# Proportional PDC Design-Based Robust Stabilization and Tracking Control Strategies for Uncertain and Disturbed T-S Model

Authors: Chekib Ghorbel; Amira Tiga; Naceur Benhadj Braiek
Journal: Complexity
 (2020)

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## Abstract

This paper presents a proportional parallel distributed compensation (PPDC) design to the robust stabilization and tracking control of the nonlinear dynamic system, which is described by the uncertain and perturbed Takagi–Sugeno (T-S) fuzzy model. The proposed PPDC control design can greatly reduce the number of adjustable parameters involved in the original PDC and separate them from the feedback gain. Furthermore, the process of finding the common quadratic Lyapunov matrix is simplified. Then, the global asymptotic stability with decay rate and disturbance attenuation of the closed-loop T-S model affected by uncertainties and external disturbances are discussed using theH∞ synthesis and linear matrix inequality (LMI) tools. Finally, to illustrate the merit of our purpose, numerical simulation studies of stabilizing and tracking an inverted pendulum system are presented.


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## Body

## 1. Introduction

During the last decade, the fuzzy logic control has attracted rapidly growing attention from both the academic and industrial communities [1–3]. Specially, the Takagi–Sugeno (T-S) fuzzy model has been extensively used to investigate nonlinear control systems [4, 5]. This model is described by a set of fuzzy If-then rules with fuzzy sets in the antecedents and linear dynamics models in the consequent [6–8]. The overall model of the complex system is achieved by fuzzy interpolating these linear models through nonlinear fuzzy membership functions. Moreover, the stability study of this class of systems has been usually based on the use of the Lyapunov direct method. The obtained stability conditions are in general given in terms of linear matrix inequalities (LMI), which can efficiently be solved by convex programming techniques [9–11]. The overall controller of the T-S fuzzy model used the parallel distributed compensation (PDC) approach which used multiple linear state feedback controllers corresponding to the local models via fuzzy rules [12, 13].Throughout this work, the PPDC control scheme is employed to design the fuzzy controllers from the T-S fuzzy model. On the one hand, most of the results obtained have focused on the stabilization and the tracking problems by using the normal PDC approach and very few of them are concerned with the stability problem of satisfying the decay rate and the disturbance attenuation problem [14–17]. On the other hand, for r rules, the total number of unknown parameters is reduced to r+mn, compared with the normal PDC rmn where r, m, and n are the numbers of the rules, the inputs, and the state variables, respectively [18].In addition, uncertainties, unknown parameters, and external disturbances are frequently a source of instability and encountered in various complex systems [19–21]. In particular, the design of the robust fuzzy control of the T-S fuzzy model has received considerable interest in the literature [3, 22–24]. Based on the PDC approach, some researchers treated the stability analysis, stabilization, and tracking control problems of this class of systems [25–27].In this study,H∞ state feedback synthesis and reference model tracking control schemes are expanded to include nonlinear systems described as uncertain and disturbed the T-S fuzzy model by using the PPDC approach. However, we consider both the stability problem of satisfying the decay rate and the disturbance attenuation. Hence, we obtain sufficient conditions, expressed in LMI terms, for the existence of robust fuzzy controllers. The main contributions of this paper are summarized as follows:(i)
Compared with the existing results obtained with the normal PDC approach [9, 13, 22, 23, 28–31], our control design is carried out based on the T-S fuzzy model via the PPDC scheme, which can significantly reduce the number of parameters in PDC.(ii)
In the proposed PPDC control design, the proportional coefficients are first assigned. Then, the common quadratic Lyapunov matrix and the feedback gains are directly obtained from the LMI constraints that consider not only stability but also other control performances such as speed of response, attenuation of the disturbances effect, and structured uncertainties. As a result, the solution of the LMI is more flexible and the controller design is more feasible.This paper is organized as follows: Section2 introduces the uncertain T-S fuzzy model and the proportional PDC-based robust fuzzy H∞ stabilization design scheme. In Section 3, sufficient stability conditions are developed to ensure the stability of the augmented system with a reference model tracking. A simulation example is considered in Section 4 to illustrate the merit of the designed H∞ controllers.


## 2. Proportional PDC-Based Robust FuzzyH∞ Stabilization Design Scheme

We consider a nonlinear affine system:(1)x˙=fx+gxu,where f and g are nonlinear functions of the state x∈ℜn and u∈ℜm is the control input. Then, its fuzzy dynamic model can be described by fuzzy If-then rules which represented local linear input-output relations [7, 8, 32, 33]. The ith rule is of the following form.


### 2.1. Plant Rulei

(2)Ifx1isMi1x1and…andxnisMinxn,thenx˙=A¯ix+B¯1iu+B¯2iwandz=Cix+D1iu+D2iw,for i=1,2,…r, where w∈ℜq is the disturbance input, z∈ℜpis the controlled output, Mij is the fuzzy set, and r is the number of If-then rules.A¯i∈ℜn×n, B¯1i∈ℜn×m, B¯2i∈ℜn×q, Ci∈ℜp×n, D1i∈ℜp×m, and D2i∈ℜp×q are real matrices verifying(3)A¯i=Ai+ΔAit,B¯1i=B1i+ΔB1it,B¯2i=B2i+ΔB2it,in which ΔAit, ΔB1it, and ΔB2it represent the time-varying parameter uncertainties defined as follows:(4)ΔAitΔB1itΔB2it=HiFtE1iE2iE3i,where Hi, E1i, E2i, and E3i are known constant real matrices of appropriate dimensions. Ft is a matrix function, which is bounded by: FTtFt≤I where I is the matrix identity of an appropriate dimension.The resulting statex˙ and the final output z of the T-S fuzzy model are inferred by using the center of gravity method for defuzzification as follows:(5)x˙=∑i=1rhixA¯ix+B¯1iu+B¯2iw,z=∑i=1rhixCix+D1iu+D2iw,where hi=μix/∑i=1rμix and μi=∏j=1nMijxj, for i=1,2,…,r.It should be noted that(6)∑i=1rhix=1,hix≥0.For the fuzzy controller design, we supposed that all the states are measurable and the studied system is locally controllable. Then, we applied a proportional PDC-based compensator for each local fuzzy model (2) as follows.


### 2.2. Control Rulei

(7)Ifx1isMi1x1and…andxnisMinxn,thenu=−kiKx,where K∈ℜm×n is the local state feedback matrix and ki, for i=1,2,…,r, represent the proportional adjustable coefficients which differ with different control rules.Then, the overall fuzzy controlleru is defined by(8)u=−∑i=1rhixkiKx.Substituting controller (8) into the T-S fuzzy system given by (5), the final closed-loop model is described by(9)x˙=∑i=1r∑j=1rhixhjxG¯ijx+B¯2iw,z=∑i=1r∑j=1rhixhjxFijx+D2iw,where G¯ij=Gij+ΔGijt, Gij=Ai−kjB1iK, ΔGijt=ΔAit−kjΔB1itK, and Fij=Ci−kjD1iK.Consequently, we presented in Theorem1 sufficient stability conditions that guarantee the stability of the considered T-S model (9) and achieve a prescribed level of disturbance attenuation γ for all admissible uncertainties such that(10)Tzw∞=supw2≠0z2w2<γ,where z2=∫0∞zTzdt.Theorem 1.
The equilibrium of the closed-loop fuzzy model (9) described by using the proportional PDC controller (8) is globally asymptotically stable with decay rate α and satisfying the performance objective (10) if there exists a common positive definite matrix X, a matrix R, and positive constants τ1∼τ3 verifying the LMI formulation:(11)maximizeX,R,γαsubject to:λ11∗∗∗∗B2iT−γ2I+τ1E3iTE3i∗∗∗CiX−kjD1iRD2i−I∗∗E1iX00−τ2−1I∗kjE2iR000−τ3−1I<0,where λ11=AiX−kjB1iR+∗+2αX+τ1−1+τ2−1+τ3−1HiHiT.
Furthermore, the common state feedback matrixK, shown in (8), is given by(12)K=RP.Proof 1.
In order to prove Theorem1, we used the quadratic Lyapunov function given by(13)Vx=xTPx,and verifying the control performance(14)V˙x≤−2αVx,where P=PT>0 and α>0 is the largest Lyapunov exponent.Additionally, the stability of the closed-loop T-S fuzzy model is satisfied under theH∞ performance, given in (10), with the attenuation index γ if(15)V˙x+2αVx+zTz−γ2wTw<0.According to equations (9), (13), and (14), the development of the above matrix inequality leads to(16)∑i=1r∑j=1rhixhjxxwTϖijxw<0,where xw=xw and ϖij=PG¯ij+∗+2αP+FijTFij∗B¯2iTP+D2iTFijD2iTD2i−γ2I.As allhix∈0,1, condition (16) gives ϖij<0. Then, by denoting X=P−1, pre- and postmultiplying to it by the positive definite matrix diagX,I, respectively, and using the Schur complement, see Appendix A, we obtain(17)G¯ijX+∗+2αX∗∗B¯2iT−γ2I∗FijXD2i−I<0.It is clear that the matrix inequality (17) contains certain Ψij and uncertain ΔΨij parts. Thereby, it can be transformed into the following form:(18)Ψij+ΔΨijt<0,with Ψij=GijX+∗+2αX∗∗B2iT−γ2I∗FijXD2i−I and ΔΨijt=ΔGijtX+∗∗∗ΔB2iTt0∗000.Using the uncertainties defined in (3), ΔΨij becomes(19)ΔΨijt=HiFtE1i−kjE2iK+∗∗∗E3iTFTtHiT0∗000.It is worth pointing out thatΔΨij contained antidiagonal terms. However, to transform them into diagonal terms, we used the appropriate lemma, as presented in Appendix B. By denoting R=KX, it follows that(20)ΔΨijt≤η11∗∗0τ1E3iTE3i∗000,where η11=τ1−1+τ2−1+τ3−1HiHiT+τ2XE1iTE1iX+τ3kj2RTE2iTE2iR.Furthermore, according toΨij and (20), the matrix inequality (18) can be transformed as(21)GijX+∗+2αX+η11∗∗B2iT−γ2I+τ1E3iTE3i∗FijXD2i−I<0.After some manipulations using the Schur complement, we complete the proof and we get an optimization problem involving LMI, as illustrated in (11).


## 2.1. Plant Rulei

(2)Ifx1isMi1x1and…andxnisMinxn,thenx˙=A¯ix+B¯1iu+B¯2iwandz=Cix+D1iu+D2iw,for i=1,2,…r, where w∈ℜq is the disturbance input, z∈ℜpis the controlled output, Mij is the fuzzy set, and r is the number of If-then rules.A¯i∈ℜn×n, B¯1i∈ℜn×m, B¯2i∈ℜn×q, Ci∈ℜp×n, D1i∈ℜp×m, and D2i∈ℜp×q are real matrices verifying(3)A¯i=Ai+ΔAit,B¯1i=B1i+ΔB1it,B¯2i=B2i+ΔB2it,in which ΔAit, ΔB1it, and ΔB2it represent the time-varying parameter uncertainties defined as follows:(4)ΔAitΔB1itΔB2it=HiFtE1iE2iE3i,where Hi, E1i, E2i, and E3i are known constant real matrices of appropriate dimensions. Ft is a matrix function, which is bounded by: FTtFt≤I where I is the matrix identity of an appropriate dimension.The resulting statex˙ and the final output z of the T-S fuzzy model are inferred by using the center of gravity method for defuzzification as follows:(5)x˙=∑i=1rhixA¯ix+B¯1iu+B¯2iw,z=∑i=1rhixCix+D1iu+D2iw,where hi=μix/∑i=1rμix and μi=∏j=1nMijxj, for i=1,2,…,r.It should be noted that(6)∑i=1rhix=1,hix≥0.For the fuzzy controller design, we supposed that all the states are measurable and the studied system is locally controllable. Then, we applied a proportional PDC-based compensator for each local fuzzy model (2) as follows.


## 2.2. Control Rulei

(7)Ifx1isMi1x1and…andxnisMinxn,thenu=−kiKx,where K∈ℜm×n is the local state feedback matrix and ki, for i=1,2,…,r, represent the proportional adjustable coefficients which differ with different control rules.Then, the overall fuzzy controlleru is defined by(8)u=−∑i=1rhixkiKx.Substituting controller (8) into the T-S fuzzy system given by (5), the final closed-loop model is described by(9)x˙=∑i=1r∑j=1rhixhjxG¯ijx+B¯2iw,z=∑i=1r∑j=1rhixhjxFijx+D2iw,where G¯ij=Gij+ΔGijt, Gij=Ai−kjB1iK, ΔGijt=ΔAit−kjΔB1itK, and Fij=Ci−kjD1iK.Consequently, we presented in Theorem1 sufficient stability conditions that guarantee the stability of the considered T-S model (9) and achieve a prescribed level of disturbance attenuation γ for all admissible uncertainties such that(10)Tzw∞=supw2≠0z2w2<γ,where z2=∫0∞zTzdt.Theorem 1.
The equilibrium of the closed-loop fuzzy model (9) described by using the proportional PDC controller (8) is globally asymptotically stable with decay rate α and satisfying the performance objective (10) if there exists a common positive definite matrix X, a matrix R, and positive constants τ1∼τ3 verifying the LMI formulation:(11)maximizeX,R,γαsubject to:λ11∗∗∗∗B2iT−γ2I+τ1E3iTE3i∗∗∗CiX−kjD1iRD2i−I∗∗E1iX00−τ2−1I∗kjE2iR000−τ3−1I<0,where λ11=AiX−kjB1iR+∗+2αX+τ1−1+τ2−1+τ3−1HiHiT.
Furthermore, the common state feedback matrixK, shown in (8), is given by(12)K=RP.Proof 1.
In order to prove Theorem1, we used the quadratic Lyapunov function given by(13)Vx=xTPx,and verifying the control performance(14)V˙x≤−2αVx,where P=PT>0 and α>0 is the largest Lyapunov exponent.Additionally, the stability of the closed-loop T-S fuzzy model is satisfied under theH∞ performance, given in (10), with the attenuation index γ if(15)V˙x+2αVx+zTz−γ2wTw<0.According to equations (9), (13), and (14), the development of the above matrix inequality leads to(16)∑i=1r∑j=1rhixhjxxwTϖijxw<0,where xw=xw and ϖij=PG¯ij+∗+2αP+FijTFij∗B¯2iTP+D2iTFijD2iTD2i−γ2I.As allhix∈0,1, condition (16) gives ϖij<0. Then, by denoting X=P−1, pre- and postmultiplying to it by the positive definite matrix diagX,I, respectively, and using the Schur complement, see Appendix A, we obtain(17)G¯ijX+∗+2αX∗∗B¯2iT−γ2I∗FijXD2i−I<0.It is clear that the matrix inequality (17) contains certain Ψij and uncertain ΔΨij parts. Thereby, it can be transformed into the following form:(18)Ψij+ΔΨijt<0,with Ψij=GijX+∗+2αX∗∗B2iT−γ2I∗FijXD2i−I and ΔΨijt=ΔGijtX+∗∗∗ΔB2iTt0∗000.Using the uncertainties defined in (3), ΔΨij becomes(19)ΔΨijt=HiFtE1i−kjE2iK+∗∗∗E3iTFTtHiT0∗000.It is worth pointing out thatΔΨij contained antidiagonal terms. However, to transform them into diagonal terms, we used the appropriate lemma, as presented in Appendix B. By denoting R=KX, it follows that(20)ΔΨijt≤η11∗∗0τ1E3iTE3i∗000,where η11=τ1−1+τ2−1+τ3−1HiHiT+τ2XE1iTE1iX+τ3kj2RTE2iTE2iR.Furthermore, according toΨij and (20), the matrix inequality (18) can be transformed as(21)GijX+∗+2αX+η11∗∗B2iT−γ2I+τ1E3iTE3i∗FijXD2i−I<0.After some manipulations using the Schur complement, we complete the proof and we get an optimization problem involving LMI, as illustrated in (11).


## 3. Design of Robust FuzzyH∞ Tracking Control

In this section, we treated the robust fuzzyH∞ tracking problem for the considered global model (5) to the following reference model [11, 13, 30]:(22)x˙e=Aexe+Bee,where Ae∈ℜn×n is an asymptotically stable matrix, Be∈ℜn is an input matrix, xe∈ℜn is the state of the reference model, and e∈ℜ is a bounded reference input. We applied then the following local control law.


### 3.1. Control Rulei

(23)Ifx1isMi1x1and…andxnisMinxn,thenu=−liLε,where ε=xe−x is the tracking error. Then, the final fuzzy controller u is defined as(24)u=−∑i=1rhixliLε.Substituting controller (24) into the state x of the T-S model (5), we obtained the following augmented model:(25)x˙a=∑i=1r∑j=1rhixhjxϒ¯ijxa+Φ¯iφ,where xa=εxe, φ=we, ϒ¯ij=ϒij+Δϒijt, Φ¯i=Φi+ΔΦit, ϒij=Ai+ljB1iLAe−Ai0Ae, Δϒijt=ΔAit+ljΔB1itL−ΔAit0Ae, Φi=−B2iBe0Be, and ΔΦit=−ΔB2it000.Consequently, we presented in Theorem2 sufficient stability conditions in terms of LMI that guarantee the stability of the augmented T-S model (25) and ensure a good H∞ tracking performance as(26)Txaφ∞=supφ2≠0xa2φ2<γ.Theorem 2.
The equilibrium of the augmented closed-loop fuzzy model (25) is globally asymptotically stable, and the H∞ tracking control performance, shown in (26), is guaranteed if there exist symmetric positive definite matrices X1 and P2, a matrix V, and positive constants γ and μ1∼μ4 verifying the LMI formulation:(27)maximizeX1,P2,V,γαsubject to:θ11∗∗∗∗∗∗AeT−AiTθ22∗∗∗∗∗−B2iT0−γ2I+μ2E3iTE3i∗∗∗∗BeTBeTP20−γ2I∗∗∗X1000−Q−1∗∗E1iX10000−μ3−1I∗ljE2iV00000−μ4−1I<0,with(i)
θ11=AiX1+ljB1iV+∗+2αX1+μ1−1+μ2−1+μ3−1+μ4−1HiHiT,(ii)
θ22=P2Ae+∗+2αP2+μ1E1iTE1i.
Furthermore, the common feedback matrixL, shown in (24), is given by(28)L=VP1.Proof 2.
In order to prove Theorem2, we considered the following candidate Lyapunov function:(29)Vxa=xaTPxa,and verifying the control performance(30)V˙xa≤−2αVxa,where P=PT>0 and α>0.Then, the corresponding closed-loop model (25) is globally asymptotically stable and the decay rate is at least α if(31)V˙xa+2αV˙xa+xaTQ¯xa−γ2φTφ<0,where Q¯=Q000. The control problem is to minimize(32)∫0∞εTQε≤γ2∫0∞φTφ.Using (25), (29), and (30), the condition above becomes(33)∑i=1r∑j=1rhixhjxxaTΩijxa<0,where Ωij=Pϒ¯ij+∗+2αP+Q¯∗Φ¯iTP−γ2I.As allhix∈0,1, the matrix inequality (33) gives Ωij<0. Then, by assuming that P=diagP1P2, we obtain(34)Λ11+ΔΛ11t∗∗∗AeT−A¯iTP1P2Ae+∗+2αP2∗∗−B¯2iTP20−γ2I∗BeTP2BeTP20−γ2I<0,with(i)
Λ11=P1Ai+ljB1iL+∗+2αP1+Q,(ii)
ΔΛ11t=P1ΔAit+ljΔB1itL+∗.Thereafter, the inequality matrix (34) can be transformed into the following form:(35)Σij+ΔΣijt<0,with(i)
Σij=Λ11∗∗∗AeT−A¯iTP1P2Ae+∗+2αP2∗∗−B2iTP20−γ2I∗BeTP2BeTP20−γ2I,(ii)
ΔΣijt=ΔΛ11t∗∗∗−ΔAiTtP10∗∗−ΔB2iTtP100∗0000.Using the uncertainties defined in (3), the matrix ΔΣijt can be rewritten as(36)ΔΣijt=P1HiFtE1i+ljE2iL+∗∗∗∗−E1iTFTtHiTP10∗∗−E3iTFTtHiTP100∗0000.By using the lemma, as presented in AppendixB, ΔΣij is increased as follows:(37)ΔΣijt≤ξ11∗∗∗0μ1E1iTE1i∗∗00μ2E3iTE3i∗0000,where ξ11=μ1−1+μ2−1+μ3−1+μ4−1P1HiHiTP1+μ3E1iTE1i+μ4lj2LTE2iTE2iL.Therefore, according toΣij and (37), the matrix inequality (35) can be transformed as(38)Λ11+ξ11∗∗∗AeT−A¯iTP1P2Ae+∗+2αP2+μ1E1iTE1i∗∗−B2iTP20−γ2I+μ2E3iTE3i∗BeTP2BeTP20−γ2I<0.By denotingX1=P1−1 and V=LX1 and pre- and postmultiplying to (38) by the positive definite matrix diagX1,I,I,I, respectively, we obtain(39)υ11∗∗∗AeT−A¯iTP1P2Ae+∗+2αP2+μ1E1iTE1i∗∗−B2iTP20−γ2I+μ2E3iTE3i∗BeTP2BeTP20−γ2I<0,where υ11=AiX1+ljB1iV+∗+2αX1+X1QX1+μ1−1+μ2−1+μ3−1+μ4−1HiHiT+μ3X1E1iTE1iX1+μ4lj2VTE2iTE2iV.Using then the Schur complement, we complete the proof and we get an optimization problem involving LMI, as illustrated in (27).Remark.
It should be mentioned that the above results on the design problem for T-S fuzzy models are based on an implicit assumption that the controller will be implemented exactly. However, uncertainties or inaccuracies do occur in the implementation of a designed filter or controller. In recent years, there are considerable studies on the robust nonfragileH∞ filtering problem of T-S fuzzy models [20, 21, 24].


## 3.1. Control Rulei

(23)Ifx1isMi1x1and…andxnisMinxn,thenu=−liLε,where ε=xe−x is the tracking error. Then, the final fuzzy controller u is defined as(24)u=−∑i=1rhixliLε.Substituting controller (24) into the state x of the T-S model (5), we obtained the following augmented model:(25)x˙a=∑i=1r∑j=1rhixhjxϒ¯ijxa+Φ¯iφ,where xa=εxe, φ=we, ϒ¯ij=ϒij+Δϒijt, Φ¯i=Φi+ΔΦit, ϒij=Ai+ljB1iLAe−Ai0Ae, Δϒijt=ΔAit+ljΔB1itL−ΔAit0Ae, Φi=−B2iBe0Be, and ΔΦit=−ΔB2it000.Consequently, we presented in Theorem2 sufficient stability conditions in terms of LMI that guarantee the stability of the augmented T-S model (25) and ensure a good H∞ tracking performance as(26)Txaφ∞=supφ2≠0xa2φ2<γ.Theorem 2.
The equilibrium of the augmented closed-loop fuzzy model (25) is globally asymptotically stable, and the H∞ tracking control performance, shown in (26), is guaranteed if there exist symmetric positive definite matrices X1 and P2, a matrix V, and positive constants γ and μ1∼μ4 verifying the LMI formulation:(27)maximizeX1,P2,V,γαsubject to:θ11∗∗∗∗∗∗AeT−AiTθ22∗∗∗∗∗−B2iT0−γ2I+μ2E3iTE3i∗∗∗∗BeTBeTP20−γ2I∗∗∗X1000−Q−1∗∗E1iX10000−μ3−1I∗ljE2iV00000−μ4−1I<0,with(i)
θ11=AiX1+ljB1iV+∗+2αX1+μ1−1+μ2−1+μ3−1+μ4−1HiHiT,(ii)
θ22=P2Ae+∗+2αP2+μ1E1iTE1i.
Furthermore, the common feedback matrixL, shown in (24), is given by(28)L=VP1.Proof 2.
In order to prove Theorem2, we considered the following candidate Lyapunov function:(29)Vxa=xaTPxa,and verifying the control performance(30)V˙xa≤−2αVxa,where P=PT>0 and α>0.Then, the corresponding closed-loop model (25) is globally asymptotically stable and the decay rate is at least α if(31)V˙xa+2αV˙xa+xaTQ¯xa−γ2φTφ<0,where Q¯=Q000. The control problem is to minimize(32)∫0∞εTQε≤γ2∫0∞φTφ.Using (25), (29), and (30), the condition above becomes(33)∑i=1r∑j=1rhixhjxxaTΩijxa<0,where Ωij=Pϒ¯ij+∗+2αP+Q¯∗Φ¯iTP−γ2I.As allhix∈0,1, the matrix inequality (33) gives Ωij<0. Then, by assuming that P=diagP1P2, we obtain(34)Λ11+ΔΛ11t∗∗∗AeT−A¯iTP1P2Ae+∗+2αP2∗∗−B¯2iTP20−γ2I∗BeTP2BeTP20−γ2I<0,with(i)
Λ11=P1Ai+ljB1iL+∗+2αP1+Q,(ii)
ΔΛ11t=P1ΔAit+ljΔB1itL+∗.Thereafter, the inequality matrix (34) can be transformed into the following form:(35)Σij+ΔΣijt<0,with(i)
Σij=Λ11∗∗∗AeT−A¯iTP1P2Ae+∗+2αP2∗∗−B2iTP20−γ2I∗BeTP2BeTP20−γ2I,(ii)
ΔΣijt=ΔΛ11t∗∗∗−ΔAiTtP10∗∗−ΔB2iTtP100∗0000.Using the uncertainties defined in (3), the matrix ΔΣijt can be rewritten as(36)ΔΣijt=P1HiFtE1i+ljE2iL+∗∗∗∗−E1iTFTtHiTP10∗∗−E3iTFTtHiTP100∗0000.By using the lemma, as presented in AppendixB, ΔΣij is increased as follows:(37)ΔΣijt≤ξ11∗∗∗0μ1E1iTE1i∗∗00μ2E3iTE3i∗0000,where ξ11=μ1−1+μ2−1+μ3−1+μ4−1P1HiHiTP1+μ3E1iTE1i+μ4lj2LTE2iTE2iL.Therefore, according toΣij and (37), the matrix inequality (35) can be transformed as(38)Λ11+ξ11∗∗∗AeT−A¯iTP1P2Ae+∗+2αP2+μ1E1iTE1i∗∗−B2iTP20−γ2I+μ2E3iTE3i∗BeTP2BeTP20−γ2I<0.By denotingX1=P1−1 and V=LX1 and pre- and postmultiplying to (38) by the positive definite matrix diagX1,I,I,I, respectively, we obtain(39)υ11∗∗∗AeT−A¯iTP1P2Ae+∗+2αP2+μ1E1iTE1i∗∗−B2iTP20−γ2I+μ2E3iTE3i∗BeTP2BeTP20−γ2I<0,where υ11=AiX1+ljB1iV+∗+2αX1+X1QX1+μ1−1+μ2−1+μ3−1+μ4−1HiHiT+μ3X1E1iTE1iX1+μ4lj2VTE2iTE2iV.Using then the Schur complement, we complete the proof and we get an optimization problem involving LMI, as illustrated in (27).Remark.
It should be mentioned that the above results on the design problem for T-S fuzzy models are based on an implicit assumption that the controller will be implemented exactly. However, uncertainties or inaccuracies do occur in the implementation of a designed filter or controller. In recent years, there are considerable studies on the robust nonfragileH∞ filtering problem of T-S fuzzy models [20, 21, 24].


## 4. Numerical Simulation

In this section, the proposed control strategies are verified for an inverted pendulum system. Its mathematical model is described by the following nonlinear equations [11, 13, 34]:(40)x˙1=x2,x˙2=m+M−dx2+mglsinx1−m2l2/2x22sin2x1m+MJ+ml2−m2l2cos2x1−mlcosx1m+MJ+ml2−m2l2cos2x1u,where x1=θrad is the angle from the vertical position and x2=θ˙rad/s is the angular velocity, g=9.81m/s2 is the gravity constant, M=2.4kg is the mass of the cart, m=0.23kg is the mass of the pendulum, l=0.36m is the pole length, J=0.099kgm2 is the moment of inertia of the pole, and d=0.005Nms/rad is the pendulum damping coefficient.For simplicity, we suppose thatm2l2cos2x1≪m+MJ+ml2 and we use two rules:Plant rule1:ifx1is about0,thenx˙=A¯1x+B¯11u+B¯21wPlant rule2:ifx1is about±π/3,thenx˙=A¯2x+B¯12u+B¯22wwith(41)A1=01mglJ+ml2−dJ+ml2,B11=0−mlm+MJ+ml2,B21=11,A2=01mglsinπ/3J+ml2−dJ+ml2,B12=0mlcosπ/3m+MJ+ml2,B22=11,ΔA1t=H1FtE11,ΔA2t=H2FtE12,Ft=sint,H1=0−m2l2m+MJ+ml2,H2=0−m2l2sin2π/32m+MJ+ml2,E11=E12=a0,ΔB11=ΔB12=ΔB21=ΔB22=00,a=maxx22=0.05.Using Theorem1 with γ=0.5, τ1=2.9, τ2=1.7, τ3=0.9, k1=0.62, and k2=0.85, we obtain α=1.4106, P=4.16560.75110.75110.1546, R=−151.3583137.6572, and K=−527.0948−92.4086.The responses of the state variablesx1t=θtrad and x2t=θ˙trad/s are shown in Figures 1 and 2, respectively. The evolution of the force FtN is depicted in Figure 3, while the disturbance input signal is given by wt=−0.055≤t≤70otherwise.Figure 1
Response of the state variablex1t=θt.Figure 2
Response of the state variablex2t=θ˙t.Figure 3
Evolution of the control signalFt.It is shown from the simulation results that the designed controlleru=−∑i=12hixkiKx is efficient and guarantees the rapid global stability of the closed-loop fuzzy model (9).In the following, we present a comparative study between the obtained results related to designing robust fuzzy tracking controllers using the proposed PPDC approach and the original PDC as developed in [13].By using Theorem2 with γ=0.5, μ1=2.9, μ2=1.7, μ3=0.9, μ4=0.5, l1=0.47, l2=0.83, Q=0.35000.03, Ae=01−15/4−4, and Be=11, we obtain α=1.2984, P1=5.05721.19141.19140.3065, P2=4.7786−0.1363−0.13634.4632, R=157.6185−69.5819, and L=714.2054166.4582.The designed robust fuzzy controlleru=−∑i=12hixliLε quadratically stabilizes the uncertain and perturbed augmented model (25) and ensures a good H∞ tracking performance, as presented in (26).For comparison, by solving the LMI constraints in [13], we obtain L1=264.520965.0914, L2=264.520965.0914, P1=3.72280.97390.97390.2871, and P2=4.6364−0.9691−0.96913.1186.The responses of the state variablesx1t=θtrad and x2t=θ˙trad/s, the control signal FtN, and the quadratic error tracking are depicted in Figures 4–7, respectively, with as initial conditions x10=0.05 and x20=0. The disturbance input signal is a random number chosen from an uniform distribution on the interval 0,0.1 and given by wt=−0.0530≤t≤500otherwise.Figure 4
Angle tracking trajectories using PDC and PPDC approaches.Figure 5
Angular velocity trajectories using PDC and PPDC approaches.Figure 6
Control signals using PDC and PPDC approaches.Figure 7
Quadratic error tracking using PDC and PPDC approaches.It is observed that in contrast with the normal PDC approach, the PPDC one can rapidly achieve desired responses despite the presence of uncertainties and disturbances. Moreover, magnitudes of the quadratic error tracking using the PDC design are larger than those of PPDC.


## 5. Conclusion

This paper has presented the robust state feedback synthesis and model reference tracking control for the nonlinear dynamic system described as the T-S fuzzy model affected by uncertainties and external disturbances using the proportional PDC design. Its interest is to reduce the number of adjustable parameters in the normal PDC one. Based on the quadratic Lyapunov function with a decay rate, sufficient stability conditions have been obtained using theH∞ criterion and LMI tools that guarantee the stability of the closed-loop T-S model and ensure a good robust performance. Moreover, the solution of finding the common positive definite matrix was simplified and proportional coefficient design was separated from the feedback matrix parameters. Finally, an inverted pendulum system was considered to show the effectiveness of the designed H∞ fuzzy controllers.



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