Improvement of system performance bythe use of time-delay elements

J.E. Marshall, B.Sc, M.Sc, PhD., C.Eng., F.I.M.A., M.lnst. P., M.I.E.E., andS.V. Salehi, M.A., M.Sc, Ph.D., Mem. I.E.E.E.

Indexing term: Control theory

Abstract: The paper explores the possibility of performance improvement by the use of added delay elements.The counterintuitive phenomenon of 'improvement by mismatch' has been noted by several authors in thecontext of predictor control. This phenomenon is illustrated using a second-order example, and, in the lightof this, it is demonstrated how it also may be improved by the use of delay elements. A third-order exampleof delay-free system improvement is described. A technique for producing good first-order estimates ofdesign parameters is presented.

1 Introduction

Many commonly used methods for the control of a time-delaysystem involve prediction [1—5]. These methods require agood system model including a model of the plant delay.There has been a renewed interest in such methods owing tothe increasing ease with which pure delays may now berealised: digital storage techniques offer significant improve-ments over continuous operational amplifiers with theattendant difficulties of Pade approximants.

It has been found that improvement in performance mayresult when the plant delay and model delay disagree [6—10].This 'temporal mismatch' may be deliberate or unavoidable.More general mismatch involving the delay-free part of theplant may give further improvement, but this is not exploredin this paper. The investigation of the mechanism for theseemingly counterintuitive improvement has led to theextension of the method to delay-free systems, wheredeliberate addition of delay elements has been found to giveimprovement.

Performance improvement, in the sense that it is used here,is defined in the following Section. In Section 3, a second-order plant with delay is improved by an overestimation ofthe delay. It has been noted informallyt that improvement, ifpossible, will probably occur through an overestimation ofdelay. In the delay-free example of Section 4, the improve-ment that occurs corresponds to an underestimation of theplant delay for the predictor scheme from which the exampleis derived.

2 Parametric optimisation

The performance of a feedback control system is oftenmeasured from the system's time response to a unit step input.Parametric optimisation, namely minimisation of the integralof some weighted function of the error signal with respect tocertain adjustable parameters, systematises the design[11 — 13]. It also results in a compromise between conflictingdesign requirements such as fast response and small over-shoot. In this paper, the integral square error (1SE) serves asthe performance-index criterion. Graham and Lathrop [14]have noted that the same parameter values come close tooptimising several different error criteria.

If one of the parameters in the optimisation is the loopgain K, it is possible to obtain increasingly lower values of theperformance index for increasingly higher values of K. How-

tGawthrop, P.J., and Clarke, D.W.: Private communication

Paper 2084D, first received 16th November 1981 and in revised form12th March 1982The authors are with the School of Mathematics, University of Bath,Claverton Down, BA2 7AY, Bath, England

ever, very large values of K are physically impracticable orundesirable. Moreover, unless a sufficient number of para-meters is available, the system may be destabilised by large K.Consequently optimisation is carried out for admissible gainsK < Kmax for some Kmax < °°. This yields a 'practicalminimum' which in general has a damped oscillatory stepresponse.

The intention of this paper is to show how systems forwhich the practical minimum has been achieved by conven-tional parametric optimisation may have yet better perfor-mance if additional delay elements are used. A secondoptimisation with respect to the additional parameters yields asecond minimum, which is less than the first (practicalminimum). Further performance improvement may bepossible if optimisation were carried out for all the parameterssimultaneously, but this is not pursued here.

The first optimisation can be achieved analytically with theaid of Parseval's theorem; however, the second optimisationrequires numerical methods. It will be shown how satisfactoryapproximate values of the second set of parameters may besimply obtained with the aid of simulation. This graphicaltechnique also demonstrates how the improvement occurs.

3 Performance improvement for plants with delay

Most, if not all, plants contain delays. If the delays are smallenough, then a delay-free model may be used for controlsystem design with subsequent iterative tuning on the actualplant. There are, however, many plants containing delayswhich are not negligible. This Section is concerned with thatimportant subset of such plants for which the delay isseparable: i.e. plants having transfer functions which are aproduct of a delay-free transfer function with the transferfunction of a delay. Important problems include unavoidabledelays occurring in measurement, for example in rolling steelor rubber, and in control, for example acoustic signals inremote control.

Predictor methods, for controlling time-delay systems, allhave a similar structure, the principal difference being thechoice of feedback elements in the outer loop. These methodsinvolve the plant together with a good model. In what follows,performance improvement will be illustrated through thepredictor method of O.J.M. Smith [ I ] , which involves unitynegative feedback.

3.1 Mismatch problem of predictor controlSmith's control scheme for a plant with delayed measurementis shown in Fig. 1. The delayed control problem results in asimilar control structure. In the Figure, the transfer functionof the delay-free subplant is denoted by Gp(s). The plantmeasurement delay of rp seconds corresponds to a plant delaytransfer function Tp(s) = exp(— STP). Gm(s) and Tm(s) =

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exp(— STm) represent, respectively, the transfer functions ofthe subplant model and delay model, where rm is the model'sdelay. It is important to observe that, in general, Gp(s) =£Gm(s) and Tp(s) =£ Tm(s) owing to modelling assumptionsand the use of nominal parameter values. C(s) denotes thetransfer function of the series controller element. The Laplacetransforms of the input and output are given by X(s) and Y(s),respectively, where the input (i.e. the desired output) is takento be a unit step function.

X(S)=1/S • plant Y(s)

Fig. 1 Time-delay system control scheme

The transfer function of the controlled system is given by

Y CGnTn

Y \+CGm+C(GpTp-GmTm)

where the argument (s) has been omitted for simplicity. WhenGp ~ Gm and Tp = Tm, the characteristic equationbecomes 1 + CGm - 0, which is the same as that of a delay-free plant Gp with a series controller C and unity negativefeedback. Then the output of the 'matched' scheme is just apostponed version of the output of the delay-free case (inaccordance with Smith's principle, for which accuratemodelling is assumed). This assumption simplifies the designof C by making it a delay-free design.

In practice, the exact matching of plant and model does notoccur. In other words, the 'mismatch term' C(GpTp — GmTm)is not zero, although it may be negligible if plant parameterestimates are good. The non-neglect of the mismatch termintroduces exponential terms into the characteristic equation

1 + CGm+C(GpTp-GmTm) = 0

This results in a system behaviour which is qualitativelydifferent from that of the matched case. The interestingconsequences are illustrated in the following Section for thestep response of a second-order example with mismatch.

3.2 Second-order exampleIn this example, a second-order subplant is first optimisedparametrically. The delay-free design is used to derive aSmith controller for a plant with this optimal second-ordersubplant together a series delay in measurement. The delaymodel is then deliberately mismatched, and a second para-metric optimisation with respect to mismatch in delay yields asmaller value of the ISE. The performance improvement isillustrated in terms of a comparison of the step responses forthe matched and mismatched cases.

Consider the second-order system with transfer function

loop system is assured, Parseval's theorem holds and

^ + a2

J =2Ka

Optimisation of / with the practical constraint K <Kmax, forsome Kmax <°°, yields Kopt = Kmax, aopt = \/Kmax andJOvt = /~'opt In this second-order example, the optimalparameter choice is equivalent to choosing a time scale relatedto Kmax. In general, the time-scaling analogy can not beexploited, because not all the coefficients in the characteristicpolynominal are freely assignable for design purposes.

For the purposes of illustration, set Kmax = 1. Then theSmith optimal design for a plant delay of TP = 1.0 s isGm(s) = Gp(s) = \/{s(s+ 1)} with C(s) = 1 and rm = TP.If the model's delay is 'overestimated' by 5 seconds, i.e.rm = 1.0 + 5, then a second optimisation with respect to5 yields 8opt = 0.33 s. This represents a 4% reduction in theISE (not including the unavoidable initial error due to theplant delay rp = 1). Further reduction may well result if thetwo stages of optimisation were combined, but at the expenseof greater computational complexity. The seemingly smallreduction in the ISE actually corresponds to a significantimprovement in the step response. From Fig. 2, it is apparentthat overshoot has been reduced from about 17 to 6%.

V .time 10

Fig. 2 Step responses for a controlled system with measurementdelay

a Without mismatch b With mismatch

3.3 Improvement mechanismConsider a general plant with series delay and temporal mis-match rm = TP + 5(5 > 0). From Fig. 1, it follows that

CGPTP

X \+CGm+C(GpTp-GmTm)

CGPTP

CGP + CGPTP (1 - exp( - s8))-since Gm = G,

1 +HTP(\ - e x p ( - s 5 ) )

where the transfer function of the delay-free part of thematched system has been written H = CGp/(l + CGP).Expanding the exponential term yields

G(s) =K

s(s + a)

with unity negative feedback. The parameters K and a arechosen to minimise the ISE, J = f£e2(t)dt, where e(t) =x(0 ~y(0 is the system error for a step input (i.e. desired out-put) x(t). If both K and a are positive, stability of the closed-

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- = HTP- H2T2ps8+ 0(52)

Hence, the Laplace-transform E8 of the error for a step inputX = \ Is is given by

IEs = X-Y = - ( 1 -

s t) + H2T2p8+0(82)

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Writing h2(t) for the impulse response of fP(s) and recog-nising the first term (1 —HTp)/s to be the error for thematched case, it follows that

where eo(t — TP) is the inverse Laplace transform of(1 —HTp)ls and the higher-order terms in 0(52) are delayedby at least 3TP. If the final term 0(52) can be neglected, then asatisfactory approximation to 8opt may be found from agraphical plot of eo(t) and h2{t).

For example, in Fig. 3, which corresponds to the system ofthe previous Section, if 5 = 0.37, then the first peak of1— eo(t — TP) cancels with the first peak of h2(t — 2TP).Not only does this yield a satisfactory approximation to&opt = 0.33, but it also indicates the mechanism ofimprovement. Note that these curves have been obtained froman analogue computer simulation of the delay-free system.

Fig. 3 Estimate of mismatch in delay: 6 = B/A = 0.37

3.4 Technique for further improvementConsider Fig. 4, which shows plots of 7(5) for the system ofSection 3.2 for various values of plant delay TP. Clearly,maximum improvement occurs for the plant which has delayTP and mismatch 8P. For a plant with general delay TP, furtherimprovement is thus possible by artificially increasing theplant delay, provided that TP <TP. Fig. 5 shows a modifiedSmith scheme in which the extra delay TP — TP in the outerfeedback loop enables TP to be increased in the characteristicequation without delaying the system output [8]. Note thatsuch further improvement is not possible when TP > TP, sinceunrealisable terms would be involved.

4 Performance improvement for delay-free plants

Initial study of Fig. 4 would appear to indicate that, when theplant delay is zero, i.e. for delay-free plants with rp = 0,there is no possibility of performance improvement by mis-

1.08

056

Fig. 4 Variation of cost with respect to model delay rm for variousvalues of plant delay rp

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match, at least, not for the second-order example of Section3.2.

However, modification of the outer feedback loop asindicated in Fig. 5 does lead to improvement

X(s)=i/s

Fig. 5 Smith scheme with additional outer-loop delay

4.1 Delayed-derivative controlThe transfer function of the system of Fig. 5, when Gp = Gm

with TP = 0 and Tm - Tp + dp, is given by

Y = CGp

X 1 + CGp + CGp exp ( - STI) [1 - exp ( - s8*)]

Writing H = CGp/(\ + CGP) for the matched closed-looptransfer function and 1 — exp( — s5p) = s8p + 0(5p2), then

Y = H

To a first approximation in 8P, this corresponds to the con-trolled delay-free system with transfer function H(s) togetherwith negative delayed-derivative feedback with gain 8P anddelay TP, as shown in Fig. 6.

X(s)=1/s Y(s)

exp(-s7j)

Fig. 6 Delayed-derivative feedback scheme

Note that the gain of the delayed derivative is related totemporal mismatch in Smith schemes.

4.2 Third-order exampleIn this example, a third-order system is first optimised para-metrically. Then the scheme with added delayed-derivativefeedback as exp(— ST) is optimised with respect t o a G R andT. The reduction in the ISE is illustrated via a comparison ofthe step responses for the controlled system with and withoutdelay.

Consider the third-order delay-free system with transferfunction

G(s) =K

s(s+ I)2

under local velocity feedback and overall unity negative feed-back. The closed-loop transfer function is given by

Kv ' s3+2s2+(l+aK)s + K

where K is the loop gain and a is the gain of the velocity feed-back. Provided that 2(1 + aK) > K, stability is assured and the

179

ISE is given by

1 +aK2K

r2aK-K

Optimisation of J with the practical constraint K < Kmax, forsome Kmax < °°, yields

K opt = K,

K,

Earlier remarks about time scaling are not appropriate herebecause the coefficient of s2 in the characteristic equation isfixed.

For the purposes of illustration, consider Kmax = 20.Then the optimal (for delay-free control) closed-loop transferfunction is

20

s3 + 2s2 + 16.325 s + 20

and the ISE /(a, T) is given by /(0, 0) = 0.5659. A secondoptimisation, this time with respect to a and r, yields aopt

= - 0 . 1 9 , Topt = 0.56 with / ( - 0 . 1 9 , 0.56) = 0.5255.

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

8 10 12

Fig. 7 Step responses for a controlled third-order systema Without delayb With delay

This represents a 7.1% reduction in the ISE. From Fig. 7, inwhich the step responses for optimal control with and with-out delay are portrayed, it is apparent that overshoot has beenreduced from — 24 to 7.3%, while settling time, to within 3%of the steady-state value, has been reduced from 6.2 to 1.7 s.

Note that otopt is negative. This corresponds to an 'under-estimate' of model delay in the Smith scheme.

4.3 Impro vemen t mechanismA calculation similar to that of Section 3.3 gives

e<*. T ( 0 = eOiO (0 + <*h2 (t - r) + 0(a2)

where eaT (t) and e00 (t) are the step-response errors for acontroller with and without delay, respectively. h2(t) is theimpulse response of H2(s). The 0(a2) term is delayed by atleast 2T.

As before, if the 0(a2) term can be neglected, then satis-factory approximations to otopt and Topt can be obtainedfrom the graphical plots of eQ0 (t) and h2 (t).

In Fig. 8, the functions eQ0 (t) and h2(t) for the exampleof Section 4.2 were plotted with the aid of an analoguecomputer. The values a = —0.22 and T = 0.58 result in thefirst trough of 1 — e00 (t) being cancelled by the first peak inh2(t). These values compare favourably with aopt = —0.19and Topt = 0.56. The mechanism for improvement in thisexample differs from that of the previous example because ofthe sign of Oiopt-

Fig. 8 Estimate of delayed-derivative parameters

a - —B/A - —0.22T = t, — t. = 0.58

5 Conclusions

Improvement in performance can occur for temporal mis-match, although the mismatch need not be an overestimate.The improvement is more noticeable in terms of step-responsecharacteristics, such as overshoot and settling time, than interms of percentage reduction in the ISE performance index.The mechanism for improvement is closely related to thenature of the step response. Whether the plant delay needs tobe over- rather than underestimated depends on whether ornot the first peak of the step response is the largest peak.Thus, for second-order plants, the delay must be over-estimated; no general statement can be made for higher-orderplants.

In the context of delay-free systems, temporal mismatch,together with the technique of Section 3.4, approximates todelayed-derivative feedback. Again, the same observationsapply as far as the mechanism of improvement is concerned.Since the gain of the delayed-derivative feedback correspondsto the mismatch in delay, over- and underestimation oftemporal mismatch correspond to positive and negativefeedback, respectively.

6 References

1 SMITH, O.J.M.: 'Feedback control systems' (McGraw Hill, 1958)2 FULLER, A.T.: 'Optimal non-linear control of systems with pure

delay', Int. J. Control, 1968, 8, pp. 145-1683 FRANK, P.M.: 'Entwurf von Regalkreisen mit Vorgeschriebenem

Verhalten' (G. Braun, Karlsruhe, 1974)4 MARSHALL, J.E., CHOTAI, A., and GARLAND, B.: 'A survey of

time-delay system control methods'. IEE Conf. Puhl. 194, 1981,pp. 316-322

5 MEE, D.H.: 'An extension of predictor control for systems withcontrol time delays', Int. J. Control, 1973, 18, pp. 1151-1168

6 MARSHALL, J.E.: 'The control of time-delay systems' (PeterPeregrinus, 1979)

7 GARLAND, B., and MARSHALL, J.E.: 'On the applicability of

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O.J.M. Smith's principle', in 'Recent theoretical developments in 11 ASTROM, K.J.: 'Introduction to stochastic control theory' (Aca-control' (Academic Press, 1978), pp. 307-325 demic Press, 1970)

8 CHOTAI, A.: 'Parameter uncertainty and time-delay system control'. 12 NASLIN, P.: 'Essentials of optimal control' (Uliffe, 1968)Third IMA conference on control theory, 1980, pp 729-750 13 JACOBS, O.L.R.: introduction to control theory'(Clarendon Press,(Academic Press, 1981) 1974)

9 'The damping effect of a time-lag', The Engineer, 1937, 163, p. 459 14 GRAHAM, D., and LATHROP, R.C.: 'The synthesis of optimum10 BYRON, R., COX, C.S., and BALL, D.J.: 'An application of a transient response', Trans. Am. Inst. Electr. Eng. Pt II, 1953,72,

Smith controller in a cinter plant'. IEE Colloquium Digest, 1979/38 pp. 273-288

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