Abstract—Some new results for stabilization design of

discrete-time (DT) nonlinear systems with ∞H performance are proposed in this paper. DT nonlinear systems can be constructed by using the Takagi-Sugeno (T-S) fuzzy model. First, we propose the modified fuzzy static output feedback (MFSOF) controller. The stabilization problem becomes to stabilize a new discrete-time fuzzy descriptor (DTFD) system. Such that, the proposed MFSOF control will stabilize the DT nonlinear systems. Furthermore, model reference fuzzy tracking control (MRFTC) design for the DTFD systems is investigated. The stability conditions can be certificate by using Lyapunov theorem. A practical system is given to verification the proposed formula.

I. INTRODUCTION

During the past 30 years, fuzzy logic theory is widely used by many researchers. The reason is they can provide an effect solution for controlling more complex nonlinear systems. The Takagi-Sugeno (T-S) fuzzy model is currently good models for fuzzy systems because it can be used to approximate complex nonlinear systems. In T-S fuzzy model, local dynamics can be represented as linear systems. The final system model is achieved by fuzzy “blending” of these fuzzy models. The control design is implemented by using the parallel distributed compensation (PDC) scheme. Recently, there has been a growing interest on the part of many researchers in investigating stabilizing method of fuzzy control systems. Such as, state feedback fuzzy control [1-5], output feedback fuzzy control [6-12], variable structure control [13-16], adaptive fuzzy control [17-19], etc. The tools for stability analysis and control design are based on the solution for linear matrix inequalities (LMI) [20]. In [1], Tanaka et al. used the sector nonlinearity method and showed that the nonlinear system can be represented exactly as the T-S fuzzy model.

Over the past several years, descriptor systems have been studied extensively in [21-23]. Another name of Singular systems is called descriptor systems. A new positive real analysis and control for uncertain discrete time descriptor systems has been presented in [21]. Using the descriptor representation approach, some sufficient conditions for the

Tzuu-Hseng S. Li* is with the aiRobots Laboratory, Department of

Electrical Engineering, National Cheng-Kung University, Tainan 701, Taiwan (corresponding author to provide e-mail: thsli@mail.ncku.edu.tw).

Kuo-Jung Lin was with the aiRobots Laboratory, Department of Electrical Engineering, National Cheng-Kung University, Tainan 701, Taiwan (e-mail: kuo5jung@seed.net.tw).

observer-based ∞H control problem of DT fuzzy systems with disturbance have been proposed in [22].

In the past 15 years, the tracking control problem has been one of most popular topics for investigation [7-9], [13-14], [24-28]. In [7], a MRFTC design has been proposed for stabilization of DT time-delay nonlinear systems. In [8], the output tracking problem for nonlinear systems is solved via the observer based fuzzy control design. Esfahani [9] has been discussed the problem of output feedback tracking control design for T-S fuzzy systems with immeasurable premise variables. To find the optimal tracking control law, the authors supply a simple numerical method. Furthermore, the output tracking problem for nonlinear system is via variable structure control (VSC) technique is examined in [13]. In [14], an adaptive fuzzy output tracking control design has been proposed for stabilization of uncertain multiple-input multiple-output nonlinear systems with a block-triangular structure. The stabilization of the fuzzy tracking control problem can be also determined by using the LMI problem [25].

In this paper, we propose using the MFSOF control to stabilize the DT nonlinear systems via the fuzzy descriptor systems (FDS) approach. Furthermore, stabilization of the DTFD system is adopted to realize the MRFTC design. The schematic of this paper is described as follows. In Section II, we give the definition of the DTFD system. The MFSOF control design via the DS approach is demonstrated in Section III. MRTFC design via stabilization of the DTFD system is presented in Section IV. In Section V, a Duffing forced-oscillation system is given to verification the proposed formula. In Section VI, we made a conclusion.

II. MODEL DESCRIPTION

Let the DT nonlinear systems [27] by the following:

)())(),(()1( kEwkukxkx +Ξ=+ (1a)

))(),(()( kukxgky = (1b)

where ))(),(( kukxΞ and ))(),(( kukxg are nonlinear

functions, 1)( ×∈ nRkx denote state vectors, 1)( rRku ∈ is the

control input, The unknown disturbances 2)( rRkw ∈ and

τ≤||)(|| kw , τ is a positive real number. 2rnRE ×∈ is the

constant matrix, and 13)( ×∈ rRky is the output signal.

Stabilization and Model Reference Tracking Control of Discrete-time Nonlinear Systems via Modified Fuzzy

Static Output Feedback Design Kuo-Jung Lin and Tzuu-Hseng S. Li*

Following, we use the Takagi-Sugeno (T-S) fuzzy model to represent the DT nonlinear systems. The i th rule of the Takagi-Sugeno (T-S) fuzzy model is described as follows, Plant Rule i: IF )(1 kq is 1Si and … and )(kqp is ipS

THEN )()()()1( kEwkuBkxAkx ii ++=+ (2a)

)()()( kuDkxCky ii += (2b)

where ri ,...2,1= , nn

i RA ×∈ , 1rn

i RB ×∈ , nr

i RC ×∈ 3 , and 13 rr

i RD ×∈ . The final form of DT fuzzy system is inferred as follows:

)}()()({)1(1

kEwkuBkxAkxr

iiii ++=+

=η (3a)

})()({)(1

=+=

r

iiii kuDkxCky η (3b)

where

=

=≡ r

ii

iii

kq

kqkq

1

))((

))(())((

μ

μηη ,

)](...)()([)( 21 kqkqkqkq p= , ∏=

=p

jjiji kqkq

1

))((S))((μ ,

rikq

kq

i

r

ii ,...,2,1,

0))((

0))((1 =

≥

>=

μ

μ. For all k, ))((S kqij is the

grade of membership of )(kq j in ijS . From (3) we have

1))((1

==

kqr

iiη , 0))(( ≥kqiη , and ri ,...,2,1= . By using the

sector nonlinearity concept [1], the DT nonlinear systems (1) can be represented exactly as the T-S fuzzy model (3). From (3), one can propose the following observer.

Observer Rule i: IF )(1 kq is 1iS and … and )(kqp is ipS

THEN )](ˆ)([)()(ˆ)1(ˆ kykyGkuBkxAkx iii −++=+ (4a)

)()(ˆ)(ˆ kuDkxCky ii += (4b)

where iG denotes the observer gain matrices. Then, the

overall fuzzy observer is

)]}(ˆ)([)()(ˆ{)1(ˆ1

kykyGkuBkxAkx iii

r

ii −++=+

=η (5a)

])()(ˆ[)(ˆ1

=+=

r

iiii kuDkxCky η (5b)

For the fuzzy control design, we propose using the MFSOF controller to stabilize the DT fuzzy system (3). The i th rule of the MFSOF controller is represented as follows: Control Rule i: IF )(1 kq is 1iS and … and )(kqp is ipS

THEN )()(ˆ)( kyFkxKku ii += (6)

where iK and iF are constant gain matrices with appropriate

dimension. Then, the overall MFSOF control is described as the following:

)}()(ˆ{)(1

kyFkxKku ii

r

ii +=

=μ (7)

Remark 1: We add the state estimator terms in (7), and the state gain

matrices are iK . We named the controller is the MFSOF

controller. If 0=iK , the MFSOF controller becomes the

general fuzzy static output feedback controller.

Define )(ˆ)()( kxkxkO −= , Substituting (3b), (5b), (7) into (5a) yields the closed-loop form of fuzzy observer as follows:

)]}()()(ˆ)({)1(ˆ1

kOCGkyFBkxKBAkx jijijii

r

ii +++=+

=η (8)

Subtract (5a) from (3a) yields

)}()(){()1(1 1

kEwkOCGAkO jii

r

i

r

jji +−=+

= =ηη (9)

If (3b) is converted into the following

= =

++−=+⋅r

i

r

jjiijir kxKDCkyky

1 13 )(){)()1(0

ηη

)}()( kOCkyFD iji ++ (10)

where 33

30 rr

r R ×∈ denotes zero matrix. From Eqs. (7), (8) and (10), we obtain the DTFD system as follows:

)]()()()[()1(1 1

kwEkOLkKBAk ijjii

r

i

r

jji ++Λ+=+ΛΤ

= =ηη (11)

where

=Λ

)(

)(ˆ)(

ky

kxk ,

=Τ

300

0

r

I,

−

=IC

AA

i

i

i

0,

=

ii

ii

i DD

BBB ,

=

j

j

j F

KK

0

0,

=

i

ji

ij C

CGL and

=

0

EE .

Property 1 [29]: If ζ denotes positive define matrix, Ξ and Ω are any matrices then

ΩΩ+ΞΞ≤ΞΩ+ΩΞ ζζζζ TTTT (12)

In the following, we can obtain gain matrices iG ,

ri ...,,2,1= , via Lemma 1.

Lemma 1: If there exists a positive positively defined symmetry matrices

1P , oR and some constant matrices iN that satisfy the

following LMIs:

05.0 11

11 <

−−

−+−PCNAP

NCPARP

jii

T

i

T

j

T

io (13)

where ....,,2,1, rji = ii NPG 1

1

−= . Then, the DT dynamical system (9) is global asymptotically stable.

Proof:

Let the Lyapunov function be defined as follows )()(][ 1 kOPkOkV T=

Then ][]1[][ kVkVkV −+=Δ

ij

r

i

r

ji

T

i

T

j

T

ij

r

i

r

ji

T APPNCPAkO 11 1

1

111 1

([)5.0)]([(){( ηηηη = =

−

= =−≤

)(})] 1 kOPCN ji −− )()()2( 1min kwkwEPE TTλ+

If

ij

r

i

r

ji

T

i

T

j

T

ij

r

i

r

ji APPNCPA 1

1 1

1

111 1

([)5.0)]([( ηηηη = =

−

= =−

oji RPCN −<−− 1)] (14)

Applying Schur complement formula [20], the above inequality becomes:

05.0 11

11

1 1

<

−−

−+−

= = PCNAP

NCPARP

jii

T

i

T

j

T

ior

i

r

jjiηη (15a)

If (13) holds, then (15a) holds. We obtain )()()2()()(][ 1min kwkwEPEkORkOkV TT

o

T λ+−≤Δ

)()()2()()(][]1[ 1min kwkwEPEkORkOkVkV TT

o

T λ+−≤+

Then, we obtain the following ∞H control performance. )()()2()()()()( 1min1 kwkwEPEkOPkOkORkO TTT

o

T λ+< (15b)

In the next section, the MFSOF controller will be designed such that the DT fuzzy system (3) is stable.

III. MFSOF DESIGN VIA THE FDS APPROACH

In the following, the MFSOF controller (7) will be design such that the DT fuzzy descriptor system is stable. First, we

define

Φ

Φ=Φ

j

j

j

2

1

0

0, PK jj Φ= ,

=

22

11

0

0

P

PP ,

=

22

11

0

0

Q

QQ , 01111 >= TPP , 02222 >= TPP , 1−= PQ ,

=

2

1

0

0

R

RR ,

=

30

00

rIT and I=+TT . It is obvious

that we have the following property. We consider the ∞H performance as follows:

)()()()()()( kOkOkTPTkkPRPk TTT Ψ+ΛΛ<ΛΛ

)()(][4 max kwkwEPE TTλ+ (16)

where ||}][)2]([max{||1 11 1

ij

r

i

r

jji

T

ij

r

i

r

jji LPL

= == ==Ψ ηηηη and

0)(lim =∞→

kOk

. A sufficient condition for stabilizing of the

DTFD system (11) under the MFSOF controller (7) can be obtained by the following theorem:

Theorem 1: If there exist positive positively defined symmetry matrices

11Q , 22Q , 1R and 2R , constant matrices j1Φ and j2Φ that

satisfy the following LMIs:

0

0000

00000

0025.00

00025.0

05.00

0005.0

2222

11

22222111

112111

222222222

111111111

<

−−

−Φ+−Φ+−ΦΦ+

Φ+−Φ+−Φ+Φ++−

QQ

Q

QDQDQC

QBBQA

QDQBRQ

DCQBAQRQ

jijii

jijii

TT

i

T

j

T

i

T

j

Ti

Tj

Ti

Ti

Tj

Ti (17)

rji ,...2,1, = , where

Φ

Φ=Φ

j

j

j

2

1

0

0, PK jj Φ= ,

=

22

11

0

0

P

PP and

=

22

11

0

0

Q

QQ . Then, the DTFD system

(11) is regular, causal and stable. Proof:

Define the Lyapunov function as )()()]([ kTPTkkV TT ΛΛ=Λ

Then ))(())1(()]([ kVkVkV Λ−+Λ=ΛΔ

)()()1()1( kTPTkkTPTk TTTT ΛΛ−+Λ+Λ=

According to Property 1 and Schur complement formula [20], if (17) hold, we obtain

)()()()()]([ kOkOkPRPkkV TT Ψ+ΛΛ−≤ΛΔ

)(])4()[( kwEPEkw TT+

From the above inequalities, we obtain the ∞H control performance (16).

IV. MRFTC DESIGN

In the following, we address the MRFTC design for the DTFD system. First, we consider a reference model as follows:

)()()1( krkxAkx mmm +=+ (18a)

)()( kCxky mm = (18b)

From (18), we obtain

)()1()1( krBkXAkXT mmmm ++=+ (19)

where

=

300

0

r

IT ,

=

)(

)()(

ky

kxkX

m

m

m ,

−

=IC

AA m

m αα0

,

=

0

IBm and

=

0

)()(

krkr . α is a nonzero real constant

such that the matrix mA is stable.

Now we consider the following fuzzy controller for the MRFTC design. The i th rule of the fuzzy controller is represented as follows:

Control Rule i: IF )(1 kq is 1iS and … and )(kqp is ipS

THEN

)]()([ˆ)]()(ˆ[ˆ)( kykyFkxkxKku mimi −+−= (20)

Then, the final output of fuzzy controller is described as follows:

)]}()([ˆ)]()(ˆ[ˆ{)(1

kykyFkxkxKku mimi

r

ii −+−=

=η (21)

From (3b) and (21), we obtain

)(ˆ)()ˆ{()()1(01 1

3 kyFDkxKDCkyky ji

r

ijiij

r

jir +++−=+⋅

= =ηη

)}(ˆ)(ˆˆD kOKDkyFDxK jimjimji −−− (22)

From (3a), (21) and (22), we obtain the DTFD system as follows:

)(~

)()~

[()1(1 1

kXKBkKBAkT mjijii

r

i

r

jji −Λ+=+Λ

= =ηη

)]()( kwEkOLij ++ (23)

where

=Λ

)(

)()(

ky

kxk ,

=

300

0

r

IT ,

−

=IC

AA

i

i

i

0,

=

ii

ii

i DD

BBB ,

=

j

j

jF

KK ˆ0

0ˆ~,

−−=

ii

jiij

KD

KBL

ˆ

ˆ and

=

0

EE . From (19) and (23), we obtain the following form

)}(~

)(~

)(~~

{)1(~

1 1

kwEkOLkAk ijij

r

i

r

jji ++Λ=+ΛΘ

= =μμ (24)

where

ΓΓ

=Θ0

0, [ ]TTT

m kkXk )()()(~ Λ=Λ ,

+−=

jiiji

m

ij KBAKB

AA ˆˆ

0~,

=

ij

ij LL

0~,

=

)(

)()(

kw

krkw and

=

E

BE m

0

0~. Following, we define

=

P

PP

0

0, 0>P ,

1−= PQ ,

=

Q

QQ

0

0,

=Θ

T

T

0

0, I=Θ+Θ ,

=

2

1

0

0

R

RR , 0>R , PK jj Φ= ~~

and

ΦΦ

=Φj

j

j

2

1

ˆ0

0ˆ~

Now, we consider the following ∞H control performance:

)()()(~

)(~

)(~

)(~

kOkOkPkkPRPk TTT Ψ+ΛΘΘΛ<ΛΛ

)()(]~~

[2 max kwkwEPE TTλ+ (25)

where ||}]~

[)4](~

[max{||1 11 1

ij

r

i

r

jji

T

ij

r

i

r

jji LPL

= == ==Ψ ηηηη . Then,

a sufficient condition for stabilizing of the DTFD system (23) under the MRFTC design can be obtained by the following theorem:

Theorem 2: The DTFD system (23) is regular, causal and stable, if there exist positive positively defined symmetry matrices Q , 1R

and 2R , there are constant matrices jΦ~ that satisfy the

following LMIs:

0

0000

0000

0025.00~~

00025.00

0~

05.00

0~

05.0

2

1

<

−−

−Φ+Φ−−

Φ++−Φ−+−

QTQ

QTQ

QBQAB

QQA

QTBAQRQ

QTBAQRQ

jiiji

m

TT

i

T

j

T

i

TT

i

T

j

T

m

(26)

rji ,...2,1, = .

Proof: Define the Lyapunov function as

)(~

)(~

)](~

[ kPkkV TT ΛΘΘΛ=Λ

According to Property 1, we obtain

))(~

())1(~

()](~

[ kVkVkV Λ−+Λ=ΛΔ

]~

)[4](~

{[)(~

1 11 1

QAPAQPk ijj

r

i

r

ji

T

ijj

r

i

r

ji

T ηηηη = == =

Λ≤

)()()(~

)2(~

)()(~

} kOkOkwEPEkwkPQPQ TTTT Ψ++ΛΘΘ−

Because,

QPQQQPQ TT ΘΘ+−≤ΘΘ− 5.0

Then the above inequalities become

]~

)[4](~

{[)(~

1 11 1

QAPAQPkV ijj

r

i

r

ji

T

ijj

r

i

r

ji

T ηηηη = == =

Λ≤Δ

)(~

)2(~

)()(~

}5.0 kwEPEkwkPQPQQ TTT +ΛΘΘ+−

)()( kOkOT Ψ+ (27)

If

RQPQQQAPAQ T

ijj

r

i

r

ji

T

ijj

r

i

r

ji −<ΘΘ+−

= == =5.0]

~)[4](

~[

1 11 1

ηηηη (28)

Applying Schur complement formula [20] to the inequality (28) implies

0

0

025.0~

~5.0

1 1

<

−Θ−

Θ+−

= =

r

i

r

jij

TT

ij

ji

QQ

QQA

QAQRQ

ηη (29)

If (26) hold, then (29)holds. From (27) and (28), we have

)()()(~

)(~

))(~

())1(~

( kOkOkPRPkkVkV TT Ψ+ΛΛ−Λ≤+Λ

)(]~

)2(~

)[( kwEPEkw TT+ (30)

From (30), we obtain the ∞H control performance (25). This completes the proof.

V. SIMULATION EXAMPLE

Consider the Duffing forced-oscillation system [31] for the MRFTC design, the state equation of such system is given by

)()( 21 txtx = (31a)

)()()(1.0)()( 2

3

12 twtutxtxtx ++−−= (31b)

)()( 1 txty = (31c)

where )cos(12)( ttw = . Using Euler’s method [30] to

(31a)-(31c), one can obtain the following discrete-time

nonlinear system, )()()1( 211 kTxkxkx +=+ (32a)

)()()()1.01()()1( 2

3

12 kTwkTukxTkTxkx ++−+−=+ (32b)

)()( 1 kxky = (32c)

where sT 01.0= is the sampling time. Assume that ]5,5[)(1 −∈kx . According to the sector nonlinearity

procedure as in [1], the following T-S fuzzy model is equivalent to the system (32). Plant Rule i: IF )(kxi is iS

THEN )()()()1( kEwkuBkxAkx ii ++=+

)()( kxCky i= , .2,1=i

where [ ]TTT kxkxkx )()()( 21= ,

=

999.00

01.011A ,

−

=999.025.0

01.012A ,

==

01.0

021 BB , [ ]0121 == CC ,

=

01.0

0E and )cos(12)( kTkw = . Furthermore,

25

)(1S

2

111

kx−== η and 25

)(S

2

122

kx== η . The reference

model is given by

)(01.0

0)(

98.003.0

01.01)1( krkxkx mm

+

−

=+ (33a)

[ ] mm xky 01)( = (33b)

where )cos(5)( kTkr = . If 01.0=α , from (19), we obtain

−−=

01.0001.0

098.003.0

001.01

mA . Then from Theorem 2, we solve

−=Φ=Φ

0001.000

00373.00017.0~~21

,

×

×= −

−

0007.000

00012.0104755.3

0104755.30005.05

5

Q,

−

−×=

4135.100

08335.00553.0

00553.09120.1

103P ,

==

001.00

0001.021 RR ,

[ ]1459.314024.5ˆˆ21 −== KK , and 2096.0ˆˆ

21 == FF . We

apply the proposed MRFTC (21) to the DT nonlinear system

(32). We set up the initial values of

−

=1

2)0(x and

−=

1

5.2)0(mx . Simulation results are shown in Fig. 1.

Furthermore, the proposed MRFTC (21) via zero order hold (ZOH) to reconstruct the continuous signal and is applied to the continuous-time nonlinear system (31). Simulation results of the tracking conditions are shown in Figure 2. Obviously, the output signal can keep track of the model reference signal under the disturbance )cos(12)( ttw = .

Fig. 1 Time responses of )(kw , )(ky and )(kym .

Fig. 2 Time responses of )(tw , )(ty and )(tym .

VI. CONCLUSION

Stabilization of DT nonlinear systems via the FDS approach and MFSOF control design has been presented in this paper. We can determine all the MFSOF controller gain matrices via some LMIs condition which can be derived by the Lyapunov stability theorem. The proposed MFSOF controller will stabilize DT nonlinear systems. Furthermore, we use MRFTC design for stabilization of DT nonlinear systems via the FDS approach and MFSOF control design. A practical numerical example is applied to verification the proposed formula. Simulations demonstrate the proposed formula is correct.

ACKNOWLEDGMENT

This work was supported in part by the Ministry of Science and Technology, Taiwan, ROC, under Grant MOST 104-2221-E-006-228-MY2 and in part by the Ministry of Education, Taiwan, within the Aim for the Top University Project through National Cheng Kung University (NCKU), Tainan, Taiwan.