Qu_ijcccv11n5.pdf


INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL

ISSN 1841-9836, 11(5):708-719, October 2016.

Fuzzy H2 Guaranteed Cost Sampled-Data Control of
Nonlinear Time-Varying Delay Systems

Z.-F. Qu, Z.-B. Du

Zifang Qu

Shandong Institute of Business and Technology
Yantai, Shandong, 264005, China
quzifang@163.com

Zhenbin Du*

Yantai University
Yantai, Shandong, 264005, China
*Corresponding author: zhenbindu@126.com

Abstract: We present and study a delay-dependent fuzzy H2 guaranteed cost
sampled-data control problem for nonlinear time-varying delay systems, which is
formed by fuzzy Takagi-Sugeno (T-S) system and a sampled-data fuzzy controller
connected in a closed loop. Applying the input delay approach and stability theorem
of Lyapunov-Krasovskii functional with Leibniz-Newton formula, the H2 guaranteed
cost control performance is achieved in the sense that the closed-loop system is asymp-
totically stable. A new sufficient condition for the existence of fuzzy sampled-data
controller is given in terms of linear matrix inequalities (LMIs). Truck-trailer system
is given to illustrate the effectiveness and feasibility of H2 guaranteed cost sampled-
data control design.
Keywords: fuzzy T-S system; sampled-data; nonlinear systems; time-varying delay;
H2 guaranteed cost control

1 Introduction

Fuzzy Takagi–Sugeno(T-S)models [1] are used to describe nonlinear systems by a set of IF–
THEN rules which gives a local linear representation. Since the work of Tanaka and Sugeno [2] on
stability analysis and stabilization being published, many efforts have been made in developing
systematic theory for such systems.

Because of the fast development of the digital circuit technology, using computers to design
controller to reduce the implementation cost and time is more and more popular. The system of
control is a sampled-data system. In sampling period,its control signals are constant. The overall
control system becomes a sampled-data system, where the control signals are kept constant during
the sampling period. It’s a popular trend to study the analysis and synthesis of fuzzy sampled-
data systems in many papers, see, for instance, [3–12] and the references therein. Of these works,
stability analysis [3], stabilization [11], H∞ control [4,6,7,9], H2 GC control [8,10], fault-tolerant
control [12] and tracking control [5] are researched, respectively.

Stability and robust stability theory was adopted in sampled-data time-delay systems [3,4,
7, 9]. In industrial systems and information networks, it’s popular to use time-delay systems.
So, we should study time-delay systems and design some controllers for them. There have
two ways for the stability analysis and synthesis of time-delay fuzzy T–S systems, i.e. delay-
independent and delay-dependent approaches. With no respective of the size of the delay, we use
delay-independent approach to assure stable conditions. The delay-dependent approach, contrast
with the delay-independent approach, is complex in design procedure. So, it always have more
conservative results. The delay-dependent approach supplies an upper bound of the time-delay.

Copyright © 2006-2016 by CCC Publications



710 Z.-F. Qu, Z.-B. Du

It deals with the size of the time-delay, as a consequence, it usually provides less conservative
results.

Among these works [3, 4, 7, 9], [7] is delay-independent and [3, 4, 9] are delay-dependent,
where time delay is assumed to be constant. However, in practical engineering systems, the
occurrence of time delay phenomena is often time-varying. Thus, fuzzy sampled-data control for
time-varying delay systems is more appealing. In fuzzy sampled-datacontrol, there is no report
aboutH2 guaranteed cost controlproblem for the nonlinear time-varying delay systems.

In this paper, we consider the delay-dependent sampled-data H2 guaranteed costperformance
problem of the nonlinear time-varying delay system represented by a fuzzy T-S model. A
Lyapunov-Krasovskii functional with Leibniz-Newton formula is employed to obtain new suf-
ficient conditions in terms of linear matrix inequalities (LMIs) to the fuzzy H2 guaranteed cost
control performance. Based on the stability condition, the guaranteed cost control is minimized
for the closed-loop system. We use truck-trailer system to prove the effectiveness and the feasi-
bility of the proposed method.

The main contributions and advantages of the present paper are summarized as follows: (i)
The H2 design via fuzzy sampled-data control for nonlinear systems with time-varying delay is
first obtained. (ii) Fuzzy sampled-data control algorithm is less conservative. Comparing with
the existing works, the dimension of the LMIs in this paper is simplified, which adds the existence
of feedback gains and lowers the implementation time. Experimental results illustrate that the
fuzzy sampled-data controller has a larger sampling interval.

Notations: Throughout this paper, if not explicitly stated, we assumed that matrices have
compatible dimensions. The notation P>0(< 0) is used to denote a positive (negative) definite
matrix. The transpose of a matrix P is denoted by P T . The symbol ∗ stands for the transposed
element in symmetric positions.

2 Problem formulation

Consider the following nonlinear time-varying delay system:

ẋ(t) = f(x(t), x(t − d(t)), u(t)), (1)

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control input vector, f is a nonlinear
function, and d(t) is time-varying delay.

The following fuzzy T-S model with time-varying delay described by IF–THEN rules is used
to represent nonlinear time-varying delay system:

IF ξ1(t) is Mi1 and · · · and ξp(t) is Mip, THEN

ẋ(t) = Aix(t) + Aidx(t − d(t)) + Biu(t), i = 1, · · · , L, (2)

where Ai, Bi and Aid are constant matrices of appropriate dimensions. L is the number of IF–
THEN rules, Mij are fuzzy sets and ξ1, . . . , ξp are premise variables, ξ(t) = [ξ1 . . . ξp]

T , and ξ(t)
is assumed to be given or a measurable function vector. We consider the following two cases for
the time-varying delay.

Case 1: d(t) is a differentiable function satisfying for all t ≥ 0:

0 ≤ d(t) ≤ dM and ḋ(t) ≤ dD,

where dM and dD are constants.



Fuzzy H2 Guaranteed Cost Sampled-Data Control of
Nonlinear Time-Varying Delay Systems 711

Case 2: d(t) is a continuous function satisfying for all t ≥ 0:

0 ≤ d(t) ≤ dM,

where dM is a constant.
By fuzzy blending, the overall fuzzy model is inferred as follows:

ẋ(t) =

L
∑

i=1

λi(ξ(t))[Aix(t) + Aidx(t − d(t)) + Biu(t)], (3)

where λi(ξ(t)) =
βi(ξ(t))

∑
L
i=1

βi(ξ(t))
, βi(ξ(t)) =

p
∏

j=1
Mij(ξj(t)) and Mij(.) is the grade of the membership

function of Mij. βi(ξ(t)) ≥ 0, i = 1, 2, . . . , L,
L
∑

i=1
βi(ξ(t)) > 0 for any ξ(t), λi(ξ(t)) ≥ 0, i =

1, 2, . . . , L,
L
∑

i=1
λi(ξ(t)) = 1.

We design the following fuzzy sampled-data controller for (3):

IF ξ1(tk) is Mj1 and · · · and ξp(tk) is Mjp, THEN

u(t) = Kjx(tk), j = 1, 2, . . . , L,

where Kj is the sate feedback gain, the time tk is the sampling instant satisfying 0 < t1 < t2 <
· · · < tk < · · · , and sampling interval is a constant, i.e. tk+1 − tk = hk = h. The overall fuzzy
sampled-data controller is as follows:

u(t) =
L
∑

j=1

λj(ξ(tk))Kjx(tk). (4)

By using input delay approach, (4) is equivalent to (5)

u(t) =
L
∑

j=1

λj(ξ(tk))Kjx(t − τ(t)). (5)

The closed-loop system (3) with (5) is given by

ẋ(t) =
L
∑

i=1

L
∑

j=1

λi(ξ(t))λj(ξ(tk))[Aix(t) + Aidx(t − d(t)) + BiKjx(t − τ(t))]. (6)

The following H2 guaranteed costcontrol performance

J =

∫

∞

0
(xT (t)Qx(t) + uT (t)Ru(t))dt. (7)

must be minimized, where the weighting positive-definite matrices Q and R are specified before-
hand according to the design purpose.

Determine a sampled-data state feedback controller such that the closed-loop system (6) is
asymptotically stable and the upper bound of H2 guaranteed cost function is minimized.

Lemma 2.1 (Gu et al. [13]). For any positive definite symmetric constant matrix M ∈ Rn×n,
scalars r1, r2 satisfying r1 ≤ r2, if ̟ : [r1, r2] → R

nis a vector function such that the integrations

concerned are well defined, then

(
∫ r2

r1

̟(s)ds

)T

M

(
∫ r2

r1

̟(s)ds

)

≤ (r2 − r1)

∫ r2

r1

̟T (s)M̟(s)ds. (8)



712 Z.-F. Qu, Z.-B. Du

Remark 1: The premise variables ξ
1
, . . . , ξ

p
can be function of measurable state variables x(t)

and x(t − d), or combination of measurable state variables. The limitation of design of fuzzy
T–S approach is that some state variables must be measurable to construct fuzzy controller.
Remark 2: It should be noted that the control signal u(t) holds constant during the period of
tk ≤ t ≤ tk+1.

3 Fuzzy H2 Guaranteed Cost Sampled-Data Control

In this section, we present a H2 guaranteed cost sampled-data control scheme of the fuzzy
system and minimization of the upper bound of (7).

Here, we give some sufficient conditions for the stability of the closed-loop system (6) in terms
of LMIs.

Theorem 1. Suppose that, under case 1, for given matrices Q > 0, R > 0, scalars h > 0,
dM > 0, dD > 0, µ > 0,there exist matrices P > 0, R1 > 0, R2 > 0, R3 > 0, such that the
following LMIs hold for all i, j = 1, 2, · · ·, L,

Σij =

























Σij11 Σij12 Σij13 0 Σij15 Σij16

∗ Σij22 0 0 0 0

∗ ∗ Σij33 Σij34 Σij35 0

∗ ∗ ∗ Σij44 0 0

∗ ∗ ∗ ∗ Σij55 Σij56

∗ ∗ ∗ ∗ ∗ Σij66

























< 0, (9)

where

Σij11 = AiP + PA
T
i + R1 − R2 − R3, Σij12 = P, Σij13 = BiKj + R2, Σij15 = µPA

T
i ,

Σij16 = AidP + R3, Σij22 = −Q
−1, Σij33 = −R2, Σij34 = K

T

j , Σij35 = µK
T

j B
T
i ,

Σij44 = −R
−1, Σij55 = −2µP + h

2R2 + d
2
MR3, Σij56 = AidP,

Σij66 = −(1 − dD)R1 − R3.

Then there exists a sampled-data controller (4) with Kj = KjP
−1

(j = 1, 2, · · ·, L) such that
H2 guaranteed cost control performance (7) is minimized in the sense that the closed-loop system
(6) is asymptotically stable.

Proof. Choose the Lyapunov-Krasovskii functional:

V (xt) = V1(x) + V2(xt) + V3(xt) + V4(xt), (10)

where

V1(x) = x
T (t)Px(t), V2(xt) =

∫ t

t−d(t)
xT (s)R1x(s)ds,

V3(xt) = h

∫ 0

−h

∫ t

t+θ
ẋT (s)R2ẋ(s)dsdθ, V4(xt) = dM

∫ 0

−dM

∫ t

t+θ
ẋT (s)R3ẋ(s)dsdθ

and P > 0, R1 > 0, R2 > 0,R3 > 0.

The derivative of V along the trajectories of the system (6) is computed as follows:



Fuzzy H2 Guaranteed Cost Sampled-Data Control of
Nonlinear Time-Varying Delay Systems 713

V̇1(x) = ẋ
T (t)Px(t) + xT (t)Pẋ(t)

=

L
∑

i=1

L
∑

j=1

λi(ξ(t))λj(ξ(tk))[x
T (t)ATi Px(t) + x

T (t − d(t))ATidPx(t)

+xT (t − τ(t))KTj B
T
i Px(t) + x

T (t)PAix(t) + xT (t)PAidx(t − d(t))

+xT (t)PBiKjx(t − τ(t)). (11)

V̇2(xt) = x
T (t)R1x(t) − (1 − ḋ(t))x

T (t − d(t))R1x(t − d(t))

≤ xT (t)R1x(t) − (1 − dD)x
T (t − d(t))R1x(t − d(t)). (12)

By using Lemma 2.1, we have

−h

∫ t

t−h

ẋT (s)R2ẋ(s)ds ≤ −τ(t)

∫ t

t−τ(t)
ẋT (s)R2ẋ(s)ds ≤ −

(

∫ t

t−τ(t)
ẋ(s)ds

)T

R2

(

∫ t

t−τ(t)
ẋ(s)ds

)

.

(13)
Leibniz-Newton formula is

∫ t

t−h

ẋ(s)ds = x(t) − x(t − h). (14)

Applying (13) and Leibniz-Newton formula, we have

V̇3(xt) = h
2ẋT (t)R2ẋ(t) − h

∫ t

t−h

ẋT (s)R2ẋ(s)ds

≤ h2ẋ(t)T R2ẋ(t) − (x(t) − x(t − τ(t)))
T R2(x(t) − x(t − τ(t)))

T

= h2ẋT (t)R2ẋ(t) − x
T (t)R2x(t) + x

T (t − τ(t))R2x(t) + x
T (t)R2x(t − τ(t))

−xT (t − τ(t))R2x(t − τ(t)). (15)

Similarly, by Lemma 2.1 and Leibniz-Newton formula, we have

V̇4(xt) ≤ d
2
Mẋ

T (t)R3ẋ(t) − x
T (t)R3x(t) + x

T (t − d(t))R3x(t)

+xT (t)R3x(t − d(t)) − x
T (t − d(t))R3x(t − d(t)). (16)

From (6), for a given µ > 0,

0 = −2µẋT (t)Pẋ(t) + µẋT (t)P{

L
∑

i=1

L
∑

j=1

λi(ξ(t))λj(ξ(tk))[Aix(t) + Aidx(t − d(t))

+BiKjx(t − τ(t))]} + µ{
L
∑

i=1

L
∑

j=1

λi(ξ(t))λj(ξ(tk))[Aix(t) + Aidx(t − d(t))

+BiKjx(t − τ(t))]}
T Pẋ(t)

= −2µẋT (t)Pẋ(t) +
L
∑

i=1

L
∑

j=1

λi(ξ(t))λj(ξ(tk))[µẋ
T (t)PAix(t)

+µẋT (t)PAidx(t − d(t)) + uẋ
T (t)PBiKjx(t − τ(t)) + uxT (t)AiT Pẋ(t)

+µxT (t − d(t))ATidPẋ(t) + µx
T (t − τ(t))KTj B

T
i Pẋ(t)]. (17)



714 Z.-F. Qu, Z.-B. Du

From (11-12) and (15-17), we obtain

V̇ (xt) + x
T (t)Qx(t) + uT (t)Ru(t) ≤

L
∑

i=1

L
∑

j=1

λi(ξ(t))λj(ξ(tk))x̃
T (t)S′ijx̃(t) (18)

where

x̃(t) = [ xT (t) xT (t − τ(t)) ẋT (t) xT (t − d(t) ]
T ,

S′ij =













S′ij11 S′ij12 S′ij13 S′ij14

∗ S′ij22 S′ij23 0

∗ ∗ S′ij33 S′ij34

∗ ∗ ∗ S′ij44













(19)

with

S′ij11 = A
T
i P + PAi + R1 − R2 − R3 + Q, S′ij12 = PBiKj + R2, S′ij13 = µA

T
i P,

S′ij14 = PAid+R3, S′ij22 = −R2 + K
T
j RKj, S′ij23 = µK

T
j B

T
i P,

S′ij33 = −2µP + h
2R2 + d

2
MR3, S′ij34 = PAid, S′ij44 = −(1 − dD)R1 − R3.

Pre-andpost-multiplying thematrix S′ij in (19)by diag
[

P−1 P−1 P−1 P−1
]

withP =

P−1,R1 = P
−1R1P

−1,R2 = P
−1R2P

−1,R3 = P
−1R3P

−1,Q̄ = P−1QP−1,K̄j = KjP
−1 (j =

1, 2, · · ·, L), we have

Sij =















∑

ij11 +Q̄
∑

ij13

∑

ij15

∑

ij16

∗
∑

ij33 +K̄
T
j RK̄j

∑

ij35 0

∗ ∗
∑

ij55

∑

ij56

∗ ∗ ∗
∑

ij66















. (20)

If (9) is satisfied, then Σij < 0 is equivalent to Sij < 0 in (20) by using the Schur complement.
And, Sij < 0 in (20) is equivalent to S′ij < 0 in (19). Thus,

V̇ (xt) + x
T (t)Qx(t) + uT (t)Ru(t) < 0. (21)

Integrating both sides of (21) from t= 0 tot = ∞, we obtain

V (xt(∞)) − V (xt(0)) +

∫

∞

0
(xT (t)Qx(t) + uT (t)Ru(t))dt < 0. (22)

Thus, we have

J < V (xt(0)) = x
T (0)Px(0). (23)

Now, we provide a stability condition for the fuzzy T–S system (6) under case 2.

Theorem 2. Suppose that, under case 2, for given matrices Q > 0, R > 0, scalars h > 0,
dM > 0 , µ > 0, there exist matrices P > 0, R1 > 0, R2 > 0such that the following LMIs hold



Fuzzy H2 Guaranteed Cost Sampled-Data Control of
Nonlinear Time-Varying Delay Systems 715

for all i, j = 1, 2, · · ·, L

Σ̄ij =

























Σ̄ij11 Σ̄ij12 Σ̄ij13 0 Σ̄ij15 Σ̄ij16

∗ Σ̄ij22 0 0 0 0

∗ ∗ Σ̄ij33 Σ̄ij34 Σ̄ij35 0

∗ ∗ ∗ Σ̄ij44 0 0

∗ ∗ ∗ ∗ Σ̄ij55 Σ̄ij56

∗ ∗ ∗ ∗ ∗ Σ̄ij66

























< 0, (24)

where

Σ̄ij11 = AiP + PA
T
i − R1 − R2, Σ̄ij12 = P, Σ̄ij13 = BiKj + R1, Σ̄ij15 = µPA

T
i ,

Σ̄ij16 = AidP + R2, Σ̄ij22 = −Q
−1, Σ̄ij33 = −R1, Σ̄ij34 = K

T

j , Σ̄ij35 = µK
T

j B
T
i ,

Σ̄ij44 = −R
−1, Σ̄ij55 = −2µP + h

2R1 + d
2
MR2, Σ̄ij56 = AidP, Σ̄ij66 = −R2.

Then there exists a sampled-data controller (4) with Kj = KjP
−1

(j = 1, 2, · · ·, L) such that
H2 guaranteed cost control performance (7) is minimized in the sense that the closed-loop system
(6) is asymptotically stable.

Proof. Choose the following Lyapunov-Krasovskii functional:

V (xt) = V1(x) + V2(xt) + V3(xt), (25)

where

V1(x) = x
T (t)Px(t), V2(xt) = h

∫ 0

−h

∫ t

t+θ
ẋT (s)R1ẋ(s)dsdθ,

V3(xt) = dM

∫ 0

−dM

∫ t

t+θ
ẋT (s)R2ẋ(s)dsdθ

and P > 0, R1 > 0, R2 > 0 are to be determined. Then following the similar line in Theorem 1,
we can obtain Theorem 2.
If there is no time delay, then we have the following Corollary 1.
Corollary 1. Suppose that, for given matrices Q > 0, R > 0, scalars h > 0, µ > 0, there

exist matrices P > 0, R1 > 0, such that the following LMIs hold for all i, j = 1, 2, · · ·, L

¯̄Σij =



















¯̄Σij11
¯̄Σij12

¯̄Σij13 0
¯̄Σij15

∗ ¯̄Σij22 0 0 0

∗ ∗ ¯̄Σij33
¯̄Σij34

¯̄Σij35

∗ ∗ ∗ ¯̄Σij44 0

∗ ∗ ∗ ∗ ¯̄Σij55



















< 0, (26)

where

¯̄Σij11 = AiP + PA
T
i − R1,

¯̄Σij12 = P,
¯̄Σij13 = BiKj + R1,

¯̄Σij15 = µPA
T
i ,

¯̄Σij22 = −Q
−1,

¯̄Σij33 = −R1,
¯̄Σij34 = K

T

j ,
¯̄Σij35 = µK

T

j B
T
i ,

¯̄Σij44 = −R
−1, ¯̄Σij55 = −2µP + h

2R1.

Then there exists a sampled-data controller (4) with Kj = KjP
−1

(j = 1, 2, · · ·, L) such that
H2 guaranteed cost control performance (7) is minimized in the sense that the closed-loop system
(6) is asymptotically stable.



716 Z.-F. Qu, Z.-B. Du

In the following, we give the design procedure of fuzzy sampled-data controller.

The H2 guaranteed cost sampled-data fuzzy control problem can be formulated as the fol-
lowing optimization problem:

min
P̄

Trace(J)

s.t.(9)and

[

J xT (0)

∗ P̄

]

> 0. (27)

Design Procedure: The delay-dependentH2 guaranteed cost sampled-datacontrol for fuzzy
time-varying delay system is summarized as follows.

Step 1: Select membership functions and fuzzy rules in (1).

Step 2: Give the upper bound of sampling interval h > 0 and a scalar µ > 0.

Step 3: Solve the LMIs (27) to obtain Kj(j = 1, 2, · · ·, L) and P .Thus,Kj = KjP
−1

(j = 1, 2, · · ·, L) can also be obtained.

Step 4: Increaseh, and repeat Step 3 until Kj(j = 1, 2, · · ·, L) and P can not be found.

Step5: Confirmfuzzy H2 guaranteedcostsampled-datacontrol and stability of the closed-loop
system, substitute P ,Kj(j = 1, 2, · · ·, L), µ and h into (19) and verify Sij < 0.

Step 6: Construct the fuzzy sampled-data controller (4).

4 Simulation example

To test the effectiveness and feasibility of the proposed method, we consider the following
truck-trailer system [14]

ẋ1(t) = −a
vt
Lt0

x1(t) − (1 − a)
vt
Lt0

x1(t − td) +
vt
lt0

u(t)

ẋ2(t) = a
vt
Lt0

x1(t) + (1 − a)
vt
Lt0

x1(t − td)

ẋ3(t) =
vt
Lt0

sin(x2(t) + a(vt/2L)x1(t) + (1 − a)(vt/2L)x1(t − td)),

(28)

where l = 2.8 L = 5.5, v = −1.0, a = 0.7, t̄ = 2.0, t0 = 0.5. x1(t) ∈ [−π/2, π/2],

ẋ1(t) ∈ [−3, 3], x2(t) ∈ [−π/2, π/2], ẋ2(t) ∈ [−2, 2].x(t) = [x1(t) x2(t) x3(t)]
T ,

[x1(0) x2(0) x3(0)] =
[

1.5 −2 5

]

.

The nonlinear truck-trailer system is modeled by two-rule fuzzy T-S system.

Rule 1: IF θ(t) = x2(t) + a(vt/2L)x1(t) + (1 − a)(vt/2L)x1(t − td) is about 0,

Thenẋ(t) = A1x(t) + Ad1x(t − τd) + B1u(t). (29)

Rule 2: IF θ(t) = x2(t) + a(vt/2L)x1(t) + (1 − a)(vt/2L)x1(t − td) is about π or −π,

Thenẋ(t) = A2x(t) + Ad2x(t − τd) + B2u(t). (30)

where

A1 =









−a vt
Lt0

0 0

a vt
Lt0

0 0

a v
2t

2

2Lt0
vt
t0

0









, Ad1 =









−(1 − a) vt
Lt0

0 0

(1 − a) vt
Lt0

0 0

(1 − a) v
2t

2

2Lt0
0 0









, B1 =









vt
lt0

0

0









,



Fuzzy H2 Guaranteed Cost Sampled-Data Control of
Nonlinear Time-Varying Delay Systems 717

A2 =









−a vt
Lt0

0 0

a vt
Lt0

0 0

adv
2t

2

2Lt0
dvt
t0

0









, Ad2 =









−(1 − a) vt
Lt0

0 0

(1 − a) vt
Lt0

0 0

(1 − a)dv
2t

2

2Lt0
0 0









, B2 =









vt
lt0

0

0









,

and d = 10t0/π.
The membership functions are defined as

λ1(θ(t)) =

(

1 −
1

1 + exp(−3(θ(t) − 0.5π))

)

×

(

1

1 + exp(−3(θ(t) + 0.5π))

)

,

λ2(θ(t)) = 1 − λ1(θ(t)).

A two-rule sampled-data fuzzy controller is employed to stabilize the truck trailer system.
The sampled-data fuzzy controller is designed as follows:

u(t) =

2
∑

j=1

λj(θ(tk))Kjx(tk).

First, we assume that time delay d(t) = 0. Applying various methods of [8] ( H2 control), [10]
( H2 control) and Corollary 1, the dimensions of the LMIs are given in Table 1. It is seen from
Table 1 that the dimension of the LMIs is greatly simplified in the proposed method of this
paper.

Table 1: The comparison for the dimensions of LMIs (Corolarry 1)

Method [8] [10] Corollary 1

Dimension 25 28 13

Next, we assume that the delay is time-invariant, i.e. dD = 0. By using various the methods
of [3](H2 control) and Theorem2, the dimensions of the LMIs are given in Table 2. It is seen
from Table 2 that the dimension of the LMIs is simplified in the proposed method of this paper,
which adds the existence of feedback gains and lowers the implementation time.

Table 2: The comparison for the dimensions of LMIs (Theorem 2)

Method [3] Theorem 2

Dimension 20 16

By using various methods of [3] and Theorem 2, the maximum allowable upper bounds of
sampling interval are given in Table 3, which show that Theorem 2 of this paper can get a larger
sampling interval. This implies that the proposed method achieves a better performance.

Table 3: The maximum allowable upper bounds of sampling interval

Method [3] Theorem 2

hmax(td = 0.5) 0.374 0.562

hmax(td = 1) 0.315 0.471

hmax(td = 2) 0.251 0.283

Finally, we consider the control design for time-varying delay td = 1 + sin t. The maximum
allowable upper bound of sampling interval that is obtained by Theorem 1 is 0.295.When the
design parameters are given by µ = 1 , dM = 2 , dD = 1 with the sampling interval h =
0.295,Theorem 1 gives the fuzzy state feedback control gains

K1 = [1.0319 -0.1019 0.0009] , K2 = [1.0319 -0.1019 0.0009] .



718 Z.-F. Qu, Z.-B. Du

Figure 1: State response x1

Figure 2: State response x2

Figure 3: State response x3

Figure 4: State response x4



Fuzzy H2 Guaranteed Cost Sampled-Data Control of
Nonlinear Time-Varying Delay Systems 719

When time-varying delay td is 1.5 + 1.5 sin t, Theorem 1 gives the maximum allowable upper
bound of sampling interval 0.172.With the design parameters h = 0.172 , µ = 1.5 , dM = 3 ,

dD = 1.5 , Q = diag{ 1 10 0.1 }× 10
−6,R = 10−5, Theorem 1 gives the fuzzy state feedback

control gains

K1 = [0.9656 -0.0657 0.0006] , K2 = [0.9656 -0.0657 0.0006] .

The sampled-data fuzzy controller with the above control gains is applied to the truck trailer
system, the results on the state responsesx1, x2, x3 and control lawuare shown in Figures 1- 4.
Simulation results illustrate the fuzzy H2 guaranteed cost sampled-data control design is

effective and feasible. Figs. 1-3 show the system stability, and Fig.4 shows the sampled-data
control signal for the system (28).

5 Conclusion

This work considers the fuzzyH2 GC sampled-data control problem for nonlinear systems
with time-varying delay. It should be pointed that this problem is more complicated and harder
to deal with due to the coexistence of feedback delay and sampled-data control. A new sufficient
condition for the existence of fuzzy sampled-data controller is given in terms of LMIs.

To better demonstrate our results, a truck-trailer system with sampled-data control is given.
Simulation results show the effectiveness and feasibility of sampled-data control design. Further-
more, this method could be extended to H∞ control.

Acknowledgment

This work was supported by the National Natural Science Foundation of China(61203320)
and Shandong Natural Science Foundation (ZR2014FL023).

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