Identification and Control of a Headbox - DiVA portal18200/FULLTEXT01.pdf · 2006-03-20 · Identification and Control of a Headbox Examensarbete utfört i Reglerteknik vid Tekniska

Identification and Control of a Headbox

Examensarbete utfort i Reglerteknikvid Tekniska Hogskolan i Linkoping

av

Carl Magnus Tjeder

Reg nr: LiTH-ISY-EX-3197

Identification and Control of a Headbox

Examensarbete utfort i Reglerteknikvid Tekniska Hogskolan i Linkoping

av

Carl Magnus Tjeder

Reg nr: LiTH-ISY-EX-3197

Supervisor: Marko Hyensjo, Metso PaperPasi Virtanen, Metso AutomationFredrik Tjarnstrom, LiTH

Examiner: Svante Gunnarsson, LiTH

Linkoping, 28th February 2002.

Abstract

The purpose of this thesis is to investigate an alternative control strategy for amulti-variate non-linear process in a paper machine called the headbox. The pro-posed solution was intended to be able to be adopted on two different headboxtypes, currently controlled by different concepts.

The methodology was to first create black-box models of the two different sys-tems based on measurements, at one working point. Secondly, various controlstrategies were investigated. A more sophisticated multi-input multi-output con-troller MPC, or model predictive control, and a less sophisticated one, a single-input single-output, decentralised PI-controller. With help of simulations the per-formances of the both strategies were tested. Finally, only the decentralised controlsolution was implemented and evaluated through trial runs on a pilot machine.

The main issue surrounding the decentralised controller was the input-outputpairing. Since the multi-variate system had four outputs and only three inputs,analysis had to be made in order to select three of those four, to form a squaresystem. This analysis was based on the relative gain array (RGA).

The resulting performance of the decentralised controller showed stability andadequate response times, surpassing the older system and making one componentobsolete through the pairing changes. The MPC controller showed even betterperformance during simulations and shall also be taken into account if furtherinvestigation is possible.

i

Acknowledgements

First of all I would like to thank Metso Paper Karlstad AB and Metso Automa-tion for offering me this project and facilitating its implementation. FurthermoreI would like to direct my gratitude to all persons supporting me in various waysduring this time. Foremost the examiner Dr. Svante Gunnarsson and the super-visors Pasi Virtanen, Lic.Eng. Marko Hyensjo, and Lic.Eng. Fredrik Tjarnstrom.I also want to thank Lic.Eng. Johan Lofberg for helping me with everything con-cerning MPC. Finally I want to thank Per Sjostedt for his invaluable help duringset-up for process data collection and the machine operators Per-Inge Johanssonand Mattias Scott.

iii

Notation

Symbols

M Number of samples of the output horizon, used for MPC.L Number of samples of the input horizon, used for MPC.N Number of samples used for parameter estimation.θ Parameter vector.y(t|θ) One-step ahead Predictor.KP Time continues gain.TI Time continues integration time.TS Sample time. Real part of a complex number.I Identity matrix.V0 Scaled volume of headbox air-cushion.n Number of states in a system.p Number of outputs from a system.m Number of inputs to a system.

Operators and functions

q Time discrete delay operator.A ∗ B Element by element multiplication of matrices A and B.A∗ Transpose and complex conjugate of A.A† Pseudo inverse of a A.C(·) Covariance of stochastic variable.E(·) Expectation value of stochastic variable.

Abbreviations

ARX AutoRegressive with eXternal input.ARMAX AutoRegressive Moving Average with eXternal input.

v

DCS Distributed Control System.MPC Model Predictive Control.MIMO Multi-Input Multi-Output.RGA Relative Gain Array.PI Proportional Integral.SISO Single-Input Single-Output.

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Process description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4.1 The Paper and board machine . . . . . . . . . . . . . . . . . 21.4.2 The forming section and the short circulation . . . . . . . . . 31.4.3 Air-cushion damped hydraulic headbox . . . . . . . . . . . . 31.4.4 Hose damped hydraulic headbox . . . . . . . . . . . . . . . . 51.4.5 Headbox control . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theory 92.1 Fundamental system descriptions . . . . . . . . . . . . . . . . . . . . 92.2 System identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Properties of input signals . . . . . . . . . . . . . . . . . . . . 102.2.2 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Model structure . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.4 Prediction and model adaptation . . . . . . . . . . . . . . . . 122.2.5 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 SISO control design . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 MIMO control design . . . . . . . . . . . . . . . . . . . . . . 15

3 Experimental set-up 193.1 Software environment . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.3 Data collection trial description . . . . . . . . . . . . . . . . . 203.2.4 Sample interval . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.5 Practical ARX identification procedure . . . . . . . . . . . . 21

3.3 Control design evaluation trial description . . . . . . . . . . . . . . . 22

vii

viii Contents

4 Results and discussion 234.1 Headbox identification . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 Case I ARX structure . . . . . . . . . . . . . . . . . . . . . . 234.1.2 Case II ARX structure . . . . . . . . . . . . . . . . . . . . . . 294.1.3 Case II state-space structure . . . . . . . . . . . . . . . . . . 30

4.2 Headbox control design . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.1 Decentralised control Case I . . . . . . . . . . . . . . . . . . . 324.2.2 Decentralised control Case II . . . . . . . . . . . . . . . . . . 334.2.3 Model predictive control of Case II . . . . . . . . . . . . . . . 35

4.3 Control design evaluation trials . . . . . . . . . . . . . . . . . . . . . 35

5 Headbox attenuating characteristics 395.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.1.2 Damping simulation’s layout . . . . . . . . . . . . . . . . . . 40

5.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.1 Headbox configuration: no overflow tank attached . . . . . . 405.2.2 Headbox configuration: overflow tank attached . . . . . . . . 405.2.3 Headbox configuration: hose damped hydraulic headbox . . . 415.2.4 Active damping characteristics . . . . . . . . . . . . . . . . . 435.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Conclusions 456.1 Headbox attenuating characteristics . . . . . . . . . . . . . . . . . . 456.2 Headbox identification and control . . . . . . . . . . . . . . . . . . . 45

A Collected data 49

B Model predictions 53

C SISO simulations 59

D MPC simulations 65D.1 Simulink circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . 65D.2 Simulation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66D.3 Implementation of MPC . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter 1

Introduction

This chapter is commenced with the background, purpose, and assignment of thisthesis found in sections 1.1, 1.2, and 1.3. In Section 1.4 a brief process description isgiven. Finally, in Section 1.5 an outline of the remainder of the thesis is presented.

1.1 Background

Metso Paper Inc. is a company manufacturing paper and board machines. Itshead office is situated in Jyvaskyla, Finland and one of the production units inKarlstad, Sweden. A sister company, Metso Automation, delivers complete con-trol solutions for the machines. Its research and development centre is situated inTampere, Finland. Recently an issue regarding the control system for two versionsof an important sub-process has arisen. Metso Automation would like to find outwhether or not two different control concepts are mergeable into one and if, dueto control system changes, a certain component can be made obsolete. Necessaryparts of such a project, such as process identification and control design evaluation,requires the utilisation of the pilot machine at Metso Paper Karlstad. The super-visors involved are Pasi Virtanen, Metso Automation and Marko Hyensjo at MetsoPaper Karlstad AB. Participants from Linkopings Universitet are the supervisorFredrik Tjarnstrom and the examiner Svante Gunnarsson.

1.2 Purpose

The objective of this Masters of Science Thesis is to constitute a foundation forfurther decision making concerning the control issue. The joint project may alsohave the effect to strengthen the co-operation and competence sharing between thetwo sister companies.

1

2 Introduction

1.3 Assignment

The assignment can be divided into two separate yet related parts.The first and least extensive task is to investigate the attenuating characteristics

of an air-cushion damped hydraulic headbox with respect to total air volume.The second and most extensive task concerns identification and control of a

hydraulic headbox in a paper and board machine. Today the control of the paperand board machine headboxes are handled differently. As mentioned earlier thereexists a wish to merge the two control designs into one. Hence the assignmentfocuses around the possibility to merge the two control concepts into one, i.e., toanalyse the different approaches in control of a hydraulic headbox.

1.4 Process description

The notions and processes described in the following sections are a compilationfrom [11], [4], and business know-how at Metso Paper Karlstad AB.

Usually we tend to think of papermaking as ”low-tech”. On the contrary, pa-permaking is a highly advanced and complex process involving modern controlsystems. Within the field of paper manufacturing there exist two different notions.The notion of paper and the notion of board, where the former is the most com-monly known. These two have distinctly separated fields of application. Typicalpaper products are newsprint and magazine paper, while examples of typical boardproducts are containers and liquid packages. These separated fields of applicationyield in different property demands. For paper the most important property oftenis printability and other optical properties, while the mechanical properties, suchas structural strength, are most important for board. Structural strength highlydepends on the grammage. The grammage of most boards are higher than 150g/m2, while most paper grades range from 10 to 80 g/m2. Hence grammage is acrucial property, having impact on how a machine is built.

1.4.1 The Paper and board machine

A modern paper machine can be divided into several sections. The first one is theforming section. It consists of a headbox and a wire. The purpose of the headboxis to distribute the pulp, mixture of fibre and water, evenly onto the wire. On thewire the fibres in the suspension take on a preferred forming and are then beingdewatered to form a wet web at the end of the wire section. Following this, thewet web enters the press section, where more water is removed, this time with theaid of press nips. Leaving the press section the paper has about 40–50 percentmoisture. The final stage is the dryer section. Here the moisture level is so reducedso that a new notion, dryness is introduced cf. [2]. Exiting the dryer section thepaper has 92–93 percent dryness. Finally the paper is rolled up on a large reel.

1.4 Process description 3

1.4.2 The forming section and the short circulation

There are mainly two different types of wire section solutions for board and papergrades. The traditional fourdrinier, cf. Figure 1.1, and the twin-wire, cf. Figure 1.2.The fourdrinier employs a one-sided dewatering facility, while the twin-wire hasa two-sided dewatering facility, i.e., dewatering with two wires, from below andabove. The higher the machine speed is, the longer the wire has to be. At high

Figure 1.1. Fourdrinier forming, headbox on the left-hand side. The suspension travelshorizontally along the forming table and is dewatered from below.

machine speeds the fourdrinier suffers from unstable liquid surface, due to thefriction between the liquid and the surrounding air. On the twin-wire the liquid isencapsulated between the two wires. Hence the twin-wire solution is more suitablefor high machine speeds.

Figure 1.2. Twin-wire forming, headbox in upper left corner. The suspension almostimmediately enters a nip between two wires, hence the double sided dewatering.

The short circulation is the flow circuit from the pulp feeding system to thewire section through the headbox, through the wire to the wire pit and back againto the pulp feeding system, consisting of the mixing pump, screen and thick stockinflow. In this sub-process, water and residuals are re-circulated and new stock iscontinuously added to re-circulated backwater to shape the right consistency.

1.4.3 Air-cushion damped hydraulic headbox

As already mentioned the headbox’s main objective is to distribute the suspensionevenly onto the wire. This is one of the most important sub-processes in the papermachine. The air-cushion damped hydraulic headbox, seen in Figure 1.3, consists

4 Introduction

of the manifold, cf. balloon 1, the middle chamber, cf. balloon 2, with an air-cushion, cf. balloon 3, on top of it, a turbulence package, cf. balloon 4 and the lipchannel, cf. balloon 5.

In the manifold the stock flow is changed from being transported in a pipingsystem to being evenly distributed over the whole machine width. This is quitecomplicated and in order to achieve the same pressure in every width section themanifold tightens and at the end there is a small 10 percent re-circulation outflow.An important feature of the headbox is to reduce pressure variations from the

Figure 1.3. An air-cushion damped hydraulic headbox. Dark colour shows the pulpflow. Balloons 1-5 show the manifold, middle chamber, air-cushion, turbulence package,and the lip channel.

process, e.g., mixing pump, pressure screen and sometimes vibrations from piping.This is especially important on the fourdrinier former, which has an amplifyingeffect on the pressure variations called barring, cf. [9] and [10]. The importanceof the pressure variation reduction depends on the their proneness to generatevariations in the machine direction basis weight. In order to achieve some sortof attenuating effect in the headbox, i.e., not being a totally stiff system, an aircushion is situated on top of the liquid’s surface generated from an opening in themiddle chamber. For easier control of this level there is an overflow, leading thesuspension away from the headbox, back into the re-circulation system. On boardmachines there is often a tank installed immediately after the exiting overflow,providing further deaeration of the suspension.

After the middle chamber comes the turbulence package. This is a package oftubes with a one-step increase in tube diameter on the outflow side. The generatedturbulence breaks up fibre flocs in the suspension, which in turn makes the smallscale variations (formation) smaller, thereby the wet web more uniform. On the

1.4 Process description 5

outflow side of the turbulence generator is the lip channel. This is where the flow isaccelerated to form the proper jet to match wire speed with necessary rush or drag,i.e., the difference between the wire speed and the jet velocity. The nominal flowcan be controlled through changing the lip opening in order to change grammage.

From now on the air-cushion damped hydraulic headbox can also be referredto as hydraulic headbox or simply headbox.

1.4.4 Hose damped hydraulic headbox

Apart from the air-cushion damped hydrulic headbox the fourdrinier former alsoemploys another type of headbox, namely the hose damped hydraulic headbox.Instead of an air-cushion acting as the passive damper this headbox has rubberhoses transferring the flow from the manifold directly to the turbulence package,cf. balloon 1, 2 and 3 in Figure 1.4. The attenuating effect is generated from theexpansion of the hoses. Hereby the whole system becomes much simpler.

Figure 1.4. A hose damped hydraulic headbox. Balloons 1-3 show the rubber hoses,manifold, and the turbulence package.

1.4.5 Headbox control

The need for control of the headbox is obvious. The machine is operated with adistributed control system where machine speed and rush–drag is chosen. Rush–drag is the velocity difference between the headbox outflow jet and the wire. Tobe able to produce a high quality product it is imperative to be able to accuratelycontrol the jet speed. The most common strategy to control the headbox, whichis a non-linear multi-variable system, is with the aid of simple SISO (single inputsingle output) PI-controllers.

6 Introduction

For the time being Metso Automation has different control strategies on thepapermaking headbox and boardmaking one. The main reason for this has beendifferences in the short circulation. The papermaking headbox transfers its overflowthrough a pipe directly connected to the secondary side of the overflow. Since thismay cause air bubbles to enter the stock flow system, the short circulation hasbeen fitted with a deaeration facility. In the boardmaking solution however, dueto lack of deaeration possibilities, the secondary overflow goes to the wire pit via atank, called the overflow tank, cf. Figure 1.5. Common for both designs is that aprimary loop is the one controlling the total pressure in the lip channel cf. balloon4 in Figure 1.5, with the mixing pump as the actuator.

On the board making headbox there are two further loops. The first of themis the primary headbox level cf. balloon 1 in Figure 1.5, with in- and outflow airvalves fitted on top of the air-cushion as actuators. The second of them controlsthe overflow tank level cf. balloon 3 in Figure 1.5, with a valve fitted to the tankoutlet.

The papermaking headbox on the other hand only has one additional loopcontrolling the secondary headbox level cf. balloon 2 in Figure 1.5.

Figure 1.5. Sketch of a hydraulic headbox with an overflow tank attached. Dark colouris pulp. Bright colour above 1, 2 ,and 3 represents air. Balloons 1-4 show the statesprimary level, secondary level, overflow tank level, and total pressure.

Now the question has arisen whether or not the overflow tank can be madeobsolete and if one single control strategy can be adopted in both the paper- andthe boardmaking case.

1.5 Thesis outline 7

1.5 Thesis outline

To present the contents of the thesis in a brief and perspicuously way the chaptersand appendices are given here.

Chapter 2 Theory, fundamental and current theory is presented to consti-tute a base for further discussions.

Chapter 3 Experimental set-up, experiments and simulations performedare here described.

Chapter 4 Results and discussion, revolves around the reasoning basedon measurements and simulated models, together with discussions when re-sults are presented.

Chapter 5 Headbox attenuation characteristics, Separately treats thetask concerning the passive headbox damping characteristics.

Chapter 6 Conclusions, summarizes the results and points out importantremarks.

Appendix A, figures of collected data for black-box model construction.

Appendix B, predictions of the black-box models.

Appendix C, relevant figures of decentralized control concepts.

Appendix D, MPC simulations, together with Matlab code.

8 Introduction

Chapter 2

Theory

In this chapter, theory relevant to the thesis will be presented. All theory given willbe in the discrete time domain. In Section 2.1 important system representations areintroduced. Furthermore, in Section 2.2 system identification is generally treated.The theory concerning control design is concluding this chapter in Section 2.3.

2.1 Fundamental system descriptions

Before any other theoretical discussions two common system representations haveto introduced.

The transfer function describe the relationship between the inputs u(k) and theoutputs y(k) at different samples by introducing the delay operator q

A(q)y(k) = B(q)u(k) (2.1)

A system with numerator polynomial one power less than the denominator polyno-mial implies that a change in the input at time t, does not affect the output untiltime t+1. Such a system, a transfer function from u(k) to y(k), can be representedas

G(q) =B(q)A(q)

=b1q

n−1 + b2qn−2 . . . + bn

qn + 1 + a1qn−1 + . . . + an(2.2)

where q is the delay operator. For a multi variable system with p outputs and minputs, G(q) is a matrix of transfer functions with p rows and m columns.

The state-space representation form contains a system memory, called state,along with the ordinary input and output signals. The state-space form also handlesmulti variable systems without any expansion necessary. For an arbitrary lineartime-invariant system the state-space representation can be given as

x(k + 1) = Ax(k) + Bu(k)y(k) = Cx(k) (2.3)

where x(k), u(k), and y(k) are the state, input, and output vectors at sample k.For further reading on these system descriptions cf. [3].

9

10 Theory

2.2 System identification

System identification is a both analytical and experimental way to build a modelsof systems and is generally treated in [6]. It is mainly carried out when the systemhas such characteristics that it is hard or impossible to physically describe it, or onehas, after conventional physical model construction unknown system parameterswhich have to be determined. Identification which is based on measurements ofinput signals and output signals, can be divided into three different principals ofwhom the second will be more closely studied later on in this chapter.

1. Linear non-parametric model construction through estimation of the impulseresponse and frequency function of the system.

2. Linear parametric model construction through black-box models with esti-mated parameters.

3. Linear parametric model-construction through ”tailor made” models, i.e, acombination of physical model construction and identification.

2.2.1 Properties of input signals

The goal in the process of choosing a proper input-signal is that it should contain agreat variety of frequencies. A good choice, when possible, is to let the input-signalrandomly change between two levels, a so called telegraphic signal. This can beseen as steps in the input-signal with different length of hold-time afterwards, a longhold-time makes it possible for a slow system to show it’s characteristics and viceversa. It is however not always possible to experiment with the process as wanted.Normally one is forced to collect data under normal operational conditions. Thisoften implies that the process is being more or less controlled, which can lead todifficulties collecting proper information from the process.

2.2.2 Data processing

The first decision when collecting data, is at what sampling rate the collectionshall be made. Sampling considerably faster than the system dynamics can lead todata redundancy and little value of information. Sampling essentially slower thanthe system dynamics on the other hand leads to lost information and difficulty indetermining the parameters which describe the fast dynamics. Generally it can besaid that it is better to sample too fast than too slow. Then it is always possibleto resample the collected data when one is more ripe for making such a decision.In practice prefiltering of data is required before picking every n’th sample, at aresampling with the factor n.

As soon as the data has been collected it is advisable to make a plot whereoddities like unexplainable spikes, measurement disruptions, and such can be dis-covered. The data collected often originates from signal measurements relative to acertain level of equilibrium. Hence one should, if not working with absolute signal

2.2 System identification 11

levels, subtract the mean level and possible existing trends from each measuredsignal respectively.

2.2.3 Model structure

Black-box models are often used when one is forced to do the modeling withoutany fundamental physical principles. To create a model of the process there area number of standard linear models available. The idea is to determine a set ofparameters, without any physical interpretation, to adapt the model to the process.This is done by using experimental data.

A linear sampled system can be written

y(t) = G(q, θ)u(t) + H(q, θ)e(t) (2.4)

where G(q, θ) is a rational function with the delay operator q

G(q, θ) =B(q)F (q)

=b1q

−nk + b2q−nk−1 + ... + bnb

q−nk−nb+1

1 + f1q + ... + fnfq−nf

(2.5)

and H(q, θ)e(t) is the term of disturbance, where e(t) is zero mean white noise andH(q, θ) its dynamics. The disturbance model can be written

H(q, θ) =C(q)D(q)

=1 + c1q

−1 + ... + cncq−nc

1 + d1q−1 + ... + dndq−nd

(2.6)

The black-box model above is commonly known as the Box-Jenkins or the BJ-model. A commonly used model is called ARMAX. The physical difference be-tween ARMAX and BJ is that the noise and the input passes through the samedenominator in the ARMAX case, i.e., D = F = A. This gives the ARMAX modelthe following appearance.

y(t|θ) =B(q)A(q)

u(t) +C(q)A(q)

e(t) (2.7)

A special case of ARMAX when C(q) ≡ 1 is called ARX. The identification ap-plied later on in this thesis is based on the multi variable extension of the ARXmodel. The only difference from the SISO ARX is that the different parametersare expanded to matrices with parameters in each element.

Iy(t) + A1y(t − T ) + . . . + Anay(t − naT ) =

B1u(t − nkT ) + . . . + Bnbu(t − (nk + nb − 1)T ) + e(t) (2.8)

The matrices Ai, i = 1, . . . , na represents the parameters for each time delay. Thesecan be gathered into a matrix A(q), where each time delay is represented by a cor-responding polynomial order of the delay operator q. In the same way the matricesBi, i = 1, . . . , nb can be gathered in B(q). The dimension of A(q) and B(q) are p×pand p × m, respectively. The polynomial order of each element in A(q) and B(q)

12 Theory

can be gathered in the matrices na, nb, and nk, with dimensions p × p, p × m andp × m, respectively. p being the number of outputs and m the number of inputs.To illustrate the ARX structure a simple example follows.

Example: The order matrices are the following for a system with 2 outputs and2 inputs

na =[

1 21 0

], nb =

[1 11 2

], nk =

[1 11 2

]

These imply that A(q) and B(q) shall be given as

A(q) =[

y1 + A111 y1q

−1 y1 + A121 y1q

−1 + A122 y1q

−2

y2 + A211 y2q

−1 y2

]

B(q) =[

B111 u1q

−1 B121 u2q

−1

B211 u1q

−1 B221 u2q

−2 + B222 u2q

−3

]

To facilitate for control design based on the state-space representation and tostraightforwardly handle multi variable systems, a parameter estimation based onthe state-space structure can be employed

x(t + 1) = A(θ)x(t) + B(θ)u(t) + K(θ)e(t) (2.9)y(t) = C(θ)x(t) + e(t) (2.10)

The dimensions of A, B, K, and C are n×n, n×m, n× p, and p×n, respectively,where n is the number of states, m the number of inputs, and p the number ofoutputs. There is a clear relation between the polynomial form and the state-spaceform, since (2.9)–(2.10) can be rewritten as

y(t) = C(qI − A)−1Bu(t) + [C(qI − A)−1K + I]e(t) (2.11)

2.2.4 Prediction and model adaptation

The principle of adapting the parametric model to data is by letting the modelpredict the output in the next sample and then minimizing the prediction error.The prediction of the ARX model is based on old input signals and old outputsignals. The unknown disturbance e(t) is predicted with its expected value. Sincee(t) is white noise with zero mean, the expected value is zero. The prediction inthe general case then becomes

y(t|θ) = [1 − H−1(q, θ)]y(t) + H−1(q, θ)G(q, θ)u(t) (2.12)

This prediction is generally a rather complex function. In the ARX case the pre-dictor simplifies to a linear regression

y(t|θ) = θT ϕ(t) (2.13)

2.3 Control design 13

where θ is a column vector consisting of the unknown parameters

θ = [a1 a2 . . . anab1 b2 . . . bnb

]T

and ϕ(t) is a column vector generated by old in- and outputs

ϕ(t) = [−y(t − 1) . . . − y(t − na) u(t − nk) . . . u(t − nk − nb + 1)]T

The prediction error is achieved by the following subtraction

ε(t, θ) = y(t) − y(t|θ) (2.14)

The parameter vector minimising the sum of the squared prediction error is

θN = arg minθ

VN = arg minθ

1N

N∑t=1

ε2(t, θ) =

= [1N

N∑t=1

ϕ(t)ϕ(t)T ]−1 1N

N∑t=1

ϕ(t)y(t) (2.15)

The covariance matrix for the estimated parameters θ is given by

C(θN ) = P = E(θN − θ0)(θN − θ0)T (2.16)

where θ0 is the θ assumed to minimise (2.15). The diagonal elements in P representthe variance of the estimated parameters θN . The square root of the variance of aparameter is the standard deviation. The parameter value divided by its standarddeviation constitutes an indication of the reliability of the estimation.

2.2.5 Model validation

Validation of a model is to evaluate if the model is satisfactory. In order to doso one first has to task oneself what constitutes a good model? The concept ofa good model can vary depending on whether one for example wants to use themodel for simulation or to design a controller. In the case of controller design onepreferably wants to keep the parameter order as low as possibly to facilitate thetransfer of parameters from model to controller. As summary it can be said thatit often appears to be a compromise between order and accuracy of the model.However some techniques are available. To see the prediction ability of the modelits predicted data can be compared with the data of a measured set of validationdata. Additionally the perfection of the estimated parameters can be evaluatedby studying the quotient between the parameter value and its standard deviation,which is retrieved from the corresponding element in the covariance matrix.

2.3 Control design

According to [3] the process of developing a control system for a process mayin many cases be both complicated and time demanding. The main issues are

14 Theory

configuration, dimensioning, and structuring. Where the questions asked may bewhat signals to measure and what new transmitters that have to be acquired. Thiswhole process is very much dependent on the experience of the persons involvedand it is difficult to present any systematic fundamental methods.

In this thesis we pay special attention to multi variable systems, since this iswhat the headbox constitutes. The difficulty of multi variable systems lies withinthe cross-couplings between inputs and outputs, if we change an input signal, sev-eral output signals are affected. The stronger the cross-couplings are, the harderit becomes to control the system.

2.3.1 SISO control design

Decentralised control

The simplest approach to control a multi variable system is by ignoring the cross-couplings and handle one circuit at a time as if no other input- and output-signalsare present. Take for example a system with outputs, yj , j = 1, . . . , p and inputsui, i : 1, . . . ,m. The j’th output is controlled to follow its set-point rj by

ui = F irrj − F i

yyj (2.17)

where Fr is referred to as the prefilter and Fy as the feedback filter. An exampleof a feedback filter is the PI-controller

y(t) =q(KP + TS

TI) − KP

q − 1u(t) (2.18)

where KP is the steady-state gain, TS the sampling time, and TI the integrationtime.

To form well performing closed-loop circuits one has to determine the mostsuitable input for each output. This is known as the ”pairing problem”. Sincedecentralised multi variable controllers always are square-matrices, an additionalproblem occurs if the number of inputs and outputs not are the same. One sup-portive tool for solving that complex of problems is the relative gain array, RGAcf. [3] and [12].

RGA

To measure the degree of cross-couplings or interaction in a system one may use therelative gain array (RGA). For an invertible square system A, the RGA is definedas

RGA(A) = A ∗ (A−1)T (2.19)

where * denotes element-by-element multiplication. The deviation of RGA from theidentity matrix can be taken as a measure of the strength of a cross-coupling. Fora clarifying reasoning cf. [3, page 245]. One important property of the RGA is thatpermutations of rows and columns in A, i.e., changes in input and output pairings,


corresponds to the same permutations of rows and columns in the RGA(A). Hencethe ”pairing problem” becomes permuting the rows and columns of A to thatRGA(A) resembles the identity matrix as much as possible.

For an arbitrary system, i.e., both square and non-square, the RGA is definedas

RGA(A) = A ∗ (A†)T (2.20)

where A† represents the pseudo inverse of A, defined as (A∗A)−1A∗ or A∗(AA∗)−1,provided that the indicated inverse exists. For the case of many canditate mea-sured outputs, i.e., a system having more rows than columns, one may consider analternative RGA methodology, consisting of two parts. Firstly a subset of outputshas to be selected to form a square system. Secondly a proper pairing of the squaresystem has to be done according to the discussion in the previous paragraph.

To form the subset of outputs one may consider not using those outputs cor-responding to rows in the RGA where the row-sum is smaller than 1, a proof ofthis can be found in [12]. Although the RGA is an efficient screening tool, it mustbe used with some caution. Since the result of the row-sum analysis sometimescontradicts the discussion around making the RGA resembling the identity matrixas much as possible.

Decoupled control

To further enhance the performance of a decentralised controller a decoupling ma-trix W1 can be constructed. This modifies the original decentralised diagonal con-troller Fy to the decoupled controller Fy with feedback in those elements accordingto the linear combination with W1

Fy = W1Fy (2.21)

W1 is according to [3] commonly chosen as the inverse of the steady-state or cut-offfrequency response of the system.

2.3.2 MIMO control design

Model Predictive Control (MPC)

MPC or Model predictive control is a family name for a group of controllers withslightly different methodology. Fundamental and common for all are that theyemploy the ability of process models to predict future outputs from known inputs.The control law generating the inputs is the result from an optimized objectivefunction subject to various constrains. Some of the advantages of the MPC isthat it explicitly handles input constrains and it is straightforwardly applicableon multi-variable systems. Additionally there are almost unlimited possibilities toexpand the basic algorithm by introducing additional constraints or by modifyingthe criterion to be optimized, cf. [7] and [1].

16 Theory

The models utilized for MPC will in this thesis be on the state-space form,cf. [8]

x(k + 1) = Ax(k) + Bu(k)y(k) = Cx(k) (2.22)

Additionally it will be assumed that C = I, i.e., no state estimation is necessary.If not, an observer, e.g., a Kalman filter must be used to estimate the states. Thek-step ahead prediction, based on the state-space model, is given by

y(t + k|t) = Cx(t + k|t) = C[Akx(t) +M∑i=1

Ai−1Bu(t + k − i|t)] (2.23)

The prediction length of the outputs, here referred to as the horizon M , shall beset to cover the transient time for the system. The input horizon L is often set tobe lower, since it affects the complexity of the optimization problem. On a vectorform future outputs can be written as

y(k + 1)y(k + 2)

...y(k + M)

= Hx(k) + S

u(k)u(k + 1)

...u(k + L − 1)

(2.24)

where S and H are given by

H =

CACA2

...CAM

, S =

CB 0 . . . 0CAB CB . . . 0

......

. . ....

CAM−1B CAM−2B . . . CAM−LB

(2.25)

We are now ready to formulate the criterion to be minimized

M∑j=1

||y(k + j) − r(k + j)||2Q1+

L∑i=1

||u(k + i) − u(k + i − 1)||2Q2(2.26)

where set-point tracking is achieved by the difference between the output y andthe reference value r in the first term. Integral effect is introduced in the secondterm, by the difference between the current input u and the one sample older u.Q1 and Q2 are weighting matrices for the first and second term, respectively

Q1 =

Q111

Q221

. . .QMM

1

, Q2 =

Q112

Q222

. . .QLL

2

(2.27)


To be able to write the difference between the inputs in (2.26) on vector form weintroduce the matrix Ω and the vector δ consisting of old inputs

u(k) − u(k − 1)u(k + 1) − u(k)

...u(k + L) − u(k + L − 1)

=

I−I I

. . . . . .−I I

U −

u(k − 1)0...0

= ΩU − δ (2.28)

The whole criterion on vector form is given by

V (U) = (Hx(k) + SU −R)T Q1(Hx(k) + Su−R) + (ΩU + δ)T Q2(ΩU + δ) (2.29)

Since we do not introduce any constrains the minimization problem

minU

V (U) (2.30)

can be solved analytically by grouping together all in front of U raised to the secondpower in F , and all in front of U raised to the first power in 2GT

minU

V (U) = UT (ST Q1S + ΩT Q2Ω︸︷︷︸F

)U + 2(ST QT1 Hx(k) − ST QT

1 R + ΩT Q2δ︸︷︷︸GT

)U

The U∗ minimising the criterion is the U making the gradient zero

∂(UT FU + 2GT U)∂U

= 0⇔

2FU + 2GT = 0⇔

U∗ = −F−1GT (2.31)

Thus the control law U∗ becomes

U∗ = −(ST Q1S + ΩT Q2Ω)−1(ST QT1 Hx(k) − ST QT

1 R + ΩT Q2δ) (2.32)

If constraints on inputs and outputs were to be introduced, the optimisation prob-lem becomes more difficult and would have to be solved at each sample, cf. [1].

18 Theory

Chapter 3

Experimental set-up

In this chapter practical parts and procedures of this thesis will be explained. InSection 3.1 the software employed is presented. Section 3.2 goes through the datacollection experiment and practical identification procedure. Finally in Section 3.3a detailed description of the trials for control design evaluation is given.

3.1 Software environment

The work in this thesis has mainly been carried out with the aid of computerprograms. Process simulations were made in Matlab. Identification in the Sys-tem Identification Toolbox and control design and control simulations in Matlab’stoolbox Simulink.

3.2 Identification

In this section details concerning the identification experiment will be presented,e.g., data logging equipment, definition of outputs and inputs, and trial description.

3.2.1 Data collection

The data collection trials on the pilot machine at Metso Paper Karlstad were carriedout in order to build an identification based model of the system. The necessarymeasurements on the headbox were supplied from pressure and pressure differencetransmitters to measure pressure in the lip channel and primary, secondary andoverflow tank level. Flow measurements were collected with an induction basedtransmitter. All these five measurements were ranged between 4-20 mA and con-nected to the distributed control system. The input signals were the commandedpump speed and valve opening in percentage. These signals were also collectedin the cross-coupling room. Logging was done with an eight channel Intab logger,cf. [5].

19

20 Experimental set-up

3.2.2 Signals

The signals that were of interest during the data collection, and hence logged, cf.Figure 3.1, are listed below.

• u1 =pump-speed [%]

• u2 =air-valve [%]

• u3 =overflow-valve [%]

and the corresponding outputs. Of course there are several unknown cross couplingsbetween them.

• y1 =primary level [mm]

• y2 =secondary level [mm]

• y3 =overflow tank level (when used) [mm]

• y4 =total pressure [kPa]

• y5 =flow through overflow valve [litre/min]

where u represents inputs and y represents outputs of the system, cf. Figure 3.1.The last output, y5, is not used in either of the two control concepts today, butmay prove to be a good alternative, hence it will also be logged.

3.2.3 Data collection trial description

To receive information from the system, trials were carried in order to collect data.The headbox construction was modified to be able to generate the two cases, withand without overflow tank. With simple valve adjustments the overflow tank couldbe re- and disconnected, i.e., flow could go left via the overflow tank, Case I, orright via the simple piping, Case II, cf. Figure 3.1.

Step changes were made as randomly as possible in the three input signals, oneat a time and some at the same time. It should be noticed that data collection wasonly carried out at one working point.

3.2.4 Sample interval

According to [6] the sampling interval should be chosen so that there are aboutfour to eight sample points on the interesting part of the step response slope.Examination of the step response in lip channel pressure, assumed to be the onewith the shortest time constant, showed that a 0.3-second sample interval wouldbe adequate. To be on the safe side the sample rate was set to 0.05 seconds. Sothat a resampling could be made later on.

3.2 Identification 21

Figure 3.1. Sketch of the headbox after trial modifications. Dark colour represents pulpand bright colour represents air. Pointed out are inputs u1-u3 and outpus y1-y5. Primarypulp flow enters headbox at u1 and exits the headbox at y4. Recirculation pulp flow goesback over y1 and y2 and out, via the overflow tank (Case I) or the simple piping (CaseII), to the wire-pit.

3.2.5 Practical ARX identification procedure

To describe what practically has to be done to identify a system model on the ARXstructure, a brief description will be given here.

The ARX structure has a system description on the form

Iy(t) + A1y(t − T ) + . . . + Anay(t − naT ) =

B1u(t − nkT ) + . . . + Bnbu(t − (nk + nb − 1)T ) + e(t) (3.1)

where T is the sample time and A1, . . . , Anaare matrices consisting of the as yet

unknown, desirable set of parameters belonging to y. Each index represents thetime displacement in samples. The matrices A1, . . . , Ana

can be grouped into thematrix A(q), where the delay is represented by the order of the delay operator q.These orders are to be determined and gathered in the matrix na, before given asinput. The B1, . . . , Bnb

matrices, consisting of the parameters belonging to u canalso be grouped into a matrix B(q), consisting of polynomials of the delay operatorq. The order of the polynomial range from the time displacement nk samples tonk + nb − 1. Consequently the matrices na, nb, and nk are to be determined andgiven as inputs.

When so is done the estimation of the parameters follows. The resulting modelis on a transfer function form.

22 Experimental set-up

3.3 Control design evaluation trial description

To be able to evaluate the preferred control algorithms of Case II, i.e., withoutoverflow tank, trials were made. The experiment design was to make steps in thedifferent controlled outputs. The steps were made by changing the correspondingset-point value of that particular closed loop and study the response behavior. Inorder to do so the control algorithms had to be implemented as programs and neces-sary changes had to be made to the distributed control system. Steps were carriedout at three different working points. At jet-speed 500, 700, and 900 m/min. Datacollection was made in the same way as earlier. Sample time was set to 0.2 seconds.Set-points for levels were chosen with respect to the height of the threshold, e.g.,around 90–110. It should be pointed out that primary and secondary level havedifferent zero-heights and can therefore show different values in the measurementsand still be the same in reality. The set-point for flow was taken without furthercalculations or thoughts. This set-point was around 40–50 l/min during the trials.

Chapter 4

Results and discussion

In this chapter the results based on practical work and reasoning are presented.First introduced in Section 4.1 is the identification of the two headbox cases. Thencontrol design of the headboxes follows, with emphasis on the second case, Case II,in Section 4.2. Finally in, Section 4.3 an evaluation of the preferred controller ispresented.

4.1 Headbox identification

From the initial sample interval 0.05 seconds it was determined that a suitablesample time at which identification could be performed would be ts = 0.2 seconds.This is the shortest possible sample time which may be used in the Metso Au-tomation DCS. Both cases, cf. Section 3.2.3, were to be identified using the ARXmodel structure, resulting in transfer functions describing the input-output rela-tions. For the second case an additional identification was performed. This time astate-space model structure was employed to provide a natural system descriptionfor later state-feedback control design, MPC. These all together sum up to threedifferent system models. Worth pointing out is that identification was only to bemade at one working-point, hence the latter control design analysis is only validfor this working-point.

4.1.1 Case I ARX structure

Case I represents the headbox configuration with the overflow tank attached, i.e.,the present configuration on board machines. Identification of this configuration ismainly done to be able to construct and simulate the system in order to comparethat to the reality as a form of verification.

An important thing to point out is that the overflow tank level during the log-ging was unstable. Because of this, steps in the different inputs were manipulatedto avoid the overflow tank level from hitting its limits. In turn this may have leadto a too short maximum hold time after each manipulated step, i.e., the slowest

23

24 Results and discussion

time constants were not allowed to settle. This may have negative effects on thesteady-state gains from the identification.

The first thing that has to be done is to examine the time plots for each signal,plots can be found in Appendix A. This can help preventing incorrect data interferewith the identification. It was found that the air-valves had been in automatic modethe first seconds. Hence that part was removed.

The next thing to do is to determine matrices nk, nb, and na, according toSection 3.2.5. We start with the nk matrix, i.e., time delay from each input toevery output. This is done by carefully examining the plots. Due to lack of impactin the output of total pressure and flow from input air-valves, these time-delaysare difficult to determine. Apart from them (marked with question marks) thefollowing matrix, consisting of the time-delays, may be constructed

nk =

1 8 43 11 811 7 42 ? 59 ? 10

where the first element, 1, represents the time delay in samples from input one tooutput one, for convention see Section 3.2.2. A way to evaluate the reliability ofthe time-delays is to examine the magnitude between the resulting parameters inB(q) divided by its standard deviation. To get a better overview of the magnitudesthey can be gathered in the corresponding matrix

Magn. ≈

20 6 26 10 12 3 78 < 1 < 13 < 1 7

where high magnitude can to some extent ensure a reliable and correct input-outputtime delay. The time delays marked < 1 are clearly not satisfying and could notbe determined from the time-plots. An alternative method is to raise the ordernb to 10 and set nk to 1. Now it is possible to exclude a nk value with very lowmagnitude between the resulting parameter value and its standard deviation. Thisway we can close in on a good nk, as well as nb, by successively decreasing the nb

number each time we increase the nk value. The resulting nk matrix then becomes

nk =

1 8 43 11 811 7 42 4 59 3 10

Now we can go on to determine the nb matrix. To do so we set the orderto 2 except for those two already determined. Where the magnitude between

4.1 Headbox identification 25

the resulting parameter value divided by its standard deviation is low we simplydecrease the order to one. Where not we keep the order at 2.

To determine the na matrix, corresponding to the polynomial order of theelements in the A(q) matrix, we first set all elements in the na matrix to 4. Wemake the model and study how it validates. It seems as no order higher than 2is necessary for outputs one, two, and five. For outputs three and four order 1 issufficient. This is preferable since a lower order is less complex.

All matrices which are to given to Matlab as inputs are now determined

na =

2 2 2 2 22 2 2 2 21 1 1 1 11 1 1 1 12 2 2 2 2

nb =

1 1 12 1 22 1 12 2 21 2 2

nk =

1 8 43 11 811 7 42 4 59 3 10

When the order of the model is determined, the software estimates the parameters,i.e., the black-box model.

To examine model’s performance it can be set to either predict, or to simulatethe outputs. We choose to let the model predict the outputs and set the predictionhorizon to 50 samples, corresponding to 10 seconds with the current sample interval.As shown in Figure 4.1, the proposed model shows good ability to predict an output.Worth noticing are the spikes in the validation data, which are caused by some sortof measurement disruption. All output predictions can be found in Appendix B.The choice of model is based on the compromise between its ability to fit thevalidation and its parameter complexity.

1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

−10

−5

0

5

10

15

Time

Measured and 50 step predicted output, primary level

Figure 4.1. Measurement and 50 step prediction of output primary level (y1) generatedby the Case I model.

To examine the dynamic behavior of the produced system model, we study thestep responses for each transfer function. The first output, primary-level shows


the same behavior as expected, cf. Figure 4.2. The step response from inputpump-speed (u1) shows an expected impulse shaped response. The responses fromair-valve (u2) seem to show a non-minimum phase behavior. This may be derivedfrom the fact that one of the two air valves was significantly slower than the other.

0 20 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

To: Y

1

From: U1

0 20 40−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

From: U2

seconds0 50 100

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

From: U3

Figure 4.2. Step responses from the Case I model. Output primary level (y1).

From the step responses for the second output, secondary level, seen in Fig-ure 4.3, one can see that the response from input air valve (u2) also shows a non-minimum phase behavior. This may also be the result from the non-synchronisedair valves.

0 50 100−1

0

1

2

3

4

5

6

7

8

9

From: U1

To: Y

2

0 2000 4000−0.2

0

0.2

0.4

0.6

0.8

1

1.2

seconds

From: U2

0 200 400−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

From: U3

Figure 4.3. Step responses from the Case I model. Output secondary level (y2).


From the Figure 4.4 it can be seen that the time constants for the outputoverflow tank level are significantly longer. Interesting is also that the step responsefrom overflow valve (u3) shows a non-minimum phase behavior and that the steady-state gain is positive and remarkably low. It does seem strange that an opening ofthe overflow valve would increase the overflow tank level. One cause for this may bethat the inadequate hold time during the data collection, leads to inability for theslow time constants to settle properly. The other cause can be of physical nature.This behavior has been detected in other studies made by Metso Automation.Hence it is hard to determine whether the step responses of the output overflowtank correspond to the real process behavior or not.

0 2000 40000

100

200

300

400

500

600

700

800

To: Y

3

From: U1

0 2000 40000

100

200

300

400

500

600

From: U2

seconds0 2000 4000

−6

−4

−2

0

2

4

6

8

10

From: U3

Figure 4.4. Step responses from the Case I model. Output overflow tank level (y3).

The fourth output, pressure, seen in Figure 4.5, shows expected behavior. Thestep response from input pump-speed (u1) is a first order system. The character-istics of the other responses are more difficult to derive. Since their steady-stategains are of such a small magnitude, their impact on total pressure are negligibleand hence need no further attention.

Finally we take a look at the fifth and last system output, flow, cf. Figure 4.6.The step responses show simple second order behaviors. One explanation for thelower gain for the air valve (u2) response might be that even though an opening inthe air valve raises the primary and secondary levels it also decreases the differencebetween the pressure above and below the overflow tank volume.

As a complement to the step response figures a frequency response matrix atsteady state can be constructed. Again we can point out that the most remarkablething is the sign and the magnitude of the gain from overflow valve to overflow


0 100 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

From: U1

To: Y

4

0 100 200−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

seconds

From: U2

0 100 200−0.016

−0.014

−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

From: U3

Figure 4.5. Step responses from the Case I model. Output total pressure (y4).

0 100 200 300−0.5

0

0.5

1

1.5

2

2.5

3

3.5

From: U1

To: Y

5

0 100 200 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

From: U2

seconds0 100 200

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

From: U3

Figure 4.6. Step responses from the Case I model. Output flow (y5).

tank level, element (3, 3) in matrix GI(0).

GI(0) =

0.1029 0.1579 0.41090.9564 0.1747 1.6980

718.4654 506.2871 8.69390.8476 −0.1262 −0.01872.5309 0.7879 4.2175


4.1.2 Case II ARX structure

This case constitutes the headbox configuration without overflow tank. This is theconfiguration currently used on paper machines. The identification is carried out inthe same way as in previous section with the same sample time, ts = 0.2 seconds.Since the overflow tank is excluded, only four outputs are present. Polynomialorders are obtained with the same method as in preceding section.

The Case II model is estimated, based on the measurements found in Ap-pendix A, with the following polynomial order matrices

na =

2 2 2 22 2 2 22 2 2 22 2 2 2

nb =

1 1 11 1 11 2 11 2 1

nk =

2 7 14 8 33 16 714 16 4

A quick validation through comparing predicted outputs from the model to mea-sured data shows a satisfying prediction with the horizon set to 50 samples, cf.Appendix B.

Although the prediction of the model is satisfying a comparison with a simplermodel with the nk and nb matrices consisting of only ones shows a similar or evenbetter result. This phenomenon might originate from the short sampling timecausing inadequate samples for time delays nk, i.e., the nk parameter values are sowrong so they have little or no impact on the system compared to ones.

Because of this result the decision fell on using the latter of the two models,the simpler one, in the following simulations.

na =

2 2 2 22 2 2 22 2 2 22 2 2 2

nb =

1 1 11 1 11 1 11 1 1

nk =

1 1 11 1 11 1 11 1 1

To examine the model’s dynamics we study the step responses. The first outputto study are the step responses of output primary level, cf. Figure 4.7. It can beseen that the transient time from air valve (u2) is significantly longer than thetransient time from pump speed (u1). The step responses of secondary level, foundin Figure 4.8, show integral behavior which agrees with the real process. The stepresponse from overflow valve (u3) indicates a non-minimum phase behavior. Thiscan be derived from the initial decrease in secondary level, due to valve opening.The following increase in air volume decreases the pressure and the primary levelrises and a new equilibrium for secondary level is established. From Figure 4.9we can see the expected pressure response from input pump speed u1. The initialeffects from an opening are not easily explained, but might depend on the responsein primary level, cf. Figure 4.2. Finally we study the step responses of outputflow. Figure 4.10 shows reasonable step responses, with the exception of the non-minimum phase behavior from input pump speed (u1) , which is difficult to derive.


0 100 200 300−0.5

0

0.5

1

1.5

2

2.5

3

To: Y

1

From: U1

0 200 400 600−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

seconds

From: U2

0 200 400 6000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

From: U3

Figure 4.7. Step responses from the Case II model. Output primary level (y1).

0 100 200 3000

5

10

15

20

25

From: U1

To: Y

2

0 100 200 3000

2

4

6

8

10

12

seconds

From: U2

0 100 200 300−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

From: U3

Figure 4.8. Step responses from the Case II model. Output secondary level (y2).

4.1.3 Case II state-space structure

In order to provide a state-space representation of the system another identificationmethodology has to be used. To identify the model the number of states has tobe determined. It was decided that 8 states were sufficient to create a model withsatisfying prediction capability at sample time 0.2 seconds.

4.2 Headbox control design 31

0 20 40 600

0.2

0.4

0.6

0.8

1

1.2

1.4

From: U1

To: Y

4

0 100 200 300−7

−6

−5

−4

−3

−2

−1

0

From: U2

seconds0 20 40 60

−0.016

−0.014

−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

From: U3

Figure 4.9. Step responses from the Case II model. Output total pressure (y4).

0 200 4000

0.5

1

1.5

2

2.5

From: U1

To: Y

5

0 200 400−2.5

−2

−1.5

−1

−0.5

0

From: U2

seconds0 100 200

0

0.5

1

1.5

2

2.5

3

3.5

4

From: U3

Figure 4.10. Step responses from the Case II model. Output flow (y5).

4.2 Headbox control design

In this section we will design appropriate controllers for the two headbox cases.Case I will be studied in order to achieve additional system model verification,hence the same controller as the one being used today will be employed. For CaseII we will try one SISO controller solution and one MIMO controller solution. TheSISO controller requires a square system. Since Case II has four outputs and threeinputs, one output has to be left out in order to form a square system. The MIMOcontroller is based on the control algorithm called Model Predictive Control (MPC).


4.2.1 Decentralised control Case I

This case is solely studied to obtain additional verification of the identification. Ifthe same output input selection is used as the one on the pilot machine and thesimulation shows same system behavior, this could indeed be an adequate modelverification. The first thing to do is to take the subset of outputs corresponding tothe actual process configuration, i.e., let

• input pump-speed (u1) control total pressure (y4),

• input air-valves (u2) control primary level (y1),

• input overflow-valve (u3) control overflow tank level (y3).

in other terminology create the square system G413I

The system is simulated with feedback over the time discrete controller FI

with similar parameters as used during the data collection. In continues time theycorrespond to gain Kp = 0.4 and integration times TI = 2, 40, and 100 seconds.

FI =

0.41z−0.4z−1 0 00 0.405z−0.4

z−1 00 0 0.402z−0.4

z−1

The methodology is to introduce one feedback at a time, starting with total pres-sure. As soon as the feedback of overflow tank level is introduced, the systembecomes unstable. This same behavior was also observed in the real process duringthe data collection. Hence this can be said to be somewhat of a verification. Thesame problem that occurred during the data collection also occurs in the simulation.

The cause of this problem may be the presence of a non-minimum phase be-havior. A cure for this problem seem to be, to let the PI-controller in the feedbackloop of over flow tank level be negative

FI =

0.41z−0.4z−1 0 00 0.405z−0.4

z−1 00 0 −0.41z+0.4

z−1

This compensates the non-minimum behavior and the system becomes stable enoughto decrease the integration time on the overflow tank level feedback, cf. Figure C.2.

To see how see current pairing matches the RGA requirement, the RGA for thesquare system is created

RGA(G413I (0)) =

0.7031 −0.0005 −0.0031

0.0181 −0.0025 0.3450−3.9143 512.1061 −0.0526

The dissimilarity between the RGA(G413I (0)) and the identity matrix implies, ac-

cording to the RGA methodology, that this system pairing could be difficult tocontrol.

4.2 Headbox control design 33

4.2.2 Decentralised control Case II

Input-output pairing

This case represents the headbox configuration without overflow tank. Since theobjectives are to make the overflow tank obsolete this is the case which we will focuson. There is from Metso Automation a recommendation that the decentralisedcontroller should consist of three circuits. Since decentralised control is based onSISO circuits, disregarding possible cross-couplings, the controller is a diagonalmatrix. Therefore a subset of three outputs must be chosen. The selection ofoutputs may be aided by the RGA-analysis. First we study the RGA and itsrow-sums of the system at steady-state

RGA(GII(0)) =

0.0391 −0.0494 0.18140.0989 0.8353 0.06570.8157 0.0005 0.01850.0463 0.2137 0.7343

row-sum =

0.17110.99990.83470.9943

According to the theory we should choose the three outputs with their row-sumsclosest to one. This implies that row two, three, and four, corresponding outputssecondary level, total pressure and flow, or y2, y4, and y5, should be chosen, cf. 3.2.2.Hence these outputs manipulated with the on beforehand given inputs form thesquare matrix G245

II . To ensure controllability RGA(G245II ) is studied.

RGA(G245II (0)) =

0.0177 0.9083 0.0740

0.9771 0.0006 0.02240.0052 0.0912 0.9036

Here we can see that this constellation of input output pairing does not fulfill theRGA controllability requirement, to have its diagonal elements as close to one aspossible. Hence we change row one and row two and obtain G425

II instead of G245II

RGA(G425II (0)) =

0.9771 0.0006 0.0224

0.0177 0.9083 0.07400.0052 0.0912 0.9036

fulfilling the RGA requirements better. This configuration corresponds to letting


• input air-valves (u2) control secondary level (y2),

• input overflow-valve (u3) control flow (y5).

To find out if this appearance is the same at higher frequencies the procedure iscarried out at ω = 15 rad/s a little below the Nyquist frequency.

(RGA(GII(eiω))) =

0.2467 0.2498 0.24840.1893 0.3109 0.16170.5497 0.3694 0.05240.0026 0.0580 0.5356

(row-sum) =

0.74490.66190.97140.5962


The by RGA recommended outputs have now changed. Since there is no pointcontrolling both the primary and secondary level and that the difference betweenthem, see row-sum, is small, we keep the previous recommended pairing. Thus thechoice of input-output pairing becomes letting


• input air-valves (u2) control secondary level (y2),

• input overflow-valve (u3) control flow (y5).

corresponding to system model G425II .

Control simulations

To be able to test the decentralised controller’s ability to control this pairing weconstruct the diagonal time discrete controller

FII =

0.48z−0.4z−1 0 00 0.402z−0.4

z−1 00 0 0.402z−0.4

z−1

(4.1)

and let it control G425II . Figure C.4 shows the different simulated outputs. The

length of the transient states are increasing seen from output total pressure tooutput flow.

To try to improve performance of this control design we can introduce a decou-pling matrix. It can be constructed as the inverse of the steady-state gain matrixof G425

II

W1 = G425II (0)−1 =

0.9340 0.0008 0.0036

−1.8456 0.0844 −0.0422−1.4197 0.0483 0.2410

(4.2)

Simulations with W1 show an increase in transient state time for secondary levelbut a decrease in transient time for flow, cf. Figure C.5. Since secondary level hashigher priority, this decoupling does not yield in any improved performance and istherefore discarded.

An interesting question now arises. Is it necessary to control flow? The con-trol design in the papermaking headbox only had two loops, cf. Section 1.4.5, thetotal pressure and the secondary level. To liken the papermaking control designwe simulate the G425

II system with feedback over FII , cf. Figure C.6. But, insteadof letting the third input, overflow valve, be in automatic mode we set it to 0 andstudy the step response behavior. With reference values set to 1 on total pressureand secondary level the flow seems to behave stable and settle at 6, cf. Figure C.7.

Question: Is the flow control loop in Case II necessary?

In order to decide if this third loop is necessary or not we will have to exam-ine whether or not the overflow valve positions are same with identical set-points

4.3 Control design evaluation trials 35

at different working points. If the valve positions are different it implies that anautomatic control is preferable and vice versa. We revisit this matter in Section 4.3by studying collected data.

4.2.3 Model predictive control of Case II

To construct a MPC controller according to the theory in Section 2.3.2, i.e., theoptimization criterion remains the same, i.e., ability to follow a reference signal,integral action, and no constrains on the input signals.

First we have to create matrices S and H. Further the Ω matrix formingthe subtraction from one sample old input signal needs to be implemented alongwith the weighting matrices Q1 and Q2, cf. D.3. The analytical solution in (2.32) isemployed to form the control law. Horizons are set to the same value, since only off-line calculations are necessary, 40 samples to be sure to cover all system transients.The complete code for the MPC implementation can be found in Appendix D.3.

To examine the performance of this controller we employ our state-space modelof the headbox and create a circuit diagram in Simulink, cf. Figure D.1. Todecrease the necessary effort of simulating the system we also assume that allstates are measurable. This implies that we do not have to involve any kind ofobserver. Set-points for the controlled outputs secondary level, total pressure, andflow are two, three, and four, respectively. In Figure D.2 we can the see smoothstep responses, without internal interference. Even though output total pressure isslow, this performance can be considered improved with respect to the precedingSISO controller.

4.3 Control design evaluation trials

Because of limited, time an implementation of the MPC controller was not possible.Instead evaluation trials were made on the SISO PI-controller concept with machineconfiguration and input output pairing according to G425

II .The first working point at 500 m/min showed stability in all controlled vari-

ables. A change in secondary level can be seen in Figure 4.11. Worth noticing isthat the change in secondary level does not affect the primary level, which is good.The next step change was a pressure change. Figure 4.12 shows the disturbancein the primary and secondary levels. Note that the primary level, even thoughuncontrolled, shows a shorter disturbance transient than secondary level. A de-creasing step change in the last controlled variable, flow, seems to affect primarylevel, seen in the slight dip in Figure 4.13. At 900 m/min an expected event takesplace. After a step change in total pressure the primary and secondary levels keepincreasing. As seen in Figure 4.14, the actuator input gain seems to be to low,resulting in the input hitting its lower saturation at 0 percent. This problem canbe derived from the pressure reduction valve fitted to the air feed system before theactuator air-valves. With increasing headbox pressure this valve has to be turnedup in order to achieve higher available air feed pressure.


1000 1200 1400 1600 1800 2000 2200 2400104

106

108

110

112

114

116

118

120

122

s

Primary level

Secondary level

Figure 4.11. Plot from measurements from the control design evaluation trials at work-ing point 500 m/min. The figure shows a step change in output secondary level (y2) andthe unaffected uncontrolled primary level (y1).

1600 1650 1700 1750 1800 1850 1900 1950 2000 2050 2100

40

50

60

70

80

90

100

110

120

130

140

s

Secondary level

Primary level

Pump speed

Figure 4.12. Plot from measurements from the control design evaluation trials at work-ing point 500 m/min. Seen in the figure are the effects on output secondary level (y2) andthe uncontrolled primary level (y1) caused by a step change in input pump speed (u1).

All together performance of this control concept is at satisfactory. Worth point-ing out is the short transient state for primary level, which is good, since it is moreimportant to keep the primary level stable than the secondary.

Now it remains to answer the Question given in Section 4.2.2. In order todo that we study output flow and its input overflow valve at 500 m/min and 700m/min. From Figure 4.15 and Figure 4.16 we can see that the input differs eventhough the set-point remains the same at two different working points. Thus itis preferable to have this loop controlled as well in order to make it easier for themachine operator, having one thing less to think of.

The final remark on the evaluation trial results is how the flow set-point shouldbe chosen. One solution can be to make an empirical formula based on the liquid

4.3 Control design evaluation trials 37

2100 2150 2200 2250 2300 2350 2400 2450 2500 2550 2600

30

40

50

60

70

80

90

100

110

s

Primary level Secondary level

Total pressure

Flow

Figure 4.13. Plot from measurements from the control design evaluation trials at work-ing point 500 m/min. Seen in the figure are the effects on output total pressure (y4),secondary level (y2) and uncontrolled primary level (y1), caused by a step change inoutput flow (y5).

4600 4700 4800 4900 5000 5100 5200 5300 5400 5500 5600

0

20

40

60

80

100

120

140

160

180

200

s

Secondary level

Air valve

Total pressure

Pump speed

Figure 4.14. Plot from measurements from the control design evaluation trials at work-ing point 900 m/min. Seen in the figure are the effects on output secondary level (y2),caused by a step change in total pressure (y4). The figure also shows the correspondinginput changes in pump speed (u1) and air valve (u2). It should be noted that input airvalve (u2) hits its lower limit at 0 percent, resulting in the unstable output secondarylevel (y2).

height over the threshold, i.e., the primary level and let it calculate the flow set-point. This way the net flow into and out of the secondary level compartment ofthe headbox becomes zero.


0 500 1000 1500 2000 2500 3000 350035

36

37

38

39

40

41

42

43Flow and overflow valve at 500 rpm

l/min

/ %

samples

Flow

Overflow valve

Figure 4.15. Plot from measurements from the control design evaluation trials at work-ing point 500 m/min. Seen in the figure are output flow (y5) and input overflow valve(u3). Flow set-point is 38 l/min.

0 500 1000 1500 2000 2500 3000 3500 4000 450031

32

33

34

35

36

37

38

39

40

41Flow and overflow valve at 700 rpm

l/min

/ %

samples

Flow

Overflow valve

Figure 4.16. Plot from measurements from the control design evaluation trials at work-ing point 700 m/min. Seen in the figure are output flow (y5) and input overflow valve(u3). Flow set-point is 38 l/min.

Chapter 5

Headbox attenuatingcharacteristics

In order to acquire information about the headbox’s damping capacity with respectto total air volume, frequency analysis was carried out for a number of differenttotal air volumes. The analysis was based on a physical model of the headbox.The study included both headbox cases and one additional, namely a hose dampedhydraulic headbox. This last model was simulated in order to retrieve a validation,since there were earlier frequency analysis results based on measurements availablefor this configuration.

5.1 Experimental set-up

In order to acquire sufficient information about the damping characteristics of theair-cushion in the headbox, a physical model of the process was created in Matlab.

5.1.1 The model

The headbox model was originally created by Pasi Virtanen and then further devel-oped and adapted to the two headbox set-up cases. In the first case the model wasintended to simulate the paper making headbox, i.e., no overflow tank attached. Inthe second case the overflow tank was included to simulate the board making head-box. In both cases initial conditions were set up and the different compartments inthe headbox were modeled as pipe sections. Appropriate differential equations wereset up to model air in- and out-flow, the air volume and pressure and the differentlevels. The pipe sections were then divided into smaller finite elements and theirequations were solved by the characteristic method described in [13]. Boundaryconditions of the fluid system were the manifold, lip opening, and the pipe endat the wire pit. The air systems boundary condition was set to be the constant

39

40 Headbox attenuating characteristics

pressure from the feeding system. The models were considered to be stiff systems,since they contained several different time constants.

5.1.2 Damping simulation’s layout

To be able to see the dampening capacity at different frequencies the boundarycondition at the manifold was set to a pure sinusoid with known amplitude. Thefrequency was gradually increased from 0.5 Hz to 20 Hz. Then the amplitude into the manifold was divided with the pressure amplitude out from the lip channel.This was done for headbox configurations, with and without overflow tank. Fivedifferent air-cushion volumes were used, the original volume scaled as V0, twosmaller 0.2V0 and 0.6V0 and finally two larger, 2.6V0 and 5.8V0.

To validate the model a third configuration was modeled and simulated. It wasthe configuration with a hose damped hydraulic headbox. Frequency analysis wasdone from the model and a comparison was made with earlier frequency analysisbased on measurements.

5.2 Results and discussion

To study the influence of total air-cushion volume on the attenuation characteristicsof the headbox, frequency response analysis of the process simulation model wasemployed, cf. Section 5.1.1. The two headbox configurations, with and withoutoverflow tank were studied. To receive some sort of verification of the model’saccuracy a third configuration, a hose damped hydraulic headbox was studied, cf.Section 1.4.4. The simulation results could be compared to frequency responseanalysis results based on measurements, and thereby form a validation.

5.2.1 Headbox configuration: no overflow tank attached

From Figure 5.1 it is clear that the peak damping capacity, at a certain nominalflow through the headbox, is displaced towards lower frequencies with greater air-cushion volumes. This implies that it is possible to design the air-cushion volumein order to maximise the passive damping ability at a certain frequency.

A comparison between two different nominal flows at a certain volume showsthat the damping capacity is decreased with greater nominal flow through theheadbox at higher frequencies, but remains the same at lower, cf. Figure 5.2. Thisimplies that a higher nominal flow yields in a loss of passive damping ability athigher frquencies.

5.2.2 Headbox configuration: overflow tank attached

The differences between this configuration and the previous are mainly due to alarger total air volume, since the overflow tank is attached. Figure 5.3 shows thatthe trend of displaced peak damping capacity remains.

5.2 Results and discussion 41

0 1 2 3 4 5 6 7 80.2

0.3

0.4

0.5

0.6

0.7

0.8

Hz

Gai

n

0.2V0

0.6V0

V0

2.6V0

5.2V0

Figure 5.1. Gain at different frequencies for different total air volumes at nominal flow180 l/s, no overflow tank attached.

0 2 4 6 8 10 12 14 16 18 203

4

5

6

7

8

9

10

Hz

Qnom=240

Qnom=180

Figure 5.2. Comparison between dampening characteristics at a certain volume at nom-inal flow 180 l/s and 240 l/s, no overflow tank attached. Y-axis does not show absolutedamping ability. It is only the comparison between the to flows that is of interest.

5.2.3 Headbox configuration: hose damped hydraulic head-box

To form a validation, a certain headbox configuration with available measurementswas chosen. The frequency response analysis in Figure 5.4(b), is based on anaverage of multiple measurements. The corresponding graph generated by thesimulations can be found in Figure 5.4(a). Except for the peak at around 12 Hz in


0 1 2 3 4 5 6 70.2

0.3

0.4

0.5

0.6

0.7

0.8

Hz

Gai

n

Figure 5.3. Gain at different frequencies for different total air volumes at nominal flow180 l/s, overflow tank attached. Largest volume is on the left hand side and decreasingaccording to Figure 5.1.

the graph based on measurements (origin unknown) the similarity is striking. Thiscould indeed be an adequate validation of the model.

0 5 10 15 20 25 30 35 40 45 500.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Hz

Gai

n

(a) Model based frequency analysis (b) Measurement based frequency analy-sis

Figure 5.4. Frequency analysis plots for a hose damped headbox configuration. Fig-ure 5.4(a) shows the model based frequency analysis. Figure 5.4(b) shows frequencyanalysis based measurements.

5.2 Results and discussion 43

5.2.4 Active damping characteristics

As a complement to the headbox’s natural tendency to dampen pressure variationswe already introduced automatic pressure control. To verify how well this con-troller dampens the low frequency pressure variations we introduce pure sinusoiddisturbances, cf. Figure 5.5.

Figure 5.5. Block circuit diagram of a controlled system G controlled by F . Referencevalue is r, disturbance w, and output y.

To study the gain at different frequencies from disturbance w on input channelpump speed (u1) to output total pressure (y4), we create the closed loop transferfunction

Gyw = (I + G425II FII)−1G425

II (5.1)

To do a frequency analysis of this transfer function one normally turns to a Bodeplot. Due to numerical problems with the model this turned out to be impossible.Instead, simulations had to be done with reference values set to zero and a puresinusoid disturbance with amplitude set to one. Output amplitude was noted. Tosee the results of the frequency analysis cf. Figure 5.6. From this figure we can seethat there exists a worst case frequency, at around 0.1 Hz.

5.2.5 Summary

The sample rate of the Metso Automation DCS is 5 Hz. Thus only disturbances upto 2.5, i.e., the Nyquist frequency, can be actively dampened by the pressure control.Disturbances with higher frequencies are supposed to be passively dampened by theair-cushion. Hence the optimal dampening capability can be chosen to be slightlyabove the Nyquist frequency. Of course, the optimum shall be placed with respectto the most common disturbance frequencies. This is however not always knownat the time of headbox construction.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

frequency analysis of Gyw

Gai

n

Hz

Figure 5.6. Frequency analysis of the closed loop transfer function from a pressuredisturbance w to output total pressure (y4), Gyw.

Chapter 6

Conclusions

This chapter summarizes the results and points out important remarks. In Sec-tion 6.1 the headbox damping studies are handled and in Section 6.2 the main workconcerning identification and control of the headbox is discussed.

6.1 Headbox attenuating characteristics

The results from the frequency analysis showed that the passive attenuation fromthe air cushion suits to dampen higher frequencies than the active damping achievedfrom the controller. The preferred peak damping ability can be chosen as a con-sequence of the total air-cushion volume. Together they can cooperate to dampenpressure variations in the headbox over a wide range of frequencies.

6.2 Headbox identification and control

The simulations and trials performed with the decentralised controller FII and thepreferred pairing in Case II, G425

II , shows that it is quite functional. From a controlpoint of view the overflow tank can be made obsolete.

A separate study of the significance of controlling the flow in this configurationwas done. It was found that since it is a non-linear process, a different positionof the input signal overflow valve (u3) was necessary at different working pointsto keep the output constant. This implies that this third loop controlling the flowis necessary. According to these results this control concept can be seen as anextension of the control strategy on paper machine headboxes.

One effect on the generality of the results is the fact that both the identificationof Case I and Case II showed non-minimum phase behavior. This is not the case innormal customer headboxes. Thus the results shall be considered with this remarkin mind.

Furthermore it became clear that the system has limitations on the input sig-nals. During the control design evaluation trials, one working point appeared not

45

46 Conclusions

to be controllable with the input signals available. This should also be taken intoconsideration when using the material of this thesis for further study of the headboxcontrol. Additional identifications at several working points may be necessary.

The question concerning the actual performance of the MPC controller remainsunanswered, even though the results from the simulations looked promising. Tofully evaluate this concept proper trials would be necessary. In order to commencesuch trials, the MPC would have to be generalized to be able to handle inputand output limitations. Furthermore it may be so that the MPC needs to employdifferent system models for different working points.

As a summary it can be said that a simple decentralised controller with inputoutput pairing as letting input

• pump speed (u1) control output total pressure (y4),

• air valve (u2) control output secondary level (y2),

• overflow valve (u3) control output flow (y5),

can be adopted on board machine headboxes with configuration according to CaseII, i.e., headbox without overflow tank attached.

Bibliography

[1] E.F. Camacho and C. Bordons. Model predictive control. Springer, London,1999.

[2] C. Fellers, P. Wellmar, and P. Kolseth. Unified symbols for expressing proper-ties of paper. Technical Report P 002, Skogsindustrins Tekniska Forskningsin-stitut, Stockholm, Sweden, Februari 1995.

[3] T. Glad and L. Ljung. Reglerteori : Flervariabla och olinjara metoder. Stu-dentlitteratur, Lund, Sweden, 1997.

[4] J. Gullichsen and H. Paulapuro, editors. Papermaking Science and Technology.Papermaking Science and Technology. Finnish Paper Engineers’ Associationand TAPPI, 2000.

[5] Intab Interface-Teknik AB, Stenkullen, Sweden. AAC-2: 8-channel PC-logger.

[6] L. Ljung and T. Glad. Modellbygge och simulering. Studentlitteratur, Lund,Sweden, 1991.

[7] J. Lofberg. Linear Model Predictive Control : Stability and Robustness. Li-centiate thesis LIU-TEK-LIC-2001:03, Department of Electrical Engineering,Linkoping University, Linkoping, Sweden, 2001.

[8] J. Lofberg. Modellbaserad prediktioinsreglering. Theory for laboratory prepa-ration, January 2002.

[9] C.J. Moen. Basis weight barring i. amplification. Tappi, 60(10):116–119, 1977.

[10] C.J. Moen. Basis weight barring ii. attenuation. Tappi, 60(11):154–158, 1977.

[11] B. Norman and C. Fellers. Pappersteknik. Institutionen for Pappersteknik,Kungliga Tekniska Hogskolan, Stockholm, 3rd edition, 1996.

[12] S. Skogestad and I. Postlethwaite. Multivariable feedback control : analysisand design. Wiley, 1996.

[13] E.B. Wylie, V.L. Streeter, and L. Suo. Fluid transients in systems. PrenticeHall, Englewood Cliffs, N.J, 1993.

47

48 Conclusions

Appendix A

Collected data

0 500 1000 1500 2000

70

80

90

100

110

120

output: prim. level

s

mm

0 500 1000 1500 2000

0

20

40

60

80

output: sec. level

s

mm

0 500 1000 1500 2000

600

800

1000

1200

1400

1600

output: of tank level

s

mm

0 500 1000 1500 2000

25

26

27

28

29

30

output: total pressure

s

kPa

Figure A.1. Collected measurements from identification trial run. Headbox configura-tion Case I. Signals from top to bottom, left to right are; primary level (y1), secondarylevel (y2), overflow tank level, y3), and total pressure (y4).

49

50 Collected data

0 500 1000 1500 2000

50

100

150

200

output: flow

s

litre

/min

0 500 1000 1500 2000

60

61

62

63

64

input: pump speed

s

%

0 500 1000 1500 200040

45

50

55

60input: air valve

s

%

0 500 1000 1500 200030

40

50

60

70input: of valve

s

%

Figure A.2. Collected measurements from identification trial run. Headbox configura-tion Case I. Signals from top to bottom, left to right are; flow (y5), pump speed (u1), airvalve, u2), and over flow valve (u3).

51

0 500 1000 1500 20000

20

40

60

80

100

120

output: prim. level

s

mm

0 500 1000 1500 2000

0

50

100

150

output: sec. level

s

mm

0 500 1000 1500 20000

5

10

15

20

25


kPa

s 0 500 1000 1500 20000

50

100

150

output: flow

s

litre

/min

Figure A.3. Collected measurements from identification trial run. Headbox configura-tion Case II. Signals from top to bottom, left to right are; primary level (y1), secondarylevel (y2), total pressure, y4), and flow (y5).

52 Collected data

0 500 1000 1500 200058

59

60

61

62

63

input: pump speed

s

%

0 500 1000 1500 200040

42

44

46

48

50

input: air valve

s

%

0 500 1000 1500 200030

40

50

60

70input: of valve

s

%

Figure A.4. Collected measurements from identification trial run. Headbox configura-tion Case II. Signals from top to bottom, left to right are; pump speed (u1), air valve(u2), and overflow valve (u3).

Appendix B

Model predictions

1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

−10

−5

0

5

10

15

Time


Figure B.1. 50 samples predicted output by Case I identified ARX model, showingprimary level (y1).

53

54 Model predictions

1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

−20

−10

0

10

20

30

40

Time

Measured and 50 step predicted output, secondary level

Figure B.2. 50 samples predicted output by Case I identified ARX model, showingsecondary level (y2).

1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

−300

−200

−100

0

100

200

300

400

500

600

Time

Measured and 50 step predicted output, overflow tank level

Prediction

Figure B.3. 50 samples predicted output by Case I identified ARX model, showingoverflow tank level (y3).

55

1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

−2

−1.5

−1

−0.5

0

0.5

1

1.5

Time

Measured and 50 step predicted output, total pressure

Figure B.4. 50 samples predicted output by Case I identified ARX model, showing totalpressure (y4).

1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

−40

−20

0

20

40

60

80

100

Time

Measured and 50 step predicted output, flow

Prediction

Figure B.5. 50 samples predicted output by Case I identified ARX model, showing flow(y5).


1100 1200 1300 1400 1500 1600 1700

−8

−6

−4

−2

0

2

4

6

8

Time


Figure B.6. 50 samples predicted output by Case II identified ARX model, showingprimary level (y1).

1100 1200 1300 1400 1500 1600 1700

−40

−20

0

20

40

60

Time

Measured and 50 step predicted output, secondary level

Figure B.7. 50 samples predicted output by Case II identified ARX model, showingsecondary level (y2).

57

1100 1200 1300 1400 1500 1600 1700

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time

Measured and 50 step predicted output, total pressure

Figure B.8. 50 samples predicted output by Case II identified ARX model, showingtotal pressure (y4).

1100 1200 1300 1400 1500 1600 1700

−60

−40

−20

0

20

40

60

Time

Measured and 50 step predicted output

Prediction

Figure B.9. 50 samples predicted output by Case II identified ARX model, showing flow(y5).


Appendix C

SISO simulations

Figure C.1. Simulink circuit diagram for the decentralised control of Case I.

59

60 SISO simulations

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

s


0 100 200 300 400 5000

0.5

1

1.5

2

s

output: primary level

0 100 200 300 400 5000

1

2

3

4

5

6

s


Figure C.2. Simulated outputs of square system G413I with feedback over altered decen-

tralised controller FI . All reference values set to one. From top to bottom, left to right;total pressure (y4), primary level (y1), and overflow tank level (y3).

61

0 10 20 30 40

0

0.5

1

1.5

2

s

output: primary level

10 20 30 40

0

1

2

3

4

5

6

s


Figure C.3. Zoomed outputs, primary level (y1) and overflow tank level (y3) fromFigure C.2.

62 SISO simulations

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4output: total pressure

s 0 50 100 150 2000

2

4

6

8

10output: secondary level

s

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

3.5output: flow

s 0 50 100 150 200−4

−3

−2

−1

0

1inputs

s

Figure C.4. Simulations of square system G425II with feedback over the decentralised

controller FII . All reference values set to one. From top to bottom, left to right; totalpressure (y4), secondary level (y2), flow (y5), and inputs (u1-u3).

63

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2


s 0 100 200 300 400 500−2

0

2

4

6

8

10output:secondary level

s

0 100 200 300 400 500−1

−0.5

0

0.5

1

1.5output: flow

s 0 100 200 300 400 500−5

−4

−3

−2

−1

0

1

2inputs

s

Figure C.5. Simulations of square system G425II with feedback over the decentralised

controller FII and decoupling matrix W1. All reference values set to one. From topto bottom, left to right; total pressure (y4), secondary level (y2), flow (y5), and inputs(u1-u3).

Figure C.6. Simulink circuit diagram for the square system G425II with feedback over the

decentralised controller FII. Input overflow valve (u3) set to zero.

64 SISO simulations

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2


s 0 50 100 150 2000

2

4

6

8

10output: secondary level

s

0 50 100 150 2000

1

2

3

4

5

6

7output: flow

s 0 50 100 150 200−4

−3

−2

−1

0

1inputs

s

Figure C.7. Simulations of square system G425II , according to circuit diagram in Fig-

ure C.6, with feedback over the decentralised controller FII . Reference values of totalpressure (y4) and secondary level (y2) are set to one. Input overflow valve (u3) is ma-nipulated to be constantly zero. From top to bottom, left to right; total pressure (y4),secondary level (y2), flow (y5), and inputs (u1-u3).

Appendix D

MPC simulations

D.1 Simulink circuit diagram

Figure D.1. Simulink circuit diagram for MPC design. L1, L2, and L3 constitute thecontrol law.

65

66 MPC simulations

D.2 Simulation result

0 10 20 30 40 50 60 70 80 90 100−2

0

2

4

6

8

10

12

seconds

Flow

Secondary level

Total pressure

Primary level

Figure D.2. Simulations of Case II state-space model with MPC design. Controlledoutputs are secondary level (y2), total pressure (y4), and Flow (y5). Uncontrolled outputis primary level (y1).

D.3 Implementation of MPC 67

D.3 Implementation of MPC

%n number of states%m number of inputs%p number of outputssampletime=0.2;%HorizonsM=40; %Output horizonN=M; %Input horizonsc=size(C); p=sc(1); [n,m]=size(B);q=0; i=0; k=0; l=0; S=0;S_e=0;%Reference vector for output [one two etc.]Ref_en=[1 2 3 4]; max=length(Ref_en); step=1; for d=1:M*max,

if d==1Ref=Ref_en(step);

elseRef=[Ref;Ref_en(step)];

endstep=step+1;if step > max

step = 1;endend%H-matrixfor i = 1:M,

if i==1H=C*A^i;

elseH=[H;C*A^i];

endend

68 MPC simulations

%S-matrixS_e=0; S(1:N)=0; for k=1:N,

for l=M-1:-1:0,if l==M-1

if M-k-l <0S_e=zeros(d(1),d(2));

elseS_e=C*A^(M-k-l)*B;d=size(S_e);

endelse

if M-k-l <0S_e= [S_e;zeros(d(1),d(2))];

elseS_e=[S_e;C*A^(M-k-l)*B];

endend

endif k==1

S=S_e;else

S=[S S_e];end

end%omega-matrixomega_1=eye(N*m); omega_2=-diag([ones(1,N*m-1)],-m);omega_2=omega_2(1:N*m,1:N*m); omega=omega_1+omega_2;%Q1 matrix with punishment on y-re=0; Q1_en=[0 0.1 5 0.1]; max_Q1=length(Q1_en); step_Q1=1; fore=1:M*max_Q1,

if e==1Q1=Q1_en(step_Q1);

elseQ1=[Q1 Q1_en(step_Q1)];

endstep_Q1=step_Q1+1;if step_Q1 > max_Q1

step_Q1 = 1;endend Q1=diag(Q1);%Q2-matrix punishments on u(k+1)-u(k)d21=1*ones(1,N);d22=1*ones(1,N);d23=1*ones(1,N);

Q2=diag([d21 d22 d23]);

D.3 Implementation of MPC 69

Delay=tf([1],[1 0],sampletime)*diag([ones(1,m)]);%analytical solutionF=S’*Q1*S+omega’*Q2*omega;

L1=-1*F\(S’*Q1*H);L2=-1*F\(S’*Q1);L3=-1*F\(omega’*Q2);L3=L3(1:N*m,1:m);U=eye(m,m);%inputsignals for u(k) shall only be usedU=[U zeros(m,N*m-m)];

Avdelning, Institution Division, Department

Institutionen för Systemteknik 581 83 LINKÖPING

Datum Date 2002-02-20

Språk Language

Rapporttyp Report category

ISBN

Svenska/Swedish X Engelska/English

Licentiatavhandling X Examensarbete

ISRN LITH-ISY-EX-3197-2002

C-uppsats D-uppsats

Serietitel och serienummer Title of series, numbering

ISSN

Övrig rapport ____

URL för elektronisk version http://www.ep.liu.se/exjobb/isy/2002/3197/

Titel Title

Identifiering och reglering av en inloppslåda Identification and Control of a Headbox

Författare Author

Carl Magnus Tjeder

Sammanfattning Abstract The purpose of this thesis is to investigate an alternative control strategy for a multi-variate non-linear process in a paper machine called the headbox. The proposed solution was intended to be able to be adopted on two different headbox types, currently controlled by different concepts. The methodology was to first create black-box models of the two different systems based on measurements, at one working point. Secondly, various control strategies were investigated. A more sophisticated multi-input multi-output controller MPC, or model predictive control, and a less sophisticated one, a single-input single-output, decentralised PI-controller. With help of simulations the performances of the both strategies were tested. Finally, only the decentralised control solution was implemented and evaluated through trial runs on a pilot machine. The main issue regarding the decentralised controller was the input-ouput pairing. Since the multi-variate system had four outputs and only three inputs, analysis had to be made in order to select three of those four, to form a square system. This analysis was based on the relative gain array (RGA). The resulting performance of the decentralised controller showed stability and adequate response times, surpassing the older system and making one component obsolete through the pairing changes. The MPC controller showed even better performance during simulations and shall also be taken into account if further investigatin is possible.

Nyckelord Keyword headbox, identification, control, attenuation, simulation, RGA, MPC

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Identification and Control of a Headbox - DiVA portal18200/FULLTEXT01.pdf · 2006-03-20 · Identification and Control of a Headbox Examensarbete utfört i Reglerteknik vid Tekniska