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(1)

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2010-0009743). This research was supported by the MKE(The Ministry of Knowledge Economy), Korea, under the ITRC(Information Technology Research Center) support program supervised by the NIPA(National IT Industry Promotion Agency) (C-1090-1121-0006)) & (NIPA-2011-(C1090-1111-0011)). This research was supported by World Class University program funded by the Ministry of Education, Science and Technology through the National Research Foundation of Korea (R31-10100).

Sung Hyun Kim is with the Department of Electrical Engineering, University of Ulsan (UOU), Ulsan, 680-749, Korea e-mail: shnkim@ulsan.ac.kr.

(2)

whereζ(t) =col(e(t), xr(t)),w˜(t) =col(w(t), r(t)),

˜ A(Θt) =

ï A

t) A(Θt)−Ar

0 Ar

ò

, (7)

˜

Ad(Θt) =

ï

Bu(Θt)F(Θt) 0

0 0

ò

, (8)

˜

Bw(Θt) =

ï

Bw(Θt) −I

0 I

ò

,C˜(Θt) = C(Θt) 0

Lemma 2.1: For real matrices X, Y, and S > 0 with appropriate dimensions, it is satisfied that 0 ≤ (X − SY)TS−1(X SY) and hence the following inequality holds: YTSY XTY +YTX XTS−1X. Further if X = µI, then YTSY µY +µYT µ2S−1, where µ is a scalar. On the other hand, if S < 0, then it is assured thatYTSY ≤ −µY µYT µ2S−1

III. MAIN RESULTS

A. PLMI-type condition

Choose a Lyapunov-Krasovskii functional

V(t),V1(t) +V2(t) +V3(t), (9) V1(t) =ζ(t)TP ζ(t),

V2(t) = Z t

t−d1

ζT(α)Q

1ζ(α)dα+ Z t

t−d2

ζT(α)Q

2ζ(α)dα,

V3(t) =d1 Z 0

−d1

Z t

t+β

˙ ζT(α)R

1ζ˙(α)dαdβ

+d21 Z −d1

−d2

Z t

t+β

˙

ζT(α)R2ζ˙(α)dαdβ

where P, Q1, Q2, R1, and R2 are positive definite with appropriate dimensions. Let us define an augmented state η(t) =col(ζ(t), ζ(t−d1), ζ(t−d(t)), ζ(t−d2),w˜(t))and the corresponding block entry matrices as

e1, I 0 0 0 0 , e2, 0 I 0 0 0 ,

e3, 0 0 I 0 0 , e4, 0 0 0 I 0 ,

e5, 0 0 0 0 I ,

Φt,A˜(Θt)e1+ ˜Ad(Θt)e3+ ˜Bw(Θt)e5

Time derivative of Vi(t)

˙

V1(t) =ηT(t)He(eT

1PΦt)η(t), (10)

˙

V2(t) =ηT(t) eT1(Q1+Q2)e1−eT2Q1e2−eT4Q2e4η(t), (11) ˙

V3(t) =ηT(tT t d

2

1R1+d221R2Φtη(t) +O, (12)

where O=−d1

Z t

t−d1

˙ ζT(α)R

1ζ˙(α)dα−d21 Z t−d1

t−d(t) ˙ ζT(α)R

2ζ˙(α)dα

−d21

Z t−d(t)

t−d2

˙ ζT(α)R

2ζ˙(α)dα

˙

V(t) =ηT(t)Π0η(t) +O (13) where

Π0=He(eT

1PΦt) +eT1(Q1+Q2)e1−eT2Q1e2−eT4Q2e4 + ΦT

t d21R1+d221R2Φt (14)

By employing the Jensen inequality [8] and the lower bounds lemma [9], we can obtain a upper bound ofOas follows

O ≤ −( Z t

t−d1

˙

ζ(α)dα)TR1(

Z t

t−d1

˙ ζ(α)dα)

− 1 ρ1(t)(

Z t−d1

t−d(t) ˙

ζ(α)dα)TR2(

Z t−d1

t−d(t) ˙ ζ(α)dα)

− 1 ρ2(t)(

Z t−d(t)

t−d2

˙

ζ(α)dα)TR2( Z t−d(t)

t−d2

˙ ζ(α)dα)

=−ηT(t)(e

1−e2)TR1(e1−e2)η(t)

− 1 ρ1(t)η

T(t)(e

2−e3)TR2(e2−e3)η(t)

− 1 ρ2(t)η

T(t)(e

3−e4)TR2(e3−e4)η(t) (15) where

ρ1(t) = (d(t)−d1)/d21≥0, ρ2(t) = (d2−d(t))/d21≥0,

ρ1(t) +ρ2(t) = 1 (16)

(15) is represented as the following inequality: O ≤ηT(t)Π1η(t)

− "»ρ

2

ρ1(e2−e3)η(t)

»ρ

1

ρ2(e3−e4)η(t)

#T Π2

" »ρ

2

ρ1(e2−e3)η(t)

»ρ

1

ρ2(e3−e4)η(t)

#

(17) where

Π1= (e1−e2)TR1(e2−e1) + (e2−e3)TR2(e3−e2) + (e3−e4)TR2(e4−e3) +He((e2−e3)TS(e3−e4)),

(18) Π2=

ï

R2 S

ST R

2 ò

(19) ˙

V(t)is upper-bounded by ˙

V(t)≤ηT(t)(Π0+ Π1)η(t)

− "»ρ

2

ρ1(e2−e3)η(t)

»ρ

1

ρ2(e3−e4)η(t)

#T Π2

ρ

2

ρ1(e2−e3)η(t)

»ρ

1

ρ2(e3−e4)η(t)

#

(20) Note: (Stability criterion in the H∞ sense) (V˙(t) + zT(t)z(t)γ2w˜T(t) ˜w(t)<0)

0>Π0+ Π1+ Π3,0≤Π2 (21) where

Π3=eT1C˜T(Θt) ˜C(Θt)e1−γ2eT5e5 (22) Set P = diag(P1, P2) and P¯ = P−1 = diag( ¯P1,P¯2), whereP¯1=P1−1 andP¯2=P2−1. Then,

PA˜(Θt) =

ï

P1A(Θt) P1A(Θt)−P1Ar

0 P2Ar

ò

,

PA˜d(Θ(t)) =

ï

P1Bu(Θt) ¯F(Θt)P1 0

0 0

ò

,

PB˜ω(Θt) =

ï P

1Bω(Θt) −P1

0 P2

ò

,

(3)

PΦt=XA¯(Θt)Xe1+XA¯d(Θt)Xe3+XB¯ω(Θt)e5 (23) where

X =diag(P1, I), (24)

¯ A(Θt) =

ï

A(Θt) ¯P1 A(Θt)−Ar

0 P2Ar

ò

, (25)

¯

Ad(Θt) =

ï B

u(Θt) ¯F(Θt) 0

0 0

ò

, (26)

¯

Bω(Θt) =

ï

Bω(Θt) −I

0 P2

ò

, (27)

Theorem 1: Let µ1 >0, µ2 >0 be prescribed. Suppose that there exist a scalar γ > 0; matrices F¯, and S˜; and symmetric positive definite matrices P¯1, P2, Q˜1, Q˜2, R˜1, andR˜2 such that

0>            

(1,1) 0 d1A¯(Θt) 0 d1A¯d(Θt)

0 (2,2) d21A¯(Θt) 0 d21A¯d(Θt)

(∗) (∗) (3,3) R˜1 A¯d(Θt)

0 0 (∗) (4,4) R˜2+ ˜S

(∗) (∗) (∗) (∗) (5,5)

0 0 0 (∗) (∗)

(∗) (∗) (∗) 0 0

0 0 C˜(Θt) ¯X 0 0

0 d1B¯ω(Θt) 0

0 d21B¯ω(Θt) 0

0 B¯ω(Θt) X¯C˜T(Θt)

−S˜ 0 0

˜

R2+ ˜S 0 0

(6,6) 0 0

0 −γ2I 0

0 0 −I

           

(28)

0≤ ñ ˜

R2 S˜ ˜ ST R˜

2 ô

(29) where

(1,1) =µ21R˜1+diag(−2µ1P¯1,−2µ1P2), (2,2) =µ22R˜2+diag(−2µ2P¯1,−2µ2P2), (3,3) =He( ¯A(Θt)) + ˜Q1+ ˜Q2−R˜1, (4,4) =−Q˜1−R˜1−R˜2,

(5,5) =−2 ˜R2−He( ˜S),(6,6) =−Q˜2−R˜2,

¯ A(Θt) =

ï

A(Θt) ¯P1 A(Θt)−Ar

0 P2Ar

ò

,

¯ Ad(Θt) =

ïB

u(Θt) ¯F(Θt) 0

0 0

ò

,B¯ω(Θt) =

ïB

ω(Θt) −I

0 P2

ò

,

˜

C(Θt) = C(Θt) 0

Then the closed-loop systems (6) is asymptotically stable and satisfies||z||2< γ||w˜||2for all nonzerow˜(t)∈ L2[0,∞)and for any time-varying delay d(t) satisfying d1 ≤d(t) ≤d2. Moreover the minimized H∞ performance can be achieved by the following optimization problem:minγsubject to (28) and (29). Here the control and observer gain matrices can be reconstructed as

F(Θt) = ¯F(Θt) ¯P1−1. (30)

TheH∞ stabilization condition is given as follows:

0≤Π2, (31)

0>Π0+ Π1+ Π3

=He(eT1PΦt) +eT1(Q1+Q2)e1−eT2Q1e2−eT4Q2e4 + ΦTt d21R1+d221R2Φt+ (e1−e2)TR1(e2−e1) + (e2−e3)TR2(e3−e2) + (e3−e4)TR2(e4−e3) +He((e2−e3)TS(e3−e4)) +eT1C˜T(Θt) ˜C(Θt)e1

−γ2eT5e5 (32)

By letting R¯1 = XP R¯ 1P X¯ and R¯2 = XP R¯ 2P X¯ . Then the condition (32) can be converted by (23) into

0>He(eT1XA¯(Θt)Xe1+eT1XA¯d(Θt)Xe3+eT1XB¯ω(Θt)e5)

+eT1(Q1+Q2)e1−eT2Q1e2−eT4Q2e4+ (XA¯(Θt)Xe1 +XA¯d(Θt)Xe3+XB¯ω(Θt)e5)TX d¯ 21R¯1+d221R¯2X¯ ×(XA¯(Θt)Xe1+XA¯d(Θt)Xe3+XB¯ω(Θt)e5)

+(e1−e2)TPX¯R¯1XP¯ (e2−e1) +(e2−e3)TPX¯R¯2XP¯ (e3−e2) +(e3−e4)TPX¯R¯2XP¯ (e4−e3)

+He((e2−e3)TS(e3−e4))+eT1C˜T(Θt) ˜C(Θt)e1−γ2eT5e5 (33) where X¯ = X−1. Futher, since diag( ¯X,X,¯ X,¯ X, I¯ )eT

i =

eT

iX¯, for i = 1,2,3,4 and diag( ¯X,X,¯ X,¯ X, I¯ )eT5 = eT

5 pre- and post-multiplying both sides of (33) by diag( ¯X,X,¯ X,¯ X, I¯ )and its transpose yields

0>Ψ + ( ¯A(Θt)e1+ ¯Ad(Θt)e3+ ¯Bω(Θt)e5)T

×(d21R¯1+d221R¯2)( ¯A(Θt)e1+ ¯Ad(Θt)e3+ ¯Bω(Θt)e5)

(34) whereΨ =He(eT

1A¯(Θt)e1+eT1A¯d(Θt)e3+eT1B¯ω(Θt)e5)+ eT

1( ˜Q1+ ˜Q2)e1−eT2Q˜1e2−e4TQ˜2e4+(e1−e2)TR˜1(e2−e1)+ (e2−e3)TR˜2(e3−e2) + (e3−e4)TR˜2(e4−e3) +He((e2− e3)TS˜(e

3 −e4)) + eT1X¯C˜T(Θt) ˜C(Θt) ¯Xe1 − γ2eT5e5, in which X˜ = ¯XPX¯ = diag( ¯P1, P2) > 0, Q˜1 = ¯XQ1X¯,

˜

Q2 = ¯XQ2X¯, R˜1 = ˜XR¯1X˜, R˜2 = ˜XR¯2X˜, and S˜ = ¯

XSX¯. That is, by applying the Schur complement to (34), we can obtain

0>  

−R¯−11 0 d1( ¯A(Θt)e1+ ¯Ad(Θt)e3+ ¯Bω(Θt)e5)

0 −R¯−12 d21( ¯A(Θt)e1+ ¯Ad(Θt)e3+ ¯Bω(Θt)e5)

(∗) (∗) Ψ

 

(35) Here, sinceR¯−11 = ˜XR˜−11 X˜ andR¯−12 = ˜XR˜−12 X˜, it follows from lemma 2.1 that R¯−11 ≥ 2µ1X˜ −µ21R˜1 and R¯−12 ≥ 2µ2X˜ −µ22R˜2. In this sense, it is clear that (35) holds if

0>          

(1,1)′ 0 d

1A¯(Θt) 0 d1A¯d(Θt) 0 d1B¯ω(Θt)

0 (2,2)′d

21A¯(Θt) 0 d21A¯d(Θt) 0 d21B¯ω(Θt)

(∗) (∗) (3,3)′′′ R˜

1 A¯d(Θt) 0 B¯ω(Θt)

0 0 (∗) (4,4) ˜R2+ ˜S −S˜ 0 (∗) (∗) (∗) (∗) (5,5) ˜R2+ ˜S 0

0 0 0 (∗) (∗) (6,6) 0

(∗) (∗) (∗) 0 0 0 −γ2I

         

(4)

where

(1,1)′=µ21R˜1+diag(−2µ1P¯1,−2µ1P2), (2,2)′=µ2

2R˜2+diag(−2µ2P¯1,−2µ2P2), (3,3)′′′=He( ¯A

t))+ ˜Q1+ ˜Q2−R˜1+ ¯XC˜T(Θt) ˜C(Θt) ¯X,

(4,4)′=−Q˜1−R˜1−R˜2, (5,5)′=−2 ˜R

2−He( ˜S),(6,6)′ =−Q˜2−R˜2,

There exist a non-convex term in(3,3)′′. Thus, to deal with the term, we can obtain (28) using the Schur complement. Let us pre- and post-multiply both sides of (31) bydiag( ¯X,X¯) and its transpose. Then we can obtain

0≤ ñ¯

XR2X¯ S˜ ˜

ST XR¯

2X¯ ô

(37)

B. LMI-type condition

Another representation for (28) is given as follows: 0>L(Θ(t))

,L0+

r

X

i=1

θi(t) Li+LTi

+

r

X

i=1

θ2i(t)Lii

+ r X i=1 r X

j=i+1

θi(t)θj(t)Lij+ i−1 X

j=1

θi(t)θj(t)LTij

! , (38) where L0,            

(1,1) 0 d1A¯0 0

0 (2,2) d21A¯0 0

(∗) (∗) He( ¯A0)+ ˜Q1+ ˜Q2−R˜1 R˜1

0 0 (∗) (4,4)

(∗) (∗) (∗) (∗)

0 0 0 (∗)

(∗) (∗) (∗) 0

0 0 C˜0X¯ 0

d1A¯d,0 0 d1B¯ω,0 0 d21A¯d,0 0 d21B¯ω,0 0

¯

Ad,0 0 B¯ω,0 X¯C˜0T ˜

R2+ ˜S −S˜ 0 0

(5,5) R˜2+ ˜S 0 0

(∗) (6,6) 0 0

0 0 −γ2I 0

0 0 0 −I

            , ¯ A0=

ï A

0P¯1 A0−Ar

0 P2Ar

ò

,A¯d0= ï B

u,0F¯0 0

0 0

ò

,

¯ Bω,0=

ï

Bω,0 −I

0 P2

ò

,C˜0= C0 0

Li,

           

0 0 d1A¯i 0 d1A¯d,i 0 d1B¯ω,i 0

0 0 d21A¯i 0 d21A¯d,i 0 d21B¯ω,i 0

0 0 A¯i 0 A¯d,i 0 B¯ω,i X¯C˜iT

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

            , ¯ Ai=

ïA

iP¯1 Ai−Ar

0 P2Ar

ò

,A¯d,i=

ïB

u,0F¯i+Bu,iF¯0 0

0 0

ò

,

¯ Bω,i=

ï

Bω,i −I

0 P2 ò

,C˜i=

Ci 0

Lii,

           

0 0 0 0 d1A¯d,ii 0 0 0

0 0 0 0 d21A¯d,ii 0 0 0

0 0 0 0 A¯d,ii 0 0 0

0 0 0 0 0 0 0 0

(∗) (∗) (∗) 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

            , ¯ Ad,ii=

ï B

u,iF¯i 0

0 0

ò

Lij ,

           

0 0 0 0 d1A¯d,ij 0 0 0

0 0 0 0 d21A¯d,ij 0 0 0

0 0 0 0 A¯d,ij 0 0 0

0 0 0 0 0 0 0 0

(∗) (∗) (∗) 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

            , ¯ Ad,ii=

ï

Bu,iF¯j 0

0 0

ò

By theS-procedure

0>L(Θ(t)) +N(Θ(t)) (39) where 0 ≤ N(Θ(t)) is given by (Pr

i=1θi(t) = 1, ai ≤ θi(t)≤bi, 0≤θi(t)θj(t))

N(Θ(t)) =C1+CT

1 +

r

X

i=1

C2i(Λi+ ΛTi )

+ r X i=1 r X

j=1,j6=i

C3ij(Σij+ ΣTij),

0 =C1,

     I θ1(t)I

.. . θr(t)I

     T     I −I .. . −I     

W0 W1 · · · Wr

     I θ1(t)I

.. . θr(t)I

     ,

0≤ C2i,−θ2i(t) + (ai+bi)θi(t)−aibi,

0≤ C3ij ,θi(t)θj(t),

for0<Λi+ΛTi and0<Σij+ΣTij. With some algebraic

ma-nipulations, the constraint0≤ N(Θ(t))can be represented as follows:

0≤N(Θ(t))

=N0+

r

X

i=1

θi(t)(Ni+NTi) + r

X

i=1

θi2(t)Nii

+ r X i=1 r X

j=i+1

θi(t)θj(t)Nij+ i−1 X

j=1

θi(t)θj(t)NTij

! ,

(40) whereN0=W0+W0T−Pri=1aibi(Λi+ ΛTi ),Ni= (ai+ bi)Λi−W0+Wi, Nii = −(Λi+ ΛTi)−(Wi+WiT), and

(5)

becomes 0>Γ0+

r

X

i=1

θi(t)(Γi+ ΓTi ) + r

X

i=1

θ2i(t)∆i

+

r

X

i=1

r

X

j=i+1

θi(t)θj(t)Φij+ i−1 X

j=1

θi(t)θj(t)ΦTij

! ,

(41) where

Γ0=L0+N0=L0+W0+W0T −

r

X

i=1

aibi(Λi+ ΛTi),

Γi=Li+Ni=Li+ (ai+bi)Λi−W0+Wi,

∆i=Lii+Nii =Lii−(Λi+ ΛTi)−(Wi+WiT),

Φij=Lij+Nij=Lij−(Wi+Wj) + (Σij+ Σji)

The condition (41) boils down to 0>

I θ1(t)I · · · θr(t)I

˜ L

I θ1(t)I · · · θr(t)I

T , (42) where

˜ L,

       

Γ0 Γ1 Γ2 · · · Γr

(∗) ∆1 Φ12 · · · Φ1r

(∗) (∗) ∆2 . .. ... ..

. ... . .. . .. Φ(r−1)r (∗) (∗) · · · (∗) ∆r

       

, (43)

IV. NUMERICAL EXAMPLE

In this section, Consider the following plant [11]: ˙

x1(t) =x2(t) ˙

x2(t) =−x31(t)−0.1x2(t) + 12 cost+u(t) (44) The LPV representation of the above system is as followings:

A1=

ï0 1 0 −0.1

ò

, A2=

ï 0 1

−25 −0.1 ò

,

Bu,0= ï

0 1 ò

, Bω,0= ï

0 1 ò

, Ar=

ï

0 1

−3 −2 ò

,

A0= 0, C0= 0, Bu,i= 0, Bω,i= 0(i= 1,2, ..., r)

θ1(t) = 1−x 2 1(t)

25 , θ2(t) = x2

1(t) 25 ,

µ1=µ2= 1, a1=a2= 0, b1=b2= 1,

x1(t)∈[−5, 5], r(t) = ï

0 4 sint

ò

, ω(t) = 12 cost

x(0) = [2 −1]T, x

r(0) = [−0.5 1]T

Fig. 1 displays the first state behaviors. The simulation result shows that theH∞ tracking controller on the LPV systems over a communication network has a good performance to track the reference signal.

V. CONCLUSION

Our future work is directed to proposing a method of addressing the practical case where the parameters in the LPV system and control part are asynchronous/mismatched each other.

0 5 10 15 20 25 30

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

Time (sec)

[image:5.595.61.273.494.650.2]

x1 xr1

Fig. 1. Simulation Results

ACKNOWLEDGMENT REFERENCES

[1] W.-W. Che, Y.-P. Li, and Y.-L. Wang, “H∞Tracking control for NCS

with packet losses in multiple channels case,” Int. J. Innov. Comp. Inf.

Control, vol. 7(11), pp. 6507–6521, 2011.

[2] X. Fang and J. Wang, ”Sampled-data H∞ control for networked

systems with random packet dropouts,” Asian J. Control, vol. 12(4), pp. 549–557, 2010.

[3] D. Xie, Y. Wu, and X. Chen, “Stabilization of discrete-time switched systems with input time delay and its applications in networked control systems,” Circuits Syst. Signal Process. , vol. 28(4), pp. 595–607, 2009. [4] D. Yue, E. Tian, Z. Wang, and J. Lam, “Stabilization of systems with probabilistic interval input delays and its applications to networked control systems,” IEEE Trans. Syst. Man Cybern. Paart A-Syst. Hum., vol. 39(4), pp. 939–945, 2009.

[5] X. Jiang, Q.-L. Han, S. Liu, and A. Xue, “A newH∞ stabilization

criterion for networked control systems,” IEEE Trans. Autom. Control, vol. 53(4), pp.1025–1032, 2008.

[6] P.-G. Park and D. J. Choi, “LPV controller design for the nonlinear RTAC system,” Int. J. Robust Nonlinear Control , vol. 11(14), pp. 1343– 1363, 2001.

[7] S. W. Yun, Y. J. Choi, and P.-G. Park, “State-feedback disturbance attenuation for polytopic LPV systems with input saturation,” Int. J.

Robust Nonlinear Control , vol. 20(8), pp. 899–922, 2010.

[8] Gu, K., Kharitonov, V. L., and Chen, J., Stability of time-delay systems, Birkhuser, 2003. Int. J. Robust Nonlinear Control , vol. 20(8), pp. 899– 922, 2010.

[9] P.-G. Park, J. W. Ko and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, pp. 235-238, 2011.

[10] S. H. Kim and P.-G. Park, “H∞ state-feedback control design for

fuzzy systems using Lyapunov functions with quadratic dependence on fuzzy weighting functions,” EEE Trans. Syst., Man, Cybern. B, Cybern, vol. 16(6), pp. 1655–1663, 2008.

[11] X. Jia, D. Zhang, X. Hao, and N. Zheng “FuzzyH∞Tracking Control

for Nonlinear Networked Control Systems in T-S Fuzzy Model,” IEEE

Figure

Fig. 1.Simulation Results

References

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