Load Simulation and Investigation of PID Control for Resonant Elastic Systems

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Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Load Simulation and Investigation of PID Control

for Resonant Elastic Systems

Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping

av

Sara Lundin

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

Linköping 2007

Department of Electrical Engineering Linköpings tekniska högskola

Linköpings universitet Linköpings universitet

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Load Simulation and Investigation of PID Control

for Resonant Elastic Systems

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Sara Lundin

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

Handledare: Sami Saari

ABB Mining

Daniel Petersson

Linköpings universitet

Examinator: Alf Isaksson

Linköpings universitet

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Avdelning, Institution

Division, Department

Division of Automatic Control Department of Electrical Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

Datum Date 2007-06-20 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport

URL för elektronisk version

http://www.control.isy.liu.se http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-9198 ISBN — ISRN LiTH-ISY-EX--07/3991--SE

Serietitel och serienummer

Title of series, numbering

ISSN

—

Titel

Title

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

Författare

Author

Sara Lundin

Sammanfattning

Abstract

The 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 with simulation 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 made for four types of mine hoists. Mass less springs and dampers are used to get the elastic behaviour of the ropes.

The mathematical model is implemented in Matlab and Simulink for all four hoist types to make load simulations possible. The implementation in the labora-tory HoistLab is made by modifying an existing program with the line elongation functionality. It is only done for the tower mounted friction hoist. There are several 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 will improve 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 is extended 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 the derivative part, but the performance potential is clear so the recommendation is to make further investigations.

Nyckelord

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Abstract

The 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 with simulation 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 made for four types of mine hoists. Mass less springs and dampers are used to get the elastic behaviour of the ropes.

The mathematical model is implemented in Matlab and Simulink for all four hoist types to make load simulations possible. The implementation in the labora-tory HoistLab is made by modifying an existing program with the line elongation functionality. It is only done for the tower mounted friction hoist. There are several 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 will improve 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 is extended 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 the derivative part, but the performance potential is clear so the recommendation is to make further investigations.

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Sammanfattning

Syftet med detta examensarbete är att förbättra driftegenskaperna för gruvspel. Arbetet är uppdelat i två olika delar. Den första och största delen handlar om simulering 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 i Simulink för att utföra simuleringar. I HoistLab är modellen realiserad genom att utöka ett befintligt lastsimuleringsprogram med de nya funktionerna för lintöjning. Detta är utfört enbart för den toppmonterade typen av friktionsspel. Ett flertal funktioner fick ändras för att få realistiska simuleringar.

Den andra delen av examensarbetet går ut på att göra en förstudie kring den deriverande delen i PID-regulatorer och hur den påverkar gruvspelets prestanda. För denna del är en PI-regulator som ger ett acceptabelt resultat av backgången när bromsarna släpps designad. Därefter är modellen utökad med den deriverande delen, 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 undersökningar kring D-delen.

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Acknowledgments

First of all I would like to thank ABB Mining which has made it possible for me

to do this Master Thesis work. During my work at ABB, I have obtained a lot of

technical experiences and met many nice persons.

I would also specifically direct thanks to my supervisors Sami Saari and Håkan

Selldén 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 and

Anders Daneryd atABB Corporate Researchhas assisted with Appendix A.

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

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Introduction

The purpose of this Master Thesis is to improve the drive performance of mine

hoists. The work is made atABB 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 caused by elongation in the ropes of a mine hoist. This description is implemented in Matlab and Simulink to make it possible to simulate the model. The third step is implementation of the model in a laboratory and the local program PLC Control Builder.

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 task it 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 excellence

located in Västerås. ABB is a complete supplier of mine hoists and is the only

company 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 of a real mine hoist containing a mini shaft, control and drive equipment as well as a brake system. The Master Thesis is connected to a development project which aims at improving the drive performance of the mine host and will be tested in HoistLab.

1.2 Problem Description

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

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

ABB has an important laboratory called HoistLab which is used during the

development 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 solve the problem and figure out how to minimise the back motion at start and stop behaviour.

The control of a mine hoist uses different controllers during the production cycle. In normal operation the speed control follows a predefined reference signal but at start and stop situations when the brakes are involved another control is needed. Today this is solved with a PID controller with the derivative part (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 unknown which advantages the derivative part can bring regarding the brake release. Can it provide 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 and

Simulink to make load simulations

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

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

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

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

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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 realistic values

1.5 Definitions

The definitions that will be used later on are explained in this section. Table 1.1 introduces 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 the

mathematical model and includes the equations.

Chapter 4, Implementation of Load Models: Presents the implementation

of the load model both in Simulink and HoistLab.

Chapter 5, Investigation of the Derivative Part in PID Controllers: Includes

the speed control investigation.

Chapter 6, Conclusions: Discusses the result of this Master Thesis compared

to 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 used

in 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 the

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Quantity [SI unit] Description

ms [kg] Skip mass

mΣs [kg] Total mass of skip, payload and rope

mcw [kg] Counterweight mass

mΣcw [kg] Total mass for counterweight and rope

mr [kg] Total mass of the ropes

mp [kg] Payload mass

lhr[m] Head rope length

ltr [m] Tail rope length

lnew[m] New rope length

rhr [m] Head rope radius

rtr [m] Tail rope radius

rd [m] Drum radius

vd [m/s] Drum speed

vs [m/s] Skip speed

vcw [m] Counterweight speed

vsp [m/s] The speed at the spring

vsh [m/s] The speed at the sheave

vb [m] The speed at the bottom of the rope

k [N/m] Spring constant

c [N/m] Damping constant

Jd [kgm2] Moment of inertia for the drum

Jm [kgm2] Moment of inertia for the motor

mpm [kg/m] Mass per meter, rope density

Nr [dimless] Number of ropes

Cd [dimless] Relative damping constant

Cs[dimless] Steel area constant

E [Pa] Elasticity module

F [N] Force

T [Nm] Torque

∆r[m] Rope elongation

g [m/s2_] _{Gravitation constant = 9.81 [m/s}2_]

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Chapter 2

Preliminaries

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

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

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

Figure 2.1. Drum in a tower mounted friction hoist

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2.1 Main Parts of a Mine Hoist

Mine hoists consist of several parts and the most important for this purpose are mentioned here. The drum is a big barrel with a diameter that normally can be between three to seven meters, see Figure 2.1. The drum is driven by a motor and has a separate hydraulic braking system. The skip is like an elevator, it is a box 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 of hoists there is also a counterweight to make balance with the skip. The vertical hole where the skip is driving is called a shaft. There is a conflict between having a 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 the hole. The ropes that carry the skip are heavy and could weigh many thousand kilos. The hoist cycle is the time it takes for the skip to be loaded, unloaded and ready for load again, and it is normally about 250-300 seconds.

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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 2.3 and 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 hoist in a tall small building called a "main frame". Ground mounted mine hoists have the drum directly on the ground and the ropes pull over a sheave at a high stand

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before they go down in the shaft. The reason for this is that the skip must have enough 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

skip counter

weight

(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 at the drum depending on the direction of the skip. For shallow shafts, down to about 250 meters the drum hoist is generally the best alternative because, for instance, the shaft diameter could be minimised. The drum hoist is a comparatively simple solution.

2.2.2 Friction Hoist

In friction hoists the rope is a closed circuit with constant total rope length. The rope is hanging on both sides of the drum and it is the friction between the rope and the drum that moves the rope when the drum is driven by a motor. The skip is hanging on one side and a counterweight on the other. The system is designed so that there will be balance on both sides when the skip is half loaded. The advantage 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 is fixed between the top of the skip to the top of the counterweight and the tail rope

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2.3 Introducing HoistLab 9

from the bottom of the skip to the bottom of the counterweight. The number of head 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 250 to 1500 meters. The limit of the length is the weight of the system affecting the drum.

2.3 Introducing HoistLab

This section introduces HoistLab and its equipment. HoistLab is a mini mine hoist 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. The laboratory has an important role in mine hoist development for both electrical and mechanical 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 both to do the programming in structured text and graphically with blocks like in Simulink. Here follows a list of the equipment that are used in HoistLab. An illustration of some parts is shown in Figure 2.6.

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

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• The mini shaft is 1.6 meter high and in the control equipment this distance is 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, main desk, 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 operoper-ator and maintenance information is available in the Process Portal - Compact HMI version.

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

Torque motor Load motor

Figure 2.7. The mini mine hoist model

In HoistLab there is no possibility to load and unload the skip, instead this

is simulated by a load motor and a torque motor, see Figure 2.7. These are

connected together with a common mechanical axis via a flexible coupling. They are behaving differently depending on the relations between the brake torque and the load torque. At start the torque motor is speed controlled with zero revolutions per minute as set point which results in that the model is standing still when the

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2.4 Introduction to PID Controllers 11

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

2.4 Introduction to PID Controllers

This section gives an introduction to PID controllers and some of the basics in control theory that will be used in Chapter 5. PID controllers are today the most common type of controllers that are used in industry. The abbreviation PID means Proportional, Integral and Derivative part. The input signal to the system, G, is u(t), see Figure 2.8. The difference between desired output signal and the real output signal, the control error, e(t) = r(t) − y(t), is the signal that the controller should minimise.

F

G

−1

Σ

r

e

u

_y

Figure 2.8. Control system

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

u(t) = K   e(t) + 1 TI t Z t0 e(τ )dτ − TD d dte(τ )    (2.1)

where K is the gain, TI is the integral time and TD the derivative time. The

adjustment of these parameters is often done by help of earlier experience that are based on the following characteristics: The P-part controls the gain of the system and can result in a faster system but with the disadvantage of reduced stability, i.e. too high gain can cause instability. To eliminate the static fault, the I-part is needed. It can also give a faster system but with the same drawback as for the proportional part. The D-part can improve the stability but can also give

noisy 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 two parameters to adjust instead of four as in the PID while the D-part must include also a low-pass filter [1]. The filter parameter, α, is discussed in (5.13) later on in this report.

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

G(s) = b0s m_{+ ... + b} m sn_{+ a} 1s(n−1)+ ... + an (2.2) The roots in the numerator in (2.2) are the poles and in the analogous way the roots in the denominator are called zeros. The following functions will be a help in the investigation that is treated in Chapter 5

Go= GF Gc = Go 1 − Go S = 1 1 + G0 (2.3) F is the controller, in this report of PI or PID design. The open-loop system, also

named the loop gain Go, is used to investigate the stability of the system. Gc is

called the closed-loop system and is the relation between the reference signal and the output signal. How the output signal is affected by disturbances is described by the sensitivity function S, [3].

Another way of writing expressions in control theory is in a state space form which is a system of first order differential equations. It is easy to transform equations 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 physical modelling [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-axis

which is the logarithm of the frequency. The magnitude plot has gain in dB as y-axis and the phase plot has the unit degrees, see Figure 2.9. There is lots of

information contained in Bode diagrams, e.g. the phase margin, φm, which can be

read 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 can be displaced before the system gets unstable. It is also possible to see resonance frequencies.

A common design in a control system is to use a lead-lag compensator which

has the purpose to improve the frequency response. The lead part, Flead, can

increase the phase curve at ωc so the φm gets larger. A phase margin less than

zero corresponds to an unstable system. The disadvantage with Flead is that high

frequency noise will be enlarged. The lead-part is a way of PD control with transfer function

Flead(s) = K

τDs + 1

βτDs + 1

(2.5) The lag part should be used when the stationary fault needs to be reduced. It has the following appearance

Flag(s) = K

τIs + 1

τIs + γ

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2.4 Introduction to PID Controllers 13 −200 −180 −160 −140 −120 −100 −80 Magnitude (dB) 10−1 100 101 102 103 104 105 −90 −45 0 45 Phase (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 lag part is that the phase margin gets reduced. This has to be compensated for when calculating the lead part. The complete PID controller of lead-lag design is written as, [3], F (s) = K τDs + 1 βτDs + 1 τIs + 1 τIs + γ (2.7) This controller type with γ = 0 will be used in Section 5. A low value for β can give higher phase margin and thereby a more stable system. The disadvantage is that the gain increases at high frequencies which can cause stress at some components if there is much noise.

τD can be decided when having a value for β and for the desired crossover

frequency ωc,d, see the equation below.

τD=

1 ωc,d

√

β (2.8)

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Chapter 3

Mathematical Description of

Load Disturbance

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

The mathematical description of the ropes is made by modelling them as stiff bars connected by mass less springs and dampers which simulates the elasticity in the 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 of parts is depending on the hoist type. A spring and a damper is connected to each rope section between drum and skip, drum and sheave, skip and bottom in friction hoist etc. It is possible to extend the models and to split the sections into smaller parts but that is not necessary in this task.

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

It is not the whole weight from the rope mass, mri that affects the oscillation

in the ropes. 1/3 of it can be seen as a point load at the bottom of the rope that

influence the spring force, Fspi, but the other 2/3 can be seen as an evenly divided

mass among the whole length that only affects the static force, mg. The derivation for this approximation is given in Appendix A.

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

mr= mpmlrNr (3.1)

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The spring factor, k, is equal to k = ECs(2rr) 2_N r lr (3.2) Where E is the elasticity module, r the rope radius and l the rope length. The ropes are made by many small lines which are twined around together so they are not completely solid, this relation is described by the steel factor Cs.

To get the damping coefficient, c, the spring factor is multiplied with a damping factor, Cd.

c = Cdk (3.3)

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

F = −kx − c ˙x (3.4)

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

T = rdF + J

d

dt(vd) (3.5)

∆payload is an expression for static extension, i.e. the final rope elongation when

the oscillations have ended, at load changes [7]. The equation below is used when calculating 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. This

expression works for line elongation caused both of load changes and speed changes and therefore it will be used in the mathematical description.

∆r=

Z

vsdt (3.7)

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

extra length ∆r

lnew= lr+ ∆r (3.8)

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

F = d

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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 includes one 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 of the construction and the skip speed at the bottom. The given parameters in the

calculations are the following: drum speed vd, spring constant k, damping constant

c, skip mass ms, payload mass mp and rope mass mr.

v

d F

k c

skip v_s F_s

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+

2

3mrg (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 be calculated first. It is this force that takes care of the elasticity of the ropes. The spring and damping constants are naturally used in

Fsp= cvsp+ k

Z

vspdt (3.11)

The spring speed vspthat should be integrated is the difference between the drum

speed and the skip speed.

vsp= vd− vs (3.12)

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

vs=

1 mΣs

Z

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The mass mΣs, which is the sum of the skip weight, the payload and1₃ of the rope

mass, is acting like a point load at the skip. mΣs= ms+ mp+

1

3mr (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 principle as for tower mounted one but there is ones sheave that divides the rope into two parts. 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 which means that the drum is placed over the sheave, that is shown in Figure 3.2(b). The distance between them is constant and normally much shorter than the between sheave and skip. There is also an approximation when the moment of inertia of the sheave is neglected. Even if the following equations are very similar to the ones 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 and

rope masses mri. v sh v d k 2 c 2 k 1 c 1 skip vs

(a) Layout of a ground mounted drum hoist

v d F k₁ c₁ k 2 c 2 v_sh F_sh skip v s Fs (b) Simplified layout of a ground 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 with

2

3 of the rope force. mr1is the mass for the rope between the drum and the sheave

that has constant distance.

F = Fsp1+

2

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3.2 Ground Mounted Drum Hoist 19

Also the spring force is calculated in the same way as above, the damping constant times the spring speed added with the spring factor multiplied with the integrated spring speed.

Fsp1= c1vsp1+ k1

Z

vsp1dt (3.17)

The spring speed is different compared to a tower mounted drum hoist. Now it is the 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 mounted drum hoist. Therefore this equation has the same appearance as (3.13).

vsh= 1 1 3mr1 Z Fshdt (3.19)

The force acting at the sheave is the difference between the first and the second

spring force subtracted by 1₃ 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

Z

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

Z

Fsdt (3.23)

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

expression is

mΣs= ms+ mp+

1

3mr2 (3.24)

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

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3.3 Tower Mounted Friction Hoist

The tower mounted friction hoist model has the same structure on both sides of the drum. There are two springs and dampers on each side. The total rope length is 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. This is a good approximation because the rope length will be almost the same when hanging straight as when going round. The rope that is over the drum is much smaller than the depth of the shaft. Both pictures of the rope lengths are shown in 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., when the drum speed is not equal to zero. To make the formulas more easy to follow the 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

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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 at

counterweight side A is the head rope length subtracted with head rope length at skip side

A = lhr− C (3.26)

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

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 the

expression 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 expression

for B inserted in the formula and the connection that l = lhr+ ltr the result will

be 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 normal one and the simplified, in Figure 3.4.

v d k₁ c₁ k 2 c 2 skip v_s k₃ c₃ k 4 c 4 counter weight

(a) Layout of a tower mounted friction hoist

v d F k 1 c 1 k 2 c 2 v b Fb skip v s Fs k 3 c 3 k 4 c 4 counter weight

(b) Simplified layout of a tower mounted friction hoist

Figure 3.4. Tower mounted friction hoist

The given parameters in the following calculations are: drum speed vd, spring

constants ki, damping constants ci, skip mass ms, payload mass mp, counterweight

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F = Fskipside− Fcwside (3.30)

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

with 2₃ of the static force from the head rope

Fskipside= Fsp1+

2

3mr1g (3.31)

The spring force is calculated in the same way as for drum hoists, i.e., Fsp1= c1vsp1+ k1

Z

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 the mass that acts like a point load at the skip

vs=

1 mΣs

Z

Fsdt (3.34)

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

mΣs= mp+ ms+ 1 3mr1 (3.35a) mΣcw= mcw+ 1 3mr3 (3.35b)

Then the total force that is acting at the skip has to be calculated. At this point the two spring forces work against each other so they will have different signs. Then 2₃ of the tail rope force has to be included to get the total result Fs

Fs= Fsp1− Fsp2+ 2 3mr2g − mΣsg (3.36)

The second spring force is calculated like the force at spring number one Fsp2= c2vsp2+ k2

Z

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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 the

speed for the skip. The force is integrated and divided with 1₃ of the tail rope mass

vb= 1 1 3mr2 Z Fbdt (3.39)

The last step is the force at the bottom which is the second spring force subtracted

with the 1

3 of the static force from the tail rope

Fb= Fsp2−

1

3mr2g (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 springs and dampers. In the same way as for drum hoists the angles between the drum and the sheaves are neglected. The model will be with the drum at the top and below one sheave at each side. Besides that the model has the same appearance as the tower mounted one. The figures of the friction hoists are shown below in Figure 3.5 v sh 1 v_d v_sh 2 k 2 c 2 k₃ c₃ k 5 c 5 k 6 c 6 k 1 c 1 k 4 c 4 skip vs counter weight

(a) Layout of a ground mounted friction hoist

v d F k₁ c₁ k₂ c₂ k 3 c 3 v sh 1 F sh 1 v b Fb skip v_s F s k₄ c₄ k₅ c₅ k 6 c 6 counter weight

(b) Simplified layout of a ground mounted friction hoist

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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 equations

for the ground mounted friction hoist as for the tower mounted but it is extended with 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 both sides of the drum but only the ones for skip side will be described. The differences are 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 ropes and springs will be 4 − 6 instead of 1 − 3.

The force from the skip side is

Fskipside= Fsp1+

2

3mr1g (3.42)

The first spring force is given by

Fsp1= c1vsp1+ k1

Z

vsp1dt (3.43)

The spring speed is the drum speed subtracted with the speed at the sheave. At the 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 be calculated. It is vsh= 1 1 3mr1 Z Fshdt (3.45)

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

force from 2₃ of the second head rope mass has to be added, because 1₃ is included

in (3.50a) later on Fsh= Fsp1− Fsp2+ 2 3mr2g − mr1g (3.46)

The second spring force is calculated in exactly the same way as the first one, Fsp2= c2vsp2+ k2

Z

vsp2dt (3.47)

The speed for the second spring is

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3.4 Ground Mounted Friction Hoist 25

Just like for the other hoist types the skip speed is vs=

1 mΣs

Z

Fsdt (3.49)

The expression for the mass is depending on which side of the drum it is. At the 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 the

second rope mass

mΣs= mp+ ms+ 1 3mr2 (3.50a) mΣcw= mcw+ 1 3mr4 (3.50b)

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

Fs= Fsp2− Fsp3+ 2 3mr3g − mΣsg (3.51)

The last spring force is the one at the tail rope Fsp3= c3vsp3+ k3

Z

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 1 3mr3 Z Fbdt (3.54)

Finally the bottom force is

Fb= Fsp3−

1

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Chapter 4

Implementation of Load

Models

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

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 shorter than 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. The other values are typical for real hoists.

To make the system stable when there is no payload mass in the skip the initial conditions in the integrators, see (3.32) and (3.37), at both skip and counterweight side have to be set to balance the static force. Otherwise the simulation shows oscillations in the start phase and this is not realistic. The equation that is used is found in (3.6).

4.1 Implementation in Simulink

The mathematical description should be implemented as a model in Simulink to make 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 testing models. The results from Simulink models are then compared with the results from HoistLab.

The implementation is straight forward from the mathematical equations above but some things are worth to mention. It is important to use a suitable ode-solver to 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 used several times. So if the mathematical description would be extended with more

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springs and dampers, as mentioned in Section 3 it would not be much extra work with 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 TotalMass SkipSpeed Skip Speed [m/s] RopeLength MassPerMeter NrOfRopes RopeMass

Rope Mass Calculation [kg]

ForceDiff SkipSpeed DrumSpeed SpringConstant DampingConstant Force Force Calculation DrumSpeedRpm DrumRadius DrumSpeed 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 1 DrumSpeedRpm

Figure 4.1. One part of the model in Simulink

The payload is simulated with load steps in vector form. The drum speed is also 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 by modifying an existing program to get the desired functionality with line elongation. The available program had several functions where load simulation is one part and also the one to be changed. The new function blocks are the same types as in Simulink 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 drum hoist and to edit the code to change between the models takes only a couple of minutes.

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

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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 simulated by a drive motor and not by real mass.

The PLC program had a predefined block for a simple integrator but it was not useful because it was running all the time and not only at load and unload as desired. Therefore an own block was built and to get the desired functionality. It was enough with a simple solution that takes the actual value and adds it to the previous 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 small

faults 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 start}

reset windup. The problem occurs when the skip is standing still at some specific rope lengths. It is solved by making two extra blocks in the program that is work-ing when the system is not movwork-ing. They lock the static force value when the skip is still.

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

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k = E·Cs(2r2r)

lr is calculated by dividing with the rope length. The large forces

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

vs=_m1 R Fsdt, m = mp+ ms+1₃mr1, vb= 11

3mr2R Fbdt

As shown in the equations vs is divided by a much larger mass than vb so the

elasticity module for the head rope does not have to be as much scaled as the one for tail rope. The scaling is also different between the head rope at the skip side and the counterweight side because they do not have the same masses. The critical lengths were found during testing and so was also the necessary scaling. The test was first made to manage to load and unload the payload at different lengths and then for different drum speeds. The scaling could be done theoretically but there was not enough time to do that. That means that the scaling procedure is only adapted at the specific values of the constants that are used in HoistLab, those are given in Appendix B. The scaling gives more oscillations in the ropes in the laboratory than in reality and in Simulink. This is not a problem because if the worst case gives a acceptable solutions the other will be even better.

The last problem during implementation in HoistLab is that the speed reference from the drum is varying because of disturbances and noise. That is solved by a standard block of a first order single pole low-pass filter. They are often used after analog inputs. The variable in this block is a time constant that is set to 0.1 seconds.

4.3 Results of the Implementation

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

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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 amplitude and two swings instead of one which is clear in the zoomed plot. To get a better picture 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] Force [N] normal scaled

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] Force [N] normal scaled

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When the skip is loaded it is at the length of 160 meters, see Figure 4.5. At that point no scaling is necessary so no differences will appear and no zoomed plot is needed. 3 4 5 6 7 8 9 2.5 3 3.5 4 4.5 5 x 105 Time [s] Force [N] normal scaled

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] Force [N] normal scaled

Figure 4.6. Friction hoist at length 10 meter, unloading

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

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4.3 Results of the Implementation 33

of the drum hoist. When looking at the enlarged plot in Figure 4.7 it shows almost the 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] Force [N] normal scaled

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 160 meters. At this point it is hard to see any differences. It is only when the plot get enlarged 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 1 x 105 Time [s] Force [N] normal scaled

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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] Force [N] normal scaled

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Chapter 5

Investigation of the

Derivative Part in PID

Controllers

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

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

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

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

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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 to simulate together with the controller. The states in Table 5.1 are speeds and forces and those are enough to describe the whole load model.

System Model Description

parameter parameter

x1 ω Drum speed [rad/s]

x2 vs Skip speed [m/s]

x3 vbs Bottom speed skip side[m/s]

x4 k1R (rdx1− x2)dt Force, spring 1 [N]

x5 k2R (x2− x3)dt Force, spring 2 [N]

x6 vcw Counterweight speed [m/s]

x7 vbcw Bottom speed counterweight side [m/s]

x8 k3R (−rdx1− x6)dt Force, spring 3 [N]

x9 k4R (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., compensated for the static force at both skip- and counterweight side. Otherwise the simulation shows unrealistic oscillations. The compensation is done by changing the initial conditions, 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 Description

parameter parameter

u1 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 result in the static force. The reason for having g as a signal is that the equations will be easier to analyse.

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5.1 State Space Model for Motor and Load 37

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

u1= Tm= F rd+ J ˙ω ⇒ ˙ω =

u1− F rd

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+

2

3u2(mr1− mr3) (5.7)

The equation for ˙x1 uses (5.5) and (5.7) to get the expression written with states

and input signals. ˙ x1= 1 J u1− rd c1(rdx1− x2) + x4− c3(−rdx1− x6) − x8+ 2 3(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 =                 r2 d(−c1−c3) J c1rd J 0 − rd J 0 − c3rd J 0 rd J 0 c1rd mΣs −c1−c2 ms c2 mΣs 1 mΣs − 1 mΣs 0 0 0 0 0 3c2 mr2 − 3c2 mr2 0 3 mr2 0 0 0 0 k1rd −k1 0 0 0 0 0 0 0 0 k2 −k2 0 0 0 0 0 0 −c3rd mΣcw 0 0 0 0 − c3+c4 mΣcw c4 mΣcw 1 mΣcw − 1 mΣcw 0 0 0 0 0 3c4 mr4 −3c4 mr4 0 3 mr4 −k3rd 0 0 0 0 −k3 0 0 0 0 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 the only output signals the size will be 2 × 9

C =rd 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

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

5.2 The Control System

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

Dspeedm/s Dspeed_rpm DSpeed e SSpeed SkipPos DrumPos MotorTorque SkipSpeed DSpeed Motor+Load 1 s Integrator2 1 s 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 and motor torque as a second order system with time delay. In the simulations the time

delay Td has, however, been approximated with a first order Padé approximation,

yielding the converter model

Gconv= ω2−Td 2 s + 1 (s2_{+ 2ζωs + ω}2₎Td 2s + 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 most important is similarity for both types.

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

Fa(s) = Ka 1 + 1 TIs + TDs αTDs + 1 (5.13)

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5.2 The Control System 39

when using lead-lag design

Fm(s) = Km τDs + 1 βτDs + 1 τIs + 1 τIs + 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) which

means 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 zero

which gives the controller a PI-function. The two expressions for Fa and Fmare

identified when TI = τI, Ka = Kmand γ = 0.

Inside the ACS 800 the calculations are done with scaled variables, i.e. drum speed and motor torque are represented as a percentage of their nominal values

nnom [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 to Nm. This is done by the gain factor K

K = Km

Tnom

nnom

(5.15)

If the control error is equal to the nnom [rpm] the torque will be the nominal

torque, Tnom[Nm], if K = 1.

The lead-lag design is used at the expression Fm and the following equations

may 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 1500

meters. To see the differences in the system for these lengths, with constant

payload, the Bode diagrams are shown in Figure 5.2. At constant length and variable 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 two head ropes. The two tail ropes resonance frequencies are so small that they can not 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 from a realistic torque instead of zero. The value for the pre-torque is the same as

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Bode Diagram Frequency (rad/sec) Phase (deg) Magnitude (dB) −200 −180 −160 −140 −120 −100 −80 500 m 700 m 900 m 1100 m 1300 m 1500 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) Phase (deg) Magnitude (dB) −200 −180 −160 −140 −120 −100 5000 kg7000 kg 9000 kg 11000 kg 13000 kg 15000 kg 100 101 102 103 104 105 −90 −45 0 45

Load Simulation and Investigation of PID Control for Resonant Elastic Systems

Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Load Simulation and Investigation of PID Control

for Resonant Elastic Systems

Load Simulation and Investigation of PID Control

for Resonant Elastic Systems

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Abstract

Sammanfattning

Acknowledgments

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1

Background

1.2

Problem Description

1.3

Purpose

1.4

Scope

1.5

Definitions

1.6

Thesis Structure

Chapter 2

Preliminaries

2.1

Main Parts of a Mine Hoist

2.2

Types of Mine Hoists

2.2.1

Drum Hoist

2.2.2

Friction Hoist

2.3

Introducing HoistLab

2.4

Introduction to PID Controllers

F

G

−1

Σ

r

e

u

y

Chapter 3

Mathematical Description of

Load Disturbance

3.1

Tower Mounted Drum Hoist

3.2

Ground Mounted Drum Hoist

3.3

Tower Mounted Friction Hoist

3.4

Ground Mounted Friction Hoist

Chapter 4

Implementation of Load

Models

4.1

Implementation in Simulink

4.2

Implementation in HoistLab

4.3

Results of the Implementation

Chapter 5

Investigation of the

Derivative Part in PID

Controllers

5.1

State Space Model for Motor and Load

5.2

_y