ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic CHAPTER 13 MULTIOBJECTIVE CONTROL VIA DYNAMIC PARALLEL DISTRIBUTED COMPENSATION dynamic parallel distributed compensation DPDC..
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 13
MULTIOBJECTIVE CONTROL
VIA DYNAMIC PARALLEL
DISTRIBUTED COMPENSATION
dynamic parallel distributed compensation DPDC It is often the case in the practice of control engineering that a number of design objectives have to be achieved concurrently The associated synthesis problems are formulated as
linear matrix inequality LMI problems, that is, the parameters of the DPDC controllers are obtained from a set of LMI conditions The approach in this chapter can also be applied to hybrid or switching systems
We present the performance-oriented controller synthesis of DPDCs to incorporate a number of practical design objectives such as disturbance attenuation, passivity, and output constraint Performance specifications
con-troller synthesis procedures are formulated as LMI problems In the case
of meeting multiple design objectives, we only need to group these LMI conditions together and find a feasible solution to the augmented LMI
problem 15
q
As discussed in Chapter 12, in general, the choice of a particular DPDC parameterization will be influenced by the structure of the T-S subsystems
In this chapter, we will only discuss DPDC in the quadratic parameteri-zation form It is easy to extend the results in this chapter to the cubic
259
Trang 2parameterization case Recall the quadratic parameterization is represented as
x s hŽp h Žp A x q. h B y, Ž13.1.
r
i
u s Ýh iŽp C x q D y,. c c c Ž13.2.
is1
or equivalently that
A cŽp s Ý Ý h iŽp h. jŽp A ,. c B cŽp s Ýh iŽp B ,. c
13.3
r
i
C cŽp s Ýh iŽp C ,. c D cŽp s D . c
is1
variables
This section presents LMI conditions which can be used to design DPDC controllers which satisfy a variety of useful performance criteria The presen-tation is divided into two subsections In the first subsection, we assume only
a linear parameter-dependent controller structure and derive a collection of parameter-dependent conditions expressed in inequalities Each condition corresponds to a different performance criterion In the second subsection,
we restrict our consideration to a DPDC controller structure This restriction allows us to convert the parameter-dependent inequalities to parameter-free LMIs which can be solved numerically
13.1.1 Starting from Design Specifications
equations
x Ž t s A Žp x Ž t q B Ž p w t , Ž
13.4
z t s CŽ c lŽp x. c lŽ t q D c lŽp w t , Ž
Ž
the system states or some exogenous input variables
Trang 3L Gain Performance 2
Definition 2 14 : For a casual NLTI nonlinear time-invariant operator G: e
w g L2 ᑬ ™ z g L ᑬ2 with G 0 s 0, G is L2 stable if w g L2 ᑬ
␥ G 0 if and only if
z tŽ dt F␥ w tŽ dt Ž13.5.
The well-known Bounded Real Lemma is given below 25
LEMMA 1 For system G: A c l p , B c l p , C c l p , D c l p , the L2 gain will
be less than ␥ ) 0 if there exists a matrix P s P T ) 0 such that
T
L ŽA c lŽp , P . PB c lŽp. C c lŽp.
B c lŽp. P y␥ I D c lŽp.
C c lŽp. D c lŽp. y␥ I
General Quadratic Constraint
Definition 3 15 : For a casual NLTI,G: w ™ z with G 0 s 0 Given fixed
Ž
w t need to satisfy the following constraint:
X
z tŽ U W z tŽ
T
H0 žw tŽ / žW T V/ žw tŽ /
Remark 41 15 : Many performance specifications such as L gain, passiv-2
ity, and sector constraint can be incorporated into this general quadratic
L ŽA c lŽp , P . PB c lŽp q C. c l T W
žB c lŽp P q W C. c l D W q W D q V c l c l /
T T
C c lŽp.
Trang 4X
d x c lŽ t L ŽA c lŽp , P . PB c lŽp. x c lŽ t
X
z tŽ U W z tŽ
Inequality 13.7 will result by integrating both sides of 13.9
Applying the Schur complement to 13.8 , we get the following lemma:
LEMMA 2 For system G: A c l p , B c l p , C c l p , D c l p , the general
quadratic constraint 13.7 will be satisfied if there exists a matrix P s P ) 0
such that
L ŽA c lŽp , P . PB c lŽp q C. c lŽp W. C c lŽp S.
B c lŽp P q W C. c l W D q D W q V c l c l D c lŽp S. - 0 13.10Ž
S C c lŽp. S D c lŽp. y⌺
Generalized H Performance 2
Definition 4 15 : A causal NLTI G: w ™ z with G 0 s 0 is said to have
L ŽA c lŽp , P . PB c lŽp.
T
Then drdt V x c l t - w t w t We will suppose D p s 0 In this c l
case, if the equation
P C T c lŽp.
Trang 5Ž Ž Ž Ž
LEMMA 3 For system G: A c l p , B c l p , C c l p , 0 , the generalized H per-2
T
formance will be less than if there exists a matrix P s P ) 0 such that 13.12
and 13.13 are feasible.
Constraint on System Output
Definition 5 24 : A casual NLTI G: x s A˙c l c l p x c l and z s C c l p x c l
satisfies an exponential constraint on the output if
y␣ T
Ž
equation
L ŽA , P q 2 c l . ␣P - 0 Ž13.15.
Furthermore, if the equations
P Px c lŽ 0
X
and
P C T c lŽp.
hold, then the inequality
zXŽ t z tŽ - xX Ž t Px Ž t - ey 2␣ t xX Ž 0 Px Ž 0 -2ey 2␣ t
will also be satisfied Combining these results, we have the following lemma:
LEMMA 4 For the system G: x s A˙c l c l p x c l and z s C c l p x , the expo- c l
nential constraint z T F e ,᭙T G 0, will be satisfied if there exists a
T
matrix P s P ) 0 such that 13.15 , 13.16 , and 13.17 are feasible.
Constraints on Control Input
Definition 6 24 : A casual NLTI G: x s A˙c l c l p x c l and u s K p x c l with
Ž
input if
y␣ T
Similar to the discussion for exponential constraint on the system output, we have the following lemma:
Trang 6Ž Ž
LEMMA 5 For system G: x s A˙c l c l p x c l and u s K p x , the exponential c l
constraint u T F e ,᭙T G 0, will be satisfied if there exists a matrix T
P s P ) 0 such that 13.15 , 13.16 , and
T
P K pŽ
13.1.2 Performance-Oriented Controller Synthesis
In this subsection, we consider T-S models which are represented by a set of fuzzy rules in the following form:
Dynamic Part
Rule i
IF p t is M1 i1 ⭈⭈⭈ and p t is M , l i l
THEN˙x t s A x t q B u t q B w t i i w
Output Part
Rule i
IF p t is M1 i1 ⭈⭈⭈ and p t is M , l i l
THEN
y t s C x t q DŽ i Ž w i w t ,Ž
z t s CŽ z i x t q DŽ z i u t q DŽ z w i w t Ž
Ž
performance variables of the control systems
We can simplify the expressions of the T-S model as
r
i
x s hŽp. A x q B u q B w , Ž13.20.
is1 r
z s Ýh iŽp ŽC x q D u q D w , z z z w Ž13.21.
is1 r
i
y s Ýh iŽp ŽC x q D w i w Ž13.22.
Trang 7Ž Ž
DPDC controller 13.1 and 13.2 have the form
r r
x s hŽp h Žp. A x q B w , Ž13.23.
is1 js1
r r
z s c l Ý Ý h iŽp h. jŽp ŽC x q D w , c l c l c l Ž13.24.
is1 js1
where
A q B D C i i c j B C i c B q B D D w i c w
A s c l ž B C c i j A i j c /, B s c l B D c i w j ,
C s c l z z c j z c , D s D c l z wqD D D z c w
Now, we are ready to apply the results in Section 13.1.1 to 13.23 and
Ž13.24
L Gain Performance We begin by applying a congruence transformation 2
on 13.6 using the matrix
,
0 0 I0
where the closed-loop system is defined as in 13.23 and 13.24 By utilizing
the notation in the quadratic parametrization discussed in Chapter 12, 13.6 becomes
r r
is1 js1
where
E11i j E12i j E13i j E14i j
Ž 12. 22 23 Ž 42.
Ž 13 Ž 23 Ž 43.
Ž 14. 42 43
Trang 8E s11 L ŽA , Q i 11.qB iCjqžB iCj/ , E s A q B12 i i DC q j Ai j,
T
E s B q B13 w i DD , w E s C Q q D14 ž z 11 zCj/ ,
T
E s22 L ŽA , P i 11.qBi C q j Ž Bi C j. , E s P B q23 11 w Bi D , w
E i jsC iqD i
DD jqD i
Condition 13.25 is equivalent to
r r
h Žp h Žp E q E Ž - 0 Ž13.26.
is1 js1
The inequality 13.26 will hold true according to Theorem 45 if there exist
We will express the resulting theorem using the notation in the previous section:
THEOREM 51 Gi®en a T-S model of the form 13.20 ᎐ 13.22 with DPDC
controller 13.1 and 13.2 , the L gain performance will be less than2 ␥ if the
LMI conditions 12.16 , 13.27 , and 12.85 are feasible with LMI ®ariables Q ,11
P , T ,11 i j Ai j, Bi, Ci, and D:
where
i j
E s11 L ŽA , Q i 11.qL ŽA , Q j 11.qB iCjqžB iCj/ qB jCiqžB jCi/ ,
T
i j
E s A q A q B12 i j i DC q B j j DC q 2 i Ai j ,
E s B q B q B13 w w i DD q B w j DD , w
T
E s C Q q C Q q D14 ž z 11 z 11 zCjqD zCi/ ,
E s22 L ŽA , P i 11.qL ŽA , P j 11.qBi C q j Bj C q i ž Bi C j/ qž Bj C i/ ,
Trang 9i j i j j i
E s P B q P B q23 11 w 11 w Bi D q w Bj D , w
E s C q C q D42 z z z DC q D j z DC , i
E s D43 z DD q D w z DD q D w z wqD z w
The resulting dynamic controller is gi®en by 12.87 ᎐ 12.90 where P , P , Q ,11 12 11
and Q12 satisfy the constraint P Q q P Q11 11 12 T12sI
General Quadratic Performance Similarly, we get the following theorem
by applying a congruence transform on 13.10 using the matrix
0 0 I0
Ž13.1 and 13.2 , the generalized quadratic constraint 13.7 will be satisfied ifŽ Ž
the LMI conditions 12.16 , 12.85 , and 13.28 are feasible with LMI ®ariables
Q , P , T ,11 11 i j Ai j, Bi, Ci and D
E11 E12 E13 E14
T
i j T i j T i j T i j 0
where
i j
E s11 L ŽA , Q i 11.qL ŽA , Q j 11.qB iCjqžB iCj/ qB jCiqžB jCi/ ,
T
i j
E s A q A q B12 i j i DC q B j j DC q 2 i Ai j ,
E s B q B q B13 w w i DD q B w j DD w
T
qžC Q q D z 11 zCjqC Q q D z 11 zCi/ W ,
T
E s C Q q C Q q D14 ž z 11 z 11 zCjqD zCi/ S,
Trang 10T T
E s22 L ŽA , P i 11.qL ŽA , P j 11.qBi C q j Bj C q i ž Bi C j/ qž Bj C i/ ,
E s P B q P B q23 11 w 11 w Bi D q w Bj D w
T
qžC q C q D z z z DC q D j z DC i/ W ,
T
E s C q C q D24 ž z z z DC q D j z DC i/ S,
E s 2V q W33 žD z wqD z wqD z DD q D w z DD w/
T
qžD z wqD z wqD z DD q D w z DD w/ W ,
T
E s D34 ž z wqD z wqD z DD q D w z DD w/ S,
i j
E s y244 ⌺
The controller is gi®en by 12.87 ᎐ 12.90
Generalized H Performance If we apply a congruence transform on both 2
,
we get the following theorem:
THEOREM 53 For a T-S model 13.20 ᎐ 13.22 with PDC controller 13.1
and 13.2 , the generalized H2 performance will be less than if the LMI
conditions 12.16 , 12.85 , 13.20 , 13.30 , and 13.31 are feasible with LMI
®ariables Q , P , T , S ,11 11 i j i j Ai j, Bi, Ci, and D for all i F j:
T
qD zCjqD zCi.0
T
) S ,
j
2 I
13.29
Trang 11S11 S1r
E11 E12 E13 T
ŽE13i j. ŽE23i j. E33i j 0
where
i j
E s11 L ŽA , Q i 11.qL ŽA , Q j 11.qB iCjqžB iCj/ qB jCiqžB jCi/ ,
T
i j
E s A q A q B12 i j i DC q B j j DC q 2 i Ai j ,
E s B q B q B13 w w i DD q B w j DD , w
E s22 L ŽA , P i 11.qL ŽA , P j 11.qBi C q j Bj C q i ž Bi C j/ qž Bj C i/ ,
E s P B q P B q23 11 w 11 w Bi D q w Bj D , w
i j
E s y233 I,
and
D z wqD z wqD z DD q D w z DD s 0, w ᭙i F j. Ž13.32.
The controller is gi®en by 12.87 ᎐ 12.90
Constraints on the Outputs Applying a congruence transform on 13.15
,
we get the following theorem
THEOREM 54 Consider a T-S model 13.20 ᎐ 13.22 suppose D z ws0,
D s 0 and B s 0 with DPDC controller 13.1 and 13.2 Suppose the w w
initial state is gi®en by x 0 x 0 c ⬘; then z t - e for all t G 0 if the
LMI conditions 13.33 ᎐ 13.35 and 12.85 and 13.31 are feasible with LMI
®ariables Q , P , P , T , S ,11 11 12 i j i j Ai j, Bi, Ci and D:
E11 E12
ŽE12. E220
Trang 12T
i j
E s11 L ŽA , Q i 11.qL ŽA , Q j 11.qB iCjqžB iCj/
T
T
i j
E s A q A q B12 i j i DC q B j j DC q i Ai j q2␣I,
E s22 L ŽA , P i 11.qL ŽA , P j 11.qBi C q j Bj C i
Ž
P x 011
X
Ž
x 0 P11
X
13.34
T
T
C q C q D z z z DC j
) S
j
2 I
qD CqD C 0 j 0
13.35
The controller is gi®en by 12.87 ᎐ 12.90
Constraints on the Inputs Applying a congruence transform on 13.15
,
we get the following theorem:
Trang 13Ž Ž Ž i
THEOREM 55 Consider a T-S model 13.20 ᎐ 13.22 suppose D z ws0,
D s 0, and B s 0 with PDC controller 13.1 and 13.2 Suppose the initial w w
state is gi®en by x 0 x 0 ; then u t c - e for all t G 0 if the LMI
conditions 13.33 , 13.34 , 13.36 , and 12.85 are feasible with LMI ®ariables
Q , P , T ,11 11 i j Ai j, Bi, Ci, and D
T
T
I P11 ž DC i/
The controller is gi®en by 12.87 ᎐ 12.90
To illustrate the DPDC approach, consider the problem of balancing an inverted pendulum on a cart Recall the equations of motion for the
lum 26 :
x s x ,
g sin xŽ 1.yamlx22sin 2Ž x r2 y a cos x1 Ž 1.u
˙2 4lr3 y aml cos2Žx1.
and x is the angular velocity; g s 9.8 mrs2 2 is the gravity constant, m is
a s 1r m q M We choose m s 2.0 kg, M s 8.0 kg, 2 l s 1.0 m in this
study
The control objective is to balance the inverted pendulum for the
represent the system 13.37 by a Takagi-Sugeno fuzzy model Notice that
two-rule fuzzy model as shown in Chapter 2
Model Rule 1
IF x is about 0,1
THEN˙x s A x q B u.
Trang 14Model Rule 2
IF x is about1 "r2 x - r2 ,1
THEN˙x s A x q B u.2 2
Here,
y
Membership functions for Rules 1 and 2 are shown in Figure 13.1
Now we apply the DPDC design to the pendulum system Assume that
only x is measurable, that is, y s Cx s 11 0 x We employ the following
DPDC controller:
x s hŽy h Žy A x q. h B y,
2
i
u s Ýh iŽy C x q D y.. c c c
is1
Fig 13.1 Membership functions of the fuzzy model.
Trang 15Employing Theorem 47, we obtain the following control parameters for the DPDC controller:
11
12 21
22
5.0666
1
20.8530 3.4824
2
12.5320
C s 388.9291 c 113.6926 ,
C s 794.6242 c 247.5543 ,
D s 4.4624 c
Figure 13.2 illustrates the closed-loop system response with the DPDC
mance-oriented DPDC designs have also been carried out according to the principles of Section 13.1
Fig 13.2 Angle response using the DPDC controller.
Trang 16If variable p comes from the output of the system, the dynamic feedback
controller will become a dynamic output feedback controller which is essen-tial for practical applications when only the system output is available The framework used in this chapter can also be applied to generate nonlinear controllers for uncertain systems One of the basic tools for robustness analysis of such uncertain systems is the small-gain theorem which
sufficiently small, we can guarantee the robust stability The results in this chapter are also applicable to hybrid and switching systems
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