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1 SOME REMARKS ON PAD MVajtaDepartment of Mathematical SciencesUniversity of TwentePOBox 217 7500 AE EnschedeThe Netherlandse-mail mvajtamathutwentenlABSTRACT approximations are widely used to approxi

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**3 TEMPUS-INTCOM Symposium, September 9-1**

3 TEMPUS-INTCOM Symposium, September 9-14, 2000, Veszprém, Hungary.
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SOME REMARKS ON PADÉ M.VajtaDepartment of Mathematical SciencesUniversity of TwenteP.O.Box 217, 7500 AE EnschedeThe Netherlandse-mail: m.vajta@math.utwente.nl
ABSTRACT approximations are widely used to approximate a dead-time in continuous control systems. It providesa finite-dimensional rational approximation of a dead-time. However, the standard Padé approximation(recommended in many textbooks) with equal numerator- and denominator degree, exhibits a jump at timet=0. This is highly undesirable in simulating dead-times. To avoid this phenomena we shall reconsider thePadé approximation with different numerator degrees.Keywords: Padé approximation, rational functions.1. INTRODUCTIONThere are many physical processes with dead-time. For example virtually all chemical processes involvessome time delay and all transport processes also exhibit dead-time [3,12]. Control systems with dead-timeare difficult to analyze and simulate. One of the reasons is that a closed-loop control system with dead-timeis in fact an infinite dimensional system, i.e. the closed-loop has infinite number of poles [3,6]. It is alsodifficult to determine all the system poles. One of the most widely recommended remedies to overcome thisdifficulty is to approximate the dead-time by some method and analyze the resulting system [6,8]. The step-response of a dead-time is a delayed step-signal h(t) = 1(t-T) where T denotes the dead-time. The Laplacetransform of h(t) = 1(t-T) is:sT(Among the many methods Padé approximations are the most frequently used methods to approximate adead-time by a rational function. Almost every textbook about classical control system theory provides thebasic relation, but usually only for an approximation with equal numerator and denominator degree (try forexample the subroutine pade.m in MATLAB). The most widely recommended Padé approximation is of 2ndorder with equal numerator- and denominator degree [6,8]:
222()(12()(12(sTsTsT++It is a b

2
**it puzzling to realize, that the step-re**

it puzzling to realize, that the step-response of this approximation (say, transfer function) exhibits ajump at t=0 due to the equal numerator and denominator degree. That is, instead of delaying the input signalthere appear an output signal at t=0. This seems to be quite bad. On the other hand, this approximation hasnice properties in the frequency domain. So one may ask: is it possible to modify the approximationavoiding the jump at t=0 but keeping the frequency domain properties?
3 TEMPUS-INTCOM Symposium, September 9-14, 2000, Veszprém, Hungary.
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2. APPROXIMATIONS WITH CONSTANT NUMERATORThere are many ways of approximating e by a rational function. Consider for example its Maclaurin series[1,11]. By taking only the first n-terms we can define the following approximation:
!((!()(1)((20sTsTsTnk++++=This formula is recommended in Kuo [pp.183] and in Palm [pp.509]. Although the expression seems naturalto apply, an unexpected difficulty arises as one increases the degree of approximation. The rational functionR exhibits right-half-plane poles as n increases, namely as n�4! Although the approximation's accuracyincreases as n increases in the s-domain, but as a transfer function R becomes unstable. This is a rarely
known phenomenon and makes the seemingly simple approximation useless for �n4. Consider for examplethe first 5 terms (5th order approximation):
543252060120120120(sssssR++++The poles of this rational function are: p = 0,23981± 3,12834; p = -1,44180± 2,43452 and p = -2,1806.Since there are two conjugate complex poles on the right-half plane, this "approximation" is unstable!Another method recommended in some textbooks [8, pp.521; 13, pp.216] is based on the infinite productformula of the exponential function [11]. Taking only the first n terms in the product leads to the followingapproximation:
nnnnnnn=1()(This approximation has multiple poles (with multiplicity n) at p = -n/T. In fact, equation (5) gives a ratherpoor approximation for low value of n [13,14]. Without going into more

3
** details, we can conclude, thesesimple a**

details, we can conclude, thesesimple approximations (without numerator dynamics) give poor approximations of a dead-time. One mayexpect to improve the accuracy by choosing an appropriate numerator.3. PADÉ APPROXIMATION OF e-xThe approximations given in the previous paragraph are rational functions but with zero numerator dynamics(numerator is constant). We shall now consider another kind of approximations, namely, approximationsderived by expanding a function as a ration of two power series (thus with numerator and denominatordynamics). These approximations are usually called Padé approximants. They are usually superior toTaylor expansions when functions contain poles, because the use of rational functions allows them to bewell-represented. Let us now consider the general equations of the Padé approximation. Let A(x) denote afunction having a Maclaurin series expansion [1,2]:å(kkwhich converges in some neighborhood of the origin1. The Padé approximation of order (m to A(x) isdefined to be a rational function R expressed in a fractional form:
1 If A(x) is a transcendental function then the a coefficients are given by the Taylor series about x0
(
!
(
k
3 TEMPUS-INTCOM Symposium, September 9-14, 2000, Veszprém, Hungary.
3
)()()(where P and Q are two polynomials2nnmmqp++++++((2102210The unknown coefficients p ... pm and q ... qn of R can be determined from the condition that the first(m+n+1) terms vanish in the Maclaurin series3
()()(()()(Substituting the two polynomials into this expression and equating the coefficients leads to a system ofm+n+1 linear homogeneous equation [2] which can be expressed in matrix form (assuming qúúúúúûùêêêêêêëéúúúúúûùêêêêêêëéúúúúúúûùêêêêêêëé------------++++++-mmmmmmmmmmaaaaaqqqpppaaaaaaaaaaaaaMMMLMLLLLLMLLL110211021312211100000000010000010000001Now we would like to apply this to the exponential function with the Maclaurin series:
...1)(320xxxkxekxWe conclude that the coeffi

4
**cient a = (-1)k In this case the polynom**

cient a = (-1)k In this case the polynomials P and Q of the Padéapproximation R can be expressed by the following recursive relations [4]:
å-kkm()!(!)!(!)!()(
-kk()!(!)!(!)!()(Note, that the numerator coefficients have always alternating sign and P = Q for m=n. As aconsequence, the zeros and poles of R are symmetrical to the imaginary axes!
2 Note that there is no constraint on the degree's of the polynomials. That is to say, the numerator mayhave higher degree than that of the denominator.3 Q can be multiplied by an arbitrary constant which will rescale the other coefficients, so anadditional constraint can be applied. This is usually Q
3 TEMPUS-INTCOM Symposium, September 9-14, 2000, Veszprém, Hungary.
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4. PADÉ APPROXIMATIONS OF e-sTTo determine the transfer functions of the Padé approximations with different numerator degree, one simplysubstitutes x=sT into (12) and (13). For example, the 4th order approximation with 3rd order numerator canbe expressed as [14]:
432324()((480840()(360840(sTsTsTsTsTsT+++Note, that the nth order Padé approximation has different denominator polynomials depending on thenumerator's degree. It is interesting to determine the pole-zero configuration of the approximation. Figure 1shows the pole-zero configuration of the 4th order Padé approximation with different numerator degree.Note, that all poles are on the left-half-plane and all zeros are on the right-half-plane. Notice, that the polesand zeros of the Padé approximation R are symmetrical to the imaginary axis and are close to a circle.Due to the symmetrical pole-zero configuration, the phase of R goes to -2n and its amplitude remainsconstant at all frequencies. On the other hand, the step-response of R exhibits a jump at t=0 which is notvery desirable. To avoid the jump in the step-response we recommend to use R instead of R. InTable 1 we give the transfer functions of both up to 5th order. However, there is a price to be paid: due to thelower numerator's degree

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**, the phase of R goes to -(m+n) only and**

, the phase of R goes to -(m+n) only and its amplitude goes to zero at veryhigh frequencies. But all in all, R seems to be a good compromise.Figure 2 shows the step-responses of R and R. One can easily see that R gives a betterapproximation in the time-domain [14], specially in the interval [0,T]. As a measure of the error we give inTable 2 the mean-square-errors defined in the time-domain by:
{
-2,()(Figure 1. Pole-zero configuration of the 4th order Padé approximation with different numerator degree.All poles are on the left-half-plane and all zeros are on the right-half-plane.( ? = R1,4(s); x = R2,4(s); ? = R3,4(s); o = R4,4(s) )
-8
-2
2
4
6
8
-6
-4
-2
2
4
6
8
poles-zeros of Padé approximation Rm(s) real
3 TEMPUS-INTCOM Symposium, September 9-14, 2000, Veszprém, Hungary.
5
n
(626sTsT+
2(12(12sTsTsT++
22()(3660(2460sTsTsTsT+++
232()(60120()(60120sTsTsTsTsT++
3232()((480840()(360840sTsTsTsTsTsT+++
32432()((8401680()((8401680sTsTsTsTsTsTsT++++
432432()(((840015120()((672015120sTsTsTsTsTsTsTsT+++++
4325432()(((1512030240()(((1512030240sTsTsTsTsTsTsTsTsT++++
Table 1. Transfer functions R and R of the Padé approximations.Figure 1. Step responses of the Padé approximations with different order.Figure 2. Step-responses of Padé approximationsf R and R
Table 2. Mean-square-errors of step-responses of R and R
d order Padé
R
order Padé
R
order Padétime [sec]
R
order Padétime [sec]
R
3 TEMPUS-INTCOM Symposium, September 9-14, 2000, Veszprém, Hungary.
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CONCLUSIONSWe have considered the general Padé approximation of a dead-time with transfer function e. Thepolynomials of the rational approximations are given in analytic form. The "standard" Padé approximationR exhibits a jump at t=0 in its step-response. To avoid this phenomenon we recommend the Padé R where the numerator's degree is one less than that of the denominator. This gives abetter approximation of the step-response. Applied in closed-loop, they differ due to their differentfrequency characteristics. There seems to be a clear comprom

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**ise between the use of R or Rdepending o**

ise between the use of R or Rdepending on the frequency range. One has to realize that by approximating a dead-time in control systems,we introduce modeling errors, which consequently limits the achievable bandwidth. Some consequences arediscussed in [7, pp.115]. Padé approximations can also be used for model reduction [10].REFERENCES[1] ERD. at all.: Higher Transcendental Functions,McGraw-Hill Book Company, Inc., New York, 1953.[2]FIKE,C.T.: Computer Evaluation of Mathematical Functions,Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1868.[3]FRIEDLY,J.C.: Dynamic Behavior of Processes,Prentice-Hall,Inc., Englewoods Cliffs, N.J. 1972.[4]GOLUB,G.H. and Ch.F.van LOAN: Matrix Computations,Johns Hopkins Universit Press, Baltimore, 1989.[5]KAMEN,E.W. and B.S.HECK: Fundamentals of Signals and Systems using Matlab,Prentice-Hall, Inc., Upper Saddle River, New Jersey, 1997.[6]KUO,B.C.: Automatic Control Systems,Inc., Englewoods Cliffs, New Jersey, 6th edition, 1991.[7]MORARI,M. and E.ZAFIRIOU: Robust Process Control,Prentice-Hall Int. Inc., New York, 1989.[8]PALM,W.J.III.: Control Systems Engineering, John Wiley & Sons, Inc., New York, 1986.[9]PRESS,W.H., B.P.FLANNERY, S.A.TEUKOLSKY and W.T.VETTERLINK: "Padé Approximants"§5.12 in Numerical Recipes in FORTRAN: The Art of Scientific Computing,Cambridge University Press, Cambridge, 2nd edition, pp.194-197, 1992.[10]PURI,N.N. and D.P.LAN: Stable Model Reduction by Impulse Response Error MinimizationUsing Michailov Criterion and Pade's Approximation,Journal of Dynamic Systems, Measurement and Control,Trans. of ASME, 110, (1988), pp.389-394.[11]SPIEGEL,M.R.: Mathematical Handbook, in "Schaum's Outline Series", McGraw-Hill Book Company, New York, 1968.[12]STEPHANOPOULOS,G.: Chemical Process Control,Prentice-Hall Int. Inc., New York, 1984.[13]K,R.: Control Systems,(in hungarian), Technical University Press, Budapest, 4th edition, 1998.[14]VAJTA,M.: On Padé approximations of a dead-time, Internal Report, Dept. of Mathematical Sciences, University of Twente, 2000.

1 SOME REMARKS ON PAD MVajtaDepartment of Mathematical SciencesUniversity of TwentePOBox 217 7500 AE EnschedeThe Netherlandse-mail mvajtamathutwentenlABSTRACT approximations are widely used to approxi

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