Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach
Kazuo Tanaka, Hua O. Wang
Copyright ᮊ 2001 John Wiley & Sons, Inc.
Ž. Ž .
ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic
CHAPTER 2
TAKAGI-SUGENO FUZZY
MODEL AND PARALLEL
DISTRIBUTED COMPENSATION
Recent years have witnessed rapidly growing popularity of fuzzy control
systems in engineering applications. The numerous successful applications
of fuzzy control have sparked a flurry of activities in the analysis and design
of fuzzy control systems. In this book, we introduce a wide range of analysis
and design tools for fuzzy control systems to assist control researchers and
engineers to solve engineering problems. The toolkit developed in this book
is based on the framework of the Takagi-Sugeno fuzzy model and the
so-called parallel distributed compensation, a controller structure devised in
accordance with the fuzzy model. This chapter introduces the basic concepts,
analysis, and design procedures of this approach.
This chapter starts with the introduction of the Takagi-Sugeno fuzzy
Ž.
model T-S fuzzy model followed by construction procedures of such models.
Then a model-based fuzzy controller design utilizing the concept of ‘‘parallel
distributed compensation’’ is described. The main idea of the controller
design is to derive each control rule so as to compensate each rule of a fuzzy
system. The design procedure is conceptually simple and natural. Moreover,
it is shown in this chapter that the stability analysis and control design
Ž.
problems can be reduced to linear matrix inequality LMI problems. The
design methodology is illustrated by application to the problem of balancing
and swing-up of an inverted pendulum on a cart.

Ž. Ž. Ž.
˙
ii
THEN i s 1,2,...,r.2.1
Ž.
½
y t s Cx t ,
Ž. Ž.
i
Discrete Fuzzy System: DFS
Model Rule i:
Ž. Ž.
IF ztis M and иии and ztis M ,
1 i1 pip
x t q 1 s Ax t q Bu t ,
Ž . Ž. Ž.
ii
THEN i s 1, 2, . . . , r.2.2
Ž.
½
y t s Cx t ,
Ž. Ž.
i
Ž.
n
Here, M is the fuzzy set and r is the number of model rules; x t g R is
ij
Ž.
m
Ž.

r
w z t Ax t q Bu t
Ä4
Ž. Ž. Ž.
Ž.
Ý
iii
i
s1
x t s
Ž.
˙
r
w z t
Ž.
Ž.
Ý
i
i
s1
r
s h z t Ax t q Bu t ,2.3
Ä4
Ž. Ž. Ž. Ž .
Ž.
Ý
iii
i
s1
r

Ž.
Ý
iii
i
s1
x t q 1 s
Ž.
r
w z t
Ž.
Ž.
Ý
i
i
s1
r
s h z t Ax t q Bu t ,2.5
Ä4
Ž. Ž. Ž. Ž .
Ž.
Ý
iii
i
s1
r
w z t Cx t
Ž. Ž.
Ž.
Ý
ii

Ž.
Ł
iijj
j
s1
w z t
Ž.
Ž.
i
h z t s 2.7
Ž. Ž .
Ž.
r
i
w z t
Ž.
Ž.
Ý
i
i
s1
ŽŽ.. Ž.
for all t. The term Mzt is the grade of membership of zt in M .
ij j j ij
Since
r
°
w z t ) 0,
Ž.
Ž.

for all t.
Example 1 Assume in the DFS that
p s n,
zts xt, zts xty 1 ,..., zts xty n q 1.
Ž. Ž. Ž. Ž . Ž. Ž .
12 n
Then, the model rules can be represented as follows.
Model Rule i:
Ž. Ž .
IF xt is M and иии and xty n q 1isM ,
i1 in
x t q 1 s Ax t q Bu t ,
Ž . Ž. Ž.
ii
THEN i s 1,2,...,r,
½
y t s Cx t ,
Ž. Ž.
i
Ž. w Ž. Ž . Ž .x
T
where x t s xt xty 1 иии xty n q 1.
Remark 1 The Takagi-Sugeno fuzzy model is sometimes referred as the
Ž.
Takagi-Sugeno-Kang fuzzy model TSK fuzzy model in the literature. In this
Ž. Ž.
book, the authors do not refer to 2.1 and 2.2 as the TSK fuzzy model. The
CONSTRUCTION OF FUZZY MODEL
9
reason is that this type of fuzzy model was originally proposed by Takagi and

method and the Newton-Euler method. In such cases, the second approach,
which derives a fuzzy model from given nonlinear dynamical models, is more
appropriate. This section focuses on this second approach. This approach
utilizes the idea of ‘‘sector nonlinearity,’’ ‘‘local approximation,’’ or a combi-
nation of them to construct fuzzy models.
2.2.1 Sector Nonlinearity
The idea of using sector nonlinearity in fuzzy model construction first
wx
appeared in 10 . Sector nonlinearity is based on the following idea. Consider
Ž. Ž Ž.. Ž.
a simple nonlinear system xt s fxt , where f 0 s 0. The aim is to find
˙
Ž. Ž Ž.. wxŽ.
the global sector such that xt s fxt g aaxt. Figure 2.2 illustrates
˙
12
the sector nonlinearity approach. This approach guarantees an exact fuzzy
model construction. However, it is sometimes difficult to find global sectors
for general nonlinear systems. In this case, we can consider local sector
nonlinearity. This is reasonable as variables of physical systems are always
bounded. Figure 2.3 shows the local sector nonlinearity, where two lines
Ž.
become the local sectors under yd - xt - d. The fuzzy model exactly
Ž.
represents the nonlinear system in the ‘‘local’’ region, that is, yd - xt - d.
The following two examples illustrate the concrete steps to construct fuzzy
models.
Fig. 2.2 Global sector nonlinearity.
CONSTRUCTION OF FUZZY MODEL
11

x t s x t ,
Ž. Ž.
˙
2
3 q xt xt y1
Ž. Ž.
Ž.
21
Ž. w Ž. Ž.x
T
Ž.
2
Ž. Ž Ž..
2
Ž.
where x t s xtxt and xtx tand 3 q xtxt are nonlinear
12 12 2 1
Ž. Ž.
2
Ž. Ž. Ž
terms. For the nonlinear terms, define zt' xtx t and zt' 3 q
112 2
Ž..
2
Ž.
xtxt. Then, we have
21
y1 zt
Ž.
1

zts xtx
2
t s Mzt и 1 q Mzt и y1,
Ž. Ž. Ž. Ž. Ž. Ž .
Ž. Ž.
112 11 21
zts 3 q xt x
2
t s Nz t и 4 q Nzt и 0,
Ž. Ž. Ž. Ž. Ž.
Ž. Ž.Ž.
2211222
where
Mzt q Mzt s 1,
Ž. Ž.
Ž. Ž.
11 21
Nzt q Nzt s 1.
Ž. Ž.
Ž. Ž.
12 2 2
Therefore the membership functions can be calculated as
ztq 11y zt
Ž. Ž.
11
Mzt s , Mzt s ,
Ž. Ž.
Ž. Ž.
11 21
22

IF ztis ‘‘Negative’’ and ztis ‘‘Big,’’
12
CONSTRUCTION OF FUZZY MODEL
13
Ž Ž .. Ž Ž ..
Fig. 2.4 Membership functions Mzt and Mzt.
11 21
ŽŽ.. ŽŽ..
Fig. 2.5 Membership functions Nzt and Nzt.
12 22
Ž. Ž.
THEN x t s Axt .
˙
3
Model Rule 4:
Ž. Ž.
IF ztis ‘‘Negative’’ and ztis ‘‘Small,’’
12
Ž. Ž.
THEN x t s Axt .
˙
4
Here,
y11 y11
A s , A s ,
12
4 y10y1
y1 y1 y1 y1
A s , A s .
34

h z t s Mzt = Nzt.
Ž. Ž. Ž.
Ž.
Ž. Ž.
42122
This fuzzy model exactly represents the nonlinear system in the region
wxwx
y1, 1 = y1, 1 on the x -x space.
12
wx
Example 3 The equations of motion for the inverted pendulum 21 are
xts xt,
Ž. Ž.
˙
12
g sin xt y amlx
2
t sin 2 xtr2 y a cos xt ut
Ž. Ž. Ž. Ž. Ž.
Ž. Ž . Ž.
121 1
xts ,
Ž.
˙
2
2
4lr3 y aml cos xt
Ž.
Ž.
1

21
g sin xt y xty a cos xt ut .
Ž. Ž. Ž. Ž.
Ž. Ž.
121
ž/
2
2.12
Ž.
Define
1
zt' ,
Ž.
1
2
4lr3 y aml cos xt
Ž.
Ž.
1
zt' sin xt ,
Ž. Ž.
Ž.
21
zt' xtsin 2 xt ,
Ž. Ž. Ž.
Ž.
32 1
zt' cos xt ,
Ž. Ž.
Ž.

As shown in Example 2, we replace zty zt with T-S fuzzy model
14
representation. Since
1
max zts ' q ,
␤
s cos 88Њ ,
Ž. Ž .
11
2
4lr3 y aml
␤
Ž.
xt
1
1
min zts ' q ,
Ž.
12
4lr3 y aml
Ž.
xt
1
Ž.
ztcan be rewritten as
1
2
zts Ezt q, 2.13
Ž. Ž. Ž .
Ž.

two lines bx and bx, where the slopes are b s 1 and b s 2r
␲
.
11 2 1 1 2
ŽŽ..
Fig. 2.6 sin xt and its sector.
1
TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION
16
ŽŽ..
Therefore, we represent sin xt as follows:
1
2
zts sin xt s Mz t b xt. 2.14
Ž. Ž. Ž. Ž. Ž .
Ž. Ž.
Ý
21 i 2 i 1
ž/
i
s1
w ŽŽ.. ŽŽ.. x
From the property of membership functions Mzt q Mzts 1,we
12 2 2
can obtain the membership functions
°
y1
zty 2r
␲
Sin zt

y1
Mzt s
Ž.
Ž.
1 y 2r
␲
Sin zt
Ž.Ž.
Ž.
22
2
¢
0, otherwise.
Ž. Ž. Ž Ž..
Next, consider zts xtsin 2 xt. Since
32 1
max zts
␣
' c and min ztsy
␣
' c ,
Ž. Ž.
31 3 2
Ž. Ž. Ž. Ž.
xt, xt xt, xt
12 12
Ž.
we can derive in the same way as the ztcase:
1
2

2
zts cos xt s S z td, 2.16
Ž. Ž. Ž. Ž .
Ž.
Ž.
Ý
41 ii
i
s1
where
zty ddy zt
Ž. Ž.
42 14
Szt s , Szt s .
Ž. Ž.
Ž. Ž.
14 24
d y ddy d
12 12
CONSTRUCTION OF FUZZY MODEL
17
Ž.Ž.
From 2.13 ᎐ 2.16 , we construct the following Takagi-Sugeno fuzzy model
for the inverted pendulum:
22 22
xt
Ž.
˙
1
s Ezt Mz t N z t Sz t

Ž. Ž. Ž. Ž.
Ž.Ž.Ž.Ž.
ÝÝÝÝ
i 1 j 2 k 3 l 4
i
s1 js1 ks1 ls1
= Axt q B ut . 2.17
Ž. Ž. Ž .
Ä4
ijkl ijkl
Ž.
The summations in 2.17 can be aggregated as one summation:
16
x t s h z t A*x t q B*ut , 2.18
Ž. Ž. Ž. Ž. Ž .
Ž.
Ä4
˙
Ý
␳
␳␳
␳
s1
where
␳
s l q 2 k y 1 q 4 j y 1 q 8 i y 1,
Ž.Ž.Ž.
h z t s Ezt Mz t N z t Szt ,
Ž. Ž. Ž. Ž. Ž.
Ž.

˙
22
Model Rule 3:
Ž. Ž.
IF ztis ‘‘Positive’’ and ztis ‘‘Zero’’
12
Ž. Ž.
and ztis ‘‘Negative’’ and ztis ‘‘Big,’’
34
Ž. Ž. Ž.
THEN x t s A*x t q B*ut.
˙
33
TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION
18
Model Rule 4:
Ž. Ž.
IF ztis ‘‘Positive’’ and ztis ‘‘Zero’’
12
Ž. Ž.
and ztis ‘‘Negative’’ and ztis ‘‘Small,’’
34
Ž. Ž. Ž.
THEN x t s A*x t q B*ut.
˙
44
Model Rule 5:
Ž. Ž.
IF ztis ‘‘Positive’’ and ztis ‘‘Not Zero’’
12

Ž. Ž.
IF ztis ‘‘Positive’’ and ztis ‘‘Not Zero’’
12
Ž. Ž.
and ztis ‘‘Negative’’ and ztis ‘‘Small,’’
34
Ž. Ž. Ž.
THEN x t s A*x t q B*ut.
˙
88
Model Rule 9:
Ž. Ž.
IF ztis ‘‘Negative’’ and ztis ‘‘Zero’’
12
Ž. Ž.
and ztis ‘‘Positive’’ and ztis ‘‘Big,’’
34
Ž. Ž. Ž.
THEN x t s A*x t q B*ut.
˙
99
CONSTRUCTION OF FUZZY MODEL
19
Model Rule 10:
Ž. Ž.
IF ztis ‘‘Negative’’ and ztis ‘‘Zero’’
12
Ž. Ž.
and ztis ‘‘Positive’’ and ztis ‘‘Small,’’
34

Ž. Ž.
and ztis ‘‘Positive’’ and ztis ‘‘Big,’’
34
Ž. Ž. Ž.
THEN x t s A *x t q B *ut.
˙
13 13
Model Rule 14:
Ž. Ž.
IF ztis ‘‘Negative’’ and ztis ‘‘Not Zero’’
12
Ž. Ž.
and ztis ‘‘Positive’’ and ztis ‘‘Small,’’
34
Ž. Ž. Ž.
THEN x t s A *x t q B *ut.
˙
14 14
Model Rule 15:
Ž. Ž.
IF ztis ‘‘Negative’’ and ztis ‘‘Not Zero’’
12
Ž. Ž.
and ztis ‘‘Negative’’ and ztis ‘‘Big,’’
34
Ž. Ž. Ž.
THEN x t s A *x t q B *ut.
˙
15 15
TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION

2
01 0
UU
aml
A s A s , B s B s ,
3 1121 3 1121
g и qb y и qc ya и qd
11 12 11
2
01 0
UU
aml
A s A s , B s B s ,
4 1122 4 1122
g и qb y и qc ya и qd
11 12 12
2
01 0
UU
aml
A s A s , B s B s ,
5 1211 5 1211
g и qb y и qc ya и qd
12 11 11
2
01 0
UU
aml
A s A s , B s B s ,
6 1212 6 1212

UU
aml
A s A s , B s B s ,
10 2112 10 2112
g и qb y и qc ya и qd
21 21 22
2
01 0
UU
aml
A s A s , B s B s ,
11 2121 11 2121
g и qb y и qc ya и qd
21 22 21
2
01 0
UU
aml
A s A s , B s B s ,
12 2122 12 2122
g и qb y и qc ya и qd
21 22 22
2
01 0
UU
aml
A s A s , B s B s ,
13 2211 13 2211
g и qb y и qc ya и qd
22 21 21

Ž. Ž.
Ž. Ž.
11 21
q y qqy q
12 12
sin xt y 2r
␲
zt xty zt
Ž. Ž . Ž. Ž. Ž.
Ž.
12 12
Mzt s , Mzt s ,
Ž. Ž.
Ž. Ž.
12 22
1 y 2r
␲
zt 1 y 2r
␲
zt
Ž . Ž. Ž . Ž.
22
zty ccy zt
Ž. Ž.
32 13
Nzt s , Nzt s ,
Ž. Ž.
Ž. Ž.
13 2 3
c y ccy c