3 JAYAWANT, B.V., SINHA, P.K., WHEELER, A.R., WHORLOW,R.J., and WILLSHER, J.: 'Development of 1-ton magneticallysuspended vehicle using controlled d.c. electromagnetics', ProcIEE, 1976, 123, (9), pp. 941-948

4 GOTTZEIN, E., MILLER, L., and MEISINGER, R.: 'Magneticsuspension control system for high speed ground transportationvehicles'. World Electrotechnical Congress, June 1977, Moscow

5 GOTTZEIN, E, and CRAMER, W.: 'Critical evaluation of multi-variable control techniques based on Maglev vehicle design'. Sym-posium on multivariable technological systems, IFAC 77

CorrespondenceIMPROVEMENT OF SYSTEM PERFORMANCE BYTHE USE OF TIME-DELAY ELEMENTS

I should like to make a few comments regarding paper 2084 D[IEE Proc. D, Control Theory & Appl., 1982, 129, (5), pp.177—181], by Marshall and Salehi. Regarding their third-order example (Section 4.2 of their paper), we have foundit possible to improve on their minimum value of / ( a , T)by including another delay term in the velocity feedbackpath so that the transfer function in this path becomes

a + a e x p ( - r s ) + 0 . 0 1 5 exp ( — 1.12s)

with a, a, and r as in their paper. We have also found thatit is not possible to improve on the results of Marshall andSalehi with rational functions of s, the best results in thesecircumstances occurring when the rational functioncorresponds to the particular Pade approximation for thedelay used by Marshall and Salehi. This was ratherdisappointing as such a function may have noise-rejectionproperties which are not possessed by pure delay.

I should also like to make the point that there is somedoubt as to whether the quadratic performance index is veryuseful to engineers [A], but Marshall and Salehi also considerovershoot and settling time. We have found that shortersettling time, based on 0.1% deviation, may be obtainedwith a transfer function of

a + a exp (~TS) -0.1 exp (—0.28s)

with a, a, T as before. This results in a settling time of 4 swhereas the use of the first two terms (as in Marshall andSalehi) results in a settling time of 5 s. Presumably we couldhave many terms with different gains and delays, and 'playtunes' on these to obtain minimum overshoot, minimumsettling time, or some combination of both.

A matter of some concern is the susceptibility of theapproach of Marshall and Salehi to the effects of eitherparameter variation, or simply erroneous estimation of a time-invariant parameter. If the loop gain in their example is raisedto 60, then the system is unstable if their delay term isincluded, but the system remains stable if it is left out.

Another point to be considered is that all servo amplifierssaturate when the servo experiences a large step change indemand. It is not economically or technically acceptable toincrease the dynamic range of the amplifiers to avoid this.We have found that, for a step demand of five times thesaturation voltage of the amplifier, the scheme of Marshalland Salehi results in an overshoot of 14.5% whereas the systemwithout their delay term has an overshoot of 8.7%. Thearrangement of Marshall and Salehi still results, however,in shorter settling time.

There still remains the problem of noise. In numerouspapers (e.g. Reference B) Isaac Horowitz and his colleagueshave pointed out that the more feedback and designer uses,the greater the transducer noise presented to the error channel.In extreme cases this may cause the first amplifier stage to

92

MULLER, P.: 'Design of optimal state observer and its applicationsto Maglev vehicle suspension control'. 117 AC conferenceGONDHALERKAR, V.M.: 'Control systems and dynamics of amagnetically suspended vehicle'. D.Phil, thesis, University of Sussex,1980GONDHALEKAR, V., and JAYAWANT, B.V.: 'Control and designaspects of magnetically suspended vehicles'. IEAC Congress, Kyoto,1981

saturate, a phenomenon which may be disastrous. Now velocitytransducers are fairly noisy, and additional velocity feedbackpathways will increase the noise appearing in the error channel.It may therefore be desirable to use a Pade approximation,having more poles than zeros, in preference to a pure delay,despite the consequent theoretical reduction in optimality.

24th November 1982 P.H. LANDERS

Department of Electrical & Electronic EngineeringDundee College of TechnologyBell StreetDundee DD1 1HGScotland

We are grateful to Dr. Landers for his interest in our paper,and for his supportive comments.

As mentioned in the abstract, our paper is an explorationinto the possibility of performance improvement by theaddition of delay elements. The ISE criterion is a mathemat-ically convenient method for finding an optimal compromiseregarding the overshoot and settling time of a step response.We are not surprised that further improvement is possibleby the addition of further delay(s), although the extra benefitmay be marginally worthwhile in practical cases. The remarkconcerning the Pade approximant, in the first paragraph ofthe correspondence, is interesting and indicates that thetechnique suggested in our paper is soundly based.

As References 4, 6, 7 and 8 demonstrate, we are wellaware of the susceptibility of any time-delay system to theeffects of parameter variation, nonlinearity and noise. Ifany of these are present, they should be taken into accountexplicitly in the design. Replacing a delay by its Pade approxi-mant is perhaps too ad hoc for dealing with noise, surely anexplicit filter would be preferable.

In submitting our exploratory paper, we had hoped thatit would stimulate interest in the arguably counter-intuitiveuse of time-delay elements in controllers realised by micro-processor techniques. The response of Dr. Landers in hiscomments and experimentation is encouraging.

J.E. MARSHALLS.V. SALEHI

20th January 1983

School of MathematicsUniversity of BathClaverton DownBath,BA2 7AYEngland

References

A FOSS, A.S.: 'Critique of chemical process control theory', IEEETrans., 1973, AC-18, pp. 646-652

B HOROWITZ, I.M. and SHAKED, U.: 'Superiority of transfer func-tion over state-variable methods in linear time-invariant feedbacksystem design', ibid., 1975, AC-20, pp. 84-97

DTC130D

IEE PROC, Vol. 130, Pt. D, No. 2, MARCH 1983