Master of Science Thesis in Electrical Engineering
Department of Electrical Engineering, Linköping University, 2017
Modelling and Control of a
Forklift’s Hydraulic Lowering
Function
Modelling and Control of a Forklift’s Hydraulic Lowering Function Ludvig Fri and Daniel Fahlén
LiTH-ISY-EX--17/5050--SE
Supervisor: Greger Bäckström
Jakob Johansson
Company
Du Ho
isy, Linköping university
Examiner: Martin Enqvist
isy, Linköping university
Division of Automatic Control Department of Electrical Engineering
Linköping University SE-581 83 Linköping, Sweden
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 den hydrauliska sänkningsfunktionen hos en specifik gaffeltruck. Dagens lösning an-vänder sig av öppen styrning vilket gör reglerprestandan känslig för störningar och systemförändringar. En störning av extra intresse är temperaturen av hyd-raulvätskan. Målet med detta arbete var därför att designa en regulator med ökad robusthet och prestanda.
För att lösa detta har en modellbaserad metod för regulatordesign använts där en olinjär gray-box modell härleddes, implementerades och validerades. Modellpa-rametrarna skattades genom att ställa upp och lösa ett ickelinjärt minsta-kvadrat optimeringsproblem. Den resulterande modellen fångar det mesta av systemdy-namiken och modellpassningen till uppmätt data var högre än 70% vilket ansågs bra nog för att kunna använda modellen som en bas för regulatordesign.
En PID regulator designades och regulatorparametrarna optimerades med hjälp av modellen. Regulatorn utvärderades i simuleringar och för att sedan implemen-teras den på en riktig gaffeltruck. Den föreslagna regulatorn jämfördes med den ursprungliga regulatorn i flera olika testfall. Resultaten visade ett bättre steady-state beteende och ökad robusthet mot temperaturförändringar för den designa-de regulatorn jämfört med designa-den ursprungliga regulatorn.
Abstract
Material handling and logistics are fundamental parts of today’s global society and forklifts are a crucial part of the material handling process. Making these as efficient 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 is controlled through an open-loop control scheme making the control performance sensitive to disturbances and system changes. One disturbance of special interest is the temperature of the hydraulic fluid. The goal of this thesis was therefore to design a controller with improved robustness as well as improved performance. To solve this a model-based control design approach was used and a nonlinear grey-box model was derived, implemented and validated. The model parameters were estimated using a nonlinear least-squares optimisation problem. The result-ing model captures most of the system dynamics and the model fit is higher than 70% which was deemed good enough to use for control design.
A PID controller was designed based on the estimated model and the controller parameters 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 improved steady state behaviour with enhanced temperature robustness compared to the original controller.
Acknowledgments
We would like to thank our examiner, Martin Enqvist, and our supervisor at Linköping University, Du Ho, for their guidance and interest in our work. We would also like to thank our supervisors at the company, Greger Bäckström and Jakob Johansson, for their invaluable input, knowledge and support.
Linköping, June 2017 Ludvig Fri and Daniel Fahlén
Contents
Notation xiii 1 Introduction 1 1.1 Motivation . . . 1 1.2 Purpose . . . 2 1.3 Limitations . . . 2 1.4 Method . . . 2 1.5 Literature Study . . . 3 1.6 Thesis Outline . . . 4 2 System Overview 5 2.1 Mechanical System . . . 5 2.2 Electronic System . . . 7 2.3 Hydraulic System . . . 7 2.3.1 Valves . . . 8 3 Modelling 9 3.1 Important Properties . . . 9 3.1.1 Viscosity . . . 93.1.2 Empirical Effective Bulk Modulus . . . 10
3.1.3 General Friction Model . . . 10
3.1.4 Position Sensor . . . 11
3.2 Hydraulic Submodels . . . 12
3.2.1 Flow through Valves . . . 13
3.2.2 Proportional Valve . . . 13 3.2.3 Compensator Valve . . . 14 3.2.4 Control Volume . . . 16 3.2.5 Cylinder Pressure . . . 16 3.2.6 Tank . . . 16 3.3 Mechanical Submodels . . . 17
3.3.1 Free Lift Force Balance . . . 17
3.3.2 Main Lift Force Balance . . . 18
3.3.3 Friction . . . 19
3.4 Model Structure Summary . . . 20 3.5 Data Collection . . . 21 3.6 Parameter Estimation . . . 23 3.6.1 Problem Setup . . . 24 3.7 Model Validation . . . 25 3.7.1 Validation Criteria . . . 25 4 Control 27 4.1 System Overview . . . 27 4.2 Control Strategies . . . 28
4.3 Nonlinear Compensation via Look-Up Table . . . 29
4.4 PID Control . . . 30
4.4.1 Choice of PID Parameters . . . 30
4.4.2 Anti-Windup Method . . . 30 4.4.3 Feed-Forward Control . . . 31 4.5 System Characteristics . . . 31 4.5.1 Temperature Dependency . . . 31 4.5.2 System Delays . . . 32 4.6 Reference Signal . . . 32
4.6.1 Reference Signal for Transition . . . 32
4.7 Summarised Control Structure . . . 33
4.8 Implementation . . . 33
4.8.1 Discretization . . . 33
4.8.2 Quantization . . . 34
4.8.3 Fixed Point Precision . . . 34
4.9 Stability . . . 35 5 Results 37 5.1 Data Collection . . . 37 5.1.1 Temperature Dependency . . . 38 5.1.2 Load Dependency . . . 39 5.1.3 Measurement Noise . . . 40 5.2 Parameter Estimation . . . 41
5.2.1 Free Lift Parameters . . . 42
5.2.2 Main Lift Parameters . . . 43
5.2.3 Control Parameters . . . 44
5.3 Model Validation . . . 45
5.3.1 Free Lift Validation . . . 46
5.3.2 Main Lift Validation . . . 48
5.4 Control Validation . . . 49
5.4.1 Simulated Free Lift Controller . . . 49
5.4.2 Simulated Main Lift Controller . . . 50
5.5 Stability . . . 51
5.6 Implementation and Final Testing . . . 51
5.6.1 Implemented Free Lift Controller . . . 51
Contents xi 5.6.3 Implemented Transition . . . 58 6 Conclusion 61 6.1 Results . . . 61 6.2 Method Evaluation . . . 62 6.3 Future Work . . . 63 6.4 Final Conclusion . . . 63 A Simulink Models 67
B Simulated Free Lift Controller 87
C Simulated Main Lift Controller 89
D Implemented Transition Controller 91
Notation
Abbreviations
Abbreviation Meaning
ecu Electronic control unit
pwm Pulse width modulation
pid Proportional, integral, differential (controller)
nmse Normalised mean-square-error
nrmse Normalised root-mean-square-error
prms Pseudo-random multilevel signal
lq Linear quadratic (controller)
mpc Model predictive control (controller)
1
Introduction
This master thesis concerns modelling and control of the lowering function of a specific forklift. In many technical industries using a model-based approach might be advantageous. When using models, it is possible to simulate the system which 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 which experiences 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 single cylinder 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”. The lowering of the forks is controlled via a single valve coupled with a compensator valve for each of the free lift and the main lift. The dynamics of the system are dependent on the temperature of the oil. Warmer oil has lower viscosity which leads to an increase in flow which in turn will lead to an increase in speed, for the 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 that does not consider the influence of the temperature. It is tuned to give the max-imum lowering speed when the temperature is close to its maxmax-imum allowed value to not exceed the speed limitation. This will cause the lowering speed to be slower when the oil temperature is cooler.
It is desirable to always have a lowering speed as close to the speed limitation
as possible to increase the performance of the forklift. A higher lowering speed means that the forklift can operate more efficiently, which has a direct impact on the 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 the speed cannot be accurately controlled, this leads to a nonsmooth transition when the state (load and temperature) of the system is not the one it has specifically been tuned for. It is desirable to have a smooth transition regardless of operating condition to avoid oscillations in the transition step as well as avoiding an abrupt mechanical halt of the main lift.
1.2
Purpose
The purpose of this thesis is to create a model of the lowering function of the system and investigate if it is possible, with this model, to recreate the current problems with the lowering function mentioned in Section 1.1. Furthermore, a controller with the purpose of controlling the lowering speed is to be designed based on the estimated model. A stability analysis of the controller should be done to ensure stability of the complete system. The control performance should be tested and verified on a forklift.
The requirement of the controller is to keep the lowering speed of the forks close to, without exceeding, 0.6 m/s in steady state. As mentioned in Section 1.1, the temperature affects the lowering speed, therefore it is crucial that the controller is robust to temperature changes. The controller should also be able to follow a changing reference accurately and without oscillations, to ensure smooth starts and 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 on different 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 are performed in steps. The steps are, in chronological order; data collection, mod-elling, model validation, control, and implementation on the forklift electronic control 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 from the previous iteration are used to discern in which step alterations are needed to improve the final result.
1.5
Literature Study
In [15], a general model of a proportional solenoid valve is constructed from the laws of physics and the model is also validated. This provides a starting point when 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 hydraulic control valve. This might be of interest when the nonlinearities in the valve are modelled using the laws of physics. The dynamics of the proportional valve can successfully be modelled as a second order system [12, 22, 23].
An alternative approach is shown in [16] where black-box modelling is used to model the valve. In this approach system identification is done to find a model which corresponds well with the data. However, in this paper the investigated valve 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 also presents a Simulink model of this hysteresis.
The focus of the thesis lies in analysis and design of hydraulic control systems which is also the focus of [12]. It covers (among others) first principles modelling and empirical modelling as well as hydraulic control system design. The work of [12] is therefore of interest at all stages in the thesis and in particular when modelling 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 the same model or control design as the one proposed in this thesis. In [2], modelling and control of an electro-hydraulic system with a different structure is examined. In [7, 11, 13] and [14] modelling and simulation of similar hydraulic systems are examined. Modelling and control of the lift function of a similar forklift is studied in [4].
The modelling of the nonlinearities of the proportional valve is one of the crucial points of this thesis. In [15], a proportional solenoid valve is studied which is of the same type used in this thesis. An analysis of the dynamics and nonlinearities of a hydraulic control valve is done in [23]. In [16] the method of black-box mod-elling is used for modmod-elling of a hydraulic directional control valve. A hysteresis model 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 used can be found in [3], particularly the trusted-region method which can be used to solve large-scale bound-constrained minimisation problems.
de-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 be the most widely utilised among the proposed models. This paper concludes that a steady-state model is not appropriate to use when modelling friction in hydraulic cylinders. It is also shown that the nominal LuGre friction model gives good re-sults and even better rere-sults can be achieved with a modified LuGre model. How-ever, the modified LuGre model includes more parameters that are difficult to identify.
1.6
Thesis Outline
An overview of the system is given in Chapter 2, consisting of the mechanical system, the electronic system and the hydraulic system.
Chapter 3 describes how the model was derived and validated. Important aspects of the system are presented along with models for the system components. The parameter estimation and model validation are explained.
The control design is explained in Chapter 4. Different parts of the controller along 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, measuremeasure-ments from both the implemented 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 results of the simulated free and main lift controllers, respectively. Appendix D contains the results from the implemented transition controller.
2
System 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. The mast of the forklift consists of a base attached to the chassis of the forklift and a first stage and a second stage that can slide up and down. Mounted on the second stage there is a carriage assembly which has the forks attached to it. The assembly is divided into two parts, the main lift and the free lift. The main lift is defined as the parts of the mast controlled by the left cylinder in Figure 2.1 and consists of the left hydraulic cylinder attached to the base, the first stage that is connected to the left piston rod and the second stage that is connected the first stage through a chain. The free lift is defined as the parts of the mast controlled by the right cylinder in Figure 2.1 and consists of the right hydraulic cylinder that is attached to the second stage and the forks and carriage that is connected to the right piston rod through a chain.
xML xFL xG Base First stage Second stage Forks Chains
Figure 2.1:Sketch of the mast. The left cylinder controls the main lift which
is connected to the base and first stage of the mast and the second stage of the mast trough a chain. The right cylinder controls the free lift which is connected to the forks trough a chain.
There is one position sensor measuring the distance between the base and the
first stage of the mast, called the main lift position, xML, and one position sensor
measuring the distance between the forks and the bottom of the second stage of
the mast, called the free lift position, xFL. The distance between the the bottom of
the first and second stage is not measured but is given indirectly by the relation created by the chains that connects them. The the distance between the bottom of 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 ECU controls the valves with a pulse width modulation (PWM) signal. The control signal to a valve is represented in the ECU as a current. The ECU controls the PWM signal to achieve the correct output current. Hence in the sequel the control signal 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, calculating the 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 not used for control.
PWM ECU PWM Flow Valves Flow Oil temperature Free lift pressure Main lift pressure Free lift position Main lift position Hydraulic and mechanic system 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 and mechanic 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 system for the free and main lift.
MAIN LIFT FREE LIFT T H1 H2 H3 M1 M2 Q1 Q2 C1 C2
Figure 2.3: Overview of the lowering function of the hydraulic system. Q1
and Q2 are the proportional valves where Q1 controls the main lift and Q2 controls 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. H1 and H2 are the cylinders for the main lift and H3 is the cylinder for the free lift.
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 both the free and main lift are moving and lastly only the free lift is moving until the forks 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 temperature of the hydraulic fluid have significant impact. The temperature of the hydraulic fluid affects the flow due to the temperature dependent viscosity. As previously mentioned in Section 1.1, an increase in temperature leads to lower viscosity which 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 will decrease only due to heat dissipation from the oil to the environment since there is no active cooling of the oil.
3
Modelling
The goal of the modelling is to create a model of the lowering function of the forklift that reproduces the current problems with this function mentioned in Section 1.1. This model can then be used to study the system to gain further understanding of it as well as aid in troubleshooting the system. Furthermore, a model must be available to use model-based control designs.
The model can be divided into two parts, the free and the main lift. The model for each of them was developed individually without regard for the other. To do this the measurements of the free lift model were taken only when the free lift was moving and the main lift was stationary. For the main lift the opposite setup was used. Any dynamic effects coming from the other part of the lift were neglected.
3.1
Important Properties
Important properties that must be considered when modelling the hydraulic and mechanical 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, η0is the dynamic viscosity at reference
tem-perature T0and λ is the viscosity-temperature coefficient [12].
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, which also includes the effects of entrained air and mechanical compliance, is
E(p) = a1Emaxlog (a2 p
pmax
+ a3) (3.2)
where E(p) is the pressure dependent bulk modulus, Emaxis the maximum bulk
modulus, p is the pressure, pmax is the maximum pressure and a1, a2and a3are
parameters 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. Two important aspects of friction are the pre-sliding behaviour and steady state fric-tion.
One common steady state friction model, among others, that incorporates Coulomb friction, 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 static
friction force, vst is the Stribeck velocity, n is an exponent that affects the slope
of the Stribeck curve, σ2 is the viscous friction coefficient and v is the relative
velocity 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 not always yield zero friction force. One model of the pre-sliding behaviour that incorporates 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 on the bristle model shown in Figure 3.1. The bristles on one surface are modelled as 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 of
the bristle. The bristles on the bottom surface is rigid while the bristles on the top is elastic.
The LuGre model is given by dz dt = v − σ0z gs(v) |v| (3.4) Ff r= σ0z + σ1 dz dt + σ2v (3.5)
where z is the mean deflection of the elastic bristles, σ0 is the stiffness of the
elastic 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 the
friction 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 rotary encoder. It consists of a wire wound around a rotating disc with light shining through evenly spaced slits. When the wire is pulled up and down this will cause the disc to rotate in different directions. As the disc rotates a photodiode will detect the light pulses. Since the diameter and the number of slits of the encoder are known, it is possible to calculate the distance the wire has travelled between each pulse. Each time a pulse occurs this distance is incremented. By placing two photodiodes in such a way that a 90 degrees phase shift occurs it is possible to determine which way the disc rotates. Thus, the sensor sends out two
phase-shifted pulses and these are used to calculate both the distance travelled and whether the wire is travelling up or down.
The signals of interest are the velocity and position of the forks relative to the ground. This is not what is measured by the position sensors as can be seen in
Figure 2.1. The main lift position, xML, the free lift position, xFL, and the position
of 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 are connected with chains.
The sensor used in this thesis has a diameter of 80 mm and 128 slits. This means
the accuracy of the position for the free lift, xFL, is 2 mm. Thus the accuracy for
the position relative ground, xG, is 2 mm when using the free lift. The accuracy
of the main lift position sensor, xML, is also 2 mm but since xG = 2xML+ xFLthe
accuracy for the position relative ground, xG, will be 4 mm when using the main
lift.
3.2
Hydraulic Submodels
The hydraulic system was divided into submodels that are presented here. The structures 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 whereas only one cylinder is used in the free lift as can be seen in Figure 2.3. The two cylinders in the main lift can, in the model, be considered as one cylinder with the combined area of the two. Because of this the same equations can be used to model both the free and the main lift.
The components that are considered are the cylinder, the proportional valve, the compensator 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 the entire hydraulic system can be seen in Figure 2.3 and for clarity a more detailed figure of the proportional valve coupled with the compensator valve can be seen
in Figure 3.2. The pressure pA is the pressure before the proportional valve, pB
is the pressure after the proportional valve, pC is the pressure before the
com-pensator valve, pD is the pressure after the compensator valve, pxand pyare the
pressure at the compensator control terminals. The flow qAis the flow through
the proportional valve and qB is the flow through the compensator valve. The
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 components
are 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 between the proportional and compensator valve. The pressure at different points in
the figure are represented with pi where i indicates the point. qAis the flow
through the proportional valve and qBis the flow through the compensator
valve. The dotted lines represent the connection between the compensator valve 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 r 2 ρ p (p2+ p2 cr) 1 4 (3.7)
where Q is the flow through the valve, CD is the flow discharge coefficient, A is
the opening area of the valve, ρ is the density of the hydraulic fluid and p is the
pressure differential over the valve. The parameter pcr is the minimum pressure
for turbulent flow which can be written as
pcr = ρ 2( Recrη CDDH )2 (3.8) DH = r 4A π (3.9)
where Recris 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 dynamics and the other the static behaviour.
Dynamics of the Proportional Valve
The dynamics of a proportional valve has been successfully modelled as a second order system with hysteresis [12, 22, 23]. A second order system with hysteresis
can be written as
¨
x = ω2(u −2D
ω ˙x − x − fhssign( ˙x)) (3.10)
where u is the input current to the system, x is the output current, ω is the natural
frequency, D is the damping ratio and fhsis a constant describing the hysteresis.
The hysteresis is largely caused by friction effects in the valve [12]. Modelling the
hysteresis as fhssign( ˙x) can cause numerical problems when simulating because
the function acts a discontinuity when ˙x changes sign. To solve this a hysteresis model inspired by a simplified LuGre friction model was used. The hysteresis can be described by dz dt = ˙x − σ0z fhs |˙x| (3.11) Fhs= σ0z + σ1 dz dt. (3.12)
The dynamics of the valve can then be written as ¨
x = ω2(u −2D
ω ˙x − x − Fhs) (3.13)
where Fhs is guaranteed to be a continuous function. Written in this way the
model still incorporates a hysteresis but is not affected by numerical issues to the same 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) is used for the viscosity.
Combining these equations gives
qA= CDA(x) r 2 ρ pA−pB ((pA−pB)2+ (ρ2( Recr(η0e−λ(T −T0)) CD q 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 relation between the current, x, and the valve opening area A(x). The output is the flow
qA.
3.2.3
Compensator Valve
The compensator valve is divided into three parts, one part describing the static area, 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≤pset
Amax−k(pxy−pset) pset< pxy+ pf low< pmax
Aleak pxy+ pf low≥pmax
(3.15)
where pxy = px−py is the pressure differential over the control terminals of the
pressure compensator, As(pxy) is the pressure dependent compensator opening
area in steady state, Amax is the maximum area, pset is the preset pressure, Aleak
is the minimum area, pmaxis the pressure needed to fully close the valve and k is
defined as
k = Amax−Aleak
preg
where pregis the pressure regulation range [18].
The pressure pf lowis a pressure acting on the compensator originating from the
flow force that occurs when a flow is passing through the valve. The flow force was added to the model at suggestion from the valve manufacturer. The pressure originating from the flow force acting on the compensator spool can be written as
pf low=
(pC−pD)ACDcos θ
Aspool
(3.16)
where pD is the pressure after the compensator valve, pC is the pressure before
the compensator valve, A is the valve opening area, CDis 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 dA
dt =
As(pC−pD) − A
τ (3.17)
which is a first order system where As(pC−pD) is the pressure dependent opening
area in steady state, A is the actual opening area and τ is the time constant for the first order system [18].
Flow through the Compensator Valve
The flow through the compensator valve is modelled using the same equations as for the proportional valve, that is (3.7), (3.8) and (3.9) where (3.1) is used for the viscosity. The valve opening area in steady state is modelled according to (3.15).
Combining these equations gives qB= CDA r 2 ρ (pC−pD) ((pC−pD)2+ (ρ2( Recr(η0e−λ(T −T0)) CD q 4A π )2)2)14 (3.18)
where the inputs are the pressure before the compensator valve, pC, the pressure
after the compensator valve, pD, and the temperature T . The output is the flow
qB.
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 volume can be written as
˙pV = E
V∆Q (3.19)
where pV is the pressure in the volume, E is the bulk modulus, V is the volume
and ∆Q = qA−qBis the flow difference in and out of the volume. The pressure
pV in the volume is approximated as homogeneous meaning pV = pB= pC = px.
The bulk modulus was approximated as a constant since the effect of pressure dependence 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 the
proportional valve, pA. The pressure dynamics in the cylinder can be modelled
as ˙ pA= E(pA) V (xcyl) (Q − A ˙x) (3.20)
where xcyl is the position of the cylinder, E(pA) is the pressure dependent bulk
modulus 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 the
cylinder 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 increase in the tank it is negligible compared to the pressure in the cylinder. This means
3.3 Mechanical Submodels 17
3.3
Mechanical Submodels
The model used for the mechanical systems of the free and main lift are presented here.
3.3.1
Free Lift Force Balance
By using Newton’s second law on both the cylinder and the forks, expressions for their 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 F F Ffr,c mcg
(a)Forces acting on the cylinder.
(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 can be written as
mcx¨c= pA − mcg − Ff r,c−2F (3.22)
where mcis the mass of the cylinder, ¨xcis the acceleration of the cylinder, pA is
the force arising from the pressure in the cylinder, mcg is the gravitational force,
Ff r,cis the cylinder friction force and F is the force from the chain connected to
the forks.
Using Newton’s second law on the forks and solving for F gives that F can be written as
(mf + mL) ¨xf = F − (mf + mL)g − Ff r,f (3.23)
where mLis the mass of the load on the forks and mf is the mass of the forks, ¨xf
is the acceleration of the forks, (mf + mL)g is the gravitational force and Ff r,f is
the friction force between the forks and the stand.
The relation between the acceleration of the forks and the cylinder can be written
as ¨xf = 2 ¨xc. This means that (3.22) and (3.23) can be combined to
mcx¨c= pA − mcg − Ff r,c−2((mf + mL)2 ¨xc+ (mf + mL)g + Ff r,f) (3.25) ⇔x¨c= 1 mc+ 4(mf + mL) (pA − (mc+ 2(mf + mL))g − Ff r,c−2Ff r,f) (3.26) ⇔x¨c= 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 not move relative to the second stage, then the mass of those components can be added 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 on the first stage (Figure 3.4a) and the second stage (Figure 3.4b) is shown.
pA xfs, xfs F F Ffr,c mcg mfsg Ffr,fs
(a)Forces acting on the first stage.
(mss+mL)g
Ffr,ss
F
xss, xss
(b)Forces acting on the second stage.
3.3 Mechanical Submodels 19
Using Newton’s second law on the first stage the mass times the acceleration can be 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 is
the acceleration of the first stage, pA is the force arising from the pressure in the
cylinder, (mc+ mf s)g is the gravitational force, Ff r,f s is the first stage friction
force 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 can be 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 mssis the mass of the second stage, mLis the mass of the load on the forks,
¨
xss is the acceleration of the second stage, (mss+ mL)g is the gravitational force
and Ff r,ssis the friction force of the second stage.
The relation between the acceleration of the second stage and the first stage can
be written as ¨xss= 2 ¨xf 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)2 ¨xf s+ (mss+ mL)g + Ff r,ss) (3.31) ⇔x¨f 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) ⇔x¨f 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. To
model 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 Fpris the preload force, mLis the load, fcf ris the Coulomb friction
coeffi-cient, Kbrkis the breakaway friction force increase coefficient and cvis the
transi-tion coefficient. This choice of gs(v) was inspired by [19] in which a steady state
friction model is described for a cylinder. Some modifications were made in an effort to describe all frictions in the system, not just the friction in the cylinder.
Most noticeably the friction has been made load-dependent to reflect observed system behaviour.
The load dependency is shown in Figure 3.5. The pressure in the cylinder and the velocity were measured for the free lift in steady state velocity at several different velocities. From this the combined friction force of the cylinder and the forks was
calculated using (3.25) where ¨xcwas 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 force [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 different
loads on the forks.
3.4
Model Structure Summary
To make it more clear how all equations were incorporated into the model, a block diagram of the model structure is shown in Figure 3.6. The model structure, as previously mentioned, is the same for both the free and the main lift. The inputs
to the model are the load mL, the current u and the temperature T . The equations
are incorporated in each block in the figure.
The proportional valve consists of the proportional valve dynamics block and the proportional valve flow block which incorporate (3.13) and (3.14), respectively. The compensator valve consists of the compensator area block, the compensator area 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 pressure of 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 if it is the free or main lift model.
3.5 Data Collection 21 Compensator valve Proportional valve u x Proportional valve dynamics p_A p_B x T q_A Proportional valve flow q_A q_B p Volume p_C p_D p_x p_y A_s Compensator area A_s A Compensator area dynamics p_C p_D A T q_B Compensator flow p_d Tank x_c dot_c q_A p_A Cylinder pressure p_A m_L x_c dot_x_c Mecanical system 2 T 1 u 3 m_L
Figure 3.6: Block diagram of the model structure. Each block incorporates
different equations from Chapter 3.
The model was implemented and simulated in Simulink, the complete Simulink models 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. To do 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 full list of the parameters that were estimated and the result of the estimation will be described later in Section 5.2.
3.5
Data Collection
There are several signals of interest that can be monitored: the valve current control signal, the positions of the free lift and the main lift, the pressures in the hydraulic system of the free lift and the main lift, and the temperature of the oil in the tank. In addition to this, extra sensors were added to collect more data. 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 free
and main lift cylinders that are more accurate than the internal ones. The sample frequency of the system was chosen to 50 Hz.
The measurements for the free and main lift were taken separately. When the measurements for the free lift were taken the main lift was not moving and vice versa. There is still a small cross coupling between the free and main lift but this was assumed negligible.
Two types of signals were used for data collection. The first type of signal was a pseudo-random multilevel signal (PRMS) chosen to excite the dynamic behaviour of the system. Another approach is to use a binary signal with random switching probability, 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 is shown in the time domain. The valve current is quantized to 256 steps where
0 represents 0 current and 255 represents max current (1,75 A). Note that the amplitude of the signal lies between 58 - 174 and not 0 - 255. This is because the valve has lower and upper saturation limits and it is desirable to lie in the working 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 corresponds to 20 seconds since the sampling frequency is 50 Hz. The time it takes to lower the forks down to its lowest state from the top position is shorter than 20 seconds with this kind of signal and therefore the signal time will not be a problem.
The input signal to the valve needs to be pass filtered since without the low-pass filter the valve will close too quickly and this will cause a large spike in pressure which might damage the hardware.
In Figure 3.7b the power spectral density of the pseudo-random multilevel signal is shown. The switching probability determines in which frequencies most of the power of the signal exists. With a switching probability of 0.05, as in the figure, most of the energy lies at the lower frequencies. This was chosen because the interesting phenomena occur at low frequencies. The filtered signal is also shown in Figure 3.7b. Note that there is almost no difference between the power spectral densities of the unfiltered and filtered signals. This is because the signal switching probability is chosen to excite the most interesting (low) frequencies.
Two realisations of PRMS signals were created, called PRMS1 and PRMS2, in accordance with the above stated principles. They were used as input to both the free and the main lift at different operating points. The operating points consisted of loads on the forks of 0 kg, 800 kg and 1500 kg at oil temperatures in
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 Amplitude
Pseudo-random multilevel signal
(a) Pseudo-random multilevel signal with switching probability 0.05 (blue) and the lowpass-filtered input signal to the valve (red). 0 5 10 15 20 25 Frequency [Hz] 0 20 40 60 80 100 120 140 160 180 200 PSD
(b) Power spectral density of a pseudo-random multilevel signal with 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 several evenly spaced levels. After a step was taken the signal was constant for some time 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 free lift with loads of 0 kg and 1000 kg and it was done for the main lift with loads of 0 kg, 800 kg, 1500 kg. The temperature was varying somewhat between different
measurements but was lying in the range of 20 to 40◦C. There was no particular
reason 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 many unknown parameters that need to be estimated from measurements. To do this the tool Parameter Estimation GUI built into Simulink was used. The tool formu-lates a nonlinear least squares optimisation problem that minimises the sum of the squared difference of the simulated and measured data.
A nonlinear least squares problem can be formulated as
V (x) = 1 2 N X k=1 Tk(x)k(x) (3.35)
where V (x) is a cost function, x are the design variables and
where ykare the observed values at sample k and hk(x) are the estimated values at sample k.
The goal of the optimisation is to minimise the cost function by changing the design variables, which is usually done through an iterative optimisation method where the design variables are estimated and updated according to
ˆ
x(i+1)= ˆx(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 length and search direction [17]. The trust-region algorithm approximates the cost func-tion with a simpler funcfunc-tion, usually the first terms of a Taylor expansion of the cost function, which closely resembles the cost function in a region around the design 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 boundary of the region without finding a point where the cost function is smaller, the trust region 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 initial guess. Depending on this the method might converge to completely different results.
3.6.1
Problem Setup
There are many unknown parameters and it is difficult to set up a parameter estimation problem for all parameters, using all available inputs, and get it to converge to reasonable values. For this reason the parameter estimation of the model was divided into two parts where a smaller set of parameters could be estimated.
Steady State
In the first part the steady state behaviour was of prime interest. The parts of the model relating to the cylinder pressure dynamics and the mechanical system were discarded and replaced with the static relationship
Q = 1
2Ac˙xG (3.38)
where Q is the flow through the proportional valve, Acis the cylinder area and
˙xGis the velocity of the forks relative to the ground. This is true for both the free
and 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 steady
state measurements. This optimisation gave parameter values of the parameters that 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 mostly regarded as known from the previous steady state estimation.
The remaining parameters were estimated where the inputs to the system were the current to the proportional valve, the temperature and the load on the forks. The input and output were taken from the PRMS1 measurements mentioned in Section 3.5.
Some parameters that relate to the temperature dependency of the steady state characteristics were estimated again during the second round of estimation as the PRMS1 data was regarded to excite the temperature dependency better since the measurements were taken in a wider range of temperatures.
After these optimisations were done the model was considered complete and the results 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 know how the model performs. The validation indicates where the model captures the system dynamic well by using the validation criteria below. The model validation also 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 a measure of the model fit in percentage which is defined as
Model fit = 100 · (1 −||v − ˆv||
2
||v − ¯v||2) (3.39)
where || . || indicates the 2-norm, v is the measured velocity, ˆv is the estimated
velocity 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 − ˆv||
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
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 the squaring of each term.
4
Control
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 in load and temperature and should preferably not introduce too much oscillations in 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 derivative filter. The derivative of the position sensor values are filtered with a low-pass filter and used as the estimated velocity. The output of the controller is the valve current which the ECU translates into a PWM signal.
Valve current PWM ECU PWM Flow Valves Flow Oil temperature Free lift pressure Main lift pressure Free lift position Main lift position Hydraulic and mechanic system Position sensor values Estimated velocity Discrete derivative Display Estimated velocity Reference signal Valve current Controller 1 Reference
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. Which control strategy to use depends on, amongst others, the computational power available, 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. For a relatively simple system a PID controller will often suffice whereas for more complex systems a more advanced approach might be necessary.
One interesting choice of controller which is slightly more complex than a PID is the LQ (linear quadratic) controller, which is a linear controller defined using a quadratic criterion. However, in this thesis the system is nonlinear which means that if a LQ controller is used the system might need to be linearized around an operating point. Another interesting option is an MPC (model predictive control) controller. However, since MPC solves an optimisation problem in every time instance it requires more computational power than the other options. Since the computational power and available memory in this system are limited, the implicit 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 in a different operating region where they provide satisfactory results. Scheduling variables 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 input and the output, linear control strategies can be used.
In this thesis the hardware and the authors’ time are the main limiting factors of the controller. Because of these limitations an attempt was made to keep the controller as simple as possible. This resulted in a control structure using a PID with 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 system can be described as
y = Gu0 (4.1)
u0 = f (u) (4.2)
where G is a linear system, u is the input and f is a static non linearity. If the
output from a controller is transformed as u = f−1(v), where v is the output from
a linear control algorithm, then y = Gv which means the output y is linearized with respect to the input v [6].
The system in this thesis is nonlinear which will cause a linear controller to have decreased performance compared to when controlling a linear system. One of the main nonlinearities is the input current to the flow through the proportional valve which is described in (3.14).
Ignoring the dynamics of the proportional valve and also ignoring the nonlinear properties of the mechanical system, the system can be written as above where f is the static nonlinearity from current to flow in the proportional valve, u is
current, u0is flow and G is the mechanical system.
Calculating f−1 will require the inverse to (3.14) which can be quite difficult to
calculate. One way to create an approximate inverse of this relationship is to take steady state measurements of the lowering speed for different amplitudes of the input current. These measurements can be used to create a static look-up table from current to velocity. Instead of flow, the velocity of the lift was used. As the conversion from flow to velocity has a constant relation in steady state, the velocity can be used instead without affecting the linerization. Inverting this look-up table gives a nonlinear look-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. It is mainly the pressure over the valve and the oil temperature that will vary and
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 state for different operating points. The temperature is not mechanically compensated for, therefore a look-up table measured at one operating temperature will not be valid 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 based on proportional (P), integral (I) and derivative (D) terms. The input to the PID control 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 Z 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 to minimise a criterion [8]. The criterion that was used here is
V (x) = 1 2 N X k=1 ekT(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 measured
speed and x is a vector containing the PID parameters.
The optimisation of the parameters for the PID controller is based on the same principles 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 results in a limit of the flow through the proportional valve. This means that the control signal can be saturated. In turn, this might lead to an undesirable windup of the integral part of a PID controller.
There exist several anti-windup methods, but in this thesis adjustment of the integral 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+ Ts Tt (uk−vk) (4.6) where uk = umax if vk > umax vk if umin≤vk ≤umax umin if vk < umin (4.7)
and where Ik is the state of the integral part, Ts is the sampling time, uk is the
controller 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 controller is used in combination with the look-up table mentioned in Section 4.3. The feed-forward gain is 1 which creates a nonlinear neutral feed-forward since the look-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 temperature dependency 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 not valid when the temperature differs from the operating point temperature in the table. To approximate the temperature dependence of the look-up table, the ta-ble input was scaled and the output shifted linearly with the temperature. The scaling of the input was done according to
u0= (T a + b)u (4.8)
where u is the output from the controller, u0 is the scaled input to the look-up
The shift of the output from the look-up table was done according to
i0 = (T c + d) + i (4.9)
where i is the output from the look-up table, i0 is the shifted output, T is the oil
temperature, and c and d are tuning parameters. To select the tuning parameters an optimisation problem was designed in the same way as previously mentioned in Subsection 4.4.1.
4.5.2
System Delays
The system contains various delay sources which is summarised to a total delay of around 160 ms from input current to output speed. This delay causes problems with the controller since its input is the error between the reference input and the estimated velocity. Because of the delay the estimated velocity will always lag behind 160 ms compared to the reference signal. This will cause the integral part of the controller to rise in 160 ms before the estimated velocity has caught up. The increased integral part will in turn lead to an overshoot. To avoid this the reference model was chosen to be a pure delay of 160 ms.
The system delay creates another problem. The oscillations in the system have a period of around 400 ms. The controller will try to suppress the oscillations but because 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 this problem a predictive model strategy would be needed. The solution used in this thesis is to let the feed-forward part of the controller control the fast dynamics and 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 as the operators. It is based on a ramp signal which is low-pass filtered to smoothen the 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 present for a few forklift generations. Changing the reference signal changes the forklift behaviour and the operators are conservative when it comes to changes in system behaviour.
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 by choosing a reference signal for each of the controllers. One example of such a strategy 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 ,MLis the reference
signal for the main lift. R is a ratio ∈ [0, 1] which is dependent on xGand during
the transition phase goes from zero to one.
4.7
Summarised Control Structure
Figure 4.2 shows the summarised control structure. This structure contains all parts mentioned above.
Figure 4.2: Summarised control structure containing all previously
men-tioned parts. The trajectory generator creates the reference signal from user
input. Gmis 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 algorithm needs 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 method Forward Euler. This results in
U (z) = (P + I · Ts 1 z − 1+ D · 1 Ts z − 1 z )E(z) (4.12)
