Fuzzy \(H {\infty}\) output feedback control for the discrete time system with channel fadings, sector nonlinearities, and randomly occurring interval delays and nonlinearities

R E S E A R C H

Open Access

Fuzzy

H

_∞

output-feedback control for the

discrete-time system with channel fadings,

sector nonlinearities, and randomly occurring

interval delays and nonlinearities

Xiaozheng Fan

_{, Yan Wang}

_{and Manfeng Hu}

*_{Correspondence:}

[email protected]

1_{School of Science, Jiangnan}

University, Wuxi, 214122, China

3_{Key Laboratory of Advanced}

Process Control for Light Industry, Ministry of Education, Institute of Automation, Jiangnan University, Wuxi, 214122, China

Full list of author information is available at the end of the article

Abstract

In this paper, the fuzzyH_∞output-feedback control problem is investigated for a class of discrete-time T-S fuzzy systems with channel fadings, sector nonlinearities,

randomly occurring interval delays (ROIDs) and randomly occurring nonlinearities (RONs). A series of variables of the randomly occurring phenomena obeying the Bernoulli distribution is used to govern ROIDs and RONs. Meanwhile, the measurement outputs are subject to the sector nonlinearities (i.e.the sensor

saturations) and we assume the system output isy(k) = 0,k∈ {–l,. . ., 0}. TheLth-order Rice model is utilized to describe the phenomenon of channel fadings by setting different values of the channel coefficients. The aim of this work is to deal with the problem of designing a full-order dynamic fuzzyH_∞output-feedback controller such that the fuzzy closed-loop system is exponentially mean-square stable and theH_∞

performance constraint is satisfied, by means of a combination of Lyapunov stability theory and stochastic analysis along with LMI methods. The proposed fuzzy controller parameters are derived by solving a convex optimization problem via the

semidefinite programming technique. Finally, a numerical simulation is given to illustrate the feasibility and effectiveness of the proposed design technique.

Keywords: Takagi-Sugeno (T-S) fuzzy system; fuzzyH_∞output-feedback control; channel fadings; sector nonlinearities; randomly occurring interval delays (ROIDs); randomly occurring nonlinearities (RONs)

1 Introduction

It is well known that the complexity and nonlinearity of the models are considered as ubiq-uitous in practical systems. The emergence of this fuzzy modeling approach is based on the Takagi-Sugeno (T-S) fuzzy system (see []), which provides a powerful tool for modeling complex nonlinear systems. TheH∞output-feedback control problem for the T-S fuzzy system has received considerable attention (see [–]). Nevertheless, the nonlinearity of the T-S fuzzy subsystem is inevitable in practical applications, along with the fact, that the T-S fuzzy model has been successfully used in complex nonlinear systems (see [–]). The authors of [] assumed the nonlinear function in the T-S fuzzy cellular neural networks

satisfied the -Lipschitz condition and researched the global exponential stability problem for T-S fuzzy cellular neural networks.

In the past decade, networked control systems (NCSs) have played an important role in many engineering applications such as remote militarization, remote medical service, and so on (see [–]). However, there are some unavoidable phenomena for the NCSs which may cause poor performance of the controlled systems, for instance, the signal is often transmitted through networks which might be subjected to the occurrence of the phenomenon of incomplete information. The considered incomplete information mainly includes the ROIDs (see []), RONs (see []), channel fadings (see [, ]), and sector nonlinearities (see [, ]). The nonlinear output-feedback controller design for poly-nomial system has been studied (see [, ]). A full-order dynamicH∞output-feedback controller was designed by [] for the time-varying delays case when all state variables are not available for the feedback. Further, the author of [] has researched theH∞ output-feedback controller design problem for networked systems with random communication delays by using a linear matrix inequality (LMI) approach. TheH∞output-feedback con-trol problem for a class of discrete-time systems with RONs has been investigated in [], where random variables are adopted to characterize the RONs and satisfy the binary distri-bution. The designingH∞output-feedback controller problems for the T-S fuzzy system with randomly occurring phenomena have been studied in [, ]. However, in the case when ROIDs and RONs appear simultaneously in a controlled system, the designing of the fuzzyH∞output-feedback control problem has received little attention by researchers.

In practical applications, the phenomena of the channel fadings and sector nonlinear-ities based on unreliable communication networks could occur, which should not be ig-nored. Considering the situation of signal transmission in fading channels, theH∞ output-feedback control problem with channel fadings has been studied (see []). The channel fading has been modeled as a time-varying stochastic model which can describe the trans-mitted signal’s change in both the amplitude and the phase. The channel fadings with ex-ogenous input disturbance in wireless mobile communications has not been researched extensively (see [, ]). On the other hand, the sector nonlinearities of the sensors are usually in order in practical industrial control systems, and this is the main factor that gives rise to the nonlinearity of control systems (see []). Since the sensor nonlinearity cannot be neglected and often leads to bad performance of the discrete-time control system, it has attracted the attention of researchers (see [–]).

In [], theH∞ output-feedback control problem for a class of discrete-time systems with channel fadings and sector nonlinearities has been studied, and the existence of the desired controllers has been derived via using a combination of the stochastic analysis and Lyapunov function approach. The design ofH∞ fuzzy controller problem for the fuzzy system with the probabilistic infinite-distributed delay and the channel fadings also has been investigated in [], where the channel fading model can better reflect the reality of measurement transmission especially through a wireless sensor network. So far, to the best of the authors’ knowledge, the fuzzyH∞output-feedback control problem for a class of discrete-time T-S fuzzy system with channel fadings, sector nonlinearities, ROIDs and RONs have not been investigated yet, and the main purpose of this paper is to bridge such a gap.

for describing the discrete-time fuzzy model. () Moreover, a newly fuzzy control system as well as measurement model is put forward which can account for the randomly occur-ring incomplete information phenomena of the sensor saturation and the channel fadings. () A new fuzzyH∞output-feedback controller has been designed.

Motivated by the above discussion, this paper intends to study the fuzzyH∞ output-feedback control problem for a class of discrete-time Takagi-Sugeno (T-S) fuzzy model system with channel fadings, sector nonlinearities, ROIDs, and RONs. The rest of this pa-per is organized as follows. In the next section, the problem descriptions of the discrete-time T-S fuzzy system with ROIDs, RONs, sector nonlinearities, and channel fadings are stated, and the necessary definitions and relevant lemmas are recalled. Section  presents the main results of this paper. Illustrative examples are provided in Section . Finally, con-clusions are drawn in Section .

2 Model description and preliminaries

In this paper, we consider the following discrete-time fuzzy system with RONs and ROIDs is described by the following fuzzy IF-THEN rules:

Plant Rule i: IFθ(k) isMiandθ(k) isMi, . . . , andθr(k) isMir, THEN

⎧ ⎪ ⎨ ⎪ ⎩

x(k+ ) =Aix(k) +Ai

h

mβm(k)x(k–τm(k)) +Biu(k) +Diω(k) +r(k)f(x(k)), z(k) =Eix(k) +Diω(k),

x(s) =φ(s), s= –h, –h+ , . . . , , ,

()

wherei= , . . . ,r, the system () is equivalent to a fuzzy combination ofrsubsystems.Mij

is the fuzzy set,θ(k) = [θ(k),θ(k), . . . ,θr(k)] is the premise variable vector.x(k)∈Rnis

the state vector withx() =φ(),u(k)∈Rr _{is the control input vector,z(k)∈R}q _{is the}

controlled output vector,ω(k)∈l([, +∞),Rn) is the exogenous disturbance input.φ(s) is the initial state.Ai,Ai,Bi,Di,Ei, andDi are known real matrices with appropriate

dimensions.

To characterize the phenomena of randomly occurring interval delays and randomly occurring nonlinearities, we employ the stochastic variablesβm(k) (m= , . . . ,h) andr(k)

in (), which are mutually independent Bernoulli-distributed white sequences with the following probability distribution:

Probβm(k) = 

=Eβm(k)

=β¯m,

Probβm(k) = 

=  –β¯m,

Probr(k) = =Er(k)=r¯,

Probr(k) = =  –r¯.

The variableτm(k) (m= , , . . . ,h) means the time-varying delays satisfying

d_m≤τm(k)≤ ¯dm, ()

whered_mandd¯mare real positive integers representing the lower bounds and the upper

Assumption  The nonlinear vector-valued functionf(x(k)) :Rn_→Rn_{withf() =  is}

seen as continuous and satisfies the following sector-bounded condition:

fx(k) ≤λ Gx(k)  ()

for allk∈[,N], whereλ>  is a known positive scalar andGis a known real matrix. Remark  In model (), the stochastic variablesr(k) is used for characterizing the phe-nomena of RONs. The T-S fuzzy model with RONs includes the fuzzy model with nonlin-earity in [, ] as a special case where the values ofr(k) are . Note that RONs is considered for the first time for fuzzy output-feedback control problem. On the other hand, the oc-currence of the ROIDsx(k–τm(k)) is characterized by the random variablesβm(k) in a

probabilistic way, which is more suitable for reflecting the network-induced phenomena. Meanwhile, it is worth of note that there are some results concerned with the continuous time-varying delays in [, ] and few results for randomly occurring interval delays, es-pecially when the fuzzyH∞output-feedback control problem becomes a research focus.

Let us now consider the case when the phenomena of the sector nonlinearities and the channel fadings may occur in signal transmission, where the system output is subject to sector-like bounds and the sensor signal sent to the actuator subject to channel fadings for the control purpose. The signal received by the actuator is modeled in the following:

y(k) =g(x(k)),

ξ(k) = _l=αl(k)y(k–l) +ν(k),

()

wherey(k)∈Rn_{represents the system output withy(k) = ,∀k∈ {–l, . . . , },g(x(k)) is the}

sector nonlinearity of the sensor,ξ(k)∈Rn _{is the signal from the actuator, and ν(k)∈} l([, +∞),Rn) is an external disturbance.αl(k)∈R(l= , , . . . , ) is the channel coefficient

which is independent and conform Gaussian random variables distributed with meanα¯l

and varianceα˜_l. In practice, the channel coefficients typically take values over the interval [, ].

Assumption  The nonlinear functiong(x(k)) in () represents the sector nonlinearities satisfying the following sector condition:

gx(k)–Mx(k) T

gx(k)–Mx(k)

≤, ()

whereMandM(M>M≥) are known real matrices with appropriate dimensions. Remark  In this paper, the channel fadings and sector nonlinearities of the sensors can be described simultaneously by the model () in the measurement. In [], this Rice fadings model can properly describe the phenomena of the channel fadings, time-delay, and date dropout, therefore the fadings model can be employed in this paper for the design of the

H∞fuzzy output-feedback controller of the discrete-time fuzzy system.

syn-thesis problems for a series of dynamics systems with sector nonlinearities has been in-vestigated in [, ]. Particularly, the nonlinear function lies inside the sector [M,M] in []. Furthermore, the Lipschitz condition can be concluded by the nonlinear description as a special case ifM=  orM=  in Assumption .

Assumption ([]) For technical convenience, the nonlinear functiong(x(k)) can be

decomposed into a linear and a nonlinear parts as

gx(k)=gs

x(k)+Mx(k), ()

where the nonlinear partgs(x(k)) belongs to the setGsdefined by

Gs=

gs

x(k):g_sTx(k)gs

x(k)–Mx(k)≤ ()

withM=M–M> .

In this paper, we adopt a full-order fuzzy output-feedback controller for the discrete-time system by the fuzzy IF-THEN rules as follows:

Control rule i: IFθ(k) isMiandθ(k) isMi, . . . , andθr(k) isMir, THEN

xc(k+ ) =Acixc(k) +Bciξ(k), u(k) =Ccixc(k),

()

wherexc(k)∈Rnis the state estimate of system (), andAci,Bci,Cciare appropriately

di-mensioned parameters matrices to be determined. Set

hi

θi(k)

=

r

j=Mij(θj(k))

r i=

r

j=Mij(θj(k))

, ()

whereMij(θj(k)) denotes the grade of membership ofθj(k) inMij. Obviously,hi(θ(k))≥,

andr_i=hi(θ(k)) = . To ease the presentation, we usehiinstead ofhi(θ(k)).

Above all, the T-S fuzzy system () model can be constructed as follows:

⎧ ⎪ ⎨ ⎪ ⎩

x(k+ ) =r_i=hi{Aix(k) +Ai

h

mβm(k)x(k–τm(k)) +Biu(k)

+Diω(k) +r(k)f(x(k))}, z(k) =r_i=hi{Eix(k) +Diω(k)}.

()

Furthermore, the fuzzy control system can be described by

xc(k+ ) =

r

i=hi{Acixc(k) +Bciξ(k)}, u(k) =r_i=hiCcixc(k).

Combining (), (), (), and (), the fuzzy control system can be represented by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

η(k+ ) =r_i=r_j=hihj{ ¯Aijη(k) +A¯i

h

m=β¯mZη(k–τm(k))

+A¯i

h

m=β˜m(k)Zη(k–τm(k)) +Dijω˜(k) +r¯F(η(k)) +r˜(k)F(η(k))

+ _l=α¯lB¯jZη(k–l) + l=α˜l(k)B¯(j)Zη(k–l) + _l=α¯lB˜jgv(Zη(k–l)) + l=α˜l(k)B˜jgv(Zη(k–l))}, zk=ri=

r

j=hihj{Eiη(k) +Djω˜(k)},

()

where

η(k) =xT(k),xT_c(k)T, ω˜=ωT(k),νT(k)T, Fη(k)=

f(x(k)) 

,

¯

Aij=

Ai BiCcj

¯

αBcjM Acj

, A¯i=

Ai



, Z= [In ],

¯

Bj=



BcjM

, B˜j=



Bcj

, Ei= [Ei ],

Dij=

Di  o Bcj

, Dj= [Dj ],

with

˜

α(k) =α(k) –α¯;

˜

βm(k) =βm(k) –β¯m;

˜

r(k) =r(k) –¯r.

Obviously

Eα˜(k)= , Eα˜(k)α˜_l (l= , , . . . , );

Eβ˜m(k)

= , Eβ˜_m(k)=β¯m( –β¯m)β˜m (m= , . . . ,h);

Er˜(k)= , Er˜(k)=¯r( –r¯)˜r.

Particularly, we can see from Assumption  that the nonlinear functionF(η(k)) satisfies the following formula:

FT_{η(k)Fη(k)≤λη}T_(k)GT_{Gη(k), ()}

whereG= (G, ).

To describe our main result more precisely, we first introduce the following definition and lemmas.

Definition  (Exponentially mean-square stability []) The T-S fuzzy control system

be exponentially mean-square stable if, in the case ofω˜(k) = , then exist constantsδ>  and  <<  such that

E η(k) ≤δksup

i∈K

E φ(i) , ∀k≥. ()

With Definition , the aim of this paper is to design a robustH∞output-feedback con-troller in the form of () such that the fuzzy discrete-time system () is exponentially mean-square stable and theH∞performance is satisfied or, more specifically, the follow-ing two requirements are satisfied simultaneously:

(R) The fuzzy discrete-time system () is exponentially mean-square stable. (R) Under the zero-initial condition, the controlled outputz(k) satisfies

∞

k=

E z(k) ≤γ

∞

k=

E ω˜(k)  ()

for all nonzeroω˜(k), whereγ >  is a prescribed scalar.

Lemma (Schur complement []) We have the linear matrix inequality

S=

S S

ST_ S

< ,

where S=STand S=STare equivalent to

S< , S–STS–S< .

Lemma ([]) For a symmetric positive definite S,and any real matrices X(ijp)with

ap-propriate dimensions,we can get

r

i=

r

j=

r

s=

r

t=

hihjhsht(X(ij))TSX(st)≤

r

i=

r

j=

hihj(X(ij))TSX(ij).

Lemma ([]) Given any matrices x,y,a matrix P> ,and a positive scalar,then we have

xTy≤xTPx+–yTPy. 3 Main result

In this part, the following theorem provides a sufficient condition for the discrete-time T-S fuzzy system () to be exponentially mean-square stable and the controlled output

zkto satisfy theH∞disturbance reject requirement in ().

Theorem  Let a scalarγ > and the controller parameters matrix Acj,Bcj,and Ccj(j=

, . . . ,r)be given.The fuzzy closed-loop system()is exponentially men-square stable and the controlled output z(k)satisfies(),if there exist matrices P> ,Qm> ,and Rl> 

(m= , . . . ,h;l= , , . . . , ),and a positive scalarψ> andϕ> satisfying

ϒ_ij+ϒ_ji< , ()

Proof We choose the following Lyapunov function:

+ 

+ η(k)TA¯T_ijP

Also, it can be seen that

≤E

For notational convenience, we have

≤E exponentially mean-square stable as can be seen in the same way as in [].

Now let us dispose of theH∞ performance for the system (). For this purpose, we establish a cost function

J(n) =E

Along the trajectory of the fuzzy discrete-time system () and taking () into

consid-By using the Schur complement lemma, the conclusion can be drawn from () and () that J(n) < . Lettingn→ ∞, it follows from the above inequality that

which completes the proof of Theorem .

Theorem  Let the H∞disturbance attenuation levelγ > be given.A desired controller

the controller parameters in the form of()are given in the following:

Acj=X

Kj[T–S]–X–SBiCcj

Bcj=X–Kj, Ccj=Kj[T–S]–X, ()

where the matrix Xderives from the factorization I–ST–=XYT< ,and then the

fuzzy discrete-time closed-loop system ()is exponentially mean-square stable and the controlled output zksatisfies().

Proof For the purpose of design desired controller parametersAcj,Bcj, andCcjfrom

The-orem , we partitionPandP–_as

(∗) P=

S X

XT

 X

, P–=

T– _Y 

YT

 Y

,

where the partitioning ofPandP–_{are appropriately dimensioned to be determined by}

¯

Aij,A¯i,Dij, andB¯jin ().

Define

W=

T– I YT

 

, W=

I S

 XT



,

and then we havePW=WandWTPW=WTW. Now we define the controller param-eters from () as follows:

Kj= [SBiCcj+XAcj]YTT,

Kj=XBcj, ()

Kj=CcjYTT.

By applying the congruence transformation

diag{W,W , . . . , W +

,I,I,W,Ih,I+,I,I,I,W , . . . , W

h

,W,I,I+,I+}

to () and (), we can have

⎡ ⎢ ⎣

Λ_ Ω__ij 

∗ Λ_ Ω__ii

∗ ∗ Λ_

⎤ ⎥

⎦< , ()

Λ_=diag{–P¯, –P¯ +, –I, –ϕI},

Λ_=diag–P¯, –Qh, –R , –ϕI, –ψI, –γI

,

_=diag–P¯h, –P¯, –Q¯–m, –R–, –ψI+

,

Ω__ij= ⎡ ⎢ ⎢ ⎢ ⎣

˜

Aij ¯⊗ ¯Ai ¯⊗ ¯B r¯W¯ ¯⊗ ˜B D¯ij

  ˜⊗ ¯B  ˜⊗ ˜B 

˜

Ei     Dj

ϕ√λG˜     

Ω__ii= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  ZT_T

– ZTT–⊗ ¯I 

˜

⊗ ¯ATi    

    ψMT

 rW˜    

    

    

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,

¯

P=

T– _I

∗ S

, P¯s=diag{¯P , . . . ,P¯ s

},

¯

Ei= [EiT– Ei], G¯= [GT– G], ¯Il= [I , . . . ,I

+ ],

˜

Aij=

(Aij+BiKj)T– Aij

(SAij+α¯KjM+Kj)T– SAij+α¯KjM

.

On the other hand, it follows from [T–_–H]T_T[T–_{–H]≥ that}

–T–≤–H–HT+HTTH. ()

Furthermore, again applying the congruence transformation

diag{I,I+,I,I,T,Ih,I+,I,I,I,T , . . . ,T

h

,T,Q¯m,I+,R+}

to (), we have

⎡ ⎢ ⎣

Λ

 Ωij 

∗ Λ_ Ω__ii

∗ ∗ Λ_

⎤ ⎥

⎦< , ()

whereT =diag{T,I}. By combination () and (), if () and () are satisfied, the in-equality () holds. Therefore the sufficient condition () and () of Theorem  is effec-tive.

Next, let us calculate the desired controller parameters. We can obtain fromPP–_=I

I–ST–=XY. ()

ByP> ,T> ,S> ,WT

 PW=WTW= _T– _I

I S

, and if () and () are feasible, we can inferI–ST–=XY< . SoI–ST–is nonsingular. Hence one can always find square and nonsingularXandYsatisfying () []. In this case, we can obtainAcj,Bcj, and Ccjvia solving (). Now it can be concluded from Theorem  that the fuzzy closed-loop

system () is exponentially mean-square stable and the controlled outputz(k) satisfies () with the controller parameters given by ().

in []. The results derived in this paper also contain the two theorems in [] as special cases. Moreover, randomly occurring nonlinearities have not been considered in [], and sector nonlinearities have not been studied in [].

Remark  The design of controller directly affects the stability andH∞performance of the discrete-time closed-loop system. Compared with [, , ], it should be pointed out that the fuzzy controller designing arithmetic in Theorem  has more generality than the usual controller, that is to say, the controller designing arithmetic in [, ] cannot be available for the design of fuzzyH∞ output-feedback controller for a class of discrete-time T-S fuzzy systems with channel fadings, sector nonlinearities, ROIDs, and RONs. To design the controller and complete the proof of Theorem ,PandP–_{are in the form of (∗),} which can be found in [, ]. The conditions as regardsPin [] are more conservative than ours because one not only needsP> , but alsoS>  andX> . Therefore, Theorem  has less conservatism.

Remark  In Theorem , the sufficient conditions involved in the randomly occurring nonlinearities, the probabilistic interval delays, sector nonlinearities, and channel fad-ings were first established for the desired fuzzy output-feedback controller. The fuzzy output-feedback controller is designed such that the discrete-time system () is expo-nentially mean-square stable and, under the zero-initial condition, the proposedH∞ per-formance index can be satisfied. Particularly, with the designed ofH∞fuzzy controllers, the robustness of our developed controller operation algorithms of the discrete-time fuzzy system includes the traditional controller algorithms. In other words, the tradi-tional controller algorithms means that we have the membership functionhi= ,hj= 

(i=j,j= , . . .i– ,i+ , . . . ,r) in the discrete-time fuzzy system. Obviously, the devel-oped controller algorithms work better than the traditional algorithms in dealing with the occurrence probability of randomly occurring nonlinearities, interval delays, sector non-linearities, and channel fadings, which appropriately avoid the deterioration of the H∞ performance.

4 Numerical example

In this section, we present illustrative examples to show the effectiveness of the proposed controller design approach.

Consider the following discrete-time T-S fuzzy model from (): ⎧

⎪ ⎨ ⎪ ⎩

x(k+ ) =_i=hi{Aix(k) +Ai



mβm(k)x(k–τm(k)) +Biu(k)

+Diω(k) +r(k)f(x(k))}, z(k) =_i=hi{Eix(k) +Diω(k)}.

Consider the model parameters as follows:

A=

. –. . –.

, A=

. –. . –.

,

A=

.  . –.

, A=

.  . –.

,

D=D= [. .]T,

D=D= .,

E=E= [–. .],

with the initial valueφ(s) = [, ]T_{(s= , –, –, –, –, –). The nonlinear vector-valued}

functionh(k) is as follows:

hx(k)=

_.x (k) x_(k)+ .x(k)sin(x(k))

.

Besides, it can easily be noticed that Assumption  satisfiesλ= . and

G=

.   .

,

the sensor nonlinearity is given as

gx(k)=M+M  x(k) +

M–M  sin

x(k),

where

M=

.   .

, M=

.   .

.

In the meantime, the output measurement is described as follows:

y(k) =g(x(k)),

ξ(k) =α(k)y(k) + 

l=αl(k)y(k–l) +ν(k).

Here, the order of channel fading is = , the mathematical expectations of the channel coefficients areα¯= .,α¯= . andα¯= ., and the variances of the channel coefficients areα˜_∗= .,α˜∗_= ., andα˜_∗= .. Assume thath=  for the time-varying delays,τ(k) andτ(k) are, respectively, uniformly distributed in the intervals [, ] and [, ], and the stochastic variablesβ¯= .,β¯= .. Other stochastic variables arer¯= .,r˜= ..

To further illustrate the effectiveness of the designedH∞fuzzy controller, the exogenous disturbance inputsν(k),ω(k) are assumed to be

ν(k) = .sin(k)

k , ω(k) =

.sin(k)

k .

Figure 1 Membership function.

Figure 2 The states of the system and the fuzzy controller.

Applying Theorem  and the LMI toolbox, we can obtain the desired controller param-eter matrices in the form of () such that the fuzzy system () is exponentially mean-square stable with theH∞norm boundγ = . as follows:

Ac=

. . –. –.

, Ac=

. . –. –.

,

Bc=

–. –. . .

, Bc=

–. –. . .

,

Cc= [. –.], Cc= [. –.].

Figure 3 State evolutionx(k) of uncontrolled fuzzy system.

Figure 4 State evolutionx(k) of controlled fuzzy system.

5 Conclusions

In this paper, a fuzzyH∞output-feedback controller has been designed for a class of fuzzy discrete-time systems with sector nonlinearities, channel fadings, randomly occurring in-terval delays as well as randomly occurring nonlinearities. A sufficient condition for the

H∞ robust exponential stability of the fuzzy discrete-time system has been obtained by a Lyapunov stability analysis approach and stochastic analysis theory. Moreover, by using the LMI technique, a clear expression of the desiredH∞fuzzy output-feedback controller can be obtained and the proposedH∞-norm bound constraint has been guaranteed. At last, the developed fuzzy controller design approach has been checked by a numerical simulation example. Further research topics might include the development of our results to more complex and more varied cases with sector nonlinearities and channel fadings by using a stochastic analysis approach, such as multi-agent systems based on the T-S fuzzy model, descriptor systems, and affine fuzzy systems.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to the writing of this paper. All authors read and approved the final manuscript.

Author details

1_{School of Science, Jiangnan University, Wuxi, 214122, China.}2_{School of Internet of Things, Jiangnan University, Wuxi,}

214122, China.3_{Key Laboratory of Advanced Process Control for Light Industry, Ministry of Education, Institute of}

Acknowledgements

The authors are grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. This work is supported by the National Science Foundation of China (61572238), the Provincial Outstanding Youth Foundation of Jiangsu Province (BK20160001), the National High Technology Research and Development Program of China (National 863 Program) (No:2014AA041505) and the Nature Science Foundation of Jiangsu Province of China under Grant No. BK20161126.

Received: 2 May 2016 Accepted: 10 October 2016

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