Robust H2 fuzzy output feedback control for discrete-time nonlinear systems with parametric uncertainties

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Robust H

₂

fuzzy output feedback control

for discrete-time nonlinear systems with

parametric uncertainties

Huai-Ning Wu

School of Automation Science and Electrical Engineering, Beihang University (Beijing University of Aeronautics and Astronautics), No. 37, Xue Yuan Road, Beijing 100083, PR China Received 10 March 2006; received in revised form 19 November 2006; accepted 22 November 2006

Available online 21 December 2006

Abstract

This paper deals with the robust H2fuzzy observer-based control problem for discrete-time

uncer-tain nonlinear systems. The Takagi and Sugeno (T–S) fuzzy model is employed to represent a discrete-time nonlinear system with parametric uncertainties. A fuzzy observer is used to estimate the state of the fuzzy system and a non-parallel distributed compensation (non-PDC) scheme is adopted for the control design. A fuzzy Lyapunov function (FLF) is constructed to derive a suﬃcient condition such that the closed-loop fuzzy system is globally asymptotically stable and an upper bound on the quadratic cost function is provided. A suﬃcient condition for the existence of a robust H2fuzzy observer-based controller is presented in terms of linear matrix inequalities (LMIs).

More-over, by using the existing LMI optimization techniques, a suboptimal fuzzy observer-based control-ler in the sense of minimizing the cost bound is proposed. Finally, an example is given to illustrate the eﬀectiveness of the proposed design method.

Keywords: Discrete-time nonlinear systems; Fuzzy control; H2control; Linear matrix inequality (LMI);

Para-metric uncertainty; State observer

E-mail address:[email protected]

46 (2007) 151–165

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1. Introduction

Recently, fuzzy logic control has attracted increasing attention, because it can provide an effective solution to the control of plants that are complex, uncertain, ill-defined, and have available qualitative knowledge from domain experts for their controllers design. Though the method has been practically successful, it has proved extremely difficult to develop a general analysis and design theory for conventional fuzzy control systems. It is believed that the reason for this is due to the fact that no mathematical model is avail-able for the design of the fuzzy controller.

In recent years, it becomes quite popular to adopt the so-called Takagi–Sugeno (T–S) type fuzzy models[1]to represent or approximate a nonlinear system. This fuzzy model is described by a family of fuzzy IF–THEN rules which represent local linear input–output relations of a nonlinear system. The overall fuzzy model is achieved by smoothly blending these local linear models together through fuzzy membership functions. As a result, the conventional linear system theory can be applied to analysis and synthesis of nonlinear control systems. The stability issue of T–S fuzzy control systems in nonlinear stability frameworks has been studied extensively [2–7]. Recently, many works on T–S fuzzy-model-based control for a nonlinear system were developed to stabilize the T–S fuzzy model with guaranteed H2or/and H1performance[8–12], or multiobjective control per-formance[13]. As a common belief, this fuzzy-model-based control technique is conceptu-ally simple and eﬀective for controlling complex nonlinear systems. However, these results are derived by using asingle Lyapunov function (SLF) method. The main drawback of this method is that an SLF must work for all linear models, which in general leads to con-servative results. To reduce the conservatism of using an SLF, recently, the fuzzy Lyapu-nov functions (FLFs) have been constructed for the control design of T–S fuzzy systems

[14–16].

In many practical systems, there always exist uncertainties which are frequently a source of instability or degraded performance. Thus, recently, the robust control problems have been investigated for uncertain nonlinear systems based on T–S fuzzy model[17–27]. However, most of the existing results are dedicated to the problems of robust stabilization

[17–20]and robust H1control[21–25]. Fewer results are available for the robust H2

con-trol of uncertain T–S fuzzy systems with exceptions [26,27]. The objective of robust H2

control is to design a control system that is not only robust stable but also provides an upper bound of quadratic performance for all admissible uncertainties[28,29]. In [26], a robust H2fuzzy control design is provided for continuous time uncertain nonlinear

sys-tems, where both the state-feedback and observer-based control cases are considered. In

[27], the robust H2fuzzy control problem for the discrete-time uncertain nonlinear systems

is studied, where the state-feedback case is only considered. Moreover, due to the use of an SLF, the results in [17–24,26,27] are usually conservative. It is worth noting that, more recently, an FLF has been employed to deal with the robust H1fuzzy state-feedback con-trol problem for uncertain T–S fuzzy systems in[25].

In this paper, we study the robust H2fuzzy observer-based control problem for

discrete-time uncertain nonlinear systems. The T–S fuzzy model is employed to represent a dis-crete-time nonlinear system with parametric uncertainties. Based on a fuzzy observer, which is used to estimate the state of the fuzzy system, a non-parallel distributed compen-sation (non-PDC) scheme [15] is adopted for the control design of the fuzzy system. An FLF is constructed, which not only includes a fuzzy Lyapunov matrix, but also an

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additional fuzzy slack variable. Based on this FLF, a suﬃcient condition is derived, which can guarantee the stability of the closed-loop fuzzy system and provide an upper bound for the quadratic cost function. From this condition, a suﬃcient condition for the exis-tence of a robust H2fuzzy observer-based controller is presented in the form of a set of

linear matrix inequalities (LMIs). The control matrices and observer gain matrices can be obtained by directly solving this set of LMIs. Moreover, by the existing LMI optimi-zation techniques [30,31], a suboptimal robust H2fuzzy observer-based controller in the

sense of minimizing the cost bound is obtained. Finally, it is also demonstrated, through numerical simulations on a chaotic system, that the proposed design method is eﬀective.

Notations:Throughout the paper,R,Rn,Rnmdenote the sets of real numbers,n com-ponent real vectors, and nm real matrices, respectively. For a symmetric matrix M,

M>ðPÞ0 means that it is positive deﬁnite (or positive semi-deﬁnite). The superscript T represents the transpose. The symbol * denotes the transposed elements in the symmet-ric positions of a matrix.

2. Problem formulation

Consider a discrete-time uncertain nonlinear system which can be described by the fol-lowing T–S fuzzy model:

xðkþ1Þ ¼X r

i¼1

hiðhðkÞÞfðAiþDAiðkÞÞxðkÞ þ ðBiþDBiðkÞÞuðkÞg; xð0Þ ¼x0 ð1Þ zðkÞ ¼X

r

i¼1

hiðhðkÞÞfC1ixðkÞ þD1iuðkÞg ð2Þ

yðkÞ ¼X r

i¼1

hiðhðkÞÞC2ixðkÞ ð3Þ

wherehðkÞ ¼ bh1ðkÞ h2ðkÞ hgðkÞ cis the premise variable vector,ris the number of model rules.hiðhðkÞÞis the normalized weight for theith rule,i2S,f₁;2;. . .;rg, that is, hiðhðkÞÞP0,i2SandPr_i_¼₁hiðhðkÞÞ ¼1 for allk.xðkÞ 2Rnis the state vector,uðkÞ 2Rm is the control input vector,zðkÞ 2Rpis the controlled output vector,yðkÞ 2Rqis the mea-surement output vector,x0is the initial state.Ai,Bi,C1i,C2i, and D1i,i2Sare known

constant matrices that describe the nominal system.DAiðkÞ,DBiðkÞ, i2S represent the time-varying parametric uncertainties having the following structure:

DAiðkÞ DBiðkÞ

½ ¼HFðkÞ½E1i E2i; i2S; ð4Þ

where FðkÞ 2Rab is an unknown matrix function with Lebesgue-measurable elements and satisfying

FTðkÞFðkÞ6_I _ð₅_Þ

andH,E1i,E2i,i2Sare known constant matrices with appropriate dimensions that spec-ify how the uncertain parameters inFðkÞenter the nominal matrices Ai andBi. For the convenience of notations, in the sequel, we will denote

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AðkÞ ¼X r i¼1 hiðhðkÞÞAi; BðkÞ ¼X r i¼1

hiðhðkÞÞBi; E1ðkÞ ¼X r i¼1 hiðhðkÞÞE1i; E2ðkÞ ¼X r i¼1 hiðhðkÞÞE2i; C1ðkÞ ¼X r i¼1 hiðhðkÞÞC1i; D1ðkÞ ¼X r i¼1 hiðhðkÞÞD1i; C2ðkÞ ¼X r i¼1 hiðhðkÞÞC2i:

In this study, it is assumed that all of the premise variables are available for feedback control. As in[3], the following fuzzy observer is employed to deal with the state estima-tion of the system(1)–(3):

^

xðkþ1Þ ¼AðkÞxð^ kÞ þBðkÞuðkÞ þLðkÞðyðkÞ ^yðkÞÞ; ^xð0Þ ¼0 ð6Þ

where ^

yðkÞ ¼C2ðkÞxð^ kÞ; ð7Þ

LðkÞ ¼Pri¼1hiðhðkÞÞLi, and Li2R

nq_, _i₂_S _{are the observer gain matrices to be}

determined.

As in[15], the following non-PDC fuzzy controller is proposed for the system(1)–(3):

uðkÞ ¼KðkÞG1

1 ðkÞ^xðkÞ; ð8Þ

whereKðkÞ ¼Pr_i_¼1hiðhðkÞÞKi,G1ðkÞ ¼

Pr

i¼1hiðhðkÞÞG1i, andKi2R mn_,

G1i2Rnn,i2S are the control matrices to be determined.

Denote the estimation error aseðkÞ ¼xðkÞ ^xðkÞ. From(1)–(3)and(6)–(8), the follow-ing augmented closed-loop fuzzy system can be obtained:

nðkþ1Þ ¼ðAaðkÞ þHaFðkÞEaðkÞÞnðkÞ; nð0Þ ¼n0 ð9Þ

zðkÞ ¼CaðkÞnðkÞ ð10Þ wherenðkÞ ¼ xðkÞ eðkÞ ,n0¼ x0 x0 , and AaðkÞ ¼ AðkÞ þBðkÞKðkÞG 1 1 ðkÞ BðkÞKðkÞG 1 1 ðkÞ 0 AðkÞ LðkÞC2ðkÞ " # ; Ha¼ H H ;

EaðkÞ ¼ E1ðkÞ þE2ðkÞKðkÞG1

1 ðkÞ E2ðkÞKðkÞG 1 1 ðkÞ ; CaðkÞ ¼ C1ðkÞ þD1ðkÞKðkÞG1 1 ðkÞ D1ðkÞKðkÞG11ðkÞ : In this paper, we consider the following quadratic cost function:

J¼X 1

k¼0

zTðkÞzðkÞ: ð11Þ

The objective of this paper is to seek a fuzzy observer-based controller of the form(8)

and determine a cost boundJb<þ1as small as possible such that the closed-loop fuzzy

system(9) and (10)is globally asymptotically stable and the cost function deﬁned by(11)

satisﬁesJ 6_J_b_{for all admissible parametric uncertainties.}

Before concluding this section, we recall a matrix inequality that will be used in next section.

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Lemma 1 [32]. LetA;H;E, andFbe real matrices of appropriate dimensions withFTF6_I_.

For any matrix P>0 and scalar e>0 such that PeHHT>0, then the following inequality holds:

ðAþHFEÞTP1ðAþHFEÞ6_AT_ðP_eHHT_Þ1_A_þ_e1_ET_E:

3. Robust H2fuzzy observer-based control design

In this section, we ﬁrst discuss a suﬃcient condition that not only guarantees the stabil-ity of the closed-loop fuzzy system(9) and (10), but also provides an upper bound for the cost function(11). And then, a design method for a robust H2fuzzy observer-based

con-troller is presented.

For convenience, we denote ~

PðkÞ,_GT_ð_k_ÞPð_k_ÞG1_ð_k_Þ_; _ð₁₂_Þ where GðkÞ ¼Pr_i_¼₁hiðhðkÞÞGi is non-singular, PðkÞ ¼Pr_i_¼₁hiðhðkÞÞPi>0, and Gi;Pi2 R2n2n. Let us consider the following FLF candidate[15]:

VðnðkÞÞ ¼nTðkÞPð~ kÞnðkÞ: ð13Þ

Remark 1. Observe that the FLF candidate (13) includes not only a fuzzy Lyapunov matrix PðkÞ, but also an additional fuzzy slack variable GðkÞ. Moreover, it has been shown in[15]that(13)is indeed a candidate Lyapunov function.

Along the trajectory of system(9), we have

DV ,_V_ðnð_k_þ₁_ÞÞ_V_ðnð_k_{ÞÞ ¼}_nT_ð_k_þ₁_Þ_Pð~ _k_þ₁_Þnð_k_þ₁_Þ_nT_ð_k_Þ_Pð~ _k_Þnð_k_Þ

¼nTðkÞnðkÞ ð14Þ

where

,½AaðkÞ þHaFðkÞEaðkÞT_Pð~ _k_þ₁_Þ½Aað_k_{Þ þ}_HaFð_k_ÞEað_k_Þ_Pð~ _k_Þ_: From(10) and (14), we get

DV þzTðkÞzðkÞ ¼nTðkÞ½þCT_aðkÞCaðkÞnðkÞ: ð15Þ

Obviously, if the following inequality holds:

þCTaðkÞCaðkÞ<0; ð16Þ

then we have

DV þzT_ð_k_Þzð_k_Þ₆₀_: _ð₁₇_Þ

Therefore, we have the following theorem which presents a suﬃcient condition such that the closed-loop fuzzy system (9) and (10) is globally asymptotically stable and an upper bound is provided for the cost function(11).

Theorem 1. Consider the closed-loop fuzzy system(9) and (10). If there exist a scalare>0, a time-varying matrixPðkÞ>0and a non-singular time-varying matrixGðkÞsuch that

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PðkÞ AaðkÞGðkÞ Gðkþ1Þ GT_ð_k_þ₁_{Þ þ}_Pð_k_þ₁_Þ 0 eHT_a eI EaðkÞGðkÞ 0 0 eI CaðkÞGðkÞ 0 0 0 I 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 <0; ð18Þ

then the system is globally asymptotically stable and the cost function(11)satisfies

J 6_nT

0G T_ð

0ÞPð0ÞG1_ð

0Þn0 ð19Þ

for all admissible parametric uncertainties.

Proof. From the fact ½Pðkþ1Þ Gðkþ1ÞP1_ð_k_þ₁_Þ½Pð_k_þ₁_Þ_Gð_k_þ₁_ÞT

P0, we have _P~1_ð_k_þ 1Þ ¼ Gðkþ1ÞP1_ð_k_þ 1ÞGT_ð_k_þ 1Þ6Gðkþ1Þ GTðkþ1Þ þPðkþ1Þ: ð20Þ

Thus, it follows from(18) and (20)that

PðkÞ AaðkÞGðkÞ P~1_ð_k_þ₁_Þ 0 eHT a eI EaðkÞGðkÞ 0 0 eI CaðkÞGðkÞ 0 0 0 I 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 <0: ð21Þ

Deﬁne matrixT,_diag _fG1_ðkÞ;I;I;I;Ig. Pre- and post-multiplying(21)byTTandT, respectively, and considering(12), yield

Pð~ kÞ AaðkÞ P~1_ð_k_þ₁_Þ 0 eHT_a eI EaðkÞ 0 0 eI CaðkÞ 0 0 0 I 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 <0: ð22Þ

By Schur complement, it follows that(22)is equivalent to

Pð~ kÞ þe1 ETaðkÞEaðkÞ þC T aðkÞCaðkÞ AaðkÞ P~1_ð_k_þ₁_{Þ þ}_eHaHT a " # <0 ð23Þ which is equivalent to ~ P1ðkþ1Þ eHaHT a >0 ð24Þ and AT_aðkÞ _P~1_ð_k_þ 1Þ eHaHT_a 1 AaðkÞ _Pð~ _k_{Þ þ}_e1 ET_aðkÞEaðkÞ þCT_aðkÞCaðkÞ<0: ð25Þ

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ByLemma 1, we have

AaðkÞ þHaFðkÞEaðkÞ

½ T_Pð~ _k_þ₁_Þ_½_Aað_k_{Þ þ}_HaFð_k_ÞEað_k_Þ

6_AT aðkÞ P~ 1_ð_k_þ₁_Þ_eHaHT a 1 AaðkÞ þe1 ETaðkÞEaðkÞ: ð26Þ

It follows from(25) and (26)that(16)holds, and thus(17)is fulﬁlled. It is clear from(14) and (16)thatDV <0 for anynðkÞ 6¼0. Hence, the system(9) and (10)is globally asymp-totically stable. In this case, Vðnð1ÞÞ ¼0. Summing the inequality (17) from k= 0 to

k=1, we have

Vðnð1ÞÞ Vðnð0ÞÞ ¼ nT0Pð~ 0Þn06

X1 k¼0

zT_ð_k_Þzð_k_Þ_: _ð₂₇_Þ

Thus, from(27), we have(19). h Let Pibe partitioned as

Pi¼ P1i P2i P3i

where P1i>0; P2i; and P3i>0 are nn: ð28Þ

Moreover, to obtain a LMI formulation, we letGibe the following form:

Gi¼ G1i G 1 2

0 G₂1

" #

where G1i and G2 are nn and non-singular: ð29Þ

Then, based onTheorem 1, the following result can be obtained:

Theorem 2. Consider the fuzzy system (1)–(3). For a given scalar e>0, if there exist matricesG2,G1i,P1i>0,Q2i2R

nn

,Q3i>02R nn

,Ki, andSi2Rnq,i2Ssatisfying

the following LMIs:

Ulii<0; l;i2S; ð30Þ 1 r1Uliiþ 1 2 UlijþUlji <0; l;i;j2S; i6¼j; ð31Þ where Ulij, P1i Q2i Q3i AiG1jþBiKj Ai G1lGT1lþP1l 0 GT₂AiSiC2j IþQ2l G T 2G2þQ3l 0 0 eHT eHTG2 eI E1iG1jþE2iKj E1i 0 0 0 eI C1iG1jþD1iKj C1i 0 0 0 0 I 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 ;

then there exists a fuzzy observer-based controller of the form (8)with the observer gain matrices

Li¼G₂TSi; i2S ð32Þ

such that the closed-loop fuzzy system(9) and (10)is globally asymptotically stable and the cost function(11)satisfies(19)with

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Pð0Þ ¼ P1ð0Þ G₂TQ2ð0Þ G₂TQ3ð0ÞG21 " # ; Gð0Þ ¼ G1ð0Þ G 1 2 0 G₂1 " # ð33Þ

for all admissible parametric uncertainties.

Proof. It is clear from(30)thatG1lþGT_1l>P1l>0 andGT₂ þG2>Q3l>0,l2S. Thus,

we haveG1ðkÞ þGT

1ðkÞ>0 and G T

2þG2>0 which ensure the non-singularity of G1ðkÞ

andG2. From (28) and (29), we have

PðkÞ ¼ P1ðkÞ P2ðkÞ P3ðkÞ " # ; and GðkÞ ¼ G1ðkÞ G 1 2 0 G21 " # is non-singular.

Hence,(18)can be written as

P1ðkÞ P2ðkÞ P3ðkÞ N31ðkÞ AðkÞG21 N33ðkþ1Þ 0 ALðkÞG1 2 G T 2 þP2ðkþ1Þ N44ðkþ1Þ 0 0 eHT eHT eI N61ðkÞ E1ðkÞG1 2 0 0 0 eI N71ðkÞ C1ðkÞG1 2 0 0 0 0 I 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 <0 ð34Þ where

N31ðkÞ,_Að_k_ÞG1ð_k_{Þ þ}_Bð_k_ÞKð_k_Þ_; _N61ð_k_Þ,_E1ð_k_ÞG1ð_k_{Þ þ}_E2ð_k_ÞKð_k_Þ_;

N71ðkÞ,_C_1ð_k_ÞG1ð_k_{Þ þ}_D1ð_k_ÞKð_k_Þ_; _ALð_k_Þ,_Að_k_Þ_Lð_k_ÞC2ð_k_Þ_; N33ðkþ1Þ,G1ðkþ1Þ GT 1ðkþ1Þ þP1ðkþ1Þ; N44ðkþ1Þ, _G1 2 G T 2 þP3ðkþ1Þ:

Deﬁne matrixT1 ,_diag_fI_;_G2_;_I_;_G2_;_I_;_I_;_Ig_{. Pre- and post-multiplying}₍₃₄₎_by_TT

1 andT1,

respectively, and letting

Q2i¼G T 2P2i; Q3i¼G T 2P3iG2; i2S; ð35Þ we get P1ðkÞ Q2ðkÞ Q3ðkÞ N31ðkÞ AðkÞ N33ðkþ1Þ 0 GT2ALðkÞ IþQ2ðkþ1Þ C44ðkþ1Þ 0 0 eHT eHTG2 eI N61ðkÞ E1ðkÞ 0 0 0 eI N71ðkÞ C1ðkÞ 0 0 0 0 I 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 <0 ð36Þ

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whereQ2ðkÞ ¼Pr_i_¼1hiðhðkÞÞQ2i,Q3ðkÞ ¼ Pr i¼1hiðhðkÞÞQ3i, and C44ðkþ1Þ ¼ GT 2 G2þQ3ðkþ1Þ: Letting Si¼GT₂Li; i2S; ð37Þ

then(36)can be written as

Xr l¼1 hlðhðkþ1ÞÞX r i¼1 Xr j¼1 hiðhðkÞÞhjðhðkÞÞUlij<0: ð38Þ

By using Theorem 2.2 in[5], if the conditions(30) and (31)hold, then(38)is fulﬁlled. From

(35), we have P2i¼G₂TQ2i; P3i¼G T 2 Q3iG 1 2 ; i2S: ð39Þ

Moreover, it follows from(30)that

P1i

Q_2i Q_3i

>0; i2S: ð40Þ

Pre- and post-multiplying(40)by diagfI;G2Tgand diagfI;G 1

2 g, respectively, and

consid-ering(39), we have P1i P2i P3i ¼ P1i G₂TQ2i G T 2 Q3iG 1 2 >0; i2S: ð41Þ

This implies that there exist a matrixPðkÞ ¼ P1ðkÞ G₂TQ2ðkÞ G₂TQ3ðkÞG1 2 >0 and a non-singular matrixGðkÞ ¼ G1ðkÞ G 1 2 0 G₂1

satisfying(18). Therefore, it follows fromTheorem 1that the closed-loop fuzzy system (9) and (10)is globally asymptotically stable and the cost function(11)satisﬁes(19)with(33)for all admissible parametric uncertainties. Fur-thermore, by(37), we have(32). h

Theorem 2 provides an LMI-based condition for the existence of a robust H2 fuzzy

observer-based controller for the fuzzy system(1)–(3). Different from the two-stage proce-dure of[12,20,24,26], where the resolution needs to separate the inequalities into two sets, one for the observer gain matrices, the second for the control gain matrices, the method proposed in this paper can give the control matrices and observer gain matrices of the fuzzy controller(8)by directly solving the LMIs of(30) and (31)for a fixede> 0. More-over, the resulting fuzzy controller can guarantee that the cost function(11)satisfies(19)

with(33)for all admissible uncertainties.

Remark 2. Notice that in the proof ofTheorem 2, we have to face the following problem: Find the less conservative sufficient conditions so that the inequality (38) is fulfilled. A trivial solution would be for a fixed l:Ulii<0, UlijþUlji<0, l;i;j2S, j>i [3]. However, it has been shown that this result is too conservative [4–7,11]. To outperform this result, several relaxation techniques have been proposed, for example[4–7,11]. These

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techniques can be applied to this study straightforwardly. However, the methods in

[4,6,7,11] must introduce a huge number of additional variables, which may be not compatible with the actual LMI solvers. Thus, we prefer adopting the relaxation technique without additional variables in[5]to give the less conservative sufﬁcient conditions for(38)

in this study.

To minimize the upper bound on the cost function(11), we seek to minimize the follow-ing upper bound ofJb ,nT0G

T_ð₀_ÞPð₀_ÞG1_ð₀_Þn 0: Jb6nT0Vn0 ð42Þ whereV , V1 V2 V3 ,Vi2Rnn,i¼1;2;3 satisfying GTð0ÞPð0ÞG1_ð 0Þ<V: ð43Þ

Remark 3. Note that the upper bound of the cost function(14)given by(42)depends on the initial conditionx0. To avoid this dependence, we will assume thatx0 is a zero mean

random variable satisfying Efx0xT0g ¼I where Efg denotes the expectation operator.

Then, from(19) and (42), we have the following cost bound:

J ,_E_f_J_g6_E_f_J_bg6 _trace V1 V2 V3 2 4 3 5 Efx0x T 0g Efx0xT 0g Efx0x T 0g 2 4 3 5 8 < : 9 = ; ¼ trace fV1þV2þVT2þV3g: ð44Þ

By Schur complement, it follows that(43)is equivalent to

V

I Gð0ÞP1_ð₀_ÞGT_ð₀_Þ

<0: ð45Þ

From the factPð0Þ Gð0ÞP1_ð₀_Þ_Pð₀_Þ_Gð₀_ÞT _P_{0, we have} Gð0ÞP1_ð

0ÞGT_ð

0Þ6_Gð₀_Þ_GT_ð₀_{Þ þ}_Pð₀_Þ_: _ð₄₆_Þ

Obviously, if the following inequality holds:

V

I Gð0Þ GT_ð₀_{Þ þ}_Pð₀_Þ

<0 ð47Þ

which can be written as

Xr i¼1 hiðhð0ÞÞ V1 V2 V3 I 0 G1iGT_1iþP1i 0 I GT 2 þP2i G 1 2 G T 2 þP3i 2 6 6 6 4 3 7 7 7 5<0; ð48Þ

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then(45)or(43)holds. By considering(39), it is clear that if V1 V2 V3 I 0 G1iGT_1iþP1i 0 I G2TþG T 2 Q2i G 1 2 G T 2 þG T 2 Q3iG 1 2 2 6 6 6 4 3 7 7 7 5<0; i2S: ð49Þ

then(48)holds, and thus(43)holds. Deﬁne matrixT2 ,_diag_fI_;_I_;_I_;_G2g_{. Pre- and} post-multiplying(49)byTT₂ andT2, respectively, yield the following LMIs:

V1 V2 V3 I 0 G1iGT_1iþP1i 0 GT₂ IþQ2i G2G T 2 þQ3i 2 6 6 6 4 3 7 7 7 5<0; i2S: ð50Þ

Therefore, for some givene> 0, a suboptimal robust H2fuzzy observer-based control

problem can be addressed in the sense of minimizing the bound in (44)as the following LMI optimization problem:

minX traceV1þV2þVT₂ þV3 subject to the LMIs of ð30Þ;ð31Þ;andð50Þ ð51Þ

where X,_fV1_;_V2_;_V3_;_G2_;_G1i_;_P1i_>₀_;_Q

2i;Q3i>0;Ki;Si;i2Sg. By using the existing

LMI optimization techniques[30,31], the problem(51)can be eﬃciently solved.

Remark 4. Note that the LMIs of(30) and (31)inTheorem 2are not strictly linear due to the fact that it involves a tuning parametere> 0. Obviously, different values ofecan give rise to different optimized bounds trace V1þV2þVT2 þV3 by solving the problem

(51). Thus, one can still optimize this bound by an enumeration method or a one-dimensional search fore> 0 to obtain a suboptimal H2fuzzy observer-based controller. 4. Numerical example

In this section, a numerical example is presented to demonstrate the eﬀectiveness of the proposed design method. Consider the following second-order discrete-time chaotic sys-tem[33]: x1ðkþ1Þ ¼ax1ðkÞ x3 1ðkÞ þx2ðkÞ þuðkÞ ð52Þ x2ðkþ1Þ ¼bx1ðkÞ ð53Þ zðkÞ ¼½0:5x1ðkÞ 0:5uðkÞT ð54Þ yðkÞ ¼x1ðkÞ ð55Þ

where the nonlinear term isx3

1ðkÞ. It is assumed that the parametersaandbare uncertain

and satisfya0da6a6a0þdaandb0db6b6b0þdb, wherea0andb0are the

nom-inal values ofaandb, respectively,daP0 and dbP0.

Suppose thatx1ðkÞ 2 ½d;dandd> 0. Using the sector nonlinearity approach[3], we can exactly represent the system(52)–(55)by the T–S fuzzy model(1)–(3)withr= 2 and

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xðkÞ ¼ x1ðkÞ x2ðkÞ ; A1¼ a0 1 b0 0 ; A2¼ a0d 2 1 b0 0 " # ; B1¼B2¼ 1 0 ; C11¼C12¼ 0:5 0 0 0 ; D11¼D12¼ 0 0:5 ; C21¼C22¼½1 0;

H ¼½da dbT; E11¼E12¼½1 0; E21¼E22¼0;

h1ðx1ðkÞÞ ¼1x2₁ðkÞ=d2; h2ðx1ðkÞÞ ¼x2₁ðkÞ=d2:

In this study, the model parameters are given as a0¼1:9,b0¼0:5,da¼0:25,db¼0:2,

andd¼2.

By using the Matlab LMI toolbox, it can be found that the problem(51)has a solution fore2 ½0:00001;0:4. The optimized bound in(51)versus the scaling parametereis tabu-lated inTable 1.

By performing a simple one-parameter-search minimization of the optimized bound, it can be found that the optimal scaling parameter is given bye¼0:1805. The corresponding optimized bound is 245.9656 and the corresponding control and observer gain matrices are obtained as K1¼3:8787108 4:0000; K2¼ 1:7881109 4:0000; G11¼ 0:1274 0:0137 0:2542 3:9757 ; G12¼ 0:1281 0:0183 0:2708 4:0346 ; L1¼ 2:5010 0:5912 ; L2¼ 1:7771 0:4317 :

Now, we apply the fuzzy observer-based controller(8) with the above matrices to the system (52)–(55) under the initial condition xð0Þ ¼ ½1:5 0:8T. Assume that a¼1:9þDaðkÞ and b¼0:5þDbðkÞ where DaðkÞ and DbðkÞ are random variables Table 1

Optimized bound versus the scaling parametere

e 0.00001 0.001 0.1 0.15 0.1805 0.2 0.4 Optimized bound 2.0144·106 _2.0201 ·104 _298.6727 _252.1023 _245.9656 _248.2712 _3.4377 ·103 0 5 10 15 20 25 30 -2 -1 0 1 2

step

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uniformly distributed in the intervals [0.25, 0.25] and [0.2, 0.2], respectively. The sim-ulation results are shown inFigs. 1–3. The solid lines inFigs. 1 and 2show the actual tra-jectories of states x1ðkÞ and x2ðkÞof the system, respectively, and the dotted lines show those of the estimated states^x1ðkÞand^x2ðkÞof the fuzzy observer, respectively. The control actionuðkÞis shown inFig. 3. The simulation results show that the resulting fuzzy control-ler stabilizes the uncertain nonlinear system(52)–(55)and provides an optimized quadratic cost bound for the expected value of cost function(11).

5. Conclusions

In this paper, based on the T–S fuzzy model, we have studied the robust H2fuzzy

obser-ver-based control problem for discrete-time nonlinear systems with parametric uncertain-ties. The suﬃcient condition for the existence of a robust H2 fuzzy observer-based

controller is presented in the form of a set of LMIs. The control matrices and observer gain matrices can be obtained by directly solving this set of LMIs. Moreover, by the exist-ing LMI optimization techniques, a suboptimal robust H2fuzzy observer-based controller

can be obtained, which can not only guarantee that the closed-loop fuzzy system is glob-ally asymptoticglob-ally stable, but also provide an optimized upper bound on the quadratic cost function for all admissible parametric uncertainties. Finally, simulation results indi-cate that the proposed method is eﬀective. One of the future research topics is to extend the result developed in this paper to the robust H1 fuzzy observer-based control design for discrete-time uncertain T–S fuzzy systems.

0 5 10 15 20 25 30 -1 -0.5 0 0.5 1 1.5

step

Fig. 3. Control actionuðkÞ.

0 5 10 15 20 25 30 -1 -0.5 0 0.5 1

step

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Acknowledgements

The author would like to thank the anonymous reviewers for their valuable comments and constructive suggestions on the improvement of the text. This work was supported by the National Natural Science Foundation of China under Grant 60674058 and Beijing Natural Science Foundation under Grant 4062021.

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