ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic CHAPTER 6 OPTIMAL FUZZY CONTROL In control design, it is often of interest to synthesize a controller to satisfy, in an optimal f
Trang 1Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach
Kazuo Tanaka, Hua O Wang Copyright 䊚 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic
CHAPTER 6
OPTIMAL FUZZY CONTROL
In control design, it is often of interest to synthesize a controller to satisfy, in
an optimal fashion, certain performance criteria and constraints in addition
to stability The subject of optimal control addresses this aspect of control system design For linear systems, the problem of designing optimal
trollers reduces to solving algebraic Riccati equations AREs , which are usually easy to solve and detailed discussion of their solutions can be found
w x
in many textbooks 1 However, for a general nonlinear system, the
tion problem reduces to the so-called Hamilton-Jacobi HJ equations, which
counterparts for linear systems, HJ equations are usually hard to solve both numerically and analytically Results have been given on the relationship between solution of the HJ equation and the invariant manifold for the Hamiltonian vector field Progress has also been made on the numerical
w x computation of the approximated solution of HJ equations 3 But few results so far can provide an effective way of designing optimal controllers for general nonlinear systems
In this chapter, we propose an alternative approach to nonlinear optimal control based on fuzzy logic The optimal fuzzy control methodology
utilizing the relaxed stability conditions The optimal fuzzy controller is designed by solving a minimization problem that minimizes the upper bound
of a given quadratic performance function In a strict sense, this approach is
a suboptimal design One of the advantages of this methodology is that the
w x design conditions are represented in terms of LMIs Refer to 8 for a more thorough treatment of optimal fuzzy control
109
Trang 26.1 QUADRATIC PERFORMANCE FUNCTION
AND STABILIZING CONTROL
The control objective of optimal fuzzy control is to minimize certain perfor-mance functions In this chapter, we present a fuzzy controller design to minimize the upper bound of the following quadratic performance function
Ž6.1 :
⬁
J sH y Ž t Wy t q uŽ Ž t Ru tŽ 4dt, Ž6.1.
0 where
r
yŽ t s Ýh z t iŽ Ž C x i Ž t
is1
The following theorem presents a basis to the optimal fuzzy control problem The set of conditions given herein, however, are not in terms of LMIs The LMI-based optimal fuzzy control design will be addressed in the next section
THEOREM 24 The fuzzy system 2.3 and 2.4 can be stabilized by the PDC
fuzzy controller 2.23 if there exist a common positi®e definite matrix P and a common positi®e semidefinite matrix Q satisfying0
where s) 1,
T
žqP A y B FŽ i i i /
y1
y 1
T
Ž i i j.
qP A y B FŽ i i j.
T
qŽA y B F j j i. P
qP A y B FŽ j j i. 0
y 1
y1
y 1
y 1
Trang 3QUADRATIC PERFORMANCE FUNCTION AND STABILIZING CONTROL 111
Then, the performance function satisfies
J - x TŽ 0 P x 0 ,Ž
TŽ Ž
where x 0 Px 0 acts as an upper bound of J.
Proof. Let us define the following new variable
yŽ t s s h z tŽ Ž xŽ t
Equation 6.1 can be rewritten as
J sH ˆy Ž t 0 R ˆyŽ t dt.
0
Assume that there exists a common positive definite matrix P and a common
complements, we have
T
0 R yF i
and
T
T
qŽA y B F j j i. P q P A y B F yŽ j j i. 2 Q0
0 R yF j
0 R yF i
From 6.6 and 6.7 , we obtain
T
Trang 4T
T
qŽA y B F j j i. P q P A y B F yŽ j j i. 2 Q0- 0. Ž6.9.
It is clear from Theorem 9 in Chapter 3 that the fuzzy control system is
globally asymptotically stable if 6.2 and 6.3 hold
Next, it will be proved that the quadratic performance function satisfies
from 6.6 , 6.7 , and the Appendix,
x Ž t Px tŽ
dt
s˙x TŽ t Px t q xŽ TŽ t Px t˙ Ž
r r
T T
s Ý Ý h z t iŽ Ž h z t jŽ Ž x Ž t ½ ŽA y B F i i j. P q P A y B FŽ i i j 5xŽ t
is1 js1
r
T
2 T
s Ýh iŽzŽ t .x Ž Žt A y B F i i i. P q P A y B FŽ i i i 4xŽ t
is1
r
T T
qÝ Ýh z t iŽ Ž h z t jŽ Ž x Ž t ½ ŽA y B F i i j. P q P A y B FŽ i i j 5xŽ t
is1 i /j
r
T
2 T
- Ýh iŽzŽ t .x Ž Žt A y B F i i i. P q P A y B FŽ i i i 4xŽ t
is1
yx Ž t ½ Ý Ýh z t iŽ Ž h z t jŽ Ž C i yF j 0 R yF j
is1 i -j
qÝ Ýh z t iŽ Ž h z t jŽ Ž C j yF i 0 R yF i 5xŽ t
is1 i -j
r
T
q2Ý Ýh z t iŽ Ž h z t jŽ Ž x Ž t Q x t0 Ž
is1 i -j
is1
yx Ž t ½ Ý Ýh z t iŽ Ž h z t jŽ Ž C i yF j 0 R yF j
is1 i -j
Trang 5QUADRATIC PERFORMANCE FUNCTION AND STABILIZING CONTROL 113
qÝ Ýh z t iŽ Ž h z t jŽ Ž C j yF i 0 R yF i 5xŽ t
is1 i -j
r
2 T
yŽs y 1 Ýh iŽzŽ t .x Ž t Q x t0 Ž
is1
r
T
q2Ý Ýh z t iŽ Ž h z t jŽ Ž x Ž t Q x t0 Ž
is1 i -j
is1
yx Ž t ½ Ý Ýh z t iŽ Ž h z t jŽ Ž C i yF j 0 R yF i
is1 i -j
qÝ Ýh z t iŽ Ž h z t jŽ Ž C j yF i 0 R yF j 5xŽ t
is1 i -j
r
2 T
yŽs y 1 Ýh iŽzŽ t .x Ž t Q x t0 Ž
is1
r
T
q2Ý Ýh z t iŽ Ž h z t jŽ Ž x Ž t Q x t0 Ž
is1 i -j
s yx Ž t ½ Ý Ýh z t iŽ Ž h z t jŽ Ž C i yF i 0 R yF j 5xŽ t
is1 js1
yž Žs y 1.is1Ýh iŽzŽ t .y2is1 iÝ Ý-j h z t iŽ Ž h z t jŽ Ž /x Ž t Q x t0 Ž
s yx Ž t ½ ž Ýh z t iŽ Ž C i yF i / 0 R ž Ýh z t iŽ Ž yF i / 5xŽ t
yž Žs y 1.is1Ýh iŽzŽ t .y2is1 iÝ Ý-j h z t iŽ Ž h z t jŽ Ž /x Ž t Q x t0 Ž
T
0 R
yž Žs y 1.is1Ýh iŽzŽ t .y2is1 iÝ Ý-j h z t iŽ Ž h z t jŽ Ž /x Ž t Q x t0 Ž
T
0 R
Trang 6x Ž t Px tŽ - yy tˆ Ž 0 R ˆyŽ t dt
J sH0 y Ž t 0 R ˆyŽ t dt - yx t Px tŽ Ž 0
Since the fuzzy control system is stable,
o
Q.E.D
TŽ Ž
Remark 18 The above design procedure guarantees J - x 0 Px 0 for all
Theorem 24 are reduced to the following conditions:
X X TŽ Ž
6.2 OPTIMAL FUZZY CONTROLLER DESIGN
We present a design problem to minimize the upper bound of the perfor-mance function based on the results derived in Theorem 24 As shown in the
T
of Theorem 24 The optimal fuzzy controller to be introduced is in the strict
T
sense a ‘‘sub-optimal’’ controller since x 0 Px 0 will be minimized instead
of J in the control design procedure The following theorem summarizes the
design conditions for such scheme
THEOREM 25 The feedback gains to minimize the upper bound of the performance function can be obtained by sol®ing the following LMIs From the solution of the LMIs, the feedback gains are obtained as
F s M Xy 1
TŽ Ž
for all i Then, the performance function satisfies J - x 0 Px 0 -
Trang 7OPTIMAL FUZZY CONTROLLER DESIGN 115
X , M , , M , Y1 r 0
subject to
X)0, Y G 00 ,
T
xŽ 0 X
ˆ
ˆ
where s) 1,
T
XA q A X i i
XC i yM i
T T
žyB M y M B i i i i /
ˆ
y 1
y 1
T
XA q A X i i
T T
yB M y M B i j j i
T
qXA q A X j j
yB M y M B j i i T T j 0
ˆ
y 1
y1
y 1
y 1
TŽ Ž
Proof. The main idea here is to transform the inequality J - x 0 Px 0 - and the conditions of Theorem 24 into LMIs:
Trang 8w x w x
T
XA q A X i i
XC i yM i
T T
žyB M y M B i i i i /
s
y1
y 1
qŽs y 1.⭈ block-diag XQ X 0 0Ž 0 .
ˆ
sU q i i Žs y 1 Y ,. 3
where
Y s XQ X0 0
We obtain the following condition as well:
T
XA q A X i i
T T
yB M y M B i j j i
T
qXA q A X j j
yB M y M B j i i T T j 0
s
y1
y 1
y 1
y 1
ˆ
sV y i j 2 Y 4
Then, the quadratic performance function satisfies
J - x T
Ž
the initial values x 0 are assumed known If not so, Theorem 25 is not
Ž
Ž
Trang 9OPTIMAL FUZZY CONTROLLER DESIGN 117
that is,
l
xŽ 0 s Ý x 0 , k kŽ
ks1
G 0,k
l
s 1,
Ý k
ks1
x kŽ 0 g R n Theorem 25 can be modified as follows to handle this case
THEOREM 26 The feedback gains to minimize the upper bound of the performance function can be obtained by sol®ing the following LMIs From the solution of the LMIs, we obtain
F s M X i i y1
TŽ Ž
for all i Then, the performance function satisfies J - x 0 P x 0 -:
minimize
X , M , , M , Y1 r 0
subject to
X) 0, Y G 00 ,
T
) 0, k s 1,2, , l,
x kŽ 0 X
ˆ
U q i i Žs y 1 Y. 3- 0,
ˆ
Remark 19 An alternative approach to handle the uncertainty in initial
w condition is to employ the initial condition independent design see Chapter
3, equation 3.56
An interesting and important theorem is given below
THEOREM 27 The following statements are equi®alent.
Ž 1 There exist a common positi®e definite X and a common positi®e
nite Y satisfying 3.23 and 3.24
Trang 10Ž 2 There exist a common positi®e definite X s XrX and a common positi®e
semidefinite Y satisfying 6.12 and 6.13 , where0 ) 0
Proof. 1 ´ 2 Assume that 3.23 is satisfied Since
0 R yM i
XAž T iqA X y B M y M i i i i T B i TqŽs y 1 Y. 0
for i s 1, 2, , r The above condition is equivalent to
T
XA q A X i i
žyB M y M B i i i T T i / XC i y M i
y 1
y1
q s y 1 Y - 0,Ž 3 i s 1, 2, , r
Since XXs X, M iXs M , and Y i 3 Xs Y can be regarded as new X, M ,3 i
We can obtain the condition 6.13 from 3.24 as well
X Ž .
The theorem above says that there exists a common X satisfying 6.12
and 6.13 for any W and R if conditions 3.23 and 3.24 hold The optimal
fuzzy controller design in Theorem 25 is feasible if the stability conditions
Ž3.23 and 3.24 hold Ž
A design example for optimal fuzzy control will be discussed in detail in Chapter 7
APPENDIX TO CHAPTER 6
COROLLARY A.1
yC i T WC y C i T j WC F yC j i T WC y C j j T WC i,
where
W) 0.
Trang 11REFERENCES 119
Proof. It is clear
COROLLARY A.2
where
Proof. From Corollary A.1, we have
s yC i T WC y F i j T RF y C j j T W C y F j i T RF i
F yC i T WC y F j j T RF y C i j T W C y F i i T RF j
Q.E.D
REFERENCES
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Cliffs, NJ, 1970.
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Ž 1995
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