Institutionen för systemteknik - Linköping University · Institutionen för systemteknik Department of Electrical Engineering Examensarbete Load Simulation and Investigation of

Institutionen för systemteknikDepartment of Electrical Engineering

Examensarbete

Load Simulation and Investigation of PID Controlfor Resonant Elastic Systems

Examensarbete utfört i Reglerteknikvid Tekniska högskolan i Linköping

av

Sara Lundin

LiTH-ISY-EX--07/3991--SE

Linköping 2007

Department of Electrical Engineering Linköpings tekniska högskolaLinköpings universitet Linköpings universitetSE-581 83 Linköping, Sweden 581 83 Linköping

Load Simulation and Investigation of PID Controlfor Resonant Elastic Systems

Examensarbete utfört i Reglerteknikvid Tekniska högskolan i Linköping

av

Sara Lundin

LiTH-ISY-EX--07/3991--SE

Handledare: Sami SaariABB Mining

Daniel PeterssonLinköpings universitet

Examinator: Alf IsakssonLinköpings universitet

Linköping, 20 June, 2007

Avdelning, InstitutionDivision, Department

Division of Automatic ControlDepartment of Electrical EngineeringLinköpings universitetSE-581 83 Linköping, Sweden

DatumDate

2007-06-20

SpråkLanguage

� Svenska/Swedish� Engelska/English

�

�

RapporttypReport category

� Licentiatavhandling� Examensarbete� C-uppsats� D-uppsats� Övrig rapport�

�

URL för elektronisk versionhttp://www.control.isy.liu.se

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-9198

ISBN—

ISRNLiTH-ISY-EX--07/3991--SE

Serietitel och serienummerTitle of series, numbering

ISSN—

TitelTitle

Lastsimulering och undersökning av PID-reglering av resonanta elastiska systemLoad Simulation and Investigation of PID Control for Resonant Elastic Systems

FörfattareAuthor

Sara Lundin

SammanfattningAbstract

The purpose of this Master Thesis is to improve the driving performance of minehoists. The work is divided into two parts. The first and main part deals withsimulation of the rope elongation that occurs at load changes in the mine hoist.A mathematical load model of the elongation in the ropes at a mine hoist is madefor four types of mine hoists. Mass less springs and dampers are used to get theelastic behaviour of the ropes.

The mathematical model is implemented in Matlab and Simulink for all fourhoist types to make load simulations possible. The implementation in the labora-tory HoistLab is made by modifying an existing program with the line elongationfunctionality. It is only done for the tower mounted friction hoist. There areseveral functions that are modified to make the simulations realistic.

The task for the second part of this Master Thesis is to do a pilot study to de-cide if it is worth making further investigations about how the derivative part willimprove the drive performances. A PI controller is designed and gives an accept-able rollback as result when the brakes are released. Then the controller model isextended with the derivative part, D-part, which improves the results essentially.It is still too uncertain how sensitive the system will be for noise when using thederivative part, but the performance potential is clear so the recommendation isto make further investigations.

NyckelordKeywords Load Simulation, PID Control, Resonant Elastic Systems

http://www.control.isy.liu.se

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-9198

AbstractThe purpose of this Master Thesis is to improve the driving performance of mine

hoists. The work is divided into two parts. The first and main part deals withsimulation of the rope elongation that occurs at load changes in the mine hoist.A mathematical load model of the elongation in the ropes at a mine hoist is madefor four types of mine hoists. Mass less springs and dampers are used to get theelastic behaviour of the ropes.

The mathematical model is implemented in Matlab and Simulink for all fourhoist types to make load simulations possible. The implementation in the labora-tory HoistLab is made by modifying an existing program with the line elongationfunctionality. It is only done for the tower mounted friction hoist. There areseveral functions that are modified to make the simulations realistic.

The task for the second part of this Master Thesis is to do a pilot study to de-cide if it is worth making further investigations about how the derivative part willimprove the drive performances. A PI controller is designed and gives an accept-able rollback as result when the brakes are released. Then the controller model isextended with the derivative part, D-part, which improves the results essentially.It is still too uncertain how sensitive the system will be for noise when using thederivative part, but the performance potential is clear so the recommendation isto make further investigations.

v

SammanfattningSyftet med detta examensarbete är att förbättra driftegenskaperna för gruvspel.

Arbetet är uppdelat i två olika delar. Den första och största delen handlar omsimulering av den lintöjning som uppkommer vid lastförändringar i gruvspel. Ma-tematiska modeller för detta är framtagna för fyra olika sorters typer av gruvspel.Elasticiteten i linorna är modellerad genom masslösa fjädrar och dämpare.

De matematiska sambanden är implementerade i Matlab och som modeller iSimulink för att utföra simuleringar. I HoistLab är modellen realiserad genom attutöka ett befintligt lastsimuleringsprogram med de nya funktionerna för lintöjning.Detta är utfört enbart för den toppmonterade typen av friktionsspel. Ett flertalfunktioner fick ändras för att få realistiska simuleringar.

Den andra delen av examensarbetet går ut på att göra en förstudie kring denderiverande delen i PID-regulatorer och hur den påverkar gruvspelets prestanda.För denna del är en PI-regulator som ger ett acceptabelt resultat av backgångennär bromsarna släpps designad. Därefter är modellen utökad med den deriverandedelen, D-delen, vilket ger väsentligt bättre resultat. Det är dock osäkert hur brus-känsligt systemet blir när den deriverande delen används men eftersom förbätt-ringspotentialen är tydlig är rekommendationen att göra vidare undersökningarkring D-delen.

vi

AcknowledgmentsFirst of all I would like to thank ABB Mining which has made it possible for meto do this Master Thesis work. During my work at ABB, I have obtained a lot oftechnical experiences and met many nice persons.

I would also specifically direct thanks to my supervisors Sami Saari and HåkanSelldén at ABB for all coaching and all interesting discussions during this time.Mats Tallfors at ABB Rolling Mills has been a great help during the last week andAnders Daneryd at ABB Corporate Research has assisted with Appendix A.

Finally I would like to thank my supervisor at Linköping University DanielPetersson and my examiner Alf Isaksson for the support throughout the work.Also a huge thank to my opponent Stina Wahnström for putting time and effortinto reading my report and giving me valuable feedback.

vii

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Preliminaries 52.1 Main Parts of a Mine Hoist . . . . . . . . . . . . . . . . . . . . . . 62.2 Types of Mine Hoists . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Drum Hoist . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Friction Hoist . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Introducing HoistLab . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Introduction to PID Controllers . . . . . . . . . . . . . . . . . . . . 11

3 Mathematical Description of Load Disturbance 153.1 Tower Mounted Drum Hoist . . . . . . . . . . . . . . . . . . . . . . 173.2 Ground Mounted Drum Hoist . . . . . . . . . . . . . . . . . . . . . 183.3 Tower Mounted Friction Hoist . . . . . . . . . . . . . . . . . . . . . 203.4 Ground Mounted Friction Hoist . . . . . . . . . . . . . . . . . . . . 23

4 Implementation of Load Models 274.1 Implementation in Simulink . . . . . . . . . . . . . . . . . . . . . 274.2 Implementation in HoistLab . . . . . . . . . . . . . . . . . . . . . . 284.3 Results of the Implementation . . . . . . . . . . . . . . . . . . . . . 30

5 Investigation of the Derivative Part in PID Controllers 355.1 State Space Model for Motor and Load . . . . . . . . . . . . . . . . 365.2 The Control System . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 Results of the Controllers . . . . . . . . . . . . . . . . . . . . . . . 41

5.3.1 Results of the PI Controller . . . . . . . . . . . . . . . . . . 415.3.2 Results of the PID Controller . . . . . . . . . . . . . . . . . 43

5.4 Advantages and Disadvantages with theDerivative Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

ix

x Contents

5.5 Future for the PID Controller Investigation . . . . . . . . . . . . . 49

6 Conclusions 516.1 Conclusions of the Rope Elongation Part . . . . . . . . . . . . . . . 516.2 Conclusions of the Controller Part . . . . . . . . . . . . . . . . . . 52

Bibliography 53

A Rope Weights 55

B Implementation 58B.1 Values of Constants at Implementation . . . . . . . . . . . . . . . . 58B.2 Values for the Scaling Factors . . . . . . . . . . . . . . . . . . . . . 59

C State Space Form 60

D Plots from the Results in Section 5.3 62

Contents xi

List of Figures2.1 Drum in a tower mounted friction hoist . . . . . . . . . . . . . . . 52.2 Skip close to the load section . . . . . . . . . . . . . . . . . . . . . 62.3 Tower mounted friction hoist . . . . . . . . . . . . . . . . . . . . . 72.4 Ground mounted drum hoist . . . . . . . . . . . . . . . . . . . . . 72.5 Example layouts of mine hoists . . . . . . . . . . . . . . . . . . . . 82.6 The drive systems for HoistLab . . . . . . . . . . . . . . . . . . . . 92.7 The mini mine hoist model . . . . . . . . . . . . . . . . . . . . . . 102.8 Control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.9 Example of Bode diagrams . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Tower mounted drum hoist . . . . . . . . . . . . . . . . . . . . . . 173.2 Ground mounted drum hoist . . . . . . . . . . . . . . . . . . . . . 183.3 Rope lengths of a friction hoist . . . . . . . . . . . . . . . . . . . . 203.4 Tower mounted friction hoist . . . . . . . . . . . . . . . . . . . . . 213.5 Ground mounted friction hoist . . . . . . . . . . . . . . . . . . . . 23

4.1 One part of the model in Simulink . . . . . . . . . . . . . . . . . 284.2 One part of the model in PLC Control Builder . . . . . . . . . . . 294.3 Drum hoist at length 10 meter, unloading . . . . . . . . . . . . . . 314.4 Zoomed figure of drum hoist at length 10 meter, unloading . . . . 314.5 Drum hoist at length 160 meter, loading . . . . . . . . . . . . . . . 324.6 Friction hoist at length 10 meter, unloading . . . . . . . . . . . . . 324.7 Zoomed figure of friction hoist at length 10 meter, unloading . . . 334.8 Friction hoist at length 160 meter, loading . . . . . . . . . . . . . 334.9 Zoomed figure of friction hoist at length 160 meter, loading . . . . 34

5.1 Overview of the controller design . . . . . . . . . . . . . . . . . . . 385.2 Bode diagrams for different lengths and constant payload . . . . . 405.3 Bode diagrams for different payload and constant lengths . . . . . 405.4 Speed and rollback results for the PI controller . . . . . . . . . . . 425.5 Speed and rollback results for the PI controller . . . . . . . . . . . 425.6 Bode diagrams for Go for the PI controller . . . . . . . . . . . . . 435.7 Speed and rollback results for the PID controller . . . . . . . . . . 445.8 Speed and rollback results for PID controller . . . . . . . . . . . . 445.9 Bode diagrams for Go for the PID controller . . . . . . . . . . . . 455.10 Position comparison between PI and PID . . . . . . . . . . . . . . 465.11 Control error affected of noise . . . . . . . . . . . . . . . . . . . . . 465.12 Motor torque affected of noise . . . . . . . . . . . . . . . . . . . . 475.13 Drum and skip speed affected of noise . . . . . . . . . . . . . . . . 475.14 Drum and Skip position affected of noise . . . . . . . . . . . . . . 48

A-1 Definitions for calculation of rope weight . . . . . . . . . . . . . . 55A-2 How big part of the rope weight that affects oscillations . . . . . . 57

D-1 Bode plot for Gc for the PI controller . . . . . . . . . . . . . . . . 62

xii Contents

D-2 Bode plot for S for the PI controller . . . . . . . . . . . . . . . . . 63D-3 Bode plot for Gc for the PID controller . . . . . . . . . . . . . . . 63D-4 Bode plot for S for the PID controller . . . . . . . . . . . . . . . . 64

List of Tables1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5.1 System and model parameters . . . . . . . . . . . . . . . . . . . . 365.2 System input signals . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A-1 Definitions for rope mass calculation . . . . . . . . . . . . . . . . . 56

B-1 The predefined values at constants in HoistLab . . . . . . . . . . . 58B-2 Scale values for elasticity module at different rope lengths . . . . . 59

Chapter 1

Introduction

The purpose of this Master Thesis is to improve the drive performance of minehoists. The work is made at ABB Miningin Västerås and is divided into two parts.The first and main part deals with rope elongation and consists of three steps.The first is to make a mathematical description of the load disturbance causedby elongation in the ropes of a mine hoist. This description is implemented inMatlab and Simulink to make it possible to simulate the model. The third stepis implementation of the model in a laboratory and the local program PLC ControlBuilder.

The second part is an investigation about how drive performance will be af-fected if the derivative part in the speed controller is used. More specific the taskit is to look into the situations at start and stop.

1.1 Background

ABB Miningis the leading supplier of mine hoists and has its centre of excellencelocated in Västerås. ABB is a complete supplier of mine hoists and is the onlycompany on the market that has both electrical and mechanical design and devel-opment. A good help during this process is HoistLab which is a mini version ofa real mine hoist containing a mini shaft, control and drive equipment as well asa brake system. The Master Thesis is connected to a development project whichaims at improving the drive performance of the mine host and will be tested inHoistLab.

1.2 Problem Description

When the skip is loaded with mass, for instance rock, there will be rope elongationsand oscillations caused by the mass change. These can not be measured since thebrakes are always active when the skip is standing still, and then the motor does

1

2 Introduction

not feel any mass changes. This results in the problem that the skip position afterthe unbalance torque is unknown. It could be possible to use a scale weighingmachine to find out the weight or sensors to get position signals but that are tooexpensive solutions. In production hoists the weight of the payload is well definedand the motor torque can compensate for the stretch to avoid rollback, i.e. whenthe skip moves in the wrong direction. The skip in a service hoist can be loadedwith different masses, e.g. trucks or people, which makes it impossible to knowthe mass in advance and to compensate for the load change. This can result inrollback. To be able to solve or minimise the problem a mathematical model ofthe rope elongation is needed. To make it easy to simulate, the model should beimplemented in Simulink.

ABB has an important laboratory called HoistLab which is used during thedevelopment process. It is a mini mine hoist model that simulates a real mine hoist.At this moment it is not possible to simulate rope elongation in the laboratory.With a mathematical model including stretch it would make it easier to solvethe problem and figure out how to minimise the back motion at start and stopbehaviour.

The control of a mine hoist uses different controllers during the productioncycle. In normal operation the speed control follows a predefined reference signalbut at start and stop situations when the brakes are involved another controlis needed. Today this is solved with a PID controller with the derivative part(D-part) set to zero. The problem is that no investigation about how the D-part will affect the performance of the hoist drive has been made. It is unknownwhich advantages the derivative part can bring regarding the brake release. Can itprovide essebtuak difference with faster control without significant disadvantages?

1.3 Purpose

The purpose of this Master Thesis is to:

• Make a mathematical model of the elongation in the ropes at a mine hoist

• Implement the mathematical model of the rope elongation in Matlab andSimulink to make load simulations

• Implement the mathematical model of the rope elongation in a local programto make it possible to make load simulations in the laboratory

• To design a PI controller that results in an acceptable rollback when thebrakes are released

• To design a PID controller that results in an acceptable rollback when thebrakes are released

• To do a pilot study to decide if it is worth further investigating about whetherthe derivative part will improve the drive performance

1.4 Scope 3

1.4 Scope

The scope of the Master Thesis is to:

• Make mathematical models for four types of mine hoists

• Make models in Simulink for four types of mine hoists

• Make model in PLC-control builder for tower mounted drum hoist and fric-tion hoist

• Do the PID investigation for a tower mounted friction hoist within realisticvalues

1.5 Definitions

The definitions that will be used later on are explained in this section. Table 1.1introduces some variables and constants that are used in the mathematical model.

1.6 Thesis Structure

The structure for this Master Thesis is briefly described below.

Chapter 2, Preliminaries: Includes facts about mine hoists and an introduc-tion to the control theory that is needed.

Chapter 3, Mathematical Description of Load Disturbance: Describes themathematical model and includes the equations.

Chapter 4, Implementation of Load Models: Presents the implementationof the load model both in Simulink and HoistLab.

Chapter 5, Investigation of the Derivative Part in PID Controllers: Includesthe speed control investigation.

Chapter 6, Conclusions: Discusses the result of this Master Thesis comparedto the purpose.

Bibliography: Presents the literature that has been used in this report.

Appendix A, Rope Weights: Explains the rope weight calculation.

Appendix B, Implementation: Includes the values and constants that are usedin the implementation.

Appendix C, State Space Form: Presents the state space equations.

Appendix D, Plots from the Results in Section 5.3: Includes some of theplots from the results in Section 5.3.

4 Introduction

Quantity [SI unit] Descriptionms [kg] Skip massmΣs [kg] Total mass of skip, payload and ropemcw [kg] Counterweight massmΣcw [kg] Total mass for counterweight and ropemr [kg] Total mass of the ropesmp [kg] Payload masslhr [m] Head rope lengthltr [m] Tail rope lengthlnew [m] New rope lengthrhr [m] Head rope radiusrtr [m] Tail rope radiusrd [m] Drum radiusvd [m/s] Drum speedvs [m/s] Skip speedvcw [m] Counterweight speedvsp [m/s] The speed at the springvsh [m/s] The speed at the sheavevb [m] The speed at the bottom of the ropek [N/m] Spring constantc [N/m] Damping constantJd [kgm2] Moment of inertia for the drumJm [kgm2] Moment of inertia for the motormpm [kg/m] Mass per meter, rope densityNr [dimless] Number of ropesCd [dimless] Relative damping constantCs [dimless] Steel area constantE [Pa] Elasticity moduleF [N] ForceT [Nm] Torque∆r [m] Rope elongationg [m/s2] Gravitation constant = 9.81 [m/s2]

Table 1.1. Definitions of variables and constants

Chapter 2

Preliminaries

To understand the mathematical equations and models later on some specificknowledge about mine hoists is needed and gives completeness. Like all busi-nesses some special vocabulary is used and the most important for this purposeis explained in the section below. There are also descriptions of the two types ofmine hoists that will be treated later.

The laboratory where the mathematical model is implemented for simulationis called HoistLab and will be introduced in this chapter.

The last part of this chapter includes general information about PID controllersand control theory that will be used later on in Chapter 5.

Figure 2.1. Drum in a tower mounted friction hoist

5

6 Preliminaries

2.1 Main Parts of a Mine Hoist

Mine hoists consist of several parts and the most important for this purpose arementioned here. The drum is a big barrel with a diameter that normally can bebetween three to seven meters, see Figure 2.1. The drum is driven by a motorand has a separate hydraulic braking system. The skip is like an elevator, it is abox that transports goods, e.g. ore or people, see Figure 2.2. The mass in the skip

Figure 2.2. Skip close to the load section

is called payload and usually weighs between ten to forty tons. In some types ofhoists there is also a counterweight to make balance with the skip. The verticalhole where the skip is driving is called a shaft. There is a conflict between havinga big skip to carry as much as possible and to have as small a shaft as possible.Both characteristics are desired for reducing costs while it is expensive to dig thehole. The ropes that carry the skip are heavy and could weigh many thousandkilos. The hoist cycle is the time it takes for the skip to be loaded, unloaded andready for load again, and it is normally about 250-300 seconds.

2.2 Types of Mine Hoists 7

2.2 Types of Mine Hoists

Mine hoist can be mounted either in a tower or at ground level, see Figures 2.3and 2.4.

Figure 2.3. Tower mounted friction hoist

Figure 2.4. Ground mounted drum hoist

Tower mounted means that the drum is placed at the top of the mine hoistin a tall small building called a "main frame". Ground mounted mine hoists havethe drum directly on the ground and the ropes pull over a sheave at a high stand

8 Preliminaries

before they go down in the shaft. The reason for this is that the skip must haveenough space left to empty the load.

Two main types of mine hoists will be analysed, drum hoists and friction hoists,see Figure 2.5. Both types will be described in the sections below.

skip

(a) Layout of a drum hoist

skipcounterweight

(b) Layout of a friction hoist

Figure 2.5. Example layouts of mine hoists

2.2.1 Drum Hoist

In drum hoists the rope length will vary because the rope is wrapped up or down atthe drum depending on the direction of the skip. For shallow shafts, down to about250 meters the drum hoist is generally the best alternative because, for instance,the shaft diameter could be minimised. The drum hoist is a comparatively simplesolution.

2.2.2 Friction Hoist

In friction hoists the rope is a closed circuit with constant total rope length. Therope is hanging on both sides of the drum and it is the friction between the ropeand the drum that moves the rope when the drum is driven by a motor. The skipis hanging on one side and a counterweight on the other. The system is designedso that there will be balance on both sides when the skip is half loaded. Theadvantage is that the motor only has to drive the actual load, i.e., the payload,instead of the entire mass from skip, ropes and payload as in drum hoists.

The rope is divided in two parts, head rope and tail rope. The head rope isfixed between the top of the skip to the top of the counterweight and the tail rope

2.3 Introducing HoistLab 9

from the bottom of the skip to the bottom of the counterweight. The number ofhead ropes is often larger than the tail ropes but then the rope diameter is smaller.The task of the tail ropes is to make balance in the system.

Normally friction hoist are possible to use when the depth of the shaft is 250to 1500 meters. The limit of the length is the weight of the system affecting thedrum.

2.3 Introducing HoistLab

This section introduces HoistLab and its equipment. HoistLab is a mini minehoist model. It is used for development and tests of new and improved functions,training, demonstrations and as a help in finding faults at real mine hoists. Thelaboratory has an important role in mine hoist development for both electrical andmechanical parts, such as control, drive, hydraulic brake system, etc.

ACS 800

Hoist Control Cubicle

PG

Figure 2.6. The drive systems for HoistLab

The programming language in HoistLab is called PLC Control Builder AC 800M.PLC is an abbreviation for Programmable Logic Controller. It is possible bothto do the programming in structured text and graphically with blocks like inSimulink. Here follows a list of the equipment that are used in HoistLab. Anillustration of some parts is shown in Figure 2.6.

• The drive system is built up of ACS 800 drives (frequency converters) andtwo small AC motors connected to the mini shaft. One motor runs the modelskip and the other simulates the load.

10 Preliminaries

• The mini shaft is 1.6 meter high and in the control equipment this distanceis scaled 1:100, i.e., it represents a real depth of 160 meters.

• Micro switches are used as shaft switches for check points and synchronisa-tion of the pulse tachos on the motors.

• The control system hardware consists of the AHC control cubicles, maindesk, I/O box, cage level box and the Hoist Monitor AHM 800.

• The HoistLab has the Process Panel, placed in the main desk, as basic oper-ator interface. Additional operator and maintenance information is availablein the Process Portal - Compact HMI version.

• The brake operates as in a real mine hoist, both as normal stop brake andemergency brake. All braking is synchronised with start and stop of the skipin the mini shaft.

Torque motorLoad motor

Figure 2.7. The mini mine hoist model

In HoistLab there is no possibility to load and unload the skip, instead thisis simulated by a load motor and a torque motor, see Figure 2.7. These areconnected together with a common mechanical axis via a flexible coupling. Theyare behaving differently depending on the relations between the brake torque andthe load torque. At start the torque motor is speed controlled with zero revolutionsper minute as set point which results in that the model is standing still when the

2.4 Introduction to PID Controllers 11

drive motor gets its pre-torque. When the hoist torque is greater than the braketorque the torque should be controlled with a torque reference that correspondsto a real hoist torque. In the opposite situation, when the brake torque is greaterthan the hoist torque the motor is again speed controlled with zero revolutionsper minute. With this idea the model will be load simulated as it would be a realhoist.

2.4 Introduction to PID Controllers

This section gives an introduction to PID controllers and some of the basics incontrol theory that will be used in Chapter 5. PID controllers are today the mostcommon type of controllers that are used in industry. The abbreviation PID meansProportional, Integral and Derivative part. The input signal to the system, G, isu(t), see Figure 2.8. The difference between desired output signal and the realoutput signal, the control error, e(t) = r(t)− y(t), is the signal that the controllershould minimise.

F G

−1

Σr e u y

Figure 2.8. Control system

The equation for a PID controller is as follows, [1]

u(t) = K

e(t) +1TI

t∫t0

e(τ)dτ − TDddt

e(τ)

(2.1)

where K is the gain, TI is the integral time and TD the derivative time. Theadjustment of these parameters is often done by help of earlier experience thatare based on the following characteristics: The P-part controls the gain of thesystem and can result in a faster system but with the disadvantage of reducedstability, i.e. too high gain can cause instability. To eliminate the static fault, theI-part is needed. It can also give a faster system but with the same drawback asfor the proportional part. The D-part can improve the stability but can also givenoisy signals. Usually the PID-controllers are used as PI, e.g. TD is set to zero.The reason is that it is much easier to install this system because it is only twoparameters to adjust instead of four as in the PID while the D-part must includealso a low-pass filter [1]. The filter parameter, α, is discussed in (5.13) later on inthis report.

12 Preliminaries

A transfer function of a system is a practical way of representing its dynamicbehaviour. An example a transfer function is given by

G(s) =b0s

m + ... + bm

sn + a1s(n−1) + ... + an(2.2)

The roots in the numerator in (2.2) are the poles and in the analogous way theroots in the denominator are called zeros. The following functions will be a helpin the investigation that is treated in Chapter 5

Go = GF Gc =Go

1−GoS =

11 + G0

(2.3)

F is the controller, in this report of PI or PID design. The open-loop system, alsonamed the loop gain Go, is used to investigate the stability of the system. Gc iscalled the closed-loop system and is the relation between the reference signal andthe output signal. How the output signal is affected by disturbances is describedby the sensitivity function S, [3].

Another way of writing expressions in control theory is in a state space formwhich is a system of first order differential equations. It is easy to transformequations between these forms, especially when using programs such as Matlab.One advantage with the state space form is that it often is a result of physicalmodelling [2]. The general formula for a linear state space model is

x = Ax + Bu

y = Cx + Du (2.4)

Bode diagrams are charts where both the logarithm of the magnitude, |Go|,and phase, arg Go, are shown as separate graphs. Both have the same x-axiswhich is the logarithm of the frequency. The magnitude plot has gain in dB asy-axis and the phase plot has the unit degrees, see Figure 2.9. There is lots ofinformation contained in Bode diagrams, e.g. the phase margin, φm, which can beread at the frequency ωc where the loop gain crosses the logarithm 1 (=0). φm

shows the stability margin of the system, i.e. how much the amplitude curve canbe displaced before the system gets unstable. It is also possible to see resonancefrequencies.

A common design in a control system is to use a lead-lag compensator whichhas the purpose to improve the frequency response. The lead part, Flead, canincrease the phase curve at ωc so the φm gets larger. A phase margin less thanzero corresponds to an unstable system. The disadvantage with Flead is that highfrequency noise will be enlarged. The lead-part is a way of PD control with transferfunction

Flead(s) = KτDs + 1βτDs + 1

(2.5)

The lag part should be used when the stationary fault needs to be reduced. It hasthe following appearance

Flag(s) = KτIs + 1τIs + γ

(2.6)

2.4 Introduction to PID Controllers 13

−200

−180

−160

−140

−120

−100

−80M

agni

tude

(dB

)

10−1

100

101

102

103

104

105

−90

−45

0

45

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 2.9. Example of Bode diagrams

With γ = 0 this is a PI controller. The negative effect that comes with the lagpart is that the phase margin gets reduced. This has to be compensated for whencalculating the lead part. The complete PID controller of lead-lag design is writtenas, [3],

F (s) = KτDs + 1βτDs + 1

τIs + 1τIs + γ

(2.7)

This controller type with γ = 0 will be used in Section 5. A low value for β can givehigher phase margin and thereby a more stable system. The disadvantage is thatthe gain increases at high frequencies which can cause stress at some componentsif there is much noise.

τD can be decided when having a value for β and for the desired crossoverfrequency ωc,d, see the equation below.

τD =1

ωc,d

√β

(2.8)

The presented information is used in the following chapter of this Master Thesis.

14 Preliminaries

Chapter 3

Mathematical Description ofLoad Disturbance

When the skip is loaded the ropes gets stretched and oscillates. The mathematicaldescription of this should result in an expression for the force, F , acting on thedrum. In drum hoists it is only one total force but in friction hoists there is oneforce for each side of the drum, Fskipside and Fcwside.

The mathematical description of the ropes is made by modelling them as stiffbars connected by mass less springs and dampers which simulates the elasticity inthe rope. This is a way of modelling that is common, see for instance [5] or [6].To make the model simple the mine hoist is divided into parts, the number ofparts is depending on the hoist type. A spring and a damper is connected to eachrope section between drum and skip, drum and sheave, skip and bottom in frictionhoist etc. It is possible to extend the models and to split the sections into smallerparts but that is not necessary in this task.

The mathematical descriptions of load disturbance will be calculated for fourdifferent models: tower mounted single drum hoist, ground mounted single drumhoist, tower mounted friction hoist and ground mounted friction hoist.

It is not the whole weight from the rope mass, mri that affects the oscillationin the ropes. 1/3 of it can be seen as a point load at the bottom of the rope thatinfluence the spring force, Fspi, but the other 2/3 can be seen as an evenly dividedmass among the whole length that only affects the static force, mg. The derivationfor this approximation is given in Appendix A.

The following calculations will be used in the force and speed equations lateron and are therefore introduced here. The total rope mass is the density mass permeter multiplied with the length and the number of ropes

mr = mpmlrNr (3.1)

15

16 Mathematical Description of Load Disturbance

The spring factor, k, is equal to

k =ECs(2rr)2Nr

lr(3.2)

Where E is the elasticity module, r the rope radius and l the rope length. Theropes are made by many small lines which are twined around together so they arenot completely solid, this relation is described by the steel factor Cs.To get the damping coefficient, c, the spring factor is multiplied with a dampingfactor, Cd.

c = Cdk (3.3)

Both the spring- and the damping factors are used in the equations for forcecalculations later on and the expression is as follows [7]

F = −kx− cx (3.4)

The torque acting at the drum is the perpendicular distance from the force’s pointof attack, i.e. the drum radius, multiplied with the magnitude of the force. Themoment of inertia is equal to the acceleration times the inertia. The total torqueis the sum of the two torques

T = rdF + Jddt

(vd) (3.5)

∆payload is an expression for static extension, i.e. the final rope elongation whenthe oscillations have ended, at load changes [7]. The equation below is used whencalculating the initial conditions in Chapter 4.

∆payload =mg

k(3.6)

When the skip speed, vs, is integrated the result is the rope elongation, ∆r. Thisexpression works for line elongation caused both of load changes and speed changesand therefore it will be used in the mathematical description.

∆r =∫

vsdt (3.7)

The rope length after elongation is the default rope length, lr, added with theextra length ∆r

lnew = lr + ∆r (3.8)

Newtons second law, often called law of acceleration, is used to calculate the force,[7],

F =ddt

(mv) (3.9)

3.1 Tower Mounted Drum Hoist 17

3.1 Tower Mounted Drum Hoist

The mathematical model of the tower mounted drum hoist is simple. It includesone spring and one damper placed in the middle of the rope, see Figure 3.1.The speed of the spring is the difference between the drum speed at the top ofthe construction and the skip speed at the bottom. The given parameters in thecalculations are the following: drum speed vd, spring constant k, damping constantc, skip mass ms, payload mass mp and rope mass mr.

vd F

kc

skip vs

Fs

Figure 3.1. Tower mounted drum hoist

The total force F is the spring force added with 23 of the static rope force, as

shown in

F = Fsp +23mrg (3.10)

The reason for this is that 13 of the rope force already is calculated in the total

mass, see (3.14).To get the result of the total force the expression for the spring force has to becalculated first. It is this force that takes care of the elasticity of the ropes. Thespring and damping constants are naturally used in

Fsp = cvsp + k

∫vspdt (3.11)

The spring speed vsp that should be integrated is the difference between the drumspeed and the skip speed.

vsp = vd − vs (3.12)

The drum speed is known but the skip speed needs to be calculated. To do thatthe connection F = ma is used. When the skip force divided by the mass isintegrated the result is

vs =1

mΣs

∫Fsdt (3.13)


The mass mΣs, which is the sum of the skip weight, the payload and 13 of the rope

mass, is acting like a point load at the skip.

mΣs = ms + mp +13mr (3.14)

The force acting at the skip is

Fs = Fsp −mΣsg (3.15)

3.2 Ground Mounted Drum Hoist

The mathematical model of the ground mounted drum hoist has the same principleas for tower mounted one but there is ones sheave that divides the rope into twoparts. Therefore it will be two springs and dampers, see Figure 3.2(a). To simplify,the model the angle between the drum and the sheave is set to 180 degrees whichmeans that the drum is placed over the sheave, that is shown in Figure 3.2(b). Thedistance between them is constant and normally much shorter than the betweensheave and skip. There is also an approximation when the moment of inertia ofthe sheave is neglected. Even if the following equations are very similar to theones from the previous section they are all presented here for completeness.

The given parameters in the calculations are the following: drum speed vd,spring constants ki, damping constants ci, skip mass ms, payload mass mp andrope masses mri.

vsh

vd

k2

c2

k1

c1

skip vs

(a) Layout of a ground mounted drum hoist

vd F

k1

c1

k2

c2

vsh

Fsh

skip vs

Fs

(b) Simplified layout of aground mounted drum hoist

Figure 3.2. Ground mounted drum hoist

Just like in the previous section the total force, F , is the spring force added with23 of the rope force. mr1 is the mass for the rope between the drum and the sheavethat has constant distance.

F = Fsp1 +23mr1g (3.16)

3.2 Ground Mounted Drum Hoist 19

Also the spring force is calculated in the same way as above, the damping constanttimes the spring speed added with the spring factor multiplied with the integratedspring speed.

Fsp1 = c1vsp1 + k1

∫vsp1dt (3.17)

The spring speed is different compared to a tower mounted drum hoist. Now it isthe drum speed subtracted with the sheave speed

vsp1 = vd − vsh (3.18)

In this section the sheave has the same position as the skip in the tower mounteddrum hoist. Therefore this equation has the same appearance as (3.13).

vsh =1

13mr1

∫Fshdt (3.19)

The force acting at the sheave is the difference between the first and the secondspring force subtracted by 1

3 of the static force from the rope

Fsh = Fsp1 − Fsp2 −mr1g

3(3.20)

The following expression is the same as (3.17) but for the second spring

Fsp2 = c2vsp2 + k2

∫vsp2dt (3.21)

The speed for spring number two is the difference between two other velocities,the sheave speed subtracted by the skip speed

vsp2 = vsh − vs (3.22)

Also the following speed expression is of the same type as earlier

vs =1

mΣs

∫Fsdt (3.23)

Even in this case the mass mΣs is acting like a point load at the skip and theexpression is

mΣs = ms + mp +13mr2 (3.24)

Finally the skip force is the difference between the second spring force and thestatic force.

Fs = Fsp2 −mΣsg (3.25)


3.3 Tower Mounted Friction Hoist

The tower mounted friction hoist model has the same structure on both sides ofthe drum. There are two springs and dampers on each side. The total rope lengthis constant but the length of the head rope and the tail rope on each side will vary.To get a simple model the tail rope is split into two parts at the bottom. Thisis a good approximation because the rope length will be almost the same whenhanging straight as when going round. The rope that is over the drum is muchsmaller than the depth of the shaft. Both pictures of the rope lengths are shownin Figure 3.3.

C

D

A

B

(a) Rope lengths

C

D

A

B

(b) Simplified layout of rope lengths

Figure 3.3. Rope lengths of a friction hoist

The calculation of rope lengths is used when the system is in motion, i.e., whenthe drum speed is not equal to zero. To make the formulas more easy to followthe following terms will be used:

• A = Head rope length counterweight side

• B = Tail rope length counterweight side

• C = Head rope length skip side

• D = Tail rope length skip side

• lhr = Head rope length, A + C

• ltr = Tail rope length, B + D

• l = Total rope length, lhr + ltr

3.3 Tower Mounted Friction Hoist 21

C, lhr, ltr and l are known and A, B and D will be calculated. The head rope atcounterweight side A is the head rope length subtracted with head rope length atskip side

A = lhr − C (3.26)

The following expression uses the fact that the both sides of the drum have thesame total length. (3.26) is used after the first implication to solve B in knownvariables

A + B = C + D ⇔ lhr − C + B = C + D (3.27)

Then B is added on the both the left and right side. While B + D = ltr theexpression for B is as follows

lhr − C + B + B = C + B + D ⇔ B =ltr + 2C − lhr

2(3.28)

Finally the tail rope length at skip side is determined. With the previous expressionfor B inserted in the formula and the connection that l = lhr + ltr the result willbe

D = ltr −B = ltr −ltr + 2C − lhr

2=

l − 2C

2(3.29)

There are two sketches of the tower mounted friction hoist, both the normalone and the simplified, in Figure 3.4.

vd

k1

c1

k2

c2

skip vs

k3

c3

k4

c4

counterweight

(a) Layout of a towermounted friction hoist

vd F

k1

c1

k2

c2

vb

Fb

skip vs

Fs

k3

c3

k4

c4

counterweight

(b) Simplified layout of a towermounted friction hoist

Figure 3.4. Tower mounted friction hoist

The given parameters in the following calculations are: drum speed vd, springconstants ki, damping constants ci, skip mass ms, payload mass mp, counterweightmass mcw and rope masses mri. Firstly define F as


F = Fskipside − Fcwside (3.30)

It is the same principle for the calculation at both sides of the drum but onlythe ones for skip side will be described here. The differences are only that thedrum speed is defined negative at counterweight side, the mass m would not havethe same contents, see (3.35), and the index at ropes and springs will be 3 and 4instead of 1 and 2. The force at skip side is the force from the first spring addedwith 2

3 of the static force from the head rope

Fskipside = Fsp1 +23mr1g (3.31)

The spring force is calculated in the same way as for drum hoists, i.e.,

Fsp1 = c1vsp1 + k1

∫vsp1dt (3.32)

In the previous equation the speed for the first spring is needed. It is

vsp1 = vd − vs (3.33)

The skip speed is the result from the integration of the skip force divided with themass that acts like a point load at the skip

vs =1

mΣs

∫Fsdt (3.34)

There will be different expressions for the mass depending on which side of thedrum the calculations are made. The reason is that only the skip side can beloaded with payload. Both expressions are therefore shown below and the first isfor the skip side and the second for the counterweight side

mΣs = mp + ms +13mr1 (3.35a)

mΣcw = mcw +13mr3 (3.35b)

Then the total force that is acting at the skip has to be calculated. At this pointthe two spring forces work against each other so they will have different signs.Then 2

3 of the tail rope force has to be included to get the total result Fs

Fs = Fsp1 −(

Fsp2 +23mr2g

)−mΣsg (3.36)

The second spring force is calculated like the force at spring number one

Fsp2 = c2vsp2 + k2

∫vsp2dt (3.37)

3.4 Ground Mounted Friction Hoist 23

The spring speed is a difference between two other speeds, skip and bottom

vsp2 = vs − vb (3.38)

The speed at the bottom of the rope is calculated with the same principle as thespeed for the skip. The force is integrated and divided with 1

3 of the tail rope mass

vb =1

13mr2

∫Fbdt (3.39)

The last step is the force at the bottom which is the second spring force subtractedwith the 1

3 of the static force from the tail rope

Fb = Fsp2 −13mr2g (3.40)

3.4 Ground Mounted Friction Hoist

The difference between tower mounted and ground mounted friction hoists is sim-ilar as for drum hoists. There are two extra sheaves and thereby two more springsand dampers. In the same way as for drum hoists the angles between the drumand the sheaves are neglected. The model will be with the drum at the top andbelow one sheave at each side. Besides that the model has the same appearanceas the tower mounted one. The figures of the friction hoists are shown below inFigure 3.5

vsh

1

vd

vsh

2k

2c

2

k3

c3

k5

c5

k6

c6

k1

c1

k4

c4

skip vs

counterweight

(a) Layout of a ground mounted friction hoist

vd F

k1

c1

k2

c2

k3

c3

vsh

1

Fsh

1

vb

Fb

skip vs

Fs

k4

c4

k5

c5

k6

c6

counterweight

(b) Simplified layout of a groundmounted friction hoist

Figure 3.5. Ground mounted friction hoist


The given parameters in the calculations are the following: drum speed vd,spring constants ki, damping constants ci, skip mass ms, payload mass mp, coun-terweight mass mcw and rope masses mri. There are almost the same equationsfor the ground mounted friction hoist as for the tower mounted but it is extendedwith four more. To get a complete view they are all described below.The first equation defines F , the force difference


The equations are like in the tower mounted friction hoist almost the same for bothsides of the drum but only the ones for skip side will be described. The differencesare also this time that the drum speed is defined negative at counterweight side,the mass m will not have the same contents, see (3.50), and the indices at ropesand springs will be 4− 6 instead of 1− 3.The force from the skip side is

Fskipside = Fsp1 +23mr1g (3.42)

The first spring force is given by

Fsp1 = c1vsp1 + k1

∫vsp1dt (3.43)

The spring speed is the drum speed subtracted with the speed at the sheave. Atthe counterweight side the drum speed would be negative

vsp1 = vd − vsh (3.44)

In the previous equation the drum speed is known but the sheave speed has to becalculated. It is

vsh =1

13mr1

∫Fshdt (3.45)

The expression for the force acting at the sheave is next to be calculate. Thetwo spring forces work against each other and have different signs. The staticforce from 2

3 of the second head rope mass has to be added, because 13 is included

in (3.50a) later on

Fsh = Fsp1 −(

Fsp2 +23mr2g

)−mr1g (3.46)

The second spring force is calculated in exactly the same way as the first one,

Fsp2 = c2vsp2 + k2

∫vsp2dt (3.47)

The speed for the second spring is

vsp2 = vsh − vs (3.48)

3.4 Ground Mounted Friction Hoist 25

Just like for the other hoist types the skip speed is

vs =1

mΣs

∫Fsdt (3.49)

The expression for the mass is depending on which side of the drum it is. Atthe skip side the payload mass is included but not on the counterweight side.Otherwise it is the same with skip or counterweight mass added with 1

3 of thesecond rope mass

mΣs = mp + ms +13mr2 (3.50a)

mΣcw = mcw +13mr4 (3.50b)

The formula for the skip force has the same appearance as the sheave force butwith other indices. That depends on that both forces acts at a fixed point incontrast with the spring forces

Fs = Fsp2 −(

Fsp3 +23mr3g

)−mΣsg (3.51)

The last spring force is the one at the tail rope

Fsp3 = c3vsp3 + k3

∫vsp3dt (3.52)

The spring speed that is needed in previous equation is

vsp3 = vs − vb (3.53)

At the bottom of the tail rope the speed is

vb =1

13mr3

∫Fbdt (3.54)

Finally the bottom force is

Fb = Fsp3 −13mr3g (3.55)


Chapter 4

Implementation of LoadModels

This chapter includes the implementation part, both in Simulink and HoistLab.They are based on the mathematical model that has been explained in the previouschapter.

The parameter values that are used in the simulation are shown in Appendix B-1.The lengths of the ropes are 170 and 180 meters respectively which is much shorterthan the common lengths that normally are between 500 to 1500 meters. The rea-son for the short ropes is the scaling of the shaft in HoistLab that is 1:100. Theother values are typical for real hoists.

To make the system stable when there is no payload mass in the skip the initialconditions in the integrators, see (3.32) and (3.37), at both skip and counterweightside have to be set to balance the static force. Otherwise the simulation showsoscillations in the start phase and this is not realistic. The equation that is usedis found in (3.6).

4.1 Implementation in Simulink

The mathematical description should be implemented as a model in Simulink tomake it possible to simulate. The constants are written in a text file in Matlab.Simulink is a common program used during the development process when testingmodels. The results from Simulink models are then compared with the resultsfrom HoistLab.

The implementation is straight forward from the mathematical equations abovebut some things are worth to mention. It is important to use a suitable ode-solverto be able to run the simulations. In this case ode-23tb is chosen.

The different blocks in Simulink are built in a general way so they can be usedseveral times. So if the mathematical description would be extended with more

27

28 Implementation of Load Models

springs and dampers, as mentioned in Section 3 it would not be much extra workwith the implementation. A picture of one Simulink model is shown in Figure 4.1.

3

SkipSpeed

2

DrumSpeed

1

TotalForce

RopeMass

SkipMass

PayloadMass

StaticForce

2/3RopeForce

TotalMass

Static Force and Total Mass

ElasticityModule

RopeRadius

RelativeDamping

RopeLength

SteelConst

NrOfRopes

SpringConstant

DampingConstant

Spring Constant Calculation [N/m]

Force

TotalMassSkipSpeed

Skip Speed [m/s]

RopeLength

MassPerMeter

NrOfRopes

RopeMass

Rope Mass Calculation [kg]

ForceDiff

SkipSpeed

DrumSpeed

SpringConstant

DampingConstant

Force

Force Calculation

DrumSpeedRpm

DrumRadiusDrumSpeed

Drum Velocity Calculation [m/s]

Add

11

DrumRadius

10

RopeLength

9

NrOfRopes

8

MassPerMeter

7

SteelConst

6

RelativeDamping

5

PayloadMass

4

SkipMass

3

RopeRadius

2

ElasticityModule

1DrumSpeedRpm

Figure 4.1. One part of the model in Simulink

The payload is simulated with load steps in vector form. The drum speed isalso written in the same form so it is easy to edit and change.

4.2 Implementation in HoistLab

The implementation of the mathematical model in HoistLab laboratory is done bymodifying an existing program to get the desired functionality with line elongation.The available program had several functions where load simulation is one part andalso the one to be changed. The new function blocks are the same types as inSimulink and are shown in Figure 4.2.

In HoistLab the drum type is a tower mounted friction hoist so the implemen-tation is done for that type. The program is also prepared for a top mounted drumhoist and to edit the code to change between the models takes only a couple ofminutes.

The result from the simulation can be compared with the ones from Simulinkbut then it is important to remind that Simulink is a very good mathematicalprogram that gives ideal results. There are some differences from the simulationenvironment in Simulink compared to HoistLab. The code for the model shouldbe written in another programming language and there are also several things that

4.2 Implementation in HoistLab 29

Figure 4.2. One part of the model in PLC Control Builder

have to be modified to make it work. One reason is that the skip load is simulatedby a drive motor and not by real mass.

The PLC program had a predefined block for a simple integrator but it wasnot useful because it was running all the time and not only at load and unload asdesired. Therefore an own block was built and to get the desired functionality. Itwas enough with a simple solution that takes the actual value and adds it to theprevious value. Then this is multiplied with the time factor, dt.

Matlab and Simulink calculates with much higher accuracy than the pro-gram in HoistLab. This results in problem in the laboratory because the smallfaults from the rounding of multiplications with large forces at the size of 106

to 107 which leads to a significant fault. This fault can in some situations startreset windup. The problem occurs when the skip is standing still at some specificrope lengths. It is solved by making two extra blocks in the program that is work-ing when the system is not moving. They lock the static force value when the skipis still.

Another problem with the implementation was the large forces when the ropelengths are short. That depends of the fact that the spring factor k, see (3.2),


k =E·Cs(2r2

r)lr

is calculated by dividing with the rope length. The large forcesgive large speeds and fast changes and therefore it will be problem because of theslow execution time of 50 milliseconds in the program. It can not follow the quickchanges and therefore calculates incorrect values, which can lead to reset windup.The problem is solved by scaling the elasticity module E depending on what typeof rope and how long it is, see (3.34), (3.35) and (3.39) which are repeated below

vs = 1m

∫Fsdt, m = mp + ms + 1

3mr1, vb = 113 mr2

∫Fbdt

As shown in the equations vs is divided by a much larger mass than vb so theelasticity module for the head rope does not have to be as much scaled as the onefor tail rope. The scaling is also different between the head rope at the skip sideand the counterweight side because they do not have the same masses. The criticallengths were found during testing and so was also the necessary scaling. The testwas first made to manage to load and unload the payload at different lengths andthen for different drum speeds. The scaling could be done theoretically but therewas not enough time to do that. That means that the scaling procedure is onlyadapted at the specific values of the constants that are used in HoistLab, thoseare given in Appendix B. The scaling gives more oscillations in the ropes in thelaboratory than in reality and in Simulink. This is not a problem because if theworst case gives a acceptable solutions the other will be even better.

The last problem during implementation in HoistLab is that the speed referencefrom the drum is varying because of disturbances and noise. That is solved bya standard block of a first order single pole low-pass filter. They are often usedafter analog inputs. The variable in this block is a time constant that is set to0.1 seconds.

4.3 Results of the Implementation

In this section are plots of the differences between the normal model and themodel with scaling factor for a tower mounted drum hoist and a tower mountedfriction hoist during load and unload. The plots and simulations are made inSimulink because this program has more user friendly printing possibilities thanthe program in HoistLab. The lengths and scaling factors are adjusted to thevalues in HoistLab and are written in Table B-2 in Appendix B.

4.3 Results of the Implementation 31

The first Figure 4.3 shows a drum hoist at the length of 10 meters at unloading.As mentioned above the scaled model gives more oscillating result than the other.This difference is not insignificant because the scaled line shows higher amplitudeand two swings instead of one which is clear in the zoomed plot. To get a betterpicture of the difference Figure 4.4 shows a zoomed plot of the same result.

13 14 15 16 17 18

2

2.5

3

3.5

4

x 105

Time [s]

For

ce [N

]

normalscaled

Figure 4.3. Drum hoist at length 10 meter, unloading

14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16

2

2.2

2.4

2.6

2.8

3

3.2

x 105

Time [s]

For

ce [N

]

normalscaled

Figure 4.4. Zoomed figure of drum hoist at length 10 meter, unloading


When the skip is loaded it is at the length of 160 meters, see Figure 4.5. At thatpoint no scaling is necessary so no differences will appear and no zoomed plot isneeded.

3 4 5 6 7 8 92.5

3

3.5

4

4.5

5

x 105

Time [s]

For

ce [N

]

normalscaled

Figure 4.5. Drum hoist at length 160 meter, loading

13 14 15 16 17 18

−16

−14

−12

−10

−8

−6

−4

−2

0

2

x 104

Time [s]

For

ce [N

]

normalscaled

Figure 4.6. Friction hoist at length 10 meter, unloading

Then the same tests are done for the tower mounted friction hoist. Figure 4.6shows the unloading at the head rope length at 10 meters for the skip side. Whentaking a closer look to the y-axis the force is negative when the payload is set tozero. This depends on the hoist construction with tail ropes. There are the sametype of difference between the scaled and the non scaled force that in the picture

4.3 Results of the Implementation 33

of the drum hoist. When looking at the enlarged plot in Figure 4.7 it shows almostthe same size at the difference as for the drum hoist.

14.5 15 15.5 16

−1.6

−1.4

−1.2

−1

−0.8

−0.6

x 105

Time [s]

For

ce [N

]

normalscaled

Figure 4.7. Zoomed figure of friction hoist at length 10 meter, unloading

Finally Figure 4.8 shows the comparison at the friction hoist at loading at 160meters. At this point it is hard to see any differences. It is only when the plot getenlarged as in Figure 4.9 that it is possible to see two lines instead of one.

3 4 5 6 7 8 9

−1

−0.5

0

0.5

1x 10

5

Time [s]

For

ce [N

]

normalscaled

Figure 4.8. Friction hoist at length 160 meter, loading


5.265 5.27 5.275 5.28 5.285 5.29

9.1

9.15

9.2

9.25

9.3

9.35

9.4

9.45

x 104

Time [s]

For

ce [N

]

normalscaled

Figure 4.9. Zoomed figure of friction hoist at length 160 meter, loading

Chapter 5

Investigation of theDerivative Part in PIDControllers

This part of the Master Thesis is different from the previous chapters. It has thesame purpose which is to improve driving performance of mine hoists, but it is aninvestigation about how the derivative part in PID controllers will influence theoperation performance. More precisely it is speed control on a drive motor duringstart when the brakes are released.

The control of a mine hoist uses different controllers during the different partsof the production cycle. In normal running the speed control follows a predefinedreference signal but at start and stop situations when the brakes are involvedanother controller is needed. When the skip is standing still in those positions thebrakes are active and the motor is not. While the brakes are released the motortorque increases until it balances the load torque. During this period the skip ismoving in an unwanted way, called rollback. The coming sections describe how thederivative part in a PID-controller can reduce this rollback and thereby improvethe performance.

The simulation of brake release is done by a step instead of a ramp as inthe reality. Normally the ramp takes between 0.5 and 1 second so it will give asmoother motion than in simulations.

The hoist type that is investigated in this part of the Master Thesis is the towermounted friction hoist but it is the same principle for all four hoist types. Thevalues of the rope lengths and masses are limited to realistic values when lookingat for instance stability for the load system.

35

36 Investigation of the Derivative Part in PID Controllers

5.1 State Space Model for Motor and Load

The load system is written in the state space form, see (2.4), to make it easy tosimulate together with the controller. The states in Table 5.1 are speeds and forcesand those are enough to describe the whole load model.

System Model Descriptionparameter parameterx1 ω Drum speed [rad/s]x2 vs Skip speed [m/s]x3 vbs Bottom speed skip side[m/s]x4 k1

∫(rdx1 − x2)dt Force, spring 1 [N]

x5 k2

∫(x2 − x3)dt Force, spring 2 [N]

x6 vcw Counterweight speed [m/s]x7 vbcw Bottom speed counterweight side [m/s]x8 k3

∫(−rdx1 − x6)dt Force, spring 3 [N]

x9 k4

∫(x6 − x7)dt Force, spring 4 [N]

Table 5.1. System and model parameters

As in Chapter 4, the system should be in balance at start, i.e., compensatedfor the static force at both skip- and counterweight side. Otherwise the simulationshows unrealistic oscillations. The compensation is done by changing the initialconditions, which by default are set to zero, for the states that affects the force.They are

x40 = mΣsg (5.1)

x50 = mr2g (5.2)

x80 = mΣcwg (5.3)

x90 = mr4g (5.4)

The input signals to the system are described in Table 5.2.

System Model Descriptionparameter parameteru1 Tm Motor torque [Nm]u2 g Gravitation [m/s2]

Table 5.2. System input signals

The second input signal for u is the gravitation that together with mass will resultin the static force. The reason for having g as a signal is that the equations willbe easier to analyse.

5.1 State Space Model for Motor and Load 37

The motor torque is the signal that controls the drum speed and indirectly theother velocities as well. In the equation below the relation between the torque andthe force is described

u1 = Tm = Frd + Jω ⇒ ω =u1 − Frd

J(5.5)

where J = Jm + Jd.The equation for the force, F , is known from the mathematical model in Section 3


Equations (3.31), (3.32) and (3.33) give the expressions for Fskipside and Fcwside

and the speeds and the positions are replaced by the states which gives this re-sulting formula for F

F = c1(rdx1 − x2) + x4 − c3(−rdx1 − x6)− x8 +23u2(mr1 −mr3) (5.7)

The equation for x1 uses (5.5) and (5.7) to get the expression written with statesand input signals.

x1 =1J

(u1 − rd

(c1(rdx1 − x2) + x4 − c3(−rdx1 − x6)− x8 +

23(mr1 −mr3)u2

))(5.8)

All the equations for the states and their derivatives are written in Appendix C.Here follows the A-matrix which represents the coefficients before the xi

A =

r2d(−c1−c3)

Jc1rd

J 0 − rd

J 0 − c3rd

J 0 rd

J 0c1rd

mΣs

−c1−c2ms

c2mΣs

1mΣs

− 1mΣs

0 0 0 00 3c2

mr2− 3c2

mr20 3

mr20 0 0 0

k1rd −k1 0 0 0 0 0 0 00 k2 −k2 0 0 0 0 0 0

− c3rd

mΣcw0 0 0 0 − c3+c4

mΣcw

c4mΣcw

1mΣcw

− 1mΣcw

0 0 0 0 0 3c4mr4

−3c4mr4

0 3mr4

−k3rd 0 0 0 0 −k3 0 0 00 0 0 0 0 k4 −k4 0 0

(5.9)

The B-matrix includes the coefficients before the two input signals u1 and u2

B =

[1/J 0 0 0 0 0 0 0 0

− 2(mr1−mr3)rd

3J − 2/3mr2+mΣs

mΣs−1 0 0 2/3mr2−mΣcw

mΣcw−1 0 0

]T

(5.10)The C-matrix describes the output signals. As drum speed and skip speed are theonly output signals the size will be 2× 9

C =[rd 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0

](5.11)


The matrix for D includes only zeros and is therefore not shown. The reason isthat the input signals do not directly affect the output signals. They influencethrough the derivatives of xi.

5.2 The Control System

The control system is built in a text file in Matlab and simulation blocks inSimulink, see Figure 5.1. A converter model is placed between the controller and

Dspeedm/sDspeed_rpm

DSpeed

eSSpeed

SkipPos

DrumPos

MotorTorque

SkipSpeed

DSpeed

Motor+Load

1s

Integrator2

1s

Integrator

DrumSpeedRef TorqueRef MotorTorque

Converter

DrumSpeedDiff TorqueRef

Controller

Figure 5.1. Overview of the controller design

the load model. It represents the closed-loop system between torque reference andmotor torque as a second order system with time delay. In the simulations the timedelay Td has, however, been approximated with a first order Padé approximation,yielding the converter model

Gconv =ω2(−Td

2 s + 1)

(s2 + 2ζωs + ω2)(

Td

2 s + 1) (5.12)

In the simulator the value for Td = 4 ms, ω = 497 rad/s and ζ = 0.59.The transfer function for the motor is included in the load block. It is a

simplification because it includes only a moment of inertia and not a time delay.But as the purpose is to see the differences between PI and PID control, the mostimportant is similarity for both types.

The rollback caused by the brake release, described in the introducing text, istoday controlled by a PID controller. The transfer function is given by

Fa(s) = Ka

(1 +

1TIs

+TDs

αTDs + 1

)(5.13)

The expression for the controller can also be written in another way that is suitable

5.2 The Control System 39

when using lead-lag design

Fm(s) = Km

(τDs + 1βτDs + 1

)(τIs + 1τIs + γ

)(5.14)

The indices for F , a and m, are used to separate the different controller types.The transfer function (5.13) can have complex zeros in contrast to (5.14) whichmeans that Fm always can be written as Fa. The opposite is not always possible.

As mentioned before, the D-part is often not used and thereby set to zerowhich gives the controller a PI-function. The two expressions for Fa and Fm areidentified when TI = τI , Ka = Km and γ = 0.

Inside the ACS 800 the calculations are done with scaled variables, i.e. drumspeed and motor torque are represented as a percentage of their nominal valuesnnom [rpm] and Tnom [Nm]. The controller gains (Ka or Km) are thus from %to %. The control error is in rpm and the gain factor K has to be scaled to go toNm. This is done by the gain factor K

K = KmTnom

nnom(5.15)

If the control error is equal to the nnom [rpm] the torque will be the nominaltorque, Tnom [Nm], if K = 1.

The lead-lag design is used at the expression Fm and the following equationsmay be used to convert the constants to be suitable for Fa [3]

TI = τI + (1− β)τD (5.16)

TD = τD + (τI

TI− β) (5.17)

Ka = Km(TI

τI) (5.18)

α =βTI

τI(5.19)

The interval of the rope lengths in this investigation is between 500 and 1500meters. To see the differences in the system for these lengths, with constantpayload, the Bode diagrams are shown in Figure 5.2. At constant length andvariable payload the Bode diagrams for the system will be as in Figure 5.3.Both the Bode diagram show two resonance frequencies. Those represent the twohead ropes. The two tail ropes resonance frequencies are so small that they cannot be seen, but they affect the other two anyway.

The control is done with an initial value for the torque, called a pre-torque.This is used in real mine hoists to make the control faster when it starts froma realistic torque instead of zero. The value for the pre-torque is the same as


Bode Diagram

Frequency (rad/sec)

Pha

se (

deg)

Mag

nitu

de (

dB)

−200

−180

−160

−140

−120

−100

−80500 m700 m900 m1100 m1300 m1500 m

10−1

100

101

102

103

104

105

−90

−45

0

45

Figure 5.2. Bode diagrams for different lengths and constant payload

Bode Diagram

Frequency (rad/sec)

Pha

se (

deg)

Mag

nitu

de (

dB)

−200

−180

−160

−140

−120

−1005000 kg7000 kg9000 kg11000 kg13000 kg15000 kg

100

101

102

103

104

105

−90

−45

0

45

Figure 5.3. Bode diagrams for different payload and constant lengths

5.3 Results of the Controllers 41

for the static force that is acting at the drum subtracted with the force causedby the payload. This is added to the output signal from the converter. Whenstudying the torque signal it first moves in the wrong direction for a short time,about 30 milliseconds, before it increases. This depends on that the system seesthe pre-torque signal as a disturbance and starts to control it.

5.3 Results of the Controllers

When doing the parameter settings in this task the total rope length for each sideof the drum is chosen to 1020 meters and the skip position is 1000 meters down inthe shaft. The commissioning engineers have some initial values to start with whenthey install the control system. This investigation start with those for Km and TI .Then the task is to improve the result to get the best possible performance, in thiscase a well-damped system with one overshoot. In other words, smooth curvesof the position changes is desired instead of a fast control that can minimise themotion but at the same time increase the oscillations. The reason for this priorityis that the system should have a stable appearance and that it will result in lesswear of the equipment.

The controller is designed to fulfil the desired characteristic for the drum speedand especially the drum position while the drum speed is the reference. Theappearance at the curves for both the torque and the speeds depends on the gainfactor Km that is scaled with the nominal values depending off how big the distanceis from the actual point to the reference point.

5.3.1 Results of the PI Controller

The results for the speeds and position changes in the PI controller are shown inFigure 5.4. The settings τI = 0.7 and Km = 60 gave the best appearance for thewell-damped system.The rollback for the drum is about 4.8 centimetres and the overshoot is roughly3 millimetres. It can be seen that the skip has more oscillated motion than thedrum. The motor torque signal is verified with the same settings to see that thechanges are not too fast, see Figure 5.5. The drive system is fast enough so noproblem occurs. It can also be mentioned that the pre-torque is shown in thefigure.

The Bode diagram for the open system Go can be seen in Figure 5.6. The phasemargin is 51.9 degrees which means that the system has an acceptable stabilitymargin. The crossover frequency is 1.77 rad/s. The plots for Gc and S are inAppendix D. The magnitude in the Bode diagrams for Gc is equal to 0 dB (= 1)for frequencies between 10−7 and 1 rad/s. This means that it can follow thereference signal well for these frequencies. S describes how the output signal isaffected by disturbances. At frequencies below 1 rad/s the disturbances are wellsuppressed. Between about 1 and 3 rad/s the disturbances will be amplified with amaximum of 1 dB and over 3 rad/s the disturbance will again be well suppressed.


0 1 2 3 4 5 6 7−0.1

−0.05

0

0.05

0.1

Time [s]

Spe

ed [m

/s]

Speeds for the system with PI controller

0 1 2 3 4 5 6 7−0.06

−0.04

−0.02

0

0.02

Time [s]

Leng

th [m

]

Positions for the system with PI controller

Drum speedSkip speed

Drum positionSkip position

Figure 5.4. Speed and rollback results for the PI controller

0 1 2 3 4 5 6 73

3.5

4

4.5

5

5.5x 10

5

Time [s]

Tor

que

[Nm

]

Motor Torque for the system with PI controller

Figure 5.5. Speed and rollback results for the PI controller


−100

−50

0

50

100

150

200

Mag

nitu

de (

dB)

Bode plots for Go

Frequency (rad/sec)10

−210

−110

010

110

210

310

4−180

−135

−90

−45

0

45

System: GoPhase Margin (deg): 51.9Delay Margin (sec): 0.512At frequency (rad/sec): 1.77Closed Loop Stable? YesP

hase

(de

g)

Figure 5.6. Bode diagrams for Go for the PI controller

5.3.2 Results of the PID Controller

The design for the PID controller has partly been done according to lead-lag [3].The values for the desired phase margin, φm,d, and crossover frequency, ωc,d,are decided based on the results for the PI controller. The phase margin shouldincrease with at least 20 degrees to make the system more stable. An extra phasemargin of 8 degrees is added to prepare for the lag-part. Totally 28 degrees extraphase margin gave the result β = 0.35, according to Figure 5.13 in [3].The new crossover frequency should increase to make the closed-loop system faster.The desired value was chosen to 3 rad/s. Then (2.8) is used to calculate τD = 0.56.When this lead-part is designed there are extra margins and therefore possible toadjust the values for τI and Km to reach better performance. This was done bychoosing τI = 0.5 and Km = 80.

The speed and position results of the PID designed controller are shown inFigure 5.4. The rollback for the drum is about 2.6 centimetres and the overshootis roughly 2 millimetres. It is obvious that the drum moves less than the skip.

As for the PI controller the motor torque is investigated for the PID settingsto see that there are not too fast changes, see Figure 5.8. Like in the previouscase the drive system is fast enough to manage the control. The pre-torque is alsoshown in this figure.

The Bode diagram for the open system Go can be seen in Figure 5.9. Thephase margin shows that the stability margin has indeed been increased for thiscontroller type, 87.2 degrees, and the crossover frequency is 2.71 rad/s. The plots


0 1 2 3 4 5 6 7−0.05

0

0.05

0.1

0.15

Time [s]

Spe

ed [m

/s]

Speeds for the system with PID controller

0 1 2 3 4 5 6 7−0.03

−0.02

−0.01

0

0.01

Time [s]

Leng

th [m

]

Positions for the system with PID controller

Drum speedSkip speed

Drum positionSkip position

Figure 5.7. Speed and rollback results for the PID controller

0 1 2 3 4 5 6 70

1

2

3

4

5

6x 10

5

Time [s]

Tor

que

[Nm

]

Motor Torque for the system with PID controller

Figure 5.8. Speed and rollback results for PID controller


−200

−100

0

100

200

300M

agni

tude

(dB

)

Bode plots for Go

Frequency (rad/sec)10

−210

−110

010

110

210

310

410

5−90

0

90

180

270

360

450

System: GoPhase Margin (deg): 87.2Delay Margin (sec): 0.563At frequency (rad/sec): 2.71Closed Loop Stable? Yes

Pha

se (

deg)

Figure 5.9. Bode diagrams for Go for the PID controller

for Gc and S are found in Appendix D. The appearance for Gc is almost the sameas for the PI controller but with a smoother start at the low frequencies. S showsthat the disturbances are well suppressed.

Figure 5.10 makes it easy to compare the results from the PI and the PIDcontrollers.It is obvious that the PID design gives better performance. However the derivativepart makes the system more sensitive for noise. As the simulation is implementedin Simulink and not in HoistLab the disturbances do not appear naturally. There-fore an extra noise block is inserted in Simulink to make a noise comparisonbetween the PI and the PID controller. The noise signal is added to the controlerror, see Figure 5.11 where the curve for the PI controller is on top. The resultfor the motor torque is shown in Figure 5.12 and for the drum and skip speed inFigure 5.13. The PID design gives more disturbance sensitive systems, but at thisdisturbance level it does not appear any noise at the skip speed, drum position orthe skip position, see Figure 5.14.


0 1 2 3 4 5 6 7−0.06

−0.04

−0.02

0

0.02

Time [s]

Leng

th [m

]Drum position comparison

Drum position PIDrum position PID

0 1 2 3 4 5 6 7−0.06

−0.04

−0.02

0

0.02

Time [s]

Leng

th [m

]

Skip position comparison

Skip position PISkip position PID

Figure 5.10. Position comparison between PI and PID

0 1 2 3 4 5 6 7−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time [s]

Spe

ed [m

/s]

Control error with noise

Control error PIControl error PID

Figure 5.11. Control error affected of noise


0 1 2 3 4 5 6 73

4

5

6x 10

5

Time [s]

Tor

que

[Nm

]Motor torque with noise, PI

0 1 2 3 4 5 6 70

2

4

6x 10

5

Time [s]

Tor

que

[Nm

]

Motor torque with noise, PID

Figure 5.12. Motor torque affected of noise

0 1 2 3 4 5 6 7−0.06

−0.04

−0.02

0

0.02

0.04

Time [s]

Spe

ed [m

/s]

Drum speed with noise

Drum speed PIDrum speed PID

0 1 2 3 4 5 6 7−0.06

−0.04

−0.02

0

0.02

0.04

Time [s]

Spe

ed [m

/s]

Skip speed with noise

Skip speed PISkip speed PID

Figure 5.13. Drum and skip speed affected of noise


0 1 2 3 4 5 6 7−0.06

−0.04

−0.02

0

0.02

0.04

Time [s]

Spe

ed [m

/s]

Drum position with noise

0 1 2 3 4 5 6 7−0.06

−0.04

−0.02

0

0.02

0.04

Time [s]

Spe

ed [m

/s]

Skip position with noise

Drum position PIDrum position PID

Skip position PISkip position PID

Figure 5.14. Drum and Skip position affected of noise

5.4 Advantages and Disadvantages with theDerivative Part

As described in Section 2.4 the derivative part can improve the stability of thesystems. This makes it possible to change the other parameters, τI and Km to geta better performance, in this case less rollback. This is naturally a big advantageand as mentioned before shown in Figure 5.10.

There are several disadvantages with the derivative part and the main is thatthe system gets more sensitive for noisy signals when the D-part is used. This hasalso clearly been shown in the previous section. Another negative part is that it ismore difficult and takes more time to do the parameter tuning when the derivativepart is used, since the PID controller needs four parameters to be found insteadof two. Almost all mine hoist systems are different so it is not possible to use thesame default parameter settings.

Finally it could be a disadvantage that the derivative part hardly ever hasbeen used before in mine hoist systems while the existing rollback was found goodenough. It is easier to do the same way as before than thinking in new ways.

5.5 Future for the PID Controller Investigation 49

5.5 Future for the PID Controller Investigation

The PID controller investigation presented in this chapter has taken the form of apilot study. The potential for better control when the derivative part is used hasbeen shown. To get this work in the reality a more automatic system for parametertuning is needed. This can, for example, be solved as an optimisation problem.The model that gives the drum speed and skip speed is already made but it has tobe completed with an optimisation routine that minimises the expression for theabsolute value of the skip speed, |

∫vsdt|, while it is the position error that should

be as small as possible.


Chapter 6

Conclusions

In this chapter the conclusions for the Master Thesis will be presented. Since thereport is divided in to two separate parts the sections below will follow the samestructure.

6.1 Conclusions of the Rope Elongation Part

The purpose of the first part of this Master Thesis is fulfilled. A mathematicalmodel of the elongation in the ropes at a mine hoist has been made for four typesof mine hoists. Mass less springs and dampers are used to get the elastic behaviourof the ropes. One third of the rope masses are counted as a point load and the restis only affecting the static force. The mathematical description gives a sufficientlygood result even though several approximations were made.

The mathematical model is implemented in Matlab and Simulink for allfour hoist types to make load simulations possible. The constants are written ina text file in Matlab. The different blocks in Simulink are built in a generalway so they can be used several times. If the mathematical description would beextended with more springs and dampers it would not be much extra work withthe implementation.

The implementation of the mathematical model in HoistLab laboratory wasmade by modifying an existing program with the line elongation functionality.It is only done for the tower mounted friction hoist. The new function blocksare of the same types as in Simulink. There were several functions that had tobe modified to make the simulations realistic, for instance a scale factor for theelasticity module was introduced to prevent problems occurring caused by too highforces and too slow execution speed.

51

52 Conclusions

6.2 Conclusions of the Controller Part

The first part of the purpose with the PID investigation chapter was to design aPI controller that results in an acceptable rollback when the brakes are released.It has been done and the result is shown in Section 5.3.1. The size of the rollbackis about 4.8 centimetres. It is obvious that the drum moves less than the skipwhich depends on that it is the drum speed that is the reference while the skipspeed, as mentioned before, can not be measured.

When the PI design was done the next step was to design a PID controllerfor the same purpose. This has been done by partially using lead-lag design. Theresults are presented in Section 5.3.2. The rollback is less than before, 2.6 cen-timetres. It is hence obvious that the PID design gives better performance. Butas mentioned before the derivative part has the disadvantage that the system getsmore sensitive to noise.

To fulfil the whole purpose the task was to decide if it is worth doing furtherinvestigations about how the derivative part will improve the drive performances.The recommendation that can be made with consideration to the results is yes. Ithas been shown that there can be significant improvements but it is still too uncer-tain which problems the noise may cause. This can be checked through implemen-tation in HoistLab. The potential of the derivative part can be more investigatedby for instance finding an optimisation routine, mentioned in Section 5.5.

Bibliography

[1] T. Glad, S. Gunnarsson, L. Ljung, T. McKelvey, A. Stenman, and J. Löfberg.Digital styrning kurskompendium. Institutionen för systemteknik, Linköping,2003. (In Swedish).

[2] T. Glad and L. Ljung. Reglerteori Flervariabla och olinjärametoder. Studentlitteratur, Linköping, second edition, 2003.www.studentlitteratur.se/6547-02 (In Swedish).

[3] T. Glad and L. Ljung. Reglerteknik Grundläggande teori. Studentlitter-atur, Linköping, fourth edition, 2006. www.studentlitteratur.se/1789-04(In Swedish).

[4] B. Åkesson, H. Tägnefors, and O. Johannesson. Böjsvängande balkar ochramar. Awe/Gebers, Stockholm, second edition, 1977. (In Swedish).

[5] C. Kumpulainen. Studium av lastpendlingar vid styrning av gruvspel.Linköpings tekniska högskola, Linköping, 1994. (In Swedish).

[6] J. Löfmark. Rope Oscillation Control for Hoists. Royal institute of Technology,Stockholm, 1998.

[7] A. Pytel and J. Kiusalaas. Engineering Mechanics Dynamics. Thomson Learn-ing, second edition, 2001.

53

http://www.studentlitteratur.se/6547-02

http://www.studentlitteratur.se/1789-04

54 Bibliography

Appendix A

Rope Weights

The following calculations are made to decide the relation between the part of ropemass that affects the spring force, and thereby the rope oscillations, and the partthat only influences the static force. The figure below illustrates a tower mounteddrum hoist which is a good and simple example to base the following equationsupon.

E, A L, ml

skip ms

Figure A-1. Definitions for calculation of rope weight

Some definitions are needed to understand the coming calculations and those arelisted below.The first equation gives the relation between the skip mass and the rope mass. Ituses the fact that the rope mass is the rope mass per meter multiplied with thelength.

ms = αml = αmL (A-1)

According to Åkesson et.al [4] the resonance frequency, ω comes from the equationbelow. Both ω and ν represents the frequency but they have different units [rad/s]respective [Hz].

55

56 Rope Weights

Definitions DescriptionA Cross section areaE Elasticity moduleρ DensityL Rope lengthm = ρA Rope mass per meterα Relation between skip and rope mass

Table A-1. Definitions for rope mass calculation

ω = ν

√EA

mL2(A-2)

The following equation has the same source as above and gives a relation betweenthe frequency ν and the mass relation between rope and skip α, see [4]

tan ν − 1αν

= 0 (A-3)

The spring constant k is calculated in the same way as in Chapter 3 but this timethe steel constant is ignored

k =EA

L(A-4)

The expression for k put into (A-2) gives the following result

ω =

√ν2k

mL(A-5)

As the task is to calculate the relation between the part of rope mass that affectthe spring force and the part that only influence the static force, a spring with themass M hanging below is needed. The equation for this resonance frequency is asfollows

ω =

√k

M(A-6)

When the two expressions for ω are compared it results in an equation for themass

M =mL

ν2(A-7)

The mass M is the total point load mass hanging under the spring, i.e. the wholeskip load and the part of the rope mass that affects the spring force, β. (A-1) isused for the relation of the skip mass ms

M = ms + βml = (α + β)mL (A-8)

57

The expression for α + β using (A-7) is as follows.

α + β =M

mL=

1ν2⇒ β =

1ν2− α (A-9)

To make this connection easy to understand a plot of β as a function of α is shownin Figure A-2. The normal value for α in mine hoists is about ten, which meansthat 1/3 is a good approximation for the relation between the part of rope massthat affects the spring force and the part that only influences the static force.

0 10 20 30 40 50 60 70 80 90 1000.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0.41

0.42

α

β

β(α)

Figure A-2. How big part of the rope weight that affects oscillations

Appendix B

Implementation

B.1 Values of Constants at Implementation

The values in Table B-1 are used in Chapter 4, Implementation of Load Models.

Constants Valuems 25 300 [kg]mcw 38 300 [kg]mp 15 000 [kg]lhr 170 [m]ltr 180 [m]rhr 0.03 [m]rtr 0.03 [m]rd 3.5 [m]Jd 41 120 [kgm2]Jm 12 400 [kgm2]mpm,hr 5.862 [/m]mpm,tr 8.175 [/m]Nrhr 4Nrtr 3Cd 0.05Cs 0.405Ehr 1.1 E+11 [Pa]Etr 1.1 E+11 [Pa]Nnom 50 [rpm]Tnom 400000 [Nm]

Table B-1. The predefined values at constants in HoistLab

58

B.2 Values for the Scaling Factors 59

B.2 Values for the Scaling Factors

It is assumed that the skip will be loaded at 10 meters and unloaded at 160 meters.Below are the scaling factors, S, at this lengths for a top mounted drum hoist, D,and friction hoist, F .

Hoist lHRs lTRs lHRcw lTRcw SHRs STRs SHRcw STRcw

D 10 - - - 0.2 - - -D 160 - - - 1.0 - - -F 10 160 165 15 0.2 1.0 0.18 0.0015F 160 10 15 165 1.0 0.4 0.0015 0.15

Table B-2. Scale values for elasticity module at different rope lengths

Appendix C

State Space Form

This section includes all the equations for the states and their derivatives.

x1 = ω (C-1)x2 = vs = /(3.34), (3.36)/ =

=1

mΣs

∫ (c1(rdx1 − x2) + x4 − c2(x2 − x3)− x5 −

(23mr2 + mΣs

)u2

)dt

(C-2)x3 = vbs = /(3.39), (3.40), (3.37)/ =

=1

13mr2

∫ (c2(x2 − x3) + x5 −

13mr2u2

)dt

(C-3)

x4 = k1

∫(rdx1 − x2) dt (C-4)

x5 = k2

∫(x2 − x3) dt (C-5)

x6 = vcw = /(3.34), (3.36)/ =

=1

mΣcw

∫ (c3(−rdx1 − x6) + x8 − c4(x6 − x7)− x9 −

(23mr4 −mΣcw

)u2

)dt

(C-6)x7 = vbcw = /(3.39), (3.40), (3.37)/ =

=1

13mr4

∫ (c4(x6 − x7) + x9 −

13mr4u2

)dt

(C-7)

x8 = k3

∫(−rdx1 − x6) dt (C-8)

x9 = k4

∫(x6 − x7) dt (C-9)

60

61

The derivatives of the states are as follows

x1 =1J

(u1 − rd

(c1(−rdx1 − x2) + x4 − c3(rdx1 − x6)− x8 +

23(mr1 −mr3)u2

))(C-10)

x2 =1

mΣs

(c1(rdx1 − x2) + x4 − c2(x2 − x3)− x5 −

(23mr2 + mΣs

)u2

)(C-11)

x3 =1

13mr2

(c2(x2 − x3) + x5 −

13mr2u2

)(C-12)

x4 = k1(rdx1 − x2) (C-13)x5 = k2(x2 − x3) (C-14)

x6 =1

mΣcw

(c3(−rdx1 − x6) + x8 − c4(x6 − x7)− x9 −

(23mr4 −mΣcw

)u2

)(C-15)

x7 =1

13mr4

(c4(x6 − x7) + x9 −

13mr4u2

)(C-16)

x8 = k3(−rdx1 − x6) (C-17)x9 = k4(x6 − x7) (C-18)

Appendix D

Plots from the Results inSection 5.3

This appendix includes some of the plots from the results Section 5.3. Figure D-1shows Gc and Figure D-1 shows S and for the PI controller. Figure D-3 shows Gc

and Figure D-4 shows S for the PID controller.

−60

−40

−20

0

20

Mag

nitu

de (

dB)

10−6

10−4

10−2

100

102

104

−180

−135

−90

−45

0

45

Pha

se (

deg)

Bode plots for Gc

Frequency (rad/sec)

Figure D-1. Bode plot for Gc for the PI controller

62

63

−100

−80

−60

−40

−20

0

20M

agni

tude

(dB

)

10−2

10−1

100

101

102

103

104

−45

0

45

90

135

180

Pha

se (

deg)

Bode plots for S

Frequency (rad/sec)

Figure D-2. Bode plot for S for the PI controller

−200

−150

−100

−50

0

50

Mag

nitu

de (

dB)

10−6

10−4

10−2

100

102

104

−180

−90

0

90

180

270

360

Pha

se (

deg)

Bode plots for Gc

Frequency (rad/sec)

Figure D-3. Bode plot for Gc for the PID controller

64 Plots from the Results in Section 5.3

−100

−80

−60

−40

−20

0

20

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

103

104

−45

0

45

90

135

180

Pha

se (

deg)

Bode plots for S

Frequency (rad/sec)

Figure D-4. Bode plot for S for the PID controller

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Institutionen för systemteknik - Linköping University · Institutionen för systemteknik Department of Electrical Engineering Examensarbete Load Simulation and Investigation of