Content uploaded by Robert Fonod

Author content

All content in this area was uploaded by Robert Fonod on Oct 18, 2017

Content may be subject to copyright.

State Control Design for Linear Systems with

Distributed Time Delays

Daniel Gontkoviˇ

c and Róbert Fónod

Technical University of Košice, Faculty of Electrical Engineering and Informatics

Department of Cybernetics and Artiﬁcial Intelligence

Košice, Slovakia

e-mail: damiel.gontkovic@tuke.sk; robert.fonod@student.tuke.sk

Abstract—This paper is concerned with the problem of

stabilization for continuous-time systems with distributed

time delays. Using an extended form of the Lyapunov-

Krasovskii functional the controller design conditions are

derived with respect to application of structured matrix

variables in linear matrix inequalities. The result giving a

sufﬁcient condition for stabilization of the system with dis-

tributed time delays is illustrated with a numerical example

to note reduced conservatism in the system structure.

Keywords — Linear matrix inequality, systems with dis-

tributed time delays, Lyapunov-Krasovskii functional, state

control, asymptotic stability.

I. INTRODUCT IO N

Control systems are used in many industrial applica-

tions, where time delays can take a deleterious effect

on both the stability and the dynamic performance in

open and closed-loop systems. Therefore the stability and

control of the dynamical systems involving distributed

time delays is a problem of large practical interest where

intensive activity are done to develop control laws for

systems stabilization.

During the last decades, considerable attention has been

devoted to the problem of stability analysis and con-

troller design for systems with time-delay. The existing

stabilization results for time-delay systems can be de-

lay independent or delay-dependent. The delay-dependent

stabilization is concerned with the size of the delay and

usually provides an upper bound of the delay such that

the closed loop system is stable for any delay less than

the given upper bound. On the other side, the delay-

independent stabilization provides a controller, which can

stabilize given system irrespective of the size of the delay.

The use of Lyapunov method for stability analysis of

the time delay systems has been ever growing subject of

interest, starting with the pioneering works of Krasovskii

[9], [10]. Usually nowadays for the stability issue some

modiﬁed Lyapunov-Krasovskii functionals are used (e.g.

see [4], [5]) to obtain delay-independent stabilization and

the results based on these functionals are applied to

controller synthesis and observer design. This time-delay

independent methodology and the bounded inequality

techniques are sources of a conservatism that can cause

higher norm of the state feedback gain. Progres review in

this research ﬁeld can be found e.g. in [18], [19], and the

references therein.

Despite the signiﬁcance, as the controllers are usually

digitally implemented, systems with distributed time de-

lays have not been paid due attention, and in contradiction

to results given e.g. in [13], [15] there didn’t exist

much structures to solve this problem, formulated with

respect to LMI ([1], [6], [8], [16]). However, standard

schemes are not applicable to systems with distributed

time delays and new design conditions have to be derived

[7]. By introducing triple integral terms into Lyapunov-

Krasovskii functional [17], the conservatism of condi-

tions is further reduced but the design task still state

singular. The presented LMI approach is based on the

form of Lyapunov-Krasovskii functional used in [2] but

the stability conditions as well as the controller design

condition are reformulated with respect the application

of structured matrix variables in LMI solution. Generally,

since Lyapunov-Krasovskii functional is used only suf-

ﬁcient conditions for system stability are obtained. Used

modiﬁcation was motivated by [3], in here presented form

enables to design systems with standard structures. It

seems, another applications based on the bounded real

lemma are immediate.

II. SYSTE M MOD EL

The systems under consideration are understood as

multi-input and multi-output linear (MIMO) dynamic

systems with distributed time delay. Without lose of

generalization this class of systems can be represented

in a state-space form by the set of equations

˙

q(t) = Aq(t) + Ah

t

Z

t−h

q(s)ds+Bu(s)(1)

y(t) = Cq(t) + Du(t)(2)

with the initial condition

q(θ) = ϕ(θ),∀θ∈ h−(h+h

m),0i(3)

where h > 0represents the system distributed delay,

m > 0is a partitioning factor, q(t)∈IRn,u(t)∈IRr,

and y(t)∈IRpare vectors of the state, input and

output variables, respectively, and matrices A∈IRn×n,

Ah∈IRn×n,B∈IRn×r,C∈IRp×n, and D∈IRp×r

are real matrices. Throughout the paper it is assumed that

the couple (A,B) is controllable.

SAMI 2012 • 10th IEEE Jubilee International Symposium on Applied Machine Intelligence and Informatics • January 26-28, 2012 • Herl’any, Slovakia

– 97 –

978-1-4577-0197-9/11/$26.00 ©2011 IEEE

Using a linear memoryless state feedback controller

u(t) = −Kq(t)(4)

where the matrix K∈IRr×nis the gain matrix, problem

of the interest is to design Ksuch that the closed-loop

system

˙

q(t) = (A−BK )q(t) + Ah

t

Z

t−h

q(s)ds(5)

is asymptotically stable for given h.

III. BASI C PRELIMINARIES

Proposition 1: If Nis a positive deﬁnite symmetric

matrix, and Mis a square matrix of the same dimension

then

M−TN M −1≥M−1+M−T−N−1(6)

Proof: Since Nis positive deﬁnite then it yields

(M−1−N−1)TN(M−1−N−1)≥0(7)

M−TN M −1−M−T−M−1+N−1≥0(8)

respectively, and evidently (8) implies (6). This concludes

the proof.

Proposition 2: (Schur Complement) Let S,Q=QT,

R=RT,det R6= 0 are real matrices of appropriate

dimensions, then the next inequalities are equivalent

Q S

STR>0⇔Q−SR−1ST0

0 R >0

m

Q−SR−1ST>0,R>0

(9)

Proof: (see e.g. [11]) Let the linear matrix inequality

takes form Q S

STR>0(10)

then using Gauss elimination principle it yields

I−SR−1

0 I Q S

STR I 0

−R−1STI=

=Q−SR−1ST0

0 R (11)

Since

det I−SR−1

0 I = 1 (12)

it is evident that this transform doesn’t change positivity

of (10), and so (11) implies (9). This concludes the proof

(compare e.g. [11]).

Proposition 3: (Symmetric upper-bounds inequalities)

Let f(x(p)),x(p)∈IRn,X=XT>0,X∈IRn×nis

a real positive deﬁnite and integrable vector function of

the form

f(x(p)) = xT(p)Xx(p)(13)

such that there exist well deﬁned integrations as following

Z0

−bZt

t+r

f(x(p))dpdr > 0(14)

Zt

t−b

f(x(p))dp > 0(15)

with b > 0,b∈IR,t∈ h0,∞), then

0

R

−b

t

R

t+r

xT(p)Xx(p)dpdr≥

≥2

b2

0

R

−b

t

R

t+r

xT(p)dpdrX

0

R

−b

t

R

t+r

x(p)dpdr

(16)

t

Z

t−b

xT(p)Xx(p)dp≥1

b

t

Z

t−b

xT(p)dpX

t

Z

t−b

x(p)dp(17)

Proof: (see e.g. [12]) Since with (13) it can be written

xT(p)Xx(p)−xT(p)Xx(p) = 0 (18)

and according to Schur complement (9) it is true that

xT(p)Xx(p)xT(p)

x(p)X−1= 0 (19)

then the double integration of (19) leads to

0

R

−b

t

R

t+r

xT(p)Xx(p)dpdr

0

R

−b

t

R

t+r

xT(p)dpdr

0

R

−b

t

R

t+r

x(p)dpdr

0

R

−b

t

R

t+r

X−1dpdr

≥0(20)

Using the equalities

Zt

t+r

X−1dp=−rX−1,

0

Z

−b

−rX−1dr=r2

2X−1(21)

inequality (20) can be rewritten as

0

R

−b

t

R

t+r

xT(p)Xx(p)dpdr

0

R

−b

t

R

t+r

xT(p)dpdr

0

R

−b

t

R

t+r

x(p)dpdrr2

2X−1

≥0(22)

It is evident, that (22) implies (16).

Analogously using (19) it yields

t

R

t−b

xT(p)Xx(p)dp

t

R

t−b

xT(p)dp

t

R

t−b

x(p)dp

t

R

t−b

X−1dp

≥0(23)

and since

Zt

t−b

X−1dp=bX−1(24)

the following is obtained

t

R

t−b

xT(p)Xx(p)dp

t

R

t−b

xT(p)dp

t

R

t−b

x(p)dp bX−1

≥0(25)

which implies (17). This concludes the proof.

D. Gontkovic and R. Fónod • State Control Design for Linear Systems with Distributed Time Delays

– 98 –

IV. STAB ILITY O F TH E AUTO NO MOUS SYS TE M

Theorem 1: The autonomous system of (1) is asympto-

tically stable if for given h > 0,m > 0there exist

symmetric positive deﬁnite matrices P,U,V∈IRn×n,

W∈IRmn×mn such that

P=PT>0,U=UT>0,V=VT>0,W=WT>0(26)

P◦=TT

AP TI+TT

IP TA+

+TT

UU◦TU+TT

VV◦TV+TT

WW◦TW<0(27)

where

TU="qh

mIn

pm

hIn#"In00···00

0In0···00#(28)

TV="h

√2mIn

√2m

hIn#"A AhAh···Ah0

h

mIn−In0··· 00#(29)

TW="0wImn 0w

0w0wImn #(30)

U◦=U

−U,V◦=V

−V,W◦=W

−W(31)

TA=A AhAh·· · Ah0(32)

TI=In00·· · 00(33)

In∈IRn×n,Imn ∈IRmn×mn are identity matrices, 0∈

IRn×n,0w∈IRmn×nare zero matrices, respectively, and

U◦,V◦∈IR2n×2n,W◦∈IR2mn×2mn are structured

matrix variables.

Proof: Deﬁning Lyapunov-Krasovskii functional candi-

date as follows

v(q(t)) =

t

R

t−h

m

pT(s)W p(s)ds+

+qT(t)P q(t) +

0

R

−h

m

t

R

t+ϑ

qT(s)Uq(s)dsdϑ+

+

0

R

−h

m

0

R

ϑ

t

R

t+λ

˙

qT(s)V˙

q(s)dsdλdϑ+

(34)

with

pT(s) = pT

1(s)pT

2(s)(35)

pT

1(s) = Zt

t−h

m

qT(s)ds(36)

pT

2(s) = "t−h

m

R

t−2h

m

qT(s)ds···

t−(m−1) h

m

R

t−h

qT(s)ds#(37)

then evaluating the derivative of v(q(t)along a solution

of (1) it can be obtained

˙v(q(t)) = ˙v1(q(t)) + ˙v2(q(t))+

+˙

qT(t)P q(t) + qT(t)P˙

q(t)+

+pT(t)W p(t)−pT(t−h

m)W p(t−h

m)

(38)

where

˙v1(q(t)) =

=

0

R

−h

m

(t

R

t

qT(s)Uq(s)dϑ−

t

R

t+ϑ

qT(s)Uq(s)dϑ

)ds=

=

0

R

−h

m

qT(t)Uq(t)ds−

0

R

−h

m

t

R

t+ϑ

qT(s)Uq(s)dϑds=

=h

mqT(t)Uq(t)−

t

R

t−h

m

qT(s)Uq(s)ds

(39)

˙v2(q(t)) =

=

0

R

−h

m

0

R

ϑ(t

R

t

˙

qT(s)V˙

q(s)dλ−

t

R

t+λ

˙

qT(s)V˙

q(s)dλ

)dsdϑ=

=

0

R

−h

m

−ϑ˙

qT(t)V˙

q(t)dϑ−

0

R

−h

m

t

R

t+ϑ

˙

qT(s)V˙

q(s)dsdϑ=

=1

2h

m2˙

qT(t)V˙

q(t)−

0

R

−h

m

t

R

t+ϑ

˙

qT(s)V˙

q(s)dsdϑ

(40)

and subsequently, using (16), (17), it yields

˙v1(q(t)) ≤

≤h

mqT(t)Uq(t)−m

h

t

R

t−h

m

qT(s)dsU

t

R

t−h

m

q(s)ds=

=h

mqT(t)Uq(t)−m

hpT

1(t)Up1(t)

(41)

˙v2(q(t)) ≤1

2h

m2˙

qT(t)V˙

q(t)−

−2m

h20

R

−h

m

t

R

t+ϑ

˙

qT(s)dsdϑV

0

R

−h

m

t

R

t+ϑ

˙

q(s)dsdϑ=

=1

2h

m2˙

qT(t)V˙

q(t)−

−2m

h20

R

−h

m

t

R

t+ϑ

˙

qT(s)dsdϑV

h

mq(t)−

−

t

R

t−h

m

q(s)ds

=

=1

2h

m2˙

qT(t)V˙

q(t)−

−2m

h2(h

mqT(t)−pT

1(t))V(h

mq(t)−p1(t))

(42)

Thus, using the notation

q◦T(t) = qT(t)pT

1(t)pT

2(t)pT

3(t)(43)

pT

3(t) = Zt−h

t−h−h

m

qT(s)ds(44)

then with (32), (33) it can be written

Aq(t) + Ah

t

Z

t−h

q(s)ds=TAq◦(t)(45)

q(t) = TIq◦(t)(46)

and ˙

qT(t)P q(t) + qT(t)P˙

q(t) =

=q◦T(t)(TT

AP TI+TT

IP TA)q◦(t)(47)

In the same sense using (28)–(31) it can be obtained

pT(t)W p(t)−pT(t−h

m)W p(t−h

m) =

=q◦T(t)TT

WW◦TWq◦(t)(48)

SAMI 2012 • 10th IEEE Jubilee International Symposium on Applied Machine Intelligence and Informatics • January 26-28, 2012 • Herl’any, Slovakia

– 99 –

h

mqT(t)Uq(t)−m

hpT

1(t)Up1(t) =

=q◦T(t)TT

UU◦TUq◦(t)(49)

1

2h

m2˙

qT(t)V˙

q(t)−

−2m

h2(h

mqT(t)−pT

1(t))V(h

mq(t)−p1(t)) =

=q◦T(t)TT

VV◦TVq◦(t)

(50)

Thus, with P◦given in (27) it yields

˙v(q(t)) ≤q◦T(t)P◦q◦(t)<0(51)

and it is obvious that P◦has to be negative deﬁnite.

V. CONTROL LAW PARAME TE R DESIGN

Theorem 2: The closed-loop system (1) controlled by

the control law (4) is asymptotically stable if for given

h > 0,m > 0there exist symmetric positive deﬁnite

matrices Y,U•,V•∈IRn×n,W•∈IRmn×mn, and a

matrix Z∈IRr×nsuch that

Y=YT>0

U•=U•T>0,V•=V•T>0,W•=W•T>0(52)

"P•∗

TAY⋄−2m

h2V•#<0(53)

where

P•=Y⋄TT•T

ATI+TT

IT•

AY⋄+

+TT

UU⋄TU+TT

V2V•

2TV2+TT

WW⋄TW

(54)

U⋄=U•

−U•,W⋄=W•

−W•,V•

2=V•−2Y

(55)

TV2=√2m

hh

mIn−In0··· 00(56)

T•

A=[A−B]AhAh·· · Ah0(57)

Y⋄=diag Y

ZYY·· · YY(58)

Y⋄∈IR(n(m+2)+r)×n(m+2) ,W⋄∈IR2rn×2r n,U◦∈

IR2n×2nare structured matrix variables, and TU,TW,

and TIare used as in (28), (30), (33), respectively.

Now, the control gain is given as

K=ZY −1(59)

Hereafter, ∗denotes the symmetric item in a symmetric

matrix.

Proof: Using Schur complement property then (27) can

be rewritten as

"P⋄TT

V01

TV01 −2m

h2V−1#="P⋄TT

A

TA−2m

h2V−1#<0

(60)

where

P⋄=TT

APTI+TT

IP TA+

+TT

UU◦TU+TT

V2V◦TV2+TT

WW◦TW

(61)

Then deﬁning the congruence transform matrix

TC=diag TC1In=

=diag P−1P−1P−1···P−1P−1In(62)

and pre-multiplying right-hand side and left-hand side of

(60) by (62) gives the next result

"TC1P⋄TC1TC1TT

A

TATC1−2m

h2V−1#<0(63)

Using notation P−1=Ythen (63) implies

TATC1=TAY⋄(64)

Y⋄=T⋄

C1=diag Y Y Y·· · YY(65)

TC1(TT

AP TI+TT

IP TA)TC1=

=Y⋄TT

ATI+TT

ITAY⋄(66)

TC1TT

UU◦TUTC1=TT

UU⋄TU,U•=Y UY (67)

TC1TT

WW◦TWTC1=TT

WW⋄TW,

W•=diag Y·· · YWdiag Y··· Y(68)

and denoting V−1=V•then (6) implies

TC1TT

V2V TV2TC1≤TT

V2V•

2TV2(69)

Replacing the matrix Ain (32) by the closed-loop

system matrix Ac=A−BK results in

AcY=AY −BK Y (70)

and with the notation KY =Z(64) can be replaced by

T•

AY⋄.

Writing now compactly P•=TC1P⋄TC1as given in

(54), then (63) implies (53). This concludes the proof.

VI. ILLUST RATI VE EX AMPLE

To demonstrate the algorithm properties it was assumed

that system is given by (1), (2), where h= 6

A=

2.6 0.0−0.8

1.2 0.2 0.0

0.0−0.5 3.0

,C=121

110

Ah=

0.00 0.02 0.00

0.00 0.00 −1.00

−0.02 0.00 0.00

,B=

1 3

2 1

1 1

Setting m= 3 and solving (52), (53) with respect the LMI

matrix variables Y,Z,U•,V•, and W•using Self-

Dual-Minimization (SeDuMi) package [14] for Matlab

[6], the gain matrix problem was solved as feasible giving

K=−4.6371 −2.8106 19.8292

4.1418 1.9946 −12.9338

Ac=

−5.1883 −3.1732 18.1722

6.3324 3.8266 −26.7247

0.4953 0.3160 −3.8954

and the stable eigenvalue spectrum of the closed-loop

system matrix eig(A

c) = {−0.2110 −0.9606 −4.0855}.

To characterize the steady-state control properties the

extended closed-loop system matrix A

ce =A+A

h−BK

was computed, where

Ace =

−5.1883 −3.1532 18.1722

6.3324 3.8266 −27.7247

0.4753 0.3160 −3.8954

D. Gontkovic and R. Fónod • State Control Design for Linear Systems with Distributed Time Delays

– 100 –

−6 0 5 10 15 20 25 30

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

qh(t)

t[s]

qh1(t)

qh2(t)

qh3(t)

Fig. 1. Distributed delay state response of the system

This matrix eigenvalue spectrum is also stable since

eig(A

ce) = {−0.2380 −1.1129 −3.9062}

In the presented Fig. 1 the example is shown of the

unforced closed-loop system state response, where the

initial state was qT

h(−6) = [0.1 0.0−0.1]. It is possible to

verify that closed-loop dynamic properties for this unsta-

ble autonomous time-delay system are less conservative.

VII. CONCLUDING REMARKS

Modiﬁed design conditions, explained with respect to

special forms of structured matrix variables and based

on an extended version of the Lyapunov-Krasovskii func-

tional, are given in the paper. Obtained formulation is

a convex LMI problem where the manipulation is ac-

complished in that manner that produces the closed-loop

system asymptotical stability. Presented illustrative exam-

ple conﬁrms the effectiveness of proposed control design

techniques. In particular, with the use of an extended

version of Lyapunov-Krasovskii functional, it was shown

how to adapt the standard approach to design optimal

matrix parameters of state controller for systems with

distributed time delays.

ACK NOW LEDGM EN T

The work presented in this paper was supported by

Grant Agency of Ministry of Education and Academy

of Science of Slovak Republic VEGA under Grant No.

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

REF ERENC ES

[1] D. Boyd, L. El Ghaoui, E. Peron, and V. Balakrishnan, Linear

Matrix Inequalities in System and Control Theory, Philadelphia,

SIAM Society for Industrial and Applied Mathematics, 1994.

[2] Z. Feng and J. Lam, "Integral partitioning approach to stability

analysis and stabilization of distributed time delay systems",

Preprints of the 18th IFAC World Congress, Milano, Italy, 2011,

pp. 5094-5099.

[3] A. Filasová and D. Krokavec, "Uniform stability guaranty control

of the discrete time-delay systems", Journal of Cybernetics and

Informatics, vol. 10, 2010, pp. 21-28.

[4] A. Filasová and D. Krokavec, "Delay-dependent control of lin-

ear time-delay systems", Proceedings of the 12th International

Carpathian Control Conference ICCC’2011, Velké Karlovice,

Czech Republic 2011, pp. 111-114.

[5] E. Fridman, "New Lyapunov-Krasovskii functionals for stability

of linear returded and neutral type systems", Systems and Control

Letters, vol. 43, 2001, pp. 309-319.

[6] P. Gahinet, A. Nemirovski, A.J. Laub, and M. Chilali, LMI Control

Toolbox User’s Guide, Natick, The MathWorks, Inc., 1995.

[7] K. Gu, "An improved stability criterion for systems with dis-

tributed delays", International Journal of Robust & Nonlinear

Control, vol. 13, 2003, p. 819 ˝

U831.

[8] G. Herrmann, M.C. Turner, and I. Postlethwaite, "Linear matrix

inequalities in control", Mathematical Methods for Robust and

Nonlinear Control, Berlin, Springer-Verlag, 2007, pp. 123-142.

[9] N.N. Krasovskii, "On the application of Lyapunov’s second

method for equations with time delays", Prikladnaja matematika i

mechanika, vol. 20, 1956, pp. 315-327. (in Russian)

[10] N.N. Krasovskii, Stability of Motion: Application of Lyapunov’s

Second Method to Differential Systems and Equations with Delay,

Standford, Standford University Press, 1963.

[11] D. Krokavec and A. Filasová, Discrete-Time Systems, Košice,

Elfa, 2006. (in Slovak)

[12] D. Krokavec and A. Filasová, "Exponential stability of networked

control systems with network-induced random delays", Archives

of Control Sciences, vol. 20, 2010, No. 2, pp. 165-186.

[13] S.I. Niculescu, E.I. Veriest, L. Dugard, and J.M. Dion, "Stability

and robust stability of time-delay systems: A guided tour", Stabil-

ity and Control of Time-delay Systems, Berlin, Springer–Verlag,

1998, pp. 1-71.

[14] D. Peaucelle, D. Henrion, Y. Labit, and K. Taitz, User’s Guide for

SeDuMi Interface 1.04, Toulouse, LAAS-CNRS, 2002.

[15] U. Shaked, I. Yaesh, and C.E. De Souza, "Bounded real criteria

for linear time systems with state-delay", IEEE Transactions on

Automatic Control, vol, 43, 1998, pp. 1116–1121

[16] R.E. Skelton, T. Iwasaki, and K. Grigoriadis, A Uniﬁed Algebraic

Approach to Linear Control Design, London, Taylor & Francis,

1998.

[17] J. Sun, J. Chen, G. Liu, and D. Rees, "On robust stability of

uncertain neutral systems with discrete and distributed delays",

Proceedings of American Control Conference, St. Louis, MO,

USA, 2009, pp. 5469-5473.

[18] M. Wu, Y. He, J.J. She, and G.P. Liu, "Delay-dependent criteria

for robust stability of time-varying delay systems", Automatica,

vol. 40, 2004, pp. 1435-1439.

[19] Q.C. Zhong, Robust Control of Time-delay Systems, London,

Sprin-ger-Verlag, 2006.

SAMI 2012 • 10th IEEE Jubilee International Symposium on Applied Machine Intelligence and Informatics • January 26-28, 2012 • Herl’any, Slovakia

– 101 –