Identification and Control for Web Forming Processes

7c-Ol 2

Copyright © 19961FAC 13th Triennial World Congress, San Francisco, USA

IDENTIFICATION AND CONTROL FOR WEB FORMING PROCESSES P.E. Wellstead l , w. P. Heatb 2 and A.P. Kjaer3. I . Control Systems Centre. UMlSl: P. O. Box 88. Manchester MW IQD. U.K. and Infrared Enxineerinf, Ud. Galliford Road. The Causeway. Maldoll. &sex. CM97XD.. UK. (e-mail peter@ .csc.umist.(lc.llk). 2. Lucas AVSD. Shirley. SolihuLL. West Midlands. H904JJ. UK 3. Del f)all.l'ke Staalvalsevauk AlS. DK-33(XJ. Frederihvaerk. Denmark.

Abstract: The paper concerns web forming processes such as paper making, polymer extrusion and related coating processes. Discrete orthogonal polynomials arc used in the identification and constrained optimal control of the cross direction (CD) profile. The paper shows (i) the use of discrete orthonormal Chebyshev polynomials to parametelise the CD behaviour, (ii) a Chebyshev domain estimator for CD profile response and (iii) the use of an iterative method for the constrained quadratic optimisation in the Chebyshev domain of the CD profile. Keywords: Identificalion algorithms, Constraints, Optimal control , Paper industry, Plastics industry.

I. INTRODUCTION 1.1. General

The paper concerns the control of web forming processes. Examples of web forming and related coating processes are encountered in paper and textile making, plastic extrusion, steel rolling and food manufacturing. The aim of this paper is to describe the development and application of efficient procedures for the identification and control of the cross direction (CD) profile of such systems. The motivation for the research described here is that the type of material forming and coating processes, known generically as web forming processes, is a special dass of dynamical control systems which have particular features and dilTiculties. Il is therefore necessary to develop special estimation and control solutions which arc specific to web-forming. The methods described here are a result of a university/industry research collaboration at the Control Systems Centre, UMIST. The aim of the collaboration is to develop new control and signal processing methods for two dimensional systems (Wcllstead and Zarrop, 19(1) and apply them to manufacturing processes, such as web forming, which involve two-dimensional dynamical phenomena. The

particular web forming application considered is the cross directional(CD) profile control problem. The paper brings together a number of theoretical strands as follows: (i) general orthogonal polynomial representation procedures, (ii) their use in direct and parsimonious estimation of CD characteristics and (iii) constrained optimisation for profile control or web-forming pnx:esses with constraint" upon the CD actuators. 1.2. Previous Work

The paper builds upon previous research on CD control in the paper industry (Heaven et al. 1994), and elseWhere. In particular, the pioneering work of Wilhelm and Fjeld ( 1983) on CD representation and control is of great significance. More recently, Halouskova, Karny and Nagy (1993) showed how special basis functions (splines) could be used to provide a basis sel for CD representation. Duncan (1989) used Fourier methods for the same purpose and analysed the controllability properties. Heath (1993) generalised his results to other basis functions sueh as the Chebyshev polynomials considered here . The Chebyshev polynomials used in this paper arc the same as the Gmm polynomials used by Kristinsson and Dumont (1993), and

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of the

L objects:

orthonormal basis functions. Because the actuator response is bandlimited, and 12 can be represented by a small number of spectral functions, (equation 4.), then r(t) can

N -1

y(t)= ~ Yi(t)f i i=(l N p -1_

S (t) = -p

L

~

i=(J -

.(t)f.

p,1

-I

(3)

N -1

~s(t)= ~

es,i(tH i

i=()

Now, just in the same way that a few Fourier coefficients may be used to represent certain waveforms, then similar bandwidth concepts (Heath, 1993), can be used to show that the interaction matrix can be represented by a finite number of such basis functions. This can be written as the matrix sum of k outer products: k

be separated into two components: the first with low spectral content and the second with high spectral content. If the low spectral content is described by k spectral functions L,{i =l, ... ,k}. It follows that the higher spectral component of r(t) is uncontrollable. The low spectral component is controllable if and only if the interaction matrix has rank k. Usually, there are many morc actuators than k, indeed in some cases symmetries in the actuator response require this before the rank is k. As a result the square matrix, !! T!! used to obtain the unconstrained optimal steady state input will be illconditioned, and the control input will be large. The use of constrained control in section 4 is a consequence of this observation, (Heath, 1993).

T

3. System Identification

B=~f.f1. -

~-I-I

i=()

The object {Yi(t),

(4)

i= 1, ... ,Npl in equation (3) can be

viewed, by analogy with discrete Fourier analysis, a<; the discrete orthonormal Chebyshev transform (DCT) of r(t). The advantage of using the DCT to represent the cross direction behaviour IS that the number k of relevant Chebychev coefficients fJi required can be much smaller than N y . For example, for a typical bump test obtained from a polymer extruder with 5(X) data boxes only the first 30 were found to be significant. Hence k=30 would be sufficient to characterise all the information in the 500 data boxes.

The dynamical behaviour of the machine and cross machine dynamics can be estimated in any of a number of ways. However, in web forming systems the techniques lor estimation are limited by the nature of the data and the sensing process. The method used for the cross directional behaviour is to directly cstimate the Chebyshev coefficients of the intemction matrix. 3.i. interaction Matrix Estimation With some reasonable assumption on the system dynamics and process noise. (Kjaer, A.P., W.P. Heath and P.E. Wellstead, 1995), the discrete version of equation (8) can be put in the discrete incremental Chebyshev form, to give: 1 Av(t) = - - 1 BAu(t-T)+C(t)

The discrete Chebyshev transform of equation 2 can be formed and written in matrix form:

(5)

=.L

A(z- ) -

-

-

(6)

Here ~(t) is a vector of k+l elements, with the i th clement AYi(t) and

Where the objects vet) -l...

,g_p (t) and

~s (t)

.

are the DCTs of

their respective time domain counterpaIts. The matrix 12 is composed of the Chebyshev transforms of the interaction . (4». . an dh as the I.Ih row I~T matnx -i (see equatIOn 2.3. Controllability The representation of the CD profile by a finite number of of spectral functions, such as the Chebyshev objects, allows the controllability for the CD profile to be formalised. Heath (1993) showed that the Fourier transform results of Duncan (19R9) may be expressed in terms of Fourier series and generalised to other

k_

Ay(x,t)= LAYj(t)fi(x) j .. O

Also

i : (t)

is a vector of k+ 1 elements, with the i th element

Ci (t) -

1 -

Ci(t)=(1+CiZ- )ei(t) and ~ j(t) is the ith element of a vector of k + 1 Chebyshev transformed white noise sequences. Note that the number of Chebyshev terms, k + 1, is chosen to cover all controllable frequencies.

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The system parameter estimation problem can be decomposed into a set of smaller problems by a test signal being applied to single actuator. If the j th actuator, Uj (t), is selected then the ith row of equation 6 would be:

(7)

The time-invariant Chebyshev coefficients

Pi, j, for the jth

actuator can now be estimated in a series of k+ 1 SISO es ti mators, one for eac h outpu t seq uence yJt), i = 0, ... , k. In this way the set of Chebyshev coefficients is assembled and the actuator response shape found by inverse Chebyshev transformation. 4. Control 4.1. Background

The CD control strategy applied here to CD profile control is special in that it allows for the physical limits upon the deflection of the CD actuators (such as an extruder or head box lip). Boyle (1977) recognised that for a quadratic regulator to be feasible in CD profile regulation it must account for these physical limits. With modem computing power and new iterative algorithms, the quadratic control problem maybe solved in a real time implementation.

A large number of constraints are produced in this way, sometimes several hundreds. These can be handled in a way which allows real-time constrained optimal control using an iterative technique known as the mixed weight least squares method (Rossi ter and Kovaritakis 1993, Heath 1994). The iterative nature of the mixed weight least squares algorithm makes it useful in real time applications because it can be applied as a fixed number of iterations at each sample interval and hence takes a predictable amount of computing time at each interval, this is not true of other quadmtic progmmme solutions. Also it seems to converge quickly (less than 20 iterations) in this particular application where there are many more constrdints than pardmeters. An issue to note in this control algorithm is that the optimisation is performed on the discrete Chebyshev coefficients of the data box array. This reduces the size of the optimisation problem and means that the representation, parameter estimation and controller synthesis are all performed in the Chebyshev domain. 5. Results 5.1. SimulaTion Trials

The following simulated example illustrates the Chebyshev domain estimation and constrained optimal control. The simulated web forming process had an output vector r(t) with 100 data boxes and an actuator vector l!(t) with 25 elements. A first order AR model was used to describe the

4.2. ConSTrained Optimal COlllro/ in The Chebyshev Domain

The control method used here (Heath, 1993) works directly with the identified Chebyshev transform model. For example, if the cost function to be minimised is

(8) then using the properties of orthonormal functions the cost function can be re-expressed in terms of the Chebyshev coefficients Yi(t + 1), i = 0, ... , N p . Only the first k of these coefficients are controllable, thus the cost function can be written in terms of just these controllable coefficients. This represents a significant reduction in the computation of the optimal control. The cost is to be minimised subject to constraints on the absolute limits upon the control, limits upon the difference between adjacent actuators and limits upon the second derivative of the actuator array (e.g. the lip cunature). Specifically,

actuator dynamics C(z -1) and the process disturbance wa<; integrated white noise with \ariance 0.1. Using a unit amplitude square wave applied to one actuator and thirty sensor scans, the Chebyshev coefficients of the actuator response were estimated in a series of SISO extended least squares estimators. Estimated Cheb she\ Coefficients

1.5

0.5

o -0.5 -I

-1.5

()

5

10 15 20 coefficient index, k

25

30

Fig. 2. E<;timated Chebyshev coefficients for a simulated web process

i=l, ... nu

Fig. 2 shows the first 30 estimated Chebyshev coefficients. Note the magnitude of the coefficients decreases with coefficient number, supporting the argument that a small number of Chebyshev coefficients are adequate to

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the extrusion line using the Chebyshev technique. Further estimation details appear in Kjaer, (1994). A similar actuator response estimate obtained from the line was used to design a constmined optimal controller for the CD profile. The controller was applied to the line and acheived stable, nat profile control of the machine for a prolonged test period. Unfortunately, no permanent records of the achieved control performance are available, but satisfactory profile control was obtained with minimal tuning and in a short commissioning time.

6. Review and Future Developments The contribution of the project described in this paper has been to develop a coherent method for chamcterising and controlling the CD behaviour of a web forming process in terms of discrete orthonormal Chebyshev polynomials. Absolute actuator constraints are included in an optimal CD regulation system, where positional, rate and second derivative constraints on the CD actuation apply. One of the most important difficulties in the control of two dimensional processes such as web forming is the sparsity of the data obtained from traditional scanning sensors. In this context an important trend is the development of special estimation methods (Kulhavy, Well stead and Zarrop, 1(92) .and hardware to increase the density of data from sensors. See for example, Kjaer ( 1(94) on the use of vision systems to sense the dry-line in a paper machine. Also, Hindle and Smith (1994) describe a multi-head sensor which greatly increases the scanning speed and density of data in an infrared scanning gauge. The techniques described here are compatible with all these algorithmic and hardware developments. Indeed, they have been tested with a multi-head gauge installed on a plastic film production line with satisfactory results. 7. Acknowledgements The authors wish to thank all colleagues at the Control Systems Centre, the directors and stall of Infrared Engineering Lld. and Prof. M. Wailer of Miami University for sUPlxxt advice and encoumgement. We also gratefully acknowledged the financial support of the EPSRC, DTI and Infrared Engineering Lld (grant GR/M216(2). NATO (grant CRG 91(435) and the Royal Society/EPSRC Industrial Research Fellowship progmmme for supporting Peter Wellstead. 8. References

Boyle, T. 1., (1977), Control of Cross-Direction Variations in Web Forming Machines, Canadian J. of Chem. Eng. 35, pp 457-461. Duncan, S. R. (1989), The Cross Directional Control (~f Web Forming Processes, Ph. D. Thesis, University of London. Heaven, E.M., I. M. 10nsson, T.M. Kean, M. A. Manness, and R.N. Vyse (1994) Recent Advances in Cross

Machine Profile Control, IEEE Control Systems Magazifw, October, pp 35 - 46. Halouskova, A., M. Kamy and I. Nagy, (1993), Adaptive Cross-Direction Control of Paper Basis Weight, Automatica, 29, pp 425-429. Heath, W.P. (1993), Orthogonal Functions for Cross Directional Control of Paper Making Machines and Plastic Film Extruders, Control Systems Centre Report 787, UMIST, Manchester, (to appear in modified and enhanced form in Automatica, February, 1(96). Heath, W.P. (1994), Constmincd Stable Predictive Control, IEH Proc Control Theory, 141, pp 274-276 Hindle, P.H. and C. R. R. Smith, Ultm-fast Web Thichness and Coating Measurements with True TwoDimensional Control, (1994) Proc. NIR '94. Iflternaliollal Conference 011 Speclroscopy, Sydney, Australia. Kristinsson, K. and G.A. Dumont (1993), Paper Machine Cross Directional Basis Weight Control Using Gram Polynomials, Proc. Second IEEE Conference Oil Control Applicatio/lS. September. Vancouver, Canada. Kjaer. A.P. (\994), Modelling. Sensing and Identification (~f Web Forming Processes, PhD Thesis, Control Systems Centre, UMIST, Manchester. Kjaer. A.P., W.P. Heath and P.E. Wellstead, (1995), Identification or Cross Directional Behaviour in Web Production: Techniques and Experiences, IFAC Journal of CofltroJ f:ngilleering Practice, 2, pp129-139 Kulhavy, R., P.E. Wellstead and M.B. Zarrop, (1992), On Possible Ways of Coping with Missing Data in Two Dimensional Processes, Research Note, Control Systems Centre, UMIST, Manchester. Ringw(xxl, 1. V. and M.J.Grimble, (1990), Shape Control in Scndzimir Mills using Crown and Intermediate Roll Actuators, UTE Trans. on Automatic Control. 35, pp 453-459. Rossiter, 1.A. and Kovaritakls, 8. (1993), Constrained Stable Generalised Predicti ve Control, IEI·,: Proceedings-/) 140, pp243-254. Wcllstead, P. E. and Zarrop, M. B. (1991), Se{fTuning Systems: COfllrol and Signal Processing. J. Wiley, New York. Wilhelm, R. G. and M. Fjeld (1983), Control Algorithms for Cross-Directional Control: the State of the Art, Proc Slh Conference on Instrumentation and Automation in the Paper. Rubber Plastics and Poimerisation Industries, Antwerp, Belgium, pp 162-174.

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