TECHYICAL NOTES AND CORRESPOSDENCE 671

100.0 -1.235 1 . 1 2 1 I 0 . i o x 10-3

Using P and G*, (1 1) can be expressed as

F*C*E{z(r)z’(t)}C*‘=C*A*E{z(t)t‘(r)}C*’ (18)

G*= C*B*. (19)

Premultiplying and postmultiplying (16) by C* and C*‘. respectively, yields

C * A * E { z ( t ) z ’ ( r ) } C * ’ + C*E ( z ( t ) z ‘ ( t ) } A * ‘ C * ’ + C*B*NB*‘C*‘=O.

(20)

Substituting (18) and (19) into (20) yields

P C * E { z ( t ) z ’ ( t ) } C * ’ + C * E { z ( t ) r ’ ( t ) } C * ’ F * ’ + G*NG*’=O. (21)

The covariance matrix E {i( t) i’(f)} of the composite system given by (17) satisfies the following equation:

PE{i(t)i’(t)]+E{i(r)i’(r)}P’+G*NG*‘=O. (22)

Comparing (21) with (22). we can conclude that E { i ( t ) i ’ ( f ) ) equals C * E { z ( f ) r ’ ( t ) } C * ‘ ; that is. the covariance of the outputs of the reduced-order model is the same as that of the original system.

NLMERICAL EXAMPLE

Consider the system (see Galiana [ 6 ] ) given by

where 6 is a positive real coefficient. The input u ( t ) is the output of the following shaping filter:

i ( l ) = - d u ( t ) + d c ( r )

where c(r ) is zero-mean white noise with variance

E { u’(z)} = I .

Table I shows results of approximation for this system by a first-order model of the form

If 6>1, then f - - 1. This result is reasonable. in the table. the mean square errors of output are also shown. We can say that the reduced- order model gives a satisfactory approximation.

CONCLUSIONS

The problem of modeling a linear time-invariant system by a reduced-order model is considered. An expression for a reduced-order model is given in the case when a time-invariant random input is used. The method proposed here has several advantages. First, i t is easy to

calculate the parameters of the reduced-order model. Second, the para- meters are determined uniquely when the condition in the theorem is satisfied. Third. the covariance matrix of outputs is identical for both the reduced-order model and the original system.

REFERENCES

[ I ] E. J. Davison, “A method for simplifying linear dynamic systems,” IEEE Tram.

(21 41. Aoki, “Control of large scale dynamic systems by aggregation,” IEEE Tram. Automar. Conrr., vol. AC-11. pp. 93-101. Jan. 1966.

[3] 41. R. Chidambara, “Ta.0 slmple techniques for the simplification of large dynamic Auromar. Conrr.. vol. AC-13, pp. 24&253. June 1968.

[4] 1. Meler and D. G. Luenberger, ”Approximatton of linear constant systems,” IEEE systems.” In Pror. I969 Joinr Auromric Conrrol Con! . pp. 66S674.

[ 5 ] D. A. Wilson, ”Optimum solution of model-reductton problem.” Proc. Imr . Elec. Tram. Auromar. Conrr., vol. AC-12, pp. 585-588, Oct. 1967.

[ 6 ] F. G. Galiana. “On the approximation of multiple input-multiple output constant Eng., vol. 117. pp. 1161-1 165. June 1970.

[7] G. Oblnata and H. Inooka, “A method for modeling linear tlme-invanant systems by linear systems.” In!. 3. Conrrol, vol. 17, pp. 1313-1324, June 1973.

linear systems of low order.“ IEEE Tram. Auromar. Conrr.. vol. AC-21, pp. 602403, Aug. 1976.

A Note on Selecting a Loworder System by Davison’s Model Simplification Technique

G. B. MAHAPATRA

A6sfmcr-A criterion is presented in this correspondence to select the low-order system after simplifying the given high-order system by Davi- son’s model simplification technique. The criterion is judged by a numeri- cal example.

I. INTRODUCTIOK

The idea of retaining dominant eigenvalues to simplify a high-order system to a low-order plant was presented for the first time in [ I ] . The state-variables of the low-order system were chosen such that they form large percentages of the eigenvectors corresponding to the retained dominant-eigenvalues [l , p. 951. It is not yet clear how small the approximate model can be and yet accurately represent the process. This aspect is discussed in this note for a stable system having distinct eigenvectors. It is seen that the smallness of the approximate model representing the original plant accurately can be decided in terms of the largest eigenvalue neglected, the sizes of the original plant, and the reduced plant. This criterion is judged by comparing the time responses of various low-order systems illustrated by Davison [I].

11. DEVELOPMENT OF THE CRITERION

Consider the linear time invariant system

Y = A Y + B u ( t ) , t > O

with Y(O)=O and ~ ( t ) being unit step input.

A and B are of sizes ( n X n) and ( n X r ) , respectively. The solution of (1) and (2) is given by

Y ( Z ) = P , ~ ~ ~ ~ ( D , ~ ) - ~ ] D ~ ~ ~ V , , B , + VI$,]

Manuscript received November 16, 1976. T h e author is with the Department of Electrical Engineering, Roorkee University,

Roorkee, India.

678

where the eigenvalue matrix D equals

0

0

IEEE TRANSACTIOSS ON AUTOMATIC CONTROL, VOL. AC-22, NO. 4. AUGUST 1977

TABLE I ERROR CRITERIOI: FOR DAVISON'S EXAMPLE

3 0.098154 25-00 4 0.23753 9.70 5 0.32726 6.15 6 2.9418 0.59

The corresponding modal matrix P. its inverse V-matrix. D-matrix. and B-matrix are partitioned as follows:

~

n - m ) diagonal matrices, respectively. V I , ( m X m). k . , ? ( n - 171 x m). V , , ( F m X 117). and V,,(n - m x n - m ) are the partitioned matrices

_ _ _ _

of V-matrix. B,(n1x r ) and B z ( G X r ) are partitioned matrices of B ( n x r ) matrix. If first m-eigenvalues are considered 10 be dominant. the approximate solution of (3) is given by

~~

z ( ~ ) = P , [ ~ x ~ ( D , ~ ~ - I ] D ~ ~ [ v , , B , + I'~,B,]. (5)

The error involved b> neglecting higher modes X,,, ,: . . .A, in state equation ( 3 ) is given by

E(f)=Po[exp(Dor)-I]D, ' [ b',,B,+ 1.',,8,] (6)

t IE(r)t l<11P~1tItexp(D~r)-I I l11D,~11[11V~~I IBJ+ I IL '~~ : 1B2111 (7)

For the given system ( 1). let

Therefore

Equation (11) shows that for a given system. the error in states due to neglecting the higher modes depends upon L'n - m /Ihm+,l . Therefore, the error can be made small when

111. ILLUSTRATIVE EXAhiPLE

Consider the numeric example of Davison [ I , p. 961. A boiler system described by a 9 X 9 matrix is simplified to a 3 X 3 , 4 X 4, 5 X 5, and 6 X 6 system of matrices. The maximum error in states due to neglecting higher modes is calculated from (11) and is summarized in Table I. It is

seen that the factor dn - m / l A m + l l is the lowest for m = 6 . This is also the conclusion of Davison [ I , p. 971. It is also seen that as &- m /!A,,,+ I I is reduced, the time responses of approximate states approach those of original states.

IV. REMARKS ON CFUTERIOX

From (12), if m+n, then the error is reduced to zero, this being an obvious fact. If for a system l ~ + , I > \ / ~ , then the error will be reduced. However, if the eigenvalues IX,, ,I < 6% . then the time responses of approximate model may not tend to those of the original system and it may also be difficult to decide the order of the simplified model.

\'. COKCLUDIXG REMARKS

A criterion is developed in this note to decide the size of the low-order system using Davison's model simplification procedure. This is verified using the illustrative example of Davison [I] .

REFERENCES

[ I ] E. J. Davison. "A method of simplifying linear dynamic systems," IEEE Tram. Auromar. Conrr.. vol. AC-11. pp. 93-101, 1966.

Structure Analysis of Periodically Controlled Nonlinear Systems via the Stroboscopic Approach

MASAKAZU MATSUBARA AND KATSUAKI ONOGI

Abstract-It is shown that an efficient structure analysis of periodically controlled nonlinear systems is possible by use of the computer simulation technique coupled with the stroboscopic approach. The existence and stability of every periodic state can be determined together with the dynamic behavior in the neighborhood of these periodic states. Especially, the proposed technique seems to be the only nay for finding out periodic states of the saddle mode.

1. INTRODUCTION

Suppose that a system

i = f ( x , u ) . x ( t ) E R " . u ( t ) E Q ~ R ' (la)

is subject to a certain specific periodic control u such that

U ( f + T ) = U ( f ) , T > o . (Ib)

The u will be so chosen that the system (1) is optimized with respect to a certain objective function. Since any steady control u( t )= U (=const.) is included in the class of controls satisfying (Ib). the specific u may be a

Manuscnpl received October 12, 1976. The authors are u.ith the Depmment of Chemical Engineering. Nagoya University.

Nagoya. Japan.