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70 Acta Electrotechnica et Informatica, Vol. 11, No. 3, 2011, 70–74, DOI: 10.2478/v10198-011-0032-9

CONTROL AND STABILITY ANALYZING OF THE TIME-DELAY SYSTEMS WITH

TIME-VARYING DELAYS

Daniel GONTKOVI ˇ

C, R´

obert F ´

ONOD

Department of Cybernetics and Artiﬁcial Intelligence, Faculty of Electrical Engineering and Informatics, Technical University of

Koˇ

sice, Letn´

a 9, 042 00 Koˇ

sice, Slovak Republic, tel.: +421 55 625 2749, e-mail: daniel.gontkovic@tuke.sk,

robert.fonod@student.tuke.sk

ABSTRACT

Systems with delays are a speciﬁc group of dynamic systems. The fact that some delays can be short and some can be long

makes system analysis and control design much more complex. In this paper we present a proposal to solve the problem of stability

and control design of continuous time systems with time-varying delays. Based on the Lyapunov-Krasovskii functional the stability

condition is derived using the linear matrix inequality (LMI) techniques, and convex optimization principle to ﬁnd LMI variables which

simultaneously satisﬁes the limitations given by the theory of Lyapunov-Krasovskii functionals. Obtained solution is the feasible convex

LMI problem for the static controller design. Finally the design method is demonstrated using a system model example.

Keywords: Lyapunov-Krasovskii functional, LMI, Lyapunov function, Time-delay systems, Time-varying delays, Schur complement

1. INTRODUCTION

Systems with delays frequently appear in engineering,

where time delays are the property of a physical system.

Whenever material, information or energy is physically

transmitted from one place to another, there is a delay asso-

ciated with the transmission. The value of the delay is de-

termined by the distance and the transmission speed, where

some delays are short, some are very long. Since delays

might lead to a system destabilization, additive conditions,

based on the assumption that in the system there is informa-

tion on delays states, are taking into account in design task

formulation. Such systems are called systems with delayed

state variables. However, the presence of long delays makes

system analysis and control design substantially much more

complex.

The study of functional differential equations started

long before 1900, but the mathematical formulation of the

problems were developed in the 20th century. Thus, the

notion of a functional differential equation was introduced

by Myshkis [13] in 1949. A further progress have been

made to study systems with delays in the past 50 years

( [14], [2] and the references therein); main ideas concern-

ing the work in this area can be found in e.g. Krasovskii [8]

(time-domain approach, extension of the Lyapunov second

method to functional differential equations). Burton [1] (re-

ﬁnements of the Lyapunov-Krasovskii theory and periodic

solutions) During the last decades, we have witnessed sig-

niﬁcant development in the control of time-delay systems.

Some recent comprehensive introductions to the problem

are in Greeki [5], Marshall et al. [12], Kolmanovskii and

Myshkis [7] and Hale and Lunel [6].

In this paper the design task of the stabilizing con-

troller for the closed-loop system is transmuted into LMI

framework and solved. There are two ways to use the

second method of Lyapunov for time-delay systems. The

ﬁrst is based on the theory of Lyapunov-Krasovskii func-

tionals and the other is based on the theory of Lyapunov-

Razumikhin functions. The paper extends the control

design and stability analyzing methods based on theory

of Lyapunov-Krasovskii functionals with limited speed of

change of time delays on the interval β=h0,1).

2. SYSTEM MODEL

Equations of linear dynamic systems with time delay

are as follows

˙

q(t) = Aq(t) + Adq(t−τ(t)) + Bu(t)(1)

y(t) = Cq(t)(2)

0≤τ(t)≤τm<∞, 0 <˙

τ(t)≤β<1, τm,β∈IR

with the initial condition

q(ϑ) = ϕ(ϑ),∨ϑ∈ h−τm,0i(3)

where τ(t)>0 is the time-varying delay of state, q(t)∈IR n

is a vector of the state, u(t)∈IR rand y(t)∈IR mare vectors

of the input and output variables, A∈IR n×nis the nominal

system matrix, Ad∈IR n×n,B∈IR n×r,C∈IR m×n.

Problem of the interest is to design stable close-loop

system with the linear memoryless state feedback controller

of the form

u(t) = −Kq(t),(4)

where t≥0 and K∈IR r×nis the controller gain matrix.

3. BASIC PRELIMINARIES

Working with the linear matrix inequalities the con-

cept of symmetric positive deﬁnite matrix or positive semi-

deﬁnite matrix is principally used. The necessary and suf-

ﬁcient conditions for positive deﬁniteness of a symmetric

matrix Pare:

– there is a regular matrix Q, such as that

P=QTQ,

– all eigenvalues of the matrix Pare positive,

– all main subdeterminants of the matrix Pare positive

p11 >0,

p11 p12

p21 p22

>0, . . . , det(P)>0.

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Acta Electrotechnica et Informatica, Vol. 11, No. 3, 2011 71

Working with the norm, in the next k · k denotes the stan-

dard Euclidean norm, and k·k∞the H∞norm.

Proposition 3.1. (Schur Complement)

If Q=QT>0,R=RT,Sare real matrices of appro-

priate dimensions, then the next inequalities are equivalent

Q S

STR>0⇔Q−SR−1ST0

0 R >0

m

Q−SR−1ST>0,R>0.

(5)

Proof. (e.g. see [9, 10])

Let the linear matrix inequalities takes form

Q S

STR>0 (6)

than using Gauss elimination it yields

I−SR−1

0 I Q S

STR I 0

−R−1STI=

=Q−SR−1ST0

0 R

(7)

and it is evident that this transformation does not change

positivity (6), and so (7) implies (5).

Deﬁnition 3.1. (Lyapunov function [11, 15])

For some τ(t)>0the equilibrium 0of system (1), (2)

is:

– uniformly stable if there exists a positive deﬁ-

nite continuous functional v(qt(θ)) whose derivative

˙v(qt(θ)) is negative semi-deﬁnite functional;

– uniformly asymptotically stable if there exists a pos-

itive deﬁnite upper-bounded continuous functional

v(qt(θ)), whose derivative ˙v(qt(θ)) is negative deﬁ-

nite functional;

– uniformly exponentially stable if there exist a posi-

tive deﬁnite continuous functional v(qt(θ)) and pos-

itive real constants α,β,γ,δsuch that αkqt(θ)k ≤

v(qt(θ)) ≤βkqt(θ)k,˙v(qt(θ)) ≤ −γkqt(θ)k, and

|vqt(θ1)) −v(qt(θ2))|≤δkqt(θ1)−qt(θ2)k.

Proposition 3.2. (Lyapunov-Krasovskii Ineqality [3,4])

The linear time delay system of form (1), (2) is sta-

ble with quadratic performance kC(sI−A)−1Bk∞=λfor

˙

τ(t)≤βif exists a symmetric positive deﬁnite matrix Pand

Qand a scalar variable γ>0such that

ATP+PA +Q PAdPB CT

∗ −(1−β)Q 0 0

∗ ∗ −γIr0

∗ ∗ ∗ −Im

<0

P=PT>0,Q=QT>0,γ>0

(8)

Hereafter, *denotes the symmetric item in a symmetric ma-

trix.

The problem is reduced to solving LMI i.e. to ﬁnd

matrices Pand Q,P∈IR n×n,Q∈IR n×n, and a scalar vari-

able γ>0, which simultaneously satisﬁes the boundary (8).

Proof. Deﬁning Lyapunov-Krasovskii functional candidate

as follows

v(q(t),t) = qT(t)Pq(t) +

t

R

t−τ(t)

qT(r)Qq(r)dr+

+

t

R

0

(yT(r)y(r)−γuT(r)u(r))dr>0

P>0,Q>0,γ>0

(9)

Using Laplace transform, let

˜

y(s)= G(s)˜

u(s),kG(s)k∞=kC(sI−A)−1Bk∞=λ

(10)

when ˜

y(s),˜

u(s)stand for Laplace transform of mdimen-

sional output vector and rdimensional input vector signal,

respectively and G(s)is the transfer function matrix.

Then Parcevall’s theorem implies

"∞

R

0

yT(t)y(t)dt#1

2

"∞

R

0

uT(t)u(t)dt#1

2

=ky(t)k

ku(t)k≤λ.(11)

Since H∞norm can be interpreted as a maximum gain in

any direction and at any frequency then there exists a scalar

γ λ 2>0 such that Lyapunov-Krasovskkii functional candi-

date be positive deﬁnite.

The result of the functional derivative of v(q(t),t)is

˙v(q(t),t) =

=˙

qT(t)Pq(t) + qT(t)P˙

q(t) + qT(t)Qq (t)−

−qT(t−τ(t))Qq (t−τ(t)) (1−˙

τ(t))+

+yT(t)y(t)−γuT(t)u(t)<0

(12)

Inserting (1) into (12) gives

(Aq(t) + Adq(t−τ(t)) + Bu(t))TPq(t)+

+qT(t)P(Aq(t) + Adq(t−τ(t)) + Bq(t))+

+qT(t)Qq(t)−(1−˙

τ(t))qT(t−τ(t))Qq(t−τ(t))+

+qT(t)CTCq(t)−γuT(t)u(t)<0

(13)

where

yT(t)y(t) = qT(t)CTCq(t).(14)

Creating a composite vector q◦(t)

q◦T(t) = qT(t)qT(t−τ(t)) uT(t)(15)

the inequality (13) can be rewritten in a form

q◦T(t)

∆•

11 PAdPB

∗ −(1−β)Q 0

∗ ∗ −γIr

q◦(t)(16)

where ∆•

11 =ATP+PA +Q+CTC.

Using Schur complement property it is possible to write

ATP+PA +Q PAdPB CT

∗ −(1−β)Q 0 0

∗ ∗ −γIr0

∗ ∗ ∗ −Im

<0.(17)

It is obvious that (17) implies (8).

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72 Control and Stability Analyzing of the Time-Delay Systems with Time-Varying Delays

4. PARAMETER DESIGN

Theorem 4.1. For system (1), (2) the necessary and suf-

ﬁcient condition for the stable nominal control (4) with

quadratic performance kC(sI−A)−1Bk∞=λis that there

exist positive deﬁnite symmetric matrices X,Z, matrix Y

and a scalar variable γ>0, such that the following LMIs

are satisﬁed

∆11 X AdZ B XCT

∗ −Z 0 0 0

∗ ∗ −(1−β)Z 0 0

∗ ∗ ∗ −γIr0

∗ ∗ ∗ ∗ −Im

<0

X>0,Z>0,γ>0

(18)

where ∆11 =XAT+AX −BY −YBT.

The control law gain matrix is then given as

K=YX−1.(19)

Proof. The linear state feedback control law, deﬁned in (4)

gives rise to the closed-loop system as follows

˙

q(t) = (A−BK)q(t) + Adq(t−τ(t)).(20)

Subsituting in (8) gives for P=PT>0 and Q=QT>0

∆•

11 PAdPB CT

∗ −(1−β)Q 0 0

∗ ∗ −γIr0

∗ ∗ ∗ −Im

<0 (21)

where ∆•

11 = (A−BK)TP+P(A−BK) + Q.

Deﬁning the congruence transform matrix

T1=T1T=diagP−1III(22)

and premultiplying (21) from the left and right side by T1

gives

∆◦

11 AdB P−1CT

∗−(1−β)Q 0 0

∗ ∗ −γIr0

∗ ∗ ∗ −Im

<0 (23)

where ∆◦

11 =P−1(A−BK)T+ (A−BK)P−1+P−1QP−1.

Substituting P−1=Xand Y=KP−1then (23) can be

rewriten such that

∆

11 AdB XCT

∗ −(1−β)Q 0 0

∗ ∗ −γIr0

∗ ∗ ∗ −Im

<0 (24)

where ∆

11 =XAT+AX −BY −YBT+XQX.

Using the Schur complement (24) can be modiﬁed as

∆11 X AdB XCT

∗ −Q−10 0 0

∗ ∗ −(1−β)Q 0 0

∗ ∗ ∗ −γIr0

∗ ∗ ∗ ∗ −Im

<0 (25)

where ∆11 =XAT+AX −BY −YBT.

Deﬁning the next congruence transform matrix

T2=T2T=diagIIQ−1IrIm(26)

and then premultiplying (25) from the right and left side by

(26) modify it in the form

∆11 X AdQ−1B XCT

∗ −Q−10 0 0

∗ ∗ ∆33 0 0

∗ ∗ ∗ −γIr0

∗ ∗ ∗ ∗ −Im

<0 (27)

where ∆33 =−(1−β)Q−1. The substitution Q−1=Zleads

to inequality

∆11 X AdZ B XCT

∗ −Z 0 0 0

∗ ∗ −(1−β)Z 0 0

∗ ∗ ∗ −γIr0

∗ ∗ ∗ ∗ −Im

<0.(28)

It is obvious that (28) implies (18).

5. ILLUSTRATIVE EXAMPLE

To demonstrate the algorithm properties it was assumed

that system is given by (1), (2), where

A=

−2.6 0.0 0.8

−1.2 0.2 0.0

0.0 0.5−3.0

,C=

1 1

2 1

1 0

,

Ad=

0.00 0.02 0.00

0.00 0.00 −1.00

−0.02 0.00 0.00

,B=

4 0

7 1

1 0

.

Solving (18) with respect to the LMI matrix variables

X>0, Z>0, Yand a scalar variable γ>0 using SeDuMi

package for Matlab, given task was feasible with

X=

1.3892 −0.5363 −0.0136

−0.5363 0.9210 −0.2723

−0.0136 −0.2723 1.1760

,

Z=

3.9383 −0.0842 0.0013

−0.0842 3.8726 −0.0286

0.0013 −0.0286 3.0627

,

Y=−0.1671 2.0496 −0.2288

−0.1352 0.4042 0.5026 ,γ=4.6235,

where P−1=Xand Q−1=Z. The controller gain ma-

trix K1can be obtained from the relation (19) for β=0.1,

where

K1=1.0341 2.9774 0.5068

0.1885 0.7253 0.5975 ,

ρ(Au) = ρ(A−BK1) = {−11.0867,−2.0799,−2.7745}

It is evident that the eigenvalues spectrum ρ(Au)of the

closed control loop is stable.

Setting β=0.5 the gain matrix K2takes the form

K2=1.1799 3.3692 0.5428

0.1997 0.8179 0.6076 ,

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Acta Electrotechnica et Informatica, Vol. 11, No. 3, 2011 73

ρ(Au) = ρ(A−BK2) = {−12.4348,−2.0624,−2.7705}

and the spectrum of the eigenvalues ρ(Au)of the closed

control loop is stable, too.

Solving with respect to the controller gain matrix K3

and β=0.99 results

K3=2.3169 5.3545 1.2816

0.3283 1.1451 −0.8732 .

Even for this case is the eigenvalues spectrum ρ(Au)of the

closed control loop is stable, since

eig(Au) = {−18.8092,−1.4764,−4.2613}.

5.1. Simulation

Time-delay linear dynamic system (1), (2) is raised up

to the steady state starting with the nonzero initial condi-

tions [−1,0.5,3]T.

In Figure 1 to 3 is shown the time response of the

closed-loop system with respect to the stability region con-

ditioned by the delay time changes interval 0 <˙

τ(t)≤1.

Simulations were made with the prescribed values β=0.1,

β=0.5, β=0.99 from above deﬁned interval. Note, the

closed-loop system is t∈ h−τ(t),0)unstable, because one’s

eigenvalue of Adpositive. Closed-loop system reaches

the steady state values of output variables approximately

in time instants less then 11s. It can be easily seen that

the criterion proposed in this paper to design the memory-

less feedback controller gives acceptable solutions. More-

over, the limited range of the delay time changes interval

0<˙

τ(t)≤1 is given by the necessary negativeness of the

Lyapunov-Krasovskii functional derivative.

6. CONCLUDING REMARKS

Time delays can appear as a part of the dynamics in

many technological processes. Presence of delays in the

system is making the analysis and control design more com-

plex because it has a negative impact on the stability of sys-

tem. Tendency to instability generally grows with respect

to the size of the time delay. Therefore, the proposal of en-

suring stability and control design systems with delays is a

very interesting topic in the scientiﬁc community.

The method uses the standard LMI numerical optimiza-

tion procedures to manipulate the system feedback gain

matrix as the direct design variable. Since it is necessary

to propose an asymptotically stable closed-loop system, to

solve the control design problem in the sense of delay-

independent criteria, we typically use such modiﬁcations

of Lyapunov-Krasovskii functional as shown above. An ex-

ample, illustrating the design method certiﬁes the presented

principle is applicable.

Fig. 1 Response of the closed-loop system for β=0.1

Fig. 2 Response of the closed-loop system for β=0.5

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74 Control and Stability Analyzing of the Time-Delay Systems with Time-Varying Delays

Fig. 3 Response of the closed-loop system for β=0.99

ACKNOWLEDGEMENT

The work presented in this paper was supported by

VEGA, Grant Agency of Ministry of Education and

Academy of Science of Slovak Republic under Grant No.

1/0256/11. This support is very gratefully acknowledged.

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[3] FILASOV ´

A, A. – KROKAVEC, D.: Asymptotically

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Strbsk´

e Pleso, Slovak Re-

public, pp. 358–362.

[4] FILASOV ´

A, A. – KROKAVEC, D.: Global asymp-

totically stable control design for time-delay systems.

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[5] G ´

ORECKI, H. et al.: Analysis and Synthesis of Time

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ory of Functional Differential Equations, Dordrecht:

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Received March 4, 2011, accepted July 8, 2011

BIOGRAPHIES

Daniel Gontkoviˇ

cwas born in 1984 in Poprad, Slovakia.

He graduated in automation in 2009 from the Faculty of

Electrical Engineering and Informatics, Technical Univer-

sity of Koˇ

sice, Slovakia. Since 2009 he is a PhD. student

with the Department of Cybernetics and Artiﬁcial Intelli-

gence, FEI TU Koˇ

sice. His scientiﬁc research is focusing

on time-delay systems and system control.

R´

obert F´

onod was born in 1987 in Veˇ

lk´

y Krt´

ıˇ

s, Slovakia.

He received BSc. degree in Cybernetics in 2009 from the

Faculty of Electrical Engineering and Informatics, Techni-

cal University of Koˇ

sice. Since 2009 he is a MSc. student

in the same study branch with the Department of Cybernet-

ics and Artiﬁcial Intelligence, FEI TU Koˇ

sice. His interest

is in dynamic system fault diagnostics.

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