modeling and temperature control of an industrial furnace941679/fulltext01.pdf · abstract a linear...

Master of Science Thesis in Electrical EngineeringDepartment of Electrical Engineering, Linköping University, 2016

Modeling and temperaturecontrol of an industrialfurnace

Hampus CarlborgHenrik Iredahl

Master of Science Thesis in Electrical Engineering

Modeling and temperature control of an industrial furnace

Hampus CarlborgHenrik Iredahl

LiTH-ISY-EX–16/4978–SE

Supervisor: Urban JohanssonSandvik AB

Fredrik SandbergSandvik AB

Kamiar Radnosratiisy, Linköpings universitet

Examiner: Johan Löfbergisy, Linköpings universitet

Division of Automatic ControlDepartment of Electrical Engineering

Linköping UniversitySE-581 83 Linköping, Sweden

Copyright © 2016 Hampus CarlborgHenrik Iredahl

Abstract

A linear model of an annealing furnace is developed using a black-box systemidentification approach, and used when testing three different control strategiesto improve temperature control. The purpose of the investigation was to see if itwas possible to improve the temperature control while at the same time decreasethe switching frequency of the burners. This will lead to a more efficient processas well as less maintenance, which has both economic and environmental bene-fits.

The estimated model has been used to simulate the furnace with both the ex-isting controller and possible new controllers such as a split range controller anda model predictive controller (MPC). A split range controller is a control strategywhich can be used when more than one control signal affect the output signal,and the control signals have different range. The main advantage with MPC isthat it can take limitations and constraints into account for the controlled pro-cess, and with the use of integer programming, explicitly account for the discreteswitching behavior of the burners.

In simulation both new controllers succeed in decreasing the switching and theMPC also improved the temperature control. This suggest that the control ofthe furnace can be improved by implementing one of the evaluated controllers.

iii

Acknowledgments

First of all we want to thank Sandvik for the opportunity for us to do this thesiswith you. More specifically, we want to thank Urban Johansson and Fredrik Sand-berg at Sandvik for the support during this time.

We want to thank our supervisor Kamiar for the comments on the report. Atremendous thanks to our examiner Johan Löfberg for the advices which madethis thesis and report a lot better.

Finally, we want to thank our families, fellows and friends for these five yearsat Linköpings University.

Linköping, May 2016Hampus Carlborg and Henrik Iredahl

v

Contents

Notation ix

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 System Description 52.1 Furnace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Burner system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Control design theory 113.1 Split range control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Model Predictive control . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Description of the model predictive controller . . . . . . . 143.2.3 Reference Tracking . . . . . . . . . . . . . . . . . . . . . . . 153.2.4 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . 153.2.5 Relaxed Constraints . . . . . . . . . . . . . . . . . . . . . . . 163.2.6 Complete model . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 DMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Model estimation 194.1 Black-box modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Discrete-time state space . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4.1 Simplification and constraints . . . . . . . . . . . . . . . . . 224.5 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.5.1 Cross validation . . . . . . . . . . . . . . . . . . . . . . . . . 234.5.2 Residual analysis . . . . . . . . . . . . . . . . . . . . . . . . 23

vii

viii Contents

4.5.3 k steps prediction . . . . . . . . . . . . . . . . . . . . . . . . 244.6 Final model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.6.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.6.2 Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Control design 375.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2.1 Split range controller . . . . . . . . . . . . . . . . . . . . . . 385.2.2 MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2.3 DMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Conclusion and Future work 51

A Parameters in state space model 55A.1 Values for A-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 56A.2 Values for B-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B Detailed figures of temperature zones 59

C Residuals 63

Bibliography 73

Notation

Abbreviations

Notation Meaning

mpc Model Predictive Controlpid Proportional, Integral, Sifferential (Controller)dmpc Distributed Model Predictive Controlpem Prediction Error Methodmiso Multi Input Single Outputmimo Multi Input Multi Output

ix

1Introduction

The purpose with this thesis was to evaluate the control of the temperature in anannealing furnace, both the existing control and possibly new control strategies.Two main goals were produced, the first one was to reduce how often the burnersswitch between on and off. The life span of the burner can be increased if thefrequency of turning on and off the burners can be decreased. In addition, main-tenance costs will most likely be reduced as well. The second goal was to achievea smoother temperature of the furnace.

To accomplish this the work was divided into two parts, model estimation andcontrol design. The model estimation part entailed development of an accurateand reliable model of the temperature behaviour in the furnace. The model willbe used to compare different control designs and also in the implementation of amodel based control design.

1.1 Background

The work for this master thesis has been performed at Sandvik Materials Technol-ogy, in Sandviken. Sandvik is a global industrial company and Sandvik MaterialsTechnology is the business area which mainly develop and manufacture stainlesssteel products. In the production of steel, furnaces are used for heat treatment.One kind of heat treatment is annealing, where steel bars are heated up and thencooled down fast. The purpose of the process is to get the steel more ductile andreduce the hardness. These properties of the steel are decided by the temperateprofile during the annealing process. Small changes of the temperature can inthe worst case cause an entire batch of product go to waste, which cost a lot ofmoney. If this happens to many times it could lead to the buyers start losingfaith and start searching for other opportunities. Since the steel industry having

1

2 1 Introduction

tough times nowadays is it more important than ever to have cost effective pro-cesses with good profitability.

The problem today is the temperature distribution in the furnace. For some sec-tions in the furnace the temperature is too high even when the burners are off.

Every year the goal is to have maintenance for the whole furnace at a specificweek and nothing during the other weeks. Some of the parts that wear the mostin the furnace are the burners, as every switch between on and off causes wear onthem. By reducing the switching, the maintenance and replacement of burnerscan hopefully be reduced. The big saving is in that the furnace can operate longerwithout interruption since it takes a long time for the furnace to cool down andheat up for the maintenance.

1.2 Related work

There exist several reports about modelling an annealing furnace, but there ap-pear to be no standard, since there are differences between the furnaces in thelayout, the numbers and kinds of burners, kind and shape of the material, andfuel. Those difference make it difficult to use the same approach from other workof modeling furnaces since the properties of the furnaces differ and the assump-tions are not valid. In [8], a comprehensive mathematical model of an annealingfurnace is developed, the model takes both radiation and convective heat transferin consideration for the components in the furnace. The different components inmodel are the flue gas, the insulation and the product (strip). In [7], a 3D finite el-ement model is developed using COMSOL Multiphysics software to calculate the3D temperature distribution from radiative heat transfer. The result are used toimprove a simplified model in 2D. In this annealing furnace is the material alsoa strip. In [1], the developed mathematical model is based radiation heat transferin the furnace. In the furnace is slab reheated and the temperature field insidethe slabs is part of the model. The flue gas in this furnace is non-participatingin this model unlike the flue gas in [8]. The main difference compared to the thework presented in this thesis is our use of black-box modelling, compare to thementioned references where grey/white-box modelling was used.

1.3 Limitations

The data collection was limited since the furnace was used in production. There-fore, it was not possible to specify the input during the data collection and it wasdone during feedback.

It was not possible during the thesis to either add or modify the position of theburners. No investigation of modifying the burners has been done either.

1.4 Method 3

1.4 Method

The first part was to do a pilot study and find related work. This part was per-formed in order to find opportunities for both the modeling and the controllingpart.

This was followed by data collection in the program IBA Analyser which is usedto log the different signals in the furnace. The collected data was imported intoMATLAB to estimate and validate a model of the furnace.

The last part was to take the model into SIMULINK to test different controllers.Three different control strategies were applied, Split range, model predictive con-trol (MPC) and distributed model predictive control (DMPC). The MPC prob-lems were formulated by using the toolbox YALMIP [5] and were solved with thesolver MOSEK [2]. All strategies were then evaluated, in terms of temperatureand numbers of switches for the burners.

1.5 Thesis outline

The thesis is organized as follows:

• Chapter 2 presents the furnace.

• Chapter 3 gives the background to the control design.

• Chapter 4 presents the model estimation.

• Chapter 5 presents the control design and the result of the simulation.

• Chapter 6 presents what can be done in the future based on the results fromthe thesis.

2System Description

The furnace that has been analysed during this thesis is an annealing furnace.Annealing is a process in which the material (steel bars in our case) heats up toa certain temperature to obtain specific material properties and then cools downfast to capture the properties. This type of process is well known and commonfor steel products. Since the temperature profile determines which properties thematerial will have, it is important that the furnace achieves a given reference tem-perature. A too big temperature difference can cause a whole batch of processedmaterial to be wasted. The furnace was divided into three different temperaturezones, where the reference temperature is highest for the zone where the barsexits and lowest where the bars enter the furnace.

The work order specifies how long the bars should be in the furnace and thereference temperature for the batch. A walking beam will push the bar throughthe temperature zones. In connection with the walking beam is used, the doorsopened before / after to take out / in bars from the oven. The material that goesinto the furnace were cylindrical steel bars so the material could rotate in the fur-nace.

For this thesis the furnace for heating up the steel bars has been analysed andnot the cooling process. Since the furnace has been in use during the thesis, allthe data has been collected under real production. This made it impossible toperform any specific tests.

5

6 2 System Description

2.1 Furnace

The furnace is divided into three different temperature zones, where the firstzones had the lowest reference temperature and the last zone had the highestreference temperature. To achieve a larger temperature difference between zone 1and 2 a separator was mounted in the roof between the zones. Every temperaturezone was divided into three sections. It was necessary to divide the zones intosections since the heat flow between sections in the same zone was big and causetemperature differences inside the zones. If the temperature had been the samefor each zone the sections would not have been necessary. Figure 2.1 shows asketch of the furnace. The arrows indicate where the bars enter and exit thefurnace. It can be seen that the bars enter into the furnace through zone 1 westand exit from zone 3 west. Every time a bar enter or exit, a door needs to openedwhich causes a temperature difference for some sections. When the bars enter thefurnace it has the same temperature as the room.

Figure 2.1: This is a sketch of the layout of the furnace seen from above, thearrows indicate where the bars enter and exit, the flames indicates where theburners are placed. The numbers indicate the number of the sections.

The width of the furnace is about 7 meters, length 9.25 m and height 3.3m. The bars are transported horizontally into the west section of zone one, anddepending on the length of the bars it could occupy one to all three sections. Allthe burners for each zone is in line with each other. It would be a too big effortto change the place for the burners so the numbers and the position of them canbe seen as constant. The furnace has 27 compartments, which each can containone bar. For bars with large diameters they cam only be placed in every secondcompartment. The bars are transported to the next compartments by a walkingbeam. Figure 2.2 shows the measured temperature for a batch. It is easy to seethe zig-zag behaviour in the temperature, with a period of roughly 750 seconds,caused when the walking beam are transporting the bars forward.

2.1 Furnace 7

0 500 1000 1500 2000 2500 3000 3500940

960

980

1000

1020

1040

1060

1080

1100

1120Original Temperatures

Tem

pera

ture

(cel

sius

)

Time (seconds)

Z1 WZ1 MZ1 EZ2 WZ2 MZ2 EZ3 WZ3 MZ3 E

Figure 2.2: The measured temperature for a certain order. Note the zig-zagbehaviour in the temperature, with a period of roughly 750 seconds, causedwhen the walking beam are transporting the bars forward.

In sections west and east in zone 1 there is an exhaust for the flue gas whichcaused heat transfer from zone 3 into zone 1. The heat streams have a large im-pact on the temperature in zone 1 which cause the temperature to remain toohigh even when the burners are off in east and west zone 1. For each sectionthere was different numbers of burners, varying from one to four as illustrated inFigure 2.1.

Every section of the furnace had a PI-controller that controls the temperaturein the section. The burners had two modes, being either on or off. The controlsignal from the PI-controller gave a percentage on which capacity the burnersin the section would operate on. The sensor for measuring the temperature inevery section was mounted 0.4 meters from the roof. When a new order with adifferent temperature was requested it could take a couple of hours to set it updepending of how large the temperature difference will be. The temperature ref-erence is ramped slowly to the new level. The main point of the slow change ofthe reference is to ensure that the whole section has the right temperature andnot only close to the sensor.


2.2 Burner system

The burner system had a separate control system consisting of PI-controllers. Thenumber of burners vary between sections, e.g. in zone 1 Mid the PI-control con-trols four burners and in zone 2 West it was only one burner. The controlledburners had two modes either on or off. Since the burners only operated withtwo modes the control signal was converted to a burning time for the burners.The PI-controllers were implemented as series PI-controllers.

F(s) = K(1 + I ∗ 1s

)

The parameters were K = 3.36 and I = 193.46 . The burning time was given by

the cycle time multiplied by the control signal and the numbers of burners in thesection. To minimize the frequency of switching between on and off the burningtime was set to zero if it was less than 5 seconds and to maximum burning time ifit was greater than maximum burning time minus 10 seconds. The burner systemcan be summarized in the following algorithm:

Algorithm for burner system:

1. Wait until a cycle time have passed.

2. Calculate the burn time as the cycle time multiplied withthe controlsignal and the number of burners in the section.

3. If the burn time is less than 5 seconds set it to zero.

4. If the burn time is greater than maximum burn time minus 10 sec-onds set it to maximum burn time.

5. Set the next burner to burn for the burn time.

6. Update next burner, whose burning time shall be calculated the nexttime, to the last burner to have it burn time set.

7. Repeat from step one.

If a section had two burners and the control signal was 50 %, the burnersburned on each half of the time, or if there were two burners with the signal 25% half of the time no burner would burn. Figure 2.3 shows the burner systemfor section 2, which has 4 burners, with the control signal at 25 %, the burnersoperates in series, when the first one is done the next starts and so on. Figure 2.4shows the same section for 60%, the burner are now working more in parallel. Inthe system today none of the burners in zone 1 and 2 east are on since it is alreadyto hot there. There is small delay before the burner starts to burn.

2.2 Burner system 9

0 50 100 150 200 250 300 350Time

-0.50

0.51

1.5Bu

rner

1

0 50 100 150 200 250 300 350Time

-0.50

0.51

1.5

Burn

er 2

0 50 100 150 200 250 300 350Time

-0.50

0.51

1.5

Burn

er 3

0 50 100 150 200 250 300 350Time

-0.50

0.51

1.5

Burn

er 4

Figure 2.3: How the burner system act with an input of 25%, a one representthat the burner is on and a zero that it is off.


0 50 100 150 200 250 300 350Time

-0.50

0.51

1.5Bu

rner

1

0 50 100 150 200 250 300 350Time

-0.50

0.51

1.5

Burn

er 2

0 50 100 150 200 250 300 350Time

-0.50

0.51

1.5

Burn

er 3

0 50 100 150 200 250 300 350Time

-0.50

0.51

1.5

Burn

er 4

Figure 2.4: How the burner system act with an input of 60%, one representthat the burner is on and zero that it is off.

3Control design theory

The purpose of this chapter is to give the background for the different controldesigns that were implemented in this thesis. The different controllers that wereused were split range control, MPC and DMPC.

3.1 Split range control

Split range control is a simple control strategy which can be used when there aremore than one input that controls the same output. The basic idea is to dividethe control signal into different ranges for the input signals. In the case of twoburners, only one will burn in the range 0-50 % and the other is off, and in therange 50-100 % one will burn all the time and the other is burning according tothe control signal. A setup for a system with a split range controller can be seenin Figure 3.1

Burner1

Burner2

PI-Control Split-range

ref

temp%

u1

u2

Figure 3.1: A sketch for the setup of the system with the split range con-troller implemented.

A split range control can decrease frequency the burners turn on and off since

11

12 3 Control design theory

all the burners in a section, except one is burning at 100 or 0 %. The split rangecontrol have two parameters, swap time and cycle time. Swap time decides thetime between the swap of the range for the burners. The burners have differentpositions inside the section, and therefore affect the temperature in its own sec-tion and nearby sections differently.

The use of swap time is to minimize the effect of the burners position such asuneven temperature within the section. The cycle time is used in the same wayas in the current burner system, i.e., to convert the control signal to a burn timeby multiplying the control signal with the cycle time. An example of the signalfrom the burners can be seen in Figure 3.2. The split range controller can besummarized with the following algorithm:

Algorithm for split range:

1. Wait until it is time to calculate the burn time.

2. If the time that has passed since the last swap is greater than theswap time.

• Change the operating range for the burners.

3. Calculate and set the burn time for each burner.

4. Repeat from step one.

3.2 Model Predictive control 13

0 50 100 150 200 250 300 350−0.5

00.5

11.5

Burn

er 1

Time

0 50 100 150 200 250 300 350−0.5

00.5

11.5

Burn

er 2

Time

0 50 100 150 200 250 300 350−0.5

00.5

11.5

Burn

er 3

Time

0 50 100 150 200 250 300 350−0.5

00.5

11.5

Burn

er 4

Time

Figure 3.2: This shows how the burners act when the split range controllerhave an input signal of 60 %, 1 represent that the burner is on and 0 that isoff.

3.2 Model Predictive control

In this section the MPC is introduced. We first present the background and a ba-sic MPC formulation. An advantage of the MPC is its flexibility and that the basicformulation can be extended to include reference tracking, feedforward controland relaxed constraints, as described. For further reading see [6].

3.2.1 Background

Model predictive control is based on solving an optimization problem for eachsample instant. The solution to the optimization problem is a sequence of controlsignals to be used now and in the future. Of the sequence of control signals,only the first is applied to the controlled system and the rest is discarded. Thisprocedure is repeated for each sample instant and achieves a feedback controlas the optimization problem is performed when new measurements are madeavailable. If the whole sequence would be applied it would be open loop control.The model of the controlled system is a part of the optimization problem and isused to predict the future states. The MPC can be seen as it performing an openloop optimization for each sample instant.


3.2.2 Description of the model predictive controller

This section will introduce the model predictive controller with a theoretical ex-ample. The objective for the MPC is to bring the system states and the input tozero, subject to the system dynamics and limitations. In the basic setup the costfunction is quadratic and the constraints are linear.

minimizeu( · )

N∑j=1

||x(k + j)||2Q1+ ||u(k + j − 1)||2Q2

subject to

x(k + i) = Ax(k + i) + Bu(k + i)

xmin ≤ xi ≤ xmaxumin ≤ ui ≤ umax

(3.1)

Here N represents the prediction horizon which determine how many steps for-ward the controller predicts. The norm ||x||2Q is the euclidean norm weighted bythe weight matrix Q, xTQx. The difference between the weight matrices Q1 andQ2 determinate the behaviour for the controller. If Q1 is large in comparison toQ2 will the system states will converge to zero fast, at the cost of large controlinputs. If Q1 is small in comparison to Q2 will the control inputs be small, at thecost of that, the system states will converge to zero slow.

The equality constraints represent the system dynamics from the model of thecontrolled system. The inequality constraints are bounds for the system statesand/or the control signal, which are typically for safety regulations or saturation.

In summary the MPC can be explained by the following algorithm:

Algorithm for MPC:

1. Measure the system state x(k)

2. Compute the control signal sequence u( · ) for the problem in (3.1)

3. Use the first element u(k) in the sequence from the previous stepduring one sample

4. Time update, k = k + 1

5. Repeat from step 1

3.2 Model Predictive control 15

3.2.3 Reference Tracking

In the formulation in (3.1) the system will be driven to a steady state at the origin.In order to steer the system to any other state than the origin, the MPC formula-tion must be extended to include reference tracking. This can partially be accom-plished by minimizing the difference between a state and reference. This gives aa conflict in the MPC formulation which is to have the states follow a referenceand try to keep the input zero. One solution is to minimize the increment of theinput signals instead of its amplitude. The reformulation (3.2) gives the MPCcontroller reference tracking.

minimizeu( · )

N∑j=1

||x(k + j) − r(k + j)||2Q1+ ||u(k + j − 1) − u(k + j − 2)||2Q2

subject to

x(k + i) = Ax(k + i) + Bu(k + i)


(3.2)

3.2.4 Feedforward Control

In feedforward control the disturbance is measured and with used in the controlcomputation to remove some of the disturbance effect on the system. It thus hasan advantage compared to feedback control which has to wait for the disturbanceto effect the system before it can take suitable action. The effectiveness of feedfor-ward control is limited by the model of the measurable disturbance to the output,and any remaining unmeasurable disturbances. Therefore feedforward controlis implemented together with feedback control. The feedforward control part re-moves most of the measurable disturbance effect and feedback control take careof the rest and also any unmeasured disturbance. Feedforward control is imple-mented in MPC by including the disturbance effect in the predicted future out-puts, which can be seen in (3.3). The optimizer then accounts for the disturbancewhen it computes the control signal.


minimizeu( · )

N∑j=1

||x(k + j)||2Q1+ ||u(k + j − 1)||2Q2

subject to

x(k + i) = Ax(k + i) + Bu(k + i) + Bddm(k + i)


(3.3)

3.2.5 Relaxed Constraints

A serious problem for the MPC is when the optimization problem is infeasible.This can happen because an unexpected large disturbance occurred and it is im-possible for the controller to prevent that a state breaks its constraint. Thereforeit is important that the MPC controller has a strategy to handle infeasible prob-lem. A systematic strategy is to make relaxed constraints which can be broken ifthere is no admissible solution [6]. The constraints on the input signal are usually"hard" and cannot be broken since it commonly represent a physical limitationon the actuator. Therefore it is usually the constraints on the output which arerelaxed.

A constraint is relaxed by introducing a new variable which is non-zero whenthe constraint is violated. Such a variable is called slack variable. To avoid viola-tion of constraints when unnecessary, a penalty term is added to the objective inorder to try to obtain a zero slack. The optimization problem then becomes

minimizeu( · )

N∑j=1

||x(k + j)||2Q1+ ||u(k + j − 1)||2Q2

+ λ|ε|

subject to

x(k + i) = Ax(k + i) + Bu(k + i)

xmin − ε ≤ xi ≤ xmax + ε

umin ≤ ui ≤ umax0 ≤ ε

(3.4)

The reason for not using the euclidean norm on ε in the cost function is thatall active constraint will be violated for all finite λ, even when it is not necessary.From the true constrained x∗ there exist a move to an infeasible solution x∗ + dεfor some vector d which will decrease the cost with O(ε). With the euclidean

3.3 DMPC 17

norm will the cost of violating the constraint be O(ε2) which gives a net reduc-tion in the cost for small ε. With the 1-norm in the cost function a sufficient λwill ensure that violation occur only when the original problem is infeasible.

3.2.6 Complete model

The different modifications of the basic formulation can be used together andgives the following optimization problem, which will be the form used later.

minimizeu( · )

N∑j=1

||x(k + j) − r(k + j)||2Q1+ ||u(k + j − 1) − u(k + j − 2)||2Q2

+ λ|ε|

subject to


xmin − ε ≤ xi ≤ xmax + ε

umin ≤ ui ≤ umax0 ≤ ε

(3.5)

3.3 DMPC

Another way to use the function of MPC is to enlarge the system with more MPCcontrollers. This type of control strategy is called distributed model predictivecontrol. The meaning of DMPC is to split the problem into sub systems, whereevery MPC controller computes a control input locally. This will make it easierfor the solver, as the optimization problems will be smaller. However, when do-ing this, the interaction between the different controllers and parts of the systemswill not be part of the models used locally in every MPC controller. One approx-imation, which we will use, is to assume that the states from other zones will beconstant over the prediction horizon. For further reading about DMPC, see [9].

4Model estimation

This chapter is about the identification and validation of the model for the systemdescribed in Chapter 2. The workflow for the model estimation was an iterativeprocess; it started with a simple model that was extended until no improvementscould be found. The main idea with the model was to keep it simple with rele-vant relations, which capture the dominant behaviour of the system.

4.1 Black-box modelling

There are three different approaches of deriving models; White-box, Grey-boxand Black-box. The White-box model is based only on known information aboutthe system. If all information about the system is known and the model has theright structure, the model and the system will be identical. In the case of the fur-nace much of the information is missing, such as the heat transfer between thesections and heat losses through the walls. It could be possible to model thoserelations based on thermodynamics, but many parameters are unknown such asthe flow of the flue gas and need to be estimated. A White-box model of the fur-nace would be very complex and some relations are hard to describe with physics.For example the temperatures zig-zag behaviour caused by the walking beam.

The advantage with a Black-box model is that it can be used when no informa-tion about the system is known. The Black-box model describes the relationsbetween the input and output signals without having a structure based on phys-ical relations of the system. The model can be in the form of equations, tables orgraphs. A possible problem could be that the model only performs for preciselythose cases for which it was derived.

19

20 4 Model estimation

The Grey-box model can be seen as a hybrid between Black-box and White-boxmodel. The part of the system whose information is known is modelled as aWhite-box and the rest of the system as a Black-box. This model was neglectedwith the same reason as the White-box, it is too hard to model the thermodynam-ics.

The Black-box model was chosen because it is easier to implement and the modelis a tool in this thesis. The possible improvements by using a Grey-box was con-sidered low, as said before a large part of the system is unknown.

4.2 Discrete-time state space

A broad definition of a system is an object or a collection of objects whose prop-erties are observed. A model can be used as a tool to examine the system withoutdoing any experiment on it. To experiment with a new controller in the furnaceis ineffective and will cost money in the lost production. The advantage with amodel is that it gives us the possibility to try different control strategies withoutimplementing them in the furnace.

A common way to represent a system is the model in the following form

x = f (x, u)

y = h(x, u)(4.1)

where u is the input signal, y is the output signal and x is called the state. Theequations describe the whole system’s behaviour. This structure is called statespace. Specializing to the linear case, and discretizing the continuous-time dy-namics using, e.g., a first-order hold, gives us a discrete-time state space model

x(kTs + Ts) = Ax(kTs) + Bu(kTs)y(k) = Cx(kTs) + Du(kTs)

(4.2)

where the matrices A,B,C and D describes how the future state depends on thecurrent state and the input as well as how the output depends on the state andthe input. For the furnace, the states are the temperatures; the input signals arethe burners, the position of the walking beam and the position (open/close) ofthe doors. The output signals are the measurements of the temperatures. Sincethe furnace is an unknown system the objective is to estimate the unknown pa-rameters in the matrices. The furnace itself is a nonlinear system and thereforethe linear model is only a rough approximation of the furnace.

4.3 Data collection 21

4.3 Data collection

The data of the furnace is collected during production. Thus the data was col-lected during feedback. For the furnace there is a logging system called IBAANALYSER which logs all sorts of data. The collection lasted for 1.5 hours witha sampling rate of 5 seconds. There exist different methods of handling data col-lection during feedback such as the direct approach, the indirect approach andthe joint input-output approach [3]. In this thesis, the direct approach was cho-sen where the effect of the feedback is ignored. Since the priority was the controldesign and the model as said before was a tool.

A problem with the data collection was that the temperatures were too high forsome sections. The burners in those sections did not burn, and thus there is no in-formation in the collected data set on the relation between these burners and theresulting temperature. Those burners are still included in the model but theirparameter values are just an approximation. The reason for including approxi-mate values of these parameters anyway was to enable us to use these burners insimulation.

4.4 Parameter estimation

To estimate the parameters for a model one of many methods is the predictionerror method [4]. The vector θ includes the parameters that will be estimated,in our case A,B,C,D . It will be used by the prediction y(t|θ) which depends ony(t-1). The prediction error ε(t, θ) is the difference between the output y(t) andy(t|θ).

ε(t, θ) = y(t) − y(t|θ) (4.3)

The loss function for a data set with N samples is written as

VN (θ) =1N

N∑t=1

ε2(t, θ) (4.4)

θN is given as the parameter which minimizes the loss function

θN = arg minθVN (θ) (4.5)


To perform the prediction error method in MATLAB we used PEM from theSYSTEM IDENTIFICATION TOOLBOX. The input to the function is the estima-tion data from the system and the model with the parameters that should beestimated. P EM has an option on estimation focus which weigh the loss functionfor specified frequencies. Two different focuses were used, prediction and sim-ulation. For prediction has a loss function which minimizes the one step aheadprediction. For simulation which has a loss function that favours the frequencyrange where the input has the most power.

4.4.1 Simplification and constraints

The state space model in (4.2) is a multiple input multiple output (MIMO) systemwhich made the parameter estimation difficult since it was an optimization prob-lem with many parameters. To make it easier to optimize the system was splitinto nine different multiple input single output (MISO) systems, then the MISO-systems were converted back to a MIMO-system. The advantage of convertingit back was that it was easier to implement in SIMULINK and include it in theMPC formulation. The MISO-systems have only one state, which is the temper-ature in its section. The input signal was expanded to include the temperaturesfrom nearby sections. An example is given below for how section one was treatedas a MISO-system.

T1(k + 1) = aT1 + [b1 · · · b7][T2 T4 B1 B2 W D1 D2]T

T1 is the temperature in section 1, T2 in section 2, T4 in section 4, B1 is theburner in section 1, B2 is the burner in section 2 which is closest to section 1, Wis the walking beam, D1 the door in section 1 and D2 the door in section 9. Theparameters for the MISO system were then included into the corresponding placein the A and B-matrix in the MIMO system, which is given in an example below.

A1,1:9 = [a b1 0 b2 0 0 0 0 0 0]

B1,1:20 = [b3 b4 0 · · · · · · 0 b5 b6 b7]

To avoid running into a local minimum, constraints were added to the parametersthat have a physical interpretation. The estimation was first done without theconstraints, but then some parameters were negative for the burners, which hadcaused a temperature decrease when they were on. A burner which is burningcan not decrease the temperature. All the bounds of the parameters can be seenin Table 4.1. The parameters that involve the doors and the walking beam haveno constraints since it is hard to interpret its effect on the temperature. When adoor is opened there can be an increase in heat transfer between two sections, thetemperature in one section will increase and the other will decrease.

4.5 Model Validation 23

Table 4.1: The constraints for the parameters.

Name Lower bound Upper boundTemperature from their own section 0.7 ∞Temperature from burner in section 0.1 ∞Temperature from burner in nearby section 0 0.4

4.5 Model Validation

To validate how good the models were three different methods were used. Thefirst method was cross validation, which gives a percentage of how well the out-put of the model represents the output from real system. Number two was resid-ual analysis, which shows the correlation between input and prediction error andalso the autocorrelation of the prediction error. The third method was the k-stepprediction; it shows how good the model predicts the future for the k steps ahead.

4.5.1 Cross validation

Two thirds of the data was used for estimation while the remaining third wasused for validation. The use of different data in validation and estimation iscalled cross-validation. It is used to prevent overfit where the model would de-scribe the noise instead of underlying relations.

One way to validate a model is to compare its output with the measured inputagainst the measured output. The model fit, in percentage, is given below

Mf = (1 −

∣∣∣∣∣∣ε(t, θ)∣∣∣∣∣∣∣∣∣∣∣∣y(t) − y(t)∣∣∣∣∣∣ ) ∗ 100 (4.6)

ε(t, θ) is the model error from (4.3), y(t) is the measured output and y(t) is itsaverage, || · || is the L2 norm. The model fit is an easy way to compare differentmodels. The model fit is a measure of how well the model represents the realsystem.

4.5.2 Residual analysis

Another way to validate the model is to look at the residuals of the model. Oneresidual is the cross correlation between prediction error and the inputs. In bestcase the input signal would be independent of the prediction error, if not themodel is missing some dynamics from the system.


Rεu(τ) =1N

N∑t=1

ε(t + τ)u(t), |τ | ≤ M (4.7)

The auto correlation of the prediction error is given as,

Rε(τ) =1N

N∑t=1

ε(t)ε(t + τ), |τ | ≤ M (4.8)

If the auto correlation is outside the 99% confidence interval for some τ thenthe output depends on the output τ step before.

4.5.3 k steps prediction

The k-step prediction is useful to test the model predictive power k steps aheadin time. The model output is predicted at k steps ahead based on the informationat time k0, which is given by

Xk0+k+1|k0= AXk0+k|k0

+ BUk0+k , k = 0, 1, ...

Yk0+k|k0= CXk0+k|k0

(4.9)

The predicted output Yk0+k|k0is compared against the measured output Yk0+k the

same way as in the cross validation in (4.6) where ε is replaced. In this thesis apredictive controller is used and therefore it is important that it can predict thesignals and thereby choose the best control signal.

4.6 Final model

In this section the structure of the final model is presented. The A-matrix repre-sents how future state depends on the current state, in this case the temperature.The diagonal in the A-matrix determines how much temperature every sectioncontains from the previous sample, i.e., the decay in temperature given no in-flux or loss of heat to surrounding sections, or addition of heat using burners.The other parameters represent the effect of the temperature from other sections.The final values can be found in Appendix A.1.

4.6 Final model 25

A =

a1,1 0 0 a1,4 0 0 0 0 0a2,1 a2,2 0 0 a2,5 0 0 0 00 a3,2 a3,3 0 0 a3,6 0 0 0a4,1 0 0 a4,4 a4,5 0 a4,7 0 00 a5,2 0 a5,4 a5,5 a5,6 0 a5,8 00 a6,2 a6,3 0 a6,5 a6,6 0 0 a6,90 0 0 a7,4 0 0 a7,7 a7,8 00 0 0 0 a8,5 0 a8,7 a8,8 a8,90 0 0 0 0 a9,6 0 a9,8 a9,9

(4.10)

The B-matrix represents the effect on the temperature from the burners, the doorsand the walking beam. The values are found in Appendix A.2 for the differentmodels.

BT =

b11 0 0 0 0 0 0 0 0b21 b22 0 0 0 0 0 0 00 b32 0 0 0 0 0 0 00 b42 b43 0 0 0 0 0 00 b52 b53 0 0 0 0 0 00 0 b63 0 0 0 0 0 00 0 0 b74 0 0 b77 0 00 0 0 b84 b85 0 0 0 00 0 0 0 b95 0 0 b98 00 0 0 0 b10,6 b10,7 0 b10,8 b10,90 0 0 0 0 b11,6 0 0 00 0 0 0 0 0 b12,7 0 00 0 0 0 0 0 b13,7 b13,8 00 0 0 0 0 0 0 b14,8 00 0 0 0 0 0 0 b15,8 00 0 0 0 0 0 0 b16,8 b16,90 0 0 0 0 0 0 0 b17,9

−b18,1 −b18,2 −b18,3 b18,4 b18,5 0 0 0 0b19,1 −b19,2 0 0 b19,5 0 0 0 0−b20,1 −b20,2 0 −b20,4 −b20,5 0 0 0 0

(4.11)

Since every state could be measured the C-matrix only will apply as a unit ma-trix.


C =

1 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1

(4.12)

4.6.1 Validation

The simulated output and the 12-step ahead output can be seen in figure 4.1-4.6

4.6 Final model 27

200 400 600 800 1000 1200 1400 1600 1800960

965

970

975

980

985

990

995

y1

Validation data (y1)sys1_prediction: 63.18%sys1_simulation: 55.77%

12-Step Predicted Response Comparison

Time (seconds)

Ampl

itude

(a) Validation for zone 1 west. Notice themodels’ smaller amplitude.

200 400 600 800 1000 1200 1400 1600 1800940

945

950

955

960

965

y1



Time (seconds)

Ampl

itude

(b) Validation for zone 1 middle. Notice thedifference between the two models.

200 400 600 800 1000 1200 1400 1600 1800998

1000

1002

1004

1006

1008

1010

1012

1014

1016

y1



Time (seconds)

Ampl

itude

(c) Validation for zone 1 east. The models havethe peaks at the right time, but not the rightamplitude.

Figure 4.1: Validation for zone 1 with 12 steps ahead prediction.


200 400 600 800 1000 1200 1400 1600 18001056

1058

1060

1062

1064

1066

1068

1070

y1



Time (seconds)

Ampl

itude

(a) Validation for zone 2 west. Notice the mod-els’ good performance.

200 400 600 800 1000 1200 1400 1600 18001055

1056

1057

1058

1059

1060

1061

1062

1063

1064

y1



Time (seconds)

Ampl

itude

(b) Validation for zone 2 middle. The modelsmiss some of the peaks.

200 400 600 800 1000 1200 1400 1600 18001063

1064

1065

1066

1067

1068

1069

1070

y1



Time (seconds)

Ampl

itude

(c) Validation for zone 2 east. Notice that themodels have the peaks at the right time, butnot the right amplitude


4.6 Final model 29

200 400 600 800 1000 1200 1400 1600 18001093

1094

1095

1096

1097

1098

1099

1100

1101

y1



Time (seconds)

Ampl

itude

(a) Validation for zone 3 west. Notice the smallvariation of the temperature.

200 400 600 800 1000 1200 1400 1600 18001095

1096

1097

1098

1099

1100

1101

y1



Time (seconds)

Ampl

itude

(b) Validation for zone 3 middle. The modelshave the peaks, but not at the right amplitude.

200 400 600 800 1000 1200 1400 1600 18001094

1095

1096

1097

1098

1099

1100

1101

y1



Time (seconds)

Ampl

itude

(c) Validation for zone 3 east. Notice the simi-larities between the models.



200 400 600 800 1000 1200 1400 1600 1800960

965

970

975

980

985

990

995

y1


Simulated Response Comparison

Time (seconds)

Ampl

itude

(a) Validation for zone 1 west. Notice themodels’ lack of the smaller peaks.

200 400 600 800 1000 1200 1400 1600 1800940

945

950

955

960

965

y1



Time (seconds)

Ampl

itude

(b) Validation for zone 1 middle. Notice howthe models’ performance gets worse over time.

200 400 600 800 1000 1200 1400 1600 1800998

1000

1002

1004

1006

1008

1010

1012

1014

1016

y1



Time (seconds)

Ampl

itude

(c) Validation for zone 1 east. Notice that themodels have the peaks at the right time, butnot the right amplitude

Figure 4.4: Validation for zone 1 with simulation response.

4.6 Final model 31

200 400 600 800 1000 1200 1400 1600 18001056

1058

1060

1062

1064

1066

1068

1070

y1



Time (seconds)

Ampl

itude

(a) Validation for zone 2 west. Notice the mod-els’ good performance.

200 400 600 800 1000 1200 1400 1600 18001055

1056

1057

1058

1059

1060

1061

1062

1063

1064

y1



Time (seconds)

Ampl

itude

(b) Validation for zone 2 middle. The modelslack some of the peaks.

200 400 600 800 1000 1200 1400 1600 18001063

1064

1065

1066

1067

1068

1069

1070

y1



Time (seconds)

Ampl

itude

(c) Validation for zone 2 east. The models havethe peaks, but not at the right amplitude.



200 400 600 800 1000 1200 1400 1600 18001093

1094

1095

1096

1097

1098

1099

1100

1101

y1



Time (seconds)

Ampl

itude

(a) Validation for zone 3 west, notice the smallvariations of the temperatures.

200 400 600 800 1000 1200 1400 1600 18001095

1096

1097

1098

1099

1100

1101

y1



Time (seconds)

Ampl

itude

(b) Validation for zone 3 middle, both modelshave the similar model fit.

200 400 600 800 1000 1200 1400 1600 18001095

1096

1097

1098

1099

1100

1101

y1



Time (seconds)

Ampl

itude

(c) Validation for zone 3 east, the models cap-ture the peaks, but not with the same ampli-tude.


4.7 Discussion 33

4.6.2 Residuals

Below two of the eighteen residuals is presented, since the residuals have almostthe same behavior the rest are found i Appendix C.

0-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

e@y1

AutoCorr

0

XCorr (u1)

0

XCorr (u2)

0

XCorr (u3)

0

XCorr (u4)

0

XCorr (u5)

0

XCorr (u6)

0

XCorr (u7)

0

XCorr (u8)

0

XCorr (u9)

0

XCorr (u10)Residue Correlation

Lag

Ampl

itude

(a) The residual for Zone 1 Middle with pre-diction model.

0-0.2

0

0.2

0.4

0.6

0.8

1

e@y1

AutoCorr

0

XCorr (u1)

0

XCorr (u2)

0

XCorr (u3)

0

XCorr (u4)

0

XCorr (u5)

0

XCorr (u6)

0

XCorr (u7)

0

XCorr (u8)

0

XCorr (u9)

0


Lag

Ampl

itude

(b) The residual for Zone 2 Middle with pre-diction model.

Figure 4.7: The figures show two residuals with estimating from prediction.

4.7 Discussion

It is hard to model the furnace with thermodynamics since it will be a very com-plex model and information about the system is missing. Hence, we chose tomodel the furnace as a Black-box model. One main advantages of a Black-boxmodel is its easy implementation. We believe that the difference between a Black-box and a Grey-box in model quality are small since much information is un-


known. We chose to model the system as a State-space linear model since wethought it would capture the most important aspects of the system. For a MIMOsystem, it can be difficult to determine which inputs that influence which out-puts, especially when the inputs are binary and can be similar. Therefore, toensure that the model had only relevant relations, a simple model was chosen. Arisk with a too advanced model is that it performs better in a specific case, but notin the general case. For the parameter estimation some restrictions were added.The point of this was to ensure reasonable parameters, such as enforcing thatburners only can increase temperature

The effect of the temperature when the walking beam is in use is strange andhard to explain. It causes a zig-zag behaviour in zone 1 but also in zone 2 butof smaller degrees. The peaks can be caused by an increase in the heat radiationfrom below or that the heat flows are changed. We think the drop is caused whenthe walking beam is in the end position and air is leaking in from below.

A large difference between the model and the furnace is that the furnace is anonlinear system. The temperature in the model is smoother and it lacks some ofthe peaks and drops. In general the peaks have smaller amplitude in the model.The heat losses in the model is seen as the diagonal in the A-matrix since A < 1.

As mentioned before the temperature was too high in three sections during thedata collection. We chose to include the burners for those sections in the model,but we could not estimate the parameters for them. With simulation with otherreference temperature or different controller it could be a case when these burn-ers should be on. The parameter values for the burners are approximations basedon the parameters for other burners.

The model fit for some sections were really low for the different models. Somereasons can be that the furnace is a nonlinear system and that the data collectionwas done during feedback. The problem with closed loop data is the correlationbetween the input and the noise, which typically cause bias in the parameter es-timation. For zone 3 it was hard to model the temperatures since it was close tothe given reference temperature and the large part of variation in signal could becaused by noise. Therefore, it was more important to focus on good models ofthe other zones, since those zones have the biggest opportunity to improve thetemperature control. The result from the residual analysis were similar for bothmodels with the worst result in the cross correlation with the burners, it suggeststhat it exists more information about the system from the burners. In the endthere was a small difference between the models and the prediction model waschosen since the model was better for zone 1, as the potential for improvement isgreatest in the zone first.

To improve the model one opportunity could be to include the temperature of thebars in the model. The bars had probably have a large impact on the temperaturein the furnace since they will cause the temperature in the furnace to decrease

4.7 Discussion 35

when they are heated. The temperature of the material during its time in the fur-nace could perhaps be estimated with a Kalman filter. Today, the temperature ofthe material is measured at the end of the furnace. Another improvement couldbe if the model includes the exhausts, as this could give an advantage of how theheat flow is moving from zone 3 to 1. If this information was known, the dif-ference between the Black-box and the Grey-box could be larger, and Grey-boxcould be the better option.

Some other improvements that will be harder to integrate could be to add moretemperature sensors. This will give more information for the estimation. If somesensors are mounted between the zones it could make it easier to estimate a betterheat exchange between the zones. With more sensors in each section, it will givea better picture of temperature in it. The sensor configuration used today doesprobably not give a complete picture of the whole section. Another way couldbe to perform specific identification experiments on the furnace. For example,have the same temperature for the whole furnace, and try different burners andsee their effect. This would more precisely show the effect of each burner on thetemperature in the system and thus most likely allow for a better model.

5Control design

This chapter will show the final controllers and which specification that wasgiven for the controllers.The different controllers were simulated with the modelthat was derived in Chapter 4. The result from the simulation will be presentedwith both tables and figures. Finally, the chapter ends with a discussion aboutthe different controllers and their result in the simulation.

5.1 Specification

The aim with a new controller design was to decrease the frequency in whichthe burners switch between on and off as well as a better temperature control.There are no specific requirements on the frequency of the switching, but anyreduction would increase the life span for the burners, which allows them to beused longer before replacement. For the temperature control it is a requirementthat the deviation from reference temperature should be less than 12 °C. At themoment the temperatures in zone 2 and 3 are within the requirement. There aredifficulties in zone 1, in section east the temperature is too high and the effects ofthe walking beam cause large temperature drops in section west and middle. Thelargest possibility for improving the temperature control will thus be in zone 1.

5.2 Controller

In this section the implementation of the different controllers whose theory wasgiven in Chapter 3 is presented. It starts with the split range controller, then theMPC controller and finally the DMPC controller.

37

38 5 Control design

5.2.1 Split range controller

The split range controller has a PI-controller, which has been tuned and the con-version of the control signal to the burners are different than for the current con-troller. The split range controller has two different parameters, cycle time andswap time. A longer cycle time or swap time would give less switches, but at thecost of worse temperature control. Both cycle time and swap time were chosento 45 seconds multiplied with the numbers of burners in the section which is thesame as the cycle time in the current controller. For the section in which there isonly one burner it is not possible to implement split range controller and in thosesections the current control will remain.

5.2.2 MPC

The MPC formulation presented in Chapter 3 has been on the fundamental MPCcontroller with linear model, quadratic cost function and linear constraint. Theflexibility of MPC allows the formulation to be modified to have a nonlinear in-ternal model, a more advanced cost function and non-linear constraints. Thedifference is that the optimization problem can become much more complex, butthe underlying idea is still the same.

The MPC and DMPC problem were formulated by using the toolbox YALMIP[5] and was solved with the solver MOSEK [2]. To use the MPC formulation in(3.5) it needs to be some modification. The MPC controller will use the modelderived in Chapter 4. The model needs to be rewritten to separate the controlledinputs, i.e. the burner from the disturbance inputs, the doors and the walkingbeam. The disturbance inputs are known beforehand since the walking beam isused according to a specified time and if there is a bar to be charged/dischargedthe doors will open before/after the walking beam is used. No constraints fortemperatures in zone 1 were included in the MPC formulation since the MPCcontroller could not calculate control signals with the constraints.

For the current controller the control signal is converted by the burner systemdescribed in 2.2. It would be possible to let the control signal from the MPCcontroller be converted the same way as for the PI controllers. For the MPCcontroller the burners are directly controlled. The first reason is that the perfor-mance of the MPC controller would be limited by the burner system. The secondreason is that the burner system would be a part of the MPC controller’s internalmodel of the system which would increase the complexity of the optimizationproblem, especially when the burner system is a time-variant system. The con-straint on the input signal needs to be modified since the burners are binary. TheMPC formulation for the furnace can be written as

5.2 Controller 39

minimizeu( · )

N∑j=1

||x(k + j) − r(k + j)||2Q1+ ||u(k + j − 1) − u(k + j − 2)||2Q2

+ λ|ε|

subject to


xmin − ε ≤xi ≤ xmax + ε

ui ∈ {0,1}0 ≤ ε

(5.1)

In the cost function (5.1) the term for the inputs will be zero if there were noswitches during the prediction horizon. In the case of a switch for a burner theterm will always be 1 since (0 − 1)2 = (1 − 0)2 = 1. This makes it possible to setthe controller to optimize for a specific relation between a switch and deviationfrom the reference temperature.

Some of the parameters in (5.1) was given as xmin = r − 12, xmax = r + 12 andλ = 109 since λ has to be large. The other parameters were decided by tryingdifferent configurations. It was done by simulating the system with the MPCcontroller for a half hour. This simulation was done with the same model as theinternal model in the MPC controller and in the furnace. The prediction horizonN was chosen by testing different N with Q19,9

= I9×9 and Q217,17= I17×17 and

the result can be seen in

Table 5.1: The performance for the different prediction horizon.

N Mean temperature deviation Switches Max timeN = 6 57.38 179 0.25N = 12 50.31 340 0.87N = 18 50.85 217 7.09N = 24 51.39 149 20.25

For N = 18 and 24 was the max time over the sample time and therefore notchosen. Between N = 6 and N = 12 was the performance better in different areas,N = 6 had fewer switches and N = 12 better temperature control. N = 12 waschosen since it had a better potential with a for it better ratio betweenQ1 andQ2.

The parameter Q2 was chosen by testing different values while Q19,9= I9×9. The

performance can be seen in Figure 5.1. It becomes a choice between switches andmean temperature deviation and we believe that Q2 = 10 is the best choice.It gave the following parameters N = 12, xmin = r −12, xmax = r +12, Q19,9

= I9×9,Q217,17

= 10 ∗ I17×17, λ = 109

40 5 Control design

45 50 55 60 65 70 75Mean temperature deviation from reference (celsius)

50

100

150

200

250

300

350

400sw

icth

Comparsion of different Q2Q2 = 0.1

Q2 = 0.5Q2 = 1

Q2 = 5Q2 = 10

Q2 = 15

Figure 5.1: The comparison between different Q2 for MPC controller.

5.2.3 DMPC

The DMPC controller was implemented by three MPC controllers for each of thezones. The main advantage was to keep the same structure as the furnace. Theburners from the same zone have a larger impact than the burners from anotherzone. In Figure 5.2 a sketch show how each MPC controller is working. The MPCcontroller for each zone will use the same MPC formulation as in (5.1). The dif-ference is for the internal model where the temperature from a nearby zone isassumed to be constant during the prediction horizon.

5.2 Controller 41

MPC1 MPC3MPC2

Zone1 Zone2 Zone3

Figure 5.2: A sketch of how each MPC-controls get information.

The parameters were chosen in the same way as for the MPC controller. Theresult for the different prediction horizon N can be seen in the table below

Table 5.2: The performance for the different prediction horizon for DMPCcontroller.

N Mean temperature deviation Switches Max timeN = 6 52.73 90 0.11N = 12 50.65 170 0.46N = 18 50.00 159 3.64N = 24 49.83 168 12.54

N = 24 was not chosen since it took to long time to calculate the control sig-nals. Between N = 6 and N = 18 the performance was better in different areas,but N = 18 was chosen since it had a better potential with more favorable ratiobetween Q1 and Q2.

The result for the different Q2 can been seen in the Figure 5.3. It becomes achoice between switches and mean temperature deviation and we believe thatQ2 = 30 is the best choice. The following parameters was used for the DMPC

N = 18, xmin = r − 12, xmax = r + 12, Q13,3= I3×3, zone 1: Q26,6

= 30 ∗ I6×6,zone 2: Q25,5

= 30 ∗ I5×5, zone 3: Q26,6= 30 ∗ I6×6 and λ = 109

42 5 Control design

50 55 60 65 70 75Mean temperature deviation from reference (celsius)

20

40

60

80

100

120

140

160sw

icth

Comparsion of different Q2Q2 = 1

Q2 = 5Q2 = 10

Q2 = 15Q2 = 20

Q2 = 25Q2 = 30

Q2 = 35

Figure 5.3: The comparison between different Q2 for DMPC controller.

5.3 Simulation

The simulation was done in SIMULINK with the model based on prediction andwith initial temperature, reference value and disturbance input (the doors andwalking beam) from the data collection. The simulation time was one hour, whichallows the temperature to weigh in. For the MPC/DMPC there were two simula-tions, one was the ideal case and the other robustness test case. In the idealcase was the internal model for the MPC/DMPC was the same as for the fur-nace. In the other case was the internal model for the MPC/DMPC estimated thesame way as in Chapter 4 but the estimation data where the data from the lasttwo thirds instead. The input temperature for the MPC/DMPC was modified byadding Gaussian noise with variance 3. The reason for it was to represent theuncertainty with the model and that the furnace is a nonlinear system. The ro-bustness test was to see if the performance changed.

The reference temperature for each section can be seen in Table 5.3 and the tem-perature from the simulation with the different controllers can be seen in Figures5.4-5.9. More detailed picture of zone 2 and 3 for MPC and DMPC with lineswhere the temperature should be within are seen in Appendix B.

5.3 Simulation 43

Table 5.3: The reference temperature in Celsius for each section.

Zone Temp West Temp Mid Temp East1 960 955 9552 1065 1060 10603 1098 1098 1098

0 500 1000 1500 2000 2500 3000 3500940

960

980

1000

1020

1040

1060

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1100

1120Original Temperatures

Tem

pera

ture

(cel

sius

)

Time (seconds)

Z1 WZ1 MZ1 EZ2 WZ2 MZ2 EZ3 WZ3 MZ3 E

Figure 5.4: The simulated temperatures for the furnace during 1 hour.

44 5 Control design

0 500 1000 1500 2000 2500 3000 3500Time (seconds)

940

960

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1120

Tem

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(cel

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Split-range TemperaturesZ1 WZ1 MZ1 EZ2 WZ2 MZ2 EZ3 WZ3 MZ3 E

Figure 5.5: The simulated temperatures with Split-range controller during 1hour.

0 500 1000 1500 2000 2500 3000 3500Time (seconds)

920

940

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Tem

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MPC TemperaturesZ1 WZ1 MZ1 EZ2 WZ2 MZ2 EZ3 WZ3 MZ3 E

Figure 5.6: Thesimulated temperatures with MPC during 1 hour with theideal case.

5.3 Simulation 45

0 500 1000 1500 2000 2500 3000 3500Time (seconds)

920

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(cel

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MPC TemperaturesZ1 WZ1 MZ1 EZ2 WZ2 MZ2 EZ3 WZ3 MZ3 E

Figure 5.7: The simulated temperatures with MPC during 1 hour with therobustness test case.

0 500 1000 1500 2000 2500 3000 3500Time (seconds)

920

940

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(cel

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DMPC TemperaturesZ1 WZ1 MZ1 EZ2 WZ2 MZ2 EZ3 WZ3 MZ3 E

Figure 5.8: The simulated temperatures with DMPC during 1 hour with theideal case.

46 5 Control design

0 500 1000 1500 2000 2500 3000 3500Time (seconds)

920

940

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(cel

sius

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DMPC TemperaturesZ1 WZ1 MZ1 EZ2 WZ2 MZ2 EZ3 WZ3 MZ3 E

Figure 5.9: The simulated temperatures with DMPC during 1 hour with therobustness test case.

The number of times that the burners switched between on and off during thesimulation was saved. This was also done for the corresponding measurementdata. It was used to compare the controllers between each other and also thecontroller in the actual furnace. The result can be seen in Table 5.4.

Table 5.4: The number of switches each zone had during simulation for thedifferent controllers and measured data for the furnace.

Controller switches z1 Switches z2 Switches z3 Total SwitchesMeasured 83 162 243 488Simulated 63 133 179 375Split range 38 67 96 201MPC (ideal) 69 42 30 141MPC (robustness) 172 76 64 312DMPC (ideal) 28 11 5 44DMPC (robustness) 66 36 64 166

Tables 5.5 - 5.7 shows the mean deviation from the reference temperature ineach section for the last half hour. The last half hour was used since then thefluctuations in temperature had subsided, mostly for the simulation with currentcontrol in Figure 5.4. The tables were used to see the difference between thecontrollers in another way than the figures.

5.3 Simulation 47

Table 5.5: The mean deviation for the temperature in zone 1 for the last halfhour.

Controller Temp West Temp Mid Temp EastMeasured 17.6 4.4 54.2Simulated 11.3 3.4 39.8Split range 12.0 4.1 39.7MPC (ideal) 6.8 13.5 25.8MPC (robustness) 6.8 10.3 29.3DMPC (ideal) 7.0 15.8 24.9DMPC (robustness) 8.3 8.1 32.4


Controller Temp West Temp Mid Temp EastMeasured 2.0 1.9 6.1Simulated 1.9 1.3 5.6Split range 1.5 1.5 4.9MPC (ideal) 6.4 2.8 3.2MPC (robustness) 2.0 1.3 3.6DMPC (ideal) 2.1 1.3 1.0DMPC (robustness) 0.6 0.6 1


Controller Temp West Temp Mid Temp EastMeasured 1.4 1.1 2.1Simulated 1.1 0.9 1.2Split range 1.2 1.2 1.5MPC (ideal) 2.8 1.8 2.1MPC (robustness) 1.9 0.8 3.5DMPC (ideal) 0.3 1.6 1.5DMPC (robustness) 3.6 2.1 0.9

The time to calculate the control signal during simulation is seen in Table 5.8.

48 5 Control design

Table 5.8: The mean, max and min time in seconds to calculate the controlsignal for the different MPC controller.

Controller Mean Max MinMPC (ideal) 0.1260 0.8308 0.0724MPC (robustness) 0.1218 0.9530 0.0732DMPC (ideal) zone 1 0.0297 0.1221 0.0230DMPC (ideal) zone 2 0.0543 0.2398 0.0235DMPC (ideal) zone 3 0.0886 1.1384 0.0282DMPC (robustness) zone 1 0.0357 0.1063 0.0236

DMPC (robustness) zone 2 0.0906 0.4887 0.0244DMPC (robustness) zone 3 0.1258 0.8986 0.0265

5.4 Discussion

There was a difference between the measurement and the simulation of the fur-nace in both temperature and number of switches, which is reasonable since themodel differs from the furnace. Another possible explanation could be the differ-ence between the controllers, but since the PI-controllers and the burner systemare clearly specified and implemented the same way in SIMULINK, this explana-tion is unlikely.

The temperature is much smoother in the simulation compared to the measure-ment but the mean deviation from reference value is similar except zone 1. Sincethe temperatures vary considerably more in zone 1 both in absolute and relativeterms it is reasonable that zone 1 differ. The difference in both temperature andnumber of the switches are probably due to that the model is a linear simpli-fied version of the furnace. The real system is nonlinear and depends on morethan just the burners, the doors and the walking beam. The difference in numberof switches can be explained by that the temperature is smoother in simulationwhich results in the control signal being closer to 0 or 100 % and thereby the burn-ers do not alternate. The difference in number of switches between measured andsimulated was of the same magnitude across all the zones. Therefore, there is nosystematic error in any of the zones.

The split range controller gives a clear improvement in the number of switches atthe cost of a little worse temperature control. It is reasonable since for the samecontrol signal the split range controller will have fewer switches. The parameters,cycle time and swap time, in the split range controller can be used to tune thecontroller to decrease the frequency of alternation or improve the temperaturecontrol. An increase in swap time or cycle time would decrease the number ofswitches with worse temperature control or vice verse. A possible problem withthe split range controller implemented in the furnace could be that the temper-ature inside the section gets uneven since the burners is on less often comparedto the current control. It is difficult to decide if it is a real problem since we only

5.4 Discussion 49

have one temperature measurement in each section. An advantage with the splitrange controller is that it would be fairly easy to implement in the furnace bymodifying the burner system.

The simulation of the MPC/DMPC probably overestimates the controller’s perfor-mance. The difference between the furnace and model is probably much biggerthan the models used in the robustness test case. The difference between the twocases were none with respect to temperature control but the switches was worsein the robustness test case. A reason for that could be that the internal modelworks well in short term which keeps the temperatures close to its reference. Thesimulation can be seen as indicative of the MPC/DMPC possible performance.MPC/DMPC also have the advantage over the other controllers in that the burn-ers are directly controller, and the simplicity with which the known disturbancescan be used in a feed forward fashion.

A comparison between the MPC and DMPC is more easily done than with theother controllers. The biggest difference between MPC and DMPC was the num-ber of switches for which the DMPC was much better. One reason that the DMPCcontroller is better could be that it has a longer prediction horizon which allow itto choose a better control sequence. Another reason could be that the DMPC hasa larger penalty on switches. The performance of the DMPC could be improvedif its different MPC controllers could communicate. First the MPC controllers cal-culate its control sequence as before where the temperatures in the other zonesare assumed constant over the prediction horizon. For the next step the controlsequence is calculated where the previous control sequence to model the temper-ature in the other zones. This would be repeated until the performance is goodenough. Finally, with this said the MPC/DMPC performance depends on howgood the model is. To implement the MPC/DMPC in the real furnace would re-quire a more extensive validation of the model during other condition than thoseduring the data collection.

As said it is hard to compare the Split range controller with the MPC/DMPCbut one main advantage of the MPC/DMPC is the possibility of adjustment. Ifthe requirements of the steel bars would be harder and the bounds on the tem-peratures would be tighter, the MPC/DMPC can just adjust it. The Split rangecontroller can not take this into account and it can have problem to keep the tem-perature within the bounds; in worst case it could be impossible to have a Splitrange controller.

The temperature control could be improved by introducing feed forward controlfor the other controllers besides the MPC/DMPC. An easy form of feed forwardcontrol would be that the reference temperature or the control signal is temporar-ily increased before the walking beam is used, as we know the walking beamdecreases the temperature. An improvement for the MPC/DMPC is if the throt-tle in the exhausts in zone 1 could be controlled. It would then be possible topartially control the heat flow between the sections. This could probably lead

50 5 Control design

to a smoother temperature in zone 1. Another improvement by controlling theexhaust could be that the temperature in the east section would decrease sincethe heat streams between the exhaust could be more equal.

Every time a burner malfunctions the whole furnace needs to cool downs beforeany maintenance can be done. A lot of time will pass by every time the furnaceneeds to be cooled down and then heated up again. For the heating up processthe need to go from room temperature up to reference takes much energy, whichis not good for either the company or the environment. Hence, an improved con-trol strategy could be important for several reasons.

The simulation shows that both the split range controller and MPC/DMPC havepossible improvements over the current controller. This could hopefully reducethe unplanned maintenance during the years. MPC/DMPC had better tempera-ture control while more switches in the robustness test case compared to the splitrange controller, but the uncertainty of the result is greater and it would be moredifficult to implement than the split range controller.

6Conclusion and Future work

The split range controller can be improved by taking nearby section temperaturesinto account when it decides which order the burners in the section should startto burn. If the temperature in a nearby section to the left is too high is the right-most burner is chosen first. The split range controller could also be improved bya feed forward approach where the reference temperature or the control signal istemporarily increased before the temperature drop caused by the walking beam.

The model is adapted to the specific conditions during the data collection. Themodel must be validated to see if it works in different conditions such as the sizeof the bars, the operating temperature and the time it takes for the bars to gothrough the furnace. The model may have to be modified to be more general, forexample by having different models for different operating temperatures or havesome parameters depending on the properties of the bars. Since the model nowis linear, one way can be to make it nonlinear.

The performance of the MPC/DMPC depends on the quality of the model. Thereare several potential ways to improve the model. The data collection could bedone with specific input signal devised to improve the identification performance.Another option would be to use a more extensive method than the direct ap-proach to deal with data collection during feedback. With better data can theparameter estimation of the state space model could possibly be improved, or amore complex model could be used. To improve the DMPC one way could be toadd the predicted output signals from the local MPC controller into each other.This will generate a better prediction of the temperature in the other zones.

A natural continuation would be to implement one of the examined control de-sign as the controller in the furnace. It would be possible to implement the split

51

52 6 Conclusion and Future work

range controller already now by modifying the burner system. It would requiremore work to implement the MPC/DMPC but the potential in temperature con-trol is also greater than for the split range controller. Finally, all the evaluatedcontrollers should be an improvement over the current controller.

Appendix

AParameters in state space model

This Appendix contains table of the parameters in (4.10) and (4.11) from Chapter4

55

56 A Parameters in state space model

A.1 Values for A-matrix

Parameter Prediction Simulationa1,1 0.9885 0.9890a1,4 0.0104 0.0099a2,1 0.0205 0.1304a2,2 0.9601 0.7000a2,5 0.0168 0.1503a3,2 0.0278 0.0206a3,3 0.9563 0.9609a3,6 0.0157 0.0175a4,1 0.0052 0.0054a4,4 0.9641 0.9365a4,5 0.0047 0.0435a4,7 0.0253 0.0145a5,2 0.0032 0.0040a5,4 0.0441 0.0156a5,5 0.9136 0.9605a5,6 0.0031 0a5,8 0.0346 0.0193a6,2 0.0059 0.0082a6,3 0.0002 0a6,5 0.0078 0.0314a6,6 0.9703 0.8790a6,9 0.0158 0.0799a7,4 0.004 0.0077a7,7 0.9347 0.9678a7,8 0.0608 0.0241a8,5 0.0157 0.0241a8,7 0.9347 0.0246a8,8 0.0608 0.8970a8,9 0.0264 0.0545a9,6 0.0135 0.0131a9,8 0.0371 0a9,9 0.9491 0.9868

A.2 Values for B-matrix 57

A.2 Values for B-matrix

Parameter Prediction Simulationb1,1 0.1000 0.1000b2,1 0.4000 0.4000b2,2 0.1000 0.1000b3,2 0.1000 0.1000b4,2 0.3213 0.4156b4,3 0.1782 0.4463b5,2 0.1000 0.2613b5,3 0.7534 1.0034b6,3 0.1000 0.1000b7,4 0.2311 0.3968b7,7 0.4000 0.4000b8,4 0.3555 0.4000b8,5 0.1000 0.1676b9,5 0.1000 0.1000b9,8 0.0722 0.1589b10,5 0.1000 0.1290b10,6 0.2609 0.3586b10,8 0.1595 0.2909b10,9 0.2838 0.4000b11,6 0.1000 0.1000b12,7 0.4429 0.4160b13,7 0.2052 0.2411b13,8 0 0.0483b14,8 0.3463 0.4916b15,8 0.1000 0.1000b16,8 0 0.1170b16,9 0.3417 0.1211b17,9 0.3847 0.2397b18,1 0.5300 0.5048b18,2 0.3425 0.8113b18,3 0.2414 1.1900b18,4 0.0247 -0.2249b18,5 0.0527 0.0156b19,1 0.5180 0.5705b19,2 0.0003 0.0107b19,5 0.0295 0.0462b20,1 1.8265 1.8417b20,2 0.8304 0.8328b20,4 0.5987 0.8172b20,5 0.4321 0.5420

BDetailed figures of temperature zones

This appendix shows more detailed figures for zone 2 and 3 for the MPC andDMPC controllers. For zone 2 is the dash-dotted lines are the boundaries for thetemperature in section west and the dashed lines are the boundaries for temper-atures in section mid and section east. For zone 3 is the boundaries the same forall its sections and is represented by the dashed lines.

0 500 1000 1500 2000 2500 3000 3500Time (seconds)

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MPC Temperatures for zone 2

WestMidEast

Figure B.1: The temperature in the west section is inside its boundaries andtouches only the bounds for the other sections

59

60 B Detailed figures of temperature zones

0 500 1000 1500 2000 2500 3000 3500Time (seconds)

1085

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WestMidEast

Figure B.2: The temperatures is clearly inside the boundaries.

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Figure B.3: The temperatures are clearly inside the boundaries.

61

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WestMidEast

Figure B.4: The temperatures are clearly inside the boundaries.

CResiduals

This appendix contains all the residuals from Chapter 4.

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Figure C.1: Residual of zone 1 west, with prediction model.

63

64 C Residuals

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Figure C.2: Residual of zone 1 east, with prediction model.

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Figure C.6: Residual of zone 3 mid, with prediction model.

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Figure C.8: Residual of zone 1 west, with simulation model.

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Figure C.9: Residual of zone 1 mid, with simulation model.

68 C Residuals

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Figure C.10: Residual of zone 1 east, with simulation model.

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Lag

Ampl

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Bibliography

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