# Improved Approach to Robust Control for Type-2 T-S Fuzzy Systems

Authors: Bum Yong Park; JaeWook Shin
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper is concerned with the robust stability conditions to stabilize the type 2 Takagi-Sugeno (T-S) fuzzy systems. The conditions effectively handle parameter uncertainties using lower and upper membership functions. To improve the solvability of the stability conditions, we establish a multigain controller with comprehensive information of the lower and upper membership grades. In addition, a well-organized relaxation technique is proposed to fully exploit relationship among fuzzy weighting functions and their lower and upper membership grades, which enlarges a set of feasible solutions. Therefore, we derive a less conservative stabilization condition in terms of linear matrix inequalities (LMIs) than those in the literature. Two simulation examples illustrate the effectiveness and robustness of the derived stabilization conditions.


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## Body

## 1. Introduction

Over the past few decades, the type 1 Takagi-Sugeno (T-S) fuzzy model has attracted much attention because it can systematically represent nonlinear systems via an interpolation method that smoothly connects some local linear systems based on fuzzy weighting functions [1–3]. The main advantage of type 1 T-S fuzzy systems is that they allow us to apply the well-established linear system theory for the analysis and synthesis of nonlinear systems. For this reason, the type 1 T-S fuzzy model has been a popular choice not only in consumer products but also in industrial processes, such as power converters [4], motors [5], and solar power generator systems [6].For the stability analysis and synthesis of type 1 fuzzy control systems, Lyapunov stability theory is widely used [7–10]. Fundamental stability conditions in terms of linear matrix inequalities (LMIs) are derived from Lyapunov stability condition. The conditions guarantee the stability of the fuzzy control systems if there exists a solution to a set of LMIs. Many researchers introduced the stability conditions and relaxed stability conditions using parallel distributed compensation (PDC) concept [8]. Using the information of type 1 fuzzy membership functions, the stability conditions can be further relaxed [11–14].Although the type 1 fuzzy control system can effectively handle the nonlinear systems, it cannot guarantee the stability of the nonlinear systems with parameter uncertainties. Recently, type 2 fuzzy systems have attracted a lot of research attention [15] because they are better at handling uncertainties than the conventional type 1 fuzzy systems [16, 17]. Hence, for the stability analysis and controller synthesis of nonlinear systems with parameter uncertainties, it is essential to use type 2 fuzzy systems. Several researchers have researched such type 2 fuzzy systems [18–20]. However, all the aforementioned papers have seldom studied stability analysis and controller synthesis for type 2 T-S fuzzy systems. This motivates the study of the stability analysis and controller synthesis of type 2 T-S fuzzy systems.Recently, some researchers have studied stability analysis and controller synthesis for type 2 T-S fuzzy systems [21–25]. In [21], an interval type 2 T-S fuzzy controller was proposed using a common controller gain that collectively depends on the sum of lower upper membership grades. In [22], the controller design for the interval type 2 T-S fuzzy system was introduced using a membership function different from a membership function of the system.In the above studies based on the type 2 T-S fuzzy systems, the stability conditions for the design of the type 2 fuzzy controller have some tuning parameters, which can result in increasing the implementation effort. It motivates the study of the controller synthesis for type 2 T-S fuzzy system. This paper studies the robust stability conditions to stabilize type 2 T-S fuzzy systems. The conditions effectively handle parameter uncertainties using lower and upper membership functions. To improve the solvability of the stability conditions, we establish a multigain controller with comprehensive information of the lower and upper membership grades. In addition, we propose a well-organized relaxation technique that fully exploits relationship among fuzzy weighting functions and their lower and upper membership grades, which enlarges a set of feasible solutions. Therefore, we derive a less conservative stabilization condition in terms of LMIs than those in the literature. The proposed condition has a simple structure without tuning parameters. Finally, two simulation examples are given to illustrate the effectiveness and robustness of the derived stabilization condition.Notation. The notations X≥Y and X>Y mean that X-Y is positive semidefinite and positive definite, respectively. In symmetric block matrices, (∗) is used as an ellipsis for terms that are induced by symmetry. Furthermore, Sym(X)=X+XT stands for any matrix X.


## 2. System Description and Preliminaries

Let us consider the following type 2 T-S fuzzy model [21] that represents a continuous-time nonlinear system: for i∈R={1,2,…,p},(1)Plant  Rulei:IFf1xtisF~i1and…fsxtisF~is,THENx˙t=Aixt+Biut,where x(t)∈Rn, u(t)∈Rm denote the state and control input, respectively; F~ij denotes a type 2 fuzzy set of rules i corresponding to the function fj(x(t)); and p denotes the number of IF-THEN rules. The firing interval of the ith rule is as follows:(2)θ~ixt=μ_F~i1f1xt×⋯×μ_F~isfsxt,μ¯F~i1f1xt×⋯×μ¯F~isfsxt≡θiLxt,θiUxt,where μ_F~ij(fj(x(t)))∈[0,1] and μ¯F~ij(fj(x(t)))∈[0,1] denote the lower and upper grades of membership associated with the lower and upper membership functions, respectively. Here, μ_F~ij(fj(x(t)))≤μ¯F~ij(fj(x(t))), which leads to θiL(x(t))≤θiU(x(t)) for all i. The overall type 2 T-S fuzzy system is inferred as follows:(3)x˙t=∑i=1pθixtAixt+Biut,where θi(x(t))=η(x(t))θiL(x(t))+(1-η(x(t)))θiU(x(t)) denotes a fuzzy weighting function in which η(x(t))∈[0,1] is a nonlinear function and not necessary to be considered in this paper. Now, consider a multigain controller that isindividually dependent on the lower and upper membership grades such as(4)ut=∑j=1pθ_jxtK_j+θ¯jxtK¯jxt,where (5)θ_jxt≜θjLxt∑l=1pθlLxt+θlUxt,θ¯jxt≜θjUxt∑l=1pθlLxt+θlUxt,and K_j and K¯j are the controller gains associated with the lower and upper membership grades. By the above relations, θi(x(t)), θ_i(x(t)), and θ¯i(x(t)) satisfy the following conditions:(6)∑i=1pθixt=∑j=1pθ_jxt+θ¯jxt=1,(7)0≤αi≤θixt≤βi≤1,(8)0≤α_i≤θ_ixt≤β_i≤1,(9)0≤α¯i≤θ¯ixt≤β¯i≤1,(10)0≤θ_ixt+θ¯ixt≤1,(11)0≤θixtθ_jxt+θ¯jxt,(12)θ_ixt≤θ¯ixt,where αi, α_i, α¯i, βi, β_i, and β¯i are real constant values. Henceforth, for a simple description, we use the following notations: θi(x(t))≜θi, θ_i(x(t))≜θ_i, and θ¯i(x(t))≜θ¯i. The resulting closed-loop system under (4) is represented as follows:(13)x˙t=∑i=1pθiAixt+Bi∑j=1pθ_jK_j+θ¯jK¯jxt=∑i=1pθiAi+∑i=1p∑j=1pθiθ_jBiK_j+θiθ¯jBiK¯jxt.


## 3. Stabilization Conditions

Theorem 1.
Suppose that matricesR0,S0,Ri,R_i,R¯i,Si,S_i,S¯i,Xi,Yi,Zi,Πi,Ξij,Λi∈Rn×n, L_i,L¯i∈Rm×n, and a positive definite matrix P¯∈Rn×n exist such that, for i,j={1,2,…,p},(14)Xi+XiT>0,Yi+YiT>0,Zi+ZiT>0,Πi+ΠiT>0,Ξij+ΞijT>0,Λi+ΛiT>0,Σ<0,where Σ is shown as follows:(15)
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Then, the closed-loop system (13) is asymptotically stable and the corresponding fuzzy controller is given by K_j=L_jP¯-1 and K¯j=L¯jP¯-1.Proof.
Choose a quadratic Lyapunov function,V(t)=xT(t)Px(t), where P is a positive definite matrix. Then the time derivative of V(x(t)) is given by(16)V˙t=2xtTPx˙t=2xtTP∑i=1pθiAi+∑i=1p∑j=1pθiθ_jBiK_j+θiθ¯jBiK¯jxt.For the asymptotic stability of (13), the stability criterion is given as V˙(t)<0. It leads to the following condition:(17)Sym∑i=1pθiPAi+∑i=1p∑j=1pθiθ_jPBiK_j+θiθ¯jPBiK¯j<0.
Multiplying both sides of (17) by P-1 yields(18)Sym∑i=1pθiAiP¯+∑i=1p∑j=1pθiθ_jBiL_j+θiθ¯jBiL¯j<0,where P¯=P-1, L_j=K_jP¯, and L¯j=K¯jP¯. We can then represent (18) as the following form:(19)M=∑i=1pθiM0,i+M0,iT+∑i=1p∑j=1pθiθ_jM_i,j+M_i,jT+∑i=1p∑j=1pθiθ¯jM¯i,j+M¯i,jT<0,where M(0,i)=AiP¯, M_(i,j)=BiL_j, and M¯(i,j)=BiL¯j.
For the relaxation technique using the relationship among fuzzy weighting functions and their lower and upper membership grades, we can convert conditions (6)–(12) for all i∈[1,p], respectively, into(20)0≡C1=Iθ1I⋮θpIθ_1I⋮θ_pIθ¯1I⋮θ¯pITII-I0⋮⋮-I00-I⋮⋮0-I0-I⋮⋮0-IR0TS0TR1TS1T⋮⋮RpTSpTR_1S_1⋮⋮R_pS_pR¯1S¯1⋮⋮R¯pS¯pTIθ1I⋮θpIθ_1I⋮θ_pIθ¯1I⋮θ¯pI,(21)0≤C2i=-θi-αiθi-βiXi+XiT,(22)0≤C3i=-θ_i-α_iθ_i-β_iYi+YiT,(23)0≤C4i=-θ¯i-α¯iθ¯i-β¯iZi+ZiT,(24)0≤C5i=-θ_i+θ¯iθ_i+θ¯i-1Πi+ΠiT,(25)0≤C6ij=θiθ_j+θ¯jΞij+ΞijT,(26)0≤C7i=θ¯i2-θ_i2Λi+ΛiT,where condition (20) is from ∑i=1pθiI-I=∑i=1p(θ_i+θ¯i)I-I=0 and conditions (21)–(26) are from (7)–(12), respectively. Here, the matrices R0, Ri, R_i, R¯i, Xi, Yi, Zi, Πi, Ξi, and Λi are slack variables to reduce conservatism of (19).
Then, merging conditions (20)–(26) gives(27)N=C1+C1T+∑i=1pC2i+∑i=1pC3i+∑i=1pC4i+∑i=1pC5i+∑i=1p∑j=1pC6ij+∑i=1pC7i≥0.
To derive LMI condition, (27) can be represented as the following form:(28)0≤N=N0+∑i=1pθiN0,i+N0,iT+∑i=1pθ_iN_0,i+N_0,iT+∑i=1pθ¯iN¯0,i+N¯0,iT+∑i=1pθi2Ni,i+∑i=1pθ_i2N_i,i+∑i=1pθ¯i2N¯i,i+∑i=1pθ_iθ¯iT~i,i+T~i,iT+∑i=1p∑j=1pθiθ_jT_i,j+T_i,jT+∑i=1p∑j=1pθiθ¯jT¯i,j+T¯i,jT+∑i=1p-1∑j=i+1pθiθjNi,j+Ni,jT+∑i=1p-1∑j=i+1pθiθ_jN_i,j+N_i,jT+∑i=1p-1∑j=i+1pθiθ¯jN¯i,j+N¯i,jT+∑i=1p∑j=1,j≠ipθ_iθ¯jT~i,j+T~i,jT,where (29)N0=R0+S0+R0T+S0T-∑i=1pαiβiXi+XiT-∑i=1pα_iβ_iYi+YiT-∑i=1pα¯iβ¯iZi+ZiT,N0,i=Ri+Si-R0+αi+βiXi,N_0,i=R_i+S_i-S0+α_i+β_iYi+Πi,N¯0,i=R¯i+S¯i-S0+α¯i+β¯iZi+Πi,Ni,i=-Ri+RiT-Xi+XiT,N_i,i=-S_i+S_iT-Yi+YiT-Πi+ΠiT-Λi+ΛiT,N¯i,i=-S¯i+S¯iT-Zi+ZiT-Πi+ΠiT+Λi+ΛiT,T~i,i=-S¯iT+S_i-2Πi,T_i,j=-Si+R_j+Ξij,T¯i,j=-Si+R¯j+Ξij,Ni,j=-Ri+Rj,N_i,j=-S_i+S_j,N¯i,j=-S¯i+S¯j,T~i,j=-S_i+S¯j.
TheS-procedure enables the condition in (19) subject to (28) to be expressed as (30)L=M+N<0,which can be converted into(31)L=ΘTΣΘ<0,where (32)Θ=I∣θ1⋯θp∣θ_1⋯θ_p∣θ¯1⋯θ¯pT.Remark 2.
The proposed LMIs are not always feasible for all type 2 T-S fuzzy system. However, the proposed method can guarantee a larger feasible solution set than the previous studies because we use the relationship among fuzzy weighting functions and their lower and upper membership grades and the proposed controller isindividually dependent on the lower and upper membership grades.


## 4. Numerical Examples

Example 1.
Let us consider a simple fuzzy model [21] in the following format:(33)PlantRulei:IFx1tisF~i1,THENx˙t=Aixt+Biut,i=1,2,3,where (34)A1=2.78-5.630.010.33,A2=0.2-3.220.350.12,A3=-a-6.630.450.15,B1=2-1,B2=80,B3=-b+6-1,where 14≤a≤30 and 20≤b≤35. The above model is assumed operating in the domain of x1(t)∈-1010T. The lower and upper membership functions are chosen as (35)θ1L=μ_F~11x1t=0.95-0.9251+e-x1+3.5/8,θ3L=μ_F~31x1t=0.025+0.9251+e-x1-3.5/8,θ2L=μ_F~21x1t=1-θ1U-θ3U,θ1U=μ¯F~11x1t=0.95-0.9251+e-x1+4.5/8,θ3U=μ¯F~31x1t=0.025+0.9251+e-x1-4.5/8,θ2U=μ¯F~21x1t=1-θ1L-θ3L.
For Theorem1, αi = 0, βi=1, α_1=α_3=0.154, α_2=0.164, β_1=β_3=0.65, β_2=0.25, α¯1=α¯3=0.169, α¯2=0.204, β¯1=β¯3=0.67, β¯2=0.3. Figure 1 shows the stability regions for the proposed conditions and previous conditions in Example  1 of [21]. From Figure 1, we can clearly see that the proposed conditions outperform previous one [21] in terms of a larger stability region. It is an important observation that the proposed conditions with a multigain controller and a well-organized relaxation technique achieve much better performance than those of the previous study [21].Figure 1
Stability region for the proposed (∘) and published conditions (×).Example 2.
Let us consider an inverted pendulum model subject to parameter uncertainties, which is adapted from [26]: (36)x˙2t=gsin⁡x1t-ampLx22tsin⁡2x1t/2-acos⁡x1tut4L/3-ampLcos2x1t,where x1(t) is the angular displacement of the pendulum; g=9.8m/s2; mp and Mc are uncertain in mpmin=2≤mp≤5=mpmax and Mcmin=8≤Mc≤18=Mcmax, respectively; a=1/(mp+Mc); 2L=1m; and u(t) is the force applied to the cart. From [21], we can obtain a plant rule to describe the inverted pendulum subject to parameter uncertainties in the following format:(37)PlantRulei:IFx1tisF~i1,IFx1tisF~i2THENx˙t=Aixt+Biut,i=1,2,3,4,where (38)A1=A2=01f1min0,A3=A4=01f1max0,B1=B3=0f2min,B2=B4=0f2max,xt=x1tx2tT,x1t∈x1minx1max=-5π125π12,x2t∈x2minx2max=-55,f1min=8.81≤f1xt≤f1max=20.7,f2min=-0.18≤f2x1t≤f2max=-0.017,f1xt=g-ampLx22tcos⁡x1t4L/3-ampLcos2x1tsin⁡x1tx1t,f2x1t=-acos⁡x1t4L/3-ampLcos2x1t.
The type 2 T-S fuzzy model is represented as follows:(39)x˙t=∑i=14θiAixt+Biut=∑i=14ηxtθiL+1-ηxtθiUAixt+Biut,where, for all i, (40)θiL=μ_F~i1xt×μ_F~i2xt,θiU=μ¯F~i1xt×μ¯F~i2xt,and the lower and upper grades of membership for each rule are defined as follows: (i)
x2(t)=x2max, mp=mpmax, and Mc=Mcmin:(41)μ_F~11xt=μ_F~21xt=f1max-f1xtf1max-f1min,μ_F~31xt=μ_F~41xt=f1xt-f1minf1max-f1min.(ii)
x2(t)=0, mp=mpmax, and Mc=Mcmin:(42)μ¯F~11xt=μ¯F~21xt=f1max-f1xtf1max-f1min,μ¯F~31xt=μ¯F~41xt=f1xt-f1minf1max-f1min.(iii)
mp=mpmin and Mc=Mcmin:(43)μ_F~12x1t=μ_F~32x1t=f2max-f2x1tf2max-f2min,μ_F~22x1t=μ_F~42x1t=f2x1t-f2minf2max-f2min.(iv)
mp=mpmax and Mc=Mcmax:(44)μ¯F~12x1t=μ¯F~32x1t=f2max-f2x1tf2max-f2min,μ¯F~22x1t=μ¯F~42x1t=f2x1t-f2minf2max-f2min.For Theorem1, αi = 0, βi=1, α_i=0, β_i=0.5, α¯1=0, β¯i=0.5. Let us demonstrate the validity of the proposed conditions for the type 2 T-S fuzzy model. Figure 2 shows the trajectories of the states with mp=mpmin and Mc=Mcmin under various initial conditions. Figure 3 shows the trajectories of the states with mp=mpmax and Mc=Mcmax. From Figures 2 and 3, we can clearly see that the proposed controller can stabilize the inverted pendulum with different parameter values and is robust to parameter variations in the plant model. In addition, the proposed conditions lead to less conservative results because we use the larger mass ranges than those of [21].Figure 2
The trajectories of state (mp=mpmin,Mc=Mcmin).Figure 3
The trajectories of state (mp=mpmax,Mc=Mcmax).


## 5. Conclusion

In this paper, we proposed robust stability conditions to stabilize type 2 T-S fuzzy systems. The conditions effectively handled parameter uncertainties using lower and upper membership functions. Furthermore, by applying a multigain controller and a well-organized relaxation technique, we derived a less conservative stabilization condition in terms of LMIs than those in the literature. Our simulation results showed the effectiveness and robustness of the derived stabilization conditions.



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