In this and next chapters, a systematic treatment is given for two advanced and important issues of control performance, namely, robustness and optimality, in fuzzy control system design
Trang 1Kazuo Tanaka, Hua O Wang Copyright 䊚 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic
CHAPTER 5
ROBUST FUZZY CONTROL
w x This chapter deals with the issue of robust fuzzy control 1᎐3 In general, there exist an infinite number of stabilizing controllers if the plant is stabilizable The selection of a particular controller among this group of available controllers is often decided by certain specifications of control performance Fuzzy control designs which guarantee a number of control performance considerations were presented in Chapter 3 The LMI-based techniques ensure not only stabilization but also, for example, good speed of response, avoidance of actuator saturation, and output error constraint In this and next chapters, a systematic treatment is given for two advanced and important issues of control performance, namely, robustness and optimality,
in fuzzy control system designs The robustness issue is dictated by practical control applications in which there are always uncertainties associated with, for example, the plant, actuators, and sensors in a control system Robust control addresses these uncertainties and aims to derive the best design possible under the circumstances This chapter presents such a robust fuzzy control methodology, whereas optimal fuzzy control based on quadratic performance functions will be treated in the next chapter
This chapter defines a class of Takagi-Sugeno fuzzy systems with uncer-tainty Robust stability conditions for this class of systems are derived by applying the relaxed stability conditions described in Chapter 3 This chapter also gives a design method that selects the robust fuzzy controller so as to maximize the norm of the uncertain blocks out of the class of stabilizing PDC controllers This chapter focuses on robust fuzzy control for CFS For the
w x design of robust fuzzy control for DFS, refer to 4, 5
97
Trang 25.1 FUZZY MODEL WITH UNCERTAINTY
To address the robustness of fuzzy control systems, a first and necessary step
is to introduce a class of fuzzy systems with uncertainty For this purpose, we introduce uncertainty blocks to the Takagi-Sugeno fuzzy model to arrive at the following fuzzy model with uncertainty:
Plant Rule i
Ž Ž Ž Ž
q B q D i b i ⌬ t E u t , b i b i i s 1, 2, , r, Ž5.1.
where the uncertain blocks satisfy
1
⌬aiŽ t F , Ž5.2.
␥ai
⌬aiŽ t s⌬T aiŽ t , Ž5.3.
1
⌬b iŽ t F , Ž5.4.
␥b i
⌬b iŽ t s⌬T b iŽ t Ž5.5.
for all i The fuzzy model is represented as
r
is1
qŽB q D i b i⌬b iŽ t E b i.uŽ t 4 Ž5.6.
Ž w Ž x The fuzzy model 5.1 or 5.6 contains uncertainty in the consequent parts The robust stability for the fuzzy model with premise uncertainty was
w x w x first discussed in 6 and 7 This chapter will focus on the consequent uncertainty
5.2 ROBUST STABILITY CONDITION
To begin with, this section presents a stability condition for the uncertain
fuzzy model 5.1 i.e., 5.6 By substituting the PDC controller 2.23 into
Trang 3Ž5.6 , we have.
is1 js1
s Ýh iŽzŽ t ½A y B F q i i i ai b i 0 ⌬b i yE F b i i 5xŽ t is1
r
A y B F q A y B F q i i j j j i ai b i
⌬a j 0 E a j
q a j b j 0 ⌬b j yE F b j i 5xŽ t Ž5.7.
The following theorem presents robust stability conditions for the fuzzy
model 5.1 i.e., 5.6 with a given PDC fuzzy controller 2.23 This theorem provides a basis for the robust stabilization problem which is considered in the next section
Ž w Ž x
THEOREM 22 The fuzzy system 5.1 i.e., 5.6 is stabilized ®ia the PDC
Ž
controller 2.23 if there exist a common positi®e definite matrix P and a common
T y i j 2 Q2- 0, i - j s.t h l h / i j , Ž5.9.
T
T
2
2
yE F b i i 0 0 0 y␥ I b i
Trang 4Ž i i j.
qP A y B FŽ i i j.
PD ai PD bi PD a j PD b j E ai yF E j bi E a j yF E i b j
T
q ŽA y B F j j i. P
qP A y B FŽ j j i. 0
T
T
T
T
2
2
2
2
Q s1 block-diagŽ 0 .,
Ž
Proof. Consider the T-S fuzzy control system with uncertainty 5.1 , where
Ž Ž
⌬ t and ⌬ t are the uncertain blocks satisfying ai b i
⌬aiŽ t F , ⌬aiŽ t s⌬aiŽ t ,
␥ai
⌬b iŽ t F , ⌬b iŽ t s⌬b iŽ t
␥b i
TŽ Ž
Consider a candidate of Lyapunov functions x t Px t Then,
dt
sx˙TŽ t Px t q xŽ TŽ t Px t˙ Ž
T
s Ýh iŽzŽ t .x Ž t ½ žA y B F q i i i ai b i 0 ⌬b i yE F b i i / P
is1
⌬ai 0 E ai
qP A y B F qž i i i ai b i 0 ⌬b i yE F b i i / 5xŽ t
Trang 5T
T
ai ai
yE F
¢ ž 0 ⌬b i b i j /
E
⌬ai 0 ai
qP A y B F q i i j ai b i
yE F
T
⌬a j 0 E a j
¶
⌬a j 0 E a j •
qP A y B F qž j j i a j b j 0 ⌬b j yE F b j i / ßxŽ t
r
s Ýh iŽzŽ t .x Ž t
is1
=
T
D ai
T
T
⌬ai 0 ⌬ai 0 E ai
T T
q E ai yŽE F b i i.
0 ⌬b i 0 ⌬b i yE F b i i
T T
=ž D T b i P y 0 ⌬b i yE F b i i / ßxŽ t
r
T
Trang 6~ ŽA y B F. P q P A y B F q PŽ . D D P
ai b i
T
E
T
q E ai yŽE F b i j.
yE F
0 ⌬b i 0 ⌬b i b i j
T
T
D a j
T
qŽA y B F j j i. P q P A y B F q PŽ j j i. a j b j T P
D b j
T
⌬a j 0 ⌬a j 0 E a j
T T
q E a j yŽE F b j i.
0 ⌬b j 0 ⌬b j yE F b j i
T T
D a j ⌬a j 0 E a j
y T P y
D
= T P y xŽ t Ž5.10.
D
If
T
D
D b i
1
T T
q E ai yŽE F b i j.
1 yE F b i j
0 2 I
␥b i
T
D a j
qŽA y B F j j i. P q P A y B F q PŽ j j i. a j b j T P
D b j
1
2
␥a j
T T
q E a j yŽE F b j i. 1 yE F b j i y2 Q0- 0, Ž5.11.
0 2 I
␥
Trang 7r
T
ai T
= A y B F¢ Ž i i i. P q P A y B FŽ i i i.qP ai b i D T P
b i T
⌬ai 0 ⌬ai 0 E ai
T T
q E ai yŽE F b i i.
0 ⌬b i 0 ⌬b i yE F b i i
T T
y T P y
0 ⌬ yE F
=ž D T b i P y 0 ⌬b i yE F b i i / ßxŽ t
r
r
F Ýh iŽzŽ t .x Ž t
is1
T
ai T
= A y B F¢ Ž i i i. P q P A y B FŽ i i i.qP ai b i D T P
b i T
⌬ai 0 ⌬ai 0 E ai
T T
q E ai yŽE F b i i.
0 ⌬b i 0 ⌬b i yE F b i i
T T
=ž D T b i P y 0 ⌬b i yE F b i i / ßxŽ t
r
Trang 8s Ýh iŽzŽ t .x Ž t
is1
°
T
~
= A y B F¢ Ž i i i. P q P A y B FŽ i i i.qŽs y 1 Q. 0
T
D ai
qP ai b i T P
D b i
T
⌬ai 0 ⌬ai 0 E ai
T T
q E ai yŽE F b i i.
0 ⌬b i 0 ⌬b i yE F b i i
T T
y T P y
0 ⌬ yE F
=ž D T b i P y 0 ⌬b i yE F b i i / ßxŽ t
If
T
A y B F P q P A y B F q s y 1 Q
T
D ai
qP ai b i T P
D b i
1
2
T T
q E ai yŽE F b i i. 1 - 0, Ž5.12.
yE F b i i
0 2I
␥b i
then
dt
Trang 9Ž
at x t / 0 Since
⌬aiŽ t ⌬aiŽ t F 2I, ⌬b iŽ t ⌬b iŽ t F 2 I,
T T
y T P y
0 ⌬ yE F
T
0 ⌬ yE F
By the Schur complement, 5.12 and 5.11 are rewritten as 5.8 and 5.9 ,
When Q s 0 and Q s 0, that is, Q s 0, the relaxed robust stability1 2 0
conditions are reduced to just the robust conditions:
P) 0, S i i- 0, T i j- 0, i - j s.t h l h / i j
As a result, by utilizing the relaxed stability conditions, less conservative results can be obtained in the robust stability analysis
5.3 ROBUST STABILIZATION
We define a robust stabilization problem so as to select a PDC fuzzy
Ž controller, in the class of PDC controllers 2.23 satisfying the robust stability
Ž Ž
conditions 5.8 and 5.9 , to maximize the norm of the uncertainty blocks, or
Ž equivalently, to minimize␥ and ␥ in 5.7 The following theorem providesai b i
a solution to the robust stabilization problem
Ž
THEOREM 23 The feedback gains F that stabilize the fuzzy model 5.1 and i
r
␥ , ␥ , X , M , , M , Y ai bi 1 r 0 is1
subject to
ˆ
X ) 0, Y G 0,0 S q i i Žs y 1 Y. 1- 0, Ž5.13 ˆ
T y i j 2 Y2- 0, i - j s.t h l h / i j , Ž5.14.
Trang 10where s) 1,
T
XA q A X i i
T T
žyB M y M B i i i i /
T
ˆ
T
2
2
yE M b i i 0 0 0 y␥ I b i
T
XA q A X i i
T T
yB M y i j M B j i
D ai D b i D a j D b j XE ai yM E j b i XE a j yM E i b j
T
qXA q A X j j
yB M y M B j i i T T j 0
T
T
ˆ
T
2
2
2
2
Y s1 block-diagŽ 0 .,
where
Y s XQ X0 0
positions.
Proof. The main idea is to transform the conditions of Theorem 22 into
Trang 11block-diag S q s y 1 Q 4
= block-diag X I I I I 4
T
XA q A X i i
T T
žyB M y M B i i i i /
T
s
T
2
2
yE M b i i 0 0 0 y␥ I b i
ˆ
= block-diag X I I I I I I I I 4
T
XA q A X i i
yB M y M B i j j i
D ai D b i D a j D b j XE ai yM E j b i XE a j yM E i b j
T
qXA q A X j j
yB M y M B j i i T T j 0
T
T
s
T
T
2
2
2
2
ˆ
where
X s Py 1, M s F Py 1
i i
Trang 12The feedback gains can be obtained as
F s M Xy 1
from the solutions X and M of the above LMIs. i
A design example for robust fuzzy control will be presented in Chapter 7
number of papers considering H⬁ control for fuzzy control systems have appeared in the literature Chapters 13 and 15 give an extensive treatment of
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Ž
Control IFAC World Congress, Beijing, July 1999, pp 213 ᎐218.
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