Pressure Control of a Pneumatic ActuatorUsing On/Off Solenoid Valves

MAISAM JEDDI TEHRANI

Masters’ Degree ProjectStockholm, Sweden May 2008

XR-EE-RT 2008:013

Abstract

Nowadays a very important aspect in heavy duty vehicles is the braking sys-tem. The braking system can be divided into EBS brakes, exhaust brake andretarder, where the latter is of interest in the present Master’s Thesis. Thisthesis presents an investigation whether it is possible to substitute today’sconcept, i.e. controlling the air pressure to the retarder using a proportional-valve, with two so-called on/off-valves and a pressure sensor, which will re-duce expenses and contingently hysteresis phenomena seen in the currentsystem. A non-linear model of the Electronic Control Unit (ECU) electricaldrives, and the electrical, magnetic, mechanical, and pneumatic parts of thevalves, is designed. A Proportional-Integral-Derivative (PID)-controller isdesigned based on the derived model. Two different pulsing schemes havebeen investigated. However, just one of the approaches together with theresults from the other one is presented in this thesis. In order to improvethe control performance non-linear control and prediction methods are usedso that required time response and robustness is achieved. Finally the mod-elled current and pressure are validated against the measured data, and averification of the controller is done on the prototypes.

Acknowledgments

I would like to express my deep and sincere gratitude to my supervisor,Tomas Selling. His wide knowledge and his logical way of thinking have beenof great value for me. His understanding, encouraging and personal guidancehave provided a good basis for the present thesis.

I am also grateful to my supervisor, Professor Hakan Hjalmarsson, at theDepartment of Electrical Engineering, Royal Institute of Technology, for hisdetailed and constructive comments, and for his continuous support through-out this work.

I would also like to express my sincere thanks to Mr. Richard Riis andMr. Soren Aberg for their assistance with the prototypes used for tests andexperiments.

Finally, I wish to express my appreciation to the rest of the members ofNEST at Scania, and friends and family that have supported me in doingthis thesis.

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Contents

1 Introduction 111.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Functional Description . . . . . . . . . . . . . . . . . . . . . . 121.4 Actuation Requirements . . . . . . . . . . . . . . . . . . . . . 131.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Retarder 172.1 Scania’s Retarder . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Retarder system today . . . . . . . . . . . . . . . . . . 172.1.2 Retarder System Using On/Off Solenoid Valves . . . . 18

2.2 Dead Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Solenoid Valves . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Equipments In Experiments . . . . . . . . . . . . . . . . . . . 21

2.4.1 ECU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.2 Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Retarder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Pressure sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 242.7 Software Program . . . . . . . . . . . . . . . . . . . . . . . . . 242.8 Oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.9 Multimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Modelling 253.1 System description . . . . . . . . . . . . . . . . . . . . . . . . 263.2 ECU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 The Electrical Subsystem . . . . . . . . . . . . . . . . 283.3.2 The Magnetic Subsystem . . . . . . . . . . . . . . . . . 313.3.3 The Mechanical Subsystem . . . . . . . . . . . . . . . 323.3.4 The Pneumatic Subsystem . . . . . . . . . . . . . . . . 33

3.4 Regulating valve . . . . . . . . . . . . . . . . . . . . . . . . . 36

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3.5 Summary of state-space equations . . . . . . . . . . . . . . . . 37

4 Model validation 394.1 Experiments on prototype one . . . . . . . . . . . . . . . . . . 39

4.1.1 Duty Cycle Limits . . . . . . . . . . . . . . . . . . . . 394.1.2 Filling characteristic . . . . . . . . . . . . . . . . . . . 414.1.3 Ventilation characteristic . . . . . . . . . . . . . . . . . 43

4.2 Experiments on Prototype 2 . . . . . . . . . . . . . . . . . . . 444.2.1 Filling characteristic . . . . . . . . . . . . . . . . . . . 444.2.2 Ventilation characteristic . . . . . . . . . . . . . . . . . 46

4.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.1 Current . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.2 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Model Improvements 555.1 Parameter tuning . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1.1 Discharge coefficient Cd . . . . . . . . . . . . . . . . . 555.1.2 Air gap . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 System identification of electrical model . . . . . . . . . . . . 595.3 Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.4 Temperature dependent resistance . . . . . . . . . . . . . . . . 61

6 Control Design 636.1 Control Objectives and Background . . . . . . . . . . . . . . . 636.2 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.2.1 Principles of PID Control . . . . . . . . . . . . . . . . 646.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . 65

6.3 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.3.1 Scheme 1-Fill Valve and Ventilation Valve activated

separately for filling and ventilation. . . . . . . . . . . 666.3.2 Scheme 2-Both Valves activated simultaneously for fill-

ing and ventilation. . . . . . . . . . . . . . . . . . . . . 666.4 Results-Scheme 2 . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.4.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . 686.4.2 Test on Prototypes . . . . . . . . . . . . . . . . . . . . 69

6.5 Control Improvements . . . . . . . . . . . . . . . . . . . . . . 706.5.1 Anti-Windup . . . . . . . . . . . . . . . . . . . . . . . 706.5.2 Improved Control using Non-Linear Control . . . . . . 706.5.3 Improved Control using Prediction . . . . . . . . . . . 72

6.6 Comparison between the two approaches . . . . . . . . . . . . 73

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7 Conclusion and Future work 757.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A Appendix 79A.1 Linearizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

A.1.1 Fill valve and Ventilation valve are both activated . . . 80A.2 Calculation of smallest outlet orifice area, Ao . . . . . . . . . . 84

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List of Figures

1.1 Schematic figure showing the functional description of thevalve unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 A basic model describing the interior of a solenoid valve. . . . 192.2 Normally closed 3/2 valve - Principal . . . . . . . . . . . . . . 202.3 Normally closed 3/2 valve - Unaffected, where number 1 is the

pressure source, number 2 denotes the outlet port and number3 is the air exhaust port. . . . . . . . . . . . . . . . . . . . . . 20

2.4 Normally closed 3/2 valve - Affected, where number 1 is thepressure source, number 2 denotes the outlet port and number3 is the air exhaust port. . . . . . . . . . . . . . . . . . . . . . 21

2.5 Schematic presentation of the basics of a PWM signal . . . . . 22

3.1 System description . . . . . . . . . . . . . . . . . . . . . . . . 263.2 The whole system represented with blocks . . . . . . . . . . . 263.3 The simplified model of ECU . . . . . . . . . . . . . . . . . . 273.4 The different subsystems of a solenoid valve . . . . . . . . . . 283.5 The electrical circuit when the PWM is high . . . . . . . . . . 293.6 The electrical circuit when the PWM is low . . . . . . . . . . 303.7 A model of an ideal diode . . . . . . . . . . . . . . . . . . . . 303.8 The model used for the electrical subsystem . . . . . . . . . . 313.9 The model used for the magnetic subsystem . . . . . . . . . . 323.10 The forces inflicted upon the armature. . . . . . . . . . . . . . 323.11 The model used for the mechanical subsystem . . . . . . . . . 333.12 The pneumatic subsystem model . . . . . . . . . . . . . . . . 353.13 The pressure chamber model . . . . . . . . . . . . . . . . . . . 363.14 A schematic sketch of the regulating valve, showing the forces

acting on the piston. . . . . . . . . . . . . . . . . . . . . . . . 36

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4.1 Applying a PWM of 100% on prototype 1, with a dead volumeof 100 cm3 and an orifice diameter of 1.9 mm, the desiredpressure (26.2%Psup) is obtained within 10%Treq. Howeverif time delay is taken into account, the time for filling up to26.2%Psup is 7 samples i.e. higher than 10%Treq, which doesn’tfulfill the requirements. . . . . . . . . . . . . . . . . . . . . . 42

4.2 Applying a PWM of 100% on prototype 1, with a dead volumeof 100 cm3 and an orifice diameter of 1.9 mm, the desiredpressure (Psup) is obtained within 65 samples, which is tooslow and doesn’t meet the requirements. . . . . . . . . . . . . 42

4.3 Applying a PWM of 100% on prototype 1, with a dead volumeof 100 cm3 and an orifice diameter of 1.9 mm, it takes about 4samples to empty the dead volume from 88%Psup to 70%Psup,which is too slow. . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Applying a PWM of 100% on prototype 1, with a dead volumeof 100 cm3 and an orifice diameter of 1.9 mm, it takes about65 samples to empty the dead volume from Psup to 0 bars,which is too slow. . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5 Applying a PWM of 82% on prototype 2, with a dead volumeof 75 cm3 and an orifice diameter of 1 mm, the desired pressure(26.2%Psup) is obtained within 10.5 samples, which is too slow. 45

4.6 Applying a PWM of 82% on prototype 2, with a dead volumeof 75 cm3 and an orifice diameter of 1.9 mm, it takes about 5.3samples for filling up to 26.2%Psup, which is near the requiredtime (10%Treq). . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.7 Applying a PWM of 82% on prototype 2, with a dead volumeof 75 cm3 and an orifice diameter of 1 mm, it takes approx-imately 75 ms to go from 88%Psup to 70%Psup, which is tooslow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.8 Applying a PWM of 82% on prototype 2, with a dead volumeof 75 cm3 and an orifice diameter of 1.9 mm, it takes approx-imately 26 ms to go from 7.4 bar down to 5.9 bar, almostfulfilling the requirements. . . . . . . . . . . . . . . . . . . . . 47

4.9 Comparison between modeled and measured current, whenapplying a PWM of 75%, where the frequency, time constantand mean value. . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.10 Comparison between modeled and measured current, whenapplying a PWM of 40%,where the frequency is the same.However the amplitude differed quite a lot. . . . . . . . . . . . 49

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4.11 The upper figure shows the filling process on prototype 1, whena PWM of 82% is applied to the valve. The lower figure indi-cates the ventilation of the dead volume, when a 82% PWMis applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.12 The figure shows the filling and ventilation process on proto-type 1, when a PWM of 50% is applied to the valves. Thelower figure is the error difference between the modeled andmeasured pressure, shown in percentage. . . . . . . . . . . . . 50

4.13 The figure shows the filling and ventilation process on pro-totype 1, when a PWM of 82% is applied to the valves. Thevalves are active one at the time, where the modeled and mea-sured pressure coincides quite well. . . . . . . . . . . . . . . . 51

4.14 The upper figure shows the filling process on prototype 2, whena PWM of 82% is applied to the valve. The lower figure indi-cates the ventilation of the dead volume, when a 82% PWM isapplied. In both cases a dead volume of 75cm3 and an orificediameter of 1 mm was used. . . . . . . . . . . . . . . . . . . . 52

4.15 The upper figure shows the filling and ventilation of the pres-sure for differen PWM values, and the lower figure indicatesthe various duty cycles applied on each of the valves. . . . . . 52

5.1 Applying a PWM of 82% to the fill valve, using differentdischarge coefficient values. Where the results indicate, thegreater the discharge coefficient is the faster is the valve. . . . 56

5.2 Applying 82% PWM to ventilation valve, and altering thevalue for Cdvent, a faster system is achieved when bigger valuefor Cdvent is used. . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3 The figure indicates the change in air gap due to the armaturemovement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.4 The figure shows how big the magnetic force is compared toother forces, i.e. spring-, viscous-, and pre load-force. . . . . . 58

5.5 System identification’s circular flow. The rectangles are thecomputer’s main responsibilities, and the ovals are user’s mainresponsibilities. [3] . . . . . . . . . . . . . . . . . . . . . . . . 59

5.6 The two plots show the current and duty cycle for 40% (lowerfigure), and 75% (upper figure). In both of the figures thedelay time between when the PWM goes on until the currentmoves is recognized, which is about 15 ms. . . . . . . . . . . . 60

5.7 The temperature dependent resistance . . . . . . . . . . . . . 61

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6.1 The structure of a basic feedback control loop, where r denotesthe reference signal, u is the control signal, and y is the output. 64

6.2 Traditional Pulsing Scheme (left) and Pulsing Scheme 1 (right) 666.3 Pulsing Scheme 2 . . . . . . . . . . . . . . . . . . . . . . . . . 676.4 The above figure shows the simulated pressure and its refer-

ence signal, when using an orifice diameter of 1.3 mm anda dead volume of 75 cm3, where the effects of prediction andboosting action are concluded, moreover the lower figure showsthe control signal for each of the valves. . . . . . . . . . . . . 69

6.5 Resulting PID-controller on prototype two with orifice diame-ter of 1.3 mm, including boosting action and prediction.(Scheme2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.6 The dead volume pressure for a reference change when a PIDcontroller has been applied, where slow PID reaction can beobserved for certain cases . . . . . . . . . . . . . . . . . . . . . 71

6.7 The algorithm of the non-linear control, where it is illustratedhow the integration of the control signal is activated or deac-tivated depending on the pressure change in the dead volume. 71

6.8 Block diagram illustrating a controller based on feedback prin-ciple with an additional non-linear control. . . . . . . . . . . . 72

6.9 An illustration of how the extra control signal, uN , behaveswhen non-linear control is applied to the system . . . . . . . . 72

6.10 Resulting PID-controller on prototype two with orifice diam-eter of 1.3 mm. (Scheme 1) . . . . . . . . . . . . . . . . . . . 73

A.1 The left figure shows the structure of the valve when the ar-mature is at its start position, and the right one indicates thevalves structure when the armature has moved. . . . . . . . . 84

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List of Tables

4.1 Duty cycles for opening the fill valve and keeping it fullyopened, when the system is supplied with air pressure, whereDCmin,fill and DCmax,fill are minimum and maximum dutycycles for the fill valve, respectively. . . . . . . . . . . . . . . . 40

4.2 Duty cycles for opening the ventilation valve and keeping itfully opened, when the system is supplied with air pressure,where DCmin,vent and DCmax,vent are minimum and maximumduty cycles for the ventilation valve, respectively. . . . . . . . 40

4.3 Duty cycles for opening the fill valve and keeping it fullyopened, when no air pressure is supplied to the system, whereDCmin,fill and DCmax,fill are minimum and maximum duty cy-cles for the fill valve, respectively when no pressure is suppliedto the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Duty cycles for opening the ventilation valve and keeping itfully opened, when no air pressure is supplied to the system,where DCmin,vent and DCmax,vent are minimum and maximumduty cycles for the ventilation valve, respectively when no pres-sure is supplied to the system. . . . . . . . . . . . . . . . . . . 41

4.5 The fill and ventilation sample times for each of the valves,with different dead volume and orifice diameter. . . . . . . . . 47

A.1 Table showing calculation of Ao with two different methodsfor a specified value of xp, where the smallest Ao is chosen. . . 85

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Chapter 1

Introduction

This Master’s Thesis was conducted at the department of Powertrain ControlSystem Development Scania CV AB in Sodertalje, Sweden, from November12, 2007 to April 6, 2008.Scania is a large and global company operating in Europe, Latin America,Africa, Asia and Australia, and is one of the leading manufacturers of heavytrucks, buses, and industrial and marine engines in the world. Each yearScania offers students Master’s Thesis work at the department of researchand development. Modeling and control of retarder is a thesis work providedby Scania as a part of the development and optimization of Scania’s Retardersystem. The work aims at investigating a new method for regulating thebraking torque using two on/off valves instead of a single proportional valveused in Scania’s Retarder system today.

1.1 Background

The Scania Retarder is an integrated component of Scania’s truck brakingsystem, mounted directly on the shaft at the end of the gear box. Theretarder is an aid for reducing the speed without constant use of the regularservice brake and the exhaust brake. The actual retarder system uses aproportional valve to control an air flow that determines an oil pressurein the retarder, which results in a braking torque. The proportional valveis constructed in such a way that it is very sensitive to temperature andvibrations, and has a strong influence on the internal friction and calibration.Proportional valves are quite expensive and experience has shown that theyare affected by so called hysteresis. The cost can be reduced and some of thedisadvantages affecting the dynamics can be eliminated by using two on/offvalves, a fill and ventilation valve, to regulate the braking torque. A former

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concept study [16] performed at Scania, investigating different approachesfor controlling the retarder, confirms the advantages with the valves andconcludes that a concept using on/off valves would be the most convenientand robust method. This thesis is based on the former research.

1.2 Objectives

A system using two On/Off valves to control the air pressure will be modeledand it will be investigated, whether it is possible or not to make a controllerthat fulfills Scania’s requirements specification. The valve unit used in boththe current Retarder Control and this Master’s Thesis should be capable ofapplying, removing and regulating the braking torque created by the retarderand is therefore divided into three actuating functions; an accumulating func-tion, an emptying function and a regulating function. The main objective inthis Master’s Thesis is to investigate the regulating function. Elements thatwill be included in this Master’s Thesis are summarized in the following list:

• Modelling of the system and an implementation in Simulink containing

– Electrical Drives

– Two on/off valves’ electrical, magnetic and mechanical properties

– A pressure chamber and its pneumatic properties

– A regulating valve that balances air and oil pressure

• Investigation of how pulse-width-modulation can be applied in the con-trol

• Design and implementation of controllers based on the model

• Recommendations on parameters in the system

• Verification of model and controller by measurements in prototypes andtruck.

• Code Generation in real-time workshop

1.3 Functional Description

The valves convert the electrical signals provided by the Electronic ControlUnit (ECU) to a pneumatic pressure. Figure 1.1 shows the three actuatingfunctions i.e. the Accumulating function, the Emptying function and the

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Regulating function. The Emptying function consist of an on/off valve forquickly emptying the pressure in the system. The Regulating function is com-posed of two on/off valves, a pressure chamber and a regulating valve. Thepressure build up by the two valves inside the chamber is balanced againstthe oil pressure in the regulating valve, contributing to a braking torque.The Accumulator is active when additional air is needed to be injected intothe system. Figure 1.1 is schematic sketch of how the components in theRetarder system are connected.

Figure 1.1: Schematic figure showing the functional description of the valveunit

Due to security reasons, the ventilation valve is designed as normally open,which will be the case in a real implementation. The valve is seen downto the right in the regulating function. For experiments and tests on benchduring this Master’s Thesis, a valve that is normally closed is used, and willbe handled in Chapter 2.

1.4 Actuation Requirements

To obtain the desired performance and robustness there are several require-ments the system has to fulfill. Among these are general requirements for

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the retarder system as well as requirements for all three actuation functionsin the retarder valve unit. Only the general and regulating requirements willbe considered.

General Requirements

• A change in the input (control signal) must result in a change in theoutput (air pressure)

• Filling and ventilation should not affect each other

• A pressure supply, Psup, should be used as input to the fill

Included in the general requirement list are requirements of endurance, mark-ing, deviations, resistance to oil in drainage air, ambient temperatures, avail-able outputs from ECU, diagnostics, reliability, and testability, but will notbe included in the report since they are not of importance to this work.

Regulating Function

• The function shall include one pressure sensor and two 2/21 on/offvalves where the filling valve shall be normally closed and the ventila-tion valve normally open2.

• The regulating valve has a maximum actuating volume of Vr,max formaximum stroke.

• An output volume, including the volume in the chamber and the ac-tuator volume in the regulating valve, has to be decided ( 50 - 125cm3).

• Pressure regulating tolerance: the desired pressure ±∆PTol bar shallbe reached within Treq during all valid conditions.

• Pressure sensor tolerance: ±∆PTol bar at 0-100 C temperature range.

• Filling from 0 to 26.2%Psup shall happen in less than 10%Treq whenpressure supply is Psup.

• Ventilation from 88%Psup to 70%Psup shall happen in less than 5%Treq

when output relative pressure is 0 bar.

1A 2/2 valve has an inlet- and an outlet-port and operates in two states (on or off)2A normally closed ventilation valve will be used in the modeling and experimenting

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1.5 Notation

NotationECURM Measured Resistance ΩUPWM Pulse Width Modulation Signal Voltε Power Supply VoltD Duty Cycle -ValveElectricalLc Inductance in coil HRc Resistance in coil Ωi Current in coil A

Magneticµ0 Permeability in air Vs/AmN Number of turns in coil -Aa Area of armature m2

lg Length of air gap in valve mxp Position of armature in valve mxoff Air gap length when the valve is closed mxon Air gap length when the valve is opened md Inlet and outlet diameter of valves mPs Supply pressure BarPatm Atmospheric pressure BarPch Chamber pressure BarPu Upside pressure BarPd Downside pressure Bar

Mechanicalma Mass of armature kgks Spring coefficient N/mb Viscous friction coefficient Ns/mFprs Pressure force NFpld Preload force NFk Spring force NFb Force, viscous friction NFf Static friction N

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NotationPneumaticCd Discharge coefficient -Cd,fill Discharge coefficient fill valve -Cd,vent Discharge coefficient ventilation valve -k = Cp/Cv Specific heat ratio in air -Cv Specific heat capacity at constant volume JKg−1K−1Cp Specific heat capacity at constant pressure JKg−1K−1Rgas Gas constant -Ao Area of orifice m2

Tair Temperature of supplied air KM Mach number -mch Mass of air in chamber kgmfill Mass of filling air kgmvent Mass of ventilating air kgVch Chamber volume m3

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Chapter 2

Retarder

In this chapter the retarder is described further, and the equipment used forexperiments are presented and explained.

2.1 Scania’s Retarder

The Scania Retarder is used to create a braking torque to slow down thespeed of the vehicle, and with a maximum braking power up to 500 kW theRetarder is one of the most powerful components in Scania’s braking system.The Retarder is placed on the outgoing shaft on the gear box. Oil is pumpedin between a fixed stator and a movable rotor, and as a result of the highoil pressure caused between the two components a braking power is created.One of the main benefits of the Retarder is the reduced requirement of thewheel brakes, which results in less brake wear off. In this way, the wheelbrakes remain cool and unused, and thus are more efficient and powerfulin the need of additional braking. The Retarder can be used manually orin automatic mode. Using the Retarder in automatic mode allows even formaintaining a steady speed on descents. [1].

2.1.1 Retarder system today

The current Retarder system uses a proportional valve to control the oil pres-sure between the rotor and stator. The inlet port of the proportional valve isconnected to an air pressure supply and has two outlet ports, one connectedto a cylinder containing a plunge and another to a drain. The valve is asolenoid valve and can be activated by applying an analogue current to itscoil. Inside the valve there is a movable armature. If current is applied, amagnetic field will appear between the armature and the iron core in the

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valve. This will result in a magneto motive force which will affect the ar-mature causing it to move proportional to the current. Air will pass to thecylinder or to the drain depending on the armature’s position. According tothe air in the cylinder the plunge will move, and an oil pressure causing abreaking torque will be created.

2.1.2 Retarder System Using On/Off Solenoid Valves

As described in the introduction, a new concept using on/off-valves is to beexamined in the current Master’s Thesis. The cylinder is substituted by achamber with constant volume1. Two on/off solenoid valves are introduced,one for filling and one for ventilating the chamber. How they are connectedcan be seen in Figure 1.1. The inlet port of the filling valve is supplied withair pressure, while the outlet port is connected to the chamber and to theventing valve. To the outlet port of the ventilation valve there is a drain tothe environment where there is atmospheric pressure.

By applying current to the valves, one can fill or vent the chamber. Ba-sically the on/off valves can either be open or close, but by using pulse widthmodulation (PWM) as input signal, it might be possible to manipulate theirbehavior so that the armature in the valves can switch between the on and offposition for one single PWM period, resulting in a limited air flow throughthe valve. A change in the air pressure due to air flow into the chamberaffects on the regulating valve in the Retarder. The plunge in the regulatingvalve will move from its initial position when the air pressure increases, andwill eventually result in a higher braking torque in the Retarder that brakesdown the truck. If the pressure decreases, the regulating valve will move inthe opposite direction back to its initial position, and the braking torque willbe reduced. Mounted on the chamber is a pressure sensor. In this way thepressure can be directly measured, and a controller based on the closed-loopprinciple, with pressure as the feedback, can be designed.

2.2 Dead Volume

From Figure 1.1, it can be seen that the actuating volume in the regulationvalve, the extra volume in the chamber and the volume in the housing areconnected. This volume is denoted the dead volume and will be used as a

1Due to movement in the regulating valve there will be a varying volume in the chamber.However these changes are sufficiently small compared to the total volume of the chamberso that the volume can be considered constant.

18

reference to the total volume in the retarder through the thesis. The littleamount of air capacity in the actuating volume (21cm3), makes it difficult forthe controller to regulate the pressure. To increase the performance of thecontroller, extra chamber volume has been inserted. The total volume hasyet not been decided, but is a part of the parameters that will be investigatedand decided in the Master’s Thesis, in order to fulfill the requirements onfilling versus venting time and control performance.

2.3 Solenoid Valves

An On/Off valve is an example of a solenoid valve, which will be describedin this section. A solenoid valve is an electro mechanical valve where its dy-namic behavior has an influence on fluids (liquid or gas), and can be dividedinto two main parts: a solenoid and a movable armature.

The solenoid consists of a coil of wire wounded in the form of a cylinder. Thecoil covers the movable armature that is mounted on a spring that keeps thevalve in its initial position. The valve can be activated by applying currentto the coil, and the armature will move away from its initial position.

As it appears from Figure 2.1, the interior of a solenoid valve has a complextechnical system. It contains four subsystems that all are related to eachother; electrical, magneto-dynamic, mechanical and fluid dynamical. Whencurrent is applied to the system, a magnetic field is induced around the coilcontributing to a magnetic force. The magnetic force tries to overcome thecounteracting forces, i.e. spring force and friction forces, resulting in openingor closing of the valve, depending on if the valve is normally open or closed.Valves are divided into different groups, according to their function and use.

Armature

Pressure supply 8.4 bar

Positivex-direction

Figure 2.1: A basic model describing the interior of a solenoid valve.

19

The most common valves are 2/2 valves, 3/2 valves and 5/2 valves. As seenin Figure 2.2,a 3/2 valve has three ports (inlet port, outlet port, and a drain)and two states (on and off). The valves used in the present Master’s Thesisare 3/2 valves where the drain (port 3) has been sealed and therefore operatesas a 2/2 valve. It is important to note that Figure 2.2 shows a valve thatopens or closes by pressing a button. The principles are the same for solenoidvalves, but they are activated when a current is used. As mentioned earlier,

Figure 2.2: Normally closed 3/2 valve - Principal

a valve can be normally open or normally closed. Figure 2.3 shows a valvethat is normally closed and not affected by external forces, the flow path 1-2will be closed while the flow path 2-3 will be open. If the valve is affected

31

2

Figure 2.3: Normally closed 3/2 valve - Unaffected, where number 1 is thepressure source, number 2 denotes the outlet port and number 3 is the airexhaust port.

by external forces, see Figure 2.4, the drain will close and the path 1-2 willopen. Only in this case a flow from the inlet port to the outlet port can takeplace. For a normally open valve, in the unaffected case, path 1-2 is openand 2-3 is closed. Affecting the valve with this construct, the drain will openand path 1-2 will close.

20

31

2

Figure 2.4: Normally closed 3/2 valve - Affected, where number 1 is thepressure source, number 2 denotes the outlet port and number 3 is the airexhaust port.

On/Off-valves

Another kind of solenoid valve is an on/off-valve. The valve uses the sameflow principles as the valves already described, but it is activated by passinga current through the coil. In contrast to the proportional valves that canbe partly open, the On/Off-valves can basically be only open or closed.

2.4 Equipments In Experiments

Modeling of the complete system requires an understanding of the system’sbehavior and its characteristics. To get a satisfying model which is similarto the real system experiments have been performed on two prototypes, onecontaining a chamber with fixed volume and the other in which the volumein the chamber can be adjusted. In both prototypes an electronic controlunit (ECU) was used to generate input signals to the valves, and a laptopcontaining real-time software was used to acquire data from the pressuresensor and current outputs in the ECU.

2.4.1 ECU

To control the time at which the valves open or close, the electronic controlunit (ECU), is used. The operating conditions of the valves are establishedby a pulse width modulated (PWM) scheme generated by the ECU. Availableoutputs from the ECU relevant for the work are current and pressure. TheECU can also be used to control other electrical systems for the trucks.

Pulse Width Modulated Signal, PWM

The pulse width modulated signal is a periodic square form signal in whichthe frequency and the duty cycle can be selected. The PWM is shown belowin Figure 2.5.

21

Figure 2.5: Schematic presentation of the basics of a PWM signal

The duty cycle, D, is defined by the ratio, D = Td/T, where Td is the time forwhich the signal is high and T is the period time. Td is limited to be in [0, T ].The duty cycle is also referred to as percentage, D ∈ [0%, 100%]. Unless thecurrent in the coil is not at its maximum, it will continue to increase as longas the PWM is high. If the PWM goes low, the current will start to discharge.When experimenting on the two prototypes, the duty cycle and the frequencyfor the filling and ventilation valve can be selected separately, and is availableas variables in the real-time software used2. This is an advantage since thechoice of the duty cycle and the frequency of the PWM signal for the twovalves will probably affect the filling and ventilating times in different ways.Two other important factors affecting the filling respectively ventilating timeare the inlet and outlet diameters in the chamber and the chamber volume,and will be discussed later.

2.4.2 Prototypes

During the Master’s Thesis two prototypes have been available for experi-ments and validation purposes. The first prototype is only usable for earlytests and verification of the model. It has a fixed volume and can only beused on bench. Prototype 2 can be mounted on the real Retarder in a truck,connecting the outlet pressure to the regulating valve, also making it possi-ble to verify the controller and its performance in the real system. For bothprototypes Norgren Herion, normally closed 3/2 valves with a sealed drain,as described in Section 2.3, have been used.

2Gredi KleinKnecht is a calibration software.

22

Prototype 1

Prototype 1 consists of a pressure supply inlet, a chamber with a fixed vol-ume, V = 100cm3, two on/off valves, one for filling and the other one forventilation, and a pressure sensor. A given PWM signal can be used as in-put to the valves, which will partly or fully open the valves depending onthe duty cycle and frequency of the PWM signal. Experiments can be doneeither with pressure supply or without. If only the electrical, magnetic andmechanical part of the valve are to be studied, it is convenient to start withexperiments in which no pressure supply is connected. When the pressuresupply is connected, the complete dynamics can be studied. The valves areequipped with a fixed orifice of 1.9 mm.

Prototype 2

Prototype 2 consists of the same valves as in prototype 1. The fixed volumeof the chamber is 51cm3. Taking into account the volume in the regulatingvalve (21cm3) and the connecting channels (2.8cm3) the total dead volume isapproximately 75cm3. However, in prototype 2, extra volume can be addedto the chamber manually. This is convenient when the regulating require-ments are to be examined. The volumes available for use are then 75cm3,100cm3 and 125cm3. A large volume takes longer time to fill with air orempty than a small volume.

When a controller is used to regulate the air pressure in the volume, be-cause of the limited pressure sensor accuracy and response and the reactiontime for the ECU, it will take a certain time for the chamber to be regulated.The time delay will give the following result: if the chamber has to be filledbefore the control unit is able to react, there will be a pressure drop in thechamber. In a control context it is therefore convenient to have as large vol-ume as possible so that the pressure drop will be as small as possible. Tofulfill the time requirement of filling the entire chamber with air or emptyingit, it is more convenient to have a small volume.

2.5 Retarder

A full scale retarder has been available for experiments on prototype two.Included in the retarder is the rotor and stator, and the regulating valve. Ithas on the other hand not been connected to the gear box and no oil hasbeen present in the retarder. The oil pressure has therefore been neglectedand some model simplifications have been done.

23

2.6 Pressure sensor

To measure the air pressure in the dead volume a pressure sensor manu-factured by Denso Corporation with part number 1491406-4990007670 wasused. The operating pressure, denoted in absolute pressure, is 0.06 to 2.1MPa, but is represented by the ECU in relative pressure related to the at-mospheric pressure, i.e 0 bar on the output corresponds to 1 bar in absolutepressure. Durability of the pressure sensor and the surrounding temperatureaffect its precision. Operating temperature is -40 to 135 C and the sensorrequires a supply voltage of Vsup ±∆VTol V to work properly. The pressuresensor is assumed to fulfill Scania’s pressure sensor tolerance requirement of±∆PTol bar for the required operating environments.

2.7 Software Program

”Gredi Kleinknecht” is a calibration tool for use with the ECU. It includesfunctions to display, record and evaluate simultaneously acquired ECU inter-nal and process data [2]. In this Master’s Thesis Gredi Kleinknecht has beenused in experiments to acquire data such as current and pressure. The datahas been exported to Matlab where it can easily be examined. From Gredi,internal parameters in the ECU can be set, such as input to the valves usedin the prototypes. Among the inputs that have been possible to change arethe PWM duty cycle and the frequency.Matlab and Simulink has been usedin the modeling and simulation of the system.

2.8 Oscilloscope

A Fluke 45 Dual Display Multimeter was used to examine the dynamics ofthe ECU and the electrical part of the valves.

2.9 Multimeter

To examine the electrical circuit dynamics, a basic multimeter, Fluke 75,manufactured by John Fluke has been used.

24

Chapter 3

Modelling

In order to build a model, it is very important to know how different subsys-tems in a system work. By investigating each parts behavior a greater knowl-edge about that specific part will be gained, contributing to enhancement ofachieving a model with desired specifications. There are two approaches forbuilding a model.

The most common way is by experimenting and gathering information.Theexperimental method is a scientific principal, however it has some limitations.Sometimes it is unsuitable or even impossible to carry out an experiment.The aspects causing these kinds of problems could be that it is too expensiveto investigate random configurations, or dangerous. It may also be the casethat one does not have access to the system. A cheaper and more practicalapproach could be to build a model, which is a tool answering different sys-tem characteristics without conducting an experiment. Matlab and Simulinkare powerful model building programs. In this way the system can easily bemodelled and simulated without costing so much.

Two approaches to modelling are ”Blackbox-” and ”Whitebox-modelling”.These two types of modelling methods are very important in modelling per-spective. Whitebox-modelling is based on physical models or more precisely,mathematical equations describing the physical phenomena. This type ofmodelling requires that the physical characteristics for all subsystems areknown. If not, another approach has to be considered. In order to get anidea on how the unknown part works, the relations between the inputs andoutputs, also called Blackbox-modelling, is studied.

In this chapter the physical relations contributing to each of the subsystemsand how they are modelled is discussed. After deriving the mathematical

25

equations, a state-space model of the whole system is presented.

3.1 System description

The main parts of the system consist of two on/off- valves, a pressure chamberand a regulating valve. On/Off-valves used in this project are two solenoidvalves one of which acts as a filling valve and the other as a ventilation valve.The inlet-port of the filling valve is supplied by air pressure(Psup) and theoutlet-port conducts airflow to the pressure chamber and also to the inlet-port of the ventilation valve. The net airflow, resulting from the valves,builds up pressure in the chamber, leading the air pressure to the regulatingvalve. The regulating valve balances the oil pressure in the valve, generatingthe desired torque. An overview of the system is shown in figure 3.1, whichconsists of two on/off valves, one for filling the pressure chamber and theother one for emptying the chamber. The fill valve is provided with a supplypressure of Psup and the regulating valve balances the air pressure inside thechamber with oil pressure giving the desired torque. In this Master’s Thesis

Supply valve

Ventilation valve

8.4 bar

1 bar

chamber

+

-

+

-

Regulating valve

Figure 3.1: System description

the whole system has been divided into several blocks, see figure 3.2, whereeach block will be discussed separately in the up coming sections.

Valve ChamberECURegulating

valve

Figure 3.2: The whole system represented with blocks

26

3.2 ECU

Recall from Section 2.4.1 that the ECU is an Electric Control Unit, which isan embedded system controlling one or more of the electrical subsystems ina vehicle. A simplified model of the ECU, which generates a PWM1 signal,with a freewheeling diode inserted at the output for safety reasons, is shownin Figure 3.3.

As stated before the output signal from the ECU, which drives the valvescan be expressed as:

UPWM =

high for t < DTlow for DT < t < T

where D is the duty cycle and T is the time period of the pulse width modu-lated signal. Due to the switching characteristic of the PWM signal there aretwo different electrical circuits for the respectively generated voltage. Theelectrical circuit is discussed in more detail in Section 3.3.1.

When the PWM is low the already charged inductance (see Section 3.3.1)begins to discharge, causing the current to flow through the circuit. Thefreewheeling diode placed at the output of the ECU keeps the current fromdischarging instantly, and prevents it from short-circuiting any componentsin the ECU.

Figure 3.3: The simplified model of ECU

1Pulse Width Modulation is a square wave whose pulse width is modulated resultingin the variation of the average value of the waveform.

27

3.3 Valve

The main parts of a solenoid valve consist of Electrical, Magnetic, Mechanicaland Pneumatic sections which are depicted in Figure 3.4.

Electrical

Mechanical

Magnetic

Pneumatic

UPWM

i

FM

xp

m&

Figure 3.4: The different subsystems of a solenoid valve

The upcoming sections give more insight on how each of the subsystemswork.

3.3.1 The Electrical Subsystem

The electrical part of the solenoid valve can be modeled as a resistance inseries with an inductance. Because of the altering characteristic (switchingbetween high and low values) of the PWM signal, two different cases haveto be investigated. The first case handles the situation where the PWM ishigh, and the second case gives more insight on the circuit when the PWMis low.

Case 1:

The electrical circuit corresponding to the first case is modeled accordingto Figure 3.5. Studying the schematic sketch in Figure 3.5, the power source(24 vols), ε, drives the electrical circuit. By applying Kirchhoff’s voltage law(KVL) [4] that states,”The directed sum of the electrical potential differencesaround a closed circuit must be zero”, there is a voltage drop over the resis-

28

tance, R, also a voltage drop throughout the energization of the inductance,L, giving the following mathematical relation:

ε−Ri− εL = 0 (3.1)

where εL, is the voltage drop over the inductance. Due to the armature’smovement (back and forth) inside the coil, during the magnetization, anelectro motive force (emf) is induced. The emf is expressed by:

εL = NdΦB

dt=

d

dt(Li) = L

di

dt+ i

dL

dt(3.2)

The final equation describing the voltages around circuit 1 can be obtainedby inserting (3.2) into (3.1),

ε−Ri− Ldi

dt− i

dL

dt(3.3)

Figure 3.5: The electrical circuit when the PWM is high

Case 2:

As mentioned above, during the circumstances when the PWM is low i.e.0 volts, the power supply, ε, is neglected from the circuit. Studying Fig-ure 3.6 it can be observed that the energized inductance in Case 1 acts asthe power source. In this case the freewheeling diode, involved in the circuitfor reasons stated in Section 3.2, has been modeled as an ideal diode depictedin Figure 3.7. The figure shows that when the diode is active, it will resultin a voltage drop (Vd) of 0.7 volts. Thus applying Kirchhoff’s Voltage Lawgives the following equation:

−Ldi

dt− i

dL

dt−Ri− Vd = 0 (3.4)

where didt

< 0 due to the constant discharge of the energized inductance.

29

Figure 3.6: The electrical circuit when the PWM is low

0.7

Current (i)

Voltage (V)

Figure 3.7: A model of an ideal diode

Note that the armature’s displacement in the solenoid will cause a changein the magnetic field in the coil, resulting in a change of inductance (L(xp))value. The varying inductance has a direct relation with the position of thearmature, where the values of the inductance when the armature is at itson (Lon) or off (Loff ) position is provided by the valve manufacturer. Theinductance as a function of armature position,xp, is described by:

L(xp) = Loff +Loff − Lon

xon − xoff

xp (3.5)

Equation (3.5) shows that the value of inductance, L, is dependent on thearmature’s position, therefore the emf derived in (3.2) has to be modified,using the chain rule, before implementation. The modified (3.2) will thus beexpressed as:

30

εL = L(xp)di

dt+ i

dL(xp)

dt= L(xp)

di

dt+ i

dL(xp)

dxp

.dxp

dt(3.6)

= L(xp)di

dt+ i

dL(xp)

dxp

. v

where v denotes the armature’s velocity during displacement. Having gonethrough the two cases in the electrical part, a model (see Figure 3.8) wasdesigned with the PWM signal as input and the current as output.

ElectricalUPWM i

Figure 3.8: The model used for the electrical subsystem

3.3.2 The Magnetic Subsystem

When the current determined from the electrical subsystem flows through thecoil, a magnetic field is created in the solenoid and the magnetic flux-densityis given by [5]

B =µ0iN

lg(3.7)

where lg = xoff −xp denotes the length of the airgap, N denotes the numberof turns, and µ0 is the permeability in air. The magnetic field provided,results in a magnetic force described by [5]

F =B2

2µ0

A (3.8)

Combining (3.7) and (3.8), and considering the fact that the cross sectionalarea(Ap) of the armature in the solenoid is constant, the magnetic force canbe expressed by ([5])

FM =µ0Api

2N2

2(xoff − xp)2(3.9)

The magnetic section of the valve has been modeled as shown in Figure 3.9,where current and position are inputs and magnetic force(FM) is output.

31

Magnetic

i FM

xp

Figure 3.9: The model used for the magnetic subsystem

3.3.3 The Mechanical Subsystem

As discussed in earlier sections, the magnetic force (FM) helps to move thearmature. However, there are other forces involved in the mechanical move-ment of the armature, which are depicted in Figure 3.10. When supplying thevalves with air pressure, a pressure force Fprs is provided, resulting from thepressure difference between the upside2 and downside3 pressure, see ([10], [7]),and is given by

Fprs = π(d

2)2(Pu − Pd) (3.10)

where π(d2)2 is the area of the armature that pressure is directly inforced. Pu

and Pd are upside and downside-pressures respectively.

FM

Fprs

Positive x-direction

Armature

Fb

Fk

Fpld

Figure 3.10: The forces inflicted upon the armature.

As seen in Figure 3.10 the armature is connected to a spring that counteractsthe forces mentioned above. The spring has the spring constant ks and is preloaded with a force Fpld, both given by the manufacturer. The spring force(Fk) and the pre load force is given by ([10], [7])

Fk = ksxp Spring forceFpld = ksx0 Pre load force

(3.11)

2The upside pressure is at the point where higher pressure is located.3The downside pressure is at the point where lower pressure is located.

32

where x0 is the length of the spring at the start position. Due to the ar-mature’s connection to the valve house, static and viscous friction, Ff andFb respectively, are present. The static friction affects the system when thearmature is at its start (xp = xoff ) and end (xp = xon) positions i.e. xp = 0,where xp is the armature’s velocity. When the armature is in motion (xp 6= 0),the viscous friction (Fb) affects the movement, where the friction forces arepresented by ([10], [7])

Ff for xp = 0Fb = bxp for xp 6= 0

(3.12)

where b is the damping coefficient. After going through all the forces affectedon the armature, the equation of motion can be written as

mpxp = FM + Fprs − Fb − Fk − Fpld − Ff (3.13)

Note that the armature movement is physically limited, 0 ≤ xp ≤ xpmax, dueto the valve structure, and is defined positive for xp > 0 (see Figure 3.10).

The mechanical part of the valve has been modeled as a box shown in Fig-ure 3.11, having the magnetic force, pressure supply (Ps) and the pressurefrom the chamber (Pch) as its inputs and the armature movement (xp) asoutput.

Mechanical

FM

xpPch

Ps

Figure 3.11: The model used for the mechanical subsystem

3.3.4 The Pneumatic Subsystem

As explained in Section 3.3.3, when the armature starts to move, it will resultin the opening of the valve which in its turn results in a certain amount ofairflow. From thermodynamics ([8], [9]), the mass flow through an orifice canbe written as

m =PuCdAo√RgasTair

· ξ·Ψ(Pd

Pu

) (3.14)

where

33

ξ =√

k( 2k+1

)(k + 1)/(k − 1)

and

Ψ(Pd

Pu

) =

1 for Pd

Pu≤ Pcr√

(PdPu

)2/k−(PdPu

)(k+1)/k

k−12

( 2k+1

)(k+1)/(k−1) for 1 ≥ Pd

Pu> Pcr

where k ≈ 1.4 denotes the specific heat ratio in air, Ao is the smallest outletorifice area, Rgas is the gas constant, Tair is the temperature in the deadvolume, Cd is the discharge coefficient, and m is the air mass.

The outlet orifice area depends on the armature position. If the armatureis in its start position, no air will flow through the valve. As soon as thearmature moves, the inlet orifice area gets bigger, which is the same area theair will flow out through. The smallest outlet area can be expressed as

Ao = πdxp (3.15)

It is important to note that the outlet orifice cannot be bigger than the inletorifice. Thus, when the outlet orifice is bigger than that of the inlet orificethen the outlet orifice area is calculated from the formula

Ao = π(d

2

)2(3.16)

Appendix A.2, explains this phenomenon in more detail.The critical pressure ratio, which is the ratio between the upside and down-side pressures, can vary from one component to another, depending on theshape of the orifice. For pressure ratios lower than the critical pressure ratio,the flow is called critical flow, and for those higher than the critical pressureratio it is called under critical flow. The critical pressure ratio Pcr is givenby

Pcr = (2

k + 1)

kk−1 ≈ 0.528 (3.17)

The pneumatic part of the valve has been modeled according to Figure 3.12,having pressure supply, pressure from the chamber and the armature move-ment as its inputs, and the mass airflow as output.

34

Pneumatic

xpm&Pch

Ps

Figure 3.12: The pneumatic subsystem model

The pressure chamber inserted at the outlet port of the supply valve, andthe inlet port of the ventilation valve, gathers the net airflow

mch = mfill − mvent (3.18)

from the valves. According to the Ideal gas law the pressure in the chamberis expressed by

Pch =mchRgasTair

Vch

(3.19)

Looking at the derivative of (3.19), the pressure variation can be evaluatedas

d

dt(Pch) =

d

dt(mchRgasTair

Vch

) =Rgas

Vch

(mchTair +mchTair− VchmchTair

Vch

) (3.20)

where

Tair ≈ constant ⇒ Tair = 0and

Vch ≈ constant ⇒ Vch = 0resulting in

Pch =mchRgasTair

Vch

(3.21)

The temperature is chosen constant due to little compression time, resultingin small changes in temperature, moreover in Section 3.4 the reason for usinga constant volume is described in details. The modeled pressure chamberwith net airflow and pressure from the chamber as its input and outputrespectively, is shown in Figure 3.13.

35

Chamberm& Pch

Figure 3.13: The pressure chamber model

3.4 Regulating valve

The regulating valve is placed at one of the outlet ports in the pressurechamber(see Figure 3.1). The channel attaching the chamber to the regu-lating valve, leads the air pressure to a piston placed inside the valve. Onone side of the piston the pressurized air, and on the other side the oil pres-sure, produced from an oil pump, are gathered. The air pressure is balancedthrough the oil pressure, leading to a certain braking torque. Figure 3.14shows a schematic sketch of the regulating valve.

Fprs,Pch

Ffriction

Fprs,oil

Regulating valve

Positive x-direction

OilPch

Figure 3.14: A schematic sketch of the regulating valve, showing the forcesacting on the piston.

By studying Figure 3.14, the pressures at each side of the piston will exerta certain force. When the pressure force exceeds the magnitude of the coun-teracting forces it will move the piston. The equation of motion is describedby

F = Fprs,ch − Fprs,oil − Fk − Ffriction − Fpld (3.22)

mpistonxpiston = PchAch − PoilAoil − ksxpiston − Ffriction − Fpld (3.23)

where

Ffriction =

Fsf if xpiston = 0Fdf if xpiston 6= 0

36

where xpiston denotes the velocity of the piston during its movement, xpiston isthe pistons displacement, Ach and Aoil are the cross sectional areas where airand oil pressure affects respectively. The maximum plunge stroke i.e. fromthe start position until the end position, is about 9 mm, increasing the volumewith 8.7cm3. However during the control process the plunge displacementwould be 2 mm, increasing the volume with 1.92cm3. The increment of thevolume is so small, compared to the dead volume (see Section 2.2) whichis 100cm3, that it is neglected and the calculations are done for a constantvolume.

3.5 Summary of state-space equations

In this section the equations for the complete model have been summarizedin state-space form. First the input signals, output signals and the states aredefined, followed by the expressions for all the states.

Input signals:

u1 = uPWM,sup

u2 = uPWM,vent

Output signal:

y = Pch

States:

x1 = isupply x2 = xp,sup x3 = xp,sup x4 = Pch

x5 = ivent x6 = xp,vent x7 = xp,vent

States-Space equations:

37

x1 =u1 −Rx1

L(3.24)

x2 = x3 (3.25)

x3 =

µ0ApN2x21

2(xoff−x2)2 + π

(d0

2

)2(Psup − x4)− kx2 − bx3 − Fpld

mp

(3.26)

x4 =RgasTair

Vch

(mfill − mvent)

=

√RgasTairCdd0π

Vch

(Psupx2Ψ

(x4

Psup

)− x4x6Ψ

(Patm

x4

))(3.27)

where

Ψ

(x4

Psup

)=

√k

(2

k+1

) k+1k−1 if x4

Psup≤ 0.528√

2kk−1

((x4

Psup

) 2k −

(x4

Psup

) k+1k

)if x4

Psup> 0.528

Ψ

(Patm

x4

)=

√k

(2

k+1

) k+1k−1 if Patm

x4≤ 0.528√

2kk−1

((Patm

x4

) 2k −

(Patm

x4

) k+1k

)if Patm

x4> 0.528

x5 =u2 −Rx5

L(3.28)

x6 = x7 (3.29)

x7 =

µ0ApN2x25

2(xoff−x2)6 + π

(d0

2

)2(x4 − Patm)− kx6 − bx7 − Fpld

mp

(3.30)

38

Chapter 4

Model validation

Now that a model has been derived for the whole system, the next step is tovalidate it. A model can never fully describe a system. It is a helping tool inorder to give a fairly good realization of the system characteristics. Verifyingif a model has the desired specifications is called validation. In this chapterthe characteristics of the real system i.e. the two prototypes described inSection 2.4.2 will be compared with the obtained simulink model. Experi-ments are carried out on each of the prototypes, and the collected data arepresented here. The acquired data is compared with the simulated data forvalidation.

4.1 Experiments on prototype one

Recall from Section 2.4.2 that the first prototype is the one with a fix volumeof 100 cm3, an orifice diameter of 1.9 mm and it cannot be mounted on thetruck. During experiments it was shown that the valve reacts within a certainrange of duty cycles, causing the armature to move between its higher andlower positions. It is important to find the upper and lower duty cycle limits,due to the fact that duty cycle values indicate also how long the valves areactive, meaning how much air will flow through the valves.

4.1.1 Duty Cycle Limits

Having used the same valves in prototype one and two, the duty cycle limitsare the same for both of the prototypes.

39

Throughout experiments it have been observed that the valves react differ-ently at different duty cycles, whether pressure supply is connected or not.During the circumstances when pressure is present, the pressure force willhelp in moving the armature and less current is needed, therefore a lower dutycycle is required in order to open the valve. Examining the pressure dataacquired,when the pressure starts to increase, it indicates that the armaturehas moved. Using the described procedure, the lower duty cycle boundarywas attained. However the upper limit wasn’t as easy to find and not beingable to measure the armature position, the only way was to hear when thevalve stops to vibrate, indicating that the valve is fully opened. The dutycycle limits for both of the valves are summarized in tables 4.1 and 4.2.

Table 4.1: Duty cycles for opening the fill valve and keeping it fully opened,when the system is supplied with air pressure, where DCmin,fill and DCmax,fill

are minimum and maximum duty cycles for the fill valve, respectively.

Fill ValveDCmin,fill 28.5%DCmax,fill 82%

Table 4.2: Duty cycles for opening the ventilation valve and keeping it fullyopened, when the system is supplied with air pressure, where DCmin,vent andDCmax,vent are minimum and maximum duty cycles for the ventilation valve,respectively.

Ventilation ValveDCmin,vent 25.5%DCmax,vent 80%

The duty cycle limits when no pressure is applied to the system is obtainedprecisely the same way. However not having access to the pressure data, thelower limit was attained through listening when the valve starts to vibrate.The following tables show the duty cycle limits for fill and ventilation valveduring the circumstances where the system is not supplied by air pressure.

40

Table 4.3: Duty cycles for opening the fill valve and keeping it fullyopened, when no air pressure is supplied to the system, where DCmin,fill

and DCmax,fill are minimum and maximum duty cycles for the fill valve,respectively when no pressure is supplied to the system.

Fill ValveDCmin,fill 54.5%DCmax,fill 82%

Table 4.4: Duty cycles for opening the ventilation valve and keeping it fullyopened, when no air pressure is supplied to the system, where DCmin,vent

and DCmax,vent are minimum and maximum duty cycles for the ventilationvalve, respectively when no pressure is supplied to the system.

Ventilation ValveDCmin,vent 51.5%DCmax,vent 80%

4.1.2 Filling characteristic

As a requirement, the fill valve has to fill the dead volume up to 26.2%Psup

in less than 10%Treq. By studying Figure 4.1, the specified behavior is seen.The experiment is done for a fixed orifice diameter of 1.9 mm and a fixeddead volume of 100 cm3.

Figure 4.1, indicates that the desired specification is obtained. However ifconsidering the time delay from the time where the PWM is activated untilthe pressure changes, the total time for the pressure to reach 26.2%Psup willbe 7 samples, which is too slow. The fill valve must also reach any referencepressure in less than Treq. Applying a PWM of 100% duty cycle, the pressurechamber is filled up from 0 bar to Psup in approximately 65 samples, which ishigher than Treq, including the time delay in the beginning. This is depictedin Figure 4.2.

41

420 422 424 426 428 430 4320

0.1

0.2

0.3

Pch

/Psu

p [N

orm

aliz

ed]

Time [sample]

420 422 424 426 428 430 4320

0.2

0.4

0.6

0.8

1

PW

M d

utyc

ycle

[Nor

mal

ized

]

Pch/PsupPWM Duty Cycle

Tf = 5Td = 2

Figure 4.1: Applying a PWM of 100% on prototype 1, with a dead volume of100 cm3 and an orifice diameter of 1.9 mm, the desired pressure (26.2%Psup)is obtained within 10%Treq. However if time delay is taken into account, thetime for filling up to 26.2%Psup is 7 samples i.e. higher than 10%Treq, whichdoesn’t fulfill the requirements.

420 430 440 450 460 470 480 4900

0.2

0.4

0.6

0.8

1

Pch

/Psu

p [N

orm

aliz

ed]

Time [sample]

420 430 440 450 460 470 480 4900

0.2

0.4

0.6

0.8

1

PW

M d

utyc

ycle

[Nor

mal

ized

]

Pch/PsupPWM Duty Cycle

Figure 4.2: Applying a PWM of 100% on prototype 1, with a dead volumeof 100 cm3 and an orifice diameter of 1.9 mm, the desired pressure (Psup) isobtained within 65 samples, which is too slow and doesn’t meet the require-ments.

42

The results in Figures 4.1 and 4.2 show that the system is too slow. In orderto meet the requirements, one should either reduce the size of the pressurechamber or increase the orifice diameter.

4.1.3 Ventilation characteristic

The same kind of valves were used for both filling and ventilation. Throughanalyzing the data gathered from each of the valves it can be seen thatduring ventilation the valve opens and closes earlier than the fill valve (seetables 4.1- 4.4). It is required that the ventilation valve empties the pressureinside the dead volume from 88%Psup to 70%Psup within 5%Treq. The testwas carried out on Prototype 1 with similar specifications (100 cm3 in volumeand 1.9 mm in orifice diameter), with a PWM of 82%, which is depicted inFigure 4.3. The figure verifies that the system is too slow.

784 785 786 787 788 789 790

0.65

0.7

0.75

0.8

0.85

0.9

Pch

/Psu

p [N

orm

aliz

ed]

Time [sample]

Tv ~ 4.2 samples

Figure 4.3: Applying a PWM of 100% on prototype 1, with a dead volumeof 100 cm3 and an orifice diameter of 1.9 mm, it takes about 4 samples toempty the dead volume from 88%Psup to 70%Psup, which is too slow.

Figure 4.4 shows that using an orifice of 1.9 mm, and a dead volume of 100cm3 it takes too long time to empty the pressure from Psup to 0 bars.

43

780 790 800 810 820 830 840 850 860 8700

0.2

0.4

0.6

0.8

1

Pch

/Psu

p [N

orm

aliz

ed]

Time [sample]

780 790 800 810 820 830 840 850 860 8700

0.2

0.4

0.6

0.8

1

PW

M d

utyc

ycle

[Nor

mal

ized

]

Pch/PsupPWM Duty Cycle

Figure 4.4: Applying a PWM of 100% on prototype 1, with a dead volumeof 100 cm3 and an orifice diameter of 1.9 mm, it takes about 65 samples toempty the dead volume from Psup to 0 bars, which is too slow.

Prototype 1 was mainly used for improving the model, and from the achievedresults, in order to fulfill the requirements a smaller dead volume (75 cm3)was chosen for the second prototype with adjustable orifice diameters.

4.2 Experiments on Prototype 2

Due to adjustability of the dead volume capacity, orifice diameter, and thecapability of being mounted on the retarder, the results presented in thissection of the report are of more interest than the ones obtained for Prototype1.Due to usage of the same valves, the fill- and ventilation-valves react exactlyin the same way as the valves used in Prototype 1.

4.2.1 Filling characteristic

Using a dead volume of 75 cm3 and an orifice diameter of 1 mm, the resultfor obtaining a pressure of 26.2%Psup, is depicted in Figure 4.5.

44

364 366 368 370 372 374 376 3780

0.05

0.1

0.15

0.2

0.25

0.3

Pch

/Psu

p [N

orm

aliz

ed]

Time [sample]

364 366 368 370 372 374 376 3780

0.2

0.4

0.6

0.8

1

1.2

PW

M d

utyc

ycle

[Nor

mal

ized

]

Pch/PsupPWM Duty Cycle

Td ~1 sample Tf ~9.5 samples

Figure 4.5: Applying a PWM of 82% on prototype 2, with a dead volume of75 cm3 and an orifice diameter of 1 mm, the desired pressure (26.2%Psup) isobtained within 10.5 samples, which is too slow.

Studying the data shown in Figure 4.5, it takes 10.5 samples to achieve thepressure of 26.2%Psup. With a dead volume of 75 cm3 and 1 mm as the orificediameter, the system became very slow. In order to speed up the process, theonly way was to increase the opening diameter, because the merest volumeat hand was 75 cm3. In this case increasing the volume of the dead volumewould make the process even slower, so by applying an orifice diameter of1.9 mm the results shown in Figure 4.6 were attained.

485 486 487 488 489 490 491 4920

0.05

0.1

0.15

0.2

0.25

0.3

Pch

/Psu

p [N

orm

aliz

ed]

Time [sample]

485 486 487 488 489 490 491 4920

0.2

0.4

0.6

0.8

1

1.2

PW

M d

utyc

ycle

[Nor

mal

ized

]

Pch/PsupPWM Duty Cycle

Tf ~3.3 samplesTd ~2 samples

Figure 4.6: Applying a PWM of 82% on prototype 2, with a dead volumeof 75 cm3 and an orifice diameter of 1.9 mm, it takes about 5.3 samples forfilling up to 26.2%Psup, which is near the required time (10%Treq).

Using 82% PWM on prototype 2, with a dead volume of 75 cm3 and anorifice diameter of 1.9 mm, it takes approximately 5.3 samples to fill the

45

dead volume from 0 to 26.2%Psup (see Figure 4.6). The results show that thesystem is faster than when using a diameter of 1 mm. Comparing the resultsfrom Figure 4.6 with results achieved from Figure 4.1, the system becamefaster which is because of the smaller dead volume.

4.2.2 Ventilation characteristic

Applying the same experimental set-up, i.e. using a dead volume of 75 cm3

and an orifice diameter of 1 mm, the time it took to empty the dead volumefrom 88%Psup to 70%Psup was approximately 7.5 samples, which is quite slowand does not fulfill the requirements. The result is depicted in Figure 4.7.

869 870 871 872 873 874 875 876 877 8780.65

0.7

0.75

0.8

0.85

0.9

Pch

/Psu

p [N

orm

aliz

ed]

Time [sample]

Tv ~7.5 samples

Figure 4.7: Applying a PWM of 82% on prototype 2, with a dead volume of75 cm3 and an orifice diameter of 1 mm, it takes approximately 75 ms to gofrom 88%Psup to 70%Psup, which is too slow.

Using an orifice diameter of 1.9 mm the ventilation time improved and be-came faster. Figure 4.8 shows the ventilation time when using an orifice of1.9 mm.

46

1178 1178.5 1179 1179.5 1180 1180.5 1181 1181.5 1182 1182.5 11830.65

0.7

0.75

0.8

0.85

0.9

Pch

/Psu

p [N

orm

aliz

ed]

Time [sample]

Tv ~2.6 samples

Figure 4.8: Applying a PWM of 82% on prototype 2, with a dead volume of75 cm3 and an orifice diameter of 1.9 mm, it takes approximately 26 ms togo from 7.4 bar down to 5.9 bar, almost fulfilling the requirements.

The filling and ventilation times for both of the prototypes are summarizedin table 4.5

Table 4.5: The fill and ventilation sample times for each of the valves, withdifferent dead volume and orifice diameter.Prototype Dead Orifice Filling: Ventilation:

Volume diameter 0 → 26.2%Psup bar 88%Psup → 70%Psup barP1 100 cm3 1.9 mm 2+5 samples 4 samplesP2 75 cm3 1 mm 1+9.5 samples 7.5 samplesP2 75 cm3 1.9 mm 2+3.3 samples 2.6 samples

Analyzing the data in table 4.5, it is recommended to reduce the size of thedead volume from 100 cm3 to 75 cm3, and use at least an orifice of 1.9 mm.However in order to achieve all the requirements it is endorsed either decreasethe size of the dead volume or increase the orifice diameter. Note that orificediameters bigger than 1.9 mm have not been investigated due to lack of time.

4.3 Validation

Now that some knowledge is gained about the valves behavior, the derivedmodel was compared to the actual process, and validated. As mentioned inprevious chapters, due to limited equipment, only data for current and pres-sure could be attained. Doing numerous experiments, results have shownthat the current is one of the key aspects. Therefore the model was vali-dated by comparing the current from the model with the one obtained from

47

experiments on the prototypes. The pressure is of great importance for thecontrolling part, therefore it was also validated.

4.3.1 Current

Recalling Section 3.3.1, considering case 1, the voltage throughout the circuit,where the freewheeling diode is not present is expressed by (3.1). Solvingthe differential equation, the current can be expressed as

i(t) =UPWM

R

(1− e−

RL

t)

(4.1)

where t is the time. Equation 4.1 indicates that the current depends on timeand the PWM signal driving the valves. For example at t = 0 the currentis i(0) = 0mA, which is the lowest value that the current can maintain. Onthe other hand when a PWM of 100% is applied to the system and the timet →∞, then the current would have the maximum value of i ≈ 292mA.

In Section 3.3.1, due to the complexity of the electronics involved in theECU, a simplified model of the ECU was introduced. As a consequence ofthe simplified ECU model, some dynamics were not included in the model.Applying different PWM signals to the model and comparing them with itscorresponding measured data clearly shows that the model behaves quitesimilar (having the same frequency, time constant and mean value) to themeasured data for high PWMs, and not as good for low PWMs. This phe-nomenon is illustrated in figures 4.9 and 4.10. Having modeled the currentas explained above will affect the system at different stages, e.g. controlling,for more information the reader is referred to Chapter 6.

6.8 6.81 6.82 6.83 6.84 6.85 6.86 6.87 6.88 6.89

0

50

100

150

200

250

time [s]

duty

cyc

le [%

] and

cur

rent

[mA

]

Duty Cycle [%]Measured Current [mA]Modeled Current [mA]

Figure 4.9: Comparison between modeled and measured current, when ap-plying a PWM of 75%, where the frequency, time constant and mean value.

48

5.76 5.78 5.8 5.82 5.84 5.860

20

40

60

80

100

120

140

160

180

200

time [s]

duty

cyc

le [%

] and

cur

rent

[mA

]

Duty Cycle [%]Measured Current [mA]Modeled Current [mA]

Figure 4.10: Comparison between modeled and measured current, when ap-plying a PWM of 40%,where the frequency is the same. However the ampli-tude differed quite a lot.

4.3.2 Pressure

The braking torque is decided by the air pressure going into the regulatingvalve, therefore it is of great importance to have a good pressure model.In this section the modeled pressure and the measured one, for both of theprototypes, was validated by applying different PWM signals. Finally apressure sequence was used as a reference pressure to see how well the modelcan follow the reference pressure.

Pressure with different PWM

Figure 4.11 compares the modeled pressure with the measured one, whena PWM of 82% is applied for both fill valve and ventilation valve. Theexperiment was carried out on Prototype 1.Analyzing the characteristics of the fill valve, depicted in the upper figure inFigure 4.11, it is observed that the simulated pressure has a faster dynamicthan what the measured pressure has. However in overall the modeled pres-sure behaves quite similar to the measured pressure. The simulated pressurewas derived by tuning the discharge coefficient (see Section 5.1.1 for moreinformation on the discharge coefficient). In the same way, the lower figurein Figure 4.11 describes the ventilation process of the dead volume, wherethe modeled pressure acts very similar to the measured pressure. However,the dynamics of the model is slow.

49

410 420 430 440 450 460 470 4800

0.2

0.4

0.6

0.8

1

Time [Sample]P

ch/P

sup

[Nor

mal

ized

]

Measured PressureModeled Pressure

1260 1270 1280 1290 1300 1310 13200

0.2

0.4

0.6

0.8

1

Time [Sample]

Pch

/Psu

p [N

orm

aliz

ed]

Measured PressureModeled Pressure

Figure 4.11: The upper figure shows the filling process on prototype 1, whena PWM of 82% is applied to the valve. The lower figure indicates the venti-lation of the dead volume, when a 82% PWM is applied.

As stated in Section 4.3.1, for low PWM values the measured current differssignificantly compared to the simulated current. This would affect the clos-ing and opening reaction time for the valve, influencing the pressure whichis depicted in Figure 4.12.

300 400 500 600 700 800 900 1000 1100 12000

0.2

0.4

0.6

0.8

1

Time [Sample]

Pch

/Psu

p [N

orm

aliz

ed]

Pressure Validation

300 400 500 600 700 800 900 1000 1100 1200−20

0

20

40

60

Time [Sample]

Err

or [%

]

Error

Measured PressureModeled Pressure

Error

Figure 4.12: The figure shows the filling and ventilation process on prototype1, when a PWM of 50% is applied to the valves. The lower figure is the errordifference between the modeled and measured pressure, shown in percentage.

50

Studying Figure 4.12, it is apparent that using a PWM of 50% to drive bothof the valves would result in an undesirable behavior, where the error couldincrease up to 60%. In order to achieve a better result, more work should bedevoted to designing a better model for the ECU.

As discussed in Section 4.3.1 the simulated current coincides better with themeasured pressure for higher PWMs. Using a PWM of 82% for both of thevalves, the resulting pressure is shown in Figure 4.13.

0 200 400 600 800 1000 1200 1400 1600 18000

0.2

0.4

0.6

0.8

1

Time [Sample]

Pch

/Psu

p [N

orm

aliz

ed]

Measured PressureModeled Pressure

0 200 400 600 800 1000 1200 1400 1600 18000

0.2

0.4

0.6

0.8

1

Time [Sample]

Dut

y C

ycle

[Nor

mal

ized

]

Dutycycle fill valveDutycycle vent valve

Figure 4.13: The figure shows the filling and ventilation process on prototype1, when a PWM of 82% is applied to the valves. The valves are active oneat the time, where the modeled and measured pressure coincides quite well.

All of the results shown above were carried out on Prototype 1, where themodeled pressure behaved more like the measured one for high PWMs thanlow PWMs. The same model was also run on Prototype 2 for validation.Applying a 82% PWM on fill and ventilation valves, where a dead volume of75cm3 and an orifice of 1 mm was used. Figure 4.14 indicates the behaviorof the modeled pressure corresponding to the measured pressure.

51

360 370 380 390 400 410 420 430 4400

0.2

0.4

0.6

0.8

1

time [s]P

ch/P

sup

[Nor

mal

ized

]

Filling: Orifice 1.0 mm and Volume 75 cm3

Measured PressureModeled Pressure

860 880 900 920 940 960 980 1000 10200

0.2

0.4

0.6

0.8

1

Time [Sample]

Pch

/Psu

p [N

orm

aliz

ed]

Venting: Orifice 1.0 mm and Volume 75 cm3

Measured PressureModeled Pressure

Figure 4.14: The upper figure shows the filling process on prototype 2, whena PWM of 82% is applied to the valve. The lower figure indicates the ven-tilation of the dead volume, when a 82% PWM is applied. In both cases adead volume of 75cm3 and an orifice diameter of 1 mm was used.

Pressure with a scheme of random duty cycles as input

Having validated the fill valve and ventilation valve, a sequence of randomduty cycles is tested to show how the model reacts at different PMW signals.As discussed in Sections 4.3.1 and 4.3.2, the current model for low PWMsdiffered extensive from the measured data, affecting the closing and openingof the valves which in its turn influenced the pressure. Figure 4.15 indicateshow the modeled pressure reacts compared to the measured pressure for asequence of duty cycles.

2 4 6 8 10 12 140

2

4

6

8

time [s]

Measured pressureModel pressure

2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

time [s]

Duty cycle fill valveDuty cycle vent valve

Figure 4.15: The upper figure shows the filling and ventilation of the pressurefor differen PWM values, and the lower figure indicates the various dutycycles applied on each of the valves.

52

Analyzing Figure 4.15, it is noticeable that for some cases the pressure be-haves quite alike the measured pressure, however the pressure building upfor a desired duty cycle will affect the pressure behavior for the upcomingduty cycle. For example between 3.5-4 seconds, the modeled pressure ishigher than the measured one. In order to compensate for the high pressurea higher duty cycle is needed for emptying the pressure to reach the expectedpressure. This is impossible due to the pre-defined PWM sequence, thereforea poor result is achieved.

53

Chapter 5

Model Improvements

A models is an approximation of the real process. In a model there are severalfactors which has to be considered in order to get a fairly good model. Forexample disturbances or other parameters that have leading affect on thesystem’s behavior has to be taken into consideration. In this case the valveshave shown to react quite differently from each other, regarding how fast theyfill or empty the air pressure. One of the parameters affecting the valves isthe discharge-coefficient, Cd (see (3.14)), the discharge coefficient is furtherdiscussed in Section 5.1. Other parameters shown to be quite importantthroughout the project were: the air gap in the magnetic part of the valve,and pre-load force from the spring in the mechanical part of the valve. Exceptparameter tuning, physical disturbances, i.e. temperature and time delays,acting on the system were also investigated in order to improve the model.

5.1 Parameter tuning

With parameter tuning, the aim is to build a model by considering variousbehavior in the system. The discharge coefficient and air gap is studied inthis section.

5.1.1 Discharge coefficient Cd

In a nozzle or other restrictions for a compressible-flow, the ratio of theactual mass flow rate to the isentropic1, adiabatic mass flow rate is knownas discharge coefficient which is expressed by ([12])

1An isentropic flow is a flow which is both adiabatic and reversible, meaning that noenergy affects the system or is taken from it.

55

Cd =Mactual

Misen,adiab

=

∫ ∫A2

ρ (~v·~n) dA

Misen,adiab

(5.1)

where Cd is the discharge coefficient, n denotes the unit normal vector, v isthe velocity vector, ρ describes the density and A2 is the minimum cross-sectional area of a converging nozzle.

The geometry of the orifice affects the value of discharge coefficient. In [12]John C. Kayser and Robert L. Shambaugh, studied four types of geometriesi.e. sharp or knife-edge orifices, straight-bore orifices, quadrant or rounded-entry nozzles, and elliptical-entry nozzles. During the study, the orifice di-ameters used varied in a range of 0.9-1.9 mm. The result showed that thevalue of the discharge coefficient varied between 0.6 and 1. Cd = 1 occursduring the circumstances where the actual flow is approximately adiabatic.

With the results in [12] as starting point, the values for discharge coeffi-cient in this Master’s Thesis were chosen. Figures 5.1 and 5.2 indicate howdifferent values of discharge coefficient affect the fill and ventilation time ofthe valves.

0.8 1 1.2 1.4 1.6 1.8

0

1

2

3

4

5

6

7

8

x 105 Discharge coefficient fill valve

Time [s]

Pre

ssur

e [B

ar]

Cd 0.6Cd 0.7Cd 0.5

Figure 5.1: Applying a PWM of 82% to the fill valve, using different dischargecoefficient values. Where the results indicate, the greater the discharge coef-ficient is the faster is the valve.

Figure 5.1 shows the filling process, using 82% PWM with different Cdfill

values. Moreover the closer discharge coefficient value is to 1, the faster thevalves behavior is. The ventilation valve behaves the same as the fill valve,i.e. higher Cdvent value gives a faster system, which is depicted in Figure 5.2.

56

3.8 4 4.2 4.4 4.6 4.8 5 5.20

1

2

3

4

5

6

7

8

x 105 Discharge coefficient vent valve

Time [s]

Pre

ssur

e [B

ar]

Cd 0.5Cd 0.6Cd 0.7

Figure 5.2: Applying 82% PWM to ventilation valve, and altering the valuefor Cdvent, a faster system is achieved when bigger value for Cdvent is used.

5.1.2 Air gap

Recall from Section 3.3.2 that the magnetic force produced depends on thelength of air gap where the magnetic flux passes. Due to limited access toequipment, it is quite difficult to predict how the magnetic flux behaves,therefore different cases were investigated. The three cases that were studiedare constant air gap, varying air gap depending on the armature position,and small changes in the air gap.

The data acquired from the valve manufacturer indicates that the air gapcan vary up to 0.6 mm. Figure 5.3 shows how the armature position altersthe amount of air gap affected by the magnetic flux.

Armature

Positive x-direction

System pressure 8.4 bar

Air gap

1

2

Figure 5.3: The figure indicates the change in air gap due to the armaturemovement.

First of all a Simulink model was build in a way that the air gap could vary

57

up to 0.6 mm (which is the maximum amount of variation), depending onthe armature movement. The results showed a very strong magnetic force,FM , overcoming all the other forces mentioned in Section 3.3.3. This behav-ior made the valve to fully open at a very low PWM, which is not desirable(see Figure 5.4). In order to overcome the magnetic force e.g. a very strongspring force was needed, and the force could be achieved by using a springconstant of ks = 80000N/m, which is unacceptable comparing it with thedata received from the valve manufacturer which indicates a spring constantof ks = 1170N/m.

0.8 1 1.2 1.4 1.6 1.8 2 2.20

10

20

30

40

50

60

70

80

90

100

Time [s]

For

ce [N

]

Magnetic Force [N]Preload Force [N]Viscous Friction [N]Spring Force [N]

Figure 5.4: The figure shows how big the magnetic force is compared to otherforces, i.e. spring-, viscous-, and pre load-force.

Not having relevant information on the magnetic flux characteristic, a con-stant air gap was assumed. The places where air gap is present in the valvehave been marked which is depicted in Figure 5.3. While the armature moves,the air gap in both regions change proportionally, that is when air gap 1 in-creases, the other one decreases and vice versa. Hence, a constant air gap wasused. By modeling the air gap as constant, the magnetic force, FM , endedup in a reasonable magnitude compared with the other forces acting on thearmature. However it did not give a satisfying result so a third approach hadto be examined.

The third method changes the air gap with a factor of the armature dis-placement, which resulted in a minimal change in the air gap. By havinga small change in the air gap a more appropriate magnetic force, consider-ing its magnitude is achieved. Summarizing this section, the last option forchoosing air gap gave better result than the other two, therefore the finalmodel was based on the third approach.

58

5.2 System identification of electrical model

One of the methods for creating a model is System identification. As men-tioned in Chapter 3, a simplified model of the electronics in the ECU wasconsidered. This resulted in a poor model of the current. Using systemidentification-toolbox2, the whole electrical circuit is assumed as a black-box,where only the input (PWM signal) and the output (current) are of interest.According to the circular flow representing system identification shown inFigure 5.5, experiment is conducted and data for the input and output aregathered, where a model is chosen and accommodated to the data and at theend validated.

Construct experiment and

gather data

Polish and study data

Accommodate model to data

Does data need to be filtered?

Choose model structure

Evaluate model

Can model be accepted?

Data

Data

Model

No

Model structureNot OK

Data Not OK

Yes

Figure 5.5: System identification’s circular flow. The rectangles are the com-puter’s main responsibilities, and the ovals are user’s main responsibilities. [3]

2System Identification Toolbox software extends the Matlab computational environ-ment for estimating linear mathematical models to fit measured data from dynamic sys-tems.

59

By following the circular flow presented in figure 5.5, an ARX3 model of theelectrical part of the valve was created.Applying the ARX model to the simulink model, did not provide satisfactoryresult, because the identification was done on a specific PWM value. As dis-cussed in previous sections the current behaves differently for various PWMsignals, therefore just considering the identification based on a specific PWMwould most likely contribute to a poor model. However instead of using aconstant PWM signal, an altering one could be used, including both low andhigh PWMs. Due to limited time, this was not implemented.

5.3 Time Delay

The data gathered from experiments indicate a time delay which has to betaken into consideration to get more precise model. The time delay coulddepend on three different aspects. First of all the sampling time of the ECU,second the collected sample data by the pressure sensor and finally the timeit takes for the current to reach the desired value. By studying Figure 5.6,where current plots and its respective duty cycle have been plotted together.It can be seen that there is a time from which the PWM goes on until thecurrent starts to increase.

4.74 4.742 4.744 4.746 4.748 4.75 4.752 4.754 4.756 4.7580

50

100

150

200

time [s]

duty

cyc

le [%

] and

cur

rent

[mA

]

Duty Cycle [%]Measured Current [mA]

3.9 3.902 3.904 3.906 3.908 3.91 3.912 3.914 3.916 3.918 3.92

0

20

40

60

80

100

120

time [s]

duty

cyc

le [%

] and

cur

rent

[mA

]

Duty Cycle [%]Measured Current [mA]

Time Delay

Time Delay

Figure 5.6: The two plots show the current and duty cycle for 40% (lowerfigure), and 75% (upper figure). In both of the figures the delay time betweenwhen the PWM goes on until the current moves is recognized, which is about15 ms.

3ARX model is a type of linear parametic model, which is described by:

A(q)y(t) = B(q)u(t) + e(t) (5.2)

60

The time delay appearing in the plots has been modeled as a constant delayof 15 ms this does not represent the time delay in all cases, but is a fairlygood approximation of the time delay. The time delay is of great interestwhen applying controller on the system. The reaction time of the valvesaffect the opening and closing time, which could cause, e.g., the ventilationvalve to empty more than it should due to the valve reaction.

5.4 Temperature dependent resistance

When analyzing the measured data, a special phenomenon was observed.The current passing through the coil behaved quite oddly, it decreased withtime (see Figure 5.7). This was due to increased resistivity(ρ) in the wires ofthe coil. Resistivity is proportional to resistance(R) leading to increment ofresistance, when voltage is applied to the system.

10 20 30 40 50 60

270

275

280

285

290

295

time [s]

curr

ent [

mA

]

current [mA]PWM dutycycle [%]

Figure 5.7: The temperature dependent resistance

This characteristic had to be taken into consideration in order to get a morecorrect model. The resistance which appeared to be depending on the tem-perature was derived accordingly:

By considering the first law of thermodynamics [10], the change in internalenergy(∆Eint) is expressed as

∆Eint = Q + W (5.3)

where Q denotes the energy transfer represented by heat, and W is theamount of work done. Internal energy changes of an ideal gas [6] is written

61

as

∆Eint =nRT

γ − 1(5.4)

where R is molar gas constant, n is number of moles and γ = Cp

Cv= 1.4

The equations for energy transfer and work done due to heat are

Q = −C(T − Tenvironment) (5.5)

andW = Ri2(t) (5.6)

Combining (5.4), (5.5), (5.6) and (5.3) the following relation is obtained

T =γ − 1

nR(Ri2(t)− C(T − Tenvironment)) (5.7)

Over a limited temperature range, the resistance of a conductor varies ap-proximately linearly with temperature according to

R = R0(1 + α(T − T0)) (5.8)

Note that the unit for α is [Kelvin−1].

A model for the resistance was made according to the equations in Section5.5. Looking closer at Figure 5.7, it can be seen that the resistance changesover a long period of time (100 s). In reality the valves will not be active insuch a long time. Because of this observation the model of the temperaturedependent resistance has been neglected.

62

Chapter 6

Control Design

As described in [13], controllers are designed such that the system outputsare used to obtain the desired inputs, in order to achieve the system require-ments. Often the purpose is to follow a reference signal.

There are various types of controllers to choose from, and the most commonone is the PID-controller. The PID-controller is often used for industrialpurpose, due to its simplicity. Because of satisfying results attained usinga PID-controller in other works such as [14] carried out by Klas Hakanssonand Mikael Johansson, the main focus has been on PID.

Two different PID methods has been investigated, whereof, the first oneis implemented such that the acting valves (fill and ventilation) work one atthe time, this approach has been examined by Vidar Steinsland [15], and onlythe results from that study will be presented here. However the main focusin this report will be on the second method, which is based on controllingthe system while both of the valves are active simultaneously.

6.1 Control Objectives and Background

The aim of the Master’s Thesis is to study what performance can be achievedwith this system configuration. In terms of acceptable control error and risetime, the aim is to make the system as fast and accurate as possible. Asmentioned before, a PID-controller is using the pressure in the chamber asfeedback. Moreover, improving the controller using non-linear control andprediction have also been investigated, and this is described in the upcomingsections.

63

6.2 PID Control

The PID controller is the most common controller used on different kinds ofsystems. With this simple controller, fairly good results can be achieved byonly tuning the PID parameters. The PID controller can be implementedin various ways. However the most common ones are: error feedback andoutput feedback. In this case error feedback shown in Figure 6.1 has beenused.

-1

yuerrorrPlantController+

Figure 6.1: The structure of a basic feedback control loop, where r denotesthe reference signal, u is the control signal, and y is the output.

Tuning the parameters can be done in two ways. The easiest way is todirectly tune the PID parameters, while experimenting on the prototypes.However by doing so the model derived in Chapter 3 would loose its relevance.Therefore it is desired to develop a controller based on the model.

6.2.1 Principles of PID Control

As mentioned above an error feedback controller has been used. In continuostime this controller is described as

u(t) = Pe(t) + Dde(t)

dt+ I

t∫

t0

e(t)dt

= K

(e(t) + Td

de(t)

dt+

1

Ti

t∫

t0

e(t)dt

)(6.1)

where K denotes the gain, Td is the derivative time, and Ti is the integrationtime. In order to achieve a satisfying results, the rule of thumb for tuningthe parameters is to first fix the P-part, secondly the Derivative, and finallythe Integration part. There are other concepts for adjusting the parameters,e.g. Ziegler-Nichols Tuning Rule [18]. We now provide brief description on

64

how each part affects the system.

Proportional action

The proportional part of a PID controller helps to make the system faster.The main purpose of the P-part is to reduce the error so that the output sig-nal can reach the reference value faster. However a very large proportionalpart will end in a quite oscillative system.

Derivative action

When an oscillative system is present due to high proportional gain, a deriva-tive action is recommended. The derivation part tries to dampen the system,i.e. to reduce the oscillations, however it will slow down the process.

Integral action

If the system is stable there may be an error between the output and thereference at stationarity. By inducing the integral part, this station error canbe removed.

6.2.2 Implementation

The controller is designed based on the derived model (see Chapter 3) andhence the controller is in continuous time, which is acceptable if applied onthe simulink model. The intention is to implement the model on the proto-types, and for this the controller has to be in discrete time. Equation (6.1),can be written in discrete time as

un = K

(en + Td

en − en−1

Ts

+Ts

Ti

n∑

k=0

ek

)(6.2)

where en is the error in sample n, and Ts denotes the sampling time.

6.3 Approaches

Looking back at (6.1), the control signal depends on the behavior of the er-ror. When the pressure does not reach up to the reference value, a positive

65

error is obtained, contributing to a positive control value. However havinga pressure bigger than the reference pressure, gives a negative error followedby a negative control signal. The control signal obtained from the controllerhas to be distributed between the two valves. In this Master’s Thesis twodifferent approaches are studied. The first approach is to design a controllerwhere the control signal operates in a way that one of the fill or ventila-tion valves are active at a time. This approach is investigated in [15] andwill not be investigated in depth, but the results are shown. The secondapproach concentrates on using a control signal, where the valves are activesimultaneously, which is described in details.

6.3.1 Scheme 1-Fill Valve and Ventilation Valve acti-vated separately for filling and ventilation.

The first approach is designed in a way that the control signal from thecontroller is resolved into two individual pulsing of the two valves. Figure 6.2is a sketch of how the method works and how it distributes the control signalbetween the two valves. However this approach is investigated through aparallel work and will not be discussed in details here.

−100 −50 0 50 100

0

20

40

60

80

100

Control Signal − u [%]

Val

ve D

uty

Cyc

el [%

]

PWM Pulsing Scheme 1

−100 −50 0 50 100

0

20

40

60

80

100

Control Signal − u [%]

Val

ve D

uty

Cyc

el [%

]

Traditional Linear Scheme

Empty ValveFill Valve

Empty ValveFill Valve

Figure 6.2: Traditional Pulsing Scheme (left) and Pulsing Scheme 1 (right)

6.3.2 Scheme 2-Both Valves activated simultaneouslyfor filling and ventilation.

The pulsing scheme is based on results from former research [17], wherethe aim was to design a method to completely remove the nonlinearitiesand dead band over the entire range of control signal u. The algorithmintroduced in [17] was used as base line for building a distributer which isable to distribute the control signal to fill and ventilation valve, in order to

66

keep both of the valves working simultaneously. The distributer is designedaccording to

ufill = u(100%− 2dcmin)/2ui + 50%uvent = −u(100%− 2dcmin)/2ui + 50%

for −ui ≤ u ≤ ui

ufill = dci + (u− ui)(100%− 2dcmin)/ui

uvent = dcmin

for ui < u ≤ 100% (6.3)

ufill = dcmin

uvent = dci − (u + ui)(100%− 2dcmin)/ui

for −100% ≤ u < −ui

where dcmin is the minimum possible duty cycle where the valve will stillrespond, dci is the duty cycle corresponding to the inflection point where theduty cycle is increased by twice the slope to maintain a linear output/inputrelationship and ui is the corresponding control output. Figure 6.3 givesmore insight on how the method works.

−100 −50 0 50 1000

10

20

30

40

50

60

70

80

90

100

control signal: u [%]

duty

cyc

le [%

]

Pulsing Scheme 2

Fill ValveVentilation Valve

di,ventdi,fill

ui,ventui,fill

Figure 6.3: Pulsing Scheme 2

The inflection points in Figure 6.3 are placed at the specified positions, dueto the results obtained in a research done by R. van Varseveld and G. M.Bone [17], showing that placing the inflection points on positions which ismarked in Figure 6.3 would remove the nonlinearities over the entire rangeof the control signal u. Equation (6.3) specifies that both the fill valve andventilation valve will not have a duty cycle value which undergoes dcmin forrespective valve. Moreover both of the valves have an upper limit of 100%duty cycle.

67

A modified version of the method was used, where the saturated control sig-nal range [0%, 100%] is scaled into the active duty cycle range [dcmin,f , 82%]and [dcmin,v, 80%] for fill valve resp. vent valve. This is depicted in Figure 6.3.This is done due to the fact that in Chapter 4 it was seen that a duty cycle ofdcmax was sufficient to fully open the valves. Using a higher duty cycle overmagnetized the valves, causing the coil to discharge even slower, resultingin a longer time to close the valves when needed. However having a deadband for duty cycles beneath dcmin, would increase the demagnetization time,closing the valves later than they should. This phenomenon causes problemsduring the controlling part, therefore a tolerance of ±0.1 was added to thedesigned distributor. Whenever the error signal is within the tolerance rangethe control signal is set to zero.

6.4 Results-Scheme 2

Before applying the controller on the prototypes, the designed controllerwas applied on the model. Simulating the model and investigating how thecontroller acts on the model, the three parameters, i.e. proportional, integraland derivative gains were tuned to achieve the desired behavior. Havingfound satisfying parameters, they were used as initial values when controllingthe prototypes.

6.4.1 Simulations

A simulation is done on the model using an orifice diameter of 1.3 mm and adead volume of 75 cm3, which is depicted in Figure 6.4, having concluded pre-diction and boosting action. Studying Figure 6.4, the results are satisfying.The control signals verify that both of the valves are working simultaneously,which was the aim of the second approach. However taking into consider-ation the complexity of the distribution algorithm (equation 6.3), a reallygood PID-controller is achieved by the model.

68

2 4 6 8 10 12 14

2

4

6

8

10

Time [s]

Pre

ssur

e [B

ar]

Reference Pressure [Bar]Simulated Pressure [Bar]

2 4 6 8 10 12 140

20

40

60

80

100

Time [s]

Dut

y C

ycle

[%]

Fill Valve DC [%]Ventilation Valve DC [%]

Figure 6.4: The above figure shows the simulated pressure and its referencesignal, when using an orifice diameter of 1.3 mm and a dead volume of 75 cm3,where the effects of prediction and boosting action are concluded, moreoverthe lower figure shows the control signal for each of the valves.

6.4.2 Test on Prototypes

Having found initial values for the PID parameters, the following experimentwas carried out on Prototype 2, using an orifice diameter of 1.3 mm and 75cm3 of dead volume. Both valves being active at same time makes it moredifficult for the controller stabilizing the pressure around the reference value.This contributes oscillations in the system depicted in Figure 6.5.

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

time [s]

pres

sure

[Bar

]

Reference Air PressureMeasured Air Pressure

Figure 6.5: Resulting PID-controller on prototype two with orifice diameterof 1.3 mm, including boosting action and prediction.(Scheme 2)

69

6.5 Control Improvements

As mentioned before, due to the delays corresponding to the valves reactiontime and the ECU, it is quite difficult to close the valve at a required time.When the pressure approaches the reference pressure, it is desired to closethe valves so that the pressure will always stay around the reference value.However having the delays in the system causes a late reaction from the valve,forcing the controller to make the other valve stronger in order to compensatefor the first valve, resulting in a oscillative signal. By using prediction theproblem could be solved. While experimenting with the controller in somecases the control signal needed to be stronger. Instead of increasing theproportional gain, which could give an oscillative system, a non-linear controlwas introduced. In this thesis only control improvement using non-linearcontrol will be introduced in details, prediction is described in [15].

6.5.1 Anti-Windup

All actuators have physical limitations, a control valve cannot be more thanfully open or fully closed. The integral action of a PID controller is stableduring the time where the feed back loop is not broken. However due tosaturation the feed back loop would be broken because the output of thesaturating element will not be influenced by its input. This would cause theintegrating part to integrate more than it should giving very large values,which is not desirable. Since it will take the system a long time to recover,having a negative effect on the controlling of the system. Therefore integralanti-windup could be used in order to prevent this sort of behavior, this wasalso implemented.

Integrator windup can be avoided, by making sure that the integral is keptto a proper value when the actuator saturates, so that the controller is readyto resume action, as soon as the control error changes.

6.5.2 Improved Control using Non-Linear Control

During simulation with tuned PID parameters, it could be seen that in somecases the pressure is too slow when trying to reach the reference pressure,this phenomena is depicted in Figure 6.6. The pressure has stoped increasingbetween 2.89-2.94 seconds before reaching the reference value which is 6 bar.In this case the control signal needs to be stronger (bigger). In order to speedup the process a non-linear control was introduced.

70

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2

4

4.5

5

5.5

6

Time [s]

Rel

ativ

e P

ress

ure

[Bar

]

PrefPch

Figure 6.6: The dead volume pressure for a reference change when a PIDcontroller has been applied, where slow PID reaction can be observed forcertain cases

Figure 6.7 is a schematic sketch on how the non-linear control works. Thenon-linear control is basically an additional integrator inserted in the sys-tem. According to Figure 6.7 the whole process depends on the derivativeof the pressure inside the chamber. When there is no change in the pressurethe extra integrator is activated, generating an additional control signal, uN ,which is added to the control signal from the PID-controller, boosting upthe pressure to reach the reference pressure. However, during the integra-tion if the pressure begins to change, uN will hold its previous value andstop increasing. To prevent from boosting up the control signal too much,whenever the error is within the tolerance interval i.e. ±0.1 bar from thereference value, uN is set to zero. If there is a change in the pressure thenthe integrator will not be activated, resulting in uN = 0.

Is Pchamber

= 0 ?

YES

NOPchamber

•

Activate/Holdintegrator

Reset/deactivate integrator

Is |error|<= tolerance

?

YES

NO

•

Figure 6.7: The algorithm of the non-linear control, where it is illustratedhow the integration of the control signal is activated or deactivated dependingon the pressure change in the dead volume.

The non-linear control uses output feed-back. The block diagram is shownin Figure 6.8.

71

-1

yuFBerrorrPlantController+ +

uN

u

Non-Linear control

Figure 6.8: Block diagram illustrating a controller based on feedback princi-ple with an additional non-linear control.

Figure 6.9 shows how the non-linear control method works when it is insertedin the system. Analyzing the plot, in the beginning when the pressure isconstant the control signal uN is integrated up. Between the interval 2.5-2.9seconds where the pressure changes, uN is held at its previous value andwhen the pressure is within the tolerance the integrator resets and uN = 0.

2.5 2.6 2.7 2.8 2.9 3 3.1 3.24

4.5

5

5.5

6

Time [s]

Rel

ativ

e P

ress

ure

[Bar

]

2.5 2.6 2.7 2.8 2.9 3 3.1 3.20

2

4

6

Time [s]

Boo

st a

ctio

n: u

n [%

]

PrefPch

Figure 6.9: An illustration of how the extra control signal, uN , behaves whennon-linear control is applied to the system

6.5.3 Improved Control using Prediction

The time delay resulting from the valve reaction times could cause the valvesto open and close later than they should. This will prevent the controllerfrom controlling the system properly. Having considered the time delays,one way to overcome the phenomenon is the usage of prediction. By usingerror prediction the controller is able to estimate in advance, when the errorwill fall between the tolerance boundaries i.e. ±0.1 bar from the referencevalue, and being able to react in time to close the valve. Error estimation hasbeen applied as an improvement to the controller estimating the error, onesample or even two samples in advance by calculating the error derivativeusing Euler Backward for predicting when to close the valve.

72

For further information about prediction, it is referred to the parallel work [15],where affects of prediction and results has been investigated more in details.

6.6 Comparison between the two approaches

The Scheme 2 approach turned out to give very poor result compared toScheme 1 which was investigated in [15]. The results obtained by Scheme 1,using one valve at the time, is shown in Figure 6.10.

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

Time [s]

Rel

ativ

e P

ress

ure

[Bar

]

PchPref

Figure 6.10: Resulting PID-controller on prototype two with orifice diameterof 1.3 mm. (Scheme 1)

Comparing the results from both of the schemes, it is evident that Scheme1, works much better and it is easier to control than Scheme 2. Anotheradvantage of using Scheme 1 may be the life span of the valves. By usingScheme 1, one valve is activated at the time, increasing the durability of thevalves. Having both of the valves active at the same time making it difficultto regulate the system near the reference signal, ending up in oscillationssuch as those seen in Figure 6.5.

73

Chapter 7

Conclusion and Future work

7.1 Conclusion

The Master’s Thesis is about investigating whether it is possible to replacetwo On/Off-valves and a pressure chamber with pressure sensor with today’sProportional valve to control the air pressure that determines the brakingtorque in the retarder. A model of a pneumatic actuator consisting of twoon/off solenoid vales, a pressure chamber and a regulating valve was derived.The pressure and the current of the model were validated against the mea-sured data. The results showed that the model has similar behavior (samecurrent and pressure) as the real system for high PWM values; however itdiffered to a very great degree for low PWMs, which possibly is due to thesimplified model of the Electrical Control Unit (ECU) used in the Thesis.Moreover, different size of dead volume (the volume in the chamber, regu-lating valve and the housing) and orifice diameter were investigated in orderto obtain the specified requirements. The results from the experiments indi-cated that a dead volume of 60-75 cm3 and an orifice diameter bigger than1.9 mm will fulfill the desired requirements.

For controlling the pressure a Proportional Integral Derivative (PID) - con-troller was introduced. A distributor was used to distribute the control signalgenerated from the controller to each of the valves, which in this case twodifferent approaches were examined, denoted as Scheme 1 and Scheme 2.Scheme 1 investigated the case where one valve was active at a time, whichwas examined by Vidar Steinsland in [15]. In this Thesis Scheme 2, whereboth valves were active simultaneously, was studied. Due to the time delayspresent in the system, error prediction was applied to the controller for im-proving its performance and being able to close the valves in time so that the

75

fill and ventilation valve would not fill or empty the pressure more than theyshould respectively. Another phenomena seen in the measurements was theslow reaction of the pressure when trying to reach the reference pressure insome cases. An extra integrator was presented as a non-linear control to im-prove the controller and boost up the process such that the pressure reachesthe reference value faster. By applying the controller on the prototypes,having considered prediction and non-linear control as an improvement tothe controller, Scheme 1 gave better results than Scheme 2, which was quiteoscillative and difficult to control compared to Scheme 1.

7.2 Future work

Throughout the Master’s Thesis some improvements have been discussed andapplied on the system. However there are other aspects which have not beenconsidered in the Thesis which would be worth to look into for further im-provement of the system.

It is encouraged to obtain a good model as possible for the current, dueto its decisive effect on the other parts of the system. Therefore in the fu-ture it is recommended to consider a better model for the electronics in theECU, which probably will results in a more satisfying current model. Theexperiments applied during the project were carried out by a frequency offfreq. Discussing with more experienced engineers within the subject theysuggested an investigation with lower frequency could be an option that mayimprove the controlling of the system. An alternative approach for improv-ing the current could be to design a black-box model of the whole electricalsystem. Instead of using just one PWM signal, which is the case in thisMaster’s Thesis, a varying PWM signals is to be taken into consideration. Amodel based controller is also suggested which would give more freedom inusing different types of controllers for controlling the valves.

76

Bibliography

[1] Scania CV AB, Sodertalje. 2008. http://www.scania.com

[2] Kleinknecht Automotive GMBH. 2008. http://www.kleinknecht.com/

[3] Lennart Ljung, Torkel Glad, Modellbygge och simulering,2004.

[4] Gunnar Petersson, ”Elkretsanalys”, Alfvenlaboratoriet, KungligaTekniska Hogskolan, Stockholm, 2003.

[5] G.Petersson. ”Teoretisk Elektroteknik - Stationara fenomen”. 2004.

[6] C.Nordling, J.osterman. ”Physics Handbook for Science and Engineer-ing”. 1999.

[7] Christer Nyberg, Mekanik Grundkurs.Liber AB, 336, January 2003.

[8] Karl-Erik Rydberg, Basic Theory for Pneumatic System Design.IKP,Fluid and Mechanical Engineering Systems, 1997.

[9] Clayton T. Crowe, Donald F. Elger, John A. Roberson, EngineeringFluid Mechanics.

[10] Raymond A. Serway, John W. Jewett,Jr., Physics for Science and En-gineering with modern physics. Californa State Polytechnic University-Pomona,6th edition,2004.

[11] B.Eriksson, J. Wikander. Lecture notes in MF2007. Dynamics and Mo-tion Control. 2007.

[12] Kayser, J.C.; Shambaugh, R.L.,Discharge coefficients for compressibleflow through small-diameter orifices and convergent nozzles. ChemicalEngineering Science, v 46, n 7, 1991, p 1697-1711.

[13] Lennart Ljung and Torkel Glad. Reglerteknik: Grundlaggande teori.Preliminary edition, 2005.

77

[14] Klas Hakansson and Mikael Johansson, Modeling and Control of anElectro-Pneumatic Actuator System Using On/Off Valves. Master’sThesis, Department of Electrical Engineering, Linkoping 2007.

[15] Vidar Steinsland, Modeling and Control of Retarder using On/Offsolenoid valves, Master’s Thesis, Department of Electrical Engineering,Royal Institute of Technology, Stockholm 2008.

[16] Linda Krause and Jimmy Larsson, ”Konceptstudie: TryckstyrningAv Hydrodynamisk Broms”, Master’s Thesis, Linkoping University,Linkoping 2004.

[17] Robert B. van Varseveld and Gary M. Bone, Accurate Position Controlof a Pneumatic Actuator Using On/Off Solenoid Valves. IEEE/ASMEtransaction on mechatronics, vol. 2, NO. 3, September 1997.

[18] Katsuhiko Ogata, Modern Control Engineering. Prentice Hall, New Jer-sey, 4 th ed., 2002.

78

Appendix A

Appendix

A.1 Linearizing

According to [11] a nonlinear system x = f (x, u) can be linearized aroundsome operating point x0, u0 by considering a neighborhood around theoperating point and approximating the nonlinear model with a truncatedTaylor series. This is done by setting x = x0 + ∆x, u = u0 + ∆u andy = y0 + ∆y, then

x = f (x, u) + ∂f∂x

∣∣x=x0

u=u0

∆x + ∂f∂u

∣∣x=x0

u=u0

∆u

y = g (x, u) + ∂g∂x

∣∣x=x0

u=u0∆x + ∂g

∂u

∣∣x=x0

u=u0∆u

(A.1)

The system has been thought to be in equilibrium either when both valvesare deactivated, when only one of the valves is fully open, or when bothvalves are fully open. The equilibrium when both valves are fully activatedhas been considered. In this case the PWM signal is set to 100 % DC onboth valves, and the pressure will reach a constant pressure unless there areany disturbances disturbing the valves. The steady position will correspondto the current when the coil is fully charged, and will probably be outsidethe physical position limitation in the valve, i.e. outside the walls. In thisequilibrium, the velocity will be zero.

79

A.1.1 Fill valve and Ventilation valve are both acti-vated

Equilibrium points:x0

1 x02 x0

3 x04

x05 x0

6 x07

Linearized Model:

∆x = A∆x + B∆u

∆y = C∆x + D∆u

∆x =

x1

x2

x3

x4

x5

x6

x7

and ∆x =

x1

x2

x3

x4

x5

x6

x7

(A.2)

A = ∂f∂x

∣∣x=x0

u=u0B = ∂f

∂u

∣∣x=x0

u=u0

C = ∂f∂x

∣∣x=x0

u=u0D = ∂f

∂x

∣∣x=x0

u=u0

(A.3)

80

A=

∂f1

∂x1

∣ ∣ ∣ x=

x0

∂f1

∂x2

∣ ∣ ∣ x=

x0

00

00

0

00

10

00

0∂f3

∂x1

∣ ∣ ∣ x=

x0

∂f3

∂x2

∣ ∣ ∣ x=

x0−

bm

p−

π mp

( d0 2

) 20

00

0∂f4

∂x2

∣ ∣ ∣ x=

x0

0∂f4

∂x4

∣ ∣ ∣ x=

x0

0∂f4

∂x6

∣ ∣ ∣ x=

x0

0

00

00

∂f5

∂x5

∣ ∣ ∣ x=

x0

∂f5

∂x6

∣ ∣ ∣ x=

x0

0

00

00

00

1

00

0−

π mp

( d0 2

) 2∂f7

∂x5

∣ ∣ ∣ x=

x0

∂f7

∂x6

∣ ∣ ∣ x=

x0−

bm

p

B=

1

x0 2

(L

of

f−

Lon

xon−

xof

f

) −Lof

f

0

00

00

00

01

x0 6

(L

of

f−

Lon

xon−

xof

f

) −Lof

f

00

00

C=

[ 00

01

00

0]

81

wher

e∂f 1

∂x

1

∣ ∣ ∣ ∣ x=

x0

=−

R

x0 2

( Lof

f−L

on

xon−x

of

f

) −L

off

∂f 1

∂x

2

∣ ∣ ∣ ∣ x=

x0

=−

(u0 1−

Rx

0 1)( L

of

f−L

on

xon−x

of

f

)

( x0 2

( Lof

f−L

on

xon−x

of

f

) −L

off

) 2

∂f 3

∂x

1

∣ ∣ ∣ ∣ x=

x0

=µ

0A

pN

2x

0 1

mp(x

off−

x0 2)2

∂f 3

∂x

2

∣ ∣ ∣ ∣ x=

x0

=µ

0A

pN

2(x

0 1)2

mp(x

off−

x0 2)3−

ks

mp

∂f 4

∂x

2

∣ ∣ ∣ ∣ x=

x0

=

√R

gasT

air

Vch

d0πC

d,s

upP

sup

√k

(2

k+

1

)k+

1k−

1

∂f 4

∂x

4

∣ ∣ ∣ ∣ x=

x0

=−

√R

gasT

air

Vch

d0πC

d,v

entx

0 6

√ √ √ √2k

k−

1

( (P

atm

x0 4

)2 k

−( P

atm

x0 4

)k+

1k

)+

√2k

k−1

( −2 k

(P

2 k atm

(x0 4)2

+k

k

)+

k+

1k

( Pa

tm

x0 4

k+

1k

))

2√( P

atm

x0 4

)2 k−

( Pa

tm

x0 4

)k+

1k

x0 4

∂f 4

∂x

6

∣ ∣ ∣ ∣ x=

x0

=−

√R

gasT

air

Vch

d0πC

d,v

entx

0 4

√ √ √ √2k

k−

1

((P

atm

x0 4

2 k

)−

(P

atm

x0 4

k+

1k

))

82

∂f 5

∂x

5

∣ ∣ ∣ ∣ x=

x0

=−

R

x0 6

( Lof

f−L

on

xon−x

of

f

) −L

off

∂f 5

∂x

6

∣ ∣ ∣ ∣ x=

x0

=−

(u0 1−

Rx

0 5)( L

of

f−L

on

xon−x

of

f

)

( x0 6

( Lof

f−L

on

xon−x

of

f

) −L

off

) 2

∂f 7

∂x

5

∣ ∣ ∣ ∣ x=

x0

=µ

0A

pN

2x

0 5

mp(x

off−

x0 6)2

∂f 7

∂x

6

∣ ∣ ∣ ∣ x=

x0

=µ

0A

pN

2(x

0 5)2

mp(x

off−

x0 6)3−

ks

mp

83

A.2 Calculation of smallest outlet orifice area,

Ao

The amount of air mass flow partly depends on the outlet orifice area, Ao.Due to the armature position, the size of Ao varies. Figure A.1 shows thedifferent cases depending on the armature position.

Figure A.1: The left figure shows the structure of the valve when the arma-ture is at its start position, and the right one indicates the valves structurewhen the armature has moved.

As mentioned before depending on armature position, the smallest outletorifice area is calculated differently. E.g. the left figure in Figure A.1, thesmallest outlet orifice area is computed through:

Ao = πdxp (A.4)

where d is the orifice diameter. When the armature is in movement, at firstAo is calculated by (A.4), however when the valve opens a certain amountAo is attained through A.5.

Ao = π(d

2

)2(A.5)

Table A.1 shows how value of Ao switches between the to cases, where d =1.9mm, and xp : 0− 0.6mm. Note that the values indicated in the table arecalculated for both of the cases for a specific value of xp, and the smallestone is chosen. Hence, when xp = 0 A.4 is used.

84

Table A.1: Table showing calculation of Ao with two different methods for aspecified value of xp, where the smallest Ao is chosen.

xp [mm] Ao = πdxp [mm2] Ao = π(

d2

)2[mm2] chosen Ao

0 0 2.84 (A.4)0.1 0.6 2.84 (A.4)0.4 2.39 2.84 (A.4)0.5 2.98 2.84 (A.5)0.6 3.58 2.84 (A.5)

85