Tài liệu Hệ thống điều khiển mờ - Thiết kế và phân tích P10 ppt

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 10
FUZZY DESCRIPTOR SYSTEMS
AND CONTROL
This chapter deals with a fuzzy descriptor system defined by extending the
original Takagi-Sugeno fuzzy model. A number of stability conditions for the
fuzzy descriptor system are derived and represented in terms of LMIs.
A motivating example for using the fuzzy descriptor system instead of the
original Takagi-Sugeno fuzzy model is presented. An LMI-based design
approach is employed to find stabilizing feedback gains and a common
Lyapunov function.
The descriptor system, which differs from a state-space representation, has
generated a great deal of interest in control systems design. The descriptor
system describes a wider class of systems including physical models and
wx
nondynamic constraints 1 . It is well known that the descriptor system is
much tighter than the state-space model for representing real independent
parametric perturbations. There exist a large number of papers on the
stability analysis of the T-S fuzzy systems based on the state-space represen-
tation. In contrast, the definition of a fuzzy descriptor system and its stability
wx wx
analysis have not been discussed until recently 2 . In 2 we introduced the
fuzzy descriptor systems and analyzed the stability of such systems. This
wx
chapter presents both the basic framework of 2, 3 as well as some new
developments on this topic.
Ž.

As mentioned in Chapter 1, h l® / or denotes all the pairs i, k
ik
ŽŽ.. ŽŽ.. Ž.
excepting h z t ® z t s 0 for all z t ; h l h l® / or denotes all
ik ijk
Ž. ŽŽ..ŽŽ.. ŽŽ.. Ž.
the pairs i, j, k excepting h z thz t ® z t s 0 for all z t ; and
ijk
ŽŽ.. ŽŽ.. ŽŽ..
i - j s.t. h l h l® / or denotes all i - j excepting h z thz t ® z t
ijk i j k
Ž.
s 0, ᭙ z t .
195
FUZZY DESCRIPTOR SYSTEMS AND CONTROL
196
10.1 FUZZY DESCRIPTOR SYSTEM
wx
In 4, 5 , a fuzzy descriptor system is defined by extending the T-S fuzzy
Ž. Ž.
model 2.3 and 2.4 . The ordinary Takagi-Sugeno fuzzy model is a special
case of the fuzzy descriptor system. We derive stability conditions for the
fuzzy descriptor system, where the E matrix in the fuzzy descriptor system
is assumed to be not always nonsingular. The fuzzy descriptor system is
defined as
r
e
r
® z t Ext s h z t Ax t q Bu t ,
Ž. Ž. Ž. Ž. Ž.

Ž. Ž.
Ž.
˙
ÝÝ
kk i i i
k
s1 is1
10.1
Ž.
r
y t s h z t Cx t ,
Ž. Ž. Ž.
Ž.
Ý
ii
i
s1
where
x t g R
n
, y t g R
q
, u t g R
m
,
Ž. Ž. Ž.
r
h z t G 0, h z t s 1,
Ž. Ž.
Ž. Ž.

Ý
ii
i
s1
r
e
® z t G 0, ® z t s 1.
Ž. Ž.
Ž. Ž.
Ý
kk
k
s1
Here x g R
n
is the descriptor vector, u g R
m
is the input vector, y g R
q
is
the output vector, E g R
n=n
, A g R
n=n
, B g R
n=m
, and C g R
q=n
. The
kii i

Ž. Ž.
known premise variables zt; ztmay be functions of the states, external
1 p
disturbances, andror time.
wx wx
Remark 32 A fuzzy descriptor system was first defined in 2 . In 2 , a
Ž Ž .. Ž Ž ..
e
wx
special case, that is, h z t s® z t and r s r , was presented. In 4, 5 ,
ik
Ž.
the fuzzy descriptor system was generalized as shown in 10.1 .
Ž. w
T
Ž.
T
Ž.x
T
Ž.
By defining x* t s x t x t , the fuzzy descriptor system 10.1 can
˙
be rewritten as
rr
e
E*x* t s h z t ® z t A* x* t q B*u t ,
Ž. Ž. Ž. Ž. Ž.
Ž.Ž.
Ž.
˙

ÝÝ
ik ik i
i
s1 ks1
10.2
Ž.
r
y t s h z t C*x* t ,
Ž. Ž. Ž.
Ž.
Ý
ii
i
s1
where
0 I
I 0
E* s , A* s ,
ik
A yE
00
ik
0
C 0
B* s , C* s .
i
ii
B
i
Ž.

In the following the stability for the fuzzy descriptor system 10.2 is
considered.
STABILITY CONDITIONS
197
10.2 STABILITY CONDITIONS
Ž.
The open-loop systems of 10.2 is defined as follows:
rr
e
E*x* t s h z t ® z t A* x* t .10.3
Ž. Ž. Ž. Ž. Ž .
Ž.Ž.
˙
ÝÝ
ik ik
i
s1 ks1
Ž.
The fuzzy descriptor system 10.3 is quadratically stable if
dV x* t
Ž.
Ž.
Fy
␣
x* t ,
Ž.
2
dt
where
V x* t s x*

T
t E*
T
Xx* t ,
Ž. Ž. Ž.
Ž.
and the following conditions are satisfied:
rr
e
Ž.
1. det sE* y h z t ® z t A* / 0 and the open-loop system
Ž. Ž.
Ž.Ž.
ÝÝ
ik ik
i
s1 ks1
is impulse free.
2. There exist a common matrix X and
␣
) 0 such that
X g R
2 n=2 n
, E*
T
X s X
T
E* G 0, det X / 0.
Ž.
Theorem 33 gives a sufficient condition for ensuring the stability of 10.3 .

Ž.
THEOREM 33 The fuzzy descriptor system 10.3 is quadratically stable if
there exists a common matrix X such that
E*
T
Xs X
T
E* G 0,10.4
Ž.
A*
T
X q X
T
A* - 0, h l® / or.10.5
Ž.
ik ik i k
Proof. Consider a candidate of the quadratic function
V x* t s x*
T
t E*
T
Xx* t .
Ž. Ž. Ž.
Ž.
Then,
rr
e
U T U
TT
˙

V x* t s h z t ® z t x* t AXq XA x* t .
Ž. Ž. Ž. Ž. Ž.
Ž. Ž.Ž.
Ž.
ÝÝ
i k ik ik
i
s1 ks1
Therefore, we have the following stability conditions:
A*
T
X q X
T
A* - 0, h l® / or. Q.E.D.
Ž.
ik ik i k
FUZZY DESCRIPTOR SYSTEMS AND CONTROL
198
Ž.
Remark 33 As mentioned before, h l® / or denotes ‘‘all the pairs i, k
ik
ŽŽ.. ŽŽ.. Ž.
excepting h z t ® z t s 0 for all z t .’’ In other words, we can ignore the
ik
Ž . Ž . Ž Ž .. Ž Ž .. Ž .
condition 10.5 for the pairs i, k such that h z t ® z t s 0 for all z t .
ik
Remark 34 In Theorem 33, X is not required to be positive definite.
Corollary 5 is needed to discuss the stability of closed-loop systems.
Ž. Ž. Ž. Ž.

COROLLARY 5 The conditions 10.6 and 10.7 imply 10.4 and 10.5 ,
where S is a positi®e definite matrix:
1
S s S
T
) 0,10.6
Ž.
11
TT
ASq SA )
i 33i
- 0, h l® / or,10.7
Ž.
ik
TT
S q SAy ES yESy SE
11ik3 k 11k
Ž.
where the asterisk denotes the transposed elements matrices for symmetric
Ž. Ž
T
.
T
positions. For example, in 10.7 , it represents S q SAy ES .
11ik3
Proof. Define X as
S 0
1
X s .
SS

31
Ž. Ž.
Then, 10.6 is obtained from 10.4 as follows:
I 0 S 0 S 0
11
T
E* X ssG0,
00 SS 00
31
TT T
SS I0 S 0
13 1
T
XE* ssG0.
T
0 S 00 0 0
1
Ž.
Equation 10.7 is obtained as follows:
A*
T
X q X
T
A*
ik ik
TTT
0 AS0 SS 0 I
i 113
sq
TT

I yESS 0 SAyE
k 31 1 ik
TT TT
ASq SA Sq ASy SE
i 33i 1 i 13k
s - 0. Q.E.D.
Ž.
TT
S q SAy ES yESy SE
11ik3 k 11k
STABILITY CONDITIONS
199
Remark 35 It is stated in Remark 34 that X is not required to be posi-
tive definite. However, in Corollary 5, X is assumed to be invertible since
S 0
1
X s , where S ) 0.
1
SS
31
Next, we consider stability conditions for closed-loop systems. We propose
Ž. Ž.
a modified PDC 10.8 to stabilize the fuzzy descriptor system 10.2 :
rr
e
u t sy h z t ® z t F* x* t ,10.8
Ž. Ž. Ž. Ž. Ž .
Ž.Ž.
ÝÝ
ik ik

i
s1 ks1
F 0
where F* s . The fuzzy controller design problem is to determine
ik
ik
the local feedback gains F .
ik
Ž. Ž.
By substituting 10.8 into 10.2 , the fuzzy control system is represented as
rrr
e
E*x* t s h z thz t ® z t A* y B*F* x* t .
Ž. Ž. Ž. Ž. Ž.
Ž.Ž.Ž.
˙
Ž.
ÝÝÝ
ijk ikijk
i
s1 js1 ks1
10.9
Ž.
Ž.
Theorem 34 gives a sufficient condition for ensuring the stability of 10.9 .
Ž.
THEOREM 34 The fuzzy descriptor system 10.2 can be stabilized ®ia the
Ž.
PDC fuzzy controller 10.8 if there exist Z , Z , and M such that
13 ik

Z
T
s Z ) 0, 10.10
Ž.
11
T
yZ y Z )
33
Z q AZ
- 0,
1 i 1
T
yZE y EZ
1 kk1
ž/
yBM q EZ
iik k3
h l® / or, 10.11
Ž.
ik
T
y2 Z y 2 Z )
33
2 Z q AZ
1 i 1
F 0,
T
yBM q AZ
y2 ZE y 2 EZ
ijk j1

1 kk1
0
yBM q 2 EZ
jik k3
i - j F r s.t. h l h l® / or, 10.12
Ž.
ijk
Ž.
where the asterisk denotes the transposed elements matrices for symmetric
positions.
FUZZY DESCRIPTOR SYSTEMS AND CONTROL
200
Proof. Consider a candidate of a quadratic function
V x* t s x*
T
t E*
T
Xx* t ,
Ž. Ž. Ž.
Ž.
where
S 0
1
X s .
SS
31
Then,
rrr
e
T

˙
V x* t s h z thz t ® z t x* t
Ž. Ž. Ž. Ž. Ž.
Ž . Ž.Ž.Ž.
ÝÝÝ
ijk
i
s1 js1 ks1
=
T
T
A* y B*F* X q XA* y B*F* x* t
Ž.
Ž.Ž.
½5
ik i jk ik i jk
rr
e
2 T
s h z t ® z t x* t
Ž. Ž. Ž.
Ž.Ž.
ÝÝ
ik
i
s1 ks1
=
T
T
A* y B*F* X q XA* y B*F* x* t

Ž.Ž.Ž.
Ä4
ik i ik ik i ik
rr
e
T
q 2 h z thz t ® z t x* t
Ž. Ž. Ž. Ž.
Ž.Ž.Ž.
ÝÝÝ
ijk
i
s1 i-jks1
=
T
A* y B*F* q A* y B*F*
ik i jk jk j ik
X
½
ž/
2
A* y B*F* q A* y B*F*
ik i jk jk j ik
T
qXx* t .
Ž.
5
ž/
2
Therefore, the stability conditions are derived as follows:

E*
T
Xs X
T
E* G 0, 10.13
Ž.
G
T
X q X
T
G - 0, h l® / or, 10.14
Ž.
iik iik i k
T
G q GGq G
ijk jik ijk jik
T
X q X F 0,
ž/ž/
22
i - j F r s.t. h l h l® / or, 10.15
Ž.
ijk
where
0 I
G s A* y B*F* s ,
ijk ik i jk
A y BF yE
iijk k
F 0

F* s .
ik
ik
STABILITY CONDITIONS
201
Ž.
Equation 10.13 can be rewritten as
X
yT
E*
T
s E*X
y1
G 0.
The above inequality is
yT y1
S 0 I 0 I 0 S 0
11
sG0.
SS 00 00 SS
31 31
Therefore, we obtain
TT
Z yZI0
13
T
0 Z 00
1
I 0 Z 0 Z 0
11

ssG0,
00 yZZ 00
31
where
Z s S
y1
and Z s S
y1
SS
y1
.
11 3131
Note that the following relation holds:
S 0 Z 0 I 0
11
s .
SS yZZ 0 I
31 31
Ž.
Equation 10.14 can be rewritten as
X
yT
G
T
XX
y1
q X
yT
X
T

GX
y1
iik iik
TT TTT
Z yZ 0 A y FB
13 iiki
s
TT
0 ZI yE
1 k
Z 0
0 I
1
q
A y BF yE
yZZ
iiik k
31
T
yZ y Z )
33
Z q AZ
s - 0.
1 i 1
T
yZE y EZ
1 kk1
ž/
yBM q EZ
iik k3

Ž. Ž.
Equation 10.12 is also derived in the same way as condition 10.11 .
Ž.
Q.E.D.
FUZZY DESCRIPTOR SYSTEMS AND CONTROL
202
Ž
The fuzzy controller design problem is to determine F i s 1,2, ...,r;
ik
e
.
k s 1,2, . . . , r satisfying the conditions of Theorem 34. The feedback gains
are obtained as
F s MZ
y1
ik ik 1
S 0
1
from the solution Z and M of the above LMIs. The matrix X s is
1 ik
SS
31
obtained as S s Z
y1
and S s Z
y1
ZZ
y1
.
11 3131

Ž. ŽŽ..
Next, we derive stability conditions for 10.9 in the case of h z t s
i
ŽŽ..
e
Ž.
® z t and r s r . In this case, the fuzzy descriptor system 10.2 can be
k
rewritten as
r
E*x* t s h z t A* x* t q B*u t , 10.16
Ž. Ž. Ž. Ž. Ž .
Ž.
Ž.
˙
Ý
iii i
i
s1
where
I 00I
E* s , A* s ,
ii
00 A yE
ii
0
B* s .
i
B
i

Ž. Ž.
In this case, the PDC controller 10.17 instead of 10.8 is used:
r
u t sy h z t F*x* t , 10.17
Ž. Ž. Ž. Ž .
Ž.
Ý
iii
i
s1
w F 0 x
where F* s . In this case, Theorem 34 can be simplified as follows.
i
ii
ŽŽ.. ŽŽ..
e
THEOREM 35 Assume that h z t s® z t and r s r . Then, the fuzzy
ik
Ž. Ž.
descriptor system 10.16 can be stabilized ®ia the PDC fuzzy controller 10.17 if
there exist Z , Z , and M such that
13 i
Z
T
s Z ) 0, 10.18
Ž.
11
T
yZ y Z )
33

Z q AZ
- 0,
1 i 1
T
yZE y EZ
1 ii1
ž/
yBMq EZ
ii i3
i s 1,2,...,r , 10.19
Ž.