# PiecewiseH∞ Static Output Feedback Controller Design for Nonlinear Systems Based on T-S Affine Fuzzy Models

Authors: Xingang Zhao
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper is concerned with the problem of designingH∞ controllers via static output feedback controller for a class of complex nonlinear systems, which is approximated by continuous-time affine fuzzy models. A decomposition method is presented to divide the output-space into different operating regions and interpolation regions. Based on this partition, a novel piecewise controller with affine terms via static output feedback is designed. By using a dilated linear matrix inequality (LMI) characterization, some nonconvex conditions are converted into convex ones to make the asymptotic stability and H∞ performance of the closed-looped system. The effectiveness of the proposed method is illustrated by a numerical example.


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## Body

## 1. Introduction

As a well universal approximator, Takagi-Sugeno (T-S) fuzzy models provide an effective way to deal with nonlinearities and uncertainties in practical engineering areas since 1985 [1–3]. Some locally linear time-invariant systems in the form of IF-THEN rules are used to describe the prominent feature of T-S fuzzy models. During the past few years, the T-S fuzzy control systems have been studied extensively, and many significant advances have been achieved [4–9].Both linear fuzzy models and affine fuzzy models are usually used to describe T-S Fuzzy systems [10–12]. Compared with linear fuzzy models, T-S affine fuzzy dynamic system with offset terms has much improved function approximation capabilities [3]. This has ignited enormous research activities in search of new methodologies to investigated affine T-S fuzzy systems (see [13, 14] and the references therein).Most of the above results are mainly focused on state feedback control, which requires all state variables to be accessible. This may simplify the considered problem but cause the obtained results to be restrictive in engineering. Since only a partial information through a measured output can be available in many practical nonlinear control systems, many researchers focus their attention on the output stabilization [15, 16]. The main advantage of the static output feedback is the simplicity of its implementation and reduction of real time computational cost when implementing practical applications. Recently, static output feedback (SOF) problem for T-S fuzzy systems has attracted considerable attention of many researchers. Until now, most of the existing fuzzy SOF control results are obtained for linear fuzzy systems. Many efforts have been done to convert the nonconvex problem to convex problem, such as iterative linear matrix inequality (ILMI) approaches (see [17–19]) and linear matrix inequality (LMI) approaches [20–22].Compared with linear fuzzy systems, the constant bias terms in affine fuzzy systems bring more challenge to the controller design. For continuous-time affine fuzzy systems, most existing results about controller design via output feedback are derived in the framework of bilinear matrix inequalities (BMIs) [12, 23, 24]. In [25], some sufficient conditions for the robust H∞ output feedback controller design problem for discrete-time affine fuzzy systems are presented in the frameworks of LMIs. However, they supposed the transitions of both the plant and the controller could happen among all regions, having the controller with conservative designing conditions and heavy computations. To the best of our knowledge, few results have been developed about H∞ controller design via static output feedback for continuous-time affine fuzzy systems based on output partition.In this paper, the considered nonlinear system is described by affine fuzzy models, and the system outputs are chosen as premise variables. There is no doubt that this modeling approach may be coarse to represent a complex nonlinear system because less information can be used for modeling. However, it is more convenient and less conservative in the case where state variables are not all available, for example, static output feedback problem. Moreover, static output feedback is simply to be implemented. Based on such fuzzy models, theH∞ static output feedback control design problem is discussed in this paper. A partition is presented to divide the output-space into different operating regions and interpolation regions. Then based on this partition, a new piecewise static output feedback controller with affine terms is proposed, whose design conditions are derived in the framework of LMIs by introducing additional slack matrix variables. Finally, a numerical example is given to illustrate the effectiveness and merits of the obtained method.The rest of this paper is organized as follows. Section2 introduces the preliminaries and design objective. The H∞ static output feedback controller design problem is addressed in Section 3, and a solution of this problem is given in terms of LMIs. A numerical example is given in Section 4 to show the effectiveness of proposed method. Finally, conclusions are drawn in Section 5.The following notations are used throughout the paper. The superscript “T” stands for matrix transposition. Then notation X>0 where X is symmetric matrix indicates that X is positive definite. In block symmetric matrices or long matrix expressions, the symbol * is used to represent a term that is induced by symmetry.


## 2. Preliminaries and Problem Statement

### 2.1. System Model

The following continuous-time affine T-S model is described by fuzzy IF-THEN rules, whose collection represents the approximation of a complex nonlinear system.Plant Rulei:ify1(t) is Mi1,…,  and ys(t) is Mis, then(1)x˙t=Aixt+μi+B1iut+B2iwt,zt=C1xt,yt=C2xt,where x(t)∈Rn is the state space vector, u(t)∈Rp is the control input, w(t)∈L2[0,∞) denotes the external disturbance, y(t)=[y1(t),…,ys(t)] is the measurable premise variable, and z(t)∈Rq is the regulated output. Ai, μi, B1i, B2i, Ci, C2  (i=1,2,…,r) are constant matrices with appropriate dimension. Mi1,Mi2,…,Mis are the fuzzy sets that are characterized by membership function; s is the number of premise variables; and r is the number of IF-THEN rules.The overall affine fuzzy model achieved by fuzzy-blending of each individual plant rule (local model) is given by(2)x˙t=∑i=1r‍hiytAixt+μi+B1iut+B2iwt,zt=C1xt,yt=C2xt,where(3)wiyt=∏j=1s‍Mijyjt.Denote(4)hiyt=wiyt∑i=1r‍wiyt,and then(5)0≤hiyt≤1,∑i=1r‍hiyt=1.Mij(yj(t)) is the grade of membership of yj(t) in Mij, and hi(y(t)) is the normalized membership function.Remark 1.
Similar to [12], a constant affine term μi is included in (1) to approximate the original nonlinear system, which makes the function approximation capabilities of fuzzy systems improve substantially.Assumption 2.
For eachi from 1 to r, assume that B1i are of full column rank.
A decomposition method will be used to divide the output-space into different operating regions and interpolation regions. In a certain operating region, it followshi(y(t))=1 and hj(y(t))=0  (j≠i), and then the system can be described by x˙(t)=Aix(t)+μi+B1iu(t)+B2iw(t). Between different operating regions, that is, interpolation regions, 0<hi(y(t))<1; then a convex combination of different affine systems can describe the system.
Denote{S¯i}i∈G⊆Rs as a polyhedral partition of the output, and F as the set of region indexes. G0⊆G and G1⊆G are the set of indexes for region that contain or do not contain the origin, respectively. For each local region Si, the set K(i) contains the indexes for the system matrices used in the interpolation within that region. For operating regions, K(i) only contains a single element. By a blending of m∈K(i) subsystems, the global system in (2) is represented as follows in each cell:(6)x˙t=A~ixt+μ~i+B~1iut+B~2iwt,zt=C1xt,yt=C2xt,yt∈S¯i,i∈G,where(7)A~i=∑m∈Ki‍hmytAm,μ~i=∑m∈Ki‍hmytμm,B~1i=∑m∈Ki‍hmytB1m,B~2i=∑m∈Ki‍hmytB2m,hmyt>0,∑m∈Ki‍hmyt=1.
The range ofy(t) is represented by the following system of at most s linear inequalities while y(t)∈S¯i:fory1t, y1t≤αi1 or y1t≥βi1 or αi1≤y1t≤βi1,fory2(t), y2(t)≤αi2 or y2(t)≥βi2 or αi2≤y2(t)≤βi2,⋮forys(t), ys(t)≤αis or ys(t)≥βis or αis≤ys(t)≤βis.


### 2.2. Problem Statement

Based on output partition, the following affine piecewise controller via static output feedback is chosen:(8)ut=Kiyt+ηi,yt∈S¯i,i∈G,where Ki, ηi are the controller gains to be designed later. Here we assume that the equilibrium point of closed-loop system is located at the origin and therefore we have ηi=0 for all i∈G0. Applying the piecewise static output feedback controller (8) to system (6), the resulting closed-loop system is given by the following:(9)x˙t=A~i+B~1iKiC2xt+B~1iηi+μ~i+B~2iwt,zt=C1xt,yt∈S¯i,i∈G.Remark 3.
Since in many practical control systems, the system satesx(t) of system (6) are not completely available. Thus, the static output feedback controller is more applicable than that via state feedback controller.Problem 4.
For the closed-loop system (9), design a piecewise affine H∞ via static output feedback in the form of (8), such that the following requirements are satisfied.(1)
The closed-loop system (9) is asymptotically stable when w(t)≡0.(2)
Under zero initial conditions, theL2-norm of the operator from disturbance w(t) to the controlled output z(t) is less than γ, (10)zt2<γwt2,for any external disturbancew(t)∈L2.


## 2.1. System Model

The following continuous-time affine T-S model is described by fuzzy IF-THEN rules, whose collection represents the approximation of a complex nonlinear system.Plant Rulei:ify1(t) is Mi1,…,  and ys(t) is Mis, then(1)x˙t=Aixt+μi+B1iut+B2iwt,zt=C1xt,yt=C2xt,where x(t)∈Rn is the state space vector, u(t)∈Rp is the control input, w(t)∈L2[0,∞) denotes the external disturbance, y(t)=[y1(t),…,ys(t)] is the measurable premise variable, and z(t)∈Rq is the regulated output. Ai, μi, B1i, B2i, Ci, C2  (i=1,2,…,r) are constant matrices with appropriate dimension. Mi1,Mi2,…,Mis are the fuzzy sets that are characterized by membership function; s is the number of premise variables; and r is the number of IF-THEN rules.The overall affine fuzzy model achieved by fuzzy-blending of each individual plant rule (local model) is given by(2)x˙t=∑i=1r‍hiytAixt+μi+B1iut+B2iwt,zt=C1xt,yt=C2xt,where(3)wiyt=∏j=1s‍Mijyjt.Denote(4)hiyt=wiyt∑i=1r‍wiyt,and then(5)0≤hiyt≤1,∑i=1r‍hiyt=1.Mij(yj(t)) is the grade of membership of yj(t) in Mij, and hi(y(t)) is the normalized membership function.Remark 1.
Similar to [12], a constant affine term μi is included in (1) to approximate the original nonlinear system, which makes the function approximation capabilities of fuzzy systems improve substantially.Assumption 2.
For eachi from 1 to r, assume that B1i are of full column rank.
A decomposition method will be used to divide the output-space into different operating regions and interpolation regions. In a certain operating region, it followshi(y(t))=1 and hj(y(t))=0  (j≠i), and then the system can be described by x˙(t)=Aix(t)+μi+B1iu(t)+B2iw(t). Between different operating regions, that is, interpolation regions, 0<hi(y(t))<1; then a convex combination of different affine systems can describe the system.
Denote{S¯i}i∈G⊆Rs as a polyhedral partition of the output, and F as the set of region indexes. G0⊆G and G1⊆G are the set of indexes for region that contain or do not contain the origin, respectively. For each local region Si, the set K(i) contains the indexes for the system matrices used in the interpolation within that region. For operating regions, K(i) only contains a single element. By a blending of m∈K(i) subsystems, the global system in (2) is represented as follows in each cell:(6)x˙t=A~ixt+μ~i+B~1iut+B~2iwt,zt=C1xt,yt=C2xt,yt∈S¯i,i∈G,where(7)A~i=∑m∈Ki‍hmytAm,μ~i=∑m∈Ki‍hmytμm,B~1i=∑m∈Ki‍hmytB1m,B~2i=∑m∈Ki‍hmytB2m,hmyt>0,∑m∈Ki‍hmyt=1.
The range ofy(t) is represented by the following system of at most s linear inequalities while y(t)∈S¯i:fory1t, y1t≤αi1 or y1t≥βi1 or αi1≤y1t≤βi1,fory2(t), y2(t)≤αi2 or y2(t)≥βi2 or αi2≤y2(t)≤βi2,⋮forys(t), ys(t)≤αis or ys(t)≥βis or αis≤ys(t)≤βis.


## 2.2. Problem Statement

Based on output partition, the following affine piecewise controller via static output feedback is chosen:(8)ut=Kiyt+ηi,yt∈S¯i,i∈G,where Ki, ηi are the controller gains to be designed later. Here we assume that the equilibrium point of closed-loop system is located at the origin and therefore we have ηi=0 for all i∈G0. Applying the piecewise static output feedback controller (8) to system (6), the resulting closed-loop system is given by the following:(9)x˙t=A~i+B~1iKiC2xt+B~1iηi+μ~i+B~2iwt,zt=C1xt,yt∈S¯i,i∈G.Remark 3.
Since in many practical control systems, the system satesx(t) of system (6) are not completely available. Thus, the static output feedback controller is more applicable than that via state feedback controller.Problem 4.
For the closed-loop system (9), design a piecewise affine H∞ via static output feedback in the form of (8), such that the following requirements are satisfied.(1)
The closed-loop system (9) is asymptotically stable when w(t)≡0.(2)
Under zero initial conditions, theL2-norm of the operator from disturbance w(t) to the controlled output z(t) is less than γ, (10)zt2<γwt2,for any external disturbancew(t)∈L2.


## 3. Main Results

In this section, we will give some sufficient conditions to solveH∞ static output feedback controller design problem for the closed-loop system (9).The following lemma will be used to derive the main results of this paper.Lemma 5.
The following two statements are equivalent for the closed-loop affine fuzzy system (9) being asymptotically stable and satisfying an H∞ performance index γ.(1)
If there exist a symmetric matrixP>0 and scalars τiq≥0(q=1,2,…,n) such thatfori∈G0, m∈K(i),(11)ΥimPB2mB2mTP-γ2I<0,whereΥim=(Am+B1mKiC2)TP+P(Am+B1mKiC2)+C1TC1,and fori∈G1, m∈K(i),(12)ΔimPB2mPμm-∑q=1s‍τiqC2Tuiq*-γ2I0**-∑q=1s‍τiqviq<0,whereΔim=Υim-∑q=1s‍τiqC2TTiqC2 and Tiq, uiq, viq are similar to those in [12] such that Γiq(y(t))≡yT(t)Tiqy(t)+2uiqu)y(t)+viq<0 while y(t)∈S-i, i∈G.(2)
For large enough positive constantsα,k, if there exist P>0, matrices Fi, Wi, and scalars τij≥0  (j=1,2,…,n) such thatfori∈G0, m∈K(i)(13)Φ-2αP-1P-1+Πim+αITWiP-1+WiTΠim+αI-Wi-WiT<0,and fori∈G1, m∈K(i)(14)Φi-2kP-2P-2+Γim+kITFiP-2+FiTΓim+kI-Fi-FiT<0,where(15)Φ=C1TC100-γ2I,Φi=Φi110-∑q=1n‍τiqC2Tuiq*-γ2I0**-∑q=1s‍τiqviq,Φi11=C1TC1-∑q=1s‍τiqC2TTiqC2,P-1=P001,P-2=P00*I0**1,Πim=Am+B1mKiC2B2m00,Γim=Am+B1mKiC2B2mμm+B1mηi000000.Proof.
Firstly, we prove that if the statement (1) holds, then the closed-loop affine fuzzy system (9) is asymptotically stable and satisfying an H∞ performance index γ.
Chose a Lyapunov function as(16)Vxt=xTtPxt.Then from the derivative of V(t) along system (9), it follows(17)V˙xt+zTtzt-γ2wTtwt=xtwt1TΔi1PB~2iPμ~i*-γ2I0**0xtwt1with Δi1=(A~i+B~1iKiC2)TP+P(A~i+B~1iKiC2)+C1TC1.
Fori∈G1, it can be easily seen that the following inequality implies V˙(x(t))+zT(t)z(t)-γ2wT(t)w(t)<0. Consider(18)Δi2PB~2iPμ~i-∑q=1s‍τiqC2Tuiq*-γ2I0**-∑q=1s‍τiqviq<0,where Δi2=(A~i+B~1iKiC2)TP+P(A~i+B~1iKiC2)+C1TC1-∑q=1s‍τiqC2TTiqC2.
Then condition (12) implies (18), by expanding the fuzzy-basis functions. For the case i∈G0, condition (11) can be easily obtained without using the S-procedure, where μ~i=0.
Secondly,(1)⇒(2): for the case i∈G0 if (11) is satisfied, then rewrite (11) as Φ+P-1Πim+ΠimTP-1<0. It implies that there exists a sufficiently large positive scalar α1 such that ∀α>α1:(19)Φ+P-1Πim+ΠimTP-1+12αΠimTP-1Πim<0.
The above-mentioned inequality can be rewritten as(20)Φ-2αP-1+α2P-1+1αΠim+αITP-1P-1-1×P-1+1αP-1Πim+αI<0.Applying Schur complement, we get(21)Φ-2αP-1P-1+1αΠim+αITP-1P-1+1αP-1Πim+αI-2αP-1<0.
By takingWi=(1/α)P-1, then condition (13) is obtained. In the same way, if (12) is satisfied for the case i∈G1, we can still prove (13) holds.
(
2
)
⇒
(
1
) For the case i∈G0, multiplying (13) by [I(Πim+αI)T] and its transpose from the left side and the right one implies (11) holds. Similarly for the case of i∈G1, the conclusion that (14) implies (12) holds.
Then the proof is completed.Notice that conditions (13) and (14) in Lemma 5 are nonconvex. Next theorem will give some convex conditions for the affine H∞ controller design problem via static output feedback for (6).Theorem 6.
Consider the affine fuzzy system (6) with the designed static output feedback controller (8), for large enough positive constants α,k, if there exist P>0, matrices Wi, Fi11, Fi21, Fi22, Fi31, Fi32, and Fi33 with appropriate dimensions, where Fi11=Fi10Fi2Fi3, Wi=Wi10Wi2Wi3, Yi∈Rm×s, Vi∈Rm×1, and scalars τiq≥0  (q=1,2,…,n) such thatfori∈G0, m∈K(i),(22)C1TC1-2αP***0-γ2I-2αI**Λ31imWi1TTmB2m+αWi2T-TmTWi1-Wi1TTm*0I+αWi3T-Wi2-Wi3-Wi3T<0,and fori∈G1, m∈K(i),(23)ξ11i*****0-γ2I-2kI****ξ31i0ξ33i***ξ41imξ42imξ43im-TmTFi11-Fi11TTmξ41im*0I+kFi22TkFi32T-Fi21-Fi22-Fi22T*001+kFi33T-Fi31-Fi32-Fi33-Fi33T<0,
where(24)ξ11i=C1TC1-∑q=1n‍τiqC2TTiqC2-2kP,ξ31i=-∑q=1n‍τiquiqTC2,ξ33i=-∑q=1n‍τiqviq-2k,ξ41im=P+kFi11TTm+Fi11TTmAm+Yi0C2,ξ43im=Fi11TTmμm+Vi0+kFi31T,ξ42im=Fi11TTmB2m+kFi21T,Λ31im=P+kWi1TTm+Wi1TTmAm+Yi0C2.The closed-loop system (9) is asymptotically stable with a H∞ performance index γ. Moreover, the controller gains are given by(25)Ki=Wi1-TYi,i∈G0;Ki=Fi1-TYi,ηi=Fi1-TVi,i∈G1.Proof.
LetWi and Fi have the following form:(26)Wi=∑l∈K(i)‍θlTlTWi110Wi21Wi22,Fi=∑l∈K(i)‍θlTlTFi1100Fi21Fi22Fi23Fi31Fi32Fi33,with(27)Wi11=Wi10Wi2Wi3,Fi11=Fi10Fi2Fi3.Based on Assumption 2, we can take some invertible matrices Ti=(B1iTB1i)-1B1iTΩ for i=1,2,…,r such that(28)TiB1i=I0.Here the rows of Ω are the ones which are mutually independent and perpendicular to the columns of B1i. It can be easily seen that Ω is not unique. Denote that function θl is defined as(29)θl=1,l=m,0,else.
After some matrix computation in (13) and (14), (22) and (23) are easily obtained.Remark 7.
Whenα and k in (22) and (23) are set to be fixed parameters, the problems become convex and can be solved by employing the LMI Toolbox. To find the optimal values of corresponding parameters, in this paper, we will first solve the feasibility problem of LMIs in (22) and (23) by using LMI Toolbox under a set of initial scaling parameters. Then, for the optimization problem, apply a numerical optimization algorithm, such as the program fminsearch, and then a locally convergent solution to the problem is obtained.Whenηi≡0 in (8), a linear static output feedback controller can be given as follows:(30)ut=Kiyt,y(t)∈S¯i,i∈G.Based on Theorem6, the following corollary can be easily obtained.Corollary 8.
Consider the affine fuzzy system (6) with the designed static output feedback controller (30), for large enough positive constants α,k, if there exist P>0, matrices Wi, Fi11, Fi21, Fi22, Fi31, Fi32, and Fi33, where Fi11=Fi10Fi2Fi3, Wi=Wi10Wi2Wi3, Yi∈Rm×s, and scalars τiq≥0  (q=1,2,…,n) such that fori∈G0, m∈K(i),(31)C1TC1-2αP***0-γ2I-2αI**Λ31imWi1TTmB2m+αWi2T-TmTWi1-Wi1TTm*0I+αWi3T-Wi2-Wi3-Wi3T<0,and fori∈G1, m∈K(i),(32)ξ11i*****0-γ2I-2kI****ξ31i0ξ33i**ξ41imξ42imχ43im-TmTFi11-Fi11TTmξ41im*0I+kFi22TkFi32T-Fi21-Fi22-Fi22T*001+kFi33T-Fi31-Fi32-Fi33-Fi33T<0,whereχ43im=Fi11TTmμm+kFi31T.The corresponding closed-loop system is asymptotically stable with aH∞ performance index γ. Moreover, The controller gains are given by(33)Ki=Wi1-TYi,i∈G0;Ki=Fi1-TYi,i∈G1.Remark 9.
Notice that the results given in this paper are obtained based on the assumption that the input matrixB1i is of full column rank, which leads to much wider scope of the applicability than the common input matrix. Nevertheless, if there exist invertible matrices Ti, i=1,2,…,r, such that(34)TiB1i=Inc×nc000,fori=1,2,…,r,where nc<m, which means that the input matrices are not full column rank, then a novel piecewise affine state feedback controller is desirable with the form(35)ut=K~iyt+σ~i,yt∈S¯i,i∈F,where K~i=Kinc0, σ~i=σinc0. The corresponding static output feedback control synthesis method is proposed as follows.Corollary 10.
The continuous affine fuzzy system (6) is quadratically stabilizable at large by the piecewise controller (35) if there exist P>0, matrices Wi, Fi11, Fi21, Fi22, Fi31, Fi32, Fi33 with appropriate dimensions, where Fi11=Fi10Fi2Fi3, Wi=Wi10Wi2Wi3, and Yi∈Rnc×n, Vi∈Rnc×1, and scalars τij≥0  (i∈F;j=1,2,…,n) such that for given positive constants α,k, the linear matrix inequalities in (31) and (32) are satisfied, and the local gains of the fuzzy controller are given by(36)Kinc=Wi1-TYi,i∈F0;Kinc=Fi11-TYi,σinc=Fi11-TVi,i∈F1.


## 4. Numerical Example

In order to demonstrate the effectiveness and merits of the proposed piecewise affine static output feedback controller over the existing results, the following numerical example is considered.Ri: if y1(t) is Mi, then(37)x˙t=Aixt+μi+B1iut+B2iwt;i=1,2,3,zt=C1xt;yt=C2xt,where(38)A1=A3=1.30170.150-1.5107-0.5789-1.31712.02110.5320-0.2348,A2=1.81831-0.5362-1.2883-1-12.95491-0.5326,μ1=-μ3=-0.50311.0347-1.5836,μ2=000,B11=B13=100,B12=10.50,C1=0.510,C2=0.50.1010.51,B21=0.200.1,B22=0.10.20,B23=0.30.11.Here, the system state is assumed to be not completely available for feedback and thus cannot be chosen as the premise of the fuzzy model. The membership functions M1, M2, and M3 are depicted in Figure 1.Figure 1
Membership functions.Due to the fact that system state cannot be used for controlling the system, the existing controller design results for T-S affine fuzzy systems in [14, 24] are not suitable for this class of fuzzy systems. Besides, the existing output feedback controller design method in [25] is proposed for discrete-time affine fuzzy systems and not suitable for continuous-time affine fuzzy systems.In this simulation, both the piecewise affine static output feedback controller (8) and the piecewise linear static output feedback controller (30) are used to guarantee the resulting closed-loop system asymptotically stable with H∞ performance γ under w(t)=15e-0.1tsin(20πt) and zero initial conditions. By applying Theorem 6 and the fminsearch, the controller gains of piecewise static output feedback controller are obtained as follows:(39)K1=-14.2556-0.0095,σ1=2.1926,K2=-27.7366-0.9305,σ2=2.1872,K3=-22.7194-0.9619,σ3=0,K4=-27.3177-0.9648,σ4=-2.3762,K5=-15.9958-0.1540,σ5=-2.4800.Table1 shows the minimum H∞ performance γ based on the two abovementioned approaches (Theorem 6 and Corollary 8), respectively, with scaling parameters α=1.2559 and k=0.8062. It can be seen that the proposed piecewise affine static output feedback controller (8) can guarantee a smaller optimal H∞ performance index over the piecewise linear static output feedback controller (30). The state responses curves of the closed-loop system via the proposed static output feedback controller design method are given in Figures 2–4. The square root of ratio of the regulated output energy to the disturbance input noise energy is shown in Figure 5.Table 1
H
∞ performance γ.
Approach
Performance
Theorem6
7.6819
Corollary8
9.4937Figure 2
Response curve ofx1(t).Figure 3
Response curve ofx2(t).Figure 4
Response curve ofx3(t).Figure 5
The ratio of∫0∞‍zT(t)z(t)dt/∫0∞‍wT(t)w(t)dt.


## 5. Conclusion

This paper addressed theH∞ static output feedback controller design problem for a class of continuous-time affine fuzzy systems. Based on a partition of output-space technique, a new piecewise controller with affine terms via static output feedback is designed. The corresponding design conditions to guarantee the closed-loop system asymptotic stable with H∞ performance are derived in a dilated linear matrix inequality (LMI) characterization. Finally, an example has been given to illustrate the effectiveness of the proposed approach.



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