1006

JO - IEEE Control Systems Letters

JF - IEEE Control Systems Letters

IS - 3

M1 - 9134410

ER -