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Papermaking

Adaptive ControlÅström, Karl Johan

Published in:Systems and Control Encyclopedia

1987

Document Version:Publisher's PDF, also known as Version of record

Link to publication

Citation for published version (APA):Åström, K. J. (1987). Papermaking: Adaptive Control. In M. G. Singh (Ed.), Systems and Control Encyclopedia:Theory, Technology, Applications (pp. 3590-3591). Pergamon Press Ltd..

Total number of authors:1

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P ac ket-S w ítching N etw orks Parabolíc Systems

the timeofwriting, howevet, intemational links operateeither with LAPB or with a multiline procedureihichalso operates on several parallel lines, but which wasdeveloped before standardization of the multilinkprocedure.

X.95_specìfies rhe call progress signals of public datanetworks-for the packer mode of operation. that is,the meaning of the cause field in clear, restart and reseipackers.

X121 specifies the inter¡ational numbering plan forpublic data networks. In other words it desäbes theintemrl structure of the address fields used in call-establishment packets. Typically an address for apacket-mode DTE attached to a public packet-switchednetwolk .is expressed as a one-digit þrefix (e.g., 0,follored by a field of fou digits to idènti:$ the netúork,and_ then followed by a fewìigits to ideitify the DTÉin that network. Each network specifies theîumber ofdìgits of its DTE address, with the possibility of givingshort addresses to the same DTEs ãnd longer oñes tõothers-

9. Ftøure Prospects

Although packet switching is now govemed by a well-established set of interiationd ãtandards ánd hasbecorne a worldwide communication medim- somedevelcpment is occuring, mainly in the di¡ection ofmore interworking with other networks. Ivith the pro-gress made in modems which operate in synchronousmode over the public switched telephone network(PSTN-), extensions of X.25 for operãtion via pSTNaccess ports are commercially plmned. Thev are beinsstudie'l by CCITT, in parrimlar for teletäx applica--tions :direct communication between text-proôËssingequipment).

Twc directions of cunent research include inter-working with local area networks and integration ofpackef-switching procedures into futue integìated ser-vice digital netwotks (ISDNs) . ISDNs should úring usersan open choice between cirilit switching and þacketswitching so that each is used where it best ierves : h¡cuitswitching for long, continuous and fast transmissionbetween two-devices operating at the same speed:packet switching fo¡ slow or for spasmodic traffiõ, forcommunications between equipment with diffe¡entcharacteristics and for multifleiing large numbers ofdive¡se communications over individual network accessports.

Se€ a¿ro.- Distributed Computer Systems: An Introduction;Distributed Systems: Sl,nchronization md IntemrocessCommunication; Switching Networks: General St¡i¡ctures

work, CCITT Yellow Book, VôI. VIII.I. IntemationalTelecommunícation Union, Geneva

Intemational Telegraph and Telephone Consultative Com-mittee l98l Public Data Networks, CClll Yellow Book,Vol. VIII.2. Intemational Telecommunication Union,Geneva

R. Despres

We present in rhis article only the linear case (whereD(. , r, x) is linear for any l), afte¡ giving rhe best-knownexample of such equations, the heat equation. Finallywe-study vadational inequalities, which are linear para-bolic equations in a domain with a f¡ee boundary.

l. The Heat EquationThe heat equâlion is the "model" of linear parabolicequations. 'We present here this equation in a generalcase and we give explicit results in the simple casé wherethe evolution domain is the whole space.

Let O be an open set of Rd modelling a heât conductorbody. Assume thât the initial temperature of the bodyis equal to /(r) for all r-in Q; that the boundary i¡maintained at the temperature !t(t,x) at time t andposition r; and that there is a distributed heat sourcewhose density with respect to time and space is a func-tion g(t,:). Then the evolution of thé temperatureu(t,x) of the body at time r and position r-may bemodelled by the following equarion (where r> 0):

ôu lal ãu \i - ^Q\ Ì, *,\ut-t ç,øt) = s(r.x). x e e

(1a)

u(t,x) = 1l)ç,x), xe ðe (1b)

u(0,x)=f(x), xeo (1c)

where m.and M are strictly positive functions desc¡ibingthe conductivity of the body. This kind of equation ii:TS9q il Sect. 2, (Eqn.'(1) is a nonhomog.n"ou,Dirichlet boundary condition).

Assume now that Q = Rr (then (1b) has to bedropped) and m = M = 1. Then (1) may be written inthe very simple fom

du - d'u;- ¿ r-=g(t,x)dI i=t dxi

u(0, x) : f(x)This is a Cauchy problem.

50

(391 reels)

I

Slondcrd ¿evlotion in moisiure 6 Stoñdod deviolþn In mo¡sture ø

Papermaking: Adaptive ControlThe advanced control of important quality variables ona paper machine, such as basis weight and moisturecontent, are of major importance, since improvementsof even fractions of a percent can represent substantialrvings- The process dynamics include fime delays-Operating conditions such as machine speed and ihepressure distribution in the dryer sections may vàrysigniâcantly. Large variations may also occur in thêproperties ofpulp, and this can change drying conditionsdramatically- All these variations result in chãnges in theprocess dynamics. A regulator which is well tunèd at oneoperating condition and for one paper grade will not beoptimal for other gradæ and other operating conditionò.It is possible to compensate for variations in machinespeed by gain scheduling, but this is not feasible for thepaper grade, which is influenæd by many differentfacto¡s. This makes the paper machine an interestingcandidate for adaptive control.

The steady-state control of the quality variables canconveniently be formulated as a minimum variancecontrol problem. By reducing the variances of thequality variables, these may be brought closer to theiroptimum range, bringing savings in energy and rawmâterials and increases in production.

It was shown by Àström (1970) thar the steady-staredynamics of the process and the disturbances could bedescribed well by the model

A(q-t)y(t): B(q-t)u(t -d)+ C(q-t\e(t) (t)where y is a quality variable (basis weight or moisturecontent), a is a control variable þulp flow or steampressure) md {e(r)} is a sequence of independentmndom variables. Furthemore A, B and C aie poly-nomials of low degree in the backward shift opeiatórø-'. Good results with minimum variance controlsbased on identified model were also reported.

The model (1) is in a fom which is well suited for theapplication of self-tuning regulators. Industrial feasi-bility studies of basis weight and moisture controlusing a simple self-tuner based on minimum variancecontrol and least-squares estimation were ærried outby Borisson and Wittenmark (1974) and Cegrell andHedqvist (1975). There exist substanrial disturbanceswhich excite the proæss persistently, and thus good

Figwe IHistogram for standard deviation in moistue for aself-tuner and a conventional frxed gain regulator(Fjeld and Wilhelm 1981)

parameter estimates can be obtained and simple expo-nential forgetting of pâst data works well. Commercialpaper machine controls have also been equipped withself-tuning controllers (Fjeld and Wilhelm l98l). Theperfomance improvements obtained in comparisonwith conventional control are illustrated in Fig. l.

BibliographyÄström K J 1970 Introduction to Stochntic Control Theory.

Academic P¡ess, New YorkBorisson U, lüittenmark B 1974 An industrial application

of a self-tuning regulator. Proc. IFAC Symþ. OigítatÇoypu,ter Applications to Process Contro!. Þergari.on,Oxford

Cegrell T, Hedqvist T 1975 Successful adaptive control ofpaper machines. Automaticø 11, 53-59

Fjeld M, rlr'ilhelm R G l98l Self-tuning regulators-Thesoftware way. Control Eng- 2E,99-lOt

K. J. Åström

(2)

Parabolic SystemsAny-partial differential equation where the solution Uls a tunction of a one-dimensional variable , (the time,generally) and of a second variable x (multidinìensional,generally),.and which may be writlen (explicitly oiimplicitly) in the form

du

*+ D(u(t,x),t,x) = g(t,r) (1)

may..be called a parabolic equation if. u- D(u,t,x) ísan elllptrc- cperator. where partial derivatives of secondorder wth respect to x occur. We shall see that Eqn.(1) is."well posed" if the initiat value is given, and ifboundary conditions (at the boundary of tñe domain ofx) are given.

PRoPosrrroN 1. If f and¿-([0, +-) xRd) respectiuelybounded solution u of (2).

Indeed, let

p = p(t, x, E) = (4nt)-d/2 expf-(x - Ç), l4tl (3)

This function is called Green's function associated withthe Laplacian in Rd. It may be seen that ¡¡ is a solutionto

g are in L"(Rd) andthere ãisß a unique

BiblíographyIntemæional Telegraph andTelephone Consultative Com-

mittee 1981 Data Trarcmission ouer the Telephone Net-

3-:9=odI ¡ dxî (4)

3s90

p(0,x): ô(x - Ç)

3591

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SYSTEMS & CONTROLENCYCLOPEDIA

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