# Analysis and synthesis of discrete nonlinear passive systems via affine TS fuzzy models

**ABSTRACT** The article considers the analysis and synthesis problem for the discrete nonlinear systems, which are represented by the discrete affine Takagi–Sugeno (T–S) fuzzy models. The state feedback fuzzy controller design methodology is developed to guarantee that the affine T–S fuzzy models achieve Lyapunov stability and strict input passivity. In order to find a suitable fuzzy controller, an Iterative Linear Matrix Inequality (ILMI) algorithm is employed in this article to solve the stability conditions for the closed-loop affine T–S fuzzy models. Finally, the application of the proposed fuzzy controller design methodology is manifested via a numerical example with computer simulations.

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International Journal of Systems Science

Vol. 39, No. 8, August 2008, 809–821

Analysis and synthesis of discrete nonlinear passive systems via affine T–S fuzzy models

Wen-Jer Chang*, Cheung Chieh Ku and Wei Chang

Department of Marine Engineering, National Taiwan Ocean University, Keelung 202, Taiwan, Republic of China

(Received 29 April 2006; final version received 6 January 2008)

The article considers the analysis and synthesis problem for the discrete nonlinear systems, which are represented

by the discrete affine Takagi–Sugeno (T–S) fuzzy models. The state feedback fuzzy controller design

methodology is developed to guarantee that the affine T–S fuzzy models achieve Lyapunov stability and strict

input passivity. In order to find a suitable fuzzy controller, an Iterative Linear Matrix Inequality (ILMI)

algorithm is employed in this article to solve the stability conditions for the closed-loop affine T–S fuzzy models.

Finally, the application of the proposed fuzzy controller design methodology is manifested via a numerical

example with computer simulations.

Keywords: affine Takagi–Sugeno fuzzy model; passive systems; S-procedure; iterative linear matrix inequality

1. Introduction

Passivity

Trentrlman 2002a,b; Hill and Moylan 1976; Byrnes

and Lin 1994, 1995) gives a framework for analysis of

the nonlinear or linear systems using an input-output

description based on energy-related consideration. The

main ideal behind this is that many important physical

systems have certain input-output properties related to

the conservation, dissipation and transport of energy.

Using the energy concept to interpret the behaviours

and properties of system are described with Lyapunov

function and passivity theory. As for a storage function

(Mohseni, Yaz and Olejniczak 1998; Tan, Soh and Xie

1999; Kaneko and Fujii 2000; Lozano, Brogliato,

Egeland and Maschke 2000; Li, Wang and Shao 2002;

Shao and Feng 2004) measures the amount for energy

stored in the internal of the system, so it is naturally

described by using internal variables of the system, that

is, state variables. Lyapunov function is represented as

a quadratic function of state variables in this article.

The passivity theory is intimately related to Lyapunov

stability theory. The passive property has many types

for expressing total energy and input energy of systems

via power supply. So, how to take account of choosing

the power supply (Byrnes and Lin 1993, 1995; Wang

et al. 2002; Willems, and Trentrlman 2002a,b) is a very

interesting task for the system. In the literature (Byrnes

and Lin 1993, 1995; Mohseni et al. 1998; Tan et al.

1999; Kaneko and Fujii 2000; Li et al. 2002; Shao and

Feng 2004), the passivity theory is employed to analyse

the stability of the discrete systems. The authors of

Lozano et al. (2000) indicate that the discrete systems

theory (Willems1971;Willemsand

can also be the passive system except for the strictly

causal discrete system. And the discrete passive systems

have the counterpart to most of the results in the

continuous case. In this article, an affine T–S fuzzy

model is employed to describe the discrete nonlinear

system, which is not a strictly causal discrete system.

It is well known that the T–S fuzzy model [Tanaka

and Sano (1994); Wang, Tanaka and Griffin (1996);

Chang and Sun (2003)] provides a useful tool for

stability analysis and fuzzy controller design for

complex nonlinear systems. The T–S fuzzy model is

described by IF-THEN rules and the corresponding

fuzzy controller design is developed based on Parallel

Distribution Compensation (PDC) technique (Tanaka

and Wang 2001; Wang et al. 1996). One can approach

the trajectory of an original nonlinear system via

combination of the membership function and the linear

subsystems of the T–S fuzzy model. The pioneers, Kim

and his colleagues (Kim and Kim 2001) have provided

a sufficient condition for the stability analysis of the

affine T–S fuzzy models in sense of Lyapunov

inequality. The affine T–S fuzzy model means the

T–S fuzzy model of which consequent part is affine and

which has a constant bias term. The affine T–S fuzzy

model is more natural and appealing to human beings

than the homogeneous one. The homogeneous T–S

fuzzy model is usually studied by the researchers due to

the ease of analysis. In this article, both the stability

analysis and controller synthesis of discrete-time

affine T–S fuzzy systems are considered. In general,

the stability conditions of affine T–S fuzzy systems are

most Bilinear Matrix Inequality (BMI) problems

*Corresponding author. Email: wjchang@mail.ntou.edu.tw

ISSN 0020–7721 print/ISSN 1464–5319 online

? 2008 Taylor & Francis

DOI: 10.1080/00207720801902580

http://www.informaworld.com

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which were discussed in Kim and Kim (2001) and

Chang and Chang (2005, 2006). The BMI problem

may be difficult to solve by numerically convex

optimisation technique (Boyd, Ghaoui, Feron and

Balakrishnan 1994) and cannot be calculated via

Linear Matrix Inequality (LMI) technique. For this

reason, the Iterative Linear Matrix Inequality (ILMI)

algorithm is developed in this article to solve the BMI

conditions and to find the feasible solutions of

fuzzy controller design problems for affine T–S

fuzzy models.

In order to analyse the stability of the nonlinear

systems with external disturbances, the affine T–S

fuzzy model and passivity theory are employed in this

article. In Clacev, Gorez and Neyer (1998) and Clacev

(1998), the authors used the Mamdani fuzzy model to

simulate the trajectories of the nonlinear system and

apply the passivity theory to analyse the stability.

Besides, for describing the behaviours of the nonlinear

system clearly and for attenuating the external

disturbance energy, the authors proposed the passive

controller design technique for homogeneous T–S

fuzzy model in Li, Zhang and Liao (2005, 2006).

Based on the results of Li et al. (2005, 2006) and Uang

(2005), the homogeneous T–S fuzzy model can achieve

passivity performance and stability via the designed

fuzzy controller. For affine T–S fuzzy model with

external disturbances, the authors of Chang and Chang

(2006) use the H1control method to achieve system

stabilityandH1

Unfortunately, there is not any literature to discuss

the fuzzy controller design with passivity for the affine

T–S fuzzy systems. For controlling accurately and

attenuating the external disturbance energy, the

stability conditions are derived by passivity theory to

design the fuzzy controller to achieve the performance

for affine T–S fuzzy model in this article.

The principle structure of this article is described as

follows. The descriptions of discrete affine T–S fuzzy

models and passivity properties are presented in

Section 2. Based on PDC concept, the stability analysis

and solutions of fuzzy controller design problems are

presentedfor discrete affine

in Section 3. Applications of the proposed design

approach via numerical simulations are shown in

Section 4. Finally, concluding remarks are made in

Section 5.

performance constraints.

T–Sfuzzymodel

2. System descriptions and stability analysis

with passivity

The passivity and Lyapunov stability conditions

of discrete affine T–S fuzzy systems are discussed

based on Lyapunov criterion in this section. Besides,

the S-procedure (Boyd et al. 1994; Kim and Kim 2001)

is used to ensure the analysis and synthesis results.

The discrete-time affine T–S fuzz system considered

in this article is described as follows:

Plant part:

RulePlanti: IF x1(k) is Mi1and x2(k) is Mi2and...and

xp(k) is MipTHEN

xðk þ 1Þ ¼ AixðkÞ þ BuiuðkÞ þ BwiwðkÞ þ ai

ð1aÞ

yðkÞ ¼ CixðkÞ þ DiwðkÞ,

where Ai2 <nx?nx, Bui2 <nx?nu, Bwi2 <nx?ny, ai2 <nx,

Ci2 <ny?nxand Di2 <ny?nyare constant matrices and

x k ð Þ 2 <nxis the state vector, y k ð Þ 2 <nyis the output

vector, w k ð Þ 2 <nyis the square-integrable exogenous

input vector, u k ð Þ 2 <nuis the control input vector, Mij

is the fuzzy set, p is the premise variable number and r

is number of fuzzy model rules. Given the pair

of ðxðkÞ,uðkÞÞ, the closed-loop of the fuzzy model (1)

can be referred as follows:

Xr

i ¼ 1,...,r

ð1bÞ

xðkþ1Þ¼

i¼1!iðxðkÞÞ AixðkÞþBuiuðkÞþBwiwðkÞþai

Xr

ðÞ

i¼1!iðxðkÞÞ

ð2aÞ

yðkÞ ¼

Xr

i¼1!iðxðkÞÞ CixðkÞ þ DiwðkÞ

Xr

ðÞ

i¼1!iðxðkÞÞ

,i ¼ 1,...,r

ð2bÞ

or

xðk þ 1Þ ¼

X

? AixðkÞ þ BuiuðkÞ þ BwiwðkÞ þ ai

r

i¼1

hiðxðkÞÞ

ðÞð3aÞ

yðkÞ ¼

X

r

i¼1

hiðxðkÞÞ CixðkÞ þ DiwðkÞðÞ,i ¼ 1,...,r

ð3bÞ

where

!iðxðkÞÞ ¼ ?

p

j¼1MijðxjðkÞÞð4Þ

hiðxðkÞÞ ¼

!iðxðkÞÞ

Xr

i¼1!iðxðkÞÞ

ð5Þ

and Mij(xj(k)) is the grade of membership function of

xj(k)inMij.Itisassumed

Pr

that

i¼1hiðxðkÞÞ ¼ 1 and

!iðxðkÞÞ ? 0,

i¼1!iðxðkÞÞ40, hiðxðkÞÞ ? 0,Pr

i ¼ 1,...,r for all x(k).

810W.-J. Chang et al.

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For the nonlinear plant provided in (2) or (3), the

fuzzy controller is designed to share the same fuzzy set

with the plant. The fuzzy controller design is developed

based on the PDC technique, which can be referred to

[Wang et al. (1996)]. The discrete-time T–S fuzzy

controller has the following form.

Controller part:

RuleController i: IF x1(k) is Mi1 and x2(k) is Mi2

and...and xp(k) is MipTHEN

u k ð Þ ¼ ?FixðkÞ,i ¼ 1,...,r

ð6Þ

Based on PDC concept, the fuzzy controller can be

represented via summation as follows:

uðkÞ ¼

X

r

i¼1

hiðxðkÞÞð?FixðkÞÞ,i ¼ 1,...,r

ð7Þ

Substituting (7) into (3a), one can obtain the corre-

sponding closed-loop system such as

xðk þ 1Þ ¼

X

? AixðkÞ ? BuiFjxðkÞ þ BwiwðkÞ þ ai

X

?

X

i5j

ð

Ai? BuiFj

2

?

X

i¼j

þ BwiwðkÞ þ aiÞ

X

? GijxðkÞ þ NijwðkÞ þ ~ aij

Gij¼ ððAi? BuiFjÞ þ ðAj? BujFiÞÞ=2;

Nij¼ ðBwiþ BwjÞ=2 and ~ aij¼ ðaiþ ajÞ=2.

Assumption 1:Theregion

nx-dimensional convex polyhedron, which is known as a

cell. The cell indices are denoted by~I and union of all

cells X ? [i2~IXiis referred to as the partition. Let^I ?~I

r

i¼1

X

r

j¼1

hiðxðkÞÞhjðxðkÞÞ

??

¼

r

i¼1

X

Ai? BuiFj

r

X

?

Bwiþ Bwj

2

r

j¼1

hiðxðkÞÞhjðxðkÞÞ

?xðkÞ þ BwiwðkÞ þ ai

hiðxðkÞÞhjðxðkÞÞ

???

¼ 2

i¼1

?

Þ

r

j¼1

?

?þ Aj? BujFi

wðkÞ þaiþ aj

??

??

xðkÞ

þ

?

2

?

þ

r

i¼1

ðÞ

h2

iðxðkÞÞ Ai? BuiFi

ðÞxðkÞð

?

r

i¼1

X

r

j¼1

hiðxðkÞÞhjðxðkÞÞ

??

ð8Þ

where

Xi? <nx

is

be the set of indices for the fuzzy rules that contain the

origin and others be the set of indices for the fuzzy rules

that do not contain the origin. Here, the origin x(0)¼0 is

the equilibrium point of the discrete affine T–S fuzzy

model and it is assumed ai¼ 0 for i 2^I.

The concept of the passivity theory proposes a very

useful tool in the stability analysis and controller

synthesis for linear or nonlinear dynamic systems. In

the passivity theory, the power supply is an important

role, which has the definition as follows.

Definition 1:

disturbance input w(k) and output y(k) is said to be

strictly passive (Mohseni et al. 1998; Lozano et al.

2000) if

Thesystem(3)withexternal

2

X

kq

k¼0

yTk ð Þw k ð Þ4

X

kq

k¼0

yTk ð ÞYy k ð Þ þ ?wTk ð Þw k ð Þ

??

ð9Þ

where Y 2 <ny?ny, ? ? 0 and kq40.

In the left-hand

Pkq

hand side of the (9) represents the output energy

Pkq

parameters for determining the dissipation properties.

In physical systems, the power supply can be asso-

ciated with the concept of stored energy (Hill and

Moylan 1976) or dissipated energy (Willems 1971).

Based on these concepts, one can see that if the passive

inequality (9) is held that means the stored or

dissipated energy is bigger than the energy of external

disturbance and output. Then, one can say that the

passive system can resist the effect of the external

disturbance. Throughout this article, we are only

concerned about a strictly input passive system such

as Y¼0 and ? ?0. Thus, the inequality (9) can be

represented as

side of the(9), the term

k¼0yTðkÞwðkÞ represents the power supply to the

system from external disturbance input. And the right-

k¼0yTðkÞYyðkÞ and external disturbance energy

Pkq

k¼0?wTðkÞwðkÞ, in which Y and ? are the adjustable

2

X

kq

k¼0

yTk ð Þw k ð Þ4

X

kq

k¼0

?wTk ð Þw k ð Þð10Þ

There are many other types of passive inequality,

which can be referred to in Kaneko and Fujii (2000).

The passivity analysis problem can also be considered

as an H1analysis problem via some assumptions that

can be found in Byrnes and Lin (1993, 1994) and

Mohseni et al. (1998). The passivity conditions for the

discrete-time affine T–S fuzzy system (8) and (3b) can

be characterised by the following theorem.

Theorem 1:

P>0, scalars ? ?0 and "ijq?0 for satisfying the

If there exist a positive definite matrix

International Journal of Systems Science811

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following conditions, then the discrete-time affine T–S

fuzzy model (3) driven by fuzzy controller (7) is strictly

input passive and Lyapunov stable.

"

!11

!T

21

!21

!22

#

50,i 2^I

ð11Þ

!11?

X

!21

p

q¼1

"ijqTijq

!T

21

GT

ijP~ aij?

X

ijP~ aij

p

q¼1

"ijqnijq

!22

NT

~ aT

ijPGij?

X

p

q¼1

"ijqnT

ijq~ aT

ijPNij

~ aT

ijP~ aij?

X

p

q¼1

"ijqvijq

2

6

6

6

6

4

6

6

6

6

3

7

7

7

7

5

7

7

7

7

50,

i62^I

where

ð12Þ

!11¼ GT

ijPGij? P

ð13Þ

!21¼ NT

ijPGij? Ci

ð14Þ

!22¼ NT

ijPNij? Di? DT

iþ ?I

ð15Þ

and I is the compatible dimension identity matrix.

Besides, Tijq2 <nx?nx, nijq2 <nxand vijq2 < are defined

such as

?ijqðxðkÞÞ ¼ xTðkÞTijqxðkÞ þ 2xTðkÞnijqþ vijq? 0 ð16Þ

Proof:First, we select the case of fuzzy rules i 62^I

in following proof. Applying the Lyapunov function

VðxðkÞÞ ¼ xTðkÞPxðkÞ represents the storage function

of the passive theory for the closed-loop system (8).

By evaluating the first forward difference of VðxðkÞÞ

along the trajectories of (8), we have

?VðxðkÞÞ ¼ xTðk þ 1ÞPxðk þ 1Þ ? xTðkÞPxðkÞ

X

? ðxðkÞÞhzðxðkÞÞ GijxðkÞ þ NijwðkÞ þ ~ aij

? P GszxðkÞ þ NszwðkÞ þ ~ asz

¼

r

i¼1

X

r

j¼1

X

r

s¼1

X

r

z¼1

hiðxðkÞÞhjðxðkÞÞhs

??T

ðÞ ? xTðkÞPxðkÞ

ð17Þ

Nsz¼

where

ðBwsþ BÞwz=2 and ~ asz¼ ðasþ azÞ=2.

Choose the performance function such as

Gsz¼ ððAs? BusFzÞ þ ðAz? BuzFsÞÞ=2,

JD¼

X

kq

k¼0

?wTðkÞwðkÞ ? 2yTðkÞwðkÞ

??

ð18Þ

with zero initial condition for all w(k). Then, for any

nonzero w(k) one has

JD?

X

X

X

kq

k¼0

kq

?wTk ð ÞwðkÞ ? 2yTðkÞwðkÞ

?

?

?þ VðxðkÞÞ

¼

k¼0

kq

?wTðkÞwðkÞ ? 2yTðkÞwðkÞ þ ?VðxðkÞÞ

?

?

k¼0

Lðx,w,kÞð19Þ

According to (17) and (19), one has

Lðx,w,kÞ ¼ ?VðxðkÞÞ þ ?wTðkÞwðkÞ ? 2yTðkÞwðkÞ

X

? GijxðkÞ þ NijwðkÞ þ ~ aij

? P GijxðkÞ þ NijwðkÞ þ ~ aij

? xTðkÞPxðkÞ þ ?wTðkÞwðkÞ ? 2yTðkÞwðkÞ

X

xðkÞ

wðkÞ

1

~ aT

2

4

Applying the S-procedure (Boyd et al. 1994; Kim and

Kim 2001), one can add the LMI region of the x(k)

where each rule is activated. In the Appendix 1, the

structures of the matrices of S-procedure have been

proposed. The computation details for the correspond-

ing matrices of S-procedure can be referred to

(KimandKim2001).

S-procedure technique, the (20) can be satisfied if

there exist "ijq?0 such that

?

r

i¼1

X

r

j¼1

hiðxðkÞÞhjðxðkÞÞ

??T

?

X

2

4

xðkÞ

wðkÞ

?

¼

r

i¼1

r

j¼1

hiðxðkÞÞhjðxðkÞÞ

3

5

3

5

?

67

T

!11

!T

21

GT

ijP~ aij

NT

ijP~ aij

~ aT

ijP~ aij

!21

!22

ijPGij

~ aT

ijPNij

2

6

6

4

3

7

7

5

?

1

67

ð20Þ

Throughapplyingthe

Lðx,w,kÞ ?

X

r

i¼1

X

r

j¼1

hiðxðkÞÞhjðxðkÞÞ

xðkÞ

wðkÞ

1

3

7

2

6

6

4

3

7

xðkÞ

wðkÞ

1

7

5

T

?

!11

!T

21

GT

ijP~ aij

!21

!22

NT

ijP~ aij

~ aT

ijPGij

~ aT

ijPNij

~ aT

ijP~ aij

2

6

6

6

4

7

7

5

2

6

6

4

3

7

7

5

?5

X

p

q¼1

"ijq?ijqðxðkÞÞð21Þ

812W.-J. Chang et al.

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where ?ijqðxðkÞÞ 2 < is defined in (16). Due to "ijq?0

and ?ijqðxðkÞÞ ? 0, the (21) can be rewritten as

Obviously, if inequality (12) is held for all xðkÞ 2 Xi

and i 62^I thenPkq

JD50

k¼0Lðx,w,kÞ50. In this case, one has

ð23Þ

or

2

X

kq

k¼0

yTðkÞwðkÞ4

X

kq

k¼0

?wTðkÞwðkÞð24Þ

Since (24) is equivalent to (10), it is easy to show that

the system is strictly input passive with external

disturbance input and output.

Now, we have to show that the discrete-time affine

T–S fuzzy model (8) is Lyapunov stable. From (22),

iftheinequality (12)

Lðx,w,kÞ50. By assuming the external disturbance

input w(k)¼0, one has

isheld,thus,onehas

?V x k ð Þ ðÞ50

ð25Þ

Thus, the discrete-time affine T–S fuzzy model (3) is

Lyapunov stable and strictly input passive driven by

the fuzzy controller (7). Besides, the proof of condition

(11) of rules i 2^I is similar to (12) with setting ~ ai¼ 0

and assuming ?ijqðxðkÞÞ be zero. The proof is

completed.

In order to achieve the sufficient conditions of (11)

and (12), one must find suitable positive definite

common matrix P ¼ PT40, scalars "ijp? 0 and

? ? 0. The forms of the conditions (11) and (12) are

the BMI ones, which cannot be calculated by LMI

approach for obtaining the solution to satisfy the

conditions directly. Hence, the ILMI algorithm is

developed to solve the problem of BMI and to find the

feasible solution for the conditions (11) and (12).

By applying some transform techniques, the BMI

conditions can be transformed into LMI conditions via

the applications of auxiliary variables in the next

section.

3. Fuzzy controller design by using ILMI algorithm

In this section, an ILMI algorithm is applied to obtain

fuzzy controller to achieve the conditions of Theorem 1

for discrete-time affine T–S fuzzy model (3). In order

to solve the BMI conditions of Theorem 1, we first

hold some auxiliary variables as given matrices and

then converting it to a LMI problem. After that, the

fuzzy controller design problem can be solved by LMI

toolbox of MATLAB.

Theorem 2:

P40, X40 and scalars ?51, ? 40, "ijq? 0 for

satisfying the following conditions, then the discrete-time

affine T–S fuzzy model (3) is strictly input passive and

Lyapunov stable with fuzzy controller (7).

If there exist positive definite matrices

)ij50

XTPX ? X ? 0

(

,i 2^I

ð26Þ

and

^)ij50

XTPX ? X ? 0

(

,i 62^I

ð27Þ

where

)ij¼

??P

Gij

GT

ij

GT

ijX?1Nij? CT

0

i

?X

0

NT

ijX?1Gij? Ci

NT

ijX?1Nij? Di? DT

iþ ?I

2

6

6

4

3

7

7

5,

Lðx,w,kÞ ?

X

r

i¼1

X

r

j¼1

hiðxðkÞÞhjðxðkÞÞ

xðkÞ

wðkÞ

1

2

6

6

4

3

7

7

5

T

!11?

X

!21

p

q¼1

"ijqTijq

!T

21

GT

ijP~ aij?

X

ijP~ aij

p

q¼1

"ijqnijq

!22

NT

~ aT

ijPGij?

X

p

q¼1

"ijqnT

ijq

~ aT

ijPNij

~ aT

ijP~ aij?

X

p

q¼1

"ijqvijq

2

6

6

6

6

4

6

6

6

6

3

7

7

7

7

5

7

7

7

7

xðkÞ

wðkÞ

1

2

6

6

4

3

7

7

550

ð22Þ

International Journal of Systems Science 813

Page 6

Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008

Proof:

nique, the first condition of (27), i.e.^)ij50, can be

rewritten as

Using the Schur complement converting tech-

Obviously, if the decay rate ?51 and P40

are held, then inequality (28) is negative definite.

Multiplying (28) by ½ðxTðkÞwTðkÞ1? and ½xTðkÞwTðkÞ1?T,

respectively, and rearranging it gives

xTðkÞGT

ijX?1GijxðkÞ þ xTðkÞGT

ijX?1NijwðkÞ ? xTðkÞCT

iwðkÞ

þ xTðkÞGT

ijX?1~ aijþ wTðkÞNT

ijX?1GijxðkÞ

? wTðkÞCixðkÞ þ wTðkÞNT

ijX?1NijwðkÞ

þ ?wTðkÞwðkÞ ? wTðkÞDT

iwðkÞ ? wTðkÞDiwðkÞ

þ wTðkÞNT

ijX?1~ aijþ ~ aT

ijX?1GijxðkÞ þ ~ aT

ijX?1NijwðkÞ

þ ~ aT

ijX?1~ aij? xTðkÞPxðkÞ ?

X

p

q¼1

"ijq?ijqðxðkÞÞ

5ð? ? 1ÞxTðkÞPxðkÞð29Þ

or

GijxðkÞ þ NijwðkÞ þ ~ aij

? xTðkÞPxðkÞ þ ?wTðkÞwðkÞ ? 2yTðkÞwðkÞ

X

??TX?1GijxðkÞ þ NijwðkÞ þ ~ aij

??

?

p

q¼1

"ijq?ijqðxðkÞÞ5ð? ? 1ÞxTðkÞPxðkÞð30Þ

If there exist P40 such that XTPX ? X ? 0 is held,

one can obtain the following inequality according to

(30).

GijxðkÞ þ NijwðkÞ þ ~ aij

? xTðkÞPxðkÞ þ ?wTðkÞwðkÞ ? 2yTðkÞwðkÞ

X

??TP GijxðkÞ þ NijwðkÞ þ ~ aij

??

?

p

q¼1

"ijq?ijqðxðkÞÞ5ð? ? 1ÞxTðkÞPxðkÞð31Þ

or

^)ij¼

??P ?

X

Gij

p

q¼1

"ijqTijq

GT

ij

GT

ijX?1Nij? CT

i

GT

ijX?1~ aij?

X

p

q¼1

"ijqnijq

?X

0

00

NT

ijX?1Gij? Ci

NT

ijX?1Nij? Di? DT

iþ ?INT

ijX?1~ aij

~ aT

ijX?1Gij?

X

p

q¼1

"ijqnT

ijq

0

~ aT

ijX?1Nij

~ aT

ijX?1~ aij?

X

p

q¼1

"ijqvijq

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

:

GT

ijX?1Gij? P ?

X

p

q¼1

"ijqTijq

GT

ijX?1Nij? CT

i

GT

ijX?1~ aij?

X

p

q¼1

"ijqnijq

NT

ijX?1Gij? Ci

NT

ijX?1Nij? Di? DT

iþ ?INT

ijX?1~ aij

~ aT

ijX?1Gij?

X

p

q¼1

"ijqnT

ijq

~ aT

ijX?1Nij

~ aT

ijX?1~ aij?

X

p

q¼1

"ijqvijq

2

6

6

6

6

4

6

6

6

6

3

7

7

7

7

5

7

7

7

7

5

ð? ? 1ÞP

0

00

00

000

2

6

6

4

3

7

7

5

ð28Þ

xðkÞ

wðkÞ

1

2

6

6

4

3

7

7

5

T

!11?

X

!21

p

q¼1

"ijqTijq

!T

21

GT

ijP~ aij?

X

ijP~ aij

p

q¼1

"ijqnijq

!22

NT

~ aT

ijPGij?

X

p

q¼1

"ijqnT

ijq

~ aT

ijPNij

~ aT

ijP~ aij?

X

p

q¼1

"ijqvijq

2

6

6

6

6

4

6

6

6

6

3

7

7

7

7

5

7

7

7

7

xðkÞ

wðkÞ

1

2

6

6

4

3

7

7

5v5

xðkÞ

wðkÞ

1

2

6

6

4

3

7

7

5

T

ð? ? 1ÞP

0

00

00

000

2

6

6

4

3

7

7

5

xðkÞ

wðkÞ

1

2

6

6

4

3

7

7

5

ð32Þ

814W.-J. Chang et al.

Page 7

Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008

Obviously,

!11?

X

!21

p

q¼1

"ijqTijq

!T

21

GT

ijP~ aij?

X

ijP~ aij

p

q¼1

"ijqnijq

!22

NT

~ aT

ijPGij?

2

6

Since P40 and ?51, one has

2

6

6

4

X

p

q¼1

"ijqnT

ijq

~ aT

ijPNij

~ aT

ijP~ aij?

X

p

q¼1

"ijqvijq

2

6

6

6

6

4

6

6

6

6

3

7

7

7

7

5

7

7

7

7

5

ð? ? 1ÞP

0

00

00

000

6

4

3

7

7

5

ð33Þ

!11?

X

!21

p

q¼1

"ijqTijq

!T

21

GT

ijP~ aij?

X

ijP~ aij

p

q¼1

"ijqnijq

!22

NT

~ aT

ijPGij?

X

p

q¼1

"ijqnT

ijq

~ aT

ijPNij

~ aT

ijP~ aij?

X

~ p

q¼1

"ijqvijq

6

6

6

6

6

6

6

3

7

7

5

7

7

7

7

7

7

7

50

ð34Þ

The inequality (34) is equivalent to (12). Hence, in the

case of i 62^I, if condition (27) is satisfied, then the

discrete-time affine T–S fuzzy model is strictly input

passive and Lyapunov stable. The proof of (26) for rule

including the origin (i 2^I) is similar to (27) by setting ~ ai

and ?ijqbe all zero. Hence, the proof of (26) is omitted

here and the proof of this theorem is completed.

In order to achieve the conditions (26) and (27), an

ILMI algorithm is developed to interactively search for

P Fi, ?, "ijqand to update the auxiliary variable X until

?<1. The operation process of the present ILMI

algorithm is proposed as follows:

ILMI algorithm

Step 1:

follows:

Define the iterative auxiliary variable as

Xt ð Þ¼ P?1

t?1

ðÞ

ð35Þ

where t is iterative index. Set t¼1 and the initial P(0)

can be calculated by the following Riccati equation.

^A

TP0 ð Þ^A ? P0 ð Þ?

þ Q ¼ 0

where^A ¼Pr

^A

TP0 ð Þ^B

??

1 þ^B

TP0 ð Þ^B

???1

^B

TP0 ð Þ^A

??

ð36Þ

i¼1Ai=r,^B ¼Pr

i¼1Bui=r and Q>0. The

matrix Q is assigned by the designers. Through the

initial auxiliary variable X(1), the ILMI algorithm can

be started as following steps.

Step 2:

optimisation problem for P(t), Fi(t)and "ijq(t)from (26)

and (27) subject to minimising ?(t). That is,

Use the auxiliary variable X(t)to solve the

Minimise ?t ð Þ

Subject to Pt ð Þ40, "ijq t ð Þ40, XT

? Xt ð Þ? 0 ð26Þ for i 2^I and ð27Þ for i 62^I:

If ?t ð Þ51, then P(t), Fi(t)and "ijq t ð Þobtained in Step 2

are the feasible solutions for the conditions (26) and

(27). The iterative manner is stopped in this step.

Otherwise, go to Step 3.

t ð ÞPt ð ÞXt ð Þ

Step 3:

for P(t), Fi(t)and "ijq t ð Þby using ?(t)and X(t)determined

in Step 2.

Resolve the following optimisation problem

Minimise trace ðPt ð ÞÞ

Subject to Pt ð Þ40, "ijq t ð Þ40, XT

? Xt ð Þ? 0 ð26Þ for i 2^I and ð27Þ for i 62^I:

If kX?1

small value, then the conditions (26) and (27) are not

feasible and then stop the iterative manner. Otherwise,

set t¼tþ1 to update the auxiliary variable X(t)by

Xt ð Þ¼ P?1

back to the Step 2.

t ð ÞPt ð ÞXt ð Þ

t ð Þ? Pt ð Þk5?, where ? is a pre-determined

t?1

ðÞwith the obtained P(t?1)in Step 3 and go

According to the above ILMI algorithm, a numerical

exampleis presentedto

of proposed fuzzy controller design approach for

discrete-time affine T–S fuzzy models. Using the

proposed fuzzy controller design procedure, the

feasible solutions P>0, Fiand "ijq40 can be solved

by the LMI toolbox of MATLAB to satisfy conditions

(26) and (27) of Theorem 2.

showtheapplications

4. A numerical example

In this example, a computer simulated truck-trailer

system is considered to illustrate the applications

of proposed fuzzy controller design approach. The

dynamic equations of the nonlinear truck-trailer

system have been discussed in Tanaka and Sano

(1994) such as

x1ðk þ 1Þ ¼ 1:3636x1ðkÞ ? 0:7143uðkÞ þ 0:1wðkÞ ð37aÞ

x2ðk þ 1Þ ¼ x2ðkÞ ? 0:3636x1ðkÞð37bÞ

x3ðk þ 1Þ ¼ x3ðkÞ ? 2sin x2ðkÞ ? 0:1818x1ðkÞ½?ð37cÞ

yðkÞ ¼ x1ðkÞ þ wðkÞð37dÞ

International Journal of Systems Science815

Page 8

Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008

where x1(k) is the angle difference between truck and

trailer (note that x1(k) corresponds to two ‘jackknife’

positions, 90?and ?90?); x2(k) is the angle of the

trailer; x3(k) is the vertical position of rear end of the

trailer; w(k) is the external disturbance input provided

by a random zero-mean white noise with variance 0.1.

It can be found that the equilibrium point of (37) is

x1ðkÞ ¼ x2ðkÞ ¼ x3ðkÞ ¼ uðkÞ ¼ wðkÞ ¼ 0. In addition

to the equilibrium point, we choose other two

operating points to obtain the linearised subsystems

for the truck-trailer system (37). Thus, these three

operating points are given as follows:

ðx?,u?,w?Þoper1¼ ð?88o? 90o0j0 0 j Þð38aÞ

ðx,u,wÞoper2¼ ð0o0o0j0 0 j Þ

ð38bÞ

ðxþ,uþ,wþÞoper3¼ ð88o90o0j0 0 j Þ

The structure of affine T–S fuzzy model for the

nonlinear truck-trailer system (37) can be constructed

by linearisation method through the above three linear

operating points with the membership functions

described in Figures 1 and 2. Thus, one can represent

the discrete affine T–S fuzzy model, which is composed

by the nine fuzzy rules, as follows.

ð38cÞ

Plant part:

RulePlant1: IF x1(k) is M11and x2(k) is M12THEN

xðk þ 1Þ

¼

1:3636

?0:3636

0:1002

00

10

?0:5513 1

2

4

6

6

2

4

3

5xðkÞ

7

7

3

5wðkÞ þþ

?0:7143

0

0

3

5uðkÞ þ

?xðkÞ þ wðkÞ

7

0:1

0

0

2

4

6

0

0

1:2105

2

4

6

3

5

7

yðkÞ ¼ 1

RulePlant2: IF x1(k) is M21and x2(k) is M12THEN

2

4

?0:7143

0

6

00

?

ð39aÞ

xðk þ 1Þ ¼

1:3636

?0:3636

0

2

4

00

10

0

3

5uðkÞ þ

?xðkÞ þ wðkÞ

1

6

3

5xðkÞ

7

þ

0

7

0:1

0

0

2

4

6

3

5wðkÞ þ

7

0

0

2

2

4

6

3

5

7

yðkÞ ¼ 1

RulePlant3: IF x1(k) is M31and x2(k) is M12THEN

2

6

?0:7143

0

6

00

?

ð39bÞ

xðkþ1Þ¼

1:363600

?0:3636

?0:1002 0:5513 1

2

4

10

6

4

3

7

0:1

6

7

5xðkÞ

2

6

þ

0

6

3

7

7

5uðkÞþ

0

0

4

3

7

7

5wðkÞþ

0

0

2:9424

2

6

6

4

3

7

7

5

yðkÞ¼ 1 0 0

RulePlant4: IF x1(k) is M11and x2(k) is M22THEN

2

6

?0:7143

0

6

??xðkÞþwðkÞð39cÞ

xðkþ1Þ¼

1:3636 00

?0:3636

0:3495

2

4

10

?1:9225 1

3

7

6

4

3

7

7

5xðkÞ

0:1

7

þ

0

67

5uðkÞþ

0

0

2

6

6

4

3

7

5wðkÞþ

0

0

?0:0144

2

6

6

4

3

7

7

5

yðkÞ¼ 1 0 0

??xðkÞþwðkÞð39dÞ

1

0

M12

M32

M22

x2(k)

−90°

−45°

90°

45°

Figure 2. Membership function of x2(k).

M11

M31

M21

x1(k)

−88°

−50°

88°

50°

1

0

Figure 1. Membership function of x1(k).

816W.-J. Chang et al.

Page 9

Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008

RulePlant5: IF x1(k) is M21and x2(k) is M22THEN

2

6

?0:7143

0

6

xðk þ 1Þ ¼

1:363600

?0:3636

0:3636

2

4

10

?2

3

7

1

6

4

3

7

7

5xðkÞ

2

6

þ

0

67

5uðkÞ þ

0:1

0

0

6

4

3

7

7

5wðkÞ

yðkÞ ¼ 100

??xðkÞ þ wðkÞð39eÞ

RulePlant6: IF x1(k) is M31and x2(k) is M22THEN

2

6

?0:7143

0

6

xðkþ1Þ¼

1:3636 00

?0:3636

0:3495

2

4

10

?1:9225 1

3

7

6

4

3

7

7

5xðkÞ

0:1

7

þ

0

6

7

5uðkÞþ

0

0

2

6

6

4

3

7

5wðkÞþ

0

0

0:0144

2

6

6

4

3

7

7

5

yðkÞ¼ 1 0 0

??xðkÞþwðkÞð39fÞ

RulePlant7: IF x1(k) is M11and x2(k) is M32THEN

2

4

?0:7143

0

6

xðkþ1Þ¼

1:3636

?0:3636

?0:1002 0:5513 1

2

4

?xðkÞþwðkÞ

00

10

6

3

5xðkÞ

0:1

6

7

þ

0

3

5uðkÞþ

7

0

0

2

4

3

5wðkÞþ

7

0

0

?2:9424

2

4

6

3

5

7

yðkÞ¼ 1 0 0

?

ð39gÞ

RulePlant8: IF x1(k) is M21and x2(k) is M32THEN

2

6

?0:7143

0

6

xðk þ 1Þ ¼

1:363600

?0:3636

0

2

4

10

0

3

7

1

6

4

3

7

7

5xðkÞ

þ

0

67

5uðkÞ þ

0:1

0

0

2

6

6

4

3

7

7

5wðkÞ þ

0

0

?2

2

6

6

4

3

7

7

5

yðkÞ ¼ 100

??xðkÞ þ wðkÞð39hÞ

RulePlant9: IF x1(k) is M31and x2(k) is M32THEN

xðkþ1Þ¼

1:363600

?0:3636

0:1002

2

6

10

?0:5513 1

3

7

2

6

6

4

3

7

7

5xðkÞ

0:1

7

þ

?0:7143

0

0

6

4

7

5uðkÞþ

0

0

2

6

6

4

3

7

5wðkÞþ

0

0

?1:2105

2

6

6

4

3

7

7

5

yðkÞ¼ 1 0 0

??xðkÞþwðkÞð39iÞ

The corresponding matrices of S-procedure are pre-

sented as follows:

For Rule 111, ?88o? x1k ð Þ ? ?50o, which means

2

6

?1

6

T111¼

100

000

000

6

2

6

ð

4

3

7

7

5,

n111¼

2?50 ? 88

ðÞ?=180

½?

0

0

4

3

7

7

5and

Þ

v111¼ ?50?=180

Þ ? ?88?=180

ðð40aÞ

For Rule 331, 50o? x1k ð Þ ? 88o, which means

2

6

?1

6

T331¼

100

000

000

6

2

6

ð

4

3

7

7

5,

n331¼

250 þ 88

ðÞ?=180

½?

0

0

4

3

7

Þ

7

5and

v331¼ 50?=180

Þ ? 88?=180

ðð40bÞ

For Rule 112, ?90o? x2k ð Þ ? ?45o, which means

2

6

?1

6

T111¼

100

000

000

6

2

6

ð

4

3

7

7

5,

n111¼

2?45 ? 90

ðÞ?=180

½?

0

0

4

3

7

7

5

and v111¼ ?45?=180

Þ ? ?90?=180

ðÞð40cÞ

International Journal of Systems Science817

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For Rule 332, 45o? x2k ð Þ ? 90o, which means

2

6

?1

6

T331¼

100

000

000

6

2

6

ð

4

3

7

7

5,

n331¼

245 þ 90

ðÞ?=180

½?

0

0

4

3

7

Þ

7

5and

v331¼ 45?=180

Þ ? 90?=180

ðð40dÞ

where the notion Rule ijq means the activation region

between RulePlanti and RulePlantj of the plant part on

the xq(k), e.g., Rule 123 means the activation region

between RulePlant1 and RulePlant2 for the x3(k) on the

plant part.

For the truck-trailer system (37), the control

purpose is to realise the backward movement of the

truck-trailer system to guarantee the Lyapunov stabi-

lity and passivity for the closed-loop system. Through

the discrete affine T–S fuzzy model (39), the fuzzy

controller can be designed by applying Theorem 2 and

fuzzy controller design procedure, i.e. ILMI algorithm.

First, let us choose the dissipative rate ? ¼1 and set

the matrix Q as the identity matrix. Then, the initial

matrix P(0)can be solved via Riccati Equation (36)

as follows:

P0 ð Þ¼

51:104

?29:3354

55:9672

8:5445

?29:3354

8:5445

?14:4917

7:3738

?14:4917

2

6

6

4

3

7

7

5

ð41Þ

Thus, the matrix X(1)can be obtained from (35)

and (41).

X1 ð Þ¼ P?1

0 ð Þ¼

0:02820:0129

?0:0074

0:06810:0129 0:0217

?0:00740:06810:2781

2

6

6

4

3

7

7

5

ð42Þ

Using the initial auxiliary variable (42), one can obtain

the minimum ? from Step 2. If ?<1, then one can

obtain the feasible solutions for fuzzy controllers.

Otherwise, if ? ? 1, a new matrix P(1) should be

re-calculated from the Step 3 and then be substituted

into Step 1 to produce new auxiliary parameter X(2).

Repeating this ILMI algorithm until ?<1, one

can obtain the feasible solution of suitable fuzzy

controllers. In this example, the feasible solution is

obtained after six iterations of the ILMI algorithm.

The final ? is 0.9009 and the feasible solutions are

obtained as

2

6

0:60650:1749

6

P ¼

3:6494

?6:0895

18:3963

0:642

?6:0895

0:642

?1:8353

0:27

?1:8353

6

2

6

4

3

7

7

5,

X ¼

?0:2718

1:06520:1749 0:2185

?0:27181:0652 11:5266

4

3

7

7

5,

"111¼ "331¼ 19:8444:

Then, the PDC-based fuzzy controller can be obtained

as follows:

Controller part:

RuleController1: IF x1(k) is M11and x2(k) is M12THEN

?

RuleController2: IF x1(k) is M21and x2(k) is M12THEN

?

RuleController3: IF x1(k) is M31and x2(k) is M12THEN

?

RuleController4: IF x1(k) is M11and x2(k) is M22THEN

?

RuleController5: IF x1(k) is M21and x2(k) is M22THEN

?

RuleController6: IF x1(k) is M31and x2(k) is M22THEN

?

RuleController7: IF x1(k) is M11and x2(k) is M32THEN

?

RuleController8: IF x1(k) is M21and x2(k) is M32THEN

?

RuleController9: IF x1(k) is M31and x2(k) is M32THEN

?

Settingtheinitial

x 0 ð Þ ¼ 880

states are shown in Figures 3–5. In the simulations,

uðkÞ ¼ ?2:58142:73

?0:2568

?xðkÞð43aÞ

uðkÞ ¼ ?2:67743:0386

?0:2826

?xðkÞð43bÞ

uðkÞ ¼ ?2:91023:0744

?0:3067

?xðkÞð43cÞ

uðkÞ ¼ ?2:70732:8102

?0:2485

?xðkÞð43dÞ

uðkÞ ¼ ?2:70762:8294

?0:2485

?xðkÞð43eÞ

uðkÞ ¼ ?2:70732:8102

?0:2485

?xðkÞð43fÞ

uðkÞ ¼ ?2:91023:0744

?0:3067

?xðkÞð43gÞ

uðkÞ ¼ ?2:67723:0382

?0:2825

?xðkÞð43hÞ

uðkÞ ¼ ?2:58142:73

?0:2568

?xðkÞ

condition

ð43iÞ

as

of

600

5

??T,thesimulation results

818 W.-J. Chang et al.

Page 11

Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008

the sampling time is set as 0.1 second. For checking the

satisfaction of the strictly input passive property, let us

substitute the simulation results into the following

ratio function.

Xkq

k¼0wTðkÞwðkÞ

It is easy to show that the value of the above ratio

function (44) is large than dissipative rate ? ¼1. Hence,

from these simulation responses and (44), one can find

that the closed-loop system is Lyapunov stable and

strictly input passive via the fuzzy controller (43).

2

Xkq

k¼0yTðkÞwðkÞ

¼ 1:8315

ð44Þ

5. Conclusions

In this article, the Lyapunov function associated with

the stored function for passivity property was con-

sidered. Using the properties of the passivity, this

article has discussed stability analysis and controller

synthesis for the discrete affine T–S fuzzy models.

Through applying the affine T–S fuzzy model, the

complex nonlinear system can be controlled accurately.

Based on the auxiliary variables and LMI technique,

the ILMI algorithm was employed to obtain the

feasible solutions of BMI stability conditions. The

nonlinear truck-trailer system has been provided to

demonstratetheeffectiveness

of the proposed design approach in this article.

Through the results of the simulation, one can find

that the nonlinear truck-trailer system can achieve the

strict input passivity and Lyapunov stability via the

proposed fuzzy controller design.

and applicability

Acknowledgements

The authors would like, to express their sincere gratitude to

anonymous reviewers, who gave us some constructive

comments, criticisms and suggestions. This work was

supported by the National Science Council of the Republic

of China under Contract NSC94-2213-E-019-006.

Notes on contributors

Wen-Jer Chang received the the MS

degree in the Institute of Computer

Science and Electronic Engineering

from the National Central University

in 1990, and the PhD degree from the

Institute of Electrical Engineering of

the National Central University in

1995. Since 1995, he has been with

National Taiwan Ocean University,

Keelung, Taiwan, R.O.C. He is currently the Vice Dean of

Academic Affairs, Director of Center for Teaching and

Learning and a full Professor of the Department of Marine

05 1015 202530 3540 4550

−1

0

1

2

3

4

5

Second

X3 (Meter)

Figure 5. Responses of state x3(k).

0510 152025 3035 404550

−10

0

10

20

30

40

50

60

70

80

90

Second

X1 (Angle)

Figure 3. Responses of state x1(k).

0510 1520 253035 40 4550

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Second

X2 (Angle)

Figure 4. Responses of state x2(k).

International Journal of Systems Science 819

Page 12

Downloaded By: [Chang, Wen-Jer] At: 12:38 24 May 2008

Engineering of National Taiwan Ocean University. He is

now a life member of the CIEE, CACS, CSFAT and

SNAME. Since 2003, Dr Chang was listed in the Marquis

Who’s Who in Science and Engineering. In 2003, he also won

the outstanding young control engineers award granted by

the Chinese Automation Control Society (CACS). In 2004,

he won the universal award of accomplishment granted

by ABI of USA. In 2005, he was selected as an excellent

teacher of the National Taiwan Ocean University. His

recent research interests are fuzzy control, robust control,

performance constrained control.

Cheung-Chieh Ku received the BS and

MS degree from the Department

of MarineEngineering

National Taiwan Ocean University,

Taiwan,R.O.C.,

2006, respectively. He is currently

working towards the PhD degree

intheelectrical

the National Taiwan Ocean University,

ofthe

in 2001and

engineeringat

Taiwan R.O.C. His research interests focus on fuzzy control

and control system theory.

Wei Chang received the BS and MS

degree from the

Marine Engineering of the National

Taiwan Ocean University, Taiwan,

R.O.C. in 2003 and 2005, respectively.

Hisresearch interests

fuzzy control and dynamic system

analysis.

Departmentof

focuson

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Appendix 1. S-procedure

In this appendix, the structures of the matrices Tijq, nijqand

vijq of S-procedure defined in (16) are proposed. For

illustrating S-procedure, it is assumed that the range of x

in which the fuzzy rule i is activated (i.e., hix ð Þ 6¼ 0) is

represented by the following system of at most n linear

inequalities, where n is the number of states:

For x1,x1? ?i1or x1? ?i1or ?i1? x1? ?i1

ðA1Þ

For x2,x2? ?i2or x2? ?i2or ?i2? x2? ?i2...

ðA2Þ

For xn,xn? ?inor xn? ?inor ?in? xn? ?in

ðA3Þ

Here, we denote that q ¼ 1,...,n. The range of x can be

represented by the following quadratic inequalities.

. In casexq? ?iq,

iqx þ viq? 0, where

3

7

?T

xq? ?iq,

iqx þ viq? 0, where

3

7

?T

one has

Fiqx ð Þ ? xTTiqx þ 2mT

0

6

?

. Incase

Fiqx ð Þ ? xTTiqx þ 2mT

2

6

?

. In case ?iq? xq? ?iq, i.e., ðxq? ?iqÞðxq? ?iqÞ ? 0,

one has FiqðxÞ ? xTTiqx þ 2mT

0

.

6

6

6

?

Tiq¼ 0n?n¼

???

..

0

......

.

0

???

0

2

6

4

7

5, miq

0

¼ 0

???

1=2

qthelement

???

and viq¼ ??iq

ðA4Þ

onehas

Tiq¼ 0n?n¼

0

???

..

0

...

.

...

0

???

0

6

6

4

7

7

5, miq

???¼ 0

????1=2

qthelement

0and viq¼ ?iq

ðA5Þ

iqx þ viq? 0, where

Tiq¼

..

1

qthelementof

thediagonal

..

.

0

2

6

6

6

?

6

6

4

6

6

3

7

7

7

???

7

7

7

7

7

5

7

7

, miq

¼ 0

????1=2 ?iqþ ?iq

qthelement

?

0

?T

and viq¼ ?iq?iq

ðA6Þ

International Journal of Systems Science821