Stability of accelerating repetitivesystems

J.B. Edwards. B.Sc.(Eng), M.Sc, C.Eng., M.I.E.E.

Indexing terms: Control theory, Control systems, Accelerating repetitive systems, Mathematical techniques

Abstract: The spatial stability of a sequence of accelerating repetitive operations is investigated using theoutput spectral-density in a frequency band enclosing the first resonance peak as a stability indicator. Theoperations of the sequence are identical in dynamic structure, but subject to a constant rate of accelerationbetween operations, thus resembling the rolling of metal strip and other repetitive manufacturing processes suchas machining. The technique readily yields a value for the critical number of operations within which stabilitycan be expected to be achieved, within the chosen frequency band. Simulation of a variety of systems confirmsthe physical usefulness of this number which correlates well with the observed number of operations found tobe necessary to achieve an output profile that is adequately stable from a practical viewpoint.

List of symbols and abbreviations

Gn(s) = transfer function of nth operation in arepetitive sequence G0(s), G^s), . . . , GN_l(s)

k = static gain of second-order processfcj, /c2 = gain parameters in metal rolling process

modelLn = length of metal strip after nth operation/„ = distance from leading end of strip to slice

of interest after nth operationL = normalised strip length = Lo

/ = normalised value of /„ = /„ L0/Ln

n = integer representing number of repetitiveoperations undergone by workpiece

N = total number of operations in repetitivesequence

nc = critical value of n above which spectraldensity of output of Gn(s) (in responseto impulse applied to G0(s)) will notincrease at any frequency in bandcoa ^ co ^ cob

r = constant = I/acceleration rate of processdynamics between operations

s = Laplace variableSn(co) — spectral density of output of Gn(s)X = fixed transport delay distance in metal

rolling (between rolls and gauge sensor)Xn = n-dependent delay distance in hypothetical

rolling modelyn(l) = output from nth operation at normalised

distance /yn(s) = Laplace transform of yn(l) in s with

respect to /co = angular natural frequency in radians per

unit normalised distance /co' = lowest value of co making |Gn(jco)\ = 1.0coa = lowest value of co at which | G0(jco)\ =1.0cob = next higher value of co at which

\Go(jco)\ = 1.0con = undamped natural frequency of nth second-

order operation£ = fixed damping ratio of second-order

operation

Paper 2533D, first received 9th December 1982 and in revised form 16th March1983The author is with the Department of Control Engineering, University of Sheffield,Mappin Street, Sheffield SI 3JD, England

1 Introduction

In the course of manufacture it is common for each indi-vidual workpiece flowing along a section of productionline to be subjected to a sequence of operations G0{s),G^s), ..., Gn(s), . . . , GN_l(s), in being converted from itsinitial (rough) state to its final (finished) state. Examples ofsuch sequential processes include the machining of preci-sion components, the production of metal strip by rollingetc. In the interests of standardisation, the individual oper-ations G0(s), . . . , GN_l(s) may be extremely similar to oneanother.

Because each operation may require a significant floorspace, the cost of which is ill afforded in a depressed eco-nomic climate, there is considerable incentive to explorethe possibility of minimising the total number N of suchoperations. The consequent time saving is clearly animportant additional consideration.

One important characteristic of the operation sequenceis that the dynamics of each successive unit operation,whilst retaining the same basic form of frequency response(e.g. analogous modes having identical damping ratios),tend to become progressively faster as n increases from 0to N — 1. This is fairly readily appreciated in the case ofmetal rolling [1, 2], where the dynamics of each rollingoperation are dominated by the transport delay resultingfrom a fixed spacing X between the rolls and the output-thickness (gauge) sensor used for automatic gauge-control.Although X is fixed, owing to the progressive lengtheningof the metal strip with each rolling operation, the relativemagnitude of the delay, viewed from any given verticalslice of material, appears to progressively shorten witheach operation, i.e. as n increases. Furthermore, if rollingtakes place repetitively at constant speed through the same(or identical) rolling stand(s), the fixed dynamics of the unitrolling system (i.e. the complex spring/mass network rep-resenting the roll structure and the gauge-setting servo)will also appear to vibrate faster and faster when viewedfrom any chosen vertical slice of the metal strip: a pheno-menon again resulting from the progressive lengthening ofthe strip.

In a sequence of similar machining operations [3], thisprogressive stepwise 'acceleration' of the dynamics of eachsuccessive operation may result from the use of progress-ively lighter cutting tools, and their lighter associatedsupport structures, as the average thickness of metalremoved is reduced as the final finishing cut is approached.

Unless machines of enormous mass and rigidity are

IEE PROCEEDINGS, Vol. 130, Pt. D, No. 4, JULY 1983 183

G(s)0

y,(s)G(s)

y2(s)G (s)

n

Fig. 1 Block-diagram representation of N-stage repetitive process

employed, there will always exist some dynamic inter-action between spatial profile yn(l) produced during oper-ation n — 1 and that, i.e. yn+1(l), produced on operation n,and it is this interaction which is here represented bytransfer function Gn(s). The length co-ordinate / will clearlylie within a finite range, i.e.

0 ^ / ^ L (1)

in practical processes where L is the normalised* length ofthe workpiece, but it is usual for L to greatly exceed thewavelength of the slowest mode of Gn(s) so that, for practi-cal purposes, Gn(s), if stable, may be regarded as a contin-uous process over all time. Thus, by taking Laplacetransforms in s with respect to /, we get

= Gn(s)yn(s) - I (2)

where superscript ~ denotes the Laplace transform of theassociated spatial variable. The overall system may there-fore be represented by the block diagram of Fig. 1 whichimplies a single continuous process of transfer functionG0(s)Gi(s)G2(s) ... Gn(s) ... Gx^iis). Such a representationis appropriate provided the boundary conditions can beengineered to induce no transients at the start of eachsubprocess or if attention is confined to process variablesat distances far from either end of the workpiece.

2 Stability

Now although Gn(s) may be stable for all values of n, thesequence of processes G0(s), Gx(s), . . . , Gn(s), . . . , GN_l(s)may, in a practical sense, constitute an unstable processoverall. In particular, any impulse in yo(l) may generateoscillations in y^l), and hence in y2(l), ..., yn(l), . . . , yN(l),that grow in number and/or amplitude with increasing n:clearly an undesirable state of affairs.

2.1 Repetitive systems of constant (n -independent)dynamics

In the special case Gn{s) = Gn + l{s) = G(s), (n = 0, 1, . . . ,N — 1), then, to avoid instability in the sense describedabove, G(s) must satisfy the condition that

\G(jco)\ < 1.0 for all real to (3)

This is because the spectral density of yn(l) is | G"{jco)\2 (inresponse to a unit impulse in yo(l))

a n ^ for this to reduce,over the entire range of co, with increasing n, criterion 3must be satisfied. Fig. 2 illustrates a frequency response| G(jco) | that would produce instability in the repetitivesystem GN(s), because within the frequency band

coa ^ co ̂ cob (4)

criterion 3 is clearly contravened.In a special-case (i.e. delay dominated) metal-rolling

example:

G(s) = (5)

* In metal rolling, L would be the length of the strip prior to its first rollingoperation, and / would be a normalised quantity related to real distance /„ (of theslice in question after pass n) by the relation / = lnL/Ln, where Ln is the strip lengthafter pass n.

Fig. 2 Form of frequency response \G(J(o)\, that would produce insta-bility in N-stage sequence GN(s)

where kx, k2 are constant gain parameters and X the con-stant normalised* delay distance, it is readily shown [1]that criterion 3 reduces to, simply,

kx < 1 - /c- (6)

The result is confirmed by the computed transientresponse of Fig. 3 (for kx = 0.4, k2 = 0.5), i.e. a stable case,and Fig. 4 (for kt = 0.75, k2 = 0.5), i.e. an unstable case.

Criterion 3 has been developed more fully elsewhere[1, 2] and interpreted in a variety of different ways. The

* This example involves the rather hypothetical case of an increasing transportdelay distance, Xn, with each rolling operation, proportional to strip length Ln suchthat normalised delay X = XnL/Ln, where L = Lo, the initial strip length.

1 U

n1.0

0

~~T_r—

2

* \

•6

— i

y/o

8 10

y2d)

i i

12 1/X

1.01/X

y3(0

i.Or l/X

y( l )

0

1.0

i/x

1/X

0

1.0

0

1.0

y(l)6

l/X

y7d)

l/X

y8d)

10 12 l/X

Fig. 3 Time response of stable metal-rolling system with fixed normal-ised delay (kl = 0.4, k2 = 0.5)

184 IEE PROCEEDINGS, Vol. 130, Pt. D, No. 4, JULY 1983

2 4 6 8 10 12 14 16 18I/X

i/x

Fig. 4 Time response of unstable metal rolling system with fixed nor-malised delay (/c, = 0.75, k2 = 0.5)

spectral-density concept used above, however, is the mostimportant in the present context of systematically varyingprocesses Gn(s), n = 0,1, 2 , . . . , N — 1.

2.2 Systems of accelerating dynamics Gn(s) = G0(sr n)If the unit process transfer functions are interrelated as

GJLs) = G0(sr") (7)

where r is a fixed positive parameter less than unity, then itis clear that all subprocesses of the repetitive system havemodes that are identical in form (i.e. in damping ratio) butsubject to a uniform increase in speed (i.e. in naturalfrequency) with increase in n. This form of model (eqn. 7) isparticularly appropriate to metal rolling if r is the nominalratio of output to input strip thickness on each pass(= ratio of input to output strip length if the strip widthremains unchanged). The reasons for this have been out-lined in Section 2 and are more fully treated in Reference2. This form does have more general application in manu-facturing systems, however, as had already been men-tioned.

2.2.1 Stability-band concept: Fig. 5 illustrates the fre-quency response of the n individual subprocesses contrib-uting the sequence G0(s), G^s), . . . , G,,_i(s) when these aresubject to eqn. 7. The spectra clearly spread and shifttowards the higher-frequency domain as n increases. Had rbeen unity then all the spectra would have been identicalto | G0(jco) | and the repetitive sequence would have beenunstable for the case illustrated, as

Because of the shifting operation resulting from r < 1.0,however, it is clear from Fig. 5 that, in response to a unit

narrowing bandof instability

Fig. 5 Showing the shifting and spreading of the spectrum ofGn^l(s) asn increases

impulse in yQ(l), the output-signal spectral density Sn(co),from Gn(s), given by

(9)i = 0

will increase with increasing n, within the narrowing fre-quency band

co' co cohco' > (10)

2.2.2 Critical number of repetitions, nc: The progressiveincrease of spectral density at any frequency within theband

coa^co^cob (11)

will cease, however, when n reaches a value such that

I Gn{jcob) | ^ 1.0 = | G0{jcoa) | (12)

By combining this condition with eqn. 7 we can thereforeobtain the critical value (nc) for n beyond which no spectraldensity, in the band coa ^ <increase. This value is thus obtained by setting

cob, will continue to

\G0(jcobr")\ = \G0(jcoa)\

i.e.,

coh r" = CD,,

so that we get

log (cob/coa)

(13)

(14)

(15)

So far as signals within frequency band (expression 11) areconcerned, therefore, nc, as calculated from eqn. 15, androunded up to the nearest integer, provides a useful upperbound on the number of repetitive subprocesses which willproduce instability. Before this number of operations iscomplete, natural oscillations within the band eqn. 11 maybe expected to begin attenuation. Before contemplatinghigher-frequency effects it is profitable, at this stage, toexamine specimen simulation results.

3 Results and discussion

The foregoing analysis has presupposed a process G0{s)having a single resonance peak. Such processes may wellhave high order, but it is sensible to investigate first thesimplest of such systems, namely a second-order lagprocess:

G0(s) = kcol/{s2 + 2(coo s +

so that

(16)

(17)

where con r" = co0.Fig. 6 shows the spatial response for k = 0.65, ( = 0.3

and r = 0.9, (for which coa = 0.72cw0 and cob = 1.06co0)giving a calculated value for nc = 3.7. Clearly the processstability deteriorates only for the first 3 to 4 operations forwhich nc, as calculated, provides a good estimate, andthereafter the oscillations steadily decay towards zeroamplitude. Comparing Fig. 6 (r = 0.9) and Fig. 7 (r = 1.0),clearly demonstrates the stabilising effect of acceleratingdynamics. The instability in Fig. 7 is manifested as muchby the spread as by the amplitude of yn{l). Experimentsconducted on a wide variety of systems seem to indicatethat nc has important practical significance generally, aswell as being an easily calculated mathematical parameter.

IEE PROCEEDINGS, Vol. 130, Pt. D, No. 4, JULY 1983 185

t0.05

- 2 0 ^

y2(0 A

y7d) y8d)

yQd) yn(.)

\ / V ^

Fig. 6 t/rnr impulse response of repetitive second-order system withaccelerating dynamics (k = 0.65, £ = 0.3, r = 0.9)

f\ y,d) cos

- 2 0 —0

y d )

/x A /\ ~0

0

0

\ / \ / x-'

(i)

v -

o ^

(I)Ai /N. —

y ( i )10

A A v

y6d)

A A / \ ^J \J \f ^

-Al\o ^V

J_ «/' - -\ /̂20 V

A

0

A,yy v

Ai M

A - y i ( "1 v

y/0

N / \ A A -V y V v

9 \ W^ ^.^ ^^

A nT/ \ / \ A />•

Fig. 7 t/ml impulse response with constant dynamics {k = 0.65, C = 0J ,r = 1.0)

Fig. 8 confirms this finding for the metal rolling exampleexamined in Section 2, but here with r set to 0.95 ratherthan unity. We therefore have

Gn(s) = kl exp (- (18)

from which we deduce that coa and cob are the two lowestpositive solutions of the equation:

k\= X -i cos —

, f kj + l-kjlL 2fc, J (19)

and with /cx = 0.75 and k2 = 0.5 (as for Figs. 4 and 8) thevalues for these 'unit-gain frequencies' work out to beo)a = l ^ X " 1 and cob = 3.63AT"1, thus yielding a value fornc (via eqn. 15) of 6.2. From observation of Fig. 8, stabilityclearly begins to improve after n = 3 or 4 and is hereachieved for practical purposes by n = nc.

This example again demonstrates the practical useful-ness of parameter nc, but now on a system that has amultiplicity of resonances: the numerical values of coa andcob calculated above applying strictly to only the first res-onance peak. The results given in Fig. 8 clearly show theshift of the system's spectra towards higher and higherfrequencies, but the relative amplitude of the high-frequency ripple that develops with increasing n is obvi-ously small and of minor practical importance.Concentrating analysis on the first resonance peak wouldtherefore seem to be no less valid than neglecting all butfundamental frequencies in using describing functions fornonlinear-systems analysis [4]. This conclusion is rein-forced by the realisation that, in practice, low-pass filteringprocesses (justifiably excluded in the mathematical model-ling of G0(s) because of their high bandwidths) are likely tooccur in positions interposed in the process sequence G0(s),G^s),..., Gn(s) of Fig. 1. The effect of these filters would beto prevent the shift of system energy to ever higher fre-quency domains as n increases.

1.0y,(D

0 2 A 6 8 10 12 14 16 I/X

l/X

f y«(i)

or^0.5(

l/X

y^o

i/x

Fig. 8 Response of metal-rolling system with variable normalised delay(fc, = 0.75, k2 = 0.5)

186

4 Conclusions

The stability of an important class of repetitive processesencountered in metal rolling and manufacturing generally,and described by a sequence of transfer functions G0(s),G^s), . . . , Gn(s), . . . , Gjv-i(s), has been investigated. Theindividual subprocesses are interrelated thus: Gn(s) =G0(sr"), 0 < r < 1, and the sequence may therefore bedescribed as one of accelerating dynamics. It has beenshown that, if the first resonance peak of G0(s) exceedsunity, within the frequency band coa ̂ w ^ a>b, the spectraldensity of the output of Gn(s) (in response to a unit impulseapplied to G0(s)) will continue to increase at some fre-quencies within this band, as n increases, until n exceeds acritical value nc = [log (ojb/coj]/\og (r"1). This value istherefore an upper bound on the number of operations forwhich instability persists in the sequence output, within theband coa ^ a) ^ cob.

Simulation examples of single and multiple resonancesystems have demonstrated good correlation between theeasily calculated parameter nc and the number of oper-ations actually needed for a stable output to be produced.The parameter would therefore seem to be of considerablepractical importance, particularly in view of the economicneed to keep the total number of operations to aminimum.

The theory's disregard for oscillation in higher-frequency bands has been shown by the simulation to bevalid and in practice may be further justified by the exis-tence of high-bandwidth low-pass filtering processes inter-

IEE PROCEEDINGS, Vol. 130, Pt. D, No. 4, JULY 1983

posed between the subsystems. The approximation differslittle from that (namely, the neglect of all but fundamentalcomponents of oscillation) underlying describing-functionmethods for nonlinear-system analysis.

5 Acknowledgments

The author is grateful to Mr. M. Mazandarani and Mr. R.Bouhedda, postgraduate students in the Department ofControl Engineering at the University of Sheffield, forcomputing the time responses presented in this paper.Thanks are also due to the Head of Department, Prof. H.

Nicholson, for the use of the computing facilities requiredby this research.

6 References

1 EDWARDS, J.B.: 'Stability problems in the control of multipass pro-cesses', Proc. IEE, 1974, 121, (11), pp. 1425-1432

2 EDWARDS, J.B., and OWENS, D.H.: 'Analysis and control of multi-pass processes' (J. Wiley, Research Studies Press, Letchworth, 1982)

3 WELBOURNE, D.B., and SMITH, J.D.: 'Machine-tool dynamics'(Cambridge University Press, 1970)

4 ATHERTON, D.P.: 'Stability of nonlinear systems' (J. Wiley, ResearchStudies Press, Letchworth, 1981)

IEE PROCEEDINGS, Vol. 130, Pt. D, No. 4, JULY 1983 187