Order Reduction of Linear State-Space Models Via Optimal Approximation of the Nondominant Modes

ORDER REDUCTION STATE-SPACE MODELS APPROXIMATION NONDOMINANT

OF LINEAR VIA OPTIMAL OF THE MODES

L. Litz Institut fur Regelungs- und Steuerungssysteme, Universitat Karlsruhe, Karlsruhe, Federal Republic of Germany

Abstract. A new method of high order system simplification is proposed that yields stationary exact models of lower order. Using the Jordan-transform of the original system, first the notion 'dominant eigenvalue' is clarified by a dominance measure for each eigenvalue. By the aid of these dominance measures the dominant eigenvalues can be chosen as well as the order that the reduced system should have. After chosing the dominant eigenvalues the dominant modes are retained in the reduced model. Furthermore the lost nondominant modes are expressed as optimal linear combinations of the existing dominant motions and then they are reintroduced in the model of lower order. The optimization yields a directly programmable formula for the system matrices of the reduced model. The usefulness of the proposed method is demonstrated in detail by the application to the practical problem of a processing line whose original model is of 51th order. Keywords. System order reduction; state-space methods; control system analysis; controllability; observability; optimization; least squares approximations; metallurgical industries.

INTRODUCTION

Marshall, 1966, Chidambara, 1969, or summarizing papers as Siret, Michailesco, Bertrand, 1977, or Litz, 1979) hold a particular position. Although requiring the Jordan-transformation of the original system (1) as a considerable computational amount, this step offers some advantages. Besides an attractive simulation of step-responses, the eigenvalues and eigenvectors provide a profound insight into the system behaviour. In the following the insight will even be improved by the introduction of dominance measures. Furthermore an optimization will be given that yields a directly programmable formula for the reduced system matrices.

The modelling of large technical processes by the aid of physical laws very often leads to linearized models,

x

=

A x + B u

,

(1)

with the state-vector x e Rn, the input-vector u E RP and ;atrices A and ! of approp~iate dimensions. An-advantage to the engineer is the physical meaning of the state - as well as the input variables like temperature, concentration, torque, speed, current for example. However, the largeness of a technical process often results in a high order n of the model (1) (n = 20 . . . 100 e.g.). Therefore these models may be very difficult to employ for simulation or control design. To circumvent the drawback of high order many authors suggest the application of order reduction techniques, See Genesio, Milanese (1976) with 137 references.

GOAL AND APPROACH The system of reduced order is desired to be a state-space model, too. It must get the same inpu t vector ~ E RP as the original system (1) and a state vector whose elements should still have a physical meaning. Therefore they are to approximate the most important original state variables, m composed in the vector ~1 R ,

Among the variety of techniques which are capable of handling multi-input systems in state-space description (1), the modal methods (see the original papers of Davison, 1966,

e

195

L. Litz

196

called the essential state. It contains the future control variables, other measurable variables and perhaps some variables that have to be supervised. For the sake of simplicity i t is assumed that the variables of the original state x in Eq. (1) are arranged in such a ;ay that the essential state x takes the first part -1 of ~,

0

A1

0

I

l r~:l

•A

0

- - .-

~1

0

Q

1:.. 2

rn

0

I

0

0

A

n (8)

(2 )

x = [::]

with'

~2

n-m E R a s the remaining part.

x

The state vector € 1 duced order model~1

A ~1

R

m

This can always be achieved by changing the columns of V in Eq. (4) and the rows of G in Eq~(7). V, G and A are to be th; arranged matri~es inthe following, with A containing the dominant eigenvalues~l

of the reCorresponding to Eq. (8), G and Eq. (5) can be partitioned-into

-

~

(3 )

+ B ~

G

shall approximate the essential state ~1 •

[::]

(9)

To derive the formulas for A and B first the Jordan-transformation

x = V z has to be applied to Eq. (1),

( 10) (4)

yielding

A ~ + G ~

(5 )

A

V-lA V

(6)

G

V- 1 B

(7 )

~

in

with

with ~1' ~1 as the first m rows of G, respectively, just as y in Eq. (4) can be written as

~

V= [:11 :12] -21

In the remainder ~ in Eqs. (1) and (6) is assumed to be non defective, so that there exist n linearly independent eigenvectors ~i' being the column vectors of V in Eq. (4). Because most technical high order systems lead to non defective matrices A the case of a defective A is omitted here. It would compli~ate unnecessarily the following explanations. To continue the order reduction, the eigenvalues A., occupying the diagonal of l! in Eqs.(S), (6), have to be divided into two groups, the m dominant ones and the n-m remaining nondominant ones. This important step of order reduction will be performed in the next section by the aid of V, G, ~ in Eqs. (4), (5). After choosing the m dominant eigenvalues they can be arranged to built up the left upper part of ~, according to

( 11 )

-22

with the m x m matrix V and V , f . t - 11 . -;-1 2 0 appropr~a e a~mens~ons. Tn~s enables one to split up Eq. (5) into the dominant motions

V V -21' -22

(12 )

and the nondominant ones ( 13)

~2

Then from Eqs. (2), (4), (10) the essential state follows,

and

(11)

(14 ) as a sum of the dominant and the nondominant modes. Because only the dominant eigenvalues (those of A ) will be retained in the reduced-6rder model the nondominant motions z and consequently the nondominant mo&es Y12~2 will be lost in Eq. (14). Thereby an error arises. For its reduction the lost nondominant motions z are to be approximated by the domi~ant ones: ~2

E ~1

( 15)

Order Reduction of Linear State-Space Models This yields the approximation Y12~ ~1 of the nondominant modes and starting from Eq. (14) the approximation (16 )

Interpreting the last equation as a linear transformation and applying i t to Eq. (12) yields the result

-

~1

~1

( 19)

I

being the n x n identity matrix. con s id er x 1 (t), the response of the i-th ess~rttial state variable when exciting only the j-th input variable by a unit step. Denoting the elements of ~ by vio and those of G by g 0 0' the uni t-~tep response x 1~ (t) 6110ws from Eq. (19) : 1.J

L~ t us fir s t

0

~1

E:.

u

(17)

0

t

0

+

197

n A

B

xl

E in Eq. (15) will be derived in the section 'Optimization' to minimize the integrated squared vector norm of ( 18) Thereby the condition £(t+ oo ) = Q ensures the stationary exactness of the reduced model in Eq. (17). Before minimizing £ in Eq. (18) the dominant eigenvalues have to be found.

DOMINANCE MEASURE The term "dominant eigenvalues" commonly means the eigenvalues nearest to the imaginary axis. But retaining these eigenvalues in the reduced model would not achieve the best results in general (see Hickin, Sinha, 1975). Only few authors are dealing with this problem. Commault, Guerin (1978) proposed the calculation of an error measure for all possible subsets of m eigenvalues that can be choosen from the whole of n. For m given a priori, this provides the best subset, indeed. Because of (~) possible combinations this method is too expensive if the original system order n exceeds about twenty. In the following the "classical" notion of dominance is extended in a simple way. The distance of an eigenvalue from the imaginary axis only takes into account that a faster mode causes a faster decaying part in any time response than a slower one. In doing so the magnitude of a mode remains disregarded. As will be shown this magnitude is connected with the important properties of controllability and observability. To compute the interesting magnitudes, the step-responses of the essential state x are to be regarded. From Eqs. (10) a~d (14) the essential state can be written as

x

-1

=

and with Eq. (5)

0

1.J

A

t

v

(e

(t)

-1). (20)

v=l

To clear the influence of A all k eigenvalues are restricted to be real for the present. Then the sum in Eq. (20) has only real summands whose magnitudes can be simply compared. The k-th summand

(21)

is influenced by the eigenvalue A . k Akt The term (e -1) .describes the dynamic belonging to A . Because all k Avt the terms (e -1) in the sum of Eq. (20) are bounded 1 according to

le

A t

v -11

~ 1

o

~

t

<

00

the coefficient

(22)

which effect the dynamic part Ak (e -1) takes in the sum of Eq. (20). Therefore the absolute value Ic o o I k would be regarded as dominance 1. J index for Ak in the ij-th transmission path. deci~es

Because of the denominator A in Eq. k (22) co k 01 tends to decrease as A k increas~s; being in accordance with the "classical" dominance notion. But irrespective of the distance of A k from the imaginary axis A does not k have any influence at all in the i?-th transmission path (ciko = 0) if e1.ther v or gkj equals to iero. ik

I

I

These are the limit cases that the k-th mode is not controllable (gk o = 0) from the j-th input variableJor is not observable (vok = 0) from the i-th essential state variable serving as output (see Gilbert, 1

in the Laplace-domain

0

Only stable eigenvalues (Re{ AV} < 0) have to be regarded. Unstable ones are always dominant.

I

L. Litz

198

1963). Therefore the dominance index IC'k' I considers not only the site of 1. J A in the complex plane but also the k controllabilit~/observabilitystructure of the system.

Eq. (22) is very simple but after that a lot of information has to be inter_ preted. To do this in a simple way, two measures are to be derived for each eigenvalue A~. Starting from the p·m indices IC'k' I their maximum and 1 their sum over ail i and j will be considered. Before performing this the numbers IC'k' I have to be normalized since the9 ~riginate from various step responses (Eq.25) having different levels. If y, > 0 is the characteristic step maqnitude of the j-th input variable and ~, > 0 is the 1 normalizing number of the i-th essential state-variable (for example the maximal absolute value of all its step-responses with y, as exciting step magnitude) then the two following dominance measures are defined:

Fortunately Ic, ,I remains significant even though Ak1~~comes co~plex. To show that, A and A 1 = A are ask k k sumed to be a conjug~te complex pair. Then we have from Eq. (21)

and hence the partial sum s1'kJ' +

S, 1,

k + 1 ,J' = 2 Re{s'k'} 1 J

is real. Looking back i t is possible to assign half the real partial sum, Re{s'k,}=Re{c'k,e 1 J 1 J

A t k

}-Re{c'k'}' 1 J

M

(23)

k

(max = i=l,max -1 ... ,m j=l, ... ,p ~i

(25)

to either of the complex partners A , X . Substituting k k jct>ikj

m

i=l

we obtain

Re{ c,1

k') J

ICikjl

(24) The last equation contains Eq. (21) as special case when A , c'k' become k real. Therefore witn Rets~k'} from Eq. (24) the step response in1E~. (20) can be rewritten as n

(25 )

Re{s'k'} 1

J

\>=1 comprising real as well as eigenvalues. Corresponding the parenthesis term of Eq. describes the dyn3mics, is bounded 2 as

I

e

Ck~

Re{c cos(w

k t+4>'k,l-

1 J

Ic

complex to Eq. (21) (24) which still

ikj

ikj

I

}

~ 2.

Hence IC'k' I accounts for the influence of tn~ dynamical part caused by A in the ij-th transmission path. k With P input variables and m essential state variables there are p·m paths each one providing n dominance indices IC'k,l. Their calculation according to 1

J

2Stable eigenvalues are supposed.

.!.-(~IC'k,IY,). L ~, ~'

I c,1 k J,I e into Eq. (23)

ICikjhj)'

•

1

J

1

(26)

J

j=l

If M is small, A is nondominant in k k all paths and can surely be neglected in the reduced order system. The comparison of M and Sk reveals, k whether A is dominant in many or in k few paths. Because the magnitude of M , Sk can alter very largely for k d1fferent eigenvalues A (see example) k the dominance measures Mk , Sk are powerful tools to choose the m eigenvalues that have to be retained in the reduced order model. Furthermore there can be more or less than m eigenvalues having numbers M , Sk of k considerable magnitudes. This may be interpreted as a reference to increa~ m or to decrease it. The best results occur if the number of dominant eigenvalues accords with that of the essential state-variables.

OPTIMIZATION To calculate! in Eq. (15) the error between the nondominant motions and their approximation has to be minimized. Instead of Eq. (18) E,(t) -J

= z2,(t) - J

2 ,(t) - 2J

-

(27)

i

, as is to be considered with ~2" 2 unit step responses when eX~itina only the jth input variable. From Eqs. (12), (13), (15) we obtain A

-1

Z2 ' (t) - J

- 2

2

E -A 1

' (t)

- 2J

(e

!2

t

-I

) 02 '

(28)

--m I ) 401 J'

(29 )

-n-m

-1

4

J

!1 t (e

Order Reduction of Linear State-Space Models with ~ ., ~2' as the j-th columns of G G 1In Eq~(9). Now a weighted sum o!'th~ integrated squared errors in Eq. (27) , P

~ L-

J

s'ij

j=1

I £~E

199

as auxiliary condition. Substituting Eq. (34) into Eq. (35) and adding Eq. (36) by the aid of 'Lagrange-multipliers .!l.i yields

(30)

.dt

-J-J

0

(37) has to be minimized (* denotes conjugate complex transposed). Substituting Eqs. (28), (29) into Eq. (27) yields

Herein the constants k

...

T ~2i = (0

1i

1

,

-A-- 0 ..•

)~

m+i

£ . (t)

--1 !1 '

(38)

-J

-1

-1

+ ( -E -A1 -g 1 J. - -A2

(39)

° ,). JL 2 J

(31 )

Clearly the constant parenthesis term of Eq. (31) must vanish for all j to enable the integrals in Eq. (30) to converge. Therefore

o

are due to the integration in Eq. (35) with ~, ! given by their elements as = -

(S) ..

-~J

(40)

.+~.

A

J

m+~

(32 ) (41 ) A.+~.

becomes an auxiliary condition while minimizing J in Eq. (30) with -1 ~2t £j (t) =!2 e .9- 2j -

!

-1 !1 t e .2. 1j · (33)

~1

Furthermore the eigenvalues of ~1' ~2 are assumed to be stable. This is always satisfied for the nondominant eigenvalues of ~2. The case of unstable eigenvalues ~n ~t is easy to overcome (see Litz, 1979 if the nondominant motions are approximated as linear combinations of the stable dominant ones only. Since the i-th component of Eji(t) =

1 Am+ i

(0

~2t · e

E.

-J

T -1 ~1t 5l2j - ~i!1 e .2.1j

(34)

T

-1

T

r.

(0

-~

(42)

•••

1

-A-- 0 ••• )~2

(43)

m+i

in Eq. (37), the auxiliary condition of Eq. (36) can be shortened to -

-1.

r.

-1.

=

0

(44)

•

To minimize J i in Eq. (38) its partial derivatives with respect to Re{e.} -1. and Im{~i} have to be equal to zero yielding

.

-2~2;

+ 2K 3 e. + N - -~

n. = o. -~

(45)

To eliminate the noninteresting Lagrange-multipliers n., in Eq. (45) e. is isolated, -1.

-1.

(35) (46)

0

can be minimized with the i-th row of Eq. (32) , ~i~1 §.1-(O

-1

!1 ~1

*

[ q j IIEjil2dt, i=l, ... ,m j=1

=

NTe.

p ~

N

-

3 only depends on the i-th row e: of _E, -~ instead of J in Eq. (30) the new criterion

J.

and the diagonal matrix ~ = diag (q1, ..• ,q) coming from the weighting factors q~ in Eq. (35). By the abbreviations 1.

(t) ,

o ... ) .

J

~

1 0 ... )G -A--2 m+i

T = 2. ,

and substituted in Eq. (44) :

(36)

3Vectors written in small letters are always column vectors, row vectors have to be transposed, denoted by superior T.

Now isolating n. in the last equation and sUbstituti;~ i t in Eq.(46) yields

L. Li tz

200

e. -l

--1[ k .-N(N T--1K K N) -1 (N T--1 K k .-r.)J . -3 -2l - - - 3 - - 3 - 2 l -l

=

CAl:

Converter, armature current control

(47) Removing the abbreviations k ·, K , 2 3 N, r. by Eqs. (38), £39), (42)land (43) pro~tdes the result

/

-1

(48)

~1

with~,!

from Eqs.(40),

(41).

Three matrices in Eq. (48) have to be inverted. ~2 contains the nondominant eigenvalues that are always stable so that A- 1 exists. The m x m matrix T in Eq~i40) had full rank in all examples treated so far. Necessary conditions for T to be a regular matrix are that-the dominant eigenvalues are different and the dominant modes are controllable, satisfied by the definition of dominance*i~lthe last chapter. The matrix (~1! ~1) in Eq. (48) can only have full rank if p ~ m. This restricts the number of input variables to be less than or at most equal to the number of state variables in the reduced model. It can be shown (Litz, 1979) that the transformation matrix F in Eq. (16) with E from Eq. (48) is-the optimal one concerning the minimization of the integrated squared error E:

-x

~1

(t)

-

~1

(t)

(49)

between the original and the approximated state-variables of the essential state. The reduction of £x has also been achieved by Chidambara (1969) or Siret, Michailesco, Bertrand (1978), but without yielding explicit formulas such as Eqs. (16) and (48).

APPLICATION TO A METAL STRIP PROCESSING LINE In Fig. 1 a section of a metal strip processing line is shown. It contains 10 strip-wrapped rollers, each roller driven by a d-c machine via a gear. Because all d-c machines are fed by the same converter, the sum of all armature currents is controlled. The index value c of this control loop AL is the first input variable. The other input variables are the strip force

4 TwO FORTRAN IV subroutines can be obtained by the author. MODOM calculates the dominance measure~ M , Sk k in Eqs. (25), (26). REDUZO Ylelds E in Eq.(48) and~, Bin Eq.(17).

/

/

/

/

((({ I I I

* -1 *J ~=~; 1[~+(~2-~!-1 ~1) (~1! ~1) -1 ~1·

.!

/

Fig 1.

I

I

\ '\',

'1111 I I I

I

I

I

I

I I

I I

I

I

I I

I

Section of a Metal Strip Processing Line

f at the entry of the section and tAe strip speed s10 at the end. A linearized state-space model which describes the mechanical and electrical parts has the 51th order (including the current sum control). The essential state variables are the speed s at the entry and the strip force f1 at the end of the section, since tfi~se are the inputs of the neighbouring sections. In addition the speed s is important, i t is the 5 variable t0 be controlled by a speed control that has to be designed later. With the input vector

and the essential state

we can. try to calculate a reduced model of the 3rd order. Therefore the dominance measures M (maximum) k and S (sum) according to Eqs. (25), (26) ~ith m = 3, p = 3 are computed and listed in Table 1. (For representation purposes all measures have been multiplied by 100 in the listing). The eigenvalues in Table are ordered in descending real parts. As can be seen, the most dominant eigenvalues are A , A , since M , Sl' 1 2 l M S are larger than all others M , k 2' 2 S . To compute the reduced model of 3~d order, in addition to A1 ,A 2 a

Order Reduction of Linear State-Space Models

--

TABLE 1 K

EIGENVAllJES IP1AGINARY REAL

I 2

-.23 -.23

3

"5 6 7 6

-.2<; -.2?

20."q - 20. 'I? 272.74 - 272 • 1 11

- ;29 -.2<1

-263.57

-.2~

,

-.29

-.Zc;

10

-.2q

11 12

-.29

-.7.9

l'

-.:30

-.30 -.31 -.31 -.32

14 t5

16 17 18 lq 20 21 22 23 24

-.3?-

-.34 -.34 -.55 -.55 ..• 55 -.55 -.55 -.55

2';

26 27 28

29 30

31 32

-r

Eigenvalues and Dominance Measures

2b~.57

0.16 - ; - -T~----

OOMINANCE MEASURES ptAXIMU"'''10i) 90.11

5UP1.100 526.?2

QO.l1

C)?f,. qfl

.98 .98 .8 Cl • Rq

2.52 2.52 3.56

8.0? 12.90

3.5~

1?.9fl

~.2t

15.65

-202.20 171.99 -171.99 .137.93 -137.93

".21

lCJ.65 ) 1 .77 3\.77

lCO.78 -10J.78 61.'36

-61.36 .00 • O~ .(10

·

')~

.00 .01

8.17 B.17 10.02 10.02 lb.6J

'''.61 2e.63 28.63 .t) 0 .00 .00 .00 .00

26 .3 Q 21,. JIl

.00

.00 .01}

.00 • DO .00

.00 .00 16.61

.G 0 .00 ate.9 b

.0")

• 1J

-13.51\ -13.71

~6

-13.~4

.o~

!7

-13.'18

.00

."P,

.J6

-14.13 -14.27

• ')0

• 32

.64

40 ql

-1~.42

• on

4~

43

-99. A'I •. qc; .89

q 'f

-99.f)2

-14.58

.00

.cc

• Of)

.00 .00

.21 .25 .48 .Q7

.02

45

-Qc;.Q4

.OJ

.01 .00

q6

-99.96

.0·)

• Of)

147 48

-qq.96

.00

.ac

-Y~.?7

.00 .00

• 0,-1

I

-O.04~--f~---

\!

-.-. ---.- --...---. -_. ---.--- "'-"-', . . _.._., 1I

I

I

_.-t--\

-I

I

--\-l----~-~J---t11 \'1 \),

1

!h

1.0

!

Step-responses of the 3rd order model (eigenvalues A , A , A ) 31 l 2 X111 (t) and the original model x (t) l11

2.8 .---.-

• 91

2.80 • t2

.07

.0:< •~1

.02

-'19.97 -qQ.97

.00

.Ol)

51 -qql.82

.00

~,,33

11 .25

le?

i

1.13

.00 .00 .00 .00

50

+--

2.

.0.')

~'i

?9

I

Fig.

.co

-.53 -.55 -.55

.39 • r; 1

I

.on

.00

• Jq

t----

--r~---~_____ I' 0.2 0.4 0.6 0.8

.71 .q9 1.12 .51 • 5'~ .55

34

\

\

-- -

-0 12 · 0

.GO

.~c

.22

-- -1- -

-\- --\-1----

.on

.00 .01 .10

33

-----1

0.04

.00

-.55

-e.14

\

14.28 74.28 107.a7 la1.81

.0 Ci .011

-13.31 -13.45

I

--~- t--

0.08

3.12 8.02

248.51 -2LJB.Sl 221.90 -227.90 202.20

~------i-----

0 . 1 2 - - - ---t--'\~-t

2.12 2. 12 3 .1 2

201

.oc

0.8

real eigenvalue has to be chosen. This is A = -8.14, having the 31 l~rgest measures M , Sk of all real k e1.genvalues. It can be suspected that a model of 3rd order would not be very good because there are more than three eigenvalues having considerable measures Mk , Sk' for example A1 to A 9' that cannot be retained in ~he 3r~ order model. A comparison of step-responses of the system of 3rd order with those of the original model shows a good agreement if x (t) (input c ' outAL l11 put sl) is regarded in Fig. 2, but a rather bad agreement if regarding X t13 (t) (input s10' output sl) in Fig. 3.

t

--+

-O.4~O---O'-+2--0+J.-----l0~.6--0+-.8---+1,0---f Fig.

3.

Step-responses of the 3rd order model (eigenvalues A , A , A ) 1 2 31

~113(t)

and the original model x (t) 113 To improve the results a reduced order model of the 7th order is cal-

L.S.S.-H$

L. Litz

202

1 O.t2~ f\l---f\------: I

_./'

I

I~----I

0\

n08tt-\-1·~- /1 \----rr l\

+--

2.0-+-----++~----+-

!n ;

o.o4-H---+~----L:+----H--;---1 L\-----; 1 i 1i \ l! \: I: \ 1/

0-

---* \

---L ---+l~--ii-----J--+-'

\i

:

' 1t

i

1/

I

I

--~--{-1- ----1------.1 -i--i I 1 -0.08- ----.- :--1-->-- -j:------ j~r----i :/ II

I

-012 · 0

--- J 0.2

- --- - L_

I

,

\r -- - -

0.4

I

I

I

L_

'

---

-

0.6

-J --- --0.8

t2 0.8;.-+-----+-+-

-Q04

I

1.6

0.4----- +-~-I--+--·~+-f

j

0~--~~--+---v.:----+-__4rf---+-----+

It

I~

.l-- ---~

1.0

Fig. Fig.

4.

5.

Step responses of the 7th order model (eigenvalues Al ,2' A17 to A20 , A31 )

Xl l l x

lll

REFERENCES

(t)

and the original model (t)

culated containing Ai' A , A , A , 2 17 18 A ,A and A in Table 1. These a~~ th~Oseven ~lgenvalues with the largest measures M , Sk. The quality k of this model is demonstrated by comparison of the same step-responses as before. Now the step-responses x (t), X (t) in Fig. 4 and lll x 111 (t), x (t) i n Fig. 5 are ve ry 113 113 close to another.

CONCLUSIONS The proposed method consists of two steps. First the dominant eigenvalues are chosen by the aid of dominance measures. After that the reduced system is calculated by an optimal approximation of the nondominant modes. The reduced models computed in this way are stationary exact and can reproduce the original dynamics very well. Most computational amount is needed for the Jordan-transformation which can be efficiently done by the EISPACK-routines (see Smith and others, 1974, e.g. 25 sec CPU-time for the example of 51th order on a UNIVAC 1108 computer). Due to the Jordantransformation an attractive simulation of step-responses can be performed by programming Eq. (20). Work is in progress to use the method in the context of decomposition-aggregation techniques.

Step responses like Fig. 4., but x 113 (t) (reduced model) X «t) (original model) 11

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