Linear system reduction by the modifiedfactor division method

T.N. Lucas. B.Sc. Ph.D., A.F.I.M.A.

Indexing terms: Linear networks, Matrix algebra, Mathematical techniques

Abstract: A novel method of linear system simplification is presented based on a modified factor divisionapproach. It is seen to be more flexible than most model reduction methods in that families of reduced modelsmay be easily generated by varying a single parameter in the modified transfer function denominator. It alsoguarantees stable reduced models of stable systems and preserves initial time moments and Markov parametersof the system. An example illustrates its application.

1 Introduction

A major drawback with standard stability preservingmethods [1-4] is their lack of flexibility when the reducedmodel does not produce a good enough approximation.This may be the case even when the combination of timemoments and Markov parameters are varied to producedifferent models. The problem is that the denominator ofthe reduced model derived by these methods is fixed for aparticular reduced order, irrespective of the variation ofthe retained time moments and Markov parameters (whichaffect only the numerator). Consequently, the modes of thereduced model cannot always be changed sufficiently tomeet the requirements for a good approximation.

Some notable attempts have been made recently [5-8]to take account of this 'denominator flexibility' problem,although each have their individual drawbacks. The biasedRouth method [5, 6] can only produce reduced modelswhose orders are the same parity (odd/even) as thesystem's order (proved in Reference 6); the Routh/Pademethod [7] involves much computation and is best suitedfor systems which are not too highly ordered; the mixedstability-equation/Pade-method [8] requires finding rootsof two polynomial functions which, for systems of order 6or more, means solving cubic and higher degree equations.

It is the purpose of this correspondence to present a'modified' factor division method, extending the ideas ofLucas [9, 10], which guarantees stable reduced models ofstable systems and is able to produce families of reducedmodels of all orders by varying a single parameter in themodified transfer function denominator. Consequently, themethod is more flexible than most other stability preser-ving methods and, although computer-orientated, is simpleto implement. A numerical example follows the discussionof the method to illustrate its application.

2 Modified factor division method

The first step is to 'modify' the full transfer-functiondenominator by choosing a value of the real parameter,p > 0, such that s = — p is a root of this modified denomi-nator.

Suppose the full nth-order transfer function is given by

(1)

where N(s) is the numerator polynomial and

D(s)

Paper 48OOD (C8), first received 24th April and in revised form 30th September 1985The author is with the Department of Mathematics and Statistics, Paisley College ofTechnology, High Street, Paisley PA1 2BE, United Kingdom

D{s) = ansn

+ a0

is the denominator polynomial.Now consider the nth degree polynomial defined by

P(s) = D(s)D(p)-D(-s)D(-p) (2)

where p > 0 is any real number.By the remainder theorem, it is easily seen from eqn. 2

that P( — p) = 0, and so (s + p) is a factor of P(s). Further,it is proved in Appendix 6.1 that P(s) must be Hurwitz ifD(s) is Hurwitz (i.e. all roots in left halfplane). It followsthat the (n — l)th degree polynomial defined by

P(s)s + p

(3)

must also be Hurwitz.If the pole factor (s + p) is now taken as an approx-

imation to an 'unwanted' pole factor, in the factor divisionsense [9], of the original system G(s), then the reduced(n — l)th-order model will have Q(s) as its stable denomi-nator. The reduced numerator can then be formed by thefactor division algorithm [10] to enable certain com-binations of time moments and Markov parameters to beretained in the model.

For reduction to lower-order models, the above pro-cedure may be repeated on the successive reduced-ordertransfer functions. It should, however, be noticed thatreduction directly to a /cth-order model requires only cal-culating (n — k) successive reduced-degree denominators(given by eqn. 3) for certain values of p (discussed in fol-lowing Section) and one numerator calculation of degree(k - 1).

2.1 Choice of pThe choice of p > 0 is seen to be quite arbitrary, thus pro-viding great flexibility in deriving stable reduced denomi-nators. However, experience with numerous numericalexamples has shown that best results are usually obtainedwhen p is chosen to lie in the interval

where —a is the average pole of the system which fromeqn. 1 gives

un-l (4)

and — [I is the largest negative real part of the system poles(i.e. furthest away from the imaginary axis). It is felt thatthis range for p will be fairly reliable because the modified

IEE PROCEEDINGS, Vol. 133, Pt. D, No. 6, NOVEMBER 1986 293

system poles will tend to 'straddle' the actual system polesas explained in Appendix 6.2. Notice that if — p happens tobe an actual system pole then the method becomes, inessence, similar to that of Shamash [3].

It is often the case that the pole distribution of thesystem is not known and to obtain such information mightbe inconvenient from a computational viewpoint.Although a is easily determined by eqn. 4, obtaining /? willgenerally be more difficult. Instead of finding fi accuratelyit is proposed to use a simple estimate of this value byconsidering only the first three terms of D(s) in eqn. 1 con-taining the highest powers of s (these are the most signifi-cant for large \s\, see Reference 11). This gives thequadratic equation

ans2 _2 = 0 (5)

whose larger negative real part root is an estimate for — ft,i.e.

for

for a 2 ^ <4anan_2

(6)

How good the estimate given by eqn. 6 is for /J will ofcourse depend on the relative separation of the poles, andthere may well be systems for which this estimate is notvery accurate. However, eqns. 4 and 6 at least provide aninitial range of values of p which can be widened or madesmaller as necessary.

In an attempt to further reduce the arbitrary nature ofselecting p, it is suggested that the value

P = P)should be used in the first instance. This is because itseems reasonable to assume that a pole in this position isnot the most significant in influencing transient or steady-state behaviour [5, 8] (usually the largest and smallestnegative real parts do this), also this value of p tends tomodify the system poles in a fairly logical way due to the'straddling' effect (Appendix 6.2) and has been used withsome success in various numerical examples.

For reduction by more than one order, the estimates ofa and /? are calculated from the last reduced denominator,and so these values are not generally constant throughoutthe whole reduction process. For example, if D(s) in eqn. 1is reduced to the (n — l)th degree denominator given by

+an_2s "-2+

then a and /? for the next reduction stage are calculated byeqns. 4 and 6 with n — 1, an_l5 an_2, an-i replacing n, an,an_l,an_2, respectively.

The full system transfer function [10] is given by

10s4 + 82s3 + 264s2 + 369s + 156(S) ~ 2s5 + 21s4 + 84s3 + 173s2 + 148s + 40

(poles at -0.5, - 1 , - 2 ± 2j, -5).

Successive reduced-order models are given as:

p = 3.68,

5s3 + 20.1227s2 + 45.2145s + 20.67384(S) ~ s4 + 6.5615s3 + 17.8538s2 + 18.6682s + 5.3010

(poles at -0.4435, -1.4011, -2.3585 ± 1.7232/)

5s2 + 16.2750s + 8.4040p - 2.46, G3(s) -

(poles at -0.3922, - 2 ± 1.2157/)

1 5 H r ( \ 5s + 5 m i

p = 1.83, G2(s) = s2 + 3 l 2 5 9 s + l 3 2 8 6

(poles at -0.5074, -2.6185).

Fig. 1 compares the step responses of these reduced models

u -

10time

Fig. 1 Comparison of the step responses of the reduced models with G(s)

G(s) responseG4(s) responseG3(s) responseG2(s) response

with that of G(s) and shows that they are all very goodapproximations. It is interesting to note that, for this par-ticular system, none of the 'flexible' reduction methods[5-8] produced reduced models which were significantlybetter than those given here (indeed many were worse). Inparticular, the high degree of accuracy obtained by the2nd-order model, G2(s), could not be matched by 2nd-order models using these methods.

3 Numerical example

To demonstrate the application of the new method a 5th-order system is considered. The value of p at eachreduction stage is taken to be the one suggested pre-viously, i.e. p = |(a + P) as given by eqns. 4 and 6. Thereduced denominators are calculated by eqn. 3 and areused in the factor division algorithm [10] to produce thereduced numerators, where only one Markov parameter isretained in each reduced model for simplicity of presen-tation.

4 Conclusions

A novel method of linear system reduction has been givenwhich allows 'denominator flexibility' by appropriatechoice of the real parameter p which modifies the systempoles. Stability is guaranteed in the reduced models ofstable systems and families of models of all reduced ordersmay be generated by varying p, as well as varying the com-binations of retained time moments and Markov param-eters.

Point and interval estimates for p are suggested whichhave been found to be reliable for a large class of systems,

294 IEE PROCEEDINGS, Vol. 133, Pt. D, No. 6, NOVEMBER 1986

and indeed the numerical example shows consistency withthis viewpoint. However, choice of this parameter will ulti-mately depend on the type of approximation wanted, andas such might require a computer search for some 'best'value. This is an area for future research.

In conclusion, it may be said that the proposed methodis conceptually simple and easy to program. It enablesgreater flexibility than most model-reduction methodswhile still having the desirable properties of stability pres-ervation and system parameter retention.

5 References

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

2 CHEN, T.C., CHANG, C.Y., and HAN, K.W.: 'Reduction of transferfunctions by the Stability-Equation method', J. Franklin Inst., 1979,308, pp. 389-404

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

4 GUTMAN, P, MANNERFELT, C.F, and MOLANDER, P.: 'Con-tributions to the model reduction problem', IEEE Trans., 1982,AC-27, pp. 454-455

5 SHAM ASH, Y.: 'Stable biased reduced order models using the Routhmethod of reduction', Int. J. Sys. ScL, 1980,11, pp. 641-654

6 BISTRITZ, Y., and SHAKED, U.: 'Stable linear systems simplifica-tion via Pade approximations to Hurwitz polynomials', J. Dynam.Sys. Meas. Control, 1981, 103, pp. 279-284

7 LEPSCHY, A., and VIARO, U.: 'An improvement in the Routh-Padeapproximation techniques', Int. J. Control, 1982, 36, pp. 643-661

8 PAL, J.: 'Improved Pade approximants using stability equationmethod', Electron. Lett., 1983, 19, pp. 426-427

9 LUCAS, T.N.: 'Factor division: a useful algorithm in modelreduction', IEE Proc. D, Control Theory & Appi, 1983, 130, (6), pp.362-364

10 LUCAS, T.N.: 'Biased model reduction by factor division', Electron.Lett., 1984, 20, pp. 582-583

11 SHANNON, A.G., and CLARKE, J.H.: 'Polynomial truncations',Math. Gazette, 1983, 67, pp. 278-280

6 Appendixes

6.1 Proof that P(s) must be Hurwitz if D (s) is HurwitzThe method of proof that P(s), as given by eqn. 2, isHurwitz if D(s) is Hurwitz will be by the Routh stabilitycriterion.

Let D{p) = A and D(-p) = B, then

P(s) = ans"{A-(-irB}

+ an.lS"-'{A - (-If-'B} + ••• + ao(A - B)

For the stability of P(s), it is convenient to consider thestability of q(s) = P(s)/{A - ( - \)nB), i.e.

q(s) =

where

X =

A + B

A-BA- B

A + B

+ Aan_3sn"3 + •• •+a,,.^'1 +an-2s"-2

+ Aan_3sn"3 + • •• +

(n even)

(n odd)

ao (n even)

Xa0 (n odd)

Further, notice that A > 0 for p > 0 and \B\ < A (followsfrom definition), giving X > 0.

Forming the Routh array for q(s), for n even, gives

«n-4 ' " a2 a0

n_3 X a n _ 5 ••• Xav

an-2

Xan_

where

K-i = an-i ~ an-^

and

for i = 2, 4, 6,

for i = 3, 5, 7,bH.i = an_i - bn_i^1an_l/bn

and so on.Notice that the Routh array of q(s) is the same as that of

D(s), with every even-numbered row multiplied by X > 0.Thus the elements in the first column of this array are allpositive if D(s) is Hurwitz, completing the proof that q(s)and hence P(s) must also be Hurwitz. A similar proofshows the same result for n odd.

6.2 Poles of the modified systemThe poles of the modified system are given by the roots ofP(s), which are obtained from the solution of

AD(s)- BD(-s) = 0

where A = D(p) and B = D(-p).This may also be expressed as

AD(s)/D(-s) = B

or

H{s) = B

where H(s) = AD(s)/D(-s).Thus the real roots of P(s) are given by the intersection

of the graphs of the rational function H(s) and the straightline B. Notice that these must occur for s < 0, as proved inAppendix 6.1, and, in this range, H(s) is well behaved andcontinuous (its poles lie in the right halfplane). Further,notice that, for s < 0,

\H(s)\ <A (as \D(s)\ < \D(-s)\)

and

lim H{s) = (-!)"A, with H(0) = A

H(s),

Fig. 2 I llustrative graphs for H[s){a) n even: x system pole, # modified pole(b) n odd: x system pole, # modified pole

IEE PROCEEDINGS, Vol. 133, Pt. D, No. 6, NOVEMBER 1986 295

The roots of H(s) are also the roots of D(s), the poles of thesystem, thus illustrative graphs of H(s) are exhibited in Fig.2 for n even and odd, respectively.

It is seen from these graphs that the modified poles, oneof which must be — p, will often 'straddle' the system polesin the sense that among three consecutive poles there willbe two modified poles. Hence the reasonable representa-tion of the system poles by the modified poles when p ischosen in the suggested range of a ^ p ^ /?. For reduction

by more than one order these arguments apply to the lastreduced degree denominator, continuing the straddlingeffect on the system poles.

Although these arguments are only strictly valid for realpoles, it has also been found that complex poles are oftenwell approximated by this technique, especially for systemswith a mixture of real and complex poles (see the numeri-cal example).

296 IEE PROCEEDINGS, Vol. 133, Pt. D, No. 6, NOVEMBER 1986