Stable reduced-order models for discrete-time systems

CorrespondenceSTABLE REDUCED-ORDER MODELS FORDISCRETE-TIME SYSTEMS

In paper 4322D [ /££ Proc D, Control Theory & Appi,1986, 133, (3), pp. 137-141] the authors have presented amethod for reducing the order of a discrete-time transferfunction. The authors claim that the technique is guar-anteed to produce stable reduced-order models of stablesystems. It is the purpose of this note to show that this isnot the case and a further disadvantage of the method isalso discussed.

To derive a 'stable' reduced denominator A(z) from thefull denominator polynomial E{z), the method used bythe authors is to transform E(z) to E'(p) via the lineartransformation z = p + 1. This ensures that the poles ofthe transformed system are to the left of the new origin atp = 0. The Routh array of E'(p) is formed and the 'a sta-bility parameters' are formed in the usual way [see Refer-ence 8 of paper 4322D]. However, the reduceddenominator is then formed by

<x2 p + a3 p

or

A(z) = ocl + a2(z -

<xk pk 1 + pk

ak(z-iyi+(z-l)k

(A)

Although the a parameters in eqn. A are positive, this isnot sufficient to guarantee stability of the polynomial oreven guarantee that the roots of A(z) lie within the unitcircle centred at z = 0. This can be easily demonstratedby considering the stable 4th-order denominator poly-nomial given by

E(z) = (z - 0.9)(z - 0.8)(z - 0.7)(z - 0.6)

Putting z = p + 1, gives

E'ip) = (p + 0.1)(p + 0.2)(p + 0.3)(p + 0.4)

= P4 + P3 + 0.35p2 + 0.05p + 0.0024

Forming the Routh (alpha) table gives

0.0024 0.35 1

= 0.048

0.05 1

a2 = 0.16556<

> 0.302 1

a , = 0.36192<

a* = 0.83444<

•0.83444

1

Thus the reduced 3rd-order polynomial given by eqn. Ais

A'(p) = p3 + 0.36192p2 + 0.16556p + 0.048

with roots at -0.3171 and -0.02239 ± 0.3884i whichgives the roots of A(z) at 0.6829 and 0.97761 ± 0.3884i. Itis seen that the complex pair of roots of A(z) lie outsidethe unit circle \z\ = 1 , and will therefore lead to anunstable system.

It should also be noticed that this method has anotherserious drawback compared to other methods in that, ifthe order of the reduced model is increased to that of thefull system, as given by eqn. A, then a different systemwill result in general. This is contrary to the expectedresult with model reduction methods that the closer theorder of the approximant is to that of the full system thenthe better the model becomes. This point can again bedemonstrated by the given example; if the 4th-order'approximation' of E'(p) is formulated as suggested inpaper 4322D this results in

A\p) = p4 + 0.83444p3 + 0.36192p2 + 0.16556p + 0.048

which can be shown by the Routh criterion to haveunstable roots. Hence A(z) will have roots outside\z\ = 1 .

Conclusions

It has been shown that the method given by the authorsof paper 4322D does not possess the stability-preservingproperties claimed by the authors. Further, it is seen thatthe polynomial approximation technique used is not veryreliable as the order of the approximant approaches thatof the full system.

14th May 1986 T.N. LUCAS

Department of Mathematics & StatisticsPaisley College of TechnologyHigh StreetPaisleyRenfrewshire PA1 2BEUnited Kingdom

The authors are delighted that paper 4322D has gener-ated interest, and are also pleased that the 4th-orderexample described in this correspondence, far fromproducing an unstable reduced order model, in fact is agood example of how the method produces stablereduced-order models with a relatively simple procedure.Several examples were given in the paper to show howthe method described is applied by means of a set of afi-parameters found from the definitions:

a, ='i. 1 i= 1, ...,k+ 1 (B)

(Q

It must be emphasised that the denominator subscripts inthese two equations are k + 1, 1 and k, 1, respectively,and not i + 1, 1 and i, 1. This is an important pointbecause the latter set of indices results in the <x/?-parameters defined in the paper by Hutton and Fried-land [see Reference 8 of paper 4322D], and, as shown inthe example in this correspondence, this can easily resultin an unstable reduced-order model when the parametersare used as described by the correspondent. The differ-ence in the a/f-parameters was in fact stressed on page138 of our paper, where it was stated that, quote, 'the a,and /?,• obtained . . . bear no relationship to the a, and /?,obtained in Reference 8.' The worked examples, of whichthere were three, then showed this to be the case.

The example described in this correspondence is now

IEE PROCEEDINGS, Vol. 134, Pt. D, No. 1, JANUARY 1987 51