Second-order reciprocally convex approach to stability of systems with interval time-varying delays

Applied Mathematics and Computation 229 (2014) 245–253

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Second-order reciprocally convex approach to stability of systems with interval time-varying delays Won Il Lee, PooGyeon Park ⇑ Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, Pohang, Republic of Korea

a r t i c l e

i n f o

Keywords: Triple integral terms Reciprocally convex approach Interval time-varying delays Stability analysis

a b s t r a c t Recently, some triple integral terms in the Lyapunov–Krasovskii functional have been introduced in the literature to reduce conservatism in the stability analysis of systems with interval time-varying delays. When we apply the Jensen inequality to partitioned double integral terms in the derivation of LMI conditions, a new kind of linear combination of positive functions weighted by the inverses of squared convex parameters emerges. This paper proposes an efficient method to manipulate such a combination by extending the lower bound lemma. Some numerical examples are given to demonstrate the improvement of the proposed method. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Most recent stability analyses for time-delayed systems have been based on the Jensen inequality lemma [1–11] because such approaches require fewer decision variables than approaches based on the integral inequality lemma [12–16] while achieving identical or comparable performance behavior. In the derivation of LMI conditions, applying the Jensen inequality lemma to partitioned single integral terms yields a special type of function combination called a reciprocally convex comR thðtÞ R thðtÞ bination. In [17], an approximation on the difference between delay bounds, th2 ðh2  h1 ÞxðaÞda P th2 ðh2  hðtÞÞxðaÞ da where h1 6 hðtÞ 6 h2 has introduced to handle such combinations with little sacrifice of conservatism. However, through use of the lower bound lemma [18], this conservatism has been removed and the performance has become identical to that of approaches based on the integral inequality lemma. Recently, some triple integral terms have been introduced in the Lyapunov–Krasovskii functional to develop a less conservative stability criterion [2,15,19–21]. In the process of the derivation, the double integral term in [3] is not partitioned owing to the emergence of function combinations with squared convex parameters, which cannot be handled directly by the lower bound lemma. Consequently, in [2], a novel approach is proposed to handle such a combination by introducing some R th1 1 integral terms with delay-dependent coefficients, such as hðtÞh xðaÞ da, in the augmented vector. However, there still thðtÞ 1 remains some conservatism because the aforementioned approximation is still used in the middle stage of the derivation. In this paper, based on the reciprocally convex approach [18], we propose a new approach that directly handles function combinations arising from the manipulation of the triple integral terms. We develop a less conservative stability criterion using variations of the lower bound lemma for dealing with various kinds of function combinations. It is worth noting that the conservertism is reduced by the new simple mathematical approach instead of focusing on designing the new Lyapunov– Krasovskii functional or introducing new integral inequalities. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

⇑ Corresponding author. E-mail address: [email protected] (P. Park). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.025

246

W.I. Lee, P. Park / Applied Mathematics and Computation 229 (2014) 245–253

Notations: Throughout this paper, In and 0n are the n  n identity matrix and zero matrix, respectively. Z ¼ ½Z ij kk indicates that Z is a k  k matrix whose ði; jÞ-th component is Z ij , the symmetric term in a matrix is denoted by . 2. Problem formulation Consider the following system with time-varying delay:

_ xðtÞ ¼ AxðtÞ þ Ad xðt  hðtÞÞ; xðtÞ ¼ /ðtÞ; h2 6 t 6 0;

t P 0;

ð1Þ

where 0 6 h1 6 hðtÞ 6 h2 ; h12 ¼ h2  h1 and the initial condition /ðtÞ is a continuously differentiable function. In this brief, we define a new form of function combination with squared convex parameters as follows. Definition 1. Let U1 ; U2 ; . . . ; UN : Rm # R be a given finite number of functions that have positive values in an open subset D of Rm . Then, a second-order reciprocally convex combination of these functions over D is a function of the form

1

a21

U1 þ

1

a22

U2 þ    þ

1

a2N

UN : D # Rn ;

ð2Þ P

where the real numbers ai satisfy ai > 0 and

i

ai ¼ 1.

To end this section, we introduce some essential lemmas that will play a key role in deriving our main results. Lemma 1 [18] (lower bound lemma). Let f1 ; f2 ; . . . ; fN : Rm # R have positive values in an open subset D of Rm . Then, the reciprocally convex combination of fi over D satisfies

X1

min P

fai jai >0;

a ¼1g i i i

ai

fi ðtÞ ¼

X X fi ðtÞ þ max g i;j ðtÞ g i;j ðtÞ

i

i–j

subject to

(

"

fi ðtÞ

g i;j ðtÞ

g i;j ðtÞ

fj ðtÞ

m

g i;j : R # R; g j;i ðtÞ ¼ g i;j ðtÞ;

#

) P0 :

Lemma 2 [22]. For the symmetric matrices R > 0; X and matrix C, the following statements are equivalent: 1. X  CT RC < 0, 2. There exists an appropriate dimensional matrix P such that

"

X þ CT P þ PT C PT 

R

# < 0:

3. Main results In this section, we provide a novel delay-dependent stability criterion for system (1) as follows: Theorem 1. Given scalars 0 6 h1 6 h2 , system (1) is asymptotically stable if there exist matrices P ¼ ½P ij 33 > 0; Q 1 > 0; Q 2 > 0; Rk > 0; Z k > 0; Sk ; k ¼ 1; . . . ; 4, and P1 ; P2 with appropriate dimensions such that the following conditions hold for l ¼ 1; 2:

2

Ul

6 4 

PT1

PT2

3





7 0 5 < 0; Z 4

2Z 3 6  6 6 4 

0

S1

2



Z3

Z3 

0 2Z 3





0

3

S2 7 7 7 > 0; 05 Z3

ð3Þ 2

2Z 4 6  6 6 4  

0

S3

Z4 

0 2Z 4





0

3

S4 7 7 7 > 0; 05 Z4

ð4Þ

247

W.I. Lee, P. Park / Applied Mathematics and Computation 229 (2014) 245–253

2 4

R2 þ

h212 2

Z3

3

S5 R2 þ



h212 2

Z4



5 > 0;

R4

S6



R4

 > 0;

ð5Þ

where

2

X 2C1l Z^ 3 2C2l Z^ 4

Ul ¼ 6 4 Z 4 ¼



2Z^ 3

 3Z 4

 S3 þ S4



3Z 4

3

2

CT1l

3

2

CT1l

3T

2

CT2l

3

2

2

2

2

X11 ¼ AT P11 þ P 11 A þ P12 þ PT12 þ Q 1 þ h1 R3 þ h12 R4  R1  h1 Z 1 þ AT YA; X12 ¼ P11 Ad þ AT YAd ;

X13 ¼ P12 þ P 13 þ R1 ;

X15 ¼ PT22 þ AT P 12 þ h1 Z 1 ;

X16 ¼ PT23 þ AT P 13 ;

X22 ¼ 2R2 þ S5 þ ST5 þ ATd YAd ; X26 ¼ ATd P13 ;

X23 ¼ R2  ST5 ;

X46 ¼ PT33 ; X66 ¼ R4 ; 

C11 ¼ 

C12 ¼ 

C21 ¼ 

C22 ¼

X67 ¼ S6 ;

0n

In

0n

0n

h12 In

0n

0n

0n

0n

In

0n

0n

h12 In

0n

0n

In

0n

0n

0n

0n

0n

0n

0n

In

0n

0n

0n

0n

0n

In

0n

0n

0n

0n

h12 In

0n

0n

In

0n

h12 In

0n

0n

0n

In

0n

0n

0n

0n

0n

0n

0n

In

Z3

2 h1 R1

2 h12 R2

þ

 Z4 Z^4 ¼ 

þ

h41 4

X45 ¼ PT23 ;

X56 ¼ X57 ¼ 0;



0n

;

X35 ¼ PT22 þ P T23 þ h1 Z 2 ;

X77 ¼ R4 ;

0n



X24 ¼ R2  S5 ;

X55 ¼ R3  Z 1  Z 2 ;

0n

0

X17 ¼ PT23 þ AT P13 ;

X44 ¼ Q 2  R2 ;

0n

 Z3 Z^3 ¼  with Y ¼

X34 ¼ S5 ;

X37 ¼ PT23 þ PT33 ;

X47 ¼ PT33 ;

X14 ¼ P13 ;

X27 ¼ ATd P13 ;

X33 ¼ Q 2 þ Q 2  R1  R2  h21 Z 2 ; X36 ¼ PT23 þ P T33 ;

0

;  ;

 ;  ;



Z4

ðZ 1 þ Z 2 Þ þ

h412 4

ðZ 3 þ Z 4 Þ.

Proof. Consider the following Lyapunov–Krasovskii functional candidate:

VðtÞ ¼

11 X V i ðtÞ; i¼1

V 1 ðtÞ ¼ fT ðtÞPfðtÞ; V 2 ðtÞ ¼

Z

t

th1

3T

7 6 7 7 6 7 7 T6 T6 0 5 þ 4 02n 5P1 þ P1 4 02n 5 þ 4 02n 5P2 þ P2 4 02n 5 ; 02n 02n 02n 02n 2Z^ 4 

^ 3 and Z ^ 4 are defined as and X ¼ ½Xij 77 ; C1l ; C2l ; Z

X25 ¼ ATd P12 ;

CT2l

xT ðaÞQ 1 xðaÞ da;

 3Z 3 Z 3 ¼ 

S1 þ S2 3Z 3

 ;

248

W.I. Lee, P. Park / Applied Mathematics and Computation 229 (2014) 245–253

V 3 ðtÞ ¼

Z

th1

xT ðaÞQ 2 xðaÞ da;

th2

Z

V 4 ðtÞ ¼ h1

Z

0

Z

_ aÞ da db; x_ T ðaÞR1 xð

tþb

h1

V 5 ðtÞ ¼ h12

t

Z

h1

Z

Z

0

Z

t

xT ðaÞR3 xðaÞ da db;

tþb

h1

V 7 ðtÞ ¼ h12

Z

h1

Z

2

h1 2

Z

h1 2

h12 2 2

V 11 ðtÞ ¼

0

h12 2

Z

Z

Z

0

_ aÞ da db dc; x_ T ðaÞZ 1 xð

Z

c

t

_ aÞ da db dc; x_ T ðaÞZ 2 xð

tþb

Z

h1

Z

c

h2 h1

t

tþb

h1 h1

Z

xT ðaÞR4 xðaÞ da db;

c

h1

2

V 10 ðtÞ ¼

Z

h1

2

V 9 ðtÞ ¼

0

t

tþb

h2

V 8 ðtÞ ¼

_ aÞ da db; x_ T ðaÞR2 xð

tþb

h2

V 6 ðtÞ ¼ h1

t

Z

h2

t

_ aÞ da db dc; x_ T ðaÞZ 3 xð

tþb

c

Z

h2

t

_ aÞ da db dc; x_ T ðaÞZ 4 xð

tþb

Rt R th where fðtÞ ¼ colfxðtÞ; th1 xðaÞ da; th21 xðaÞ dag. Setting a ¼ ðhðtÞ  h1 Þ=h12 ; b ¼ ðh2  hðtÞÞ=h12 and applying the Jensen inequality lemma [23] to the time derivative of VðtÞ gives

_ V_ 1 ðtÞ ¼ 2fT ðtÞPfðtÞ;

ð6Þ

V_ 2 ðtÞ ¼ xT ðtÞQ 1 xðtÞ  xT ðt  h1 ÞQ 1 xðt  h1 Þ;

ð7Þ

V_ 3 ðtÞ ¼ xT ðt  h1 ÞQ 2 xðt  h1 Þ  xT ðt  h2 ÞQ 2 xðt  h2 Þ;

ð8Þ

2 _  h1 V_ 4 ðtÞ ¼ h1 x_ T ðtÞR1 xðtÞ

Z

th1

Z

2 _  h12 V_ 5 ðtÞ ¼ h12 x_ T ðtÞR2 xðtÞ

Z

2

1

Z

Z

2

1

a

th1

_ aÞ da  h12 x_ T ðaÞR2 xð

x_ T ðaÞ daR2

Z

Z

Z

th1

_ aÞ da  xð

thðtÞ

1 b

Z

th1

xT ðaÞR4 xðaÞ da  h12

thðtÞ

xT ðaÞ daR4

Z

Z

t

_ aÞ da; xð

ð9Þ

_ aÞ da; xð

ð10Þ

xðaÞ da;

ð11Þ

xðaÞ da;

ð12Þ

th1

x_ T ðaÞ daR2

th2

Z

thðtÞ

th2

Z

t

xT ðaÞ daR3

th1

Z

t

th1

thðtÞ

xT ðaÞR4 xðaÞ da

th2

thðtÞ th1

Z

xT ðaÞR4 xðaÞ da

th2

Z

th1

thðtÞ

xT ðaÞR3 xðaÞ da 6 h1 xT ðtÞR3 xðtÞ 

th1

x_ T ðaÞ daR1

_ aÞ da x_ T ðaÞR2 xð

2

Z

t

thðtÞ

th2

t

th1

¼ h12 xT ðtÞR4 xðtÞ  h12 2

th1

Z

_ aÞ da x_ T ðaÞR2 xð

thðtÞ

2 V_ 7 ðtÞ ¼ h12 xT ðtÞR4 xðtÞ  h12

6 h12 xT ðtÞR4 xðtÞ 

th1

thðtÞ

a

2 V_ 6 ðtÞ ¼ h1 xT ðtÞR3 xðtÞ  h1

_ aÞ da 6 h21 x_ T ðtÞR1 xðtÞ _  x_ T ðaÞR1 xð

th2

_  h12 ¼ h12 x_ T ðtÞR2 xðtÞ 2 _  6 h12 x_ T ðtÞR2 xðtÞ

t

th1

thðtÞ

xðaÞ da 

1 b

Z

thðtÞ

th2

xT ðaÞ daR4

Z

thðtÞ

th2

249

W.I. Lee, P. Park / Applied Mathematics and Computation 229 (2014) 245–253 4 2 Z 0 Z t h h _  1 _ aÞ da db V_ 8 ðtÞ ¼ 1 x_ T ðtÞZ 1 xðtÞ x_ T ðaÞZ 1 xð 4 2 h1 tþb Z 0 Z t Z 0 Z t 4 h _  _ aÞ da db; 6 1 x_ T ðtÞZ 1 xðtÞ x_ T ðaÞ da dbZ 1 xð 4 h1 tþb h1 tþb

ð13Þ

4 2 Z 0 Z tþb h h _  1 _ aÞ da db V_ 9 ðtÞ ¼ 1 x_ T ðtÞZ 2 xðtÞ x_ T ðaÞZ 2 xð 4 2 h1 th1 Z 0 Z tþb Z 0 Z tþb 4 h _  _ aÞ da db; 6 1 x_ T ðtÞZ 2 xðtÞ x_ T ðaÞ dabZ 2 xð 4 h1 th1 h1 th1 4

2

h h _  12 V_ 10 ðtÞ ¼ 12 x_ T ðtÞZ 3 xðtÞ 4 2

Z

h1

Z

th1

ð14Þ

_ aÞ da db x_ T ðaÞZ 3 xð

tþb

h2

Z th1 4 2 2 Z h1 Z th1 h12 T h h _  12 ðh2  hðtÞÞ _ aÞ da db  12 _ aÞ da db x_ ðtÞZ 3 xðtÞ x_ T ðaÞZ 3 xð x_ T ðaÞZ 3 xð 4 2 2 hðtÞ tþb thðtÞ 2 Z hðtÞ Z thðtÞ h _ aÞ da db  12 x_ T ðaÞZ 3 xð 2 h2 tþb Z Z th1 Z h1 Z th1 Z h1 Z th1 4 2 h h b th1 T 1 _  12 _ aÞ da  2 _ aÞ da db 6 12 x_ T ðtÞZ 3 xðtÞ x_ ðaÞ daZ 3 xð x_ T ðaÞ da dbZ 3 xð 4 2 a thðtÞ a hðtÞ tþb thðtÞ hðtÞ tþb Z Z Z hðtÞ Z thðtÞ 1 hðtÞ thðtÞ T _ aÞ da db;  2 ð15Þ x_ ðaÞ da dbZ 3 xð b h2 tþb h2 tþb ¼

4

2

h h _  12 V_ 11 ðtÞ ¼ 12 x_ T ðtÞZ 4 xðtÞ 4 2

Z

h1 h2

Z

tþb

_ aÞ da db x_ T ðaÞZ 4 xð

th2

Z thðtÞ 4 2 2 Z h1 Z tþb h12 T h h _  12 ðhðtÞ  h1 Þ _ aÞ da db  12 _ aÞ da db x_ ðtÞZ 4 xðtÞ x_ T ðaÞZ 4 xð x_ T ðaÞZ 4 xð 4 2 2 hðtÞ thðtÞ th2 2 Z hðtÞ Z tþb h _ aÞ da db x_ T ðaÞZ 4 xð  12 2 h2 th2 Z Z thðtÞ Z h1 Z tþb Z h1 Z tþb 4 2 h h a thðtÞ T 1 _  12 _ aÞ da  2 _ aÞ da db x_ ðaÞ daZ 4 xð x_ T ðaÞ da dbZ 4 xð 6 12 x_ T ðtÞZ 4 xðtÞ 4 2 b th2 a hðtÞ thðtÞ hðtÞ th2 thðtÞ Z Z Z hðtÞ Z tþb 1 hðtÞ tþb T _ aÞ da db:  2 ð16Þ x_ ðaÞ da dbZ 4 xð b h2 th2 h2 th2 ¼

Let us define

(

vðtÞ ¼ col xðtÞ; xðt  hðtÞÞ; xðt  h1 Þ; xðt  h2 Þ;

Z

Z

t

xðaÞda;

th1

th1

xðaÞda;

Z

)

thðtÞ

xðaÞda : th2

thðtÞ

From Lemma 1, we can infer that if there exist matrices S5 and S6 such that (5) holds, then it holds that



1

Z

a 

th1

x_ T ðaÞ daR2

Z

thðtÞ 2 h12

a

th1

_ aÞ da  xð

thðtÞ

Z

thðtÞ

x_ T ðaÞ daZ 4

2 b

th2

Z

thðtÞ th2

1 b

Z

thðtÞ

x_ T ðaÞ daR2 th2

Z

thðtÞ th2

" R th1

Z Z th1 2 h12 b th1 T _ aÞ da x_ ðaÞ daZ 3 xð 2 a thðtÞ thðtÞ # " R th1 _ xðaÞ da S5 thðtÞ R thðtÞ R2 _ aÞ da xð th2

_ aÞ da; xð

_ aÞ da xð _ aÞ da 6  RthðtÞ xð thðtÞ _ aÞ da xð th2

#T 

R2 



and



1

Z

a

th1

T

x ðaÞ daR4

thðtÞ

Z

th1

thðtÞ

1 xðaÞ da  b

Z

thðtÞ T

x ðaÞ daR4

Z

th2

thðtÞ th2

2 R th

1

thðtÞ xðaÞ da 6 4 R

xðaÞ da

thðtÞ xð th2

aÞ da

3T 5



R4 

ð17Þ

2 3  R th1 xðaÞ da 4 RthðtÞ 5: thðtÞ R4 xð a Þ d a th2 S6

ð18Þ Note that if hðtÞ ¼ h1 or hðtÞ ¼ h2 , we have

Z

th1

thðtÞ

_ aÞda ¼ xð

Z

th1

thðtÞ

xðaÞda ¼ 0 or

Z

thðtÞ

_ aÞda ¼ xð th2

respectively. So inequalities (17) and (18) still hold.

Z

thðtÞ

xðaÞda ¼ 0; th2

250

W.I. Lee, P. Park / Applied Mathematics and Computation 229 (2014) 245–253

In a similar manner, we can derive the upper bounds of the second-order reciprocally convex combinations in (15) and (16) for the matrices S1 , S2 , S3 and S4 satisfying (4) as



Z

1

a2

h1

Z

th1 tþb

hðtÞ

2 R h R th 1 1 hðtÞ

tþb

_ aÞ da db xð

h2

tþb

_ aÞ da db xð

6 4 R hðtÞ R thðtÞ

Z

x_ T ðaÞ da dbZ 3 3T 5

Z

h1

_ aÞ da db  xð

tþb

hðtÞ



th1

Z3

S1 þ S2



Z3

Z

1 b2

hðtÞ Z thðtÞ

x_ T ðaÞ da dbZ 3

tþb

h2

Z

hðtÞ Z thðtÞ

2 3  R h1 R th1 xð _ aÞ da db hðtÞ tþb 4R 5; hðtÞ R thðtÞ _ aÞ da db xð h2 tþb

_ aÞ da db xð

tþb

h2

ð19Þ

and



Z

1

a

2

h1

Z

tþb

x_ T ðaÞ da dbZ 4 thðtÞ

hðtÞ

2 R h R tþb 1 hðtÞ

thðtÞ

h2

th2

6 4 R hðtÞ R tþb

_ aÞ da db xð _ aÞ da db xð

3T 5

Z

hðtÞ



Z

h1

Z4 

tþb

_ aÞ da db  xð

thðtÞ

Z

1 2

b

hðtÞ

Z

tþb

x_ T ðaÞ da dbZ 4

th2

h2

Z

hðtÞ h2

2 3  R h1 R tþb xð _ aÞ da db S3 þ S4 hðtÞ thðtÞ 4R 5: hðtÞ R tþb Z4 _ aÞ da db xð

Z

tþb

_ aÞ da db xð

th2

ð20Þ

th2

h2

It should be noted that when hðtÞ ¼ h1 or hðtÞ ¼ h2 , we have

Z

h1

hðtÞ

Z

Z

th1

_ aÞ da db ¼ xð

tþb

h1

Z

_ aÞ da db ¼ xð

Z

tþb

tþb

_ aÞ da db ¼ 0 or xð

thðtÞ

hðtÞ

hðtÞ Z thðtÞ h2

Z

hðtÞ

Z

tþb

_ aÞ da db ¼ 0; xð th2

h2

respectively. So the relations (19) and (20) still hold. From the conditions (6), (9)–(20), it can be seen that

  Z3 _ VðtÞ 6 vT ðtÞ X  CT1 ðtÞ 

S1 þ S2



Z3

C1 ðtÞ  CT2 ðtÞ



Z4

S3 þ S4



Z4





C2 ðtÞ vðtÞ;

where



C1 ðtÞ ¼ 

C2 ðtÞ ¼



0n

0n

ðhðtÞ  h1 ÞIn

0n

0n

In

0n

0n

ðh2  hðtÞÞIn

0n

0n

0n

0n

In

0n

ðhðtÞ  h1 ÞIn

0n

0n

0n

In

0n

0n

0n

0n

ðh2  hðtÞÞIn

0n

0n

In

;  :

Furthermore, the condition that

X  CT1 ðtÞ



Z3

S1 þ S2



Z3



C1 ðtÞ  CT2 ðtÞ



Z4

S3 þ S4



Z4



C2 ðtÞ < 0

ð21Þ

is intrinsically linear in hðtÞ from the Schur complement [24] and Lemma 2 as

2

UðtÞ PT1 6 4  Z 3 



PT2

3

7 0 5 < 0; Z 4

where

2

X 2C1 ðtÞZ^3 2C2 ðtÞZ^ 4

UðtÞ ¼ 6 4



2Z^ 3

0



2Z^ 4

3

2 T 3 2 T 3T 2 T 3 2 T 3T C1 ðtÞ C1 ðtÞ C2 ðtÞ C2 ðtÞ 7 6 7 7 6 7 7 T6 T6 þ P þ P þ P þ P 5 4 02n 5 1 4 02n 5 2 1 4 02n 5 2 4 02n 5 : 02n 02n 02n 02n

Thus (21) can be treated non-conservatively by two corresponding boundary LMIs (3): one for hðtÞ ¼ h1 and the other for _ < 0. This completes the proof. h hðtÞ ¼ h2 , which imply VðtÞ Remark 1. In [2], to handle the ab and ab-dependent terms in (15) and (16), the authors have introduced an approximation as  ab 6 b; ab 6 a, which contains considerable conservatism. However, based on the relations ab ¼ 1 þ a1 and ab ¼ 1 þ 1b, it can also be regarded as one of the reciprocally convex combination, which can be effectively treated by the simple variation of Lemma 1. In (17), the tighter upper bound is obtained using the condition (5) and Lemma 1 without any conservative approximation, where

W.I. Lee, P. Park / Applied Mathematics and Computation 229 (2014) 245–253

2 qffiffi R

3T 2 h212 _ aÞ da xð 6 7 4 R2 þ 2 Z 3 0 P 4 qffiffiR 5 thðtÞ _ aÞ da  ab th2 xð 

251

3 32 qffiffi R th b 1 _ a Þ d a xð a thðtÞ 7 56 4 qffiffiR thðtÞ 5: h212 a _  a Þ d a xð R2 þ 2 Z 4 b th2

th1 a thðtÞ b

S5

Remark 2. Theorem 1 proposes a new approach for dealing with a second-order reciprocally convex combination. In (19) and (20), the relations

1

a2

2

¼ ðaþbÞ and a2

1 b2

2

¼ ðaþbÞ are employed and the proof procedure is similar to that of Lemma 1 in b2

[18], where

3T 2 qffiffi R R th1 b h1 _ aÞ da db xð 2 hðtÞ tþb a 7 6 2Z 3 7 6 7 6 b R h1 R th1 _ 6 6 a hðtÞ tþb xðaÞ da db 7 6  7 6 6 0 P 6 qffiffiR 7 6 7 6 6  hðtÞ R thðtÞ _ aÞ da db 7 6  ab h2 tþb xð 4 7 6 5 4  R hðtÞ R thðtÞ _ aÞ da db  ab h2 tþb xð 3T 2 qffiffi R R tþb b h1 _ aÞ da db xð 2 hðtÞ thðtÞ a 7 6 2Z 4 7 6 R R 7 6 b h1 tþb _ 6 6 a hðtÞ thðtÞ xðaÞ da db 7 6  7 6 6 0 P 6 qffiffiR 7 6 R 7 6 6  hðtÞ tþb a _ aÞ da db 7 6  b h2 th2 xð 4 7 6 5 4  R hðtÞ R tþb _ aÞ da db  ab h2 th2 xð

0

S1

Z3

0



2Z 3





0

S3

Z4

0



2Z 4





3 2 qffiffi R R th1 b h1 _ aÞ da db xð 3 hðtÞ tþb a 7 0 6 7 6 7 76 b R h1 R th1 _ a Þ d a db xð 7 7 6 S2 76 a hðtÞ tþb 7 7; 76 qffiffiR 7 6 hðtÞ R thðtÞ 07 _ aÞ da db 7 56  ab h2 tþb xð 7 6 5 Z3 4 R hðtÞ R thðtÞ a _ aÞ da db  b h2 tþb xð 3 2 qffiffi R R tþb b h1 _ aÞ da db xð 3 hðtÞ thðtÞ a 7 0 6 7 6 7 76 b R h1 R tþb _ 6 a hðtÞ thðtÞ xðaÞ da db 7 S4 7 7 76 7: 76 qffiffiR 7 6 hðtÞ R tþb a 07 _ aÞ da db 7 56  b h2 th2 xð 7 6 5 4 Z4 R hðtÞ R tþb a _ aÞ da db  b h2 th2 xð

In the process of obtaining the upper bound of the second-order reciprocally convex combination, no any conservative approximation is introduced by appropriately extending Lemma 1 based on the property of convex parameters. The proposed approach is very effective in reducing the conservatism of stability criterion, which will be verified by simulation results in the next section. Remark 3. For directly applying Lemma 2 to (21), two additional positivity conditions should be introduced as



Z3

S1 þ S2



Z3





> 0;

Z4

S3 þ S4



Z4



> 0:

^ 3 C1 ðtÞ and 2CT ðtÞZ ^ 4 C2 ðtÞ to the left side In this paper, before applying Lemma 2 to (21) directly, we have added 2CT1 ðtÞZ 2 of (21) and have used the Schur complement [24] appropriately. This simple derivation gives a less conservative result because it fully utilizes the condition (4) without increasing the LMI conditions.

4. Examples In this section, we will present two numerical examples to illustrate the effectiveness of Theorem 1. Example 1. Consider the system (1) with

 A¼

2:0

0:0

0:0

0:9



 ;

Ad ¼

1:0



0:0

1:0 1:0

:

ð22Þ

To demonstrate the effectiveness of our approach, the maximum upper bounds on the delay (MUBDs) are compared to the literature and are listed in Table 1 for various h1 . For the cases of small h1 , the condition provided in [18] is less conservative than the ones of [3,2], whereas it is more conservative for the cases of big h1 . However for all cases of h1 , Theorem 1 gives less conservative results than the conditions from these articles, which implies our result is competitive with those conditions. Example 2. Consider the system (1) with

A¼



0:0

1:0

1:0 2:0

 ;

Ad ¼



0:0

0:0

1:0 1:0

 :

By Theorem 1, the improvement of this paper is shown in Table 2.

ð23Þ

252

W.I. Lee, P. Park / Applied Mathematics and Computation 229 (2014) 245–253 Table 1 Allowable upper bound h2 for given h1 . h1

0

0.3

0.6

1

2

3

[3] [18] [2]

1.52 1.86 1.70

1.59 1.87 1.78

1.69 1.92 1.89

1.90 2.06 2.09

2.56 2.61 2.69

3.34 3.31 3.41

Theorem 1

1.86

1.88

1.99

2.17

2.72

3.42

Table 2 Allowable upper bound h2 for given h1 . h1

0

0.3

0.5

0.8

1.0

[3] [18] [2]

0.87 1.06 1.04

1.07 1.24 1.24

1.21 1.38 1.39

1.45 1.60 1.61

1.61 1.75 1.77

Theorem 1

1.11

1.29

1.43

1.64

1.79

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