Modeling Hydrostatic Transmission in forest vehicle.pdf

Examensarbete

Modeling Hydrostatic Transmissionin Forest Vehicle

Erik Carlsson

LITH - ISY - EX - - 06/3801 - - SE

Modeling Hydrostatic Transmissionin Forest Vehicle

Department of Electrical Engineering, Linkopings Universitet

Erik Carlsson

LITH - ISY - EX - - 06/3801 - - SE

Examensarbete: 20 p

Level: D

Supervisors: Anton Shiriaev,Deparment of Applied Physics and ElectronicsUmea UniversitetJohan Sjoberg,Control & Communication,Department of Electrical Engineering,Linkopings Universitet

Examiner: Svante Gunnarsson,Control & Communication,Department of Electrical Engineering,Linkopings Universitet

Linkoping: May 2006

Institutionen for systemteknik581 83 LINKOPINGSWEDEN

May 2006

x x

http://www.diva-portal.org/liu/undergraduate/index.xsql?lang=en

LITH - ISY - EX - - 06/3801 - - SE

Modeling Hydrostatic Transmission in Forest Vehicle

Erik Carlsson

Hydrostatic transmission is used in many applications where high torque at low speed is demanded. Forthis project a forest vehicle is at focus. Komatsu Forest would like to have a model for the pressure in thehose between the hydraulic pump and the hydraulic motor. Pressure peaks can arise when the vehiclechanges speed or hit a bump in the road, but if a good model is achieved some control action can bedeveloped to reduce the pressure peaks.

For simulation purposes a model has been developed in Matlab-Simulink. The aim has been to get thesimulated values to agree as well as possible with the measured values of the pressure and also for therotations of the pump and the motor.

The greatest challenge has been due to the fact that the pressure is a sum of two flows, if one of thesesimulated flows is too big the pressure will tend to plus or minus infinity. Therefore it is necessary todevelop models for the rotations of the pump and the motor that stabilize the simulated pressure.

Different kinds of models and methods have been tested to achieve the present model. Physical modelingtogether with a black box model are used. The black box model is used to estimate the torque from thediesel engine. The probable torque from the ground has been calculated. With this setup the simulatedand measured values for the pressure agrees well, but the fit for the rotations are not as good.

Hydrostatic transmission, Forest Vehicle, Model, Pressure, SimulationNyckelord

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vi

Abstract

Hydrostatic transmission is used in many applications where high torque at lowspeed is demanded. For this project a forest vehicle is at focus. Komatsu Forestwould like to have a model for the pressure in the hose between the hydraulicpump and the hydraulic motor. Pressure peaks can arise when the vehiclechanges speed or hit a bump in the road, but if a good model is achieved somecontrol action can be developed to reduce the pressure peaks.

For simulation purposes a model has been developed in Matlab-Simulink.The aim has been to get the simulated values to agree as well as possible withthe measured values of the pressure and also for the rotations of the pump andthe motor.

The greatest challenge has been due to the fact that the pressure is a sum oftwo flows, if one of these simulated flows is too big the pressure will tend to plusor minus infinity. Therefore it is necessary to develop models for the rotationsof the pump and the motor that stabilize the simulated pressure.

Different kinds of models and methods have been tested to achieve the presentmodel. Physical modeling together with a black box model are used. The blackbox model is used to estimate the torque from the diesel engine. The probabletorque from the ground has been calculated. With this setup the simulated andmeasured values for the pressure agrees well, but the fit for the rotations arenot as good.

Keywords: Hydrostatic transmission, Forest Vehicle, Model, Pressure, Simu-lation

Carlsson, 2006. vii

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Acknowledgements

I would like to thank several people that have supported me during the workon my thesis:

My examiner Svante Gunnarsson for valuable opinions on the report, my super-visor at Umea University Anton Shiriaev for letting me be a part of this projectand my supervisor at Linkoping University Johan Sjoberg who has helped mevery much with both knowledge and encouragement during the creation of thisreport.

Komatsu Forest for putting forward this very interesting project and a specialthanks to Goran Blomberg and Joakim Johansson for the practical help duringour experimentation with the forwarder.

Rebecka Domeij Backryd for her help with LATEX.

My coworkers at the institution of Applied Physics and Electronics, Pedro,Anders, Ian, Leonid and Uwe for making my spare time here in Umea moreinteresting with squash and lunches.

I would also like to thank my family for their support and especially my girlfriendMaria for her love and invaluable help.

Carlsson, 2006. ix

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Nomenclature

Symbols and abbreviations are described here. SI units are used throughout thereport.

Symbols

p charge pressure in the hose [Pa]preturn pressure in the return hose [Pa]∆p difference between charge and return pressure [Pa]V volume of the hose [m3]β bulks modulus [Pa]C leakage coefficient [Nm5/s]TL torque from the wheels [Nm]Td torque produced by the diesel engine [Nm]Tp torque from the pump to the engine [Nm]Tm torque from the motor to the wheels [Nm]ωe angular velocity of the engine [rad/s]Qv flow through the safety valve [m3/s]Ql leakage out from the hose [m3/s]

ωp angular velocity [rad/s] ωm angular velocity [rad/s]Jp moment of inertia [Nms2] Jm moment of inertia [Nms2]Jd m.o.i. of the engine [Nms2] Jω m.o.i. of the wheels [Nms2]Jpd Jp + Jd Jwm Jm + Jω

Qp flow from the pump [m3/s] Qm flow to the motor [m3/s]Dp displacement [m3/rad] Dm displacement [m3/rad]ip current to the pump [A] im current to the motor [A]Bp friction coefficient [Nms] Bm friction coefficient [Nms]ηvp volumetric efficiency [-] ηvm volumetric efficiency [-]ηtp torque efficiency [-] ηtm torque efficiency [-]Tfp friction torque [Nm] Tfm friction torque [Nm]µp friction coefficient [Nms] µm friction coefficient [Nms]Kp friction coefficient [Nm] Kp friction coefficient [Nm]

Carlsson, 2006. xi

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Abbreviations

MS5050 Multi-System 5050CVT Continuously Variable TransmissionSITB System Identification toolboxEVPS Earthmoving Vehicle Powertrain Simulator

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.5 Outline of the Report . . . . . . . . . . . . . . . . . . . . . . . . 2

2 System overview 3

2.1 Forest vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Transmission line . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Previous work, modeling . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Previous work, identification . . . . . . . . . . . . . . . . . . . . . 7

3 Physical model of the hydrostatic transmission 9

3.1 Introduction model . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Diesel engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Hydraulic Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Hose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.5 Hydraulic Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.6 Traction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.7 Complete model . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Identification 19

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Linear identification . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.1 Linear black box modeling . . . . . . . . . . . . . . . . . . 20

4.3 Non-Linear identification . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.1 Non-linear black box modeling . . . . . . . . . . . . . . . 22

4.3.2 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . 23

Carlsson, 2006. xiii

xiv Contents

5 Data collection 25

5.1 Multi-System 5050 . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1.1 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Test day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.1 Sensor locations . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.2 Performed tests . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.3 Modifications of the data . . . . . . . . . . . . . . . . . . 29

6 Model modifications, Simulations and Results 31

6.1 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.1.1 Swashplate angle . . . . . . . . . . . . . . . . . . . . . . . 31

6.1.2 Flows from valve and leakage . . . . . . . . . . . . . . . . 32

6.1.3 Return pressure . . . . . . . . . . . . . . . . . . . . . . . . 32

6.1.4 Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.2 Simulations for the pressure . . . . . . . . . . . . . . . . . . . . . 34

6.3 Black box models . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.3.1 Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.3.2 Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.4 Grey box models . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.4.1 Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.4.2 Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.5 Complete model . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7 Discussion and Conclusions 51

7.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8 Future work 53

Bibliography 55

A Data sheet 59

B Simulink models 61

C .m-files 65

C.1 filtrera.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

C.2 umu fetch matrix.m . . . . . . . . . . . . . . . . . . . . . . . . . 66

C.3 Hydrostatic transmission.m . . . . . . . . . . . . . . . . . . . . . 67

Chapter 1

Introduction

In this thesis, a model of the hydrostatic transmission in a forest vehicle will bedeveloped. The introduction chapter presents the background, purpose and delim-itations, the method used to solve the problem and an outline of the report.

1.1 Background

Komatsu Forest develops and manufactures forest vehicles. The production inUmea started 1961 in a small family-owned company. Since then there have beensome different owners. 2004 Komatsu Ltd bought the company and KomatsuForest was founded. The development and production have changed coursefrom Slash-bundler ”Skruven” to today’s high technology forest machines. Thedevelopment and production for Komatsu is in Umea and Wisconsin. In Umea,the production mainly consists of wheel-based machines and harvester heads.Some of the research projects in Umea have been placed at Umea Universitywhere extra time and knowledge can be found. This thesis is a part of one ofthese projects. [1]

A forest vehicle needs much more torque in comparison to for example a roadtruck. Because of that a hydrostatic transmission is used. Komatsu wants amodel of this transmission. One of the motivations for the detailed modelingcomes from the fact that shifting speed usually results in a peak of pressurethat has an undesirable effect on the system and should be avoided. Such rise(drop) in pressure cannot be explained by simple static models and dynamicalmodeling is therefore needed. If a model that matches the real values in agood way is obtained then hopefully a controller that significantly reduces thispressure peak can be developed.

1.2 Purpose

The main purpose of this thesis will be to build a model for the hydrostatictransmission in a forest vehicle using available measurement signals. If this

Carlsson, 2006. 1

2 Chapter 1. Introduction

model agrees well with reality and there still is time left, then the next task willbe to control the input signals to reduce the pressure peak.

1.3 Method

To achieve the objectives of this thesis a mathematical description of the sys-tem has been developed. For simulation purposes this model is implemented inMatlab-Simulink. Experiments were needed to estimate unknown parame-ters. These experiments have been done together with Komatsu Forest on one oftheir test vehicles. Adjustments of the model and estimation of the parametershave been made to obtain a better model.

1.4 Delimitations

When developing the model six measurement signals were used. One of themwas the rotation for the pump, which in opposite to the others also is availableonline. This signal could be used when simulating the model, but that approachhas not been tested in this thesis.

A model for the pressure of the return flow is not offered, but instead measuredvalues are used.

Another restriction made in this thesis is to only consider rotation of the hy-draulic pump and motor in one direction. This has simplified the equationsdescribing the system and made it easier to understand.

The report has been written to suit students from an engineering program withsome experience of modeling systems.

1.5 Outline of the Report

The main topics dealt with are presented in the chapters below.

Chapter 2: A forest vehicle and the transmission line are shortly described.

Chapter 3: The theoretical model for the hydrostatic transmission is pre-sented.

Chapter 4: Some identification methods are explained.

Chapter 5: The experiments and the data collection are described.

Chapter 6: Simulations are performed and the results are given.

Chapter 7: Discussions and conclusions of the work are presented.

Chapter 8: Describes possible ideas for future work.

Chapter 2

System overview

In this chapter, an overview of both a forest vehicle and its transmission will begiven. The intention of this chapter is to give the reader a better understanding ofhow a hydrostatic transmission works and why it is used in a forest vehicle.

2.1 Forest vehicle

In Sweden two types of forest vehicles are used, harvesters and forwarders. Theyare usually wheel based. The harvester is used to take trees down, remove twigsand cut the trees in logs of suitable length. The task of the forwarder is tomove the logs from the harvester area to the road where a truck will come andpick them up. The focus of this thesis will be to build a model for a forwarderalthough the difference to other vehicles is small so an extension to these vehicleswill be possible.

Figure 2.1: The forwarder used in this project: Valmet 840.2

The forwarder considered in this project has model number 840.2, see Figure 2.1.The weight is almost 14 tons and the payload 11 tons. The source of power isa 6.6 liter diesel engine, developing 170 hp. The forwarder is using energy in

Carlsson, 2006. 3

4 Chapter 2. System overview

two ways. One way is to move the crane and the other is to move the vehicleforward. Usually these two cases do not occur at the same time, but if this isthe case there will be a problem with power supply for both of them. Thatproblem is outside the scope of this thesis. [1]

2.2 Transmission line

As mentioned above it is the diesel engine that delivers the power for the ma-chine. The diesel engine is connected to a hydraulic pump via a drive shaft.Inside the pump there is a swashplate that determines whether the vehicle ismoving forward, backwards or is stationary. The angle of the swashplate alsocontrols how much oil that flows out from the pump. A short description howthis works will be offered below. To the swashplate a number of pistons areattached. An angle of the swashplate will make the pistons go back and for-ward with the rotation. An increase of the angle makes the stroke of the pistonslonger which results in larger flow of oil of the pump. Figure 2.2 shows how itworks in a simplified way.

Figure 2.2: Simplified diagram of the transformation from swashplate angle toan high pressure flow into the hose.

The flow out of the pump is a product of the rotation of the pump and thevolume/displacement of the pump, where the displacement is determined bythe swashplate angle. The flow created by the pump passes through a securityvalve that ensures that the pressure is not too high. After that it is connectedto the hydraulic motor. A hydraulic motor is basically the same thing as ahydraulic pump, but it works in the opposite way. Consequently the hydraulicmotor is converting the flow back into rotation of the wheels. The motor aswell as the pump has a swashplate. These two swashplates are used to achievethe demanded speed from the driver. If really high speed is desired then theswashplate angle as well as the volume of the motor decreases, and the flowpasses through a smaller volume and the rotation increases. To transform therotation of the motor to lower speeds the motor shaft is connected to a gearboxwith two gears. From the gearbox there is a connection to the wheels.

The flow from the hydraulic motor is connected to the input of the pump, whichmeans that it is a closed loop system. A possible problem would then be thatthere is a lack in the oil supply to the pump. To ensure that oil always isavailable there is an extra pump called charge pump connected to the same

2.3. Previous work, modeling 5

shaft as the first one. The charge pump guarantees that the pressure on theinlet side of the pump is around 30 bar. To avoid damage a safety valve isplaced on the high-pressure side, and it opens when the pressure becomes toohigh. A simple schematic overview of the hydrostatic transmission is given inFigure 2.3.

Figure 2.3: Schematic overview of the hydrostatic transmission.

A hydrostatic transmission has advantages compared to a normal mechanicaltransmission line, and some of them will now be presented. Hydrostatic trans-mission is generally used in low-speed and high-torque applications. The rea-son is the continuously variable transmission, CVT, which makes it possible toachieve desired torque. A CVT is a little bit less efficient than a mechanicaltransmission but the possibility to drive at an optimal combination of torqueand speed makes the diesel engine to work in an more efficient range, and there-fore the whole vehicle becomes more efficient [2, 3]. The choice of torque andspeed can be reached with high accuracy [4]. Another motive is that a hydraulicmotor produces up to ten times more power compared to an electrical motorwith the same dimensions [5].

2.3 Previous work, modeling

The final goal for this project is to control the pressure in the hose between thehydraulic- pump and motor. To do this, a model of the hydrostatic transmissionis needed. Development of models for the hydrostatic transmission started inthe late 1940’s. Huge progress was made by Merritt in the 1960’s [6]. His workhas then later been updated by Manring [7]. Both Merritt and Manring focusedon theoretical issues. However, I will mostly describe what has been done forphysical machines. More specifically, the discussion will be focused at the workdone by Prasetiawan et al. [8] at the University of Illinois and Lennevi et al. [9]at Linkoping University. For more details see [8, 9] and references therein. Thework done at Illinois [10, 11, 12] and Linkoping [13, 14] has resulted in manytheses and articles.

In Illinois, a laboratory setup with an Earthmoving Vehicle Powertrain Simu-lator (EVPS) has been used. The EVPS has an induction motor as its primemover for the system and three hydraulic motors. A variable displacementpump then provides flow for the three motors. The hydraulic motors have fixed


displacement and are connected to valves that determine which one to use. Aload simulator is connected to each hydraulic motor.

The setup used at Linkoping university is very similar to the one used in Illi-nois. The largest difference is that the group in Linkoping has used a variabledisplacement motor instead of three fixed displacement motors. An inductionmotor controlled by a servo-valve has been used to simulate a diesel engine anda load simulator is connected to the hydraulic motor. Figure 2.4 shows the setupused in both Illinois and Linkoping.

Figure 2.4: The setup used by the groups in Illinois and Linkoping.

To the author’s knowledge no detailed modeling using data from a machine,driven in a realistic situation, has been done. Nevala et al. [15] have developedan antislip control for a forest vehicle. Fuzzy control was used to minimize theslip and therefore no model of the hydrostatic transmission was needed.

A Matlab-Simulink package with a model of a hydrostatic transmission hasbeen developed by Jedrzykiewicz et al. [16]. This model has many similaritieswith the models developed in Illinois and Linkoping. These three models arethe basis for the mathematical model presented in Chapter 3. In that chapter,a simplified model for the transmission given by Egeland and Gravdahl [4] willalso be presented.

A phenomenon described in the literature is the influence of the fluid propertieson the efficiency of the hydrostatic transmission. This has been investigatedby Dahlen [17]. Another issue that needs to be considered if a more detailedmodel is desired is the propagation of pressure. This has briefly been describedby Egeland and Gravdahl [4] and more carefully by Weddfelt [18]. Weddfelthas also approached the problem with pressure ripple. However, I have had noimmediate use of the information found in these references.

Of the references presented in this section the references from Illinois andLinkoping are most closely related to this project, but there are some majordifferences in the basic conditions. First, the diesel engine in this project isnot very well known. This would not have been a problem if it was possible tomeasure the torque from the engine, but unfortunately this is not the case. Thesetups in both Illinois and Linkoping use induction motors for which the controlsignal is the delivered torque. Moreover, a torque sensor is used in Linkoping.

Another important difference is also due to sensor signals. The other setupshave measurement signals for the swashplate angles of the pump and the mo-tor. That is not possible to get from the forest vehicle used in this project. It is

2.4. Previous work, identification 7

however still possible to measure the control signal to the hydraulic pump andthe motor which controls the swashplate angels. The problem is to determinethe dynamics of the swashplate. In this thesis, the swashplate dynamics will beneglected and hopefully this assumption will not affect the model too much. Ifthe effect would be large it would most likely only cause a small delay in thepressure.

The last bigger difference is that the Illinois and Linkoping approaches use loadsimulators. Load simulators produce a stable torque which is rather easy toadjust to a desired level. The Linkoping group does also have a torque sensorconnected. In our case, a forest vehicle without a torque sensor is connected.The load torque when driving will be quite unpredictable.

It may seem like the difference between this project and those projects aboveis very big. Therefore, it is necessary to stress that the main part of the trans-mission line is very similar for all three setups. The hydrostatic transmissionconsisting of a hydraulic pump and a hydraulic motor is basically the same.Therefore the models will approximately be the same.

2.4 Previous work, identification

A model of the hydrostatic transmission will contain a lot of parameters. Thevalues for some of them can be found in different documents from manufactures,and some are more or less known from experiments etc. However, some param-eters are unknown and need to be estimated.

An electro-hydraulic servo system (EHSS) as well as a hydrostatic transmissioncontains nonlinear hydraulic dynamics. Therefore, it is reasonable to see whathas been presented in the literature on identification of an EHSS. An EHSSconsists of a hydraulic valve, a hydraulic cylinder and a mass. The valve receivesa control signal determining which side of the cylinder that should receive theoil flow. The pressure in the receiving side rises and the mass starts to move.See figure 2.5.

Figure 2.5: Overview of the electro-hydraulic servo system.

To model the EHSS, Reuter [19] has used bilinear canonical forms. Recursiveprediction error methods are then used to identify the parameters in the bilinear


canonical forms. This concept can cause problems with convergence if the initialvalues are not ’good enough’. To avoid this difficulty Jelali and Schwarz [20]have used a modified recursive instrumental variables algorithm. Linear inte-gral filters were used to handle derivatives of measurement. The final model forthe EHSS is represented in observer canonical form. Another way to approachthe problem with identification of a nonlinear system is to use neural networks.This has been done by Anyi et al. [21]. During the training of the networkbackpropagation was used.

A problem that can occur when identifying parameters in a model is hysteresiseffects. This has been approached by Park and Lee [22] when modeling a single-rod cylinder. To avoid these effects they used a modified signal compressionmethod to estimate different dynamics during expansion and retraction.

In this project a hydrostatic transmission should be modeled and identified. Asdescribed in Section 2.3 this problem has previously been approached by Prase-tiawan in Illinois and he has used frequency and time responses to identify thesystem. Instead of estimating the parameters in the developed model, transferfunctions were shaped using system identification toolbox (SITB) in Matlab.For example, transfer functions from the reference signal for the swashplate an-gle to the pressure and to the speed of the motor were derived. The model thathad been created before provided information about which order the transferfunctions should have. An observation made by Prasetiawan was that the modelbefore improvements had good agreement between simulated and measured val-ues for the speed of the motor but the simulated and measured values for thepressure did not have the same accuracy. This shows that it is important tobuild and identify a model that is adapted to the goal of the project. In [23],Lennevi and Palmgren have created a controller for the speed of the motor.Therefore, a good model for the speed of the motor was needed but the pressuresimulations were less important. In our case, the goal is to control the pressurein the hose.

In [24], Cidras and Carrillo describe their model of a hydrostatic transmission.To estimate the unknown parameters a method based on the Melder-Mead sim-plex algorithm was used. Cidras and Carrillo evaluated if the rotation of themotor in the hydrostatic transmission could be kept constant. The reason tobuild such a controller was that an electrical generator should be attached tothe hydrostatic transmission. Another, not so successful approach was testedby Luigi del Re [25]. He developed a controller based on a model derived byusing black box identification of the hydrostatic transmission.

Chapter 3

Physical model of thehydrostatic transmission

In this chapter a model describing the hydrostatic transmission will be presented.The model will consist of smaller blocks where each block is a model for a part ofthe system. This will make it easy to exchange parts of the model, for example, thepump without changing the other elements. It will also give a fine overview of thesystem.

3.1 Introduction model

In [4] Egeland and Gravdahl present a simple model of the hydrostatic trans-mission. To get an introduction to the system the model is described in detailbelow by (3.1) – (3.3). The model has three variables that describe the rotationof the pump ωp, the rotation of the motor ωm, and the pressure in the hosebetween the pump and the motor p respectively. Figure 3.1 shows an overviewof the system. The equation for the pump is

Jpwp = Td −Bpωp −Dpp, (3.1)

where Td is the torque from the diesel engine, Jp is the moment of inertia of thepump, Bp is a friction coefficient for the pump and Dp is the displacement ofthe pump. The displacement depends linearly of the swashplate angle. Equa-tion (3.1) shows that the pump starts to rotate when a torque from the dieselengine is present. The hydraulic motor is described by

Jmwm = −Bmωm + Dmp− TL, (3.2)

where TL is the load torque from the wheels, Jm is the moment of inertia of themotor, Bm is the friction coefficient of the motor and Dm is the displacementof the motor. The two equations above demonstrate the similarity between thepump and the motor. The pressure in the hose between the pump and themotor is described by

Carlsson, 2006. 9

10 Chapter 3. Physical model of the hydrostatic transmission

V

βp = −Dmωm + Dpωp − Cp, (3.3)

where V is the volume of the pump, β is the bulk modulus and C is a leakagecoefficient. This equation illustrates that the pressure goes up when the flowfrom the pump is greater than the flow into the motor plus the leakage.

Figure 3.1: Egeland and Gravdahls model.

Now when the basics are presented a deeper exploration of the different partsof the system will be offered.

3.2 Diesel engine

The engine was not included in the introduction model described in Section 3.1.When trying to include the diesel engine in the model it is important to considerthat the engine and the pump are connected by a shaft. This means that theyhave the same rotation speed. Prasetiawan et.al. in [10] approach the modelingproblem by assuming that the rotation produced by the engine is affected bythe torque from the pump, see Figure 3.2.

Figure 3.2: Connection between the diesel engine and the pump.

In Figure 3.2 the control signal to the engine is left out. In a private car thecontrol signal is represented by the gas pedal that affects the throttle whichcontrols the air flow into the motor. As a response the control-box for theengine injects more or less fuel to keep the air/fuel-mixture constant. This isnot the case with Komatsu’s forest machines. Instead of controlling airflowthe gas pedal is input to the control-box that affects the swashplate angles ofthe pump and the motor. Komatsu’s control-box also sends a reference to theengines control-box which rotation speed is desired. The control-box of theengine then controls air and fuel input to match the desired rotation speed. SeeFigure 3.3 for an illustration. The result of the control system described is thata drop in speed will make the engine work harder to regain the reference speedgiven by Komatsu’s control-box.

3.2. Diesel engine 11

Figure 3.3: Overview of the different control signals in the transmission thataffect the final speed of the vehicle.

To describe the rotation of the engine and the pump an equation similar to (3.1)is used,

(Jp + Jd)we = Td − Tfp − Tp, (3.4)

where Jd is the moment of inertia of the diesel engine, ωe is the angular veloc-ity of the engine, Td is the torque produced by the engine, Tfp is the frictionaffecting the engine and the pump and Tp is the torque from the pump to theengine. The relation ωe = ωp holds because of the connection between the twounits.

Tfp is the joint friction of the engine and the pump, and a reasonable assumptionis to approximate it with a coulomb and a viscous friction. Then Tfp can beexpressed as

Tfp = µpωe + Kpsign(ωe), (3.5)

where µp and Kp need to be estimated from measurements. The term sign(ωe)delivers the correct sign depending on the rotation of the diesel engine, but theengine will only rotate in one direction, and therefore this term can be removed.

It will not be necessary to take special care and model the charge pump sincethe estimate of Jd and Tfp will include the charge pump without extra effort.

One way to get an accurate value for Td is to create a good model, that requiresa lot of information about the engine. Another option is to measure the motor’sinternal signals. Unfortunately, neither of these options are available. Thismeans that some kind of black/grey-box model needs to be developed for Td,see Chapter 6. Until that problem is solved, Td will be treated as an input signalto the system.


3.3 Hydraulic Pump

The rotation of the pump is generated by the diesel engine, but more modelingfor the hydraulic pump is necessary. The torque Tp represents the torque whichthe diesel engine senses from the pump and it will be a function of displacementand difference between charge- and return-pressure. Sauer Danfoss, the devel-oper of the hydrostatic system, provides a formula for Tp in the documentationfor the pump [26]

Tp =Dp∆p

ηtp, (3.6)

where ηtp is the mechanical efficiency for the pump. This is basically the sameas the term Dpp in (3.1). The big difference is the efficiency ηtp that makes theformula more accurate. An intuitive argumentation for (3.6) is that, when thepressure rises in the hose it is harder to rotate the pump. This means that Tp

goes up and the rotation initially goes down. It also happens when the displace-ment of the pump Dp increases, then the pump should provide more flow andtherefore decreases in speed. The initial drop in speed will be counteracted bythe diesel engine that increases the torque Td in order to try to keep the speedconstant.

Equation 3.7 that describes the flow out of the pump is offered by Sauer Dan-foss in [26] and in almost the same shape by Jedrzykiewicz et al in [16]. Thedifference between the two equations is mainly notational. The equation lookslike

Qp = Dpωpηvp, (3.7)

where Qp is the flow out from the hydraulic pump and ηvp is the volumetricefficiency of the pump.

One big advantage with the Sauer Danfoss representation of the pump is theexistence of the efficiencies in a datasheet, see Appendix A. That will give extrainformation about the behavior of the pump.

Modeling the pump contains a problem which has not been addressed so far. Thedisplacement Dp is linearly dependent on the swashplate angle but to control theangle of the swashplate a current is used. The relationship between the currentand the angle is approximately known but small errors in the estimate of Dp

can result in large errors in the pressure. Also the displacement of the motorDm is controlled by a current. More about the relation between the currentsand the angles can be found in Section 6.1.1.

3.4. Hose 13

3.4 Hose

The flows of oil into and out from the hose will determine its pressure. Thereare four flows to consider. The pressure in the hose is determined by the flowfrom the pump Qp, the flow into the motor Qm, the leakage Ql, and the flowthrough the valve Qv. See Figure 3.4 for a sketch of the flows.

Figure 3.4: The signals to and from the hose.

Changes in pressure propagate in the hose with the speed of sound1. Therefore,it is realistic to assume the same pressure in the 1.2 meter long hose. Expansionof the hose is another phenomenon that will not be encountered for in thisthesis. High pressures will affect the rubber hose but hopefully not enough tomake significant changes in the volume of the hose. With these two assumptionsmade, the physics behind pressure changes can be described by

p =β

V

∫ t

0

(Qp −Qm −Qv −Ql) dτ , (3.8)

where V is the volume of the hose, β is the bulk modulus and p is the pressurein the hose. This is basically the same equation as (3.3) with some small modi-fications. As explained in Section 6.1.2, Qv and Ql can be set to zero.

It could be important to stress one property of (3.8). If the pressure in thehose is constant it means that the flow out of the hose is equal to the flow intothe hose. This is of course true in the opposite way, if the flows are equal thepressure will be constant.

3.5 Hydraulic Motor

The equations describing the hydraulic pump can almost without changes beused for the motor as well. The flow Qm looks like

Qm =ωmDm

ηvm, (3.9)

1Speed of sound in oil ≈ 1000 m/s.


where ηvm is the volumetric efficiency of the motor. The reason to have ηvm inthe denominator instead of the numerator is that the flow is the input to themotor contrary to the pump where it is the output. An analogous discussioncan be used for ηtm in the equation that express Tm

Tm = Dm∆pηtm, (3.10)

where ηtm is the mechanical efficiency for the motor and Tm is the torque thatthe motor delivers to the traction model. An overview of the signals into andout from the motor can be found in Figure 3.5.

Figure 3.5: Connections with the hydraulic motor.

3.6 Traction model

A torque from the motor is delivered to the traction model, and that torqueshould be used to calculate a rotation for both the wheels and the motor. Thetorque TL stands for the torque from the wheels and Jω is moment of inertiafor the wheels. An equation for the angular velocity ωm can be written

(Jm + Jω) wm = Tm − Tfm − TL, (3.11)

where Tfm describes the friction and could be assumed to look like

Tfm = µmωm + Kmsign(ωm), (3.12)

e.g. a Coulomb friction and a viscous friction. The coefficients µm and Km

need to be estimated. The hydraulic motor together with the wheels can rotatein both directions, and therefore the term sign(ωm) is needed.

The problem left to handle is then TL. Unfortunately TL causes a lot of trouble.It depends on several parameters that possibly could be found. It also dependson how slippery the ground is, whether it is a slope or not and if there are holesor stumps. Therefore, it is necessary to treat TL as an immeasurable input tothe system or possibly use a blackbox/greybox model, see Chapter 6.

3.7. Complete model 15

3.7 Complete model

To sum up, the full set of equations that constitute the model are presented be-low. Some modifications will be presented later because Td and TL are unknown.The pump related equations are

(Jp + Jd)wp = Td − Tfp − Tp,Tfp = µpωp + Kp,

Tp =Dp∆p

ηtp. (3.13)

The motor related equations are

(Jm + Jω) wm = Tm − Tfm − TL,Tm = Dm∆pηtm,

Tfm = µmωm + Kmsign(ωm). (3.14)

The equations describing the pressure are

p =β

V

∫ t

0

(Qp −Qm) dτ ,

∆p = p− preturn,Qp = Dpωpηvp,

Qm =ωpDm

ηvm. (3.15)

How signals and sub-models are connected for the full model are shown inFigure 3.6.

Figure 3.6: Input signals to the system together with the signals that connectdifferent subsystems.

No model has been developed for the return pressure. Instead, the return pres-sure is seen as a input to the system. The torques Td and TL will at this time be


treated as disturbances. The two last signals into the system are Dp and Dm.These are the two controllable signals primary used to avoid pressure peaks.

The equations presented above represent the model for the hydrostatic trans-mission. This model can be rewritten in a better way. Introducing two newvariables Jpd = Jp + Jd and Jmw = Jm + Jω together with substitutions of Td,Tfp, Tm, Tfm, Qp, Qm and p gives

wp =Td

Jpd− µpωp

Jpd− Kp

Jpd− Dp∆p

Jpdηtp,

wm = − TL

Jmw− µmωm

Jmw− Kmsign(ωm)

Jmw+

Dm∆pηtm

Jmw,

∆p =βDpωpηvp

V− βωmDm

V ηvm− preturn. (3.16)

The goal is to rewrite (3.16) in state space form to more clearly show the struc-tures of the model. It will also make it easier to apply identification methods.Before a state space form can be given some dependencies need to be clarified.The controllable input signals to the model (3.16) are the displacement of thepump and the displacement of the motor. These displacements are determinedby electrical signals. The functions from current to displacement are approxi-mately known, see Section 6.1.1, and will therefore not render more unknownparameters. Still Dz, where z is either p or m, will be written Dz(iz) in thefollowing state space representation to keep in mind that the displacements arefunctions of the currents. In this thesis the dynamics of the swashplates havebeen neglected, otherwise a model from iz to Dz would be necessary. Hope-fully this assumption is reasonable when no measurements for the angles of theswashplates are available.

As mentioned before the values for the efficiencies are given in a data sheetprovided by Sauer Danfoss. These values are probably not totally reliable andtherefore it can be of interest to see them as unknown parameters. It couldhappen that the efficiencies are dependent on velocities, pressures and displace-ments. Therefore, the following notation will be used,

ηtp = ηtp(ωp,∆p, Dp), ηvp = ηvp(ωp,∆p,Dp),ηtm = ηtm(ωm,∆p, Dm), ηvm = ηvw(ωm,∆p, Dm).

The pressure in the return hose is not modeled but it is measured in the tests,and therefore it will be used as input signal to the system. The following vectorsare now introduced

x =

wp

wm

∆p

, u =

Dp(ip)Dm(im)preturn

, d =(

Td

TL

),

θ =(µp µm Kp Km Jpd Jmw β θηtp

θηvpθηtm

θηvm

)T ,


where x represents the states, u are the input signals, d are the disturbancesand θ contains the unknown parameters. Because ηtp is not constant it is pa-rameterized using θηtm

which can be a vector.

The model can now be expressed as

x = f(x; θ) + g(x; θ)u + ϕ(θ)d,

f(x; θ) =

−µpx1Jpd

− Kp

Jpd

−µmx2Jmw

− Kmsign(x2)Jmw

0

,

g(x; θ) =

− x3

Jpdηtp(x,u;θηtp ) 0 0

0 x3ηtm(x,u;θηtm )

Jmw0

βx1ηvp(x,u;θηvp )

V − βx2V ηvm(x,u;θηvm ) −1

,

ϕ(θ) =

1Jpd

− 1Jmw

0

. (3.17)

To simplify this representation some assumptions has been done. The two torqueefficiencies are rather constant when the speed of the vehcile is above 0.2 km/h.Therefore, they will be approximated by a constant. The two volumetric effi-ciencies are probably most dependent of the speed of the pump and the speed ofthe motor, respectively. Because of that the pressures and displacements will beneglected in the function. A delimitation made in this thesis is that the rotationof the wheels will only be considered in one direction, and therefore sign(x2)can be removed. As a result of the discussion above new θ, f(x; θ) and g(x; θ)can be formed as

θ =(µp µm Kp Km Jpd Jmw β ηtp θηvp ηtm θηvm

)T ,

f(x; θ) =

−µpx1Jpd

− Kp

Jpd

−µmx2Jmw

− Km

Jmw

0

,

g(x; θ) =

− x3Jpdηtp

0 00 x3ηtm

Jmw0

βx1ηvp(x1;θηvp )

V − βx2V ηvm(x2;θηvm ) −1

, (3.18)

As we can see there are a lot of unknown parameters in θ. Estimation of theseare necessary, and therefore an introduction to identification will be given inthe next chapter.


Chapter 4

Identification

In this chapter an introduction to linear and non-linear identification will be given.

4.1 Introduction

A lot of research has been done on identification of systems and parameters.An introduction to the aspects of identification is given by Ljung and Glad [27].Further information can be found in [28] by Ljung and for non-linear identifi-cation in [29] by Sjoberg. The information presented in this chapter is found inthis literature.

Different kinds of models can be developed for a system. These can be catego-rized into three different groups.

- White Box models

- Grey Box models

- Black Box models

A white box model is used when all the information about the system structureis available. This is almost never the case when it comes to modeling in thereal world. Even with good knowledge about the system there will be unknownparameters that need to be estimated in some way. This kind of model is calledgrey box model. Another type of model that also falls under the category ofgrey box models is when some physical information about the system is avail-able and that knowledge is used to estimate a model of black box nature. Ablack box model is used when no information and knowledge about the systemexist.

Which kind of model that should be used depends on the knowledge of thesystem, how much time one has and what the model should be used for. Maybeit is enough to use a black box identified model and then there is no need to

Carlsson, 2006. 19

20 Chapter 4. Identification

spend more time to determine the physical relations of the system. If one hassome physical insight it should probably be used when developing the model.Then a more realistic model can be achieved and fewer parameters need to beestimated.

4.2 Linear identification

If the developed model for the system is linear but have some unknown param-eters it could be written as

x(t) = A(θ)x(t) + B(θ)u(t) + d(t)y(t) = C(θ)x(t) + D(θ)u(t) + h(t), (4.1)

where u(t) stands for the inputs to the system, y(t) for the outputs, x(t) are thestates of the system, d(t) and h(t) are disturbances and θ is a vector containingthe unknown parameters. The task is to estimate these parameters through dif-ferent experiments. How this should be done depends on the system and whichmeasurements that can be collected.

If less knowledge about the system is available, then black box modeling couldbe used to approach the problem, this will be presented below.

4.2.1 Linear black box modeling

When using an identification program to estimate a black box model, for ex-ample SITB [30], the model structure needs to be chosen. Some of the possiblestructures will be presented below and an example will be given for one of them.For the rest of this chapter discrete time models with q as the shift operatorwill be used, where q works as

qy(t) = y(t + 1), q−1y(t) = y(t− 1). (4.2)

A general model can be described by

A(q)y(t) =B(q)F (q)

u(t) +C(q)D(q)

e(t), (4.3)

where y(t) is the output, u(t) is the input and e(t) is a white noise disturbance.As can be seen in (4.3) this general model will result in some special cases. Ifan OE (output error) model is desired, B and F are used and A, C and D isset to one. Other examples are: the ARX model (C=F=D=1), the ARMAXmodel (F=D=1) and the BJ (Box-Jenkins) model (A=1).

Which model structure that should be chosen depends a lot on the behavior ofthe disturbances on the system. The question is how the disturbances enter to

4.2. Linear identification 21

the system. If for example a measurement sensor is affected by a white noisedisturbance a good choice of model would be an OE model. This is becausean OE model has dynamics for the input signal u(t) while the disturbance e(t)directly affects y(t). An ARX model can be a good alternative when the distur-bance probably enters the system in the same way as the control signal. Thenboth u(t) and e(t) will have the same pole dynamics. A problem can then bethat A(q) needs to describe the properties of disturbance as well. An ARMAXmodel can be used if some extra flexibility for the estimate of the disturbanceis desired. In an BJ model the dynamics for u(t) and e(t) to y(t) are separate.Therefore, a BJ model can be good to use when the disturbances enter late inthe process.

A deeper investigation of how to estimate the parameters in an ARX model willnow be performed. The polynomials A and B look like

A(q) = 1 + a1q−1 + a2q

−2 + · · ·+ amq−m

B(q) = b1q−1 + b2q

−2 + · · ·+ bnq−n (4.4)

where a1 to am and b1 to bn are unknown parameters and will therefore be ourθ, which can be written as

θ = [a1 a2 · · · am b1 b2 · · · bn]T . (4.5)

Using (4.3) and (4.4) the model can be written as a difference equation

y(t) + a1y(t− 1) + · · ·+ amy(t−m) =b1u(t− 1) + b2u(t− 2) + · · ·+ bnu(t− n) + e(t) (4.6)or asy(t) = −a1y(t− 1)− · · · − amy(t−m)+b1u(t− 1) + b2u(t− 2) + · · ·+ bnu(t− n) + e(t). (4.7)

Introducing ϕ(t) makes it possible to write (4.7) as

y(t) = ϕT (t)θ + e(t) (4.8)with

ϕ(t) = [−y(t− 1) · · · − y(t−m) u(t− 1) · · · u(t− n)]T . (4.9)

Because e(t) directly affects y(t) and e(t) is unpredictable white noise the bestprediction y(t|θ) for y(t) is

y(t|θ) = ϕT (t)θ. (4.10)

From (4.8) it is possible to get an estimate for θ using linear regression. Termsin ϕ(t) are called regressors and ϕ(t) is the regression vector.


The estimate θ will converge to the true value of θ when the number of samplesapproaches infinity. This is true if the system could be completely modeledby an ARX model. Another restriction is that the dynamics of the systemare excited by the input signal. When estimating a black box this is a reallyimportant issue. If the identification should have a chance to be successful it isnecessary that the system dynamics is available in the output signal. To ensurethat large variations of the input signal should be performed.

4.3 Non-Linear identification

The goal when it comes to non-linear models is the same as for linear model,that means the prediction should be as close to the real values as possible. Toachieve that a cost function is formed

Vn(θ) =1n

n∑k=1

‖y(tk)− y(tk|θ)‖2 . (4.11)

This cost function will be large when our prediction for y(t) is bad. It is nowpossible to optimize θ using Vn(θ) as a measurement of how good y(tk|θ) is.The minimization of Vn(θ) by improving θ could be performed using

θ(i+1) = θ(i) − ν(i)[V′′

n (θ(i))]−1V′

n(θ(i)), (4.12)

where ν is the step length, V′

n(θ) is the derivative of Vn(θ) with respect to θ andV

′′

n (θ) is the second derivative. Calculation of (4.12) is not trivial but the actualproblem is to define y(t|θ). It could be done using either a physical model or ablack box approach (for example neural networks).

4.3.1 Non-linear black box modeling

The prediction y(t|θ) is now a function of θ and ϕ(t), (4.10) will therefore looklike

y(t|θ) = g(ϕ(t), θ). (4.13)

Also ϕ(t) can be a function

ϕ(t) = ϕ(y(t− 1), · · · , y(t−m), u(t), u(t− 1), · · · , u(t− n). (4.14)

As before, different choices for the regression vector ϕ(t) will result in differentkinds of models, for example NARX (Non-linear ARX), NOE and NARMAX.Another choice that needs to be considered is how the nonlinear mapping func-tion g(ϕ(t), θ) should look like. To illustrate this problem and give an introduc-tion to a nowadays popular method, neural networks will be explained.

4.3. Non-Linear identification 23

4.3.2 Neural Networks

The problem that will be dealt with is how g(ϕ, θ) should take us from regressorspace to output space. A reasonable assumption is that it will work well to useparameterized functions to describe g(ϕ, θ)

g(ϕ, θ) =n∑

k=1

αkgk(ϕ), (4.15)

θ = [α1 · · · αn]T ,

where α1 to αn are parameters in the expansion and gk are called basis functions.A good choice is to use the same function for all gk but to use two parametersβ and γ to make each gk individual. Equation (4.15) can then be written as

g(ϕ, θ) =n∑

k=1

αkκ(βk(ϕ− γk)), (4.16)

θ = [α1 · · · αn β1 · · · βn γ1 · · · γn]T ,

The parameter γ locates the function κ and β is a scale parameter.

A short example of a function approximation using (4.16) will now be given.Assume that ϕ is a scalar and κ is chosen to be a unit pulse, then our predictiony will be approximated with a piece-wise constant function. The parametersα will determine the level of each step, γ the position and 1/β the length.Figure 4.1 shows an example.

1 1.5 2 2.5 3 3.5 4

0.8

1

1.2

1.4

1.6

Figure 4.1: Approximation of a function using a neural network.

If a smoother function is desirable κ can be set to be for example a Gaussianbell,

κ(x) =1√2π

e−x2/2. (4.17)

In this scalar example it is relatively easy to realize that the approximation canbe arbitrary good, but this is also true for higher dimensions. This is the mainreason why neural networks are so popular.


Chapter 5

Data collection

In this chapter the experiments and the method to collect the data will be described.There are some important issues to consider before tests can be performed. Whichsignals should be measured, which signals are even possible to measure, if a signalis not measurable can it be estimated from other signals and in that case whichtests are good for achieving unknown parameters and verifying the model?

5.1 Multi-System 5050

Development of the model and estimation of the parameters assume measure-ments from a real forest vehicle. These measurements need to be collectedsomehow. In this thesis Komatsu’s measurement tool, Multi-System 5050 [31],has been used, but in the future DSpace will probably be used. MS5050 is aproduct from the German company Hydrotechnik.

Multi-System 5050 is a hand held computer with a graphical interface, see Fig-ure 5.1. It is possible to connect up to six measurement signals. Channel oneto channel four should have analogue input. Channel five and six should havepulses as input, which will be converted to the corresponding frequency of thepulses by MS5050. If all six inputs are used with the frequency 1000 Hz (e.g.storing one value every ms) the memory can store values for approximately 2minutes. A USB-interface is available in order to get the stored values fromMS5050 into a PC. The data files are then converted to excel files which areeasily accessible from Matlab. [31]

5.1.1 Sensors

Komatsu has sensors for measuring rotation speed, pressure, and electrical sig-nals. The rotation sensor uses an infrared signal to measure the rotation. Thesignal is sent out from the sensor, then it bounces on a reflex attached to therotating part, and is then detected by the sensor. It is possible to attach morereflexes on equal distances on the rotating part to pick up changes in the rota-tion quicker. The sensor is placed on a magnet that makes it possible to put

Carlsson, 2006. 25

26 Chapter 5. Data collection

Figure 5.1: Multi-System 5050, the measurement tool used in this project.

the sensor close to the rotating part.

The forest machine has several positions where it is possible to connect pressuresensors. The cap is unscrewed and replaced by the sensor, see Figure 5.2. Thesensor delivers an analogue current signal to the MS5050 unit.

Figure 5.2: One of the possible positions where a pressure sensor can be placed.In this case it is on the hydraulic motor.

The interface of the electrical sensor has many similarities with the pressuresensor. The electrical sensor is also analogue and has contacts on the vehiclewhere it can be connected.

5.2. Test day 27

5.2 Test day

The test area was located in Jamtebole, forty kilometers northwest of Umea.

5.2.1 Sensor locations

As mentioned in Section 5.1 there are two frequency inputs on a MS5050-unit,which were used for rotation sensors. One sensor measured the rotation of thediesel engine, although that one is connected to the hydraulic pump, whichmeans they have the same speed. The other rotation sensor measured the speedof the drive shaft. The rotation ratio between the hydraulic motor and the driveshaft is 5.17:1 when the low gear is used and 1.67:1 for the high gear. These areall the interesting rotations in the forest vehicle, so there would not have beenany use for more rotation sensors.

Two electrical sensors were used. They were placed to measure the signals tothe pump and the motor that command which angle the swashplates shouldhave.

The two last positions were used for pressure sensors. One should of course beplaced to measure the charge pressure to the motor. The other was placed afterthe motor to measure the pressure of the return flow. This setup has one bigadvantage; the pressure drop over the motor is available. Some other pressurescould also have been of interest, for example to see if a valve is open or closed.If there had been more time available for testing, the other interesting pressurescould have been measured by replacing the pressure sensor for the return flow.

5.2.2 Performed tests

A test plan was made before the day of testing. Unfortunately it was not possibleto keep to the plan. One reason was the lack of time. Another unexpectedcircumstance was that the vehicle had snow chains on the rear tires which madeit impossible to test on an asphalt road. Not even a forest road was used becauseit would have made it hard for the trucks to drive there later. Instead the testswere performed in a track that had been made by other forest vehicles. Thistrack was not straight and had some badly located stumps, see Figure 5.3. Theseconditions made it hard for the driver to make the preferred inputs with the gaspedal. He did succeed well at constant velocities but had a harder time withsteps. In all, 12 test runs where made.

Test 1-7: Repetitive tests at three different speeds were performed, three forthe middle speed and two for the other speeds. The reason was to check ifthe same signals were obtained when the test was repeated. One furthermotive for these tests was that it hopefully would be easier to achieve theefficiencies for the hydraulic transmission. One example of a test is shownin Figure 5.4.


Figure 5.3: The track at which the tests where performed.

Test 8-9: In test 8 the goal was to perform velocity steps up and in test 9 stepsdown. These tests were made to get dynamic information. Previouslydescribed steps were not easily performed, but hopefully they are goodenough.

Test 10-12: These tests were done on the second gear and at the same speedsas tests 1-7. Under ideal circumstances this would give the relationshipfor the friction in the motor under different rotation speeds.

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5.2. Test day 29

5.2.3 Modifications of the data

In some of the measuring points the rotation sensor on the drive shaft missedthe reflex resulting in that the rotation value was too low. In these cases theincorrect value has been replaced with the value of the previous sample. Thiswill not affect the data because the sampling frequency 10 Hz is used and therotation of the motor is pretty constant most of the time.

Because of the rough road, some of the data can be hard to get good estimatesfrom. However, it can be interesting to see what happens in the signals whenfor example a stump is hit.

As can be seen in Figure 5.4, the pressures and currents are measured withhigh frequency, 1000 Hz. The rotations are measured with 10 Hz. The currentsignals look pretty messy, and the reason for that is that they are pulse widthmodulated. To make the signals smoother and more useful they have to befiltered. Different Butterworth filters have been used for this purpose. Figure 5.5shows the same data as in Figure 5.4 but filtered.

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1100

1150

1200

Time [s]

Rot

atio

n sp

eed

pum

p [r

pm]

Figure 5.5: Example of filtered data (Same data as in Figure 5.4).


Chapter 6

Model modifications,Simulations and Results

The essence of the work done in this thesis will be presented in this chapter. Firstsome adjustments of the mathematical model in Chapter 3 will be done, and afterthat different ways to simulate the model will be explained.

6.1 Modifications

This section will add some extra information about the forest vehicle that affectsthe model. It will also explain some previous statements.

6.1.1 Swashplate angle

The amount of oil flowing into and out from the hydraulic pump and motoris determined by the displacement. The displacement is a linearly function ofthe swashplate angle. As seen in Section 5.2.1 neither the displacements northe swashplate angles are measured. Instead the electrical control signals to thepump and the motor are measured. Because of this, the relationship betweenthe current and the displacement is needed. The swashplate angle does not onlydepend on the current but also on the pressure difference over the pump andthe rotation of the pump. For the experiments in this thesis the hydrostatictransmission was a new prototype with a new pump. This new setup was prettytolerant to changes in the rotation speed making it possible to neglect the in-fluence of the pump rotation on the swashplate angle.

A hydraulic override exists in the system. It exists to avoid damage on thetransmission in case the pressure goes up. When the pressure is over 300 barthe override decreases the angle of the pump. In test run 1-9 the pressure is al-most always under 300 bar but in test run 10-12 it is sometimes over. Since thishydraulic override is not included in the model, the time periods with severalpressure peaks well over 300 bar have been excluded. Changes in displacement

Carlsson, 2006. 31

32 Chapter 6. Model modifications, Simulations and Results

due to changes in pressure will therefore be disregarded.

Now to the displacement’s function of the currents. The discussion is basedon the experience available at Komatsu. The pump starts to move the angleof the swashplate from zero when the current exceeds 300 mA. The functionis linear to the pump’s maximum displacement at 147 cm3 which is reached at800 mA. Contrary to the pump the motor starts with maximum displacement,160 cm3, when the current is less then 330 mA. Minimum displacement for themotor, 68 cm3, is achieved at 560 mA. However, this function is not linear. Thedisplacement’s function of the currents are shown in Figure 6.1.

0.2 0.4 0.6 0.8 10

50

100

150

Current to pump [A]

Dis

plac

emen

t in

pum

p [c

m3]

0.2 0.3 0.4 0.5 0.6 0.760

80

100

120

140

160

Current to motor [A]

Dis

plac

emen

t in

mot

or [c

m3]

Figure 6.1: Illustration of the functions from currents to displacements for thepump respectively the motor. Note that the two have contrary reactions to arise in current.

6.1.2 Flows from valve and leakage

As written in Section 3.4 the flow through the security valve Qv and the leakageQl can be set to zero. The leakage Ql is really small, and definitely negligible incomparison to the normal flow through the hose. In contrast to Ql, Qv can belarge. The security valve is there to make sure that the hydrostatic transmissiondoes not break if something happens. The valve opens at 415 bar and thereforeQv can be set to zero because the high pressure starts to affect the swashplateangles before the security valve opens. Therefore that data have already beenexcluded as explained in Section 6.1.1.

6.1.3 Return pressure

A model for the charge pressure (3.8) was built in Chapter 3, but we haveno dynamic model for the pressure in the return hose. One reason for this isthat the charge pump is difficult to model and would result in more unknownparameters since much information about the charge pump and the valves aremissing. The most important reason to skip a model for the return pressureis however in this thesis that the pressure is quite constant. The pressure willhave very small variations over time and will not be noticed in comparison tothe forward pressure. Still, it happens that the return pressure makes a peak,

6.1. Modifications 33

which probably is when the forwarder hit a stump in the track. For this reasonthe measurements of the return pressure will be used directly as input instead.

6.1.4 Efficiencies

The model developed in Chapter 3 uses efficiencies. The efficiencies are pro-vided by Sauer Danfoss in a data sheet that is attached in Appendix A. Theseefficiencies are given for the old system, although it is reasonable to assume thatthe different systems have quite similar structure. Even so it must be kept inmind that this new pump has a bigger volume then the old one.

There are two kinds of efficiencies, volumetric efficiencies and torque efficiencies.The parameters ηtp and ηtm represent the torque efficiency for the pump andfor the motor respectively. From the data sheet in Appendix A it is seen thatthe torque efficiencies have fairly constant levels over the working range of thepump and the motor and will therefore be set to constants

ηtp = 0.92, ηtm = 0.97.

The volumetric efficiencies render larger problems. The variations of the effi-ciencies are larger and the efficiencies given for the pump and the motor arenot perfectly covering all driving cases. For the pump the efficiency is given asa function of displacement, but just for the rotation speed 2100 rpm. Most ofthe time the pump rotation is in the range of 1100 - 1600 rpm. For the motorit is the opposite situation, then the displacement is fixed and the function ofthe rotation speed is given. This is true except for really high speeds, becausethen the displacement decreases at the same time as the rotation increases. Toget around this problem the rotation dependency of the motor will be used toadjust the volumetric efficiency for the pump. This concept is used the otherway around as well.

To implement this, two dimensional lookup-tables are used. The tables for thepump and the motor have the same structure and only the values differ. Threelevels for the displacement are used: minimum, maximum and an appropriatelevel in between. The rotation has five levels. The lowest level is zero rotationand the highest is higher than any measured rotation. That will ensure that novalues are outside the range of the table. The other three levels are set to covermost of the driving cases.

The efficiency values for the highest rotation level are set equal to the efficiencyvalues for the second highest rotation level. This is also done for the efficiencyvalues for the lowest and second lowest rotation levels. The reason is to makeit easier to optimize the efficiency values later on. The discussion above hasresulted in two Tables, 6.1 and 6.2.


Pump, Rotation speed [rpm] 0 1100 1350 1600 2500

Displacement, 0 cm3 0.56 0.56 0.58 0.59 0.59

Displacement, 85 cm3 0.84 0.84 0.86 0.87 0.87

Displacement, 147 cm3 0.90 0.90 0.92 0.93 0.93

Table 6.1: Volumetric efficiency for the pump, ηvp, given in a table.

Motor, Rotation speed [rpm] 0 600 1200 2000 4000

Displacement, 68 cm3 0.77 0.77 0.88 0.97 0.97

Displacement, 105 cm3 0.81 0.81 0.91 0.97 0.97

Displacement, 160 cm3 0.85 0.85 0.94 0.97 0.97

Table 6.2: Volumetric efficiency for the motor, ηvm, given in a table.

6.2 Simulations for the pressure

A simulation model for the charge pressure in the hose was the main goal ofthis project. Measurements of rotations and currents that can be transformed todisplacements of the pump and the motor are available. Using this informationit should be possible to simulate the pressure using

Qp = Dpωpηvp,

Qm =ωpDm

ηvm,

p =β

V

∫ t

0

(Qp −Qm) dτ ,

where the volume of the hose V can be calculated as

V = LD2

4π ≈ 600cm3. (6.1)

The parameter L stands for the length of the hose and D is the diameter of thehose. All parameters are known with more or less accuracy except for β. Thebulk modulus β depends on which oil that is used. The range of the bulk mod-ulus is 107-1010 Bar. The flows are not affected by β, only the rate of changein pressure.

An estimate of β should make it possible to simulate the pressure for the sys-tem. Unfortunately it does not. The simulated pressure can be in the samerange as the measured pressure in one or a couple of the test runs, but for the

6.3. Black box models 35

rest of the test runs the simulated pressure goes to either plus or minus infinity.The problem is that the simulated flow out from the pump is not equal to thesimulated flow into the motor when the measured pressure is constant and themeasurement values for the rotations are used for the calculations.

One idea that could improve the simulation results would be to use some iden-tification method from Chapter 4 to estimate better values for the parameters.The two efficiency parameters ηvp and ηvm are the most uncertain ones, butalso the functions from currents to displacements are uncertain. There are twoproblems when the efficiencies should be estimated, as seen in (3.17). The ef-ficiencies probably depend on many variables and are non-linear. Some linearmodels have been tested but with negative result. Non-linear identification isnecessary to get a better estimate for the efficiencies. Due to time limits non-linear identification has not been tested in this thesis, instead the efficienciesfrom Section 6.1.4 have been used.

Even if the parameters would be improved the simulated pressure will not be ac-curate for all test runs. This statement may seem strange, but when consideringthat the pressure is an integral over time of two flows it is quite obvious why itwill not work. These flows are only dependent on parameters and measurementsand there is no feedback involved in this part of the system. Consequently, if astationary error is present, for example the simulated input flow is larger than itis supposed to be, then the pressure will go to infinity. Therefore, this approachis unrealistic since perfect models are not possible to build.

To continue it is necessary to realize that the problem is that there is no feed-back from the simulated pressure. One idea to solve the problem is to add themodels for the pump and the motor. If the pressure in the hose goes up theresistance for the pump will increase and it will therefore rotate slower. This inturn will result in a decreased flow as well as a lower pressure. On the motorside the same thing will happen, the higher pressure will rotate the motor fasterand the flow out of the hose increases, with lower pressure as a result.

To summarize, it can be realized that if only a model for the hose is used, it willmake the simulations unbounded. Hopefully models on the sides of the hosewill create that necessary feedback to make the simulated pressure stable.

6.3 Black box models

The advantage with a black box model is that no knowledge about the systemis necessary. Instead the computer has the freedom to adjust a model to fitmeasurement data. Still there are choices that need to be considered; whatkind of model should be used, what the model order should be and which datathe model should be based on. In this thesis the graphical interface for systemidentification toolbox in Matlab [32] has been used to create the black boxmodels. SITB uses signal processing to estimate the black box models with the


conditions given by the user.

As previously mentioned a data set that will be used by SITB to estimate themodel is needed. The data set used should have large variations in both inputand output signals so the behavior of the system can be captured by the blackbox model. Test run nine has fairly big variations and is therefore used in allidentification.

6.3.1 Pump

A good start is to estimate a model for the rotation of the pump. To beginwith we need to decide which regression vector that should be used. Availablemeasurement signals are: ωp, ωm, ip, im, p and preturn. It is reasonable toassume that im and ωm do not directly affect the rotation for pump. Anotherassumption is that the difference ∆p between p and preturn is more importantthen the individual values. The function from ip to Dp will be used, eventhough this function is linear it is unnecessary to estimate it. The argumentsabove have resulted in the following conclusion; the displacement Dp and thepressure difference over the pump will be used as inputs and the rotation forthe pump ωp will be used as output, see Figure 6.2.

Figure 6.2: Input and output signals for the black box model for the rotationof the hydraulic pump.

The next decision that needs to be taken is which kind of model that should beused. Because the sensors are fairly accurate it seems logical that most noisewill enter at the input side of the system. Therefore, ARX and ARMAX modelsare preferable. An ARMAX model will be used because it contains more dy-namics for the noise. Different number of poles in the ARMAX model have beentested. Two poles have been found to give good agreement between measuredand simulated values, see Figure 6.3. If higher model order is used no greaterimprovement is visible.

The reason for modeling the rotation of the pump was that it would hopefullystabilize the simulations for the pressure in the hose. If this setup works itwill demand that the black box model for the rotation of the pump will reactreasonable to an input change. To test this, steps in both Dp and ∆p wereperformed. In Figure 6.4 the measured ∆p is input but at the 20th second astep of 200 bar is added to the input. The models reaction to the step is a lowerrotation speed. This is logical because higher pressure in the hose will makeit harder for the pump to press oil into the hose and therefore the speed willdecrease. However, it seems that the response in rotation speed is too low. A200 bar step would probably slower the rotation a lot more.


10 11 12 13 14 15 16 17 18 19 201300

1350

1400

1450

1500

1550

Time [s]

Rot

atio

n pu

mp

[rpm

]

Measured valuesSimulated values

Figure 6.3: Simulated and measured values for a black box model for the pump’srotation.

18 18.5 19 19.5 20 20.5 21 21.5 22

1350

1400

1450

1500

Time [s]

Rot

atio

n pu

mp

[rpm

]


18 18.5 19 19.5 20 20.5 21 21.5 220

100

200

300

400

Time [s]

Del

ta p

ress

ure

[Bar

]

Figure 6.4: Simulated rotation of the pump reacting to a step in ∆p.


The response to a step in the displacement of the pump Dp is shown in Fig-ure 6.5. The step is positive because the displacement of the pump increases.A larger displacement should result in lower rotation speed because the pumpneeds to provide more flow into the hose. As seen in Figure 6.5 this it is notthe case for this model of the rotation of the pump.

18 18.5 19 19.5 20 20.5 21 21.5 22

1400

1500

1600

1700

1800

1900

Time [s]

Rot

atio

n pu

mp

[rpm

]

18 18.5 19 19.5 20 20.5 21 21.5 2280

90

100

110

Time [s]

Dis

plac

emen

t pum

p [c

m3 ]


Figure 6.5: Simulated rotation of the pump reacting to a step in Dp.

The estimated model for the rotation of the pump is not that good. Even ifit can describe the estimation data well the model will be useless when it doesnot react in a physical way to changes in input. However, it is possible thata model which reacts in an accurate way could be found if SITB did not usethe ARMAX model. As explained in Section 4.2.1 an ARMAX model has therestriction that the same poles are used for both the input signal and the noise.To test this hypothesis a lot of different test runs and models (for example BJ,State space and process models) have been tried without any success. All ofthem reacts incorrectly to a step in Dp.

Another hypothesis was to use different input signals. As can be seen in (3.17)Dp and ∆p affect the rotation of the pump as a combination. Therefore, theterm Dp∆p has been tested as input. Unfortunately this did not turn out welleither. If only Dp∆p was used as a input signal the agreement between simu-lated and measured values became poor. This is not so surprising since thereare probably variables inside the engine and the pump that depend on Dp and∆p as separate components. An idea then would be to use Dp, ∆p and Dp∆p asinput. Approximately the same fit as in Figure 6.3 was then achieved but whenstep responses were tested the model did not react correctly. The problem couldbe that the function from ip to Dp is not correct. To see if that was the casethe tests above have been made with ip as input instead of Dp. Unfortunatelythis did not help.

To get the explanation why no good model for the rotation of the pump hasbeen found a closer look at (3.17) is motivated. The description for the rotation


of the pump contains nonlinear part such as: ηtp(ωp,∆p, Dp; θηtp) and Dp∆p.

In (3.18) ηtp(ωp,∆p, Dp; θηtp) has been approximated with a constant. Even if

this approximation is correct the term Dp∆p is present. Because of the largeworking range it is not so strange that a linear model has problems to describethe behavior of the system.

There is also another reason why it is so hard to estimate a model for therotation of the pump. The model has to include dynamics for the diesel engine.As described in Section 3.2 the diesel engine has a feedback control to ensure acertain rotation speed for the pump. This internal control will try to keep therotation for the pump at a constant level. As discussed before, large variationsof input and output signals are necessary to estimate a good model. The dieselengines internal control will counteract the excitation of the system.

6.3.2 Motor

On the other side of the hose a model for the rotation of the hydraulic motormight be able to stabilize the pressure in the hose. To develop a model for themotor the same procedure as for the pump has been used. Displacement of themotor Dm and the pressure difference ∆p are input signals and the rotation ofthe motor ωm is output. These signals correspond to the signals that workedbest for the pump.

When choosing which kind of model structure that should be used the noisemust be considered. When developing a model for the rotation of the motorit is quite obvious that the dominating disturbance is from the ground. A nonflat surface enters as a torque disturbance to the motor, which means in thebeginning of the process. Therefore, an ARMAX structure has been used whenestimating a model for the rotation of the motor. When using model order twothe result can be seen in Figure 6.6.

0 5 10 15 20 25 30 35 400

500

1000

1500

2000

2500

Time [s]

Rot

atio

n m

otor

[rpm

]


Figure 6.6: Simulated and measured values for a black box model for the motor’srotation.

The model for the pump did not react correctly when the input signals werechanged. To see if the model for the motor has a better physical relevance steps


in both Dm and ∆p were performed. In figures 6.7 and 6.8 the responses areshown. When a step up in ∆p is made the model reacts by rotating faster. Thisis reasonable for two reasons, first the equation for the rotation of the motorin (3.18) would predict an increased rotation speed when ∆p is higher. Second,with everything else unchanged an increased pressure delivers more energy andshould therefore make the vehicle go faster. In Figure 6.8 the response of anincreased displacement of the motor is visible. When the displacement goes upthe motor demands more oil for each revolution. The same amount of oil stillreaches the motor, therefore it is logical that the rotation of the motor decreaseswhen Dm increases.

14 16 18 20 22 24 26 28 30

500

1000

1500

2000

2500

Time [s]

Rot

atio

n m

otor

[rpm

]

14 16 18 20 22 24 26 28 300

100

200

300

400

Time [s]

Del

ta p

ress

ure

[Bar

]


Figure 6.7: Simulated rotation of the motor reacting to a step in ∆p.

14 16 18 20 22 24 26 28 30

0

1000

2000

Time [s]

Rot

atio

n m

otor

[rpm

]

14 16 18 20 22 24 26 28 3050

100

150

200

Time [s]

Dis

plac

emen

t mot

or [c

m3 ]


Figure 6.8: Simulated rotation of the motor reacting to a step in Dm.

The black box model for the motors rotation seems good so far. The problemis that validation only has been made on the same data as the model has beenestimated. To do a better test the model has been applied to test run number

6.4. Grey box models 41

five. Figure 6.9 shows that the simulated values do not correspond to the mea-sured values. Consequently, the model does not work well enough because itcan not handle new data sets.

There are two reasons why it does not work. The first one is the same as for thepump, a nonlinear system is approximated by a linear model. That approachcan work if the variations are small, but in our case the model should handlevery large variations in both input and output signals. The second reason forthe poor behavior of the model is due to the bumpy ground. A simulation modelfor the rotation speed of the motor is desired, but the motor is connected tothe wheels which are affected by the ground. Therefore, the model needs toconsider how large the torque from the ground is. This could maybe be possibleif the tests had been performed at a flat surface but with a rather messy roadit is very hard.

0 5 10 15 20 25 300

200

400

600

800

1000

1200

Time [s]

Rot

atio

n m

otor

[rpm

]


Figure 6.9: Test of the black box model for the motors rotation with test runfive. The black box model is estimated with test run nine.

6.4 Grey box models

The problem with unbounded simulated pressure is still present. Some kind ofmodels for the pump and the motor are necessary. Using black box models didnot work. One way to get around this problem would be to use the informationabout the system that is presented in Chapter 3.

6.4.1 Pump

As described in Section 6.3.1 there were two reasons why the black box modeldid not work. The system is non-linear and the internal control for the enginecounteracted the variations. Therefore, the equation that describes the rotationfor the pump in (3.18) will be used

(Jp + Jd)wp = Td − µpωp −Kp −Dp∆p

ηtp. (6.2)


The problem is that the dynamics behind Td are unknown. Therefore a black boxmodel for Td will be estimated. With this approach the black box approximationdoes not have to consider the non-linear term Dp∆p

ηtp. The problem with the

internal control of the engine will also disappear because the information aboutthe internal control that previously was hidden is now available when Td is usedas output from the black box model. When using this approach a better modelfor the rotation of the pump will be available because Dp and ∆p have enteredin a more realistic way, see Figure 6.10 for an illustration of the model.

Figure 6.10: The model used in this section with a black box model representingthe diesel engine.

To estimate a black box model for the torque it is necessary to know the valuesfor Td. Therefore, probable values for Td will be calculated using a modifiedversion of (6.2). The reason to call them probable is that no measurement forTd is available. An investigation of (6.2) gives that ωp and ∆p are available frommeasurements. The displacement Dp can be determined from ip as describedin Section 6.1.1. The torque efficiency ηtp is available in Section 6.1.4 and Jp

can be found in [26]. The time derivative wp of ωp can cause some trouble tocompute from measurement data but with a low pass filter it should be possible.Even so the term (Jp +Jd)wp has been set to zero when Td is calculated because(Jp + Jd)wp is small compared to Td and will therefore not affect much. Thatleaves four terms that are unknown namely Jd, Td, µp and Kp. To simplify, thefriction coefficient Kp is set to zero, this coefficient should maybe be includedlater. From (6.2) Td can be obtained

Td = µpωp +Dp∆p

ηtp. (6.3)

An assumption for µp, makes it possible to calculate probable values of Td.When the values for Td are available a black box model for Td can be estimated.An ARMAX model with model order two and Dp and ∆p as input signals, isused to reproduce Td. The black box model has been estimated with data fromtest run nine. Figure 6.11 illustrates this concept.

Figure 6.11: How to generate a black box model for Td.


The black box model for Td together with (6.2) should be able to simulate therotation of the pump. To do this an initial guess for Jd is necessary. The valuesfor Jd and µp have been modified so the model agrees better with reality. Thenumerical values for Jd and µp are

Jd = 0.17 Nms2, µp = 10 Nms.

Figure 6.12: The model for the rotation of the pump.

To validate if this model for the rotation of the pump is accurate, the data fromtest run five have been used. The model can be seen in Figure 6.12. When thismodel will be connected to the other models the measured values for ∆p willbe exchanged to the simulated values. Two properties need to be checked, howgood is the estimate of Td and how well does the simulation result for ωp agreewith the measured values. The result of these two test are shown in Figure 6.13.

5 6 7 8 9 10 11 12 13 14 15

1000

1500

2000

2500

Time [s]Tor

que

from

die

sel e

ngin

e [N

m]

5 6 7 8 9 10 11 12 13 14 15900

1000

1100

1200

1300

1400

Time [s]

Rot

atio

n pu

mp

[rpm

]

Probable valuesSimulated values


Figure 6.13: Results of a grey box simulation for the pump. Simulated valuesfor Td and ωp are compared to the probable/real values.

The simulation of Td is surprisingly good, keeping in mind that the estimate andvalidation are done on completely different data sets. The agreement betweensimulated and real values for the rotation of the pump ωp is a little bit worsebut not bad. The constant level is good but the variations are not matchingentirely. Other test runs give approximately the same results. However, theconstant level is the most important part and the variations are probably smallcompared to those that will occur because of changes in ∆p when connectingthis model to the model for the hose.


This method to develop a model for the rotation of the pump has a drawback.Problems can occur if (6.2) is a bad model for the rotation of the pump. Thenthe measured values for the system will be forced to fulfill (6.2). Because Td iscalculated the values for Td can contain system dynamics that have not beenmodeled. However that is probably not the case because a black box estimatedmodel could describe Td well. If Td had some strange dynamics the black boxestimation would not have been that accurate.

6.4.2 Motor

The grey box method seemed to work fine for the pump, and the same procedurehas been tested on the motor. An illustration of the modeling approach can beseen in Figure 6.14. The rotation of the motor is described by

(Jm + Jω) wm = Dm∆pηtm − µmωm − TL. (6.4)

If TL was known this would probably be a good description for the rotation ofthe motor. Because TL is not known a black box model for TL has been tried.To create this black box model probable values for the torque are necessary.A modified version of (6.4) can be used for this calculation. To reduce theunknown parameters Km will be set to zero. Hopefully this approximation doesnot affect the result because Km would probably be small compared to TL.Another motivation for this simplification will also be provided further on.

Figure 6.14: The modeling approach for the rotation of the motor.

In (6.4) there are three unknown terms; Jω, µm and TL. The moment of in-ertia of the motor Jm, which can be found in [33] and the other terms areeither measured or can be derived. To simplify the calculation of TL the term(Jm + Jω) wm has been set to zero. One reason to do this simplification is thatthe estimated torque TL is rather noisy, leave out the term (Jm + Jω) wm in thecalculation makes TL a bit smoother. Equation (6.4) can now be rewritten as

TL = Dm∆pηtm − µmωm. (6.5)

An assumption for the friction coefficient µm is used when probable values ofTL is calculated. An ARMAX structure with model order two has been usedto derive a model for TL. This model together with (6.4) should simulate therotation for the motor. But to do that an initial guess for Jω is necessary. Theinitial values for Jω and µm has later been modified to

Jω = 5 Nms2, µm = 5 Nms.


To test if the achieved model for the rotation is good test run five is used.As seen in Figure 6.15 the black box model for TL is not bad, but not goodeither. Compare with Td in Figure 6.13 where a very good fit is achieved. Thedisagreement between probable and simulated values for TL is the reason to thepoor simulation results for ωm.

10 11 12 13 14 15 16 17 18 19 200

500

1000

1500

2000

Time [s]

Tor

que

from

the

whe

els

[Nm

]

10 11 12 13 14 15 16 17 18 19 200

1000

2000

3000

4000

Time [s]

Rot

atio

n m

otor

[rpm

]

Probable valuesSimulated values


Figure 6.15: Results of a grey box simulation for the motor. Simulated valuesfor TL and ωm are compared to the probable/real values.

One possible explanation why this approach did not work for the rotation of themotor is that (6.4) could be a bad description for the system. When the systemis forced to work according to (6.4) the calculation of TL can be strange. If thatis the case it is impossible estimate a black box model for TL.

There is another explanation to the bad performance that is more likely. Whenestimating the black box model SITB is supposed to derive a model for thetorque from the ground. But the road used when testing was quite bumpywhich make the torque from the road unpredictable. Therefore, a black boxmodel can not be used to simulate TL. However, this approach can have somerelevance if tests were performed on a flat surface, then maybe a good modelfor TL can be achieved.

Figure 6.16: The model for the rotation of the motor.


When no model could be developed for the torque from the ground, the calcu-lated probable values of TL will be used. Figure 6.16 shows the model. Thereason to use TL instead of the measured values for the rotation of the motorωm is that when TL is used the rotation for the motor can still increase if thepressure goes up. That was the first reason why models for the pump and themotor were necessary.

Consequently, to get the rotation for the motor, probable values of TL will becalculated with (6.5). The values for TL will be inserted into (6.4). This alsoexplains why there was no reason to include Tfm in previous calculations, Tfm

would just have been added and then subtracted again. When using (6.4) andTL as a model to simulate ωm the correct values should show up. To check thatthis is true, test run eight have been used, see Figure 6.17. The reason whysimulated and measured values for TL do not match perfectly is that the term(Jm + Jω) wm has been neglected in (6.5).

8 10 12 14 16 18 20

800

1000

1200

1400

1600

1800

Time [s]

Rot

atio

n m

otor

[rpm

]


Figure 6.17: Comparison between simulated and measured values for ωm whencalculated TL is used.

6.5 Complete model

All three sub models have been developed and can now be put together to formthe complete model. Hopefully this will solve the problems with the pressuregoing to infinity which was described in Section 6.2. To recall how the completemodel looks like see Figure 6.18.

Simulations that determine p, ωp and ωm will be performed. The torque TL andthe pressure preverse are input signals to the system together with the controlsignals Dp and Dm. In Figure 6.19 and 6.20 test run two is used to examinehow the model behaves. The Simulink files used for these simulations can beseen in Appendix B. The code files used to initiate the model are attached inAppendix C.


Figure 6.18: Overview of the complete model.

30 32 34 36 38 40 42 44 46 48 50

50

100

150

200

Time [s]

Pre

ssur

e [B

ar]

30 32 34 36 38 40 42 44 46 48 501100

1200

1300

1400

1500

Time [s]

Rot

atio

n pu

mp

[rpm

]

30 32 34 36 38 40 42 44 46 48 50

400

600

800

1000

1200

Time [s]

Rot

atio

n m

otor

[rpm

]


Figure 6.19: Simulation of the complete model.

The visible results of the simulation are: the pressure agrees incredibly fine withmeasured values, the mean value for the pump rotation is good but the variationis somewhat too large and the rotation of the motor is too small. Simulationsfor other test runs show the same relationship between simulated and measuredvalues for p, ωp and ωm.

The reason why the simulated values for the rotations are not matching themeasured values better is because the model used is not perfect. There is astrong relationship between p, ωp and ωm, and therefore it is not possible tojust decrease TL to reach a faster rotation for the motor, because that will alsoaffect p. More correct values for parameters and maybe adding more dynamicscould improve the model and get better simulation results for the rotation ofthe pump and the motor. In order to make it easier to find values for momentof inertias and bulk modulus used in this thesis an optimizer was developed butsome hands on operation have also been performed to adjust all the unknownparameters.


30 30.5 31 31.5 32 32.5 33 33.5 34 34.5 35

50

100

150

200

Time [s]

Pre

ssur

e [B

ar]


Figure 6.20: Simulation of the complete model. Zoomed in version of Fig-ure 6.19.

One thought is that the calculated torque from the ground makes the pressuresimulation so good. But before θ was optimized the pressure was not correctat all. Therefore, it is reasonable to believe that the good simulations of thepressure exist because the model describes the reality when the unknown pa-rameters have been adjusted.

It would be interesting to investigate how sensitive the system is to changes inTL. If TL was not known and had to be estimated, then a model in which asmall error in TL affects the result too much is not desired. The same data asin Figure 6.20 are used for two tests. In Figure 6.21 a white noise has beenadded to TL, the noise has a sample time of 0.5 second and a variance of 600Nm. Usually TL is around 600 Nm. The noise only has a modest effect on thepressure. The second test is a bias change. A step of 200 Nm has been addedto TL at the 32th second. As seen in Figure 6.22 it does not affect the pressurethat much.

30 30.5 31 31.5 32 32.5 33 33.5 34 34.5 35

50

100

150

200

Time [s]

Pre

ssur

e [B

ar]


Figure 6.21: The simulated pressure when white noise with a sample time of 0.5second and a variance of 600 Nm has been added to TL.

The small reaction on p that a change in TL leads to keep the hope that it ispossible to make some kind of model for the motor side.


30 30.5 31 31.5 32 32.5 33 33.5 34 34.5 35

50

100

150

200

Time [s]

Pre

ssur

e [B

ar]


Figure 6.22: The simulated pressure when a step of 200 Nm at time 36 secondswas added to TL.

A possible solution would be to exchange the grey box model on the motorside to the black box model with ωm as output. This has been tested, butunfortunately it did not work well. One reason is that the black box model forthe rotation of the motor was not sensitive enough to changes in pressure.


Chapter 7

Discussion and Conclusions

First I will comment some of the material presented, and then offer conclusions Ihave been able to draw from this work.

7.1 Discussion

After different methods and parameter estimations have been tested, it is finallypossible to simulate the pressure with good accuracy. Unfortunately a calcu-lated value of the torque from the wheel is necessary. A better model is neededto get around this problem. This improved model should be able to match thethree validation signals; ωp, ωm and p. To achieve that I believe focus will beon the hydraulic model of the hose and mainly on the volumetric efficiencies.An idea for how to improve the values for the efficiencies will be presented inChapter 8.

If a model that produces better values for ωp, ωm and p is obtained it might bepossible to make a model to estimate TL instead of using it as an input signal.An argument for this statement is that the pressure was not too sensitive tochanges in TL as seen in Section 6.5. Another reason is that the black boxmodel for Td worked very well. Although it is much less noise when modelingTd it shows that the idea has some potential.

To get a better model it is also necessary to perform tests on a flat surface. Thenit would be much easier to get accurate values for unknown parameters, becauseunder those conditions TL would have a less unpredictable behavior. That couldmake it possible to estimate the swashplate dynamics. The swashplate dynamicshave not been addressed in this thesis but should probably improve the modelif implemented.

A mistake made in this thesis, was to try to keep constant speed in too manytests. The idea for that was to find out if the tests were repetitive, but it wasnot possible because of the bumpy road. Instead large variations should havebeen performed to excite the system, and then SITB would have had an easier

Carlsson, 2006. 51

52 Chapter 7. Discussion and Conclusions

task to estimate black box models.

MS5050 has the possibility to measure six signals. If an additional signal wasto be measured, then one of the signals that we used in our experiment wouldhave to be removed. Now when the work has been carried out I know that thelocations for the sensors that were used when doing the test runs were the bestones, and that too much information would be lost if one of them was removed.

The measurement of the rotation is not very smooth and this might cause prob-lems when developing the model. One way to handle this is to attach extrareflexes for the rotation sensors. Another way to get a smother signal is todecrease the sample time of the rotation sensors.

7.2 Conclusions

The greatest challenge of this thesis has been that the pressure in the hose is asum of two flows. If the measurement signals are used for the rotation of thehydraulic pump and motor there is no feedback which affects the system if thepressure gets too high. This led to the conclusion that models of the pumpand the motor are necessary to make the model stable. Because of that theassignment got more difficult than expected, and therefore no control systemfor the pressure has been developed.

Good simulation results for the pressure have been obtained, but only when thetorque from the ground is considered as input to the system. Unfortunately thisis not an available measurement signal. To get around this problem a bettermodel is necessary. The problem with the model as it is now is that simulatedvalues for the rotation of the pump and the rotation of the motor do not matchthe measured values. This could be because some dynamics have been missed,but probably it is caused by the unknown parameters which are not estimatedin a good way.

To achieve better values for the model more tests need to be accomplished. Alsothe parameters for the efficiencies should be improved. When this has been doneit could be possible to achieve a model that can predict the pressure online, andthen the task to avoid pressure peaks can be dealt with.

Chapter 8

Future work

One of the largest problems of the model is that the flow out from the pumpis not equal to the flow into the motor when the pressure is constant and themeasurement values for the rotations are used for the calculations. To fix thisthe two efficiencies ηvp and ηvm need to be adjusted.

The idea is to find a test run which have two time points when the pressure isthe same. Between these two time points the sum of the flows in and out of thehose must be zero. Otherwise the pressure would not be the same. The flowinto the hose is determined by

Qp = Dpωpηvp,

and the flow out of the hose is

Qm =ωpDm

ηvm.

To make the sum of the flows be zero the relationship between ηvp and ηvm willbe given because Dp, Dm, ωp and ωm are measured. The problem is that theefficiencies depend on a lot of terms. At least the relationship between ηvp andηvm depends on Dp, Dm, ωp and ωm but probably also on p and possibly onpreturn. Therefore time intervals when as many terms as possible are constantneed to be found. All twelve test runs can be searched for these time intervals.Then a map of how the relationship between ηvp and ηvm depends of Dp, Dm,ωp, ωm, p and preturn can be found. Hopefully this gives a much better modelfor the pressure in the hose.

Another suggestion for better simulation results would be to use the values forthe rotation of the pump that are available online. This measurement signalcould be used to make the simulation results approach more correct values.

To achieve a better model it would be very helpful if new test runs were per-formed. One problem in this thesis has been that the test runs were made on

Carlsson, 2006. 53

54 Chapter 8. Future work

a bumpy road. Therefore, it is important that new test runs were made ona flat surface. Then the estimation of the torque from the ground would bemuch easier. Another circumstance that would help even more would be to usea sensor to measure the torque from the ground. If tests were performed withsuch sensor and maybe even a sensor for the torque from the diesel engine themodel could undergo big improvements. An additional motive for new tests isthat more variation in the input signal could be used and then it is easier toapply some identification method from Chapter 4, for example neural networks.

Bibliography

[1] Komatsu Forest. http://www.komatsuforest.com/, Acc. 2005-12-20.

[2] Lino Guzzella and Antonio Sciarretta. Vehicle Propulsion System. Springer,2005.

[3] Kalevi Huhtala. Modelling of Hydrostatic transmission - Steady State, Lin-ear and Non-Linear Models. PhD thesis, Tampere University of Technology,October 1996.

[4] Olav Egeland and Jan T. Gravdahl. Modeling and Simulation for Auto-matic Control. Marine Cybernetics, 2002.

[5] Machine Design. http://www.machinedesign.com/BDE/FLUID/bdefp6/bdefp6 6.html,Acc. 2005-10-10.

[6] H. E. Merritt. Hydraulic Control System. J Wiley & Sons, 1967.

[7] N. D. Manring. Hydraulic Control System. J Wiley & Sons, 2005.

[8] E. A. Prasetiawan. Modeling, simulation fo an earthmoving vehicle power-train simulator. Master’s thesis, University of Illinois, 2001.

[9] Jerker Lennevi. Hydrostatic Transmission Control. PhD thesis, LinkopingUniversity, November 1995.

[10] Eko A. Prasetiawan, Zhang Rong, Andrew G. Alleyne, and Tsu-Chin Tsao.Modeling and control design of a powertrain simulation testbed for earth-moving vehicles. 1999.

[11] K. Wu, Q. Zhang, and A. Hansen. Modelling and identification of hy-drostatic transmission hardware-in-loop simulator. Vehicle Design, 34(1),2004.

[12] Rong Zhang. Multivariable Robust Control of Nonlinear Systems with Ap-plication to an Electro-Hydraulic Powertrain. PhD thesis, University ofIllinois, 2002.

[13] M. Sannelius. Complex Hydrostatic Transmissions - Design of a Two-MotorConcept using Computer-Aided Development Tools. PhD thesis, LinkopingUniversity, 1999.

[14] R. Andersson. A simulation model of a hydrostatic dynamometer for im-proved controller design. Linkoping University, 1999.

Carlsson, 2006. 55

56 Bibliography

[15] Kalervo Nevala, Jari Penttinen, and Pekka Saavalainen. Developing ofanti-slip contro of hydrostatic power transmission for forest tractor andoptimasation of the power of diesel engine. IEEE, 1998.

[16] Zenon Jedrzykiewicz, Janusz Pluta, and Jerzy Stojek. Application of thematlab - simulink package in the simulation tests on hydrostatic systems.Acta Montanistica Slovaca, 1998.

[17] Leon Dahlen. Numerical and Experimental Study of Performence of a Hy-draulic Motor. PhD thesis, Lulea University of Technology, September2003.

[18] Kenneth Weddfelt. On Modelling, Simulation and Measurements of FluidPower Pumps and Pipelines. PhD thesis, Linkoping University, January1992.

[19] H. Reuter. State space identification of bilinear canonical forms. In Control,Mars 1994.

[20] M. Jelali and H. Schwarz. Nonlinear identification of hydraulic servo-drivesystems. Control Systems Magazine, October 1995.

[21] H. Anyi, R. Yiming, Z. Zhongfu, and H. Jianjun. Identification and adap-tive control for electro-hydraulic servo system using neural networks. InIEEE International Conference on Intelligent Processing Systems, October1997.

[22] M. K. Park and C. L. Lee. Identification of a hydraulic simulator usingthe modified signal compression method and application to control. InIndustrial Electronics Society, October 2000.

[23] J. Lennevi and J-O. Palmgren. Application and implementation of lq designmethods for the velocity control of hydrostatic transmissions. Journal ofSystems and Control Engineering, July 1995.

[24] Jose Cidras and Camilo Carrillo. Regulation of synchronous generators bymeans of hydrostatic transmissions. IEEE Transactions on power systems,15(2), May 2000.

[25] Luigi del Re. Nonlinear modelling and black box identification of a hydro-static transmission for control system design. In Mathematical and Com-puter Modelling, September 2002.

[26] Sauer Danfoss. Series 90 Axial Piston Pumps, Mars 2004.

[27] Lennart Ljung and Torkel Glad. Modellbygge och simulering, volume 2nded. Studentlitteratur, 2004.

[28] L. Ljung. System Identification: Theory for the User, volume 2nd. PrenticeHall PTR, 1999.

[29] J. Sjoberg. Non-Linear System Identification with Neural Networks. PhDthesis, Linkoping University, 1995.

[30] Mathworks. System Identification Toolbox 6.0, 2004.

Bibliography 57

[31] Hydrotechnik. User Manual for Multi-System 5050, 2003.

[32] Mathworks. Matlab 7.0, 2004.

[33] Sauer Danfoss. Series 51 Bent Axis Variavle Displacement Motors, Decem-ber 2003.

[34] Don Earl Carter. Load modeling and emulation for an earthmoving vehiclepowertrain. Master’s thesis, University of Illinois, 2003.

[35] I A Njabeleke, R F Pannett, P K Chawdhry, and C R Burrows. Self-organising fuzzy logic control of a hydrostatic transmission. In UKACCInternational Conference on CONTROL, September 1998.

[36] Kenneth Weddfelt. On Modelling, Simulation and Measurements of FluidPower Pumps and Pipelines. PhD thesis, Linkoping University, Mars 1992.

58 Bibliography

Appendix A

Data sheet

Carlsson, 2006. 59

60 Appendix A. Data sheet

Appendix B

Simulink models

Figure B.1: Main model for the hydrostatic transmission. This model has threesub models, which can be seen in Figure B.2, B.3 and B.4.

Carlsson, 2006. 61

62 Appendix B. Simulink models

Figure B.2: Model for the hydraulic pump.

Figure B.3: Model for the hydraulic motor.

Figure B.4: Model for the hose. The sub models are shown in Figure B.5.

63

Figure B.5: Submodels in the hose model.

Figure B.6: Model for calculation of Td and TL.

64 Appendix B. Simulink models

Appendix C

.m-files

This Appendix consists of three .m-files which are used to initilize the model. Thecode file in C.3 uses the files in C.1 and C.2.

C.1 filtrera.m

Using butterworth filters to smooth the meassurementdata.

function [matrix_ff] = filtrera(matrix)

matrix_ff(:,1) = matrix(:,1);

[b,a]=butter(1,0.05);

matrix_ff(:,2) = filtfilt(b,a,matrix(:,2));


[b,a]=butter(3,0.1);



[b,a]=butter(1,0.01);


[b,a]=butter(1,0.005);


Carlsson, 2006. 65

66 Appendix C. .m-files

C.2 umu fetch matrix.m

Makes it easier to change data set.

function [matrix_name] = umu_fetch_matrix(i)

%matris_du_vill_ha = eval(umu_fetch_matrix(i));

if i == 1

matrix_name = ’umu_01’;

elseif i == 2


elseif i == 3


elseif i == 4


elseif i == 5


elseif i == 6


elseif i == 7


elseif i == 8


elseif i == 9


elseif i == 10


elseif i == 11


elseif i == 12


end

C.3. Hydrostatic transmission.m 67

C.3 Hydrostatic transmission.m

Initilazies the model.

load umu_all_fix %load the data

%scale with this factor.

nersampling = 10;

%choose not to start from 0.01

start = 1;

% ----------------- Parameters

my_p = 10;

Tfp = 0;

eta_tp = 0.92;

my_m = 5;

Tfm = 0;

eta_tm = 0.97;

Jp = 0.0023;

Jm = 0.02;

Jd = 0.17;

Jw = 5;

Jpd = Jp + Jd;

Jmw = Jm + Jw;

Bulk = 3.7e8;

V = 1.22*(25e-3/2)^2*pi;

dis_p = [0 85 147];

varv_p = [0 1100 1350 1600 2500];

dis_m = [68 105 160];

varv_m = [0 600 1200 2000 4000];

p11 = 0.56;

p12 = 0.58;

p13 = 0.59;

p21 = 0.84;

p22 = 0.86;

p23 = 0.87;

p31 = 0.90;

p32 = 0.92;

p33 = 0.93;

m11 = 0.77;

m12 = 0.88;

m13 = 0.97;

m21 = 0.81;

m22 = 0.91;

m23 = 0.97;

m31 = 0.85;

m32 = 0.94;

m33 = 0.97;

eff_p = [p11 p11 p12 p13 p13;


p21 p21 p22 p23 p23;

p31 p31 p32 p33 p33];

eff_m = [m11 m11 m12 m13 m13;

m21 m21 m22 m23 m23;

m31 m31 m32 m33 m33];

%from datasheet

n_tp_v = [0.8830 0.9111 0.9233 0.9301 0.9324 0.9308...

0.9278 0.9242 0.9202 0.9201 0.9201];

n_vp_v = [0.5973 0.7423 0.8037 0.8378 0.8496 0.8762...

0.9060 0.9258 0.9391 0.9392 0.9392];

n_tm_v = [0.9816 0.9811 0.9803 0.9791 0.9783 0.9761...

0.9710 0.9635 0.9536 0.9241 0.8696];

n_vm_v = [0.3757 0.7013 0.8037 0.8539 0.8702 0.9035...

0.9373 0.9570 0.9687 0.9688 0.9688];

Angle_pump_v = [3 5 7 9 9.9 11 13 15 17 17 17];

Angle_motor_v = [32 32 32 32 32 32 32 32 32 19.2 12.4];

Tl_v = [1081 1080 1079 1078 1077 962 808 686 593 357 218];

Tem_v = [188 296 403 511 564 564 564 564 564 564 564];

Motor_shaft_speed_v = [64 252 439 626 718 858 1086 1318 1544 2484 3821];

Pressure_v = [430 430 430 430 430 385 325 278 243 243 243];

Displacement_motor_v = [161 161 161 161 161 161 161 161 161 100 65];

Displacement_pump_v = [22 37 52 67 74 83 98 114 130 130 130];

Vehicle_speed_v = [0.1 0.5 0.8 1.2 1.4 1.6 2.1 2.5 3.0 4.7 7.3];

%--------------------

%------- Calculate Dp, Dm, Td, TL

xm = [-0.2 0.33 0.35 0.37 0.39 0.41 0.43 0.45 0.47 0.49 0.51 0.53 0.56 0.8];

ym = [160 160 145 132 120 109 100 93 86 81 76 72 68 68 ];

Dpmatris = zeros(60001,9);

Dmmatris = zeros(60001,9);

for z = 1:9

matrix_ff_in_use = filtrera(eval(umu_fetch_matrix(z)));

time = matrix_ff_in_use(start:nersampling:...

length(matrix_ff_in_use)-1000,1);

p_return = matrix_ff_in_use(start:nersampling:...


p = matrix_ff_in_use(start:nersampling:...


i_p = matrix_ff_in_use(start:nersampling:...


i_m = matrix_ff_in_use(start:nersampling:...


n_p = matrix_ff_in_use(start:nersampling:...


n_m = matrix_ff_in_use(start:nersampling:...


Dpmatris(1:length(i_p),z) = max(0, -82 + i_p*147/0.550);

for r = 1:length(i_m)

for s = 2:length(xm)

if i_m(r) < xm(s)

Dmmatris(r,z) = ym(s-1) + (ym(s-1)-ym(s))/...

(xm(s-1)-xm(s))*(i_m(r)-xm(s-1));

break

end

C.3. Hydrostatic transmission.m 69

end

end

end

T_dmatris = zeros(6001,9);

T_Lmatris = zeros(6001,9);

for z = 1:9

matrix_ff_in_use = filtrera(eval(umu_fetch_matrix(z)));















D_p = Dpmatris(1:length(time),z);

D_m = Dmmatris(1:length(time),z);

slut = length(time)/100;

sim(’Calculate_torque.mdl’)

T_dmatris(1:length(T_d),z) = T_d;

T_Lmatris(1:length(T_L),z) = T_L;

end

%------------------------------------------

%-------- Use a special test run

w = 9;

matrix_ff_in_use = filtrera(eval(umu_fetch_matrix(w)));















D_p = Dpmatris(1:length(time),w);

D_m = Dmmatris(1:length(time),w);

slut = length(time)/100;

p_0 = p(1)*1e5;

n_p0 = n_p(1)*2*pi/60;

n_m0 = n_m(1)*2*pi/60;

T_d = T_dmatris(1:length(time),w);

T_L = T_Lmatris(1:length(time),w);


% ------------------ Kors initialt med korning 9

% % up = [D_p,p-p_return];

% % yp = [T_d];

% pump= ss(d2c(amx2220)); %where amx2220 is from ident using the following

% % code

% % % Import 09Td %09Td is yp and up

% % 09Tdd = dtrend(09Td,0)

% %

% % na = 2

% % nb = [2 2]

% % nc = 2

% % nk = [0 0]

% % amx2220 = armax(09Tdd,[na nb nc nk],,’Focus’,’Sim’)

% medel_Dp1 = mean(D_p);

% medel_Dm1 = mean(D_m);

% medel_p1 = mean(p-p_return);

% medel_n_m1 = mean(n_m);

% medel_n_p1 = mean(n_p);

% medel_Dp_dp1 = mean(D_p.*(p-p_return));

% medel_Dm_dp1 = mean(D_m.*(p-p_return));

%

% medel_T_d = mean(T_d);

% medel_T_L = mean(T_L);

%----------------------------

LINKÖPING UNIVERSITY

ELECTRONIC PRESS

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Carlsson, 2006. 71

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