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6.1990-3347

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AI AA-90-3347-CP
MODEL REDUCTION WITH
WEIGHTED MODAL COST ANALYSIS
A. Hu *
Dynacs Engineering, Co., Inc.
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347
Palm Harbor, Florida
R.E. Skelton t
Purdue University,
West Lafayette, Indiana
Abstract
sensor devices can only have finite bandwidth. Furthermore, the drawbacks of the infinite-bandwidth assumption for white noise processes were evident when the
norm of the response vector was evaluated as the infinite sum of the modal cost for certain input and output
combinations for certain simple continua [4]. For example it mas shown in [4] when the input is a torque
and output is rotation rate, the norm of the response,
or the infinite s u m of the modal costs was found not
t o converge if we choose t o model the disturbances as
white noise processes. However, in reality we know that
the norm of the response in this case does not increase
without bound. This indicates t h a t some appropriate
modifications for the MCA theory is necessary, and one
direct approach is to model the disturbances as finite
bandwidth colored noise processes inqtead of white noise
processes.
Analytical expressions are developed for the controllability Grammian rnatrix and for the modal costs
when the disturbances t o t h e flexible structures a.re
modeled as colored noise processes which have only finite bandwidth. This new algorithm is called Weighted
Modal Cost Analysis, and is the generalization of the
theory of Modal Cost Analysis for which the disturbances are assumed as white noise processes which have
infinite bandwidth. Comparing to the conventional numerical procedure, t h e closed form expression derived
for controllability Grammian and modal costs are more
efficient and more accurate and are thus quite useful
in model reduction processes, especially for high order
systems since n o interations are required to solve the
Lyapunov equation.
T h e purpose of this paper is to generalize the theory
of Modal Cost Analysis when the disturbances to the
flexible structures are modeled as colored noise. Analytical expressions for the controllability Grammian matrix, which is useful in computing the modal costs, are
derived. This is particularly efficient for higher order
sys tems.
1.O Introduction
White noise processes are assumed to have no correlations between any two distinct values of the random
processes a n d have equal power spectral densities at all
frequencies. It is a convenient model for formulating
and solving linear control problems t h a t involve rand o m disturbances a n d noise [l]. Using white noise to
model random disturbances acting on space structures,
Skelton e t al. have developed the theory of Modal Cost
Analysis (MCA) [2,3]. Modal costs represent the contribution of a mode in t h e norm of response vector, and
the theory of MCA has been found very useful as mode
truncation criteria because it allows t h e control objective to influence model reduction processes.
However, such abstract white noise processes do not
exist in t h e physical world, since any real actuator and
This paper is organized as follows. Section 2.0 briefly
reviews the necessary modal cost analysis (MCA) when
the linear system is driven by white noise. Section
3.0 provides a mathematical model for Weighted Modal
Cost Analysis, t h a t is, a model of a augmented system
including the plant and actuators with the disturhances
to the plant being modeled as colored noise. This section also derives the closed-form expressions of controllability Grammian, modal costs and other useful theorems. Section 4.0 presents a numerical example with
a simply-supported beam and compares the results for
modal costs using the new analytical approach with that
using Matlab subroutines for solving Lyapunov equations.
*Project Engineer, Member AIAA
t Professor, School of Aeronautics and Astronautics, Member
AIAA
Copyright 0 1990 American Institute of Aeronautics and
Astronautics, Inc. All rights reserved.
295
2.0 Modal Cost Analysis
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347
A brief review of necessary Modal Cost Analysis
(MCA) is included in this section in order to compare
with the results obtained using Weighted Modal Cost
Analysis to be presented in Section 3.0. The disturbances to the plant model for Modal Cost Analysis are
assumed to be white noise processes, whereas the disturbances used in the Weighted Modal Cost Analysis
are assumed t o be colored noise processes.
In equation (4), W is the noise intensity matrix and
Q is the ouput weighting in the cost function (3). If we
are interested in lightly damped continua, the second
term in equation (4) can be neglected.
3.0 Weighted Modal Cost Analysis
Let the flexible space structures driven by actuators with finite-bandwidth be modeled as two interconnected systems. The plant and noise models are assumed in the modal form, and the quantities matrices
w, wr represent white and colored noise respectively.
Consider an elastic system represented by the following equations of motion.
The plant equation is
iji
Note that w(t) represents a zero-mean white noise
with intensity W . T h a t is,
+ 2CiiwiGi +
4 w , +2Cw,w,,dw,
Ew(t) = 0
E{w(t)tuT(T)} = Wb(t - T )
=[p: , . . . l
A
2
P 3
{ }
+ R, 4,
{ }
+ [ r h ,...,
*:l
9,
Qw rn
t-a,
(5)
= bZzw, i = 1 , 2 , ...rn (7)
+w,,q,,
E , = lim E ( V ( t ) )
m
(8)
In equations (7) and (8), (qw,,dw,)T are the noise
model states, w is the zero mean white noise disturbance
as in equation (l),and p i and r i are the it* columns
of P, and R, respectively.
Rearranging equations (5) - (8) we will obtain a state
space form for the augmented system
The component costs vi, associated with each component x i , are defined by
(3)
where
= b T d , i = 1 , 2 , ...n
wt = P, qw
where 6 ( t ) is the Dirac delta function.
Now let a cost function associated with equation (1)
( 2 ) be given as
V, = E,yTQy,
W:V~
A
x i = ( V i , iilT
i =
y
It was shown in [2,3] that when the system takes the
modal term as in equations ( 1 ) and (2), the component
cost is called modal cost and can be computed a.s
=
Az+Bw
(9)
cz
where
xT E ( N
+ M ) with N =A 2n and M
The system matrix A is defined as
(4)
296
A
= 2m
between either plant or noise model states respectively,
whereas 2 describes the coupling correlations between
plant and noise model states.
Specifically, X i j , j - i j and X i j are the ( 2 x 2)
( i , j ) t h (, i + n , j + n)'* and ( i ,n j ) t h blocks of the
controllability Grammian X , and they are defined as
A A
A h [ O
A ]
with
+
AE (N+M)x(N+M)
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347
where
and
For example, the correlation between plant states
is defined as follows
i.' q
211
x=
-7112
. ..
R
XIn
"2
s,,
Xnl X n z . . .
Now that the augmented system in equation
(9) is in the standard form of a linear system driven
by white noise, its controllability Grammian X can be
obtained using various kinds of numerical methods for
solving Lynapunov equation (14). But such a numerical
approach can be criticized on the grounds that it ignores
the special structure of matrices A , A and A exhibited
in equations (9) - (13). It can be anticipated that analytical expressions for controllability Grammian A' can
be derived for the augmented system. Specifically, we
can first solve ,U in terms of noise model parameters,
and then determine ,.( in terms of the elements of 3 and
finally we can obtain ,. in terms of 2 . These results
are summarized in the following theorems.
Note t h a t matrix A represents the properties of the
plant model, Aq represents the properties of the noise
model, and A represents the couplings between the
plant and the noise models through the interconnection between actuator ouputs and plant inputs. T h e
dimensions of matrices A , A, and A are N x N ,N x M ,
and Af x M respectively.
Also
I
BT = [ o , O . . . ! 0, b,, . . . O , bwm
Theorem 1.
Let xij denote the ( 2 x 2) correlation matrix
relating noise model states { ( p w , ( t ) , yw,(t))}T and
{ ( q w J ( t ) , i w J ( t ) ) T }For
. the augmented system in equation (9), the elements of %ii depends only on the noise
model parameters and can be determined as
T h e Lynapunov equation for the augmented system
(9) is
XAT
+ A X + BWBT = 0
(14)
where the controllability Grammian X has the same
dimension of A and is defined as
x
In equation (15) the matrix partitions X , X and
have the same dimensions as those of A,A,and
respectively. Note t h a t x ,2 represent the correlations
A
297
where
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Proof.
T h e formulas for plant model correlation matrix
elements 2 i j ' s in equation (16) can be directly obtained
by writing out the four scalar equations of the ( 2 x
2 ) (i + n , n + j ) t h element of equation (14), i.e., from
the following partition.
T h e last term in equation (20) can be rewritten as
m
2 [ b T ( p i 2 i j l + rigzj) b T ( p i 2 ; : + - 1
0
0
riXzj2)
where crz"j' and
are defined in equations (19).
The resultant four scalar equations of equation (20)
are
Theorem 2
Let Xij denote the ( 2 x 2) correlation matrix corresponding t o plant state ( q i ( t ) ;7j2(t))*and noise model
state ( q w l ( t ) ,i W J ( t ) ) T
For
. the augmented system in
equations (9) - (13), the elements of X'? can be obtained
in terms of noise states correlations X,, and plant parameters, i.e.,
With direct substitution, we can obtain expressions
for plant and noise model state coupling correlation
kzj's,as provided by equation (18).
Theorem 3
Let X i j denote the (2 x 2) correlation matrix relating
plant states (77i(t) ,+(t))T and ( q j ( t ) , G j ( t ) ) T . For the
augmented system in equation (9), the element of xij
can be determined as
(1) If i # j
(l8)
where
Proof.
In order t o solve for Xij , we need the (2x 2) ( i ,j+n)th
partition of equation (14):
298
( 2 ) If i = j
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Now the four scalar equations from equation (24) are
where 6 i j ,w i j , pd'j2,
and ,f$/ are defined as follows.
By using the same procedure as before we can obtain
the expressions of X i j ' s in equations (21) and (22)
Remark
It can be easily verified that 2 and X are symmetric. For example, from equations for X i j ' s and
p i j ' s in equation (21), (22) and (23), we can show that
$3 = 2 2322 , and X2'j" = X!:, which means
X1.l
3I
23 = xr!,X??
X is symmetric.
For the lightly damped structures, further simplified
expressions for X i j can be obtained. This is summarized in Theorem 4.
Proof.
The ( 2 x 2 ) ( i , j ) t h element of equation (14) is
Theoreiii 4
Let Xij denote the (2 x 2) correlation matrix relating plant states ( q i ( t ) ,i i ( t ) ) Tand ( q j ( t ) ,l i j ( t ) ) T . For
the lightly damped structures represented by equation
(9)) the element of Xij can be determined as follows
(1) If i # j
1
X!? = pa?
'3
'3
7n
k=l
[
0
-bTpk
0
-bTrh
=0
02.
(24)
The third and the fourth terms in equation (24) can
be rewritten as
(2) If i = j
and
Proof
and they can be combined as a single ( 2 x 2) matrix
299
'3
The numerical simulation proceeds as follows. First,
we do not assume light damping and compare the results using analytical solution for controllability Grammian with that using numerical approach via Matlab
subroutines. That is, we use the formulas in Theorems
1-3 to compute the “exact” controllability Grammian
and compare the residues of Lyapunov equation (14)
using either analytical or numerical method. Secondly,
we apply light damping assumption (C -+ 0) and use
formulas in Theorem 4 to compute the “approximate”
values of controllability Grammian matrix and modal
costs based on the analytical method. In this step we
will discuss the relative errors of the modal cost based
on Theorem 4 normalized by that based on equations
(21) - (23) (without assuming light damping). Note that
both solutions are obtained using analytical method,
but the former is an approximation of the latter when
the damping ratio C approaches zero.
The above expression can be easily obtained by letting IS 0 in expressions (2 1) - (23).
-+
For ease of comparison, the corresponding results in
the standard Modal Cost Analysis is cited as follows,
that is, for the lightly damped structures when the disturbance w(t) is modeled as white noise processes, the
elements of plant state correlation matrix X arc
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347
(1) I f i
#
x
j
(2) If i = j
4.1 Analytical Approach vs. Matlab Subroutille ~~~~l~~
x,
With
X , and X expressed in analytical expressions in Theorems 1-4 we are now in a position to summarize the general steps needed to compute the controllability Grammian X and modal costs for the augmented system (7), that is,
Now let us compare the analytical solution of the controlla.bility Grammian with that using Matlab subroutines.
In this part of simulation, both the plant and noise
models are assumed to contain two modes. The system
matrices, frequencies and damping ratios are as follows.
Step I. Solve for noise state correlation X in terms of
noise model parameters, i.e., determine X i j according t o equations (16) -(17).
0
Step 2. Solve for noise and plant state correlation 3 in
terms of 2 a n d other plant parameters according
t o equations (18) - (19).
1
A= 1-w1
-2CW1
0
Step 3. Solve for plant state correlation X in terms of
2 and 2 according to equations (21) - (23).
w I. - i 2 ,
Step 4. Form controllability Grammian X according
to equation (15).
1
-w;
-2Cw2
C = 0.005,
i = 1, 2
and
Step 5 . Compute the modal cost according to
2
3
The output matrix for the plant is assumed as
uw,
= tr[XCTQC]2i-i,2i-1
+ ty[XCTQC]2i,2i
4.0 A Numerical Example
= - (i + 2 ) 2 , Cw = 0.707, i = 1, 2
c = bl, rl Pa, .21,
In this section we provide a numerical example
t o illustrate the accuracy and efficiency of analytical
method for solving controllability Grammian matrix X
as derived in the previous section. The plant used is
a simply-supported beam, which is driven by colored
noise disturbance. The results are presented in Tables
1- 5.
ri
=
Pi =
sin (0.35i)
icos(0.35i)
O
I
bi = i
The output matrix for the disturbance model is assumed as
300
4.2 Approximate Formulas in Theorem 4
for Light Damping
We now present numerical results for the approximate
formulas (25) - (26) of Theorem 4 for controllability
Grammim ,q. We assume small damping for the plant,
and investigate the errors of the modal costs.
p; =
The same beam example is used, but now three modes
are retained for both plant and disturbance models. We
vary damping ratio C for the plant from 0.001 to 0.01
and compute approximate cost matrix Vu = X,CTQC
and “exact” cost matrix Y = XCTQC. Note that X ,
is obtained using Theorem 4 with small damping assumption, but
using Theorems 1 - 3 without small
damping assumption. The error in the cost matrix
e”, is defined as e u , = (14 - V a , ) / xx 100% and is
listed in Table 5. The modal costs
and V,, are
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bw, = 5i
The augmented system matrix has the following
structure
A A
A = [ o
A ]
and the controllability Grammian is
[
x x
XT
+
A
A
defined as: V , = [ V ( 2 i - 1 , 2 i - 1) V ( 2 i ,Z i ) ] , V,, =
[Va(2i - 1 , 2 i - 1) + Ya(2i, 2i)I with Y and Vu being the
cost matrices. It should be noted that the plant matrix
A and disturbance model matrix are assumed both
of dimension (6 x 6) in this case, since three modes are
included for both plant and disturbance models.
2.1
a
The numerical results of this subsection are summarized in Tables 1-4. Table 1 contains the augmented
system matrix, and Table 2 lists the elements of the
controllability Grammian X based on the analytical solution. Note that the upper-right (4x4) partition in
Table 2 represents the correlations between the plant
states.
It is clear from Table 5 that the approximate formulas developed in Theorem 4 are adequate for modal cost
computation when the damping ratio is small. For example, when ( = 0.01, the errors in VI, V2, and V3 are
only 0.62, 0.058, and 0.044 % respectively, and when C
= 0.005, the errors in Vll V 2 ,and Vs are only 0.3745,
0.0095, and 0.011% respectively.
Next we substitute X in Table 2 obtained using the
analytical solution given by Theorems 1-3 into equation
(8) to verify whether Lyapunov equation can be satisfied. The resultant residue matrix is listed in Table 3.
Note that the largest element in Table 3 is -0.114d-12.
Table 1
The Augmented System Matrix A
But if we choose t o use subroutines in Matlab t o numerically solve the Lyapunov equation for controllabilty
Grammian X ,and then substitute it back into equation (S), the residue matrix can also be obt,ained and is
shown in Table 4. Note that the largest element in the
residue matrix in this cases is 0.28d-IO, which is greater
than that using the analytical method.
(columns 1 through 4)
0.00dt00
-. lOd+01
0.00dt00
0.00dt00
0.00dt00
0.00dt00
0.00dt00
0.10dt01
-. IOd-01
0.00dt00
O.OOd+OO
0.00dt00
0.00dt00
0.00dt00
0.00dt00 0.00dt00
It should be noted that same double precision computation has been used in the calculation for both analytical solution and Matlab subroutines so that the comparison is reasonable. Clearly, the analytical method
provides more accurate solution than the numerical
method. The reason for this is obvious: numerical errors are minimal when closed-form formulas are used.
Also, since no iterations are needed, the new analytical
method is very fast t o generate results. However, we
must admit that the drawback of this new algorithm
is that it can be only used when the system takes the
special modal form such as in equations (5) - (7).
O.OOd+OO
0.00dt00
0.00dt00
-. 16dt02
0.00dt00
O.OOd+OO
0.00dt00
0.00dt00
0.00dt00
0.00dt00
O.lOd+OI
- .40d-01
O.OOd+OO
0.00dt00
0.00dt00
O.OOd+OO
(columns 5 through 8)
0.00dt00
-. 21dtOi
O.OOd+OO
- .43d+01
O.OOd+OO
-. 36dt02
O.OOd+OO
O.OOd+OO
301
0.00dt00
-.45d+01
0.00dt00
- .90d+Ol
O.lOd+OO
- . 84dt01
O.OOd+OO
0.00dt00
0.00dt00 0.00dt00
- .39d+01 - . 62dt01
0.00dt00 O.OOd+OO
-. 78dtOI -. 12dt02
O.OOd+OO O.OOd+OO
0.00dt00 0.00dt00
O.OOd+OO O.lOd+OO
- .I ld+03 - . 15d+02
Table 2
T h e System Controllability
Grammian Matrix X
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(columns 1 through 4)
0.90d+02 O.OOd+OO
O.OOd+OO 0.93d+02
-. 17d+00 0.14dtOl
-. 14d+01 0.42d+01
- . lld+00 - .40d+00
0.40d+00 - .49d+00
- .39d-0 1 -. 33d+00
0.33d+00 0.56d+00
-. 17d+00 -. 14d+01
O.l4d+OI
0.60d+00
O.OOd+OO
- . 17d+00
0.32d+00
-.77d-01
0.58d+00
Table 4
Residue Matrix of the Lyapunov Eqn
(Using R4atlab Subroutines)
0.42d+01
O.OOd+OO
0.96d+03
-. 32d+00
-.35d+01
-. 58d+00
-. 58d-01
1.0d-10*
(columns 1 through 2)
-0.0015-
(columns 5 through 8)
- . 1 ld+OO 0.40d+00 -. 39d+00
-. 40d+00 -. 49d+00 -. 33d+00
-. 17d+00 0.32d+00 -. 77d-01
-. 32d+00 -. 35d+01 - .58d+00
0.41d-01
O.OOd+OO
0.28d-01
-. 94d-01
O.OOd+OO
0.15d+01
0.94d-01
0.18d+01
0.28d-01
0.94d-01
0.29d-0 I
O.OOd+OO
-0.0027+
-0.0004+
0.33d+00
0.56d+00
0.58d+00
- .58d-01
-. 94d-01
0.18d+01
O.OOd+OO
0.33d+01
-0.0009-0.00020.0000-0.0001-0.0031-
-0.0003-0.0014+
-0.0039
0.0181+
0.0002+
-0 .OO 170.0045+
-0.0161-
(columns 1 through 4)
O.OOd+OO O.OOd+OO O.lld+OO
-. 89d-15 0.1 Id-I3 O.OOd+OO
0.lld-13 O.OOd+OO -. Ild-12
O.OOd+OO -.lld-l2 0.16d-14
-. 22d-15 O.OOd+OO - .44d- 15
0.00dt00 - .44d-15 0.71d-I4
0.22d-15 O.OOd+OO 0.17d-15
0.36d-14 -. 13d-I5 0.14d-I3
-. 89d-15
O.OOd+OO
- .44d-15
0.71d-14
-. 22d-15
O.OOd+OO
-. 22d-15
O.OOd+OO
O.OOd+OO
0.22d-I5
O.OOd+OO
0.17d-15
0.00dt00
- .22d-15
O.OOd+OO
OfOOd+OO
0.0006i -0.0005+
0.0006i
0.0056+
0.0004i
0.01810.0013i
0.1300+
0.OOOOi -0.0033+
0.OOOOi -0.0043+
0.OOOOi 0.0036+
0.0001i -0.2870+
0.0018i
0 .OO 13i
0.0023i
0.0076i
0.OOOOi
0.0004i
0.000Oi
0.0001i
(columns 5 through 6)
-0.0002+
0.00 12-
0.0002-0.0034-
-0.0000-
-o.oooo+
-0.00000.0001+
(columns 5 through 8)
O.OOd+OO
-. 22d-15
O.OOd+OO
- .44d-15
O.OOd+OO
- .22d-15
O.OOd+OO
0.67d-15
-0.0025- 0 .OO 14i
-0.0015+ 0.000Oi
-0.00 14- 0.0006i
0.0059- 0 .OO 13i
0.0012+ 0.0011i
0.0011+ 0.0047i
0.0001+ 0.0004i
-0.0000- 0.0020i
(columns 3 through 4)
Table 3
Residue Matrix of the Lyapunov Fqn
(Using Analytical Method)
O.OOd+OO
O.OOd+OO
O.OOd+OO
0.lld-13
O.OOd+OO
-. 89d-15
O.OOd+OO
-. 67d-15
+
0.0001i
0.0013i
0.0007i
0.0024i
0.0001i
0.0037i
0.OOOOi
0.0040i
-.67d-15
0.36d-14
-. 13d-15
0.14d-I3
0.67d-15
0.00dt00
0.OOdtOO
O.OOd+OO
0.0002i
0.0011i
0.00OOi
0.0OOOi
-.OOOOi
0.0001i
0.OOOOi
0.0002i
0.0004+
0.006-0.00 17+
-0.0046-
0.0036i
0.0049i
0.0001i
-.0003i
-0.0000- 0.OOOOi
0.0001+ 0.0002i
0.0003+ 0.00OOi
0.0004+ 0.00OOi
(columns 7 through 8)
-0.0001+
0.00010.00450.0037-
-0.0000+
0.0003-0.000+
0.0009-
302
0.OOOOi -0.0031+
0.0004i -0.0000+
0.OOOOi -0.0160+
0.OOOOi
-0.28700.00OOi
0.00010.OOOOi 0.0005+
0.OOOOi 0.0009+
0.OOOOi -0.0019+
0.0039i
0.0020i
0.0001i
0.0001i
0.0002i
0.OOOOi
0.OOOOi
0.OOOOi
4. A. Hu and R. Skelton, 1988. Large Space Structures: Dynamics and Control, edited by S.N.
Atluvi and A.K. Ames. p.p. 71-94, Chapter
3. M o d a l Cost Analysis for Szmple Continua,
Table 5
Error of Modal Cost in Percent
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c
e, 1
0.0873
0.1682
0.2427
0.3114
0.3745
0.4325
0.4852
0.5334
0.009 0.5768
0.010 0.6161
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
e,,
0.0012
0.0009
0.0010
0.0045
0.0095
0.0160
0.0241
0.0338
0.0450
0.0577
Springer-Verlag .
e, 3
0.0005
0.0018
0.0040
0.0071
0.0111
0.0160
0.0217
0.0284
0.0360
0.0444
vi = [V(2i- 1,2i- 1) + V ( 2 i ,2i)l
A
v,,=A [Va(2i- 1,2i- 1) + Va(2i,a i ) ]
V = XCTQC, Ya = X,CTQC
X - Plant state correlation using Theorems 1-3
X, - Plant state correlation using Theorem 4
eu, = (K - Va,)/K x 100%
Conclusions
In standard Modal Cost Analysis theory, the disturbance is treated as white noise. The Weighted Modal
Cost Analysis, developed in this paper, models the disturbance as colored noise. Analytical expressions are
developed for the controllability Grammian matrix and
for the modal costs. Comparing to the conventional numerical procedure, the closed form expression derived
for controllability Grammian and modal costs are very
efficient and are thus quite useful in model reduction
processes, especially for high order systems since no interations are required to solve the Lyapunov equation.
References
1. H. Kwakernaak and R. Sivan, 1972,Linear Optimal
System New York: Wiley.
and
A. Yousuff, 1983,
International Journalof Control 37(2), p.p. 285304. Component Cost Analysis of Large Scale Systems.
2. R.E. Skelton
3.
R. Skelton 1981
Theory and Applications of Optimal Control in Aerospace Systems (Chapter 8)
ed. P. Kant, AGARD publication no.251,KBN 92835-1391-6,July 1981. Distributed in the United
States, by NASA Langley Field, Virginia, 23365,
attn: Report Distribution and Storage Unit. Control Design of Flexible Spacecraft.
303
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