Master of Science Thesis in Electrical EngineeringDepartment of Electrical Engineering, Linköping University, 2017

Modelling and Control of aForklift’s Hydraulic LoweringFunction

Ludvig Fri and Daniel Fahlén

Master of Science Thesis in Electrical Engineering

Modelling and Control of a Forklift’s Hydraulic Lowering Function

Ludvig Fri and Daniel Fahlén

LiTH-ISY-EX--17/5050--SE

Supervisor: Greger BäckströmJakob Johansson

CompanyDu Hoisy, Linköping university

Examiner: Martin Enqvistisy, Linköping university

Division of Automatic ControlDepartment of Electrical Engineering

Linköping UniversitySE-581 83 Linköping, Sweden

Copyright © 2017 Ludvig Fri and Daniel Fahlén

Sammanfattning

Materialhantering och logistik är viktigt för att dagens globala samhälle ska fun-gera. En grundläggande del i materialhanteringsprocessen är gaffeltruckar, där-för är det av intresse att göra gaffeltruckar så effektiva och pålitliga som möjligt.

I det här examensarbetet har ett försök gjorts till att förbättra styrningen av denhydrauliska sänkningsfunktionen hos en specifik gaffeltruck. Dagens lösning an-vänder sig av öppen styrning vilket gör reglerprestandan känslig för störningaroch systemförändringar. En störning av extra intresse är temperaturen av hyd-raulvätskan. Målet med detta arbete var därför att designa en regulator med ökadrobusthet och prestanda.

För att lösa detta har en modellbaserad metod för regulatordesign använts där enolinjär gray-box modell härleddes, implementerades och validerades. Modellpa-rametrarna skattades genom att ställa upp och lösa ett ickelinjärt minsta-kvadratoptimeringsproblem. Den resulterande modellen fångar det mesta av systemdy-namiken och modellpassningen till uppmätt data var högre än 70% vilket ansågsbra nog för att kunna använda modellen som en bas för regulatordesign.

En PID regulator designades och regulatorparametrarna optimerades med hjälpav modellen. Regulatorn utvärderades i simuleringar och för att sedan implemen-teras den på en riktig gaffeltruck. Den föreslagna regulatorn jämfördes med denursprungliga regulatorn i flera olika testfall. Resultaten visade ett bättre steady-state beteende och ökad robusthet mot temperaturförändringar för den designa-de regulatorn jämfört med den ursprungliga regulatorn.

iii

Abstract

Material handling and logistics are fundamental parts of today’s global societyand forklifts are a crucial part of the material handling process. Making these asefficient and reliable as possible are therefore of great interest.

In this master thesis, an effort has been made to improve the control of the hy-draulic lowering function of a specific forklift. Today the lowering function iscontrolled through an open-loop control scheme making the control performancesensitive to disturbances and system changes. One disturbance of special interestis the temperature of the hydraulic fluid. The goal of this thesis was therefore todesign a controller with improved robustness as well as improved performance.

To solve this a model-based control design approach was used and a nonlineargrey-box model was derived, implemented and validated. The model parameterswere estimated using a nonlinear least-squares optimisation problem. The result-ing model captures most of the system dynamics and the model fit is higher than70% which was deemed good enough to use for control design.

A PID controller was designed based on the estimated model and the controllerparameters were optimised. Furthermore, the controller was evaluated in sim-ulations and implemented in a real forklift. The proposed controller was com-pared to the original controller for various scenarios. The results reveal improvedsteady state behaviour with enhanced temperature robustness compared to theoriginal controller.

v

Acknowledgments

We would like to thank our examiner, Martin Enqvist, and our supervisor atLinköping University, Du Ho, for their guidance and interest in our work. Wewould also like to thank our supervisors at the company, Greger Bäckström andJakob Johansson, for their invaluable input, knowledge and support.

Linköping, June 2017Ludvig Fri and Daniel Fahlén

vii

Contents

Notation xiii

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Literature Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 System Overview 52.1 Mechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Electronic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Hydraulic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Modelling 93.1 Important Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Empirical Effective Bulk Modulus . . . . . . . . . . . . . . . 103.1.3 General Friction Model . . . . . . . . . . . . . . . . . . . . . 103.1.4 Position Sensor . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Hydraulic Submodels . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.1 Flow through Valves . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Proportional Valve . . . . . . . . . . . . . . . . . . . . . . . 133.2.3 Compensator Valve . . . . . . . . . . . . . . . . . . . . . . . 143.2.4 Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.5 Cylinder Pressure . . . . . . . . . . . . . . . . . . . . . . . . 163.2.6 Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Mechanical Submodels . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.1 Free Lift Force Balance . . . . . . . . . . . . . . . . . . . . . 173.3.2 Main Lift Force Balance . . . . . . . . . . . . . . . . . . . . 183.3.3 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

ix

x Contents

3.4 Model Structure Summary . . . . . . . . . . . . . . . . . . . . . . . 203.5 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 243.7 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.7.1 Validation Criteria . . . . . . . . . . . . . . . . . . . . . . . 25

4 Control 274.1 System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Nonlinear Compensation via Look-Up Table . . . . . . . . . . . . . 294.4 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4.1 Choice of PID Parameters . . . . . . . . . . . . . . . . . . . 304.4.2 Anti-Windup Method . . . . . . . . . . . . . . . . . . . . . . 304.4.3 Feed-Forward Control . . . . . . . . . . . . . . . . . . . . . 31

4.5 System Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 314.5.1 Temperature Dependency . . . . . . . . . . . . . . . . . . . 314.5.2 System Delays . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.6 Reference Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6.1 Reference Signal for Transition . . . . . . . . . . . . . . . . 32

4.7 Summarised Control Structure . . . . . . . . . . . . . . . . . . . . . 334.8 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.8.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 334.8.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.8.3 Fixed Point Precision . . . . . . . . . . . . . . . . . . . . . . 34

4.9 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Results 375.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1.1 Temperature Dependency . . . . . . . . . . . . . . . . . . . 385.1.2 Load Dependency . . . . . . . . . . . . . . . . . . . . . . . . 395.1.3 Measurement Noise . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2.1 Free Lift Parameters . . . . . . . . . . . . . . . . . . . . . . 425.2.2 Main Lift Parameters . . . . . . . . . . . . . . . . . . . . . . 435.2.3 Control Parameters . . . . . . . . . . . . . . . . . . . . . . . 44

5.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.3.1 Free Lift Validation . . . . . . . . . . . . . . . . . . . . . . . 465.3.2 Main Lift Validation . . . . . . . . . . . . . . . . . . . . . . 48

5.4 Control Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.4.1 Simulated Free Lift Controller . . . . . . . . . . . . . . . . . 495.4.2 Simulated Main Lift Controller . . . . . . . . . . . . . . . . 50

5.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.6 Implementation and Final Testing . . . . . . . . . . . . . . . . . . . 51

5.6.1 Implemented Free Lift Controller . . . . . . . . . . . . . . . 515.6.2 Implemented Main Lift Controller . . . . . . . . . . . . . . 55

Contents xi

5.6.3 Implemented Transition . . . . . . . . . . . . . . . . . . . . 58

6 Conclusion 616.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Method Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.4 Final Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A Simulink Models 67

B Simulated Free Lift Controller 87

C Simulated Main Lift Controller 89

D Implemented Transition Controller 91

Bibliography 95

Notation

Abbreviations

Abbreviation Meaning

ecu Electronic control unitpwm Pulse width modulationpid Proportional, integral, differential (controller)nmse Normalised mean-square-errornrmse Normalised root-mean-square-errorprms Pseudo-random multilevel signallq Linear quadratic (controller)mpc Model predictive control (controller)

xiii

1Introduction

This master thesis concerns modelling and control of the lowering function ofa specific forklift. In many technical industries using a model-based approachmight be advantageous. When using models, it is possible to simulate the systemwhich helps development and makes it easier to identify errors early in the pro-cess. The focus of the thesis is to model and control the lowering function whichexperiences several problems described below.

1.1 Motivation

The forklift investigated in this thesis uses a hydraulic system for lifting and low-ering of the forks. This system consists of two parts. One part uses a singlecylinder to lift the forks and is called the “free lift”. Another part uses two cylin-ders for lifting a stand connected to the forks and is called the “main lift”. Thelowering of the forks is controlled via a single valve coupled with a compensatorvalve for each of the free lift and the main lift. The dynamics of the system aredependent on the temperature of the oil. Warmer oil has lower viscosity whichleads to an increase in flow which in turn will lead to an increase in speed, forthe same valve opening area, when lowering the forks.

The forks’ lowering speed should not exceed 0.6 m/s as required by the standard[1]. Today the lowering function is controlled through an open-loop control thatdoes not consider the influence of the temperature. It is tuned to give the max-imum lowering speed when the temperature is close to its maximum allowedvalue to not exceed the speed limitation. This will cause the lowering speed to beslower when the oil temperature is cooler.

It is desirable to always have a lowering speed as close to the speed limitation

1

2 1 Introduction

as possible to increase the performance of the forklift. A higher lowering speedmeans that the forklift can operate more efficiently, which has a direct impact onthe overall efficiency of the material handling process.

Another problem which also stems from the problem with controlling the lower-ing speed occurs at the transition between the free and the main lift. Since thespeed cannot be accurately controlled, this leads to a nonsmooth transition whenthe state (load and temperature) of the system is not the one it has specificallybeen tuned for. It is desirable to have a smooth transition regardless of operatingcondition to avoid oscillations in the transition step as well as avoiding an abruptmechanical halt of the main lift.

1.2 Purpose

The purpose of this thesis is to create a model of the lowering function of thesystem and investigate if it is possible, with this model, to recreate the currentproblems with the lowering function mentioned in Section 1.1. Furthermore, acontroller with the purpose of controlling the lowering speed is to be designedbased on the estimated model. A stability analysis of the controller should bedone to ensure stability of the complete system. The control performance shouldbe tested and verified on a forklift.

The requirement of the controller is to keep the lowering speed of the forks closeto, without exceeding, 0.6 m/s in steady state. As mentioned in Section 1.1, thetemperature affects the lowering speed, therefore it is crucial that the controlleris robust to temperature changes. The controller should also be able to follow achanging reference accurately and without oscillations, to ensure smooth startsand stops of the free and main lift as well as a smooth transition between them.

1.3 Limitations

In this thesis only one specific forklift is investigated which simplifies the task.However, it is possible to use this thesis as a basis when doing similar work ondifferent forklifts and perhaps even use the derived controller, with good result,with some minor changes.

Furthermore, this thesis is only focused on the lowering function.

1.4 Method

To be able to solve the general problem, it was broken down into several sub-problems. The sub-problems can be divided into five different categories that areperformed in steps. The steps are, in chronological order; data collection, mod-elling, model validation, control, and implementation on the forklift electroniccontrol unit (ECU). Creating a model, validating the model and creating a con-

1.5 Literature Study 3

troller are iterated to improve the final result. At each iteration the results fromthe previous iteration are used to discern in which step alterations are needed toimprove the final result.

1.5 Literature Study

In [15], a general model of a proportional solenoid valve is constructed from thelaws of physics and the model is also validated. This provides a starting pointwhen the model of the valve used in the forklift is constructed. Furthermore,the work in [23] focuses on the nonlinear modelling and analysis of a hydrauliccontrol valve. This might be of interest when the nonlinearities in the valve aremodelled using the laws of physics. The dynamics of the proportional valve cansuccessfully be modelled as a second order system [12, 22, 23].

An alternative approach is shown in [16] where black-box modelling is used tomodel the valve. In this approach system identification is done to find a modelwhich corresponds well with the data. However, in this paper the investigatedvalve is a directional control valve which differs from the valve used in the thesis.In [5], the modelling of hysteresis in a hydraulic valve is done. The paper alsopresents a Simulink model of this hysteresis.

The focus of the thesis lies in analysis and design of hydraulic control systemswhich is also the focus of [12]. It covers (among others) first principles modellingand empirical modelling as well as hydraulic control system design. The workof [12] is therefore of interest at all stages in the thesis and in particular whenmodelling the valves, bulk modulus and viscosity.

There are several master theses that concern similar problems relating to mod-elling and control of hydraulic systems but none was found that contains thesame model or control design as the one proposed in this thesis. In [2], modellingand control of an electro-hydraulic system with a different structure is examined.In [7, 11, 13] and [14] modelling and simulation of similar hydraulic systemsare examined. Modelling and control of the lift function of a similar forklift isstudied in [4].

The modelling of the nonlinearities of the proportional valve is one of the crucialpoints of this thesis. In [15], a proportional solenoid valve is studied which is ofthe same type used in this thesis. An analysis of the dynamics and nonlinearitiesof a hydraulic control valve is done in [23]. In [16] the method of black-box mod-elling is used for modelling of a hydraulic directional control valve. A hysteresismodel and a Simulink model of a hydraulic valve can be found in [5].

How MATLAB solves optimisation problems is described in [17], which is of in-terest when doing parameter estimation. More details about the methods usedcan be found in [3], particularly the trusted-region method which can be used tosolve large-scale bound-constrained minimisation problems.

In [18–20] MATLABs way of modelling different hydraulic components is de-

4 1 Introduction

scribed. These are of interest when creating the hydraulic models.

In [21] different friction models and their effect on hydraulic cylinders are exam-ined. Of particular interest is the LuGre friction model which is mentioned to bethe most widely utilised among the proposed models. This paper concludes that asteady-state model is not appropriate to use when modelling friction in hydrauliccylinders. It is also shown that the nominal LuGre friction model gives good re-sults and even better results can be achieved with a modified LuGre model. How-ever, the modified LuGre model includes more parameters that are difficult toidentify.

1.6 Thesis Outline

An overview of the system is given in Chapter 2, consisting of the mechanicalsystem, the electronic system and the hydraulic system.

Chapter 3 describes how the model was derived and validated. Important aspectsof the system are presented along with models for the system components. Theparameter estimation and model validation are explained.

The control design is explained in Chapter 4. Different parts of the controlleralong with the implementation of the controller on a real forklift are presented.

The results of the modelling and control from both simulations and measure-ments are presented in Chapter 5. In particular, measurements from both theimplemented controller and the currently used controller are presented and com-pared.

Chapter 6 concludes the thesis where the results and future work are discussed.

Appendix A contains the Simulink models, Appendix B and C contain the resultsof the simulated free and main lift controllers, respectively. Appendix D containsthe results from the implemented transition controller.

2System Overview

The system can be divided into mechanical, electronic and hydraulic subsystems.Each of these subsystems are described below.

2.1 Mechanical System

A simplified drawing of the mast of the forklift can be seen in Figure 2.1. Themast of the forklift consists of a base attached to the chassis of the forklift anda first stage and a second stage that can slide up and down. Mounted on thesecond stage there is a carriage assembly which has the forks attached to it. Theassembly is divided into two parts, the main lift and the free lift. The main lift isdefined as the parts of the mast controlled by the left cylinder in Figure 2.1 andconsists of the left hydraulic cylinder attached to the base, the first stage that isconnected to the left piston rod and the second stage that is connected the firststage through a chain. The free lift is defined as the parts of the mast controlledby the right cylinder in Figure 2.1 and consists of the right hydraulic cylinderthat is attached to the second stage and the forks and carriage that is connectedto the right piston rod through a chain.

5

6 2 System Overview

xML

xFL

xG

Base

First stage

Second stage

Forks

Chains

Figure 2.1: Sketch of the mast. The left cylinder controls the main lift whichis connected to the base and first stage of the mast and the second stage ofthe mast trough a chain. The right cylinder controls the free lift which isconnected to the forks trough a chain.

There is one position sensor measuring the distance between the base and thefirst stage of the mast, called the main lift position, xML, and one position sensormeasuring the distance between the forks and the bottom of the second stage ofthe mast, called the free lift position, xFL. The distance between the the bottom ofthe first and second stage is not measured but is given indirectly by the relationcreated by the chains that connects them. The the distance between the bottomof the first and second stage thus has the same length as the distance between the

2.2 Electronic System 7

base and the first stage, xML. The position of the forks relative to the ground, xG,is not measured.

2.2 Electronic System

An overview of the currently used control system is shown in Figure 2.2. The ECUcontrols the valves with a pulse width modulation (PWM) signal. The controlsignal to a valve is represented in the ECU as a current. The ECU controls thePWM signal to achieve the correct output current. Hence in the sequel the controlsignal to the valves will be referred to as the current used in the ECU.

The control signal opens the valve and thereby the oil will pass through the valve.This flow results in movement of the free and/or the main lift. The sensor mea-surements are sampled at 50 Hz and are used for security measures, calculatingthe height of the forks, and calculating the load (via the pressure sensors) to dis-play for the benefit of the operator. The sensor measurements are currently notused for control.

PWM

ECU

PWM Flow

Valves

Flow

Oiltemperature

Freeliftpressure

Mainliftpressure

Freeliftposition

Mainliftposition

Hydraulicandmechanicsystem Display

Figure 2.2: Overview of currently used control system where the ECU con-trols the valves. The valve opening generates a flow to the hydraulic andmechanic part of the system which leads to movement of the forks.

2.3 Hydraulic System

Figure 2.3 shows an overview of the lowering function of the hydraulic systemfor the free and main lift.

8 2 System Overview

MAIN LIFT FREE LIFT

T

H1 H2H3

M1 M2

Q1 Q2

C1 C2

Figure 2.3: Overview of the lowering function of the hydraulic system. Q1and Q2 are the proportional valves where Q1 controls the main lift and Q2controls the free lift. C1 and C2 are the compensators for Q1 and Q2, re-spectively. M1 and M2 are pressure sensors for the main and free lift. H1and H2 are the cylinders for the main lift and H3 is the cylinder for the freelift.

2.3.1 Valves

The lowering speed is determined by the flow through the proportional valves.The lowering of the mast is divided into three sections starting from its top po-sition. First the main lift starts lowering, then there is a transition where boththe free and main lift are moving and lastly only the free lift is moving until theforks are at ground level.

The flow through the valve is affected by a number of variables where the pres-sure differential over the valve, the area of the valve opening and the temperatureof the hydraulic fluid have significant impact. The temperature of the hydraulicfluid affects the flow due to the temperature dependent viscosity. As previouslymentioned in Section 1.1, an increase in temperature leads to lower viscositywhich in turn increases the flow through the valve. The hydraulic fluid will in-crease in temperature due to energy losses in the system. The temperature willdecrease only due to heat dissipation from the oil to the environment since thereis no active cooling of the oil.

3Modelling

The goal of the modelling is to create a model of the lowering function of theforklift that reproduces the current problems with this function mentioned inSection 1.1. This model can then be used to study the system to gain furtherunderstanding of it as well as aid in troubleshooting the system. Furthermore, amodel must be available to use model-based control designs.

The model can be divided into two parts, the free and the main lift. The modelfor each of them was developed individually without regard for the other. Todo this the measurements of the free lift model were taken only when the freelift was moving and the main lift was stationary. For the main lift the oppositesetup was used. Any dynamic effects coming from the other part of the lift wereneglected.

3.1 Important Properties

Important properties that must be considered when modelling the hydraulic andmechanical system are the viscosity, the bulk modulus, the friction in the system,and the way the system measures position and thereby velocity.

3.1.1 Viscosity

The viscosity of the fluid changes with temperature. This temperature depen-dency can be described as

η = η0e−λ(T−T0) (3.1)

where η is the viscosity of the fluid, η0 is the dynamic viscosity at reference tem-perature T0 and λ is the viscosity-temperature coefficient [12].

9

10 3 Modelling

The temperature affects the viscosity which in turn affects the flow.

3.1.2 Empirical Effective Bulk Modulus

The bulk modulus is a measure of how incompressible the fluid is and is signifi-cantly affected by the pressure. One way of describing the bulk modulus, whichalso includes the effects of entrained air and mechanical compliance, is

E(p) = a1Emax log (a2p

pmax+ a3) (3.2)

where E(p) is the pressure dependent bulk modulus, Emax is the maximum bulkmodulus, p is the pressure, pmax is the maximum pressure and a1, a2 and a3 areparameters which can be estimated with experiment data [12].

3.1.3 General Friction Model

There are many parts of a mechanical system that are affected by friction. Twoimportant aspects of friction are the pre-sliding behaviour and steady state fric-tion.

One common steady state friction model, among others, that incorporates Coulombfriction, viscous friction and static friction is

Ff r = (Fc + (Fs − Fc)e(−(v/vst)n))sign(v) + σ2v (3.3)

where Ff r is the friction force, Fc is the Coulomb friction force, Fs is the staticfriction force, vst is the Stribeck velocity, n is an exponent that affects the slopeof the Stribeck curve, σ2 is the viscous friction coefficient and v is the relativevelocity between the two bodies in contact [21].

The pre-sliding behaviour has many effects and one of special interest is the be-haviour of the hysteresis. The hysteresis has the effect that zero velocity will notalways yield zero friction force. One model of the pre-sliding behaviour thatincorporates this effect is the LuGre friction model [21]. The LuGre model is a dy-namic model that introduces a state for the pre-sliding behavior and is based onthe bristle model shown in Figure 3.1. The bristles on one surface are modelledas rigid and the bristles on the other are modelled as elastic [21].

3.1 Important Properties 11

z

v

Figure 3.1: Bristle drawing where the state z is shown as the deflection ofthe bristle. The bristles on the bottom surface is rigid while the bristles onthe top is elastic.

The LuGre model is given by

dzdt

= v − σ0zgs(v)

|v| (3.4)

Ff r = σ0z + σ1dzdt

+ σ2v (3.5)

where z is the mean deflection of the elastic bristles, σ0 is the stiffness of theelastic bristles and σ1 is the viscous friction coefficient for the internal state z.The function gs(v) together with σ2 describe the steady state properties of thefriction force.

With the choice of gs(v) as

gs(v) = (Fc + (Fs − Fc)e(−(v/vst)n))sign(v) (3.6)

the steady state friction is as previously stated in (3.3).

3.1.4 Position Sensor

The internal position sensor measuring the height is an optical incremental rotaryencoder. It consists of a wire wound around a rotating disc with light shiningthrough evenly spaced slits. When the wire is pulled up and down this will causethe disc to rotate in different directions. As the disc rotates a photodiode willdetect the light pulses. Since the diameter and the number of slits of the encoderare known, it is possible to calculate the distance the wire has travelled betweeneach pulse. Each time a pulse occurs this distance is incremented. By placingtwo photodiodes in such a way that a 90 degrees phase shift occurs it is possibleto determine which way the disc rotates. Thus, the sensor sends out two phase-

12 3 Modelling

shifted pulses and these are used to calculate both the distance travelled andwhether the wire is travelling up or down.

The signals of interest are the velocity and position of the forks relative to theground. This is not what is measured by the position sensors as can be seen inFigure 2.1. The main lift position, xML, the free lift position, xFL, and the positionof the forks relative to the ground, xG, are related in the following way

xG = 2xML + xFL

where the factor two comes from the fact that the measured position is the rela-tive distance between the base and second stage of the mast and the way they areconnected with chains.

The sensor used in this thesis has a diameter of 80 mm and 128 slits. This meansthe accuracy of the position for the free lift, xFL, is 2 mm. Thus the accuracy forthe position relative ground, xG, is 2 mm when using the free lift. The accuracyof the main lift position sensor, xML, is also 2 mm but since xG = 2xML + xFL theaccuracy for the position relative ground, xG, will be 4 mm when using the mainlift.

3.2 Hydraulic Submodels

The hydraulic system was divided into submodels that are presented here. Thestructures of the hydraulic system for the free and main lift are nearly identical,and the only difference is that two cylinders are used in the main lift whereasonly one cylinder is used in the free lift as can be seen in Figure 2.3. The twocylinders in the main lift can, in the model, be considered as one cylinder withthe combined area of the two. Because of this the same equations can be used tomodel both the free and the main lift.

The components that are considered are the cylinder, the proportional valve, thecompensator valve, a small volume that connects the two valves and the oil tank.The pipes that connect the different components are neglected. A drawing of theentire hydraulic system can be seen in Figure 2.3 and for clarity a more detailedfigure of the proportional valve coupled with the compensator valve can be seenin Figure 3.2. The pressure pA is the pressure before the proportional valve, pBis the pressure after the proportional valve, pC is the pressure before the com-pensator valve, pD is the pressure after the compensator valve, px and py are thepressure at the compensator control terminals. The flow qA is the flow throughthe proportional valve and qB is the flow through the compensator valve. Thenotation used in Figure 3.2 will be used throughout Section 3.2.

3.2 Hydraulic Submodels 13

Q

C

V

pA pB pC pD

qA qB

px

py

Figure 3.2: Drawing of the modelled valve components. The componentsare modelled in the same way for both the free and main lift. Q is the pro-portional valve, C is the compensator valve, V is a small volume betweenthe proportional and compensator valve. The pressure at different points inthe figure are represented with pi where i indicates the point. qA is the flowthrough the proportional valve and qB is the flow through the compensatorvalve. The dotted lines represent the connection between the compensatorvalve control terminals and the rest of the hydraulic system.

3.2.1 Flow through Valves

The flow through a valve can be modelled as

Q = CDA

√2ρ

p

(p2 + p2cr )

14

(3.7)

where Q is the flow through the valve, CD is the flow discharge coefficient, A isthe opening area of the valve, ρ is the density of the hydraulic fluid and p is thepressure differential over the valve. The parameter pcr is the minimum pressurefor turbulent flow which can be written as

pcr =ρ

2(

RecrηCDDH

)2 (3.8)

DH =

√4Aπ

(3.9)

where Recr is the critical Reynolds number and η is the viscosity of the fluid [20].

3.2.2 Proportional Valve

The proportional valve is divided into two parts, one part describes the dynamicsand the other the static behaviour.

Dynamics of the Proportional Valve

The dynamics of a proportional valve has been successfully modelled as a secondorder system with hysteresis [12, 22, 23]. A second order system with hysteresis

14 3 Modelling

can be written as

x = ω2(u − 2Dωx − x − fhssign(x)) (3.10)

where u is the input current to the system, x is the output current, ω is the naturalfrequency, D is the damping ratio and fhs is a constant describing the hysteresis.

The hysteresis is largely caused by friction effects in the valve [12]. Modelling thehysteresis as fhssign(x) can cause numerical problems when simulating becausethe function acts a discontinuity when x changes sign. To solve this a hysteresismodel inspired by a simplified LuGre friction model was used. The hysteresiscan be described by

dzdt

= x − σ0zfhs|x| (3.11)

Fhs = σ0z + σ1dzdt. (3.12)

The dynamics of the valve can then be written as

x = ω2(u − 2Dωx − x − Fhs) (3.13)

where Fhs is guaranteed to be a continuous function. Written in this way themodel still incorporates a hysteresis but is not affected by numerical issues to thesame extent as before since there is no discontinuity.

Flow through the Proportional Valve

The flow through the proportional valve is modelled using (3.7) where (3.1) isused for the viscosity.

Combining these equations gives

qA = CDA(x)

√2ρ

pA − pB((pA − pB)2 + ( ρ2 ( Recr (η0e

−λ(T−T0))

CD

√4A(x)π

)2)2)14

(3.14)

where the inputs are the pressure before the valve, pA, the pressure after the valve,pB, the temperature T and the effective current output, x, from the dynamic sys-tem. A look-up table, estimated from measurements, is used to get the relationbetween the current, x, and the valve opening area A(x). The output is the flowqA.

3.2.3 Compensator Valve

The compensator valve is divided into three parts, one part describing the staticarea, one part describing the dynamics and one part describing the static flow

3.2 Hydraulic Submodels 15

behaviour.

Compensator Valve Area

The area of the compensator can be modelled as

As(pxy) =

Amax pxy + pf low ≤ psetAmax − k(pxy − pset) pset < pxy + pf low < pmaxAleak pxy + pf low ≥ pmax

(3.15)

where pxy = px − py is the pressure differential over the control terminals of thepressure compensator, As(pxy) is the pressure dependent compensator openingarea in steady state, Amax is the maximum area, pset is the preset pressure, Aleakis the minimum area, pmax is the pressure needed to fully close the valve and k isdefined as

k =Amax − Aleak

preg

where preg is the pressure regulation range [18].

The pressure pf low is a pressure acting on the compensator originating from theflow force that occurs when a flow is passing through the valve. The flow forcewas added to the model at suggestion from the valve manufacturer. The pressureoriginating from the flow force acting on the compensator spool can be writtenas

pf low =(pC − pD )ACD cos θ

Aspool(3.16)

where pD is the pressure after the compensator valve, pC is the pressure beforethe compensator valve, A is the valve opening area, CD is the discharge coefficient,θ is a valve-dependent design variable and Aspool is the spool area.

Dynamics of the Compensator Valve

The dynamics of the compensator valve is modelled as

dAdt

=As(pC − pD ) − A

τ(3.17)

which is a first order system where As(pC −pD ) is the pressure dependent openingarea in steady state, A is the actual opening area and τ is the time constant forthe first order system [18].

Flow through the Compensator Valve

The flow through the compensator valve is modelled using the same equations asfor the proportional valve, that is (3.7), (3.8) and (3.9) where (3.1) is used for theviscosity. The valve opening area in steady state is modelled according to (3.15).

16 3 Modelling

Combining these equations gives

qB = CDA

√2ρ

(pC − pD )

((pC − pD )2 + ( ρ2 ( Recr (η0e−λ(T−T0))

CD

√4Aπ

)2)2)14

(3.18)

where the inputs are the pressure before the compensator valve, pC , the pressureafter the compensator valve, pD , and the temperature T . The output is the flowqB.

3.2.4 Control Volume

The control volume is the small volume (≈ 2000mm3) located in between the com-pensator valve and the proportional valve. The dynamics of the control volumecan be written as

pV =EV∆Q (3.19)

where pV is the pressure in the volume, E is the bulk modulus, V is the volumeand ∆Q = qA − qB is the flow difference in and out of the volume. The pressurepV in the volume is approximated as homogeneous meaning pV = pB = pC = px.The bulk modulus was approximated as a constant since the effect of pressuredependence for the bulk modulus, for this submodel, is neglected.

3.2.5 Cylinder Pressure

The pressure in the cylinder is approximated to be the same as the pressure at theproportional valve, pA. The pressure dynamics in the cylinder can be modelledas

pA =E(pA)V (xcyl)

(Q − Ax) (3.20)

where xcyl is the position of the cylinder, E(pA) is the pressure dependent bulkmodulus described by (3.2), V (xcyl) is the volume of the cylinder which is depen-dent on the position and A is the cylinder area [12].

V (xcyl) can be written as

V (xcyl) = V0 + xA (3.21)

where V0 is the initial volume when the cylinder is at its bottom and A is thecylinder area.

3.2.6 Tank

The tank is modelled without any dynamics which simply makes it keep a con-stant output pressure of one bar. Even though there is a small pressure increasein the tank it is negligible compared to the pressure in the cylinder. This meanspD = constant is assumed.

3.3 Mechanical Submodels 17

3.3 Mechanical Submodels

The model used for the mechanical systems of the free and main lift are presentedhere.

3.3.1 Free Lift Force Balance

By using Newton’s second law on both the cylinder and the forks, expressions fortheir acceleration can be derived. In Figure 3.3 the forces acting on the cylinder(Figure 3.3a) and the forks (Figure 3.3b) are shown.

pA

xc, xc

FF

Ffr,c

mcg

(a) Forces acting on thecylinder.

(mf+mL)g

F

Ffr,f

xf, xf

(b) Forces acting on the forks.

Figure 3.3: Free lift force balance.

Using Newton’s second law on the cylinder, the mass times the acceleration canbe written as

mc xc = pA −mcg − Ff r,c − 2F (3.22)

where mc is the mass of the cylinder, xc is the acceleration of the cylinder, pA isthe force arising from the pressure in the cylinder, mcg is the gravitational force,Ff r,c is the cylinder friction force and F is the force from the chain connected tothe forks.

Using Newton’s second law on the forks and solving for F gives that F can bewritten as

(mf + mL)xf = F − (mf + mL)g − Ff r,f (3.23)

⇔ F = (mf + mL)xf + (mf + mL)g + Ff r,f (3.24)

18 3 Modelling

where mL is the mass of the load on the forks and mf is the mass of the forks, xfis the acceleration of the forks, (mf + mL)g is the gravitational force and Ff r,f isthe friction force between the forks and the stand.

The relation between the acceleration of the forks and the cylinder can be writtenas xf = 2xc. This means that (3.22) and (3.23) can be combined to

mc xc = pA −mcg − Ff r,c − 2((mf + mL)2xc + (mf + mL)g + Ff r,f ) (3.25)

⇔ xc =1

mc + 4(mf + mL)(pA − (mc + 2(mf + mL))g − Ff r,c − 2Ff r,f ) (3.26)

⇔ xc =1

mc + 4(mf + mL)(pA − (mc + 2(mf + mL))g − FLuGre) (3.27)

where all friction forces are combined into FLuGre.

3.3.2 Main Lift Force Balance

If an assumption is made that the free lift cylinder, forks and carriage does notmove relative to the second stage, then the mass of those components can beadded to the mass of the second stage and they can be seen as one rigid body.This will be used to simplify the calculations.

By using Newton’s second law on both the first stage and the second stage, ex-pressions for their acceleration can be derived. In Figure 3.4 the forces acting onthe first stage (Figure 3.4a) and the second stage (Figure 3.4b) is shown.

pA

xfs, xfsFF

Ffr,c

mcgmfsg

Ffr,fs

(a) Forces acting on the first stage.

(mss+mL)g

Ffr,ss

F

xss, xss

(b) Forces acting on thesecond stage.

Figure 3.4: Main lift force balance.

3.3 Mechanical Submodels 19

Using Newton’s second law on the first stage the mass times the acceleration canbe written as

(mc + mf s)xf s = pA − (mc + mf s)g − Ff r,f s − 2F (3.28)

where mc is the mass of the cylinder, mf s is the mass of the first stage, xf s isthe acceleration of the first stage, pA is the force arising from the pressure in thecylinder, (mc + mf s)g is the gravitational force, Ff r,f s is the first stage frictionforce and F is the force from the chain connected to the second stage.

Using Newton’s second law on the second stage and solving for F gives that F canbe written as

(mss + mL)xss = F − (mss + mL)g − Ff r,ss (3.29)

⇔ F = (mss + mL)xss + (mss + mL)g + Ff r,ss (3.30)

where mss is the mass of the second stage, mL is the mass of the load on the forks,xss is the acceleration of the second stage, (mss + mL)g is the gravitational forceand Ff r,ss is the friction force of the second stage.

The relation between the acceleration of the second stage and the first stage canbe written as xss = 2xf s. This means that (3.28) and 3.29 can be combined to

(mc + mf s)xf s = pA − (mc + mf s)g − Ff r,f s − 2((mss + mL)2xf s + (mss + mL)g + Ff r,ss)

(3.31)

⇔ xf s =1

mc + mf s + 4(mss + mL)(pA − (mc + mf s + 2(mss + mL))g − 2Ff r,ss − Ff r,f s)

(3.32)

⇔ xf s =1

mc + mf s + 4(mss + mL)(pA − (mc + mf s + 2(mss + mL))g − FLuGre)

(3.33)

where all friction forces are combined into FLuGre.

3.3.3 Friction

The friction model used is the LuGre model mentioned in Subsection 3.1.3. Tomodel the friction in the entire system, gs(v) can be chosen as

gs(v) = ((Fpr + mLfcf r )(1 + (Kbrk − 1)e−cv |v|))sign(v) (3.34)

where Fpr is the preload force, mL is the load, fcf r is the Coulomb friction coeffi-cient, Kbrk is the breakaway friction force increase coefficient and cv is the transi-tion coefficient. This choice of gs(v) was inspired by [19] in which a steady statefriction model is described for a cylinder. Some modifications were made in aneffort to describe all frictions in the system, not just the friction in the cylinder.

20 3 Modelling

Most noticeably the friction has been made load-dependent to reflect observedsystem behaviour.

The load dependency is shown in Figure 3.5. The pressure in the cylinder and thevelocity were measured for the free lift in steady state velocity at several differentvelocities. From this the combined friction force of the cylinder and the forks wascalculated using (3.25) where xc was approximated to zero.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Velocity [m/s]

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Friction forc

e [N

]

Friction force in steady state for different loads and velocities

No load

Load 1000 kg

Figure 3.5: Friction force in steady state for the free lift for two differentloads on the forks.

3.4 Model Structure Summary

To make it more clear how all equations were incorporated into the model, a blockdiagram of the model structure is shown in Figure 3.6. The model structure, aspreviously mentioned, is the same for both the free and the main lift. The inputsto the model are the loadmL, the current u and the temperature T . The equationsare incorporated in each block in the figure.

The proportional valve consists of the proportional valve dynamics block and theproportional valve flow block which incorporate (3.13) and (3.14), respectively.The compensator valve consists of the compensator area block, the compensatorarea dynamics block and the compensator flow block which incorporate (3.15),(3.17) and (3.18), respectively.

The volume block incorporates (3.19) and the tank block has a constant pressureof one bar as output. The cylinder pressure block incorporates (3.20), (3.21) and(3.2). The mechanical system block incorporates (3.25) or (3.31) depending on ifit is the free or main lift model.

3.5 Data Collection 21

CompensatorvalveProportionalvalve

u x

Proportionalvalvedynamics

p_Ap_BxT

q_A

Proportionalvalveflow

q_A

q_Bp

Volume

p_Cp_Dp_xp_y

A_s

Compensatorarea

A_s A

Compensatorareadynamics

p_Cp_DAT

q_B

Compensatorflow

p_d

Tank

x_c

dot_c

q_A

p_A

Cylinderpressure

p_A

m_L

x_c

dot_x_c

Mecanicalsystem

2T

1u

3m_L

Figure 3.6: Block diagram of the model structure. Each block incorporatesdifferent equations from Chapter 3.

The model was implemented and simulated in Simulink, the complete Simulinkmodels can be found in Appendix A. At this point the model structure is estab-lished but there are many unknown parameters that needs to be estimated. Todo this measurements needs to be taken on the real system and system identifi-cation methods were used, which is described in the following sections. The fulllist of the parameters that were estimated and the result of the estimation will bedescribed later in Section 5.2.

3.5 Data Collection

There are several signals of interest that can be monitored: the valve currentcontrol signal, the positions of the free lift and the main lift, the pressures inthe hydraulic system of the free lift and the main lift, and the temperature ofthe oil in the tank. In addition to this, extra sensors were added to collect moredata. These extra sensors consisted of an external position sensor for the forks’distance relative the ground, xG, and extra sensors for the pressures in the freeand main lift cylinders that are more accurate than the internal ones. The samplefrequency of the system was chosen to 50 Hz.

The measurements for the free and main lift were taken separately. When themeasurements for the free lift were taken the main lift was not moving and viceversa. There is still a small cross coupling between the free and main lift but thiswas assumed negligible.

Two types of signals were used for data collection. The first type of signal was apseudo-random multilevel signal (PRMS) chosen to excite the dynamic behaviourof the system. Another approach is to use a binary signal with random switchingprobability, but as the system is known to be non-linear a PRMS signal is prefer-able [9].

Figure 3.7 depicts an example of the PRMS signal. In Figure 3.7a the signal isshown in the time domain. The valve current is quantized to 256 steps where

22 3 Modelling

0 represents 0 current and 255 represents max current (1,75 A). Note that theamplitude of the signal lies between 58 - 174 and not 0 - 255. This is becausethe valve has lower and upper saturation limits and it is desirable to lie in theworking range of the valve.

In this example the probability of switching amplitude is 0.05 and there are 8 dif-ferent amplitude levels. The signal consists of 1000 samples which correspondsto 20 seconds since the sampling frequency is 50 Hz. The time it takes to lowerthe forks down to its lowest state from the top position is shorter than 20 secondswith this kind of signal and therefore the signal time will not be a problem.

The input signal to the valve needs to be low-pass filtered since without the low-pass filter the valve will close too quickly and this will cause a large spike inpressure which might damage the hardware.

In Figure 3.7b the power spectral density of the pseudo-random multilevel signalis shown. The switching probability determines in which frequencies most ofthe power of the signal exists. With a switching probability of 0.05, as in thefigure, most of the energy lies at the lower frequencies. This was chosen becausethe interesting phenomena occur at low frequencies. The filtered signal is alsoshown in Figure 3.7b. Note that there is almost no difference between the powerspectral densities of the unfiltered and filtered signals. This is because the signalswitching probability is chosen to excite the most interesting (low) frequencies.

Two realisations of PRMS signals were created, called PRMS1 and PRMS2, inaccordance with the above stated principles. They were used as input to boththe free and the main lift at different operating points. The operating pointsconsisted of loads on the forks of 0 kg, 800 kg and 1500 kg at oil temperatures inthe tank of approximately 20 and 60 ◦C.

3.6 Parameter Estimation 23

0 2 4 6 8 10 12 14 16 18 20

Time [s]

0

20

40

60

80

100

120

140

160

180

Am

plit

ude

Pseudo-random multilevel signal

(a) Pseudo-random multilevelsignal with switching probability0.05 (blue) and the lowpass-filtered input signal to the valve(red).

0 5 10 15 20 25

Frequency [Hz]

0

20

40

60

80

100

120

140

160

180

200PSD

(b) Power spectral density of apseudo-random multilevel signalwith switching probability 0.05(blue) and its filtered counterpart(red).

Figure 3.7: A pseudo-random multilevel signal and its power spectral den-sity with corresponding low-pass filtered signals.

The second type of signal that was used as input consisted of steps to severalevenly spaced levels. After a step was taken the signal was constant for sometime in order to reach steady state. The reason for this was to get accurate mea-surements of the steady state behaviour of the system. This was done for the freelift with loads of 0 kg and 1000 kg and it was done for the main lift with loads of0 kg, 800 kg, 1500 kg. The temperature was varying somewhat between differentmeasurements but was lying in the range of 20 to 40 ◦C. There was no particularreason for using different loads for the free and the main lift.

3.6 Parameter Estimation

The system is modelled using a grey-box approach and the model contains manyunknown parameters that need to be estimated from measurements. To do thisthe tool Parameter Estimation GUI built into Simulink was used. The tool formu-lates a nonlinear least squares optimisation problem that minimises the sum ofthe squared difference of the simulated and measured data.

A nonlinear least squares problem can be formulated as

V (x) =12

N∑k=1

εTk (x)εk(x) (3.35)

where V (x) is a cost function, x are the design variables and

εk = yk − hk(x) (3.36)

24 3 Modelling

where yk are the observed values at sample k and hk(x) are the estimated valuesat sample k.

The goal of the optimisation is to minimise the cost function by changing thedesign variables, which is usually done through an iterative optimisation methodwhere the design variables are estimated and updated according to

x(i+1) = x(i) + α(i)f (i) (3.37)

where f (i) is the search direction and α(i) denotes the step length [10].

The parameter estimation uses the trust-region algorithm to obtain the step lengthand search direction [17]. The trust-region algorithm approximates the cost func-tion with a simpler function, usually the first terms of a Taylor expansion of thecost function, which closely resembles the cost function in a region around thedesign variable. This region is called the trust region and the simplified cost func-tion is minimised over this trust region. If the algorithm reaches the boundaryof the region without finding a point where the cost function is smaller, the trustregion will shrink and a new minimisation will be tried and so forth [3].

This method needs an initial guess to execute but there is no guarantee of conver-gence. This means that the result of the optimisation is dependent on the initialguess. Depending on this the method might converge to completely differentresults.

3.6.1 Problem Setup

There are many unknown parameters and it is difficult to set up a parameterestimation problem for all parameters, using all available inputs, and get it toconverge to reasonable values. For this reason the parameter estimation of themodel was divided into two parts where a smaller set of parameters could beestimated.

Steady State

In the first part the steady state behaviour was of prime interest. The parts ofthe model relating to the cylinder pressure dynamics and the mechanical systemwere discarded and replaced with the static relationship

Q =12Ac xG (3.38)

where Q is the flow through the proportional valve, Ac is the cylinder area andxG is the velocity of the forks relative to the ground. This is true for both the freeand the main lift in steady state.

The inputs to the simplified steady state model were the current to the propor-tional valve, the cylinder pressure and the temperature. The output was the ve-locity relative the ground xG. The input and output were taken from the steadystate measurements. This optimisation gave parameter values of the parametersthat affect the steady state characteristics.

3.7 Model Validation 25

Complete Model

In the second part the complete model was used in the estimation process. Pa-rameters relating to the steady state characteristics of the model were mostlyregarded as known from the previous steady state estimation.

The remaining parameters were estimated where the inputs to the system werethe current to the proportional valve, the temperature and the load on the forks.The input and output were taken from the PRMS1 measurements mentioned inSection 3.5.

Some parameters that relate to the temperature dependency of the steady statecharacteristics were estimated again during the second round of estimation as thePRMS1 data was regarded to excite the temperature dependency better since themeasurements were taken in a wider range of temperatures.

After these optimisations were done the model was considered complete and theresults of the parameter estimation are presented in Section 5.2.

3.7 Model Validation

Part of the data from the experiments were used to validate the model. No spe-cific targets for the accuracy of the model were set but it is important to knowhow the model performs. The validation indicates where the model captures thesystem dynamic well by using the validation criteria below. The model validationalso hints if the model is good enough to use as a basis for control design.

The validation and estimation data were collected using different input signals.

3.7.1 Validation Criteria

One measure of the model performance is to use the normalised mean-square-error (NMSE). Subtracting the NMSE from one and multiplying by 100 gives ameasure of the model fit in percentage which is defined as

Model fit = 100 · (1 − ||v − v||2

||v − v||2) (3.39)

where || . || indicates the 2-norm, v is the measured velocity, v is the estimatedvelocity and v is the mean velocity.

Another similar measure of the model performance is to use the normalized root-mean-square-error (NRMSE) defined as

Model fit = 100 · (1 − ||v − v||||v − v||

) (3.40)

which simply is the square root of the NMSE.

A model fit of 0% means that the estimated velocity is not explaining the measure-ments better than a straight line of the mean velocity. Note that this definition

26 3 Modelling

means that the model fit can be negative if the estimated velocity fits the mea-sured velocity worse than a straight line.

Also note that the NMSE weights large errors heavier than small errors due to thesquaring of each term.

4Control

The goal of the control design is to synthesise a controller that controls the ve-locity of the forks accurately. The controller should be robust against changes inload and temperature and should preferably not introduce too much oscillationsin the system, as mentioned earlier in Section 1.2.

4.1 System Overview

Figure 4.1 shows an overview of the system including controller and derivativefilter. The derivative of the position sensor values are filtered with a low-passfilter and used as the estimated velocity. The output of the controller is the valvecurrent which the ECU translates into a PWM signal.

27

28 4 Control

Valvecurrent PWM

ECU

PWM Flow

Valves

Flow

Oiltemperature

Freeliftpressure

Mainliftpressure

Freeliftposition

Mainliftposition

Hydraulicandmechanicsystem

PositionsensorvaluesEstimatedvelocity

Discretederivative

Display

Estimatedvelocity

Referencesignal

Valvecurrent

Controller

1Reference

Figure 4.1: Overview of the system with added derivative filter and con-troller.

4.2 Control Strategies

There are a number of different control strategies that might be of interest. Whichcontrol strategy to use depends on, amongst others, the computational poweravailable, the expected performance of the controller, whether a model is avail-able to use as an internal model or not and the complexity of the system. Fora relatively simple system a PID controller will often suffice whereas for morecomplex systems a more advanced approach might be necessary.

One interesting choice of controller which is slightly more complex than a PID isthe LQ (linear quadratic) controller, which is a linear controller defined using aquadratic criterion. However, in this thesis the system is nonlinear which meansthat if a LQ controller is used the system might need to be linearized around anoperating point. Another interesting option is an MPC (model predictive control)controller. However, since MPC solves an optimisation problem in every timeinstance it requires more computational power than the other options. Sincethe computational power and available memory in this system are limited, theimplicit and explicit MPC controls might not be implementable.

Furthermore, there also exists different approaches for controlling nonlinear sys-tems. Two of the most well-known are gain scheduling and feedback lineariza-tion.

Gain scheduling uses a number of different linear controllers, each operating ina different operating region where they provide satisfactory results. Schedulingvariables are used to switch between the appropriate linear controllers.

4.3 Nonlinear Compensation via Look-Up Table 29

Feedback linearization is another approach when controlling nonlinear systems.The idea is to transform the nonlinear system into an equivalent linear system.Using a well chosen control input, that renders a linear relation between the inputand the output, linear control strategies can be used.

In this thesis the hardware and the authors’ time are the main limiting factorsof the controller. Because of these limitations an attempt was made to keep thecontroller as simple as possible. This resulted in a control structure using a PIDwith a neutral feed-forward from reference and a look-up table for linerization.The different parts of this control structure are described below.

4.3 Nonlinear Compensation via Look-Up Table

When the main part of a nonlinearity comes from a nonlinear actuator, the con-trol can be improved by explicitly compensating for the nonlinearity. If a systemcan be described as

y = Gu′ (4.1)

u′ = f (u) (4.2)

where G is a linear system, u is the input and f is a static non linearity. If theoutput from a controller is transformed as u = f −1(v), where v is the output froma linear control algorithm, then y = Gv which means the output y is linearizedwith respect to the input v [6].

The system in this thesis is nonlinear which will cause a linear controller to havedecreased performance compared to when controlling a linear system. One ofthe main nonlinearities is the input current to the flow through the proportionalvalve which is described in (3.14).

Ignoring the dynamics of the proportional valve and also ignoring the nonlinearproperties of the mechanical system, the system can be written as above wheref is the static nonlinearity from current to flow in the proportional valve, u iscurrent, u′ is flow and G is the mechanical system.

Calculating f −1 will require the inverse to (3.14) which can be quite difficult tocalculate. One way to create an approximate inverse of this relationship is to takesteady state measurements of the lowering speed for different amplitudes of theinput current. These measurements can be used to create a static look-up tablefrom current to velocity. Instead of flow, the velocity of the lift was used. Asthe conversion from flow to velocity has a constant relation in steady state, thevelocity can be used instead without affecting the linerization. Inverting this look-up table gives a nonlinear look-up table between the velocity and the current.

This look-up table will only be valid for the operating points it was defined at. Itis mainly the pressure over the valve and the oil temperature that will vary and

30 4 Control

affect the flow through the valve. Since there is a pressure compensator regulat-ing the pressure over the valve, the pressure will be fairly constant in steady statefor different operating points. The temperature is not mechanically compensatedfor, therefore a look-up table measured at one operating temperature will not bevalid for a different operating temperature.

4.4 PID Control

The PID control is the most common type of controller and is simple to imple-ment [6]. It takes an error value as input and calculates an output signal basedon proportional (P), integral (I) and derivative (D) terms. The input to the PIDcontrol in this case is the difference between the reference speed and the esti-mated velocity. The PID controller on parallel form can be written as

u(t) = P e(t) + I

t∫0

e(τ)dτ + Dde(t)dt

(4.3)

where u(t) is the output and e(t) is the control error [6].

4.4.1 Choice of PID Parameters

One method for choosing the parameters of a controller is to use parametric op-timisation. The basic principle behind the method is to choose the parameters tominimise a criterion [8]. The criterion that was used here is

V (x) =12

N∑k=1

eTk (x)ek(x) (4.4)

where

ek(x) = rk − v(x)k (4.5)

and where rk is the reference speed to the PID controller, v(x)k is the measuredspeed and x is a vector containing the PID parameters.

The optimisation of the parameters for the PID controller is based on the sameprinciples as the estimation of the model parameters described in Section 3.6.

4.4.2 Anti-Windup Method

There exists an upper limit of the proportional valve opening area which resultsin a limit of the flow through the proportional valve. This means that the controlsignal can be saturated. In turn, this might lead to an undesirable windup of theintegral part of a PID controller.

There exist several anti-windup methods, but in this thesis adjustment of theintegral part is used. The adjustment of the integral part in each step of execution,

4.5 System Characteristics 31

k, is done according to

Ik := Ik +TsTt

(uk − vk) (4.6)

where

uk =

umax if vk > umaxvk if umin ≤ vk ≤ umaxumin if vk < umin

(4.7)

and where Ik is the state of the integral part, Ts is the sampling time, uk is thecontroller output after the saturation, vk is the controller output before the sat-uration, umin and umax are the saturation limits and Tt is the tracking constant.The tracking constant was chosen to Tt = Ts [6].

4.4.3 Feed-Forward Control

To speed up the performance of the PID control a feed-forward link was intro-duced. This link simply adds the reference speed to the output of the PID con-troller.

The total control signal from the feed-forward branch and the PID controlleris used in combination with the look-up table mentioned in Section 4.3. Thefeed-forward gain is 1 which creates a nonlinear neutral feed-forward since thelook-up table creates an approximation of the inverse static gain of the system.

4.5 System Characteristics

The system has some characteristics which are important to take into considera-tion when designing the controller. Foremost among them are the temperaturedependency and the inherent system delays.

4.5.1 Temperature Dependency

As stated in Section 4.3, the look-up table used for linearizing the system is notvalid when the temperature differs from the operating point temperature in thetable. To approximate the temperature dependence of the look-up table, the ta-ble input was scaled and the output shifted linearly with the temperature. Thescaling of the input was done according to

u′ = (T a + b)u (4.8)

where u is the output from the controller, u′ is the scaled input to the look-uptable, T is the oil temperature, and a and b are tuning parameters.

32 4 Control

The shift of the output from the look-up table was done according to

i′ = (T c + d) + i (4.9)

where i is the output from the look-up table, i′ is the shifted output, T is the oiltemperature, and c and d are tuning parameters. To select the tuning parametersan optimisation problem was designed in the same way as previously mentionedin Subsection 4.4.1.

4.5.2 System Delays

The system contains various delay sources which is summarised to a total delay ofaround 160 ms from input current to output speed. This delay causes problemswith the controller since its input is the error between the reference input andthe estimated velocity. Because of the delay the estimated velocity will alwayslag behind 160 ms compared to the reference signal. This will cause the integralpart of the controller to rise in 160 ms before the estimated velocity has caughtup. The increased integral part will in turn lead to an overshoot. To avoid thisthe reference model was chosen to be a pure delay of 160 ms.

The system delay creates another problem. The oscillations in the system have aperiod of around 400 ms. The controller will try to suppress the oscillations butbecause the delay is around half the period, it causes a phase shift of 180 degrees.This means that the controller will only amplify the oscillations. To solve thisproblem a predictive model strategy would be needed. The solution used in thisthesis is to let the feed-forward part of the controller control the fast dynamicsand use a slow PID tuning in the controller to obtain a correct steady state value,independent of temperature and load.

4.6 Reference Signal

The reference signal used in this thesis is the same one the company uses. This sig-nal has been derived by trial-and-error and intuition of the developers as well asthe operators. It is based on a ramp signal which is low-pass filtered to smoothenthe edges.

Furthermore, it would be counterproductive to change the reference signal be-cause the operators are now used to this reference signal, which has been presentfor a few forklift generations. Changing the reference signal changes the forkliftbehaviour and the operators are conservative when it comes to changes in systembehaviour.

4.6.1 Reference Signal for Transition

To solve the problem with the transition a basic strategy is to use different con-trollers for the free and the main lift and control the transition between them bychoosing a reference signal for each of the controllers. One example of such astrategy is

4.7 Summarised Control Structure 33

xref ,FL = R(xG)xref (4.10)

xref ,ML = (1 − R(xG))xref (4.11)

where xref ,FL is the reference signal for the free lift and xref ,ML is the referencesignal for the main lift. R is a ratio ∈ [0, 1] which is dependent on xG and duringthe transition phase goes from zero to one.

4.7 Summarised Control Structure

Figure 4.2 shows the summarised control structure. This structure contains allparts mentioned above.

Figure 4.2: Summarised control structure containing all previously men-tioned parts. The trajectory generator creates the reference signal from userinput. Gm is the reference model and Ff is the feed-forward block.

4.8 Implementation

To test the controller on a real forklift, the control algorithm needs to be imple-mented on the forklift ECU. To implement this the simulated control algorithmneeds to be discretized and converted to C code. This conversion can be done au-tomatically via Simulink coder (which translates Simulink models into C code.)

4.8.1 Discretization

The PID control was transformed to discrete time with the integrator methodForward Euler. This results in

U (z) = (P + I · Ts1

z − 1+ D ·

1Ts

z − 1z

)E(z) (4.12)

where Ts is the sampling time.

34 4 Control

4.8.2 Quantization

The sampling frequency of the system is 50 Hz and the position signal is quan-tized with an accuracy of 2 mm as mentioned in Subsection 3.1.4. Due to thesetup of the position sensor as mentioned in Chapter 2, the accuracy of the forkposition is 4 mm when using the main lift. Taking the derivative of a quantizedsignal leads to abrupt changes in the resulting signal. Because of the large quanti-zation errors in the position measurements, a low pass filter is used to smoothenthese measurements.

Figure 4.3 shows the simulated velocity, the numerical derivative of the quantizedposition sensor of the main lift and the low-pass filtered version.

0 2 4 6 8 10 12

Time [s]

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Velo

city [m

/s]

Velocity comparison

Derivative of position

Derivative and low-pass filtering of position

Simulated velocity

Figure 4.3: Numerical derivative of the quantized position (blue), filterednumerical derivative (orange) and simulated velocity (yellow). The quan-tized position sensor makes the derivative have abrupt changes while thefiltered version is somewhat smoothed.

4.8.3 Fixed Point Precision

The embedded processor does not have a floating point unit capable of carryingout operations on floating point numbers. This could be solved by using a soft-ware library. However, performing floating point calculations in software takesextra processing time and since the computational power is limited the option ofusing a software library was discarded.

Instead of using floating points the controller was converted to fixed point preci-sion. The fixed point data type is simply an integer scaled by an implicit factor.The scaling chosen in this case was a binary scaling for increased computationalefficiency. The binary fixed point value is represented by integer and fractional

4.9 Stability 35

bits where the integer bits plus the fractional bits equal the word length of theprocessor.

However, when converting floating points to fixed points the value is roundedwith a certain precision (one divided by the scaling factor) which causes loss ofinformation. There can also be precision loss when calculating fixed point oper-ations since the result can contain more bits than the operands which leads totruncation.

Figure 4.4 shows how the number 100.0625 is represented with fixed point preci-sion. In this case the word length is 16, the integer bits are eight and the fractionalbits are also eight. With eight fractional bits the precision is 2−8 = 0.00390625which means that all values more precise than this will be truncated.

0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0

Integer bits Fractional bits

Figure 4.4: Fixed point representation of the number 100.0625 with the in-teger and fractional bits marked.

4.9 Stability

When designing a feedback controller it is always of interest to know if the closed-loop system is stable. Investigating the stability of a nonlinear closed-loop systemis no simple task. There exist different methods such as using a phase plane, de-scribing functions or Lyapunov functions for determining if the system is stableor not.

The phase plane visualises the system behaviour by using vectors of derivativesof the state variables with respect to different parameters at each operating point.This creates a vector field where at each operating point a vector is pointing in thedirection of the derivative at that particular point. This is particularly interestingwith an oscillative system, where the vectors either spiral out to infinity or spiralin towards the centre, since this determines whether the system dynamics arestable or not.

The describing function method tries to combine linear systems to capture thebehaviour of the nonlinear system. Although describing functions do not pro-vide a complete description of the nonlinear system it could be sufficient to drawconclusions regarding the stability of the system. However, describing functionsare best for analysing systems with relatively weak nonlinearities.

Both the phase plane and describing function approaches are difficult and timeconsuming to use when applying them to a large, complex nonlinear system.

36 4 Control

Lyapunov functions are used to prove that an equilibrium point is stable, that is,whether a dynamic system is stable or not. A function V is called a Lyapunovfunction if

V (x0) = 0

V (x) > 0 ∀x , x0

∂V∂x

f (x) ≤ 0

where x0 is the equilibrium point and f (x) is the function describing the systemx = f (x). This can be used to prove stability but it is difficult to construct aLyapunov function for a complex system. [8]

A more straightforward approach to investigate the stability of the closed-loopsystem is to use the derived model in simulations to examine how the controllerbehaves in different operating points. In this case, it can be done by changing theload and temperature and varying the reference velocity to the controller in sim-ulations. This method does not give any guarantees but it is easier to implementand still gives a hint of the stability of the controller. This approach was used inthis thesis because of its simplicity.

5Results

In this chapter the results of estimating the unknown parameters of the derivedmodel are shown, which are based on the measurements of the system. The per-formance of the designed control is also validated using simulation in Simulinkand implementation in the real folklift.

5.1 Data Collection

The input signal to the system is the current applied to the proportional valve. Asmentioned in Section 3.5 this current is filtered using a low-pass filter to avoidabrupt changes which might damage the hardware. Furthermore, to reduce fric-tion forces in the valve, and thereby the hysteresis, dither is applied to the signal.Dither is a low amplitude, high frequency signal which is superimposed over theinput signal. Figure 5.1 shows the input current called “Current”, the low-pass fil-tered signal called “Req current” and the final current applied to the valve called“Raw current”.

37

38 5 Results

0 1 2 3 4 5 6 7 8 9

Time [s]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Curr

ent

of

free

lift

Free lift currents

Current

Req current

Raw current

Figure 5.1: Input current (blue), low-pass filtered current (orange) and ap-plied current with dither (yellow).

5.1.1 Temperature Dependency

Figure 5.2 shows the velocity of the free lift at temperatures 20 and 60 ◦C withtwo different loads. Figure 5.2a and 5.2b show the velocity of the free lift at thetwo different temperatures with no load and 800 kg load, respectively. The samesignals are shown in Figure 5.2c and Figure 5.2d but with a different input signal.

In the figure it can be seen that higher temperatures increase the amplitude ofthe oscillations as well as the velocity at steady state. This data confirms theinitial statements made in Section 1.1 where it is pointed out that the system istemperature dependent.

5.1 Data Collection 39

0 1 2 3 4 5 6 7 8

Time [s]

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Velo

city [m

/s]

Velocity comparison of free PRMS1 0kg with different temperatures

20 degrees

60 degrees

(a) Velocity of the free lift withno load when the oil temperatureis at 20 and 60 ◦C.

0 1 2 3 4 5 6 7 8

Time [s]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Ve

locity [

m/s

]

Velocity comparison of free PRMS1 800kg with different temperatures

20 degrees

60 degrees

(b) Velocity of the free lift with800 kg load when the oil temper-ature is at 20 and 60 ◦C.

0 1 2 3 4 5 6 7 8

Time [s]

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Velo

city [m

/s]

Velocity comparison of free PRMS2 0kg with different temperatures

20 degrees

60 degrees

(c)Velocity of the free lift with noload when the oil temperature isat 20 and 60 ◦C.

0 1 2 3 4 5 6 7 8

Time [s]

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Ve

locity [

m/s

]

Velocity comparison of free PRMS2 800kg with different temperatures

20 degrees

60 degrees

(d) Velocity of the free lift with800 kg load when the oil temper-ature is at 20 and 60 ◦C.

Figure 5.2: Comparison of the velocity for the free lift at different tempera-tures and loads for two different input signals.

5.1.2 Load Dependency

Figure 5.3 shows the velocity of the free lift with different loads at two differenttemperatures. Figure 5.3a and 5.3b show the velocity of the free lift with differentloads and an oil temperature of 20 and 60 ◦C, respectively. The same signals areshown in Figure 5.3c and 5.3d but with a different input signal.

The figure shows that higher loads increase the amplitude of the oscillations butdo not increase the velocity at steady state. The velocity at steady state is inde-pendent of load due to the compensator valve.

40 5 Results

0 2 4 6 8 10 12 14

Time [s]

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2V

elo

city [

m/s

]Velocity comparison of free PRMS1 20temp with different weights

0 kg

800 kg

1500 kg

(a) Velocity of the free lift with aload of 0, 800 and 1500 kg whenthe oil temperature is at 20 ◦C.

-2 0 2 4 6 8 10 12

Time [s]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Ve

locity [

m/s

]

Velocity comparison of free PRMS1 60temp with different weights

0 kg

800 kg

1500 kg

(b) Velocity of the free lift with aload of 0, 800 and 1500 kg whenthe oil temperature is at 60 ◦C.

0 2 4 6 8 10 12

Time [s]

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Ve

locity [

m/s

]

Velocity comparison of free PRMS2 20temp with different weights

0 kg

800 kg

1500 kg

(c) Velocity of the free lift with aload of 0, 800 and 1500 kg whenthe oil temperature is at 20 ◦C.

0 2 4 6 8 10 12

Time [s]

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Ve

locity [

m/s

]Velocity comparison of free PRMS2 60temp with different weights

0 kg

800 kg

1500 kg

(d) Velocity of the free lift with aload of 0, 800 and 1500 kg whenthe oil temperature is at 60 ◦C.

Figure 5.3: Comparison of the velocity for the free lift at different loads andtemperatures for two different input signals.

5.1.3 Measurement Noise

One way to investigate how much noise exists in the system is to use the sameinput signal multiple times and compare the outputs as mentioned earlier in Sec-tion 3.5. In Figure 5.4 the results of these comparisons can be seen where Fig-ure 5.4a depicts the different velocities for an identical input signal. Figure 5.4bdepicts these velocities subtracted from the mean velocity to show the noise level.

The comparison indicates a low level of noise in the system (< 0.01m/s amplitudein steady state). The noise is increasing with increasing oscillations.

5.2 Parameter Estimation 41

0 2 4 6 8 10 12

Time (s)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Velo

city [m

/s]

Velocities for the same input signal

Measurement 1

Measurement 2

Measurement 3

Measurement 4

Measurement 5

(a) Measured velocities for thesame input signal.

0 2 4 6 8 10 12

Time (s)

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Velo

city [m

/s]

Velocities for the same input signal with the mean subtracted

Measurement 1

Measurement 2

Measurement 3

Measurement 4

Measurement 5

(b) Measured velocities sub-tracted from the mean velocityfor the same input signal.

Figure 5.4: Velocities for an identical input signal and the same velocitiessubtracted from the mean velocity.

5.2 Parameter Estimation

As previously mentioned in Subsection 3.6.1 the parameter estimation for themodel was done using two data sets. One set contained steady state measure-ments and the other measurements of a PRMS signal.

In the column “Source” in the tables below, “Steady” means the parameter hasbeen estimated with the steady state measurements while “PRMS” means that ithas been estimated with the PRMS measurements.

In the first data set a current to area look-up table, mentioned in (3.2.2), for thevalves was also estimated. Figure 5.5 shows the current to area look-up table forthe free lift (Figure 5.5a) and the main lift (Figure 5.5b).

0 0.2 0.4 0.6 0.8 1 1.2 1.4

i [A]

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

A [m

2]

10-5 Free lift current to area map

(a) Free lift look-up table.

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

i [A]

0

1

2

3

4

5

6

7

A [m

2]

10-5 Main lift current to area map

(b) Main lift look-up table.

Figure 5.5: Current to area look-up tables estimated from steady state mea-surements for both the free and the main lift.

42 5 Results

5.2.1 Free Lift Parameters

Table 5.1: Estimated parameter values for the free lift.

Submodel Parameter Value Source EquationGlobal Oil density (ρ) 860 [kg/m3] Given 3.14

Total mass (mL) 278 [kg] Given 3.25Gravitational constant (g) -9.82 [m/s2] Given 3.25

Viscosity Dynamic viscosity (η0) 4.45 · 10−5 [P as] PRMS 3.1Reference temperature (T0) 293 [K] Chosen 3.1Viscosity coefficient (λ) 0.0050814 [K−1] PRMS 3.1

Control Control volume (V ) 2000 · 10−9 [m3] Given 3.19volume Bulk modulus (E) 7 · 108 [P a] Chosen1 3.19Cylinder Cylinder area (A) π · (0.055/2)2 [m2] Given 3.20

Mass of cylinder (mc) 12.6 [kg] Given 3.25Initial volume (V0) 0.0055302 [m3] PRMS 3.21

Bulk Max bulk modulus (Emax) 18000 · 105 [P a] Chosen 3.2modulus Max pressure (pmax) 280 · 105 [P a] Chosen 3.2

Parameter (a1) 0.15703 [-] PRMS 3.2Parameter (a2) 11.841 [-] PRMS 3.2Parameter (a3) 367.5 [-] PRMS 3.2

Friction Preload force (Fpr ) 113.81 [N ] PRMS 3.34Breakaway coefficient (Kbrk) 3.769 [−] PRMS 3.34Transition coefficient (cv) 261.2 [s/m] PRMS 3.34Coulomb coefficient (fcf r ) 2.6445 [Ns/m] PRMS 3.34LuGre coefficient (σ0) 2517.2 [N/m] PRMS 3.5LuGre coefficient (σ1) 14342 [Ns/m] PRMS 3.5LuGre coefficient (σ2) 4137.8 [Ns/m] PRMS 3.5

Proportional Damping ratio (D) 1 [−] Chosen 3.13valve Natural frequency (ω) 34.327 [s−1] PRMS 3.13

Hysteresis function (fhs) 0.018723 [A] Steady 3.11Reynolds critical (Recr ) 5543.9 [-] Steady 3.8Discharge coefficient (CD ) 0.67 [-] Chosen 3.7LuGre coefficient (σ0) 5 · 10−1 [N/m] Chosen 3.12LuGre coefficient (σ1) 1 · 10−1 [Ns/m] Chosen 3.12

Compensator Maximum area (Amax) 100 · 10−6 [m2] Given 3.15valve Minimum area (Aleak) 10−9 [m2] Given 3.15

Spool area (Aspool) 3.14 · 10−4 [m2] Given 3.16Regulation range (preg ) 2.7 · 105 [P a] Given 3.15Preset pressure (pset) 5.4 · 105 [P a] Given 3.15Time constant (τ) 0.01 [s] Chosen 3.17Discharge coefficient (CD ) 0.84417 [-] Steady 3.16Reynolds critical (Recr ) 18546 [-] Steady 3.8Design variable (θ) 0.63222 [rad] Steady 3.16

5.2 Parameter Estimation 43

5.2.2 Main Lift Parameters

Table 5.2: Estimated parameter values for the main lift.

Submodel Parameter Value Source EquationGlobal Oil density (ρ) 860 [kg/m3] Given 3.14

Total mass (mL) 520 [kg] Given 3.31Gravitational constant (g) -9.82 [m/s2] Given 3.31

Viscosity Dynamic viscosity (η0) 4.29 · 10−5 [P as] PRMS 3.1Reference temperature (T0) 293 [K] Chosen 3.1Viscosity coefficient (λ) 0.0070604 [K−1] PRMS 3.1

Control Control volume (V ) 2000 · 10−9 [m3] Given 3.19volume Bulk modulus (E) 7 · 108 [P a] Chosen1 3.19Cylinder Cylinder area (A) π · (0.04/2)2 [m2] Given 3.20

Mass of cylinder (mc) 283 [kg] Given 3.31Initial volume (V0) 0.0054528 [m3] PRMS 3.21

Bulk Max bulk modulus (Emax) 18000 · 105 [P a] Chosen 3.2modulus Max pressure (pmax) 280 · 105 [P a] Chosen 3.2

Parameter (a1) 0.13983 [-] PRMS 3.2Parameter (a2) 5.7224 [-] PRMS 3.2Parameter (a3) 202.59 [-] PRMS 3.2

Friction Preload force (Fpr ) 269.6 [N ] PRMS 3.34Breakaway coefficient (Kbrk) 7.3149 [−] PRMS 3.34Transition coefficient (cv) 254.49 [s/m] PRMS 3.34Coulomb coefficient (fcf r ) 2.68 · 10−5 [Ns/m] PRMS 3.34LuGre coefficient (σ0) 5976.3 [N/m] PRMS 3.5LuGre coefficient (σ1) 16163 [Ns/m] PRMS 3.5LuGre coefficient (σ2) 21997 [Ns/m] PRMS 3.5

Proportional Damping ratio (D) 1 [−] Chosen 3.13valve Natural frequency (ω) 47.015 [s−1] PRMS 3.13

Hysteresis function (fhs) 0.023916 [A] Steady 3.11Reynolds critical (Recr ) 5931 [-] Steady 3.8Discharge coefficient (CD ) 0.67 [-] Chosen 3.7LuGre coefficient (σ0) 5 · 10−1 [N/m] Chosen 3.12LuGre coefficient (σ1) 1 · 10−1 [Ns/m] Chosen 3.12

Compensator Maximum area (Amax) 100 · 10−6 [m2] Given 3.15valve Minimum area (Aleak) 10−9 [m2] Given 3.15

Spool area (Aspool) 3.14 · 10−4 [m2] Given 3.16Regulation range (preg ) 2.7 · 105 [P a] Given 3.15Preset pressure (pset) 5.4 · 105 [P a] Given 3.15Time constant (τ) 0.01 [s] Chosen 3.17Discharge coefficient (CD ) 0.7342 [-] Steady 3.16Reynolds critical (Recr ) 5370.9 [-] Steady 3.8Design variable (θ) 0.17686 [rad] Steady 3.16

44 5 Results

5.2.3 Control Parameters

Table 5.3 shows the estimated parameter values for the free and main lift con-trollers. Note that in the controllers the input position is scaled to mm, the refer-ence velocity to mm/s and the output current to mA. This was done because thecurrent system is implemented in these units.

Table 5.3: Estimated parameter values for the PID controllers.

Submodel Parameter Value EquationFree lift controller P (Proportional) 0.001201 4.12

I (Integral) 1.178 4.12D (Derivative) 0.00084129 4.12a -0.0030497 4.8b 1.8571 4.8c -0.3077 4.9d -0.049189 4.9

Main lift controller P (Proportional) 0.00010892 4.12I (Integral) 2.1999 4.12D (Derivative) 0.006814 4.12a -0.0023468 4.8b 1.6415 4.8c -0.36625 4.9d -0.36953 4.9

Figure 5.6 shows the nonlinear look-up tables mentioned in Section 4.3 for thefree lift (Figure 5.6a) and the main lift (Figure 5.6b) controllers.

0 100 200 300 400 500 600 700

Velocity [mm/s]

0

200

400

600

800

1000

1200

1400

Curr

ent [m

A]

Free lift velocity to current map

(a) Free lift controller look-up ta-ble.

0 100 200 300 400 500 600 700 800 900 1000

Velocity [mm/s]

200

400

600

800

1000

1200

1400

Cu

rre

nt

[mA

]

Main lift velocity to current map

(b) Main lift controller look-uptable.

Figure 5.6: Velocity to current look-up tables for both the free and main liftcontroller.

1Chosen to avoid numerical issues

5.3 Model Validation 45

5.3 Model Validation

As previously mentioned in Section 3.6, different data sets were used for parame-ter estimation and model validation. This is done to avoid overfitting.

The model was validated foremost visually by plotting the simulated and mea-sured values of the velocity. The cylinder pressure was another measured outputsignal of interest. Figure 5.7 depicts the simulated and measured velocity (Fig-ure 5.7a) as well as cylinder pressure (Figure 5.7b) with a load of 800 kg and atemperature of 20 ◦C. The velocity and cylinder pressure are closely connectedwhich can be seen in the figure.

0 2 4 6 8 10 12

Time [s]

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city [m

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Simulation of free PRMS2 800kg 20temp velocity

Measured velocity

Simulated velocity

(a) Simulated and measured velocities.

0 2 4 6 8 10 12

Time [s]

2

4

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8

10

12

14

16

Pre

ssure

[P

a]

106 Simulation of free PRMS2 800kg 20temp pressure

Measured pressure

Simulated pressure

(b) Simulated and measured pressures.

Figure 5.7: Simulated and measured velocities and pressures for a pseudo-random multilevel signal at 800 kg and 20 ◦C.

46 5 Results

To validate the temperature and load dependencies of the model several simula-tions were compared to measurements. To measure the model performance thevalidation criteria mentioned in Subsection 3.7.1 are used. Note that this is justone way of measuring the model performance and it should therefore be usedwith care, but it still gives a hint of how well the model performs.

5.3.1 Free Lift Validation

In Table 5.4 the model fit of the free lift model is shown for different loads andtemperatures. The table shows that the model fit decreases with increasing loadas well as increasing temperature.

Table 5.4: Model fit for the free lift model for different loads and tempera-tures with the validation signal as input.

Temperature [C] Load [kg] Model fit (NMSE) [%] Model fit (NRMSE) [%]20 0 98.0 85.8

800 95.2 78.11500 93.2 74.9

60 0 96.9 82.4800 91.7 71.21500 93.0 73.6

Figure 5.8 shows the simulated and measured velocities with 0 and 1500 kg loadat a temperature of 20 (Figure 5.8a and Figure 5.8c) and 60 (Figure 5.8b andFigure 5.8d) ◦C. The model is able to capture the general behaviour of this sub-system but there exist some steady state errors. These steady state errors areespecially prominent when no load is applied and at higher temperatures. Themodel has increased oscillations with increasing load as expected.

5.3 Model Validation 47

0 1 2 3 4 5 6 7 8 9 10

Time [s]

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0

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Velo

city [m

/s]

Simulation of free PRMS2 0kg 20temp velocity

Measured velocity

Simulated velocity

(a) Simulated and measured ve-locities with zero load and 20◦C. The model fit for this sig-nal is 85.8% (NRMSE), 98.0%(NRMSE).

0 1 2 3 4 5 6 7 8 9 10

Time [s]

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Velo

city [m

/s]

Simulation of free PRMS2 0kg 60temp velocity

Measured velocity

Simulated velocity

(b) Simulated and measured ve-locities with zero load and 60◦C. The model fit for this sig-nal is 82.4% (NRMSE), 96.9 %(NRMSE).

0 2 4 6 8 10 12

Time [s]

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0

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Velo

city [m

/s]

Simulation of free PRMS2 1500kg 30temp velocity

Measured velocity

Simulated velocity

(c) Simulated and measured ve-locities with 1500 kg load and 20◦C. The model fit for this sig-nal is 74.9% (NRMSE), 93.2 %(NRMSE).

0 1 2 3 4 5 6 7 8 9 10

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Velo

city [m

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Simulation of free PRMS2 1500kg 60temp velocity

Measured velocity

Simulated velocity

(d) Simulated and measured ve-locities with 1500 kg load and 60◦C. The model fit for this sig-nal is 73.6% (NRMSE), 93.0 %(NRMSE).

Figure 5.8: Simulated and measured velocities of the free lift at 0 and 1500kg, and 20 and 60 ◦C.

48 5 Results

5.3.2 Main Lift Validation

In Table 5.5 the model fit of the main lift model is shown for different loads andtemperatures. It shows that the model fit decreases with increasing load as wellas increasing temperature.

Table 5.5: Model fit for the main lift model for different loads and tempera-tures with the validation signal as input.

Temperature [C] Load [kg] Model fit (NMSE) [%] Model fit (NRMSE) [%]20 0 94.6 76.8

800 93.6 74.81500 92.4 72.5

60 0 93.2 73.9800 90.0 68.41500 91.7 71.2

Figure 5.9 shows the simulated and measured velocities with 0 and 1500 kg loadat a temperature of 20 (Figure 5.9a and Figure 5.9c) and 60 (Figure 5.9b andFigure 5.9d) ◦C. The model is able to capture the general dynamics of the sub-system but there exist some model errors. These dynamic errors are especiallyprominent when no load is applied and at higher temperatures. The model hasincreased oscillations with increasing load as expected.

The main lift model has a lower fit in all cases compared to the free lift modelwhich is reasonable since the main lift is more oscillative than the free lift.

5.4 Control Validation 49

0 2 4 6 8 10 12

Time [s]

-1

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0

0.5

Velo

city [m

/s]

Simulation of main PRMS2 0kg 20temp velocity

Measured velocity

Simulated velocity

(a) Simulated and measured ve-locities with zero load and 20◦C. The model fit for this sig-nal is 76.8% (NRMSE), 94.6 %(NRMSE).

0 2 4 6 8 10 12

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city [m

/s]

Simulation of main PRMS2 0kg 60temp velocity

Measured velocity

Simulated velocity

(b) Simulated and measured ve-locities with zero load and 60◦C. The model fit for this sig-nal is 73.9% (NRMSE), 93.2 %(NRMSE).

0 2 4 6 8 10 12 14

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city [m

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Simulation of main PRMS2 1500kg 30temp velocity

Measured velocity

Simulated velocity

(c) Simulated and measured ve-locities with 1500 kg load and 20◦C. The model fit for this sig-nal is 72.5% (NRMSE), 92.4 %(NRMSE).

0 2 4 6 8 10 12

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city [m

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Simulation of main PRMS2 1500kg 60temp velocity

Measured velocity

Simulated velocity

(d) Simulated and measured ve-locities with 1500 kg load and 60◦C. The model fit for this sig-nal is 71.2% (NRMSE), 91.7 %(NRMSE).

Figure 5.9: Simulated and measured velocities of the main lift at 0 and 1500kg, 20 and 60 ◦C.

5.4 Control Validation

The closed-loop system with the controller and model was tested in simulationsfor both the free lift and the main lift.

5.4.1 Simulated Free Lift Controller

Figure 5.10 shows the simulated velocity of the free lift with 800 kg load, attemperatures 20 (Figure 5.10a) and 60 (Figure 5.10b) ◦C. The controller is ableto control the velocity with small errors. The amplitude of the oscillations isapproximately 0.02m/s.

50 5 Results

0 5 10 15

Time [s]

-0.1

0

0.1

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0.4

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0.6V

elo

city [

m/s

]Simulation of Free 800kg 20temp velocity

Reference

Simulated velocity

(a) Simulated velocity of the freelift with 800 kg load and 20 ◦C.

0 5 10 15

Time [s]

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0

0.1

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Ve

locity [

m/s

]

Simulation of Free 800kg 60temp velocity

Reference

Simulated velocity

(b) Simulated velocity of the freelift with 800 kg load and 60 ◦C.

Figure 5.10: Simulated velocities of the free lift at 800 kg, 20 and 60 ◦C.

All plots of the simulated free lift controller can be found in Appendix B.

5.4.2 Simulated Main Lift Controller

Figure 5.11 shows the simulated velocity of the main lift with 800 kg load, attemperatures 20 (Figure 5.11a) and 60 (Figure 5.11b) ◦C. The controller is ableto control the velocity with small errors except around the time 9 seconds for 20◦C where the velocity drops to almost zero before recovering. The amplitude ofthe oscillations is approximately 0.03m/s.

0 5 10 15

Time [s]

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0

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Ve

locity [

m/s

]

Simulation of Main 800kg 20temp velocity

Reference

Simulated velocity

(a) Simulated velocity of themain lift with 800 kg load and 20◦C.

0 5 10 15

Time [s]

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0

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Ve

locity [

m/s

]

Simulation of Main 800kg 60temp velocity

Reference

Simulated velocity

(b) Simulated velocity of themain lift with 800 kg load and 60◦C.

Figure 5.11: Simulated velocities of the main lift at 800 kg, 20 and 60 ◦C.

All plots of the simulated main lift controller can be found in Appendix C.

5.5 Stability 51

5.5 Stability

Examining the results from the previous section it can be seen that the closed-loop system is stable, at least for the tested scenarios. The tested scenarios spanwhat can be considered the working range of the system, testing different loads,temperatures and using a varying reference signal.

5.6 Implementation and Final Testing

The controller was implemented on a real forklift and measurements were taken,with two different reference signals, for both the free and the main lift. Theimplemented controller was compared to the currently used controller and theresults are shown below. To compare the controllers the resulting velocity foreach of them was plotted together with the reference velocity. Moreover, thevalidation criteria NRMSE mentioned in Subsection 3.7.1 is used to compare thecontrol performance of both controllers. The control performance is presented ateach operating point for the two different reference signals.

5.6.1 Implemented Free Lift Controller

In Table 5.6 the control performance is shown for the two different controllersat different loads and temperatures. It shows that the proposed control in thisthesis has a better performance at all operating points, especially at higher tem-peratures. However, as mentioned earlier this performance value is just a numberand does not take into account oscillations, rise time, etc. This indication of thecontrol performance should therefore be used in conjunction with other indica-tions, in this case visually examining the plotted velocities.

Table 5.6: Control performance (NRMSE) in percent for the free lift con-trollers for different loads and temperatures. The ranges in preformancemeasure indicate the ranges in preformance testing two reference signals

Temperature [C] Load [kg] Thesis controller [%] Current controller [%]20 0 71.29 - 77.84 70.15 - 72.80

800 72.91 - 73.12 67.83 - 70.101500 69.31 - 72.42 68.27 - 68.87

60 0 78.50 - 78.69 61.95 - 73.10800 72.50 - 75.35 69.58 - 70.461500 67.04 - 68.42 58.27 - 62.98

Figure 5.12 shows the measured velocities for the two controllers when control-ling the free lift with 0 kg load. Figure 5.12a and Figure 5.12b show the velocitiesat 20 and 60 ◦C, respectively. The same signals but with another reference signalcan be seen in Figure 5.12c and Figure 5.12d. The maximum steady state error ofthe current controller is approximately 0.06m/s while the steady state errors of

52 5 Results

the implemented control are small.

0 2 4 6 8 10

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

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Velo

city [m

/s]

Measurement of Free lift 0kg 20temp

Current controller

Thesis controller

Reference

(a) Measured velocities of thefree lift when using different con-trollers with 0 kg load and 20 ◦C.

0 1 2 3 4 5 6 7 8 9

Time [s]

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Velo

city [m

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Measurement of Free lift 0kg 60temp

Current controller

Thesis controller

Reference

(b) Measured velocities of thefree lift when using different con-trollers with 0 kg load and 60 ◦C.

0 1 2 3 4 5 6

Time [s]

-0.1

0

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Velo

city [m

/s]

Measurement of Free lift 0kg 20temp

Current controller

Thesis controller

Reference

(c) Measured velocities of thefree lift when using different con-trollers with 0 kg load and 20 ◦C.

0 1 2 3 4 5 6

Time [s]

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0

0.1

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Ve

locity [

m/s

]

Measurement of Free lift 0kg 60temp

Current controller

Thesis controller

Reference

(d) Measured velocities of thefree lift when using different con-trollers with 0 kg load and 60 ◦C.

Figure 5.12: Measured velocities of the free lift at 0 kg, 20 and 60 ◦C whenusing different controllers and for different reference signals.

Figure 5.13 shows the measured velocities of the free lift when using differentcontrollers with 800 kg load. Figure 5.13a and Figure 5.13b show the velocitiesat 20 and 60 ◦C, respectively. The same signals but with another reference signalcan be seen in Figure 5.13c and Figure 5.13d. The maximum steady state error ofthe current controller is approximately 0.06m/s while the steady state errors ofthe implemented control are small.

5.6 Implementation and Final Testing 53

0 2 4 6 8 10

Time [s]

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0

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Velo

city [m

/s]

Measurement of Free lift 800kg 20temp

Current controller

Thesis controller

Reference

(a) Measured velocities of thefree lift when using different con-trollers with 800 kg load and 20◦C.

0 2 4 6 8 10

Time [s]

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0

0.1

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Velo

city [m

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Measurement of Free lift 800kg 60temp

Current controller

Thesis controller

Reference

(b) Measured velocities of thefree lift when using different con-trollers with 800 kg load and 60◦C.

0 1 2 3 4 5 6

Time [s]

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0

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Velo

city [m

/s]

Measurement of Free lift 800kg 20temp

Current controller

Thesis controller

Reference

(c) Measured velocities of thefree lift when using different con-trollers with 800 kg load and 20◦C.

0 1 2 3 4 5 6

Time [s]

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0.1

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Velo

city [m

/s]

Measurement of Free lift 800kg 60temp

Current controller

Thesis controller

Reference

(d) Measured velocities of thefree lift when using different con-trollers with 800 kg load and 60◦C.

Figure 5.13: Measured velocities of the free lift at 800 kg, 20 and 60 ◦C whenusing different controllers and for different reference signals.

Figure 5.14 shows the measured velocities of the free lift when using differentcontrollers with 1500 kg load. Figure 5.14a and Figure 5.14b show the velocitiesat 20 and 60 ◦C, respectively. The same signals but with another reference signalcan be seen in Figure 5.14c and Figure 5.14d. Neither controller has any promi-nent steady state errors except at 60 ◦C for the second reference signal where thecurrent controller has a steady state error of approximately 0.15m/s.

54 5 Results

0 2 4 6 8 10 12

Time [s]

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0

0.1

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Velo

city [m

/s]

Measurement of Free lift 1500kg 20temp

Current controller

Thesis controller

Reference

(a) Measured velocities of thefree lift when using different con-trollers with 1500 kg load and 20◦C.

0 2 4 6 8 10

Time [s]

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Velo

city [m

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Measurement of Free lift 1500kg 60temp

Current controller

Thesis controller

Reference

(b) Measured velocities of thefree lift when using different con-trollers with 1500 kg load and 60◦C.

0 1 2 3 4 5 6

Time [s]

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Velo

city [m

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Measurement of Free lift 1500kg 20temp

Current controller

Thesis controller

Reference

(c) Measured velocities of thefree lift when using different con-trollers with 1500 kg load and 20◦C.

0 1 2 3 4 5 6

Time [s]

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Velo

city [m

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Measurement of Free lift 1500kg 60temp

Current controller

Thesis controller

Reference

(d) Measured velocities of thefree lift when using different con-trollers with 1500 kg load and 60◦C.

Figure 5.14: Measured velocities of the free lift at 1500 kg, 20 and 60 ◦Cwhen using different controllers and for different reference signals.

5.6 Implementation and Final Testing 55

5.6.2 Implemented Main Lift Controller

In Table 5.7 the control performance is shown for the two different controllers atdifferent loads and temperatures. The proposed control in this thesis has betterperformance in all operating points except with 1500 kg load at 20 ◦C. At 60 ◦Cthe thesis controller has a much better performance which is due to the large sta-tionary error occurring at higher temperature for the currently used controller.As mentioned earlier this performance value is just a number and the plottedvelocities should be examined to get a better understanding of the control perfor-mance. The control performance is presented at each operating point for the twodifferent reference signals.

Table 5.7: Control performance (NRMSE) in percent for the main lift con-trollers for different loads and temperatures. The ranges in preformancemeasure indicate the ranges in preformance testing two reference signals

Temperature [C] Load [kg] Thesis controller [%] Current controller [%]20 0 64.69 - 70.37 34.17 - 48.59

800 57.37 - 61.22 39.10 - 52.541500 42.56 - 54.03 45.68 - 55.15

60 0 65.86 - 69.57 28.84 - 34.03800 56.94 - 58.98 15.20 - 31.811500 47.25 - 53.65 32.37 - 40.93

Figure 5.15 shows the measured velocities of the main lift when using differentcontrollers with 0 kg load. Figure 5.15a and Figure 5.15b show the velocities at 20and 60 ◦C, respectively. The same signals but with another reference signal canbe seen in Figure 5.15c and Figure 5.15d. The steady state errors of the currentcontroller are greater than 0.1m/s and the largest error at 60 ◦C is more than0.16m/s while the steady state errors of the implemented control are small.

56 5 Results

0 2 4 6 8 10 12

Time [s]

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Velo

city [m

/s]

Measurement of Main lift 0kg 20temp

Current controller

Thesis controller

Reference

(a) Measured velocities of themain lift when using differentcontrollers with 0 kg load and 20◦C.

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Time [s]

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locity [

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Measurement of Main lift 0kg 60temp

Current controller

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Reference

(b) Measured velocities of themain lift when using differentcontrollers with 0 kg load and 60◦C.

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Measurement of Main lift 0kg 20temp

Current controller

Thesis controller

Reference

(c) Measured velocities of themain lift when using differentcontrollers with 0 kg load and 20◦C.

0 2 4 6 8 10 12

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Velo

city [m

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Measurement of Main lift 0kg 60temp

Current controller

Thesis controller

Reference

(d) Measured velocities of themain lift when using differentcontrollers with 0 kg load and 60◦C.

Figure 5.15: Measured velocities of the main lift at 0 kg, 20 and 60 ◦C whenusing different controllers and for different reference signals.

Figure 5.16 shows the measured velocities of the main lift when using differentcontrollers with 800 kg load. Figure 5.16a and Figure 5.16b show the velocitiesat 20 and 60 ◦C, respectively. The same signals but with another reference sig-nal can be seen in Figure 5.16c and Figure 5.16d. The steady state errors of thecurrent controller are greater than 0.1m/s and the largest error at 60 ◦C is ap-proximately 0.2m/s while the steady state errors of the implemented control aresmall.

5.6 Implementation and Final Testing 57

0 2 4 6 8 10 12 14

Time [s]

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Ve

locity [

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]

Measurement of Main lift 800kg 20temp

Current controller

Thesis controller

Reference

(a) Measured velocities of themain lift when using differentcontrollers with 800 kg load and20 ◦C.

0 2 4 6 8 10 12

Time [s]

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Velo

city [m

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Measurement of Main lift 800kg 60temp

Current controller

Thesis controller

Reference

(b) Measured velocities of themain lift when using differentcontrollers with 800 kg load and60 ◦C.

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Time [s]

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Velo

city [m

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Measurement of Main lift 800kg 20temp

Current controller

Thesis controller

Reference

(c) Measured velocities of themain lift when using differentcontrollers with 800 kg load and20 ◦C.

0 2 4 6 8 10 12

Time [s]

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city [m

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Measurement of Main lift 800kg 60temp

Current controller

Thesis controller

Reference

(d) Measured velocities of themain lift when using differentcontrollers with 800 kg load and60 ◦C.

Figure 5.16: Measured velocities of the main lift at 800 kg, 20 and 60 ◦Cwhen using different controllers and for different reference signals.

Figure 5.17 shows the measured velocities of the main lift when using differentcontrollers with 1500 kg load. Figure 5.17a and Figure 5.17b show the veloci-ties at 20 and 60 ◦C, respectively. The same signals but with another referencesignal can be seen in Figure 5.17c and Figure 5.17d. The steady state errors ofthe current controller are greater than 0.1m/s while the steady state errors of theimplemented control are small. The thesis controller is more oscillative than thecurrently used controller when using the first reference signal.

58 5 Results

0 2 4 6 8 10 12

Time [s]

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city [m

/s]

Measurement of Main lift 1500kg 20temp

Current controller

Thesis controller

Reference

(a) Measured velocities of themain lift when using differentcontrollers with 1500 kg load and20 ◦C.

0 2 4 6 8 10 12

Time [s]

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Measurement of Main lift 1500kg 60temp

Current controller

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Reference

(b) Measured velocities of themain lift when using differentcontrollers with 1500 kg load and60 ◦C.

0 2 4 6 8 10 12

Time [s]

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Measurement of Main lift 1500kg 20temp

Current controller

Thesis controller

Reference

(c) Measured velocities of themain lift when using differentcontrollers with 1500 kg load and20 ◦C.

0 2 4 6 8 10 12

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Measurement of Main lift 1500kg 60temp

Current controller

Thesis controller

Reference

(d) Measured velocities of themain lift when using differentcontrollers with 1500 kg load and60 ◦C.

Figure 5.17: Measured velocities of the main lift at 1500 kg, 20 and 60 ◦Cwhen using different controllers and for different reference signals.

5.6.3 Implemented Transition

The reference signal for the transition was changed slightly, compared to the cur-rently used signal, during the implementation. The reference for the free lift waschanged to start earlier because there is generally a longer delay when openingthe free lift valve than when closing the main lift valve. The free lift reference wasalso changed to reach its maximum velocity quicker since the main lift reachesits bottom which causes a sudden drop in velocity. However, this is somewhatcompensated by the increase in free lift velocity.

Figure 5.18 shows the velocities in the transition phase for both controllers with800 kg load. Figure 5.18a and Figure 5.18b show the velocities for the currentlyused controller at temperatures of 20 and 60 ◦C, respectively. Figure 5.18c and

5.6 Implementation and Final Testing 59

Figure 5.18d show the same signals but for the implemented controller. Thecontrollers behave similarly and the implemented controller manages to slightlydampen the velocity spike.

6 6.5 7 7.5 8 8.5 9 9.5 10

time [s]

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locity [

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]

Measurement of transition of current controller 800kg 20temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(a) Measured velocities of thefree and main lift when using thecurrent controller with 800 kgload and 20 ◦C.

6 6.5 7 7.5 8 8.5 9 9.5 10

time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

velo

city [m

/s]

Measurement of transition of current controller 800kg 60temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(b) Measured velocities of thefree and main lift when using thecurrent controller with 800 kgload and 60 ◦C.

11 11.5 12 12.5 13 13.5 14

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ve

locity [

m/s

]

Measurement of transition of thesis controller 800kg 26temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(c) Measured velocities of thefree and main lift when using thethesis controller with 800 kg loadand 20 ◦C.

10 10.5 11 11.5 12 12.5 13

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ve

locity [

m/s

]

Measurement of transition of thesis controller 800kg 60temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(d) Measured velocities of thefree and main lift when using thethesis controller with 800 kg loadand 60 ◦C.

Figure 5.18: Transition phase velocities at 800 kg, 20 and 60 ◦C when usingdifferent controllers. The blue and orange lines are the reference signalswhile the yellow and purple lines are the measured velocities for the free andmain lift, respectively. The green line represents the sum of the measuredvelocities which is the total velocity.

The result of the transition phase is that the thesis controller and the current con-troller are fairly similar in the transition. All plots of the implemented transitioncontroller can be found in Appendix D.

6Conclusion

In this chapter, the results, method and future work are discussed.

6.1 Results

The simulated model performs well in terms of visual inspection and the valida-tion criteria. The free lift model has a slightly better performance compared tothe main lift model. Since both are modelled the same way this indicates thatthe main lift is more difficult to model than the free lift. Overall the simulatedmodels perform well enough to be used as a basis for control design.

The proposed controller, in this thesis, is able to obtain the correct steady statevelocity for all operating points. However, in the case for the main lift where theload is 1500 kg the implemented controller, in one case, induces large oscillationscompared to the currently used controller. This might be explained by the changein dynamics, especially the natural frequency of the system, during heavier loadsthat reduces the performance of the controller at those operating points.

The overall performance, according to the validation criteria, is better for thecontroller implemented in this thesis. This indication could be misleading sinceno consideration has been taken for oscillations, rise times, etc. However, whenvisually examining the control performance, the proposed controller does notseem to worsen the oscillations (except in one special case mentioned above). Theimplemented controller is slower to reach steady state. This is not surprisingsince the implemented controller has an integration part which makes sure italways reaches correct steady state. As mentioned in Subsection 4.5.2 this PIDcontroller cannot be made too fast since that would induce oscillations in thesystem.

61

62 6 Conclusion

The performance of the implemented controller is robust to temperature changeswhereas the currently used controller has decreased performance at higher tem-peratures, especially for the main lift. To make the controller robust to tempera-ture changes is one of the main purposes of this thesis.

The free lift is generally easier to control than the main lift. No controller hasany problems controlling the free lift except some steady state errors in the cur-rently used controller at zero load. The main lift is more interesting where thecurrently used controller has trouble finding the correct steady state velocity forall operating points while the implemented controller has small and negligiblesteady state errors.

The oscillations in the system are difficult to suppress with the current controlstrategy. This is partly due to the delays in the system. A control strategy withpredictive capabilities might have a better chance to suppress the oscillations butsince the behaviour of the real system varies over time it is difficult to predict thebehaviour. This could be problematic for a predictive control design since thesevariations are not captured in the implemented models. One of the main reasonsfor this erratic behaviour is the friction which probably is the most difficult partto model and predict.

In the transition phase the controllers are similar in performance. The systemdelays and the unpredictable behaviour of the system mentioned above makeit difficult to construct a good reference signal in the transition phase. In thetransition the free lift seems to follow the reference signal in a satisfactory way forall operating points whereas the main lift differs in both rise time and responsetime.

6.2 Method Evaluation

The used method consisted of a simple divide-and-conquer approach where theprocess was divided into five different categories. The different categories wereiterated to improve the final result. In the thesis three iterations were done. Inthe first iteration, the model and controller for the free lift were designed andimplemented. In the second iteration the same procedure was done for the mainlift. In the final iteration the two models were joined and a complete model wascreated. The first iteration took around ten weeks, the second one week and thethird three weeks. Much of the work done in the first iteration could be reused inthe second iteration when modelling and controlling the main lift which is whythe second iteration could be completed in only one week.

The modelling was done component-wise. Submodels were devised for differentcomponents in the system such as the compensator valve, the proportional valve,the cylinder, etc. However, a limitation with this approach is that input outputdata were not available for all the components that were being modelled. Therewas no equipment available to measure the relevant signals and it proved difficultto get measurements from the manufacturer of the components.

6.3 Future Work 63

Each submodel contained a number of unknown parameters which were esti-mated using a parameter estimation approach. As input output data was notavailable for the submodels, this parameter estimation had to be done for all ofthe submodels at once. This may have lead to larger uncertainties of the esti-mated parameters as this could introduce more local minima into the optimisa-tion problem.

6.3 Future Work

To improve the model there are numerous aspects that could be interesting toinvestigate. When designing models and controllers there is always room forimprovements and some of the more obvious ones are brought up here.

There exists a mechanical connection between the free and the main lift. Theeffects coming from this connection were neglected in this thesis even thoughthey have been observed in measurements. A model of this connection could beused to improve the performance of the controller and in particular the transitionphase where both lifts are moving simultaneously. The reference signal used inthe transition phase could also be designed better by examining more measure-ments of the transition.

Another interesting aspect is the temperature dependency. In this thesis it is mod-elled as only affecting the viscosity. The model could be improved by expandingthe temperature dependency and thereby improve the control robustness withrespect to temperature changes. The robustness of both the model and the con-troller could further be investigated by changing parameter values, i.e. by usinganother forklift. Comparing model fit and control performance between differentforklifts would give a hint of the robustness of both the model and the controller.

More complex controllers such as LQ, MPC controllers or gain scheduling areinteresting options that could be implemented and compared to the proposedcontroller in this thesis.

To further improve both the model and the controller more measurements atdifferent temperatures, with different loads, for different forklifts could be usedto improve the parameter estimation process.

6.4 Final Conclusion

The simulated model is able to capture most of the dynamics in the system andthe model fit for different operating points is & 70%. The model was deemedgood enough to use as a basis for control design even though there are somemodel errors.

The controller was designed based on the estimated model, implemented andtested on a real forklift. In this thesis, it was concluded that the closed-loopsystem most likely is stable since it showed no signs of instability at any operating

64 6 Conclusion

point tested in simulations or on the real system. The controller implemented inthis thesis is able to reach the correct steady state velocity but is generally a bitslower than the currently used controller. The implemented controller is robustto temperature changes and generally performs better although there exist someoscillative problems when controlling the main lift with 1500 kg load at 20 ◦C.

The main purpose of the thesis, which was to create a model and use it to designand implement a temperature robust controller, has been fulfilled.

Appendix

ASimulink Models

In this appendix the Simulink models, the simulated model and controller, of thefree lift are presented. The Simulink models for the main lift are almost identical.

67

68 A Simulink Models

1Current

Q_in

Q_out

p

volume

p_x

viscoisty

p_A

p_B

Q

Com

pensator

Q_T

p_T

Tank

iviscosity

p_B

p_A

Q

FlowValve

1dot_x_f

f_mec_m

_L0Constant

Add

2m_L

3tem

p

2p_c

m_L

Q

dot_x_f

p_c

x_f

Cylinder+M

echanical

TempV

iscosity

Oil

3x_f

Figure A.1: Simulink model of the free lift.

69

f_est_nu_0

Constant4

Math

Function4

-f_est_lambda

Constant3

Add2

273

CelsiustoK

elvin

f_theta_0

313KAdd3

Product

Product1

1Tem

p1

Viscosity

Figure A.2: Simulink model of the free lift viscosity.

70 A Simulink Models

Aviscosity

p

p/sqrt(p^2+p_cr^2)

Laminar-Turbulentflow

1Q4p_A3p_B 1i

1-DT(u)

i->A

f_est_prop_alpha_d

Constant1

sqrt(2/rho)

Gain

Divide

Add

MinM

ax

10^-9

Constant

2viscosity

f_est_prop_f_hs

Constant5

Add4

ii_effective

Secondorder(dynam

ic)

Figure A.3: Simulink model of the free lift proportional flow valve.

71

1i_effective

1i

1s 2u

xdx

Integrator,S

econd-Order

Add

f_est_prop_w_v^2

Gain

2*f_est_prop_D_v/f_est_prop_w

_v

Gain1

dot_xF

LuGrefriction

FigureA.4: Simulink model of the free lift proportional flow valve dynamics.

72 A Simulink Models

Add1

Math

Function1/4

Constant2

Math

Function1

Math

Function2

rho/2

Gain1

Divide1

f_est_prop_alpha_d

Gain2

f_est_prop_Re_cr

Gain3

4/pi

Gain4

Sqrt1

Math

Function3

1 A

2viscosity

3p

1p/sqrt(p^2+p_cr^2)

D_H

Figure A.5: Simulink model of the free lift laminar/turbulent flow.

73

1p

1Q_in

2Q_out

1s

Integrator

Add

f_volume_vol

Gain

Divide

1

Constant3

Product

f_volume_bulk

Constant

Figure A.6: Simulink model of the free lift volume.

74 A Simulink Models

1p_T1Q_T

Terminator

10^5

Constant

Figure A.7: Simulink model of the free lift tank.

75

1Q

3p_A4p_B

1p_xA_b

A

Area

f_est_comp_alpha_d

Constant1

sqrt(2/rho)

Gain

Divide

Add

Add1

Math

Function

1/4

Constant2

Math

Function1

Math

Function2

rho/2

Gain1

Divide1

f_est_comp_alpha_d

Gain2

f_est_comp_R

e_cr

Gain3

Sqrt1

4/pi

Gain4

p_xp_yAp

A_b

Instantaniousarea

Math

Function3

2viscoisty

Figure A.8: Simulink model of the free lift compensator.

76 A Simulink Models

1A_b

1p_x2p_y3Af_est_com

p_alpha_d

Gain

4p

Product

cos(f_est_comp_theta)/f_com

p_A_spool

Gain1

Add

>Switch

>=

Switch1

f_comp_A

_max

Constant

f_comp_A

_leak

Constant1

Add1

f_comp_A

_max

Constant2

Add2

(f_comp_A

_max-f_com

p_A_leak)/f_com

p_p_range

Gain2

f_comp_p_control

Constant3

Figure A.9: Simulink model of the free lift compensator instantaneous area.

77

1A

1A_b

1s

Integrator

1/f_est_comp_tau

Gain

Add

MinM

ax

eps

Constant

Figure A.10: Simulink model of the free lift compensator area dynamics.

78 A Simulink Models

1dot_x_f

f_cyl_m_c

Constant

f_cyl_A_c

Constant1

Product

1s 2u

xdx

Integrator,Second-O

rderLim

ited

gGain

1m_L

Add1

2Gain1

Add

Add2

4Gain4

Divide

2Q

p_state_initx

dot_x_cQ

p

Cylinderpressuredynam

ics-1

Gain7

-(1/f_cyl_A_c)

Gain8

2p_c

2Gain3

mdot_x_cF

LuGrecylinderfriction

3x_f2Gain2

dot_x_c

Figure A.11: Simulink model of the free lift cylinder and mechanical system.

79

1p

4Q2x 3dot_x_c

Add1

f_cyl_A_c

Gain

Divide2

1sxo

Integrator

f_cyl_A_c

Gain1

Divide1

pEEffectivebulkm

odulus

1p_state_init

f_est_cyl_V_0

Constant

Add

Figure A.12: Simulink model of the free lift cylinder dynamics.

80 A Simulink Models

1E

1pAdd

f_est_cyl_a2/f_cyl_p_max

Gain

f_est_cyl_a3

Constant

Math

Function

f_cyl_E_max*f_est_cyl_a1

Gain1

Figure A.13: Simulink model of the free lift cylinder bulk modulus.

81

1F2

dot_x_cAdd

Divide

1s

Integrator

f_est_mec_sigm

a_0

Gain

Add1

f_est_mec_sigm

a_1

Gain1

1m

dot_x

mg_s

g_s(v)

Product

f_est_mec_sigm

a_2

Constant2

Figure A.14: Simulink model of the free lift cylinder and mechanical fric-tion.

82 A Simulink Models

1g_s

Relay

1dot_x

f_est_mec_F_pr

Constant

Add

Add1

1

Constant1

f_est_mec_K

_brk

Constant2

Add2

1

Constant3

Math

FunctionAbs

-f_est_mec_c_v

Gain1

Product

Product1

2mf_est_m

ec_f_mfr

Gain2

Figure A.15: Simulink model of the free lift friction function gs(v).

83

Pos

Speed

Discretederivative

Add

PID(z)

TR

PID

1-DT(u)

v->i

Add2

Saturation

Add3

Temp

Scalingv

Tempscaling

Product

Temp

Shifti

Shifti

Add4

Z -6

Delay

1

Temp

2

Position

3

Reference

1Current

-1Gain1

Saturation

Figure A.16: Simulink model of the free lift controller.

84 A Simulink Models

Add1

Z -1D

elay11/Ts G

ain

1Pos

1S

peed

0.3297

1-0.6703z -1

DiscreteFilter

Figure A.17: Simulink model of the free lift controller discrete derivative.

85

1Scalingv

1Tem

pAdd

273

Constant

free_i_gain

Gain1

Add2

free_i_const

Constant2

Figure A.18: Simulink model of the free lift controller scaling.

86 A Simulink Models

1Shifti

1Tem

pAdd

20

Constant

free_i_shift

Gain1

Add2

free_i_shift_const

Constant2

Figure A.19: Simulink model of the free lift controller shifting.

BSimulated Free Lift Controller

Figure B.1 shows the simulated velocity of the free lift with 0 kg load, at temper-atures 20 (Figure B.1a) and 60 (Figure B.1b) ◦C. The controller is able to controlthe velocity with small errors.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Free 0kg 20temp velocity

Reference

Simulated velocity

(a) Simulated velocity of the freelift with 0 kg load and 20 ◦C.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Free 0kg 60temp velocity

Reference

Simulated velocity

(b) Simulated velocity of the freelift with 0 kg load and 60 ◦C.

Figure B.1: Simulated velocities at 0 kg, 20 and 60 ◦C.

Figure B.2 shows the simulated velocity of the free lift with 800 kg load, at temper-atures 20 (Figure B.2a) and 60 (Figure B.2b) ◦C. The controller is able to controlthe velocity with small errors. The amplitude of the oscillations is approximately0.02m/s.

87

88 B Simulated Free Lift Controller

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6V

elo

city [

m/s

]Simulation of Free 800kg 20temp velocity

Reference

Simulated velocity

(a) Simulated velocity of the freelift with 800 kg load and 20 ◦C.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Free 800kg 60temp velocity

Reference

Simulated velocity

(b) Simulated velocity of the freelift with 800 kg load and 60 ◦C.

Figure B.2: Simulated velocities at 800 kg, 20 and 60 ◦C.

Figure B.3 shows the simulated velocity of the free lift with 1500 kg load, attemperatures 20 (Figure B.3a) and 60 (Figure B.3b) ◦C. The controller is ableto control the velocity with small errors. The amplitude of the oscillations isapproximately 0.04m/s.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Free 1500kg 20temp velocity

Reference

Simulated velocity

(a) Simulated velocity of the freelift with 1500 kg load and 20 ◦C.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Free 1500kg 60temp velocity

Reference

Simulated velocity

(b) Simulated velocity of the freelift with 1500 kg load and 60 ◦C.

Figure B.3: Simulated velocities at 1500 kg, 20 and 60 ◦C.

CSimulated Main Lift Controller

Figure C.1 shows the simulated velocity of the main lift with 0 kg load, at temper-atures 20 (Figure C.1a) and 60 (Figure C.1b) ◦C. The controller is able to controlthe velocity with small errors except around the time 9 seconds for 20 ◦C wherethe velocity drops to almost zero before recovering.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Main 0kg 20temp velocity

Reference

Simulated velocity

(a) Simulated velocity of themain lift with 0 kg load and 20◦C.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Main 0kg 60temp velocity

Reference

Simulated velocity

(b) Simulated velocity of themain lift with 0 kg load and 60◦C.

Figure C.1: Simulated velocities at 0 kg, 20 and 60 ◦C.

Figure C.2 shows the simulated velocity of the main lift with 800 kg load, attemperatures 20 (Figure C.2a) and 60 (Figure C.2b) ◦C. The controller is able tocontrol the velocity with small errors but has the same behaviour as previouslywhere the velocity around 9 seconds for 20 ◦C drops to zero before recovering.

89

90 C Simulated Main Lift Controller

The amplitude of the oscillations is approximately 0.03m/s.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Main 800kg 20temp velocity

Reference

Simulated velocity

(a) Simulated velocity of themain lift with 800 kg load and 20◦C.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Main 800kg 60temp velocity

Reference

Simulated velocity

(b) Simulated velocity of themain lift with 800 kg load and 60◦C.

Figure C.2: Simulated velocities at 800 kg, 20 and 60 ◦C.

Figure C.3 shows the simulated velocity of the main lift with 1500 kg load, attemperatures 20 (Figure C.3a) and 60 (Figure C.3b) ◦C. The controller is able tocontrol the velocity with small errors but has the same behaviour as previouslywhere the velocity around 9 seconds for 20 ◦C drops to zero before recovering.The amplitude of the oscillations is approximately 0.05m/s.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Main 1500kg 20temp velocity

Reference

Simulated velocity

(a) Simulated velocity of themain lift with 1500 kg load and20 ◦C.

0 5 10 15

Time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ve

locity [

m/s

]

Simulation of Main 1500kg 60temp velocity

Reference

Simulated velocity

(b) Simulated velocity of themain lift with 1500 kg load and60 ◦C.

Figure C.3: Simulated velocities at 1500 kg, 20 and 60 ◦C.

DImplemented Transition Controller

Figure D.1 shows the velocities in the transition phase for both controllers with 0kg load. Figure D.1a and Figure D.1b show the velocities for the currently usedcontroller at temperatures of 20 and 60 ◦C, respectively. Figure D.1c and Fig-ure D.1d show the same signals but for the implemented controller. In this casethe implemented controller is slightly better since it manages to keep a more evenvelocity and avoid the velocity spike occurring with the currently used controller.

91

92 D Implemented Transition Controller

6 6.5 7 7.5 8 8.5 9 9.5 10

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ve

locity [

m/s

]

Measurement of transition of current controller 0kg 20temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(a) Measured velocities of thefree and main lift when using thecurrent controller with 0 kg loadand 20 ◦C.

5 5.5 6 6.5 7 7.5 8 8.5 9

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ve

locity [

m/s

]

Measurement of transition of current controller 0kg 60temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(b) Measured velocities of thefree and main lift when using thecurrent controller with 0 kg loadand 60 ◦C.

14 14.5 15 15.5 16 16.5

time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

ve

locity [

m/s

]

Measurement of transition of thesis controller 0kg 30temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(c) Measured velocities of thefree and main lift when using thethesis controller with 0 kg loadand 20 ◦C.

11 11.5 12 12.5 13 13.5 14

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

ve

locity [

m/s

]

Measurement of transition of thesis controller 0kg 60temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(d) Measured velocities of thefree and main lift when using thethesis controller with 0 kg loadand 60 ◦C.

Figure D.1: Transition phase velocities at 0 kg, 20 and 60 ◦C when usingdifferent controllers. The blue and orange lines are the reference signalswhile the yellow and purple lines are the measured velocities for the freeand main lift respectively. The green line represents the sum of the measuredvelocities which is the total velocity.

Figure D.2 shows the velocities in the transition phase for both controllers with800 kg load. Figure D.2a and Figure D.2b show the velocities for the currentlyused controller at temperatures of 20 and 60 ◦C, respectively. Figure D.2c andFigure D.2d show the same signals but for the implemented controller. Bothcontrollers behave similarly and no controller seems to have better performancethan the other.

93

6 6.5 7 7.5 8 8.5 9 9.5 10

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ve

locity [

m/s

]

Measurement of transition of current controller 800kg 20temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(a) Measured velocities of thefree and main lift when using thecurrent controller with 800 kgload and 20 ◦C.

6 6.5 7 7.5 8 8.5 9 9.5 10

time [s]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

velo

city [m

/s]

Measurement of transition of current controller 800kg 60temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(b) Measured velocities of thefree and main lift when using thecurrent controller with 800 kgload and 60 ◦C.

11 11.5 12 12.5 13 13.5 14

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ve

locity [

m/s

]

Measurement of transition of thesis controller 800kg 26temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(c) Measured velocities of thefree and main lift when using thethesis controller with 800 kg loadand 20 ◦C.

10 10.5 11 11.5 12 12.5 13

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ve

locity [

m/s

]

Measurement of transition of thesis controller 800kg 60temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(d) Measured velocities of thefree and main lift when using thethesis controller with 800 kg loadand 60 ◦C.

Figure D.2: Transition phase velocities at 800 kg, 20 and 60 ◦C when usingdifferent controllers. The blue and orange lines are the reference signalswhile the yellow and purple lines are the measured velocities for the freeand main lift respectively. The green line represents the sum of the measuredvelocities which is the total velocity.

Figure D.3 shows the velocities in the transition phase for both controllers with1500 kg load. Figure D.3a and Figure D.3b show the velocities for the currentlyused controller at temperatures of 20 and 60 ◦C, respectively. Figure D.3c andFigure D.3d show the same signals but for the implemented controller.

94 D Implemented Transition Controller

6 6.5 7 7.5 8 8.5 9 9.5 10

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7ve

locity [

m/s

]

Measurement of transition of current controller 1500kg 20temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(a) Measured velocities of thefree and main lift when using thecurrent controller with 1500 kgload and 20 ◦C.

6 6.5 7 7.5 8 8.5 9 9.5 10

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ve

locity [

m/s

]

Measurement of transition of current controller 1500kg 60temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(b) Measured velocities of thefree and main lift when using thecurrent controller with 1500 kgload and 60 ◦C.

12.5 13 13.5 14 14.5 15

time [s]

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

velo

city [m

/s]

Measurement of transition of thesis controller 1500kg 30temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(c) Measured velocities of thefree and main lift when usingthe thesis controller with 1500 kgload and 20 ◦C.

8 8.5 9 9.5 10 10.5 11

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

velo

city [m

/s]

Measurement of transition of thesis controller 1500kg 60temp

Ref free

Ref main

Velocity free

Velocity main

Velocity total

(d) Measured velocities of thefree and main lift when usingthe thesis controller with 1500 kgload and 60 ◦C.

Figure D.3: Transition phase velocities at 1500 kg, 20 and 60 ◦C when usingdifferent controllers. The blue and orange lines are the reference signalswhile the yellow and purple lines are the measured velocities for the freeand main lift respectively. The green line represents the sum of the measuredvelocities which is the total velocity.

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