Decentralized Networked Control System Design Using ......Manuscript received March 25, 2014; accepted September 25, 2014 Recommended by Associate Editor Fernando Tadeo c Institute

International Journal of Automation and Computing 12(2), April 2015, 125-133

DOI: 10.1007/s11633-015-0879-9

Decentralized Networked Control System Design

Using Takagi-Sugeno (TS) Fuzzy Approach

Chedia Latrach1 Mourad Kchaou2 Abdelhamid Rabhi 1 Ahmed El Hajjaji11Laboratory of Modelisation, Information and System, University of Picardie Jules Verne, 7 New Mill Street, Amiens, France

2Electrical Engineering Department, National Engineering School of Sfax, University of Sfax, Tunisia

Abstract: This paper proposes a new method for control of continuous large-scale systems where the measures and control functions

are distributed on calculating members which can be shared with other applications and connected to digital network communications.

At first, the nonlinear large-scale system is described by a Takagi-Sugeno (TS) fuzzy model. After that, by using a fuzzy Lyapunov-

Krasovskii functional, sufficient conditions of asymptotic stability of the behavior of the decentralized networked control system (DNCS),

are developed in terms of linear matrix inequalities (LMIs). Finally, to illustrate the proposed approach, a numerical example and

simulation results are presented.

Keywords: Continuous large-scale systems, decentralized static output feedback fuzzy control, networked control systems (NCS),

Takagi-Sugeno (TS) fuzzy model, linear matrix inequalities (LMIs).

1 Introduction

Decentralized control of large-scale systems (also known

as interconnected systems in some books) has been investi-

gated as a branch of control theory and has received con-

siderable attention over the past three decades due to its

various applications such as power systems, aerospace sys-

tems, nuclear reactors, systems control process, etc.[1−3]

In fact, various techniques for distributed control using

linear matrix inequalities (LMIs) were recently studied[4−9].The systems consist of a large set of interconnected sub-

systems which can be far from each other. That′s whywe introduce the notion of communication network to con-

nect them, and thus it aims to ensure data transmission

and coordinating manipulation among spatially distributed

components. Compared with conventional point-to-point

control systems, the advantages of networked control sys-

tems (NCS) are less wiring, lower installation cost as well

as greater agility in diagnosis and maintenance. Because of

these distinctive benefits, typical application of these sys-

tems ranges over various fields, such as automotive, mobile

robotics, advanced aircraft, etc. It is well known that lim-

ited network resources, network-induced delays and data

packets dropout through the network, may degrade the de-

centralized networked control system (DNCS) performance

and lead to instability. It is mentioned that the communi-

cation delay, which has time-varying characteristics, is one

of the important factors to be considered in NCS analysis

and synthesis[10−19].In this paper, the decentralized static output feedback

Regular PaperSpecial Issue on Advances in Nonlinear Dynamics and ControlManuscript received March 25, 2014; accepted September 25, 2014Recommended by Associate Editor Fernando Tadeoc© Institute of Automation, Chinese Academy of Science and

Springer-Verlag Berlin Heidelberg 2015

control method for stabilization of nonlinear interconnected

system, that takes into account problems of delay and data

packets dropout in communication, is proposed. Based on

Takagi-Sugeno (TS) fuzzy system, the static output feed-

back controller is designed. The sufficient condition is of-

fered to guarantee the stability of the closed-loop system

using Lyapunov Krasovskii functional. Its constructive con-

ditions are presented in LMIs terms, taking effects of com-

munication network into account.

The paper is organized as follows. Section 2 presents

system description and preliminaries. Section 3 presents

the main results, describing the control strategy for large-

scale systems through a communication network. Section 4

shows simulation results. Finally, conclusions are given in

Section 5.

Notations. sym(W ) stands for W +WT. The symbol

(∗) within a matrix represents the symmetric entries.

2 Preliminaries and system description

Consider a large-scale system S composed of J intercon-

nected subsystems Si, i = 1, 2, · · · , J . The i-th fuzzy sub-system Si is described by the following TS fuzzy model:

Si :

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

If θi1(t) is Fli1 and θig(t) is F

lig

then ẋi(t) = Alixi(t) +B

liui(t) +

J∑

j=1

fij(xj(t))

yi(t) = C2ixi(t)

(1)

where i = 1, 2, · · · , J , l = 1, 2, · · · , ri, xi(t) denotes thestate vector, yi(t) denotes the measured output, ui(t) is

the control input, Ali, Bli and C2i are constant real matrices

with appropriate dimensions and C2i is full rank, θi1(t),

126 International Journal of Automation and Computing 12(2), April 2015

θi2(t), · · · , θig(t) are some measurable premise variables forsubsystems Si, F

liq(q = 1, 2, · · · , g) represents the linguistic

fuzzy sets of the rule, fij(xj(t)) represents the interconnec-

tion of fuzzy rules in subsystem Si and subsystem Sj , and

ri represents the number of fuzzy rules in subsystem Si.

Using the central-average defuzzifier, the TS fuzzy sys-

tem can be given as⎧⎪⎪⎨

⎪⎪⎩

ẋi(t) =

ri∑

l=1

hli(θi(t))[Alixi(t) +B

liui(t) +

J∑

j=1

fij(xj(t))]

yi(t) = C2ixi(t)

(2)

where

hli(θi(t)) =υli(θi(t))

ri∑

l=1

υli(θi(t))

υli(θi(t)) =

g∏

q=1

F liq(θiq(t))

(3)

with F liq(θiq(t)) as the grade of membership of θiq(t) in the

fuzzy set F liq . hli(θi(t)) is the membership function for each

fuzzy rule, which represents normalized grade of member-

ship, and satisfies

0 ≤ hli(θi(t)) ≤ 1, for l = 1, 2, · · · , ri,ri∑

l=1

hli(θi(t)) = 1.

(4)

We assume that system S will be controlled through net-

work. Fig. 1 represents the structure of networked control

sub system Si with induced delays, where τsci is sensor-to-

controller delay and τcai is the controller-to-actuator delay.

It is assumed that the controller computational delay can

be absorbed into either τsci or τcai.

Fig. 1 Framework of networked control subsystem Si

Assumption 1. All pairs (Ali, Bli) (i = 1, 2, · · · , J and

l = 1, 2, · · · , ri) are stabilizable.[20]Assumption 2. The interconnection fij(xj(t)) satisfies

the following conditions: fij(xj(t)) = Blifijl (xj(t)) and

‖f lij(xj(t))‖ ≤ f̄ lij‖xj(t)‖, where f̄ lii = 0, f̄ lij(i �= j) is apositive constant and Bli is a constant real matrix with ap-

propriate dimensions.[20]

Assumption 3. The sensors are clock driven, the con-

troller and actuators are event driven.

Assumption 4. Data, either from measurement or for

control, are transmitted in a single packet.

Assumption 5. The effect of signal quantization is not

considered.

Assumption 6. The real input ui(t) for each subsystem,

realized through a zero-order hold (ZOH), is a piecewise

constant function.

It is worth mentioning that the sampling period of a sen-

sor is pre-determined for control algorithm design, and thus

the sensor can be assumed to be clock driven. However, an

actuator does not change its output to the plant under con-

trol until an updated control signal is received, implying

that the actuator is event driven.

To obtain our main results, the following lemmas are

needed.

Lemma 1.[21] For each real vector ζ and ρ, it follows

that

2ζTρ ≤ ζTZζ + ρTZ−1ρ (5)with Z > 0.

Lemma 2.[20] The following inequality is verified for each

real vector νi ∈ Rn:

[m∑

i=1

νi]

T

[m∑

i=1

νi] ≤ mm∑

i=1

νTi νi. (6)

3 Main results

In this section, we are interested in the design of static

output feedback controller in order to stabilize the system.

Indeed, it is assumed that the states of the system (2) are

not all available for measurement, that is why we achieve an

output feedback control. The control scheme type parallel

distributed compensation (PDC) will be considered for each

subsystem Si. The overall fuzzy PDC networked controller

corresponding to Si can be described as

ui(tk) =

ri∑

l=1

hli(θi(tk))Kliyi(tk − τki).

From the ZOH, the input signal for each subsystem Si for

tk ≤ t ≤ tk+1 is given by

ui(t) =

ri∑

l=1

hli(θi(tk))Kliyi(tk − τki). (7)

For network-induced delay (τki), one major challenge for

NCS design is the effect of network-induced delays in a con-

trol loop. It occurs when the system components exchange

data across the network. It can degrade control perfor-

mance significantly or even destabilize the system. The de-

lays in NCS consist of a communication delay between sen-

sors and controllers τsci, a communication delay between

controller and actuators τcai, computational time in con-

troller τc which can be generally included in the controller

to actuator delay.

A natural assumption on τki can be made as

0 < τmi ≤ τki ≤ τMi. (8)

C. Latrach et al. / Decentralized Networked Control System Design Using Takagi-Sugeno (TS) Fuzzy Approach 127

Packet dropouts are network-induced effects which can

be the consequence of a link failure. They can also be gen-

erated purposefully in order to avoid congestion or to guar-

antee the most recent data to be sent. Although most net-

work protocols are equipped with transmission-retry mech-

anisms, they can only re-transmit for limited time. After

this time has expired, the packets are dropped. Normally,

feedback controllers can tolerate a certain amount of packet

losses. But the consecutive packet losses have an adverse

impact on the overall performance.

tk+1 − tk = σ̄iTe + maxi

{τ(k+1)i} − mini

{τki} (9)

where Te denotes the sampling period, tk denotes the sam-

pling instant, and σ̄i denotes the maximum number of

packet dropouts in the updating periods. Using (2) and

(7), the closed-loop networked control system can be writ-

ten for tk ≤ t ≤ tk+1 as{ẋi(t) = A(t)xi(t) +H(t)xi(tk − τki) + f(xi(t))yi(t) = C2ixi(t)

(10)

with

A(t) =

ri∑

l=1

hliAli

B(t) =

ri∑

l=1

hliBli

H(t) = B(t)

ri∑

s=1

hsiKsiC2i

f(xi(t)) =

ri∑

l=1

hli

J∑

j=1

fij(xj). (11)

Defining ηi(t) = t− tk + τki, tk ≤ t ≤ tk+1, thenτki ≤ ηi(t) ≤ σ̄iTe + max

i{τ(k+1)i}.

Thus, we get from [22] that

η1i ≤ ηi(t) ≤ η2i, η̇i(t) ≤ hdi (12)where

η1i = τmi and η2i = σ̄iTe + maxi

{τMi}.

As∑∞

k=0[tk, tk+1) = [0,∞), we have⎧⎪⎨

⎪⎩

ẋi(t) = A(t)xi(t) +H(t)xi(t− ηi(t)) + f(xi(t))yi(t) = C2ixi(t)

xi(t) = φi(t), t ∈ [t0 − η2i, t0](13)

where φi(t) can be viewed as the initial condition of the

closed-loop control system. Then based on (12), it is noted

that the NCS (13) is equivalent to a system with an interval

time-varying delay.

The controller design is based on the following prelimi-

nary result given by the Lemma 3.

Lemma 3. For given scalars η1i > 0 and η2i > 0, the

closed-loop system (13) is asymptotically stable, if there

exist positive matrices Pi, Q1i, Q2i, Q3i, Z1i, and matrices

G1i, G2i and G3i, with appropriate dimensions, such that

the following conditions hold:

Φij =

⎡

⎢⎢⎢⎢⎢⎢⎣

Φ11ij Φ12i Z1i 0 Φ15i

∗ Φ22i 0 0 Φ25ij∗ ∗ −Q2i − Z1i 0 0∗ ∗ ∗ −Q3i 0∗ ∗ ∗ ∗ Φ55i

⎤

⎥⎥⎥⎥⎥⎥⎦

< 0

(14)

Φ11ij = Q1i +Q2i +Q3i + sym(GT1iA(t)) − Z1i +GT1iG1i+

(3J

J∑

j=1

f̂2ji‖B̂j‖2)I

Φ12i = A(t)TG2i +G

T1iH(t)

Φ22i = sym(GT2iH(t))− (1 − hdi)Q1i +GT2iG2i

Φ15i = Pi −GT1i +A(t)TG3iΦ25ij = −GT2i +HT(t)G3iΦ55i = η

21iZ1i − sym(G3i) +GT3iG3i.

Proof. Let the Lyapunov-Krasovskii functional candi-

date be

V (t) =J∑

i=1

vi(t), i = 1, 2, · · · , J (15)

where vi(t) denotes the Lyapunov-Krasovskii functional

corresponding to fuzzy subsystem Si. Each vi(t) is defined

as

vi(t) = xTi (t)Pixi(t) +

∫ t

t−ηi(t)xTi (s)Q1ixi(s) ds+

∫ t

t−η1ixTi (s)Q2ixi(s) ds+

∫ t

t−η2ixTi (s)Q3ixi(s) ds+

η1i

∫ 0

−η1i

( ∫ t

t+s

ẋTi (υ)Z1iẋi(υ) dυ)ds. (16)

The corresponding time derivative of vi(t) is given by

v̇i(t) ≤ 2ẋTi (t)Pixi(t) + xTi (t)(Q1i +Q2i +Q3i)xi(t)−(1 − hdi)xTi (t− ηi(t))Q1ixi(t− ηi(t))−xTi (t− η1i)Q2ixi(t− η1i)−xTi (t− η2i)Q3ixi(t− η2i)+ẋTi (t)(η

21iZ1i)ẋi(t)−

η1i

∫ t

t−η1iẋTi (υ)Z1iẋi(υ) dυ. (17)

Denoting ψ1i = xi(t)−xi(t− η1i), by Jensen inequality, wecan obtain

−η1i∫ t

t−η1iẋTi (υ)Z1iẋi(υ) dυ ≤ −ψT1iZ1iψ1i. (18)

128 International Journal of Automation and Computing 12(2), April 2015

From (13), we construct for appropriately dimensioned ma-

trices G1i, G2i, and G3i as the following zero-value expres-

sion:

2[xTi (t)G

T1i + x

Ti (t− ηi(t))GT2i + ẋTi (t)GT3i

]×[ − ẋi(t) +A(t)xi(t) +H(t)x(t− ηi(t)) + f(xi(t))

]= 0

ΨTi (t) =[

xTi (t) xTi (t− ηi(t)) xTi (t− η1i) xTi (t− η2i) ẋTi (t)

].

(19)

According to Lemmas 1 and 2, we have

2xTi (t)GT1i

J∑

j=1

fij(xj) ≤

xTi (t)GT1iG1ixi(t) +

J∑

j=1

fTij(xj)J∑

j=1

fij(xj) ≤

xTi (t)GT1iG1ixi(t) + J

J∑

j=1

fTij(xj)fij(xj). (20)

Based on Assumptions 1 and 2, and defining f̂ij = maxl f̄lij ,

‖B̂i‖ = maxl ‖Bli‖, we have

2xTi (t)GT1i

J∑

j=1

fij(xj) ≤

xTi (t)GT1iG1ixi(t) + J

J∑

j=1

fTij(xj)fij(xj) ≤

xTi (t)(GT1iG1i + J

J∑

j=1

f̂2ji‖B̂j‖2)Ixi(t) (21)

2xTi (t− ηi(t))GT2iJ∑

j=1

fij(xj) ≤

xTi (t− ηi(t))GT2iG2ixi(t− ηi(t))+

xTi (t)J

J∑

j=1

f̂2ji‖B̂j‖2Ixi(t) (22)

2ẋTi (t)GT3i

J∑

j=1

fij(xj) ≤

ẋTi (t))GT3iG3iẋi(t) + x

Ti (t)J

J∑

j=1

f̂2ji‖B̂j‖2)Ixi(t). (23)

Considering (17)−(19) and (21)−(23), the derivative of (15)along the closed loop system (13) can be described as

V̇ (t) =

J∑

i=1

v̇i(t) ≤J∑

i=1

J∑

j=1

ri∑

l=1

ri∑

s=1

hlihsjΨ

Ti (t)ΦijΨi(t) ≤ 0.

(24)

According to Lemma 3, we have V̇ (t) < 0. So system (13)

is asymptotically stable. �The objective now is to determine the gain matrices Kli

such that the static output feedback closed-loop system is

asymptotically stable.

Theorem 1. For given scalars η1i > 0, η2i > 0, μ1,

μ2, and μ3, the closed-loop system (13) is asymptotically

stable, if there exist positive matrices P̄i, Q̄1i, Q̄2i, Q̄3i,

Z̄1i, matrices Ĝ11i > 0, Ĝ21i > 0, Ĝ22i > 0, and Ys

i , with

appropriate dimensions, such that the following conditions

hold

Φ̄llij < 0 (25)

Φ̄lsij + Φ̄slij < 0, j > i, s > l (26)

where

Φ̄lsij =⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

Φ̄11il Φ̄12ils Z̄1i 0 Φ̄15il ḠTi

∗ Φ̄22ils 0 0 Φ̄25ils 0∗ ∗ −Q̄2i − Z̄1i 0 0 0∗ ∗ ∗ −Q̄3i 0 0∗ ∗ ∗ ∗ Φ̄55i 0∗ ∗ ∗ ∗ ∗ Φ̄66ij

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

Φ̄11il = Q̄1i + Q̄2i + Q̄3i + μ1sym(AliḠi) − Z̄1i + μ21I

Φ̄12ils = μ2ḠTi (A

li)

T + μ1BliY

si C2i

Φ̄22ils = μ2sym(BliY

si C2i) − (1 − hdi)Q̄1i + μ22I

Φ̄15il = P̄i − μ1Ḡi + μ3ḠTi (Ali)T

Φ̄25ils = −μ2Ḡi + μ3CT2i(Y si )T(Bli)TΦ̄55i = η

21iZ̄1i − μ3sym(Ḡi) + μ23I

Φ̄66ij = −(3JJ∑

j=1

f̂2ji‖B̂j‖2)−1I

Ḡi = Vi

[Ĝ11i 0

Ĝ21i Ĝ22i

]

V Ti . (27)

Then, the desired controller gains are given by Ksi =

Y si WiSiĜ−111iS

−1i W

Ti , where Wi, Si and Vi are derived from

singular value decomposition (SVD) of C2i.

Proof. Under the conditions of the Theorem 1, a feasi-

ble solution satisfies the condition Φ̄55i < 0, which implies

that Ḡi is nonsingular. Define Gi = Ḡ−1i , P̄i = Ḡ

Ti PiḠi,

Q̄1i = ḠTi Q1iḠi, Q̄2i = Ḡ

Ti Q2iḠi, Q̄3i = Ḡ

Ti Q3iḠi and

Z̄1i = ḠTi Z1iḠi.

Assume that C2i is full rank, then the SVD decompo-

sition exists such that WTi C2iVi =[

Si 0]

and Ḡi =

Vi

[Ĝ11i 0

Ĝ21i Ĝ22i

]

V Ti . It is obtained that

C2iḠi = Wi[Si 0]VT

i Vi

[Ĝ11i 0

Ĝ21i Ĝ22i

]

V Ti =

Wi[SiĜ11i 0]VT

i =

WiSiĜ11iS−1i W

Ti Wi[Si 0]V

Ti = ĜiC2i.

By letting Y li = KliĜ = K

liWiSiĜ11iS

−1i W

Ti , using Schur

complement and applying a congruence transformation to

(25) and (26) by diag{Gi, Gi, Gi, Gi, Gi

}, we find that the

C. Latrach et al. / Decentralized Networked Control System Design Using Takagi-Sugeno (TS) Fuzzy Approach 129

condition (14) holds considering (4) and (11). Thus, there

exists a fuzzy controller (7) such that the closed-loop system

(13) is asymptotically stable. �

4 Simulation results

Example 1. To show the effectiveness of the proposed

approach, we consider the numerical example given in [4],

which is composed of two subsystems S1 and S2 described

respectively by

⎧⎪⎪⎨

⎪⎪⎩

ẋ1(t) =

2∑

l=1

hl1(θ1(t))[Al1x1(t) +B

l1u1(t) +

J∑

j=1

f1j(xj(t))]

y1(t) = C21x1(t)

(28)

with

A11 =

⎡

⎢⎣

−6 6 00.5 −3 10 0.2 −1

⎤

⎥⎦ , A

21 =

⎡

⎢⎣

−1 0.1 0−0.2 −2 00.3 0 −1

⎤

⎥⎦

B11 =

⎡

⎢⎣

2 1

1 1

1 1

⎤

⎥⎦ , B

21 =

⎡

⎢⎣

1 2

1 2

1 1

⎤

⎥⎦

C21 =

[1 0.1 0.1

0.1 0.2 0.1

]

, f11 = 0

f12 =

⎡

⎢⎣

0.02 0.01

0.01 0.4

0.01 0.1

⎤

⎥⎦ ||x2||

h11(x1(t)) = sin2(x11(t)), h

21(x1(t)) = cos

2(x11(t)).

For subsystem S2,

⎧⎪⎪⎨

⎪⎪⎩

ẋ2(t) =

2∑

l=1

hl2(θ2(t))[Al2x2(t) +B

l2u2(t) +

J∑

j=1

f2j(xj(t))]

y2(t) = C22x2(t)

(29)

with

A12 =

[−1 00 −1

]

, A22 =

[−2 00 −1

]

B12 =

[2

1

]

, B22 =

[2

2

]

C22 =

[1 0

0.1 1

]

, f21 =

[0.01 0.01 0.01

0.02 0.01 0.1

]

||x1||

f22 = 0, h12(x2(t)) = sin

2(x21(t))

h22(x2(t)) = cos2(x21(t)).

The network-related parameters for each subsystem Siare assumed as Te = 3ms, the minimum delay η1i = 4ms,

the maximum delay η2i = 20 ms and the maximum num-

ber of packet dropouts is σ̄i = 3. The time varying de-

lays between the sensors and controller as well as between

controller and actuator are generated randomly such as

min(τsci+τcai) ≥ η1i, and max(τsci+τcai+(σ̄i+1)Te) ≤ η2i,and packet dropouts are also generated randomly such as

max(Ne) ≤ 3, where Ne is the number of packet dropouts,hdi = 0.1, μ1 = 1 , μ2 = 0.3 and μ3 = 0.5.

By Theorem 1, we find a feasible solution as K11 =[1.2494 −3.3911−1.3490 3.2305

]

, K21 =

[0.0596 0.5105

−0.1445 −0.5467

]

for

subsystem S1, and K12 =

[

−0.0559 −0.0221], K22 =

[

−0.0311 −0.0564]

for subsystem S2.

For simulation, initial conditions are x1(0) =[

1 0.5 −1]T

and x2(0) =[

2 −2]T

.

The state variables evolution of NCSs and control inputs

are shown in Figs. 2−4 from which, we can note that allstates converge to zero. Figs. 5 and 6 show the delays in-

troduced by the network and packet loss data which are

randomly generated. Therefore, according to Theorem 1,

the closed-loop overall fuzzy large-scale system composed

of two subsystems S1 and S2 is asymptotically stable. The

simulation results are consistent with the analysis and sup-

port the effectiveness of the developed design strategy.

Fig. 2 Response of state x in the S1

Fig. 3 Response of state x in the S2

130 International Journal of Automation and Computing 12(2), April 2015

Fig. 4 Evolution of control input signals ui(t)

Fig. 5 Delay induced by communication networks

Fig. 6 Data packets dropout

Example 2. We consider the same large-scale system S

composed of three fuzzy subsystems Si, i = 1, 2, 3, as that

in [20].

For subsystem S1:

Rule 1 :

If x11(t) is small and x12(t) is big

then ẋ1(t) = A11x1(t) +B

11u1 +

3∑

j=1

f1j(xj(t))

y1(t) = C11x1(t).

Rule 2 :

If x11(t) is small and x12(t) is small

then ẋ1(t) = A21x1(t) +B

21u1 +

3∑

j=1

f1j(xj(t))

y1(t) = C11x1(t).

Rule 3 :

If x11(t) is big and x12(t) is small

then ẋ1(t) = A31x1(t) +B

31u1 +

3∑

j=1

f1j(xj(t))

y1(t) = C11x1(t).

For these rules,

A11 =

[−2 31.5 −2.2

]

, A21 =

[−4 33 −2

]

A31 =

[−2 3−6 −11

]

, B11 =

[0.15

0.1

]

B21 =

[0.6

0.4

]

, B31 =

[0.3

0.2

]

C11 =

[1 0

0.1 1

]

, f11 = 0

f12 =

[0.08

0.05

]

||x2||, f13 =[0.09

0.06

]

||x3||.

For subsystem S2:

Rule 1 :

If x21(t) is small and x22(t) is small

then ẋ2(t) = A12x2(t) +B

12u2 +

3∑

j=1

f2j(xj(t))

y2(t) = C11x2(t).

Rule 2 :

If x21(t) is big and x22(t) is small

then ẋ2(t) = A22x2(t) +B

22u2 +

3∑

j=1

f2j(xj(t))

y2(t) = C11x2(t).

For these rules,

A12 =

[−3 15 −3

]

, A22 =

[−2 13 −0.3

]

B12 =

[0.1

0.6

]

, B22 =

[0.2

1.2

]

f21 =

[0.02

0.12

]

||x1||, f22 = 0

f23 =

[0.06

0.36

]

||x3||.

For subsystem S3:

C. Latrach et al. / Decentralized Networked Control System Design Using Takagi-Sugeno (TS) Fuzzy Approach 131

Rule 1 :

If x31(t) is big and x32(t) is big

then ẋ3(t) = A13x3(t) +B

13u3 +

3∑

j=1

f3j(xj(t))

y3(t) = C11x3(t).

Rule 2 :

If x31(t) is small and x32(t) is big

then ẋ3(t) = A23x3(t) +B

23u3 +

3∑

j=1

f3j(xj(t))

y3(t) = C11x3(t).

For these rules,

A13 =

[−3 14 −2

]

, A23 =

[−2 13 −1

]

B13 =

[0.6

0.8

]

, B23 =

[0.3

0.4

]

f31 =

[0.48

0.64

]

||x1||, f32 =[0.24

0.32

]

||x2||

f33 = 0.

It is seen that all fij satisfy the matching condition (2)

with f̂111 = f̂211 = f̂

311 = 0, f̂

112 = 0.5, f̂

212 = 0.125, f̂

312 =

0.25, f̂113 = 0.6, f̂213 = 0.15 and f̂

313 = 0.3 for subsystem S1.

All fij satisfy (2) with f̂121 = 0.2, f̂

221 = 0.1, f̂

122 = f̂

222 =

0, f̂123 = 0.6 and f̂223 = 0.3 for subsystem S2, All fij sat-

isfy (2) with f̂131 = 0.8, f̂231 = 1.6, f̂

132 = 0.4, f̂

232 = 0.8 and

f̂133 = f̂233 = 0 for subsystem S3.

The membership functions of each state are shown in

Fig. 1 of [20].

The network-related parameters for each subsystem Siare assumed as Te = 5ms, the minimum delay η1i = 6ms,

the maximum delay η2i = 20 ms and the maximum num-

ber of packet dropouts is σ̄i = 2. The time varying de-

lays between the sensors and controller as well as between

controller and actuator are generated randomly such as

min(τsci+τcai) ≥ η1i, and max(τsci+τcai+(σ̄i+1)Te) ≤ η2iand packet dropouts are also generated randomly such as

max(Ne) ≤ 2, hdi = 0.1, μ1 = 1, μ2 = 0.5 and μ3 = 0.9.Applying Theorem 1, the solutions of LMIs can

be obtained as K11 =[

−4.2341 −4.6081], K21 =

[

−1.1609 −1.0426]

and K31 =[

0.6774 0.2271]

for

subsystem S1, K12 =

[

−2.2344 −2.9514]

and K22 =[

−1.8578 −1.5973]

for subsystem S2, and K13 =

[

−0.8119 −0.8278]

and K23 =[

−1.9309 −1.6042]

for

subsystem S3.

For simulation, initial conditions are x1(0) =[

1.5 −1]T

, x2(0) =[

−0.5 0.5]T

and x3(0) =[

0.7 −0.3]T

.

The state variable evolution of NCSs and control inputs

are shown in Figs. 7−10 from which, we can note that allstates converge to zero. Therefore, according to Theorem

1, the closed-loop overall fuzzy large-scale system composed

of three subsystems S1, S2 and S3 is asymptotically stable.

Thus, we have shown that the proposed decentralized static

output feedback controller makes the nonlinear intercon-

nected system in network communication exhibit asymp-

totic stability.

Fig. 7 State responses x in S1

Fig. 8 State responses x in S2

Fig. 9 State responses x in S3

Fig. 10 Control signal trajectories ui(t)

132 International Journal of Automation and Computing 12(2), April 2015

5 Conclusions

In this paper, based on Lyapunov-Krasovskii functional,

new stabilization conditions have been established for net-

worked controlled large-scale system. Furthermore, using

these conditions in presence of the delay and data packets

dropout in the network communication, networked fuzzy

static output feedback controller gains have been obtained.

The simulation results are shown to prove the advantages

of the developed method.

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Chedia Latrach received the M. Sc. de-gree in electrical engineering from the Na-tional Engineering School of Sfax, Tunisiain 2011. She is currently a Ph. D. candidatein the University of Picardie Jules Verne,France.

Her research interests include fuzzy con-trol, vehicle dynamics, networked controland decentralized control.

E-mail: [email protected] (Corresponding author)ORCIDiD: 0000-0002-0201-9819

Mourad Kchaou received the Ph. D.degree in automatic and industrial comput-ing from the National School of Engineer-ing, University of Sfax, Tunisia in 2009. Heis now an assistant professor in the High In-stitute of Applied Sciences and TechnologyUniversity of Sousse, Tunisia. He is a mem-ber of the Laboratory of Sciences and Tech-niques of Automation Control and Com-

puter Engineering, National Engineering School of Sfax, Tunisia.His research interests include fuzzy control, time-delay sys-

tems, descriptor systems, with particular attention paid to non-linear systems represented by multiple-models.

E-mail: [email protected]

Abdelhamid Rabhi received thePh. D. degree in observation and controlof nonlinear and complex systems in 2005.Since 2006, he has been an associate pro-fessor at the Faculty of Sciences, Univer-sity of Picardie Jules Verne, France, anda researcher in the Modeling, Informationand Systems (MIS) Laboratory, Amiens,France.

His research interests include fuzzy control, vehicle dynamicsand renewable energy systems.

E-mail: [email protected]

Ahmed El Hajjaji received the Ph. D.degree in automatic control from Universityof Picardie Jules Verne, France in 1993 and2000, respectively. He was an associate pro-fessor in the same university from 1994 to2003. He is currently a full professor and di-rector of the Electrical Engineering Depart-ment, University of Picardie Jules Verne,France. Since 2001, he has been the head

of the research team of Control and Vehicle in Modeling, In-formation and Systems Laboratory, University of Picardie JulesVerne, France.

His research interests include fuzzy control, vehicle dynamics,fault tolerant control (FTC), neural networks, maglev systems,and renewable energy systems.

E-mail: [email protected]