References

1 MOORE, B.C.: 'Principal component analysis in linear systems: Con-trollability, observability, and model reduction', IEEE Trans. 1981,AC-26, pp. 17-32

2 HUTTON, M.F., and FRIEDLAND, B.: 'Routh approximations forreducing order of linear, time-invariant systems', ibid., 1975, AC-20, pp.329-337

3 GUTMAN, P.O., MANNERFELT, C.F., and MOLANDER, P.:'Contributions to the model reduction problem', ibid., 1982, AC-27, pp.454-455

4 FERNANDO, K.V., and NICHOLSON, H.: 'Singular perturbationalmodel reduction of balanced systems', ibid., 1982, AC-27, pp. 466-468

5 FERNANDO, K.V., and NICHOLSON, H.: 'Singular perturbationalmodel-order reduction in the frequency domain', ibid., 1982, AC-27, pp.969-970

6 MEIER, III, L., and LUENBERGER, D.G.: 'Approximation of linearconstant systems', ibid., 1967, AC-12, pp. 585-588

7 WILSON, D.A.: 'Optimum solution of model-reduction problem',Proc. 1EE, 1970, 117, pp. 1161-1165

8 NICHOLSON, H.: 'Structure of Interconnected Systems' (Peregrinus,London, 1978)

9 FERNANDO, K.V., and NICHOLSON, H.: 'On the Cauchy index oflinear systems', IEEE Trans. 1983, AC-28, pp. 222-224

2746D

FACTOR DIVISION:A USEFUL ALGORITHM INMODEL REDUCTION

Indexing terms: Poles and zeros, Linear-system reduction

Abstract: An alternative approach is given for linear-systemreduction by Pade approximation to allow retention of dominantmodes. It avoids calculation of system time moments and thesolution of Pade equations by simply dividing out the unwantedpole factors. The resulting algorithm is conceptually simple and isseen to be of use in other methods of reduction.

Introduction

Many model-reduction techniques are concerned with pre-serving stability and matching initial time moments (andMarkov parameters) between the full and reduced systems.To preserve stability it is usual to calculate the reducedtransfer-function denominator by selecting stable poles [1]or using properties of the Routh table from the fulldenominator [2, 4]. To preserve the steady-state character-istics it is usual either to solve the Pade equations [3, 4] orinvert a continued fraction [2], which yields the reducednumerator.

A simple algorithmic approach is now presented whichobtains the reduced-model numerator in a direct way. Itavoids finding system time moments and solving the Padeequations, whilst the reduced models still retain the initialtime moments of the full system. It is shown how theconcept of factor division emerged from the method ofShamash [1] and consequently how the algorithm may beused in other methods to ease computation.

Retention of dominant poles

Consider the nth-order transfer function, whose poles areknown, given by

G(5) =(S Pn)

(1)

where p,- > 0 (/ = 1, 2, 3, ..., ri) are assumed real for initialconvenience such that px ^ p2 ^ p3 < • • • ̂ pn.

362

A reduced transfer function which retains the (n — 1)dominant poles and the first (n — 1) time moments of G(s)is obtained by dividing the factor (s + pn) into the numer-ator of G(s) from the constant term first. The resultingseries in the numerator is then truncated after the term ins"~2. This gives the reduced model

do( s + p l ) ( s + p 2 ) •••

(2)

where

d0 = bo/pn

and

d^ibi-d^J/Pn

i=U2,...,n-2 (3)

It is easily proved (Reference 5, p. 41) that G^^s) is equiv-alent to that obtained by the method of Shamash [1].

For a lower-order model, the procedure may be repeat-ed on Gn_1(s) and so on until the desired order is reached.It should be noticed that for a /cth-order model successivesets of d{ for i = 0, 1, ..., k — 1 only need be calculatedfrom eqn. 3. Alternatively, an algorithm can be used forreduction to a /cth-order model which divides the (n — k)unwanted pole factors directly into the numerator, thuscovering the case of complex poles.

The (n — k) factors to be divided are taken as

P k + l ) ( s + P k + 2 ) ••• Pn)= e0

The numerator of the reduced transfer function Gk(s) willthus be given by the series expansion about s = 0 of theexpression

e0 + el s + • • • + en_ks"

as far as the term in sk~1.This is easily achieved by a modified form of a moment-

generating algorithm [6], which uses the familiar Routhrecurrence formula to generate the 3rd, 5th, 7th, etc., rowsin the following table:

0Cn = —b2

e2

a, = —

• Q.k-2

' ek-2

rk-3Ck-3

(4)

where

<7,- = bi+l -aoei+l i = 0, 1, . . . , k - 2

r,! = Qi+i -<*iei+l i = 0, 1, . . . , k - 3

v0 = ul -ock^2ei

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The reduced transfer function is then given by

at_, s*(s + + p2) • • • (s + pk)

Example 1Consider the 4th-order model [1]

G(s) =24 + 24s + 7s2 + s3

(s + l)(s + 2)(s + 3)(5 + 4)

Eqns. 3 give the following 3rd- and 2nd-order models:

6 + 4.5s + 0.625s2

G3(s) =

and

G2(s) =

(s + l)(s + 2)(s + 3)

2 + 0.833s(s + l)(s + 2)

Reduction to a 2nd-order model directly using eqns. 4gives

I 24> 7

«, =0.833 < J °

which gives

G2(s) =2 + 0.833s

(s + l)(s + 2)

All the reduced models are identical to those obtained bythe method of Shamash [1].

Extension to other methods

Any method of reduction which relies upon calculating thereduced denominator first and then the numerator byeffectively matching time moments [2-4] can also incorpo-rate the factor-division algorithm (eqns. 4), once thereduced denominator has been found.

Suppose the full system is described by G(s) = N(s)/D(s),where N(s) and D(s) are the respective numerator anddenominator polynomials. Similarly, the reduced /cth-ordermodel is given by Gk(s) = N(s)/D(s), where D{s) has alreadybeen calculated and N{s) is to be such that the first k termsin the series expansions about s = 0 of G(s) and Gk(s) arethe same. G(s) may also be expressed as

G(s) =N(s)D(s) N(s)D(s)/D(s)

D(s)D(s) D(s)

and so N(s) will be given by the series expansion ofN(s)D(s)/D(s) as far as the term in s*"1. This may beaccomplished using the factor-division algorithm in one oftwo ways

(i) by forming the product of N(s) and £>(s) as the firstrow in (4) and D(s) as the second row (only terms up tosk~l are needed in both rows)

(ii) by expressing N(s)D(s)/D(s) as N(s)/[D(s)/D(s)'] andusing (4) twice; the first time to find the terms up to sk~x inthe expansion of D(s)/D(s) (i.e. put D(s) in the first row andD(s) in the second row, using only terms up to s*"1), andthe second time with N(s) in the first row and the expan-sion of D(s)/D(s), just generated, in the second row.

The relative orders of the full and reduced models and theavailability of a polynomial multiplication algorithm willgovern the choice of which option should be preferred.

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Example 2Consider the 4th-order system [3] given by

2400 + 1800s + 496s2 + 28s3

(5) ~ 240 + 360s + 204s2 + 36s3 + 2s4

Reduction to a 2nd-order model by the methods of Refer-ences 2 and 3 gives the denominator D(s) = 4 + 6s 4- 3s2.

Option (/)

N(s)D(s) 9600 + 21600s+ • •D(s) 240 + 360s + • •

eqns. 4 give

9600 21600240 360

7200240

giving the reduced model

„ _ 40 + 30s

Option (ii)Eqns. 4 for D(s)/D(s) gives

3604 6

Eqns. 4 forN(s)

D(s)/D(s)

gives

a0 = 40 <̂/2400\ 60

18000

60

giving the reduced model

40 + 30s

In both cases, the reduced models are identical to thoseobtained in References 2 and 3.

Conclusions

A useful algorithm for linear-system reduction has beengiven which was initially conceived as an alternativeapproach for retaining dominant poles in the reducedmodel. It avoids calculating the initial time moments of thesystem and solving the Pade equations. If the system polesare known (usually the case for a useful comparison oftime responses), or at least those to be neglected, thenfactor division is often quicker to use than the method ofShamash [1]. However, if such information is not avail-able, the poles to be neglected (usually of largestmagnitude) have to be found before division is carried out.The relative separation of these poles together with theorders of the full and reduced models will dictate whetherfactor division is quicker than the method of Reference 1.

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The algorithm is also seen to be useful in other methodswhere the reduced denominator is calculated first andinitial time moments are to be preserved. Again this avoidssolving Pade equations or inverting a continued fraction toobtain the reduced numerator.

Finally, it should be noted that Markov parameters aswell as time moments can be preserved by simply dividingthe factors from the highest powers of s first. This has theeffect of reversing the coefficients in the first two rows of (4)and terms up to s"~l have to be considered in these rows.Thus certain biased models are possible using this tech-nique, which it is hoped will be discussed in a future paper.

22nd August 1982 T.N. LUCAS

Department of Mathematics & Computer StudiesDundee College of TechnologyBell StreetDundeeScotland

References

1 SHAM ASH, Y.: 'Linear system reduction by Pade approximation toallow retention of dominant modes', Int. J. Control, 1975, 21, pp.257-272

2 HUTTON, M.F., and FRIEDLAND, B.: 'Routh approximation forreducing the order of linear, time-invariant systems', IEEE Trans.,1975, AC-20, pp. 329-337

3 SHAM ASH, Y.: 'Model reduction using the Routh stability criterionand the Pade approximation technique', Int. J. Control, 1975, 21, pp.475-484

4 PAL, J.: 'Stable reduced order Pade approximants using the Routh-Hurwitz array', Electron. Lett., 1979, 15, pp. 225-226

5 LUCAS, T.N.: 'Frequency domain approximation of linear systems',Ph.D. thesis, 1978, UWIST, Cardiff, Wales

6 LAL, M., and MITRA, R.: 'Simplification of large system dynamicsusing a moment evaluation algorithm', IEEE Trans., 1974, AC-19, pp.602-603

2761D

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