Robust controllability of linear systems with multiple

www.ietdl.org

Published in IET Control Theory and Applications Received on 10th August 2011

Revised on 15th May 2012 doi: 10.1049/iet-cta.2011.0486

ISSN 1751-8644

Brief Paper

Robust controllability of linear systems with multiple

delays in control

S.-H. Chen

1

F.-I Chou

2

J.-H. Chou

1,3

1

_{Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road,}

Kaohsiung 807, Taiwan

2

_{Mechatronics Section, Energy and Agile System Department, Metal Industries Research and Development Centre,}

1001 Kaonan Highway, Kaohsiung 811, Taiwan

3

_{Institute of System Information and Control, National Kaohsiung First University of Science and Technology,}

1 University Road, Yenchao, Kaohsiung 824, Taiwan

E-mail: [email protected]

Abstract: The considered problem is robust controllability of linear systems with both multiple delays in control and structured

parametric uncertainties. Under the assumption that the linear nominal system with multiple control delays is controllable, a

sufficient condition is proposed to preserve the assumed property when system uncertainties are introduced. The application

of the proposed sufficient condition is demonstrated in two examples.

1

Introduction

Control processes for dynamic systems are often severely

limited. One example of a dynamic system with control

delay is a control actuator. Time-delay systems have been

studied intensively in the past 20 years, most research

works have focused on stability conditions and stabilisation

problems [

1

–

8

] (see, e.g., and references therein). Although

some studies have successfully solved stability and control

problems in linear systems with multiple delays in

control [

9

–

12

], controllability remains an important line

of research in control theory because of its essential role

in dynamic control systems [

13

]. Thus, some researchers

studied the controllability problem in linear systems with

multiple control delays (see, e.g. [

14

–

21

] and references

therein). Further, obtaining accurate values for some system

parameters may be very difficult, if not impossible, because

of inaccurate measurements, inaccessible system parameters,

or variation of parameters. Such system uncertainties may

compromise the controllability property of linear systems

with multiple control delays. However, a literature review

shows no studies of the issue of robust controllability of

uncertain linear systems with multiple delays in control.

That is, the controllability problem has been studied in linear

systems with multiple delays in control but not in uncertain

linear systems with multiple delays in control.

This

study

therefore

developed

an

approach

for

investigating robust controllability in linear systems with

both multiple delays in control and structured parametric

uncertainties. For a linear nominal system in which

multiple delays in control are controllable, a sufficient

condition is proposed for preserving the assumed property

when system uncertainties are introduced. The proposed

sufficient condition preserves the assumed property by

revealing the explicit relationships among bounds on system

uncertainties. Two numerical examples are given to illustrate

the application of the proposed sufficient condition.

2

Controllability robustness

Consider the following linear system with both multiple

control delays and system uncertainties

˙x(t) = (A + A)x(t) +

p



i=0

(B

+ B

)u(t

− h

)

(1)

where x(t)

∈ R

_{is the system state vector; u(t)}

_{∈ R}

_{is the}

control input vector; 0

= h

<

h

<

h

<  <

h

denote the

time delays in control input where h

is a constant; A and B

are the n

× n and n × m constant matrices, respectively; and

A and B

are the uncertain matrices existing in system

matrix A and in the control input matrices B

, respectively,

because of inaccurate measurements, inaccessible system

parameters, or variation in system parameters.

Although many interesting problems involve only a small

number of uncertainties, the uncertainties may comprise

many entries in the system and in the input matrices

[

22

–

24

]. For example, consider the two-mass system

with an uncertain stiffness given by Sinha [

24

]. The

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