Lumped approximation of distributedsystems and controllability questions

Prof. B.D.O. Anderson, B.Sc B.E., Ph.D., Fel.Aust.Acad.Sci.,Fel.Aust.Acad.Tech.Sci., Fel.I.E.E.E., F.I.E.(Aust.), and Prof. P.C. Parks, R.D.,

M.A., Ph.D., Sc.D., C.Eng., MRAes, FIMA

Indexing terms: Mathematical techniques, Control theory

Abstract: The approximation of a distributed system by a lumped system using both exact analysis and quali-tative reasoning is studied in some detail for the lossless LC transmission line, and with much less detail for anRC line, a towed cable, and a deep-sea mining hauling pipe. We delimit, at least partly, the region of validity ofthe approximation, especially in analysing controllability questions. We also suggest why a form of nonuniformlumping may be more appropriate for the towed cable and the deep-sea mining hauling pipe.

1 Introduction

In Reference 1 the following observations are made.Suppose a uniform string under tension has a nonzeroinitial displacement of velocity, and that one end of thestring can be shaken arbitrarily; then the string can bebrought to rest in a time equal to twice the one-way pro-pagation time of waves down the string, and, for genericinitial conditions, it cannot be brought to rest in a shortertime. Suppose next that the string is approximated by adiscrete lumped mass model. Controllability of theresulting linear state-variable equation can be verified,which permits the conclusion that the lumped mass modelis controllable from an arbitrary initial state to the originin an arbitrarily short time. Since the controllability pro-perty is evidently very different, the question arises: 'whatgoes wrong?'

In this paper, we answer this question. The short answeris that the lumped model is only a satisfactory approx-imation in a frequency band extending from zero (or DC)to an upper limit which is dependent on the dimension ofthe lumped model. If one attempts to control the lumpedmodel using controls which are in some way frequencylimited, then the delay behaviour of the distributed modelis approximated, but otherwise the lumped model behav-iour will not approximate the distributed model behaviour.

Section 2 discusses this question using a uniform losslessLC transmission line, rather than a string. The equationsof both systems are essentially the same. (There is,however, electrical-engineering literature on approx-imations of lines; see, for example, References 2 and 3,which we discuss in Section 3.) The key is to comparetransfer functions of the lumped and distributed systems,and the associated frequency-dependent group delays [seepage 230 of Reference 4].

In Section 3, we discuss several aspects of these results.We regard the construction of the lumped model as involv-ing spatial sampling of the line, and the creation in theprocess of a spatial aliasing frequency. We relate this to atemporal aliasing frequency and argue that calculations on

Paper 3776D (C8), first received 1st May 1984 and in revised form 3rd January 1985Prof. Anderson is with the Department of Systems Engineering, Research School ofPhysical Sciences, Australian National University, Canberra, Australia and Prof.Parks is Chairman of the School of Management and Mathematics, Shrivenham,Swindon, Wiltshire SN6 8LA, United Kingdom

the lumped model using excitations at frequencies abovethe aliasing frequency will yield misleading informationabout the distributed system. We also relate the ideas toconstant-it filter theory [2, 3] and finally examine in moredetail the controllability properties of the lumped modelby using frequency-limited signals and by studying thebehaviour of the controllability gramian.

In Section 4, we discuss other distributed systems: anRC line (which represents the heat equation), a towedcable [5, 6], and a deep-sea mine hauling pipe [7]. Weargue why nonuniform spatial sampling should be advan-tageous, thus providing a theoretical justification for anobservation of Reference 7.

2 Transfer function analysis of a lumped lineapproximation

Let us consider the arrangement of Fig. 1, in which thereare present m inductors and m capacitors. This arrange-

Fig. 1 Lumped approximation to uniform LC line

ment is an approximation of a uniform transmission line,excited at one end with a current generator, and terminat-ed at the other end with a resistor. If the transmission linehas inductance L and capacitance C per unit length, and isof length /, then

L - ^L.i —

m

C - ^m

(1)

On the transmission line, waves travel with a velocity(LC)~1/2, and the characteristic impedance of the line is{L/C)112. If the resistive termination takes this value, noreflections of waves can occur at the resistor.

Our aim here will be to compare a transfer functionassociated with the lumped model of Fig. 1 with the corre-sponding transfer function for the transmission line.

We define a state-variable realisation as follows: theentries of the state vector in positions 1, 3, 5, ..., are the

IEE PROCEEDINGS, Vol. 132, Pt. D, No. 3, MAY 1985 89

capacitor voltages multiplied by y/C^ and the entries inpositions 2, 4, 6, . . . , are the inductor currents multipliedby yjLy. Then the state-variable equations are

x =

1

R

Suppose we are interested in the current flowing in the qthinductor from the left, namely (l/y/L)xp, where p = 2q.Thus the output equation is

(3)

To simplify the notation, let

1a = ,P = RJLX

Lemma 1: Let F, G, H define the realisation in the pre-ceding text and let Ao = 1 and

Ar(s) = det

s a— a s

s a- a s +

(4)

where the matrix is of size r x r for r ^ 1. Then, withn = 2m,

Wl(s) = H'(sl - (5)

(The subscript / is to emphasise the lumped character ofthe model)

Proof: By Cramer's rule, the p — 1 entry of (si — F )" 1 isthe (/, p) cofactor of the matrix on the right in eqn. 4. It iseasily checked that this is ap~l An_p. The forms of H andG, the fact that det (si — F) = An, and the definition of athen yield eqn. 5.

Lemma 2: For real a>, define 6(a>) by cos 6(co) =•Jl — co2/4a2, sin 9(co) = (o/2a. so that when 0 ^ co/2a < 1,0(co) is in the range [0, 2n). Then, for r even and ; = J — 1,

_ cos (r + l)6(co) . p sin r6(co)a T{J(D)~ COS 0(CO) + 7 a cos 0(a>)

Proof: A simple calculation based on eqn. 4 shows that

An(s) = s An_i(s) + a2 An_2(s)

Suppose that u(s), v(s) are the roots of x 2 — sx — a2 = 0.Then, for some y(s), d(s),

An(s) = y(s)un + d(s)v"

Noting from eqn. 4 that Ax(s) = s + /?, A2(s) = s2 + fts+ a2 leads to the identification

x +

1 1

(2)

y(s)--12

r

1 + "

S

2a/ I + s2/4a2

1 -71 +s2/4a2.

Now set s — ja>. There follows, using the definition ofcos 0(co) and sin 0(a>), and recalling that r is even,

\(jco) = exp [jrd(co)] + 8(jco) exp [ - ; >

+ ^0 w ) ] c o s

<5(/<»)]sin

sin= cos r6(a>) + j

-I- -

cos 6(co)sin rd(co)

cos (r + \)6(a>) . 0 sin

cos 6(<x>)VVV

a cos

Now recall that we are evaluating the transfer function ofeqn. 5 in which n is even by construction, and p is even byassumption. We obtain

H'(jcol — F)~1G

cos (n — p + \)6(co) +j- sin (n — p)6(a>)

cos (n + l)6(a)) +j — sin n9((D)a.

Now notice

Observation: Rx is a matching resistances/?

(6)

To get a feel for the difference between the lumped anddiscrete models, suppose temporarily that

a = P (matching condition holds)

Then the transfer function is

WJJco) = H'(jcol -F)~1G = exp [-jp0(co)]K(j(o)

(7)

= exp -JP sin ^2m K(jco) (8)

90 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 3, MAY 1985

where

1 + exp l-j(n -p+ 1)0]x j[sin (n — p)6 — sin (n — p +

1 -I- exp [—j(n + l)0];[sin nd — sin (n +

(9)

Let us compare this with the corresponding expression forthe ideal line. A current input simply propagates along theline with velocity (LC)~1/2. In the lumped model, we areconsidering an inductor effectively at distance pl/2m alongthe line. So the transfer function will be

WiO"«) = exp - ; f - (10)

(The subscript d emphasises the distributed character ofthe model)

Eqns. 8 and 10 show then that the lumped model will bea good approximation of the distributed model providedthat we can write

p sin 2m r 2m

and

K(jco) ~ 1

Now observe that for 0 ^

CO (lla)

n/2

y3

^n3 ( .

^ — (sin

whence for 0 ^ (l/2m)y/LCco ^ n/2,

col ,. _ pi

y .since — ^ sin y

n

col

(12)

and a measure of the error involved in the approximationof eqn. l l a is available. Clearly, for

co2m

ujlc (13)

eqn. l l a holds in the sense that the error termpn3/4S((col/2m)yJLC)3 is small as a percentage of {pi/2m)yfLCa). For a small absolute error, we must recognisethe multiplying effect of p . Notice that the maximum valueof p is m, and the right side of eqn. 12 will be guaranteed tobe much less than 1 for any p provided that

2m2'3

co (14)

Now eqn. 13 is the condition also that 9(co) <̂ 1, and whenthis is so, it is not hard to check using eqn. 9 thatK(jco) ~ 1, as required by eqn. 116, for all n, p . A fortiori,eqn. 14 implies K(jco) ~ 1.

Consequently eqn. 13 is the condition for the phaseerror between Wt and Wd to be small as a percentage of thetotal phase of either, and also for the amplitude error to besmall. Of course, even if the phase error is small in a per-centage sense, it may not be small as a fraction of 2n. Eqn.

14 gives a strengthened condition, guaranteeing closenessof phase in an absolute sense.

It is also interesting to compare the time delay for thetwo transfer functions Wfoco) and WJJco). The group orenvelope delay T(co) at frequency co is [2] — dcfrijcoydco,where 4>(jco) is the phase of the relevant transfer function.In this case, we get

Td(co) = ^ (15a)

For the lumped circuit, the phase shift comes fromexp [ —jp sin ~ l co/2oC] and Kijco) in eqn. 8. We obtain

4m2 J1/2

d_

dco(156)

We do not evaluate the second quantity on the right, butnote simply that it will be small when eqn. 13 holds; thefirst quantity will, of course, also approximate to Td(co)when eqn. 13 holds. Consequently, eqn. 13, rather than thestronger eqn. 14, serves to define the region of goodapproximation of delay behaviour.

Now let us consider briefly what happens when thematching condition of eqn. 7 fails. Set

1 — jg/oc1+^/oc

(16)

(Notice that \rj\ is a measure of mismatch.) Then eqn. 6yields

e x P [— — *7 e x P [ - — p8(co)~\

1 - r\ exp [ -2 /n +

= {exp l-jp6(co)'] - rj exp [-j2n + 1 - pd(co)}

x {1 -I- Y\ exp [ — 2/n + 10(co)]

+ >/2 exp [ —4/n + 10(to)] -I- • • -}Kn{jco)

The explicit form of K^ijco) is not important: what is rele-vant is that, when eqn. 13 holds, K^ijco) cz 1.

One can also verify that

. Pi

-1 exp -

x {1 + rj exp [ - / 7 0 M ] + rj2 exp [-2/70(co)] + • • •}

Both transfer functions have the following interpretation:the response is due to a wave undergoing no reflection, awave undergoing one reflection, two reflections, etc., eachreflection causing, in general, an attenuation of the waves.These conclusions, with respect to phase approximation,remain valid. Those with regard to delay approximationremain valid if we look at the delays associated with suc-cessive identifiable reflections, subject to one qualification.The more reflections there are, the greater the delayinvolved, and so the greater the error in the delays of thetwo systems. Provided though that \r\\ < 1, so that R = 0,co are disallowed, the amplitude of the successive reflec-tions gets smaller with each reflection.

IEE PROCEEDINGS, Vol. 132, Pt. D, No. 3, MAY 1985 91

3 Discussion of transmission-line results

3.1 Sampling theorem viewpointThere are at least two ways in which we would like thelumped model to approximate the distributed model. First,the lumped model should reflect the same properties in theresponse of each state variable component as the variableon the line located spatially at the nominal sampling pointto which the state variable corresponds. Secondly, thelumped model should exhibit the wave-like property of theline, wherein a sinusoidal spatial variation goes togetherwith a sinusoidal temporal variation, the relation betweenthe two frequencies being governed by the velocity of pro-pagation in the medium.

Let us now observe why simultaneous imposition ofboth these requirements creates difficulties. In the lumpedmodel, there are m spatial samples, or a spatial samplingfrequency of m/l per unit length. Now it is well known thatsampling, including spatial sampling, has the effect ofmaking frequencies higher than the aliasing frequency(here m/2l per unit length) look like, or become, indistin-guishable from a certain frequency lying within an intervaldetermined by the aliasing frequency, the interval herebeing ( — m/2l, m/2/). Thus a spatial variation at a rate0.75 m/l per unit length, for example, is indistinguishablefrom a spatial variation at a rate of 0.25 m/l per unitlength. Now if the wave-like property is preserved by thelumping, and if the velocity of propagation of the waves isc = (LC)~1/2, if then follows that temporal excitation ofthe lumped model at a frequency of 0.25c m/l Hz shouldgive the same variation of amplitudes of the state vectorcomponents as would result from excitation at a frequencyof 0.75c m/l Hz. For this variation of amplitudes in thelumped model will then, and only then, correspond to thevariation of amplitudes of (normalised) voltage and cur-rents at sampling points along the line.

Hence a nice model (in the sense of one meeting the tworequirements just nominated) would have the propertythat the amplitude (if not the phase also) of the transferfunction from the input to any entry of the state variablewould vary periodically with temporal frequency,exhibiting the same value at co + 2nc(nm/l) for all n. Such avariation can never be exhibited by a lumped model. Thevery best we could hope for then would be for the lumpedmodel to behave satisfactorily, in the sense of approx-imating the distributed line, over a limited (temporal) fre-quency range. A pointer to this frequency range can befound by taking the spatial aliasing frequency and scalingit by the wave velocity, to get a temporal aliasing fre-quency. For the line, it is

con = 2n m nm2lyjLC ly/LC

(17)

This should be compared with eqn. 13, the condition forthe model to be a good approximation.

3.2 Constant-k filter argument [3, 4]Delay lines are often constructed by cascading T sectionsof the form depicted in Fig. 2. An analysis in Reference 4,

L, /2 L, /2

Fig. 2 Prototype section of constant-k filter

92

involving assumptions on impedance matching, shows thatm such sections in cascade, excited on the left and 'appro-priately' terminated on the right, will produce a time delay

j i , provided that the excitation frequency obeys

2mco (18)

Thus we recover, by a different argument, the condition ofeqn. 13. The constant-k filter theory also notes that, forexcitation frequencies co higher than 2m/lsJLC, the lumpednetwork transfer function from input to termination willattenuate, but have frequency independent phase, so thatno group delay should be expected.

3.3 Controllability and the controllability GramianWe now see why it is that the lumped model can be instan-taneously controllable while the distributed model is not.The lumped model is not a good approximation of the dis-tributed model for excitation frequencies co of the order of2m/ly/LC or higher. Instantaneous control of the lumpedmodel requires application of those frequencies.

If we consider controlling the lumped model withsignals which are artificially restricted in frequency, wecould demand that we use piecewise constant signalschanging value no more often than m/nlyjLC times persecond. Noting that the lumped model has a 2m-dimensional state vector, this will imply that, for a genericnonzero initial state, it will take least time 2m-r- [m/nly/LC] = 2nlyfLC to bring it to zero. This is incrude agreement with the least time taken to bring the dis-tributed system to zero, which is QLC or 2ly/LC accord-ing as to whether the line is or is not terminated in itscharacteristic impedance [1].

Finally, it is instructive to look at the controllabilityGramian [8]. Consider the lumped case only. Then thecontrollability Gramian associated with x = Fx + Gu overthe interval [0, T] is

W(T)= I eJo

GG'eF' dt

and the minimum energy required to move the systemfrom state x0 at time 0 to state zero at time T is [8]X'0W~1(T)XQ. Accordingly, XminW{T) provides a measureof the worst-case control problem. Our particular interestis to study the variation with T of this quantity using thematrices F and G in eqn. 2. We also consider the variationof n = 2m, the dimension of the lumped model of eqn. 2.

Fig. 3 illustrates the results for several different values ofn. The calculation may be intrinsically numerically diffi-cult, but the general form of the curves is quite certain.One unit of normalised time is the time taken for transmis-sion one way along the ideal line. The eigenvalue normal-isation is obtained as

The clear conclusion is that, if one wishes to control in atime less than twice the one-way delay, this will be muchmore costly than controlling in a time greater than twicethe one-way delay. One would expect that, for very large n,

R./2

Fig. 3 Variation of minimum eigenvalue of controllability Gramian withcontrol interval and system order

IEE PROCEEDINGS, Vol. 132, Pt. D, No. 3, MAY 1985

the plot would be approximately zero for 0 to 2 and unitythereafter.

3.4 Discretisation in time and spaceIt may be that, for simulation or other purposes, adiscrete-time, lumped (= discretised in space) model isdesired. One can obtain such a model by passing first to acontinuous-time lumped model, and then time-discretising,or one can discretise the original partial differential equa-tion both in time and space. This latter procedure is theone usually suggested in the literature, and it is pertinentto record several standard ideas [9, 10]. Let k, h be thediscretisation intervals for time and space, respectively,and let c be the wave velocity. Then k/h > c is unaccept-able: the solution for the difference equation within a(discretised) region is defined by a set of initial conditionsover a smaller region of discretised space than for thepartial differential equation, and there is instability in theface of round-off errors. If k/h = c, the difference equationsystem is particularly simple. There is a matching of initialcondition requirements, but one is on the verge of round-off error instability. If k/h < c, then initial data over abigger region are required for the difference equation thanfor the differential equation to establish the solution at acertain point, where the difference equation is stable in theface of round-off errors. If /c—>0 and h—»0 while k/hremains constant and k/h < c, then the solution of the dif-ference equation approaches that of the differential equa-tion and the domains of dependence approach oneanother.

In terms of the earlier discussion invoking sampling-theorem ideas, the remarks in the preceding text have thefollowing interpretation. Let the temporal frequencyinduced by the spatial sampling frequency be coa, as in eqn.17, and let the temporal aliasing frequency associated withthe time sampling be cba. Then k/h > c is equivalent tocba >c~1wa.

4 Other distributed systems

4.1 The heat equation or RC line equationThe heat equation applicable for a uniform rod is essen-tially the same equation as that for the voltage on auniform RC transmission line, with resistance R andcapacitance C per unit length, namely

dv(19)

0.99

0.90

«, 081

1 0.72

£ 0.63

T3

| 0.45oE 0.36oc 0.27

0.180.10 6 8 10

normalised time12

Fig. 4 Lumped approximation to uniform RC line made from cascade ofprototype constant-k sectionscurve 1 N = 2 curve 3 — — N = 6curve 2 N = 4 curve 4 N = 8

Such a line can be approximated by an nth order lumpedsystem as depicted in Fig. 4.

Using constant-fc filter theory, as in References 3 and 4,one can establish that if

a> (20a)

or

(20b)

then the transfer function of p sections is approximately

IcoRC .exp - -

L 'with an associated group delay of

El IRC 1

n V 2 27co

The sinusoidal solutions of eqn. 19 are all of the form

v(x, t) = exp [yx + jcot)

(21)

(22)

y= + .coRC

(23)

The negative sign is clearly needed given an excitation atx = 0 with a line extending into x > 0. This result agreeswith the analysis leading to eqn. 21 and eqn. 22. So, undereqn. 20, the lumped model is a good approximation. Oncewe allow high-frequency excitation, however, we draw mis-leading conclusions; for example, we could not expect thatthe group delay associated with the lumped system andwith the line were comparable.

4.2 Towed cableLet us now consider the transverse motion of a uniformtowed cable [5, 6]. With certain simplifying assumptions,the equation becomes

dt: a(x) 20dx dt ox

C T 1 = Odt

(24)

Here, /?, b and c are constants, and cc(x) is spatially depen-dent ; because of drag on the cable, the tension in the cablevaries linearly from a maximum at the towpoint, toapproximately zero at the trailing end, and this accountsfor the spatial dependence of a ( ) . A lumped approx-imation for the equation is also given in Reference 5. Now,as described in Reference 4, j?2 — a(x) is normally positivefor all values of x of interest. This means that the equationdefined is hyperbolic, and wavelike motion is to beexpected. Because of the dependence of a( •) on x, it turnsout that waves propagate one way only [5].

Now suppose the cable has length L and is approx-imated using n uniformly spaced discrete masses. Thespatial aliasing frequency is then n/L. [Because the wavestravel only one way, the sign of the spatial frequency isautomatically determined, and we can choose the intervalof unique determination of frequency to be [0, N/L) ratherthan {-N/2L, N/2L).] The transverse motions of thelumped approximation can be expected to mimic those ofthe distributed system only for frequencies less than[n/L mino^^jr, yjfi2 — a(x)] and we would expect thataccurate representation of the distributed system by thelumped system would require significantly lower fre-quencies again.

Since a ( ) is x-dependent, one could contemplate

IEE PROCEEDINGS, Vol. 132, Pt. D, No. 3, MAY 1985 93

uneven spatial sampling which would compensate in thefollowing sense. Pick points xl5 x 2 , . . . , xn_1? xn along thecable at which mass will be presumed to be concentrated.Choose the x, so that the time for a wave on the cable totravel from x, to xI + 1 is independent of i. Thus the greateris a(x), the wider is the spacing. For the towed cable, thefunctional form of a( •) then dictates that the xf be fartherapart near the towing point. Near the towing point, thetension, and therefore the wave velocity, is greater.

4.3 Deep-sea mining hauling pipeReference 7 discusses the approximation via a lumpedmodel (for the purposes of developing a controller) of adeep-sea mining hauling pipe. The distributed parameterequation is of the type

d2w

dx dt

dw

dx

dw

It82w

(25)

[Here, u ( ) is the control, and yu y2, y3, y5 and y6 areconstant, while y4 is spatially dependent.] The only differ-ence from the towed cable, apart from the explicit inclu-sion of u (which can also be included in the cable equationif desired), is the fourth-order spatial derivative, whichaccounts for the bending moment in the pipe; this termcan be, and usually is, neglected in the towed cable. Thedependence of y4( •) on x is like that of a( •) in eqn. 24.

The author of Reference 7 compared the possibility ofuniform spatial sampling with nonuniform sampling whereeach gap is a fixed multiple of the previous one, and with amixture of the two strategies. By simulation, he found thatat the upper end of the pipe (roughly in the upper half),spacing should be uniform, and in the lower part of thepipe, the interval between sampling points should getshorter, the deeper one moved. The upper end of the pipeis analogous to the towing point of the cable, so that, inqualitative terms, his empirical observations are seen toagree with the conclusions of our argument.

5 Conclusions

The key ideas can be summed up as follows. Lumpedapproximations of distributed systems are only approx-imations over a limited, low-pass, temporal frequencyband. The size of this band can be determined by relating a

spatial aliasing frequency, readily determined from knowl-edge of the distributed system and of the procedure forobtaining the lumped model, to a temporal aliasing fre-quency. The velocity of waves in the medium provides therelationship between the two aliasing frequencies. In theevent that the wave velocity is not uniform, nonuniformspatial sampling may be required. This has proved advan-tageous in the analysis of a towed cable.

In considering the specific question of controllability, itfollows that any control actions which rely on fast actingcontrols will give rise to responses which will almost cer-tainly be different for related lumped and distributedmodels.

In considering the question of closed-loop control, thespectrum of the external inputs and the desired closed-loopbandwidth for the closed-loop system (which betweenthem determine the spectrum of the input to the plant) willbe the crucial quantities that determine the order of thelumped approximation model.

6 References

1 PARKS, P.C.: 'On how to shake a piece of string to a standstill' inBELL, D.J. (Ed.) 'Recent mathematical developments in control'(Academic Press, London, 1973), pp. 267-287

2 WEINBERG, L.: 'Network analysis and synthesis' (McGraw-Hill,New York, 1962)

3 SKILLING, H.H.: 'Electrical engineering circuits' (John Wiley, NewYork, 1957)

4 MILLMAN, J., and TAUB, H.: 'Pulse and digital circuits' (McGrawHill, New York, 1956)

5 KENNEDY, R.M., and STRAHAN, E.S.: 'A linear theory of trans-verse cable dynamics at low frequencies'. NUSC Tech. Report 6463,June 1981, Naval Underwater Systems Centre, Newport, RhodeIsland

6 LEE, C : 'A modelling study on steady-state and transverse dynamicmotions of a towed array system', IEEE J. Oceanic Eng., 1978, OE-3,pp. 14-21

7 EITELBERG, E.: 'Modellreduktion fur ein Tiefseeforderrohr', Regel-ungstechnik, 1980, 28, pp. 409-419

8 BROCKETT, R.W.: 'Finite dimensional linear systems' (John Wiley,New York, 1970)

9 MITCHELL, A.R., and GRIFFITHS, D.F.: 'The finite differencemethod in partial differential equations' (John Wiley, Chichester,1980)

10 FORSYTHE, G.E., and WASOW, W.R.: 'Finite-difference methodsfor partial differential equations' (John Wiley, Inc., New York)

11 PARKS, P.C.: 'In Fourier's footsteps—an introduction to the controlof distributed parameter systems'. Preprint for IEE Colloquium onControl of Distributed Parameter Systems, London, 1974

94 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 3, MAY 1985