# Design of an Unmatching Observer-Based Controller for Discrete-Time Fuzzy Systems with Time Delay

Authors: Zejian Zhang; Dawei Wang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

The problem of an unmatching observer-based controller design for discrete-time fuzzy systems with time delay is investigated, in which the fuzzy controller shares different membership functions from the fuzzy model. The objective is to design a state observer and unmatching fuzzy controller such that the discrete closed-loop system with time delay is asymptotically stable. A sufficient condition that contains the information of the membership functions of fuzzy model and fuzzy controller for the stabilization via an unmatching observer-based output feedback is presented. The proposed control scheme is well capable of enhancing the design flexibility, and the stability condition is less conservative. Three numerical examples are given to illustrate the effectiveness and advantages of the proposed method.


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## Body

## 1. Introduction

Time-delay phenomena intensively exist in natural science and engineering and social life, such as nonlinear and time-delay models. For example, they are in numerous dynamical systems including biology systems, mechanics, economics, chemical systems, and network systems. Generally, time delay can often lead to instability and poor performances. Therefore, it is of great significance to investigate the issue of the stability for time-delay systems. A lot of relevant research work has been reported in [1–3]. However, the results obtained are mainly only based on the linear time-delay systems. It is indeed necessary to generalize them to the nonlinear time-delay systems. As we know, the fuzzy model proposed by Takagi and Sugeno can effectively represent many nonlinear dynamic systems [4]. During the past decade, more and more researchers paid attention to the issue of the stability for the nonlinear fuzzy systems with time delay [5, 6]. The stability problem refers to the design of a controller that can guarantee the fuzzy model stability. State feedback plays an important role in handling stabilization in the control systems. However, sometimes the system states cannot be measured directly, and there are limitations of the measurement equipment in use. It is actually impossible to get all the information of the system state variables. Under this circumstance, the physical form of state feedback is difficult to realize. Therefore, state estimation and observation of nonlinear systems are important but challenging problems in modern control theory. The design of observers and controllers for the T-S fuzzy systems is a demanding topic. For example, the stability analysis and stabilization of the T-S fuzzy models for designing the observers and control laws are addressed in [7, 8]. A single-step linear matrix inequality method is developed for the observer-based controller design for discrete-time fuzzy systems in [9], which overcomes the drawbacks induced by the conventional two-step approach, and yields less conservative results. The problem of fuzzy observer-based controller design is investigated for nonlinear networked control systems subject to imperfect communication links and parameter uncertainties in [10]. Some results are generalized to the observer-based adaptive model in [11]. A novel method based on the nonuniform delay partitioning approach is put forward to analyze the stability of the time-delay T-S fuzzy system, and the observer-based feedback controller design scheme via the parallel distributed compensation (PDC) scheme is also discussed in [12]. A fuzzy functional observer method is proposed to design a controller for the observer-based fuzzy model in [13]. Furthermore, some results related to the observer-based controller design problem have been extended to the MIMO time-delay systems in [14]. Recently, the observer-based controller design problem is investigated in networked control systems [15]. In the abovementioned work, the observer-based feedback controller design problems are all based on the PDC scheme, which means that the observed-based fuzzy model and the observed-based fuzzy controller share the same membership functions. As a matter of fact, if the membership functions of the fuzzy controller in the premise of the fuzzy rules are allowed to be designed arbitrarily, a greater design flexibility can be achieved. Thus, some research studies on the control problems under imperfect premise matching have been carried out in [16–20]. As the states of the systems are sometimes difficult to be obtained or the costs of measurement are too high in many practical problems, it is critical to design observers for the systems. In addition, nonlinear and time delays are inherent and not all the states are available in most practical systems. Therefore, the observer-based stabilization control for the discrete-time T-S fuzzy systems with time delay is an important topic.In this paper, we focused on the observer-based stabilization control for discrete-time T-S fuzzy time-delay systems and a novel observer-based output feedback controller is investigated for the discrete-time T-S fuzzy time-delay systems. The main contributions of this paper can be summarized as follows:(i)
As the membership functions of the fuzzy model and the fuzzy controller are all considered in the analysis, some membership-function-dependent stability conditions are derived, which are less conservative than membership-function-independent stability conditions(ii)
As the membership functions of the controller are distinctive from those of the fuzzy model, the membership functions can be chosen arbitrarily, which makes a higher degree of design flexibility of the fuzzy controller achieved, and its structure can be further simplifiedThe remainder of this paper is organized as follows. In Section2, the considered problem is described. An unmatching observer-based controller design method is proposed in Section 3. Three numerical examples are utilized to illustrate the conservativeness, effectiveness, and superiority of the proposed method in Section 4. Finally, Section 5 concludes this paper.Notation 1.
In this paper, if not explicitly stated, matrices are assumed to have compatible dimensions. The notationM>≥,<,≤0 is used to denote a symmetric positive-definite (positive semidefinite, negative, and negative semidefinite, respectively) matrix. ⋅ denotes the Euclidean norm for vector or the spectral norm of matrix. For convenience, we use wi,mi instead of wizt,mizt.


## 2. Problem Formulation and Preliminaries

Letr be the number of the fuzzy rules describing the time-delay nonlinear plant. The ith rule can be represented as follows.Ifz1t is M1i and … and zpt is Mpi, then(1)xt+1=A1ixt+A2ixt−τ+Biut,yt=C1ixt+C2ixt−τ,t≥0,where r is the number of the fuzzy rules and Mαi, α=1,2,…,pandi=1,2,…,r, denotes the fuzzy set. zkt,k=1,2,…,r, are the known premise variables which do not depend on the input variables. xt∈Rn is the state vector, ut∈Rm is the input vector, yt∈Rq is the output vector, and A1i,A2i,Bi,C1i,andC2i are some constant matrices of compatible dimensions. Here, we assume that τ>0 is a constant.With the fuzzy inference methods, the final outputs of the fuzzy time-delay model can be formalized as follows:(2)xt+1=∑i=1rwiztA1ixt+A2ixt−τ+Biut,yt=∑i=1rwiztC1ixt+C2ixt−τ,where(3)∑i=1rwizt=1,wizt≥0,i=1,2,…r∀t,wizt=μizt∑i=1rμizt,μizt=∏k=1pμMkizkt,in which z1t,…,zpt are the premise variable and μMkizkt is the grade of membership of zkt in Mki.In engineering practices, the state information of system equation (1) is often not available. Therefore, it is necessary to construct a state observer satisfying the following formal equation to estimate the state of the discrete system.Observer rulei: if z1t is M1i and … and zpt is Mpi, then(4)x^t+1=A1ix^t+A2ix^t−τ+Biut+Liyt−y^t,i=1,2,…,r,y^t=C1ix^t+C2ix^t−τ,where Li is the fuzzy observer gain for the i subsystem.The overall fuzzy observer is represented by(5)x^t+1=∑i=1rwiztA1ix^t+A2ix^t−τ+Biut+Liyt−y^t,yt=∑i=1rwiztC1ixt+C2ixt−τ,y^t=∑i=1rwiztC1ix^t+C2ix^t−τ.With the aforementioned fuzzy observer, an unmatching fuzzy observer control law is defined as follows.Ifz1t is N1i and … and zqt is Nqi, then(6)ut=Fjx^t,j=1,2,…,r.The overall observer-based fuzzy control law is represented by(7)ut=∑j=1rmjztFjx^t.Combining the fuzzy controller in equation (7) and the fuzzy observer in equation (5) and denoting et=xt−x^t, we obtain the following system representations:(8)xt+1=∑i=1r∑j=1rwimjA1i+BiFjxt+A2ixt−τ−BiFjet,(9)et+1=∑i=1r∑k=1rwiwkA1i−LiC1ket+A2i−LiC2ket−τ.Therefore, the augmented system can be written as(10)x˜t+1=∑i=1r∑j=1r∑k=1rwimjwkGijkx˜t+M¯i+N¯ikx˜t−τ,where(11)x˜t=xtet,Gijk=A1i+BiFj−BiFj0A1i−LiC1k,M¯i=A2i00A2i,N¯ik=000−LiC2k.Remark 1.
It can be observed from equation (10) that the controller we design and the observer-based fuzzy model do not share the same membership functions. Thus, for the observer-based T-S fuzzy time-delay systems, the membership functions of the fuzzy controller can be arbitrarily selected in the design phase, which can enhance the design flexibility. Consequently, it is worthy to investigate the stability conditions of this fuzzy time-delay system.Lemma 1.
(see [21]). The linear matrix inequality HθTθR>0 is equivalent to R>0,H−θTR−1θ>0, where H=HT,R=RT, and θ is a matrix with appropriate dimension.Lemma 2.
For anyxt∈Rn, there exist σ1 and σ2 such that σ1xt2≤xTtPxt≤σ2xt2, where P∈Rn×n,P>0.Proof.
For convenience, we usex instead of xt. Define λii=1,2,…,n>0 is the eigenvalue of P. Assume that ξii=1,2,…,n are the eigenvectors of λi. Here, ξii=1,2,…,n can be regarded as a group of bases of n-dimensional Euclidean space. Assuming that ξi=xi1,xi2,…,xinT∈Rn, then for any x=∑i=1nρiξi0<ρi<1, we have x=∑i=1nρixi1,∑i=1nρixi2,…,∑i=1nρixinT, and then(12)xTPx=∑i=1nρixi1,∑i=1nρixi2,…,∑i=1nρixin∑i=1nλiρixi1∑i=1nλiρixi2…∑i=1nλiρixin=∑j=1n∑i=1nρixij∑i=1nλiρixij.
Especially, if we selectηi=0,0,0,…,xij,0,…,0T, then(13)x=ρ1η1+ρ2η2+⋯+ρnηn=0,0,…,0,∑i=1nρixij,…T.
Provided that∑i=1nρixij≠0 and P>0, we have(14)xTPx=ρ1x1j+ρ2x2j+⋯+ρnxnjλ1ρ1x1j+λ2ρ2x2j+⋯λnρnxnj>0.
Notice that∑i=1nρixij2>0and∑i=1nρixij∑i=1nλiρixij>0, so there exist mj>0 and nj>0j=1,2,…,n, such that(15)nj∑i=1nρixij2≤∑i=1nρixij∑i=1nλiρixij≤mj∑i=1nρixij2.
Definingσ1=minnjj=1,2,…,nandσ2=maxmjj=1,2,…,n, we have(16)σ1x2=σ1xTx≤xTPx≤σ2xTx=σ2x2.


## 3. Main Result

Theorem 1.
The equilibrium of the closed-loop discrete fuzzy time-delay system in equation (10) with an unmatching observer-based control law in equation (7) is asymptotically stable, if the membership functions of the fuzzy model and fuzzy controller satisfy mjzt−ρjwjzt≥0 for all j and zt, where 0<ρj<1. There exist X1>0,X2>0,S˜1>0,S2>0,Yi, Λj=ΛjT∈R3n×3n, Ηij=ΗijT∈R3n×3n, Μik=ΜikT∈R3n×3n,i,j,k=1,2,…,r, and Ri satisfying(17)Ωij−Λi<0,ρiΩii−ρiΛi+Λi<Hii,ρjΩij+ρiΩji−ρjΛi−ρiΛj+Λi+Λj≤Hij+HijT,i<j,Ψii<Μii,Ψik+Ψki<Μik+ΜikT,i<k,H=H11H12⋯H1rH21H22⋯H2r⋮⋮⋮⋮Hr1Hr2⋯Hrr<0,M=M11M12⋯M1rM21M22⋯M2r⋮⋮⋮⋮Mr1Mr2⋯Mrr<0,where(18)Ωij=−X1+S˜10∗0−S˜1∗A1iX1−BiYjA2iX1−X1,Ψik=−X2+S20∗0−S2∗X2A2i−RiC1kX2A2i−RiC2k−X2.The state feedback gains and observer gains can be constructed asFi=YiX1−1andLi=X2−1Ri.Proof.
Select a Lyapunov function as(19)Vx˜t=x˜TtPx˜t+∑σ=t−τt−1x˜TσSx˜σ,where P>0 and S≥0. By Lemma 2, there exist σ1 and σ2 such that(20)σ1x˜t2≤Vx˜t≤σ2x˜t2.
In this section, we will prove the asymptotic stability of the discrete time-delay system in equation (10) based on the Krasovskii theorem [22].(21)ΔVx˜t=Vx˜t+1−Vx˜t=x˜Tt+1Px˜t+1−x˜TtPx˜t+x˜TtSx˜t−x˜Tt−τSx˜t−τ=∑i=1r∑j=1r∑k=1rwimjwkx˜TtGijkTPGijk−P+Sx˜t+x˜TtGijkTPM¯i+N¯ikx˜t−τ+x˜Tt−τM¯i+N¯ikTPGijkx˜t+x˜Tt−τM¯i+N¯ikTPM¯i+N¯ik−Sx˜t−τ=x¯Tt∑i=1r∑j=1r∑k=1rwimjwkQijkTPQijk−Θx¯Tt,where(22)x¯t=x˜tx˜t−τ,Θ=P−S00S,Qijk=GijkM¯i+N¯ik.
If(23)∑i=1r∑j=1r∑k=1rwimjwkQijkTPQijk−Θ<0,then ΔVxt<0.
By Schur complement,(24)∑i=1r∑j=1r∑k=1rwimjwkQijkTPQijk−Θ=∑i=1r∑j=1r∑k=1rwimjwk−ΘQijkTPPQijk−P=∑i=1r∑j=1r∑k=1rwimjwk−P+S0GijkTP0−SM¯i+N¯ikTPPGijkPM¯i+N¯ik−P.
If(25)∑i=1r∑j=1r∑k=1rwimjwk−P+S0GijkTP0−SM¯i+N¯ikTPPGijkPM¯i+N¯ik−P<0,then ΔVxt<0.
It is obvious that equation (25) is equivalent to(26)∑i=1r∑j=1r∑k=1rwimjwk−X+S˜0XGijkT0−S˜XM¯i+N¯ikTGijkXM¯i+N¯ikX−X<0,where X=P−1 and S˜=XSX. Let(27)X=X100X2−1,S˜=S˜100X2−1S2X2−1,and by substituting equation (27) into equation (26) and defining Yi=FiX1, we can show equation (26) is equivalent to the following LMIs:(28)∑i=1r∑j=1rwimj−X1+S˜10∗0−S˜1∗A1iX1−BiYjA2iX1−X1<0,(29)∑i=1r∑k=1rwiwk−X2−1+X2−1S2X2−10∗0−X2−1S2X2−1∗A2iX2−1−LiC1kX2−1A2iX2−1−LiC2kX2−1−X2−1<0.
Premultiplying and postmultiplying equation (29) with diagX2,X2,X2 and denoting Ri=X2Li, we have(30)∑i=1r∑i=1rwiwk−X2+S20∗0−S2∗X2A2i−RiC1kX2A2i−RiC2k−X2<0.
For arbitraryΛi=ΛiT∈R3n×3n>0,i=1,2,…,r, we have(31)∑i=1r∑j=1rwiwj−mjΛi=∑i=1rwi∑j=1rwj−∑j=1rmjΛi=∑i=1rwi1−1Λi=0.
These terms are introduced to equation (28) to alleviate the conservativeness. From equation (28), we obtain(32)Ω=∑i=1r∑j=1rwimj−X1+S˜10∗0−S˜1∗A1iX1−BiYjA2iX1−X1=∑i=1r∑j=1rwimjΩij=∑i=1r∑j=1rwimjΩij+∑i=1r∑j=1rwiwj−mj+ρjwj−ρjwjΛi=∑i=1r∑j=1rwimjΩij+∑i=1r∑j=1rwiwj−ρjwjΛi−∑i=1r∑j=1rwimj−ρjwjΛi=∑i=1r∑j=1rwimj+ρjwj−ρjwjΩij+∑i=1r∑j=1rwiwj−ρjwjΛi−∑i=1r∑j=1rwimj−ρjwjΛi=∑i=1r∑j=1rwiwjρjΩij−ρjΛi+Λi+∑i=1r∑j=1rwimj−ρjwjΩij−Λi≤∑i=1rwi2ρiΩii−ρiΛi+Λi+∑i=1r∑j=1rwimj−ρjwjΩij−Λi+∑i=1r∑i<jwiwjρjΩij+ρiΩji−ρjΛi−ρiΛj+Λi+Λj.
Withmjzt−ρjwjzt≥0 for all j and zt, let(33)Ωij−Λi<0,ρiΩii−ρiΛi+Λi<Hii,ρjΩij+ρiΩji−ρjΛi−ρiΛj+Λi+Λj≤Hij+HijT,i<j,H=H11H12⋯H1rH21H22⋯H2r⋮⋮⋮⋮Hr1Hr2⋯Hrr<0,for all i,j=1,2,⋯,r, then Ω<0.
On the other hand, for equation (30), we have(34)Ψ=∑i=1r∑i=1rwiwk−X2+S20∗0−S2∗X2A2i−RiC1kX2A2i−RiC2k−X2=∑i=1r∑k=1rwiwkΨik=∑i=1rwi2Ψii+∑i,k=1,i<krwiwkΨik+Ψki.
Let(35)Ψii<Mii,Ψik+Ψki<Mik+MikT,i<k,M=M11M12⋯M1rM21M22⋯M2r⋮⋮⋮⋮Mr1Mr2⋯Mrr<0,for all i,k=1,2,…,r, then Ψ<0.
IfΩ<0 and Ψ<0, then equation (28) and equation (30) hold, so that equation (26) holds, then we have ΔVxt<0, which means system (10) is asymptotic stable.


## 4. Numerical Examples

In this section, three examples will be given to illustrate the less conservativeness and the effectiveness of the proposed results.Example 1.
From the proposed results, we can see the stability conditions are connected with the values ofρj. In this example, different values of ρj will be testified using the LMI based on the stability conditions in Theorem 1 and then we will illustrate the less conservativeness of the proposed results. Consider the following fuzzy time-delay model [6]:
Rulei: if x1t is M1i, then(36)xt+1=A1ixt+A2ixt−τ+Biut,i=1,2,3,where(37)A11=10.50.51−0.1,A12=0.10.250.15−0.5,B1=01,A13=a0.750.25−0.8,A21=0.10−0.250,B2=1−0.5,A22=0.2500.150,A23=0.1900.060,B3=b0.5,0≤a≤2,0≤b≤2.
It is assumed that the membership functions of the discrete fuzzy model with time delay and fuzzy controller are different and satisfymjzt−ρjwjzt≥0 for all j and zt. The feedback gains of Fj,j=1,2,3, are determined such that the eigenvalues of Ai+BiFj,i,j=1,2,3, are located at −5. The stability region given by stability conditions in Theorem 1 with ρ1=0.45,ρ2=0.5, and ρ3=0.75 (denoted by “o”) and ρ1=0.5,ρ2=0.75, and ρ3=0.85 (denoted by “×“) is shown in Figure 1. For comparison purposes, the stability region given by the stability conditions in [5, 6] (denoted by “×” and denoted by “o”, respectively) is shown in Figure 2. From Figure 1, it can be seen that the proposed stability conditions in Theorem 1 become lager with the increase in ρ1, ρ2, and ρ3. Meanwhile, the proposed stability conditions offer a larger stability region than in [5, 6], as can be seen in Figure 2. Based on the simulation, it can also be observed that when a and b take certain values, the stability conditions in [5, 6] cannot provide feasible solutions.Figure 1
Stability region of Theorem1 with ρ1=0.45,ρ2=0.5, and ρ3=0.75 (denoted by “o”) and with ρ1=0.5,ρ2=0.75, and ρ3=0.85 (denoted by “×”).Figure 2
Stability conditions in [5] (denoted by “x”) and in [6] (denoted by “o”).Example 2.
An unmatching observer-based output feedback controller is designed to backing up control of a computer-simulated truck-trailer [23]. Consider the following modified truck-trailer model with time delay:(38)x1t+1=1−vt¯Lax1t+vt¯l¯+1−ax1t−τut,x2t+1=x2t+vt¯Lax1t+1−ax1t−τ,x3t+1=x3t+vt¯sinx2t+vt¯2Lax1t+1−ax1t−τ,yt=x2t+vt¯2Lax1t+1−ax1t−τ,where x1t is the angle difference between the truck and trailer, x2t is the angle of trailer, x3t is the vertical position of the rear end of trailer, and ut is the control input. The model parameters are given as l¯=2.8,L=5.5, v=−1.0,t¯=2.0,anda=0.9.
The fuzzy truck-trailer model can be modeled by a two-rule fuzzy model.
Rulei, i=1,2: if θt=x2t+vt¯/2Lax1t+1−ax1t−τ is about 0, then(39)xt+1=A11xt+A21xt−τ+B1ut,yt=C11xt+C21xt−τ.
Ifθt=x2t+vt¯/2Lax1t+1−ax1t−τ is about ±π, then(40)xt+1=A12xt+A22xt−τ+B1ut,yt=C12xt+C22xt−τ,where(41)A11=a1−vt¯L00avt¯L10av2t¯22Lvt¯1,A21=1−a1−vt¯L001−avt¯L001−av2t¯22L00,A12=a1−vt¯L00avt¯L10adv2t¯22Ldvt¯1,A22=1−a1−vt¯L001−avt¯L001−adv2t¯22L00,B1=B2=vt¯l¯00T,C11=C12=avt¯2L10,C21=C22=1−avt¯2L00.
The fuzzy state observer for the T-S fuzzy model in equations (39) and (40) is formulated as follows.
Observer rulei: if θt=x2t+vt¯/2Lax1t+1−ax1t−τ is M1i,i=1,2, then(42)x^t+1=A1ix^t+A2ix^t−τ+Biut+Liyt−y^t,i=1,2,…,r,y^t=C1ix^t+C2ix^t−τ.
The output of equation (42) is represented as follows:(43)x^t+1=∑i=12wiθtA1ixt+A2ixt−τit+Biut,y^t=∑i=12wiθtC1ixt+C2ixt−τ.
Here, we setd=10−2/π and the membership functions are defined as(44)w1θt=1−11+exp−3θt−0.5π11+exp−3θt+0.5π,w2θt=1−w1θt.
From Theorem1, according to the abovementioned fuzzy state observer model, an unmatching fuzzy controller with two rules is proposed to control the observer model so that the dynamic system equation (43) is asymptotically stable.
Rulej,j=1,2: if θt=x2t+vt¯/2Lax1t+1−ax1t−τ is N1j, then(45)ut=Fjx^t.
The fuzzy controller is defined as(46)ut=∑j=12mjθtFjx^t.
The membership functions are selected as(47)m1θt=0.93exp−θt4×1.52,m2θt=1−m1θt.
We assume thatρ1=ρ2=0.85 such that mjθt−ρjwjθt>0 for all j and θt.
Assumingτ=2.0 and applying the MATLAB LMI toolbox to solve the LMIs in Theorem 1, we have(48)F1=0.52040.26780.0456,F2=0.08070.36360.0013.L1=261.323003.1−6.6897×107T,L2=321.363462.3−1.7345×105T.
With the initial conditionx0=−0.25π0.5π−10T and x^0=000T, the simulation results under Theorem 1 are shown in Figure 3, which illustrate the effectiveness of Theorem 1.Figure 3
State responsesxt and x^t and control action ut.
(a)(b)(c)(d)Example 3.
The same fuzzy state observer model equation (43) is also considered here. Some advantages of the proposed method are shown in this example. Suppose the membership function wiθt is chosen as follows:(49)w1θt=1−ctsinθt−451+exp−100θt31−θtcosθt21+exp−2.5θt3+θt/0.42,w2θt=1−w1θt,where ct=sinθt+1/40∈−0.050.05 and θt∈−π/2π/2, in which ct is an uncertain variable.
Comparing withwjθt (equation (44)) in Example 2, we can observe that the structure of the membership functions wjθt (equation (49)) is not only complex but also contains an uncertain variable. By equation (46), under the PDC controller design framework, the membership functions of the fuzzy controller must be selected as the same as those of the fuzzy model, which will lead to the fuzzy controller unrealized. However, based on Theorem 1, some simple and specific membership functions mjθt (equation (47)) can be selected instead of the complex and uncertain functions wjθt (equation (49)) as the membership functions of the fuzzy controller, which can ensure the closed-loop system (10) is asymptotically stable (refer to Figure 4).Figure 4
State responses.Remark 2.
Comparing with the PDC scheme, as the membership functions of the controller are distinctive from those of the fuzzy model in our paper, the membership functions can be chosen arbitrarily. Therefore, the proposed method can enhance the design flexibility of the fuzzy controller as well as retain the robustness property of the T-S fuzzy control systems.


## 5. Conclusions

In this paper, an unmatching observer-based stabilization control for discrete-time T-S fuzzy systems with time delay is investigated, in which the discrete fuzzy observer model and fuzzy controller do not share the same membership functions. The information of the membership functions of the fuzzy model and controller is considered in the observer design scheme. Furthermore, the design flexibility can be enhanced by arbitrarily selecting simple membership functions for the observer-based controller. The advantages and effectiveness of the proposed method have been illustrated using simulation examples. According to the obtained results, the proposed method only fits for the systems with constant delay. Therefore, how to extend the abovementioned method to time-varying delay systems will be considered in the future.



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