Modeling and Control of Retarder using On/Off Solenoid Valves

Transcription

1 Modeling and Control of Retarder using On/Off Solenoid Valves VIDAR STEINSLAND Masters Degree Project Stockholm, Sweden April 2008 XR-EE-RT 2008:007

2 Abstract The Retarder is one of the main components in Scania s trucks braking system and is used to brake down the truck and for maintaining a steady speed on descents. This Master s Thesis aims to investigate if the current system which uses a proportional valve to control the air pressure in the Retarder, can be replaced with two on/off solenoid valves and a pressure chamber to control the air pressure, which would result in a cheaper and more robust system. By varying the air pressure, the braking torque in the truck can be regulated. A model including electrical drives from a control unit, valves, pressure chamber and a regulating valve is derived. Using the model as reference, a controller is designed and implemented to control the valves, and thereby the pressure. Based on experience from employees at Scania and former research on on/off control of an Exhaust Gas Recirculation system, a regular PID-controller is used as the base in the control. A pulsing scheme where the valves are activated separately is used to distribute the control signal to the two valves. Different ways of applying the control signal are investigated, whether the valves run digital, i.e. 0 % or 100 %, or continuously by varying the PWM signal. A boosting action using non-linear control, and prediction are investigated in order to improve the control performance in such way that the required time response and robustness is obtained. The controller is eventually tested and verified on the real system.

3 Acknowledgments I would like to express profound gratitude to my advisor at Scania, M.Sc. Tomas Selling, for his invaluable support, perseverance, supervision and useful suggestions throughout this research work. His moral support and continuous guidance enabled me to complete my work successfully. I am also highly thankful to my supervisor Håkan Hjalmarsson, Professor at the School of Electrical Engineering, Royal Institute of Technology, for his valuable suggestions throughout this study. I appreciate the kindness of Mr. Fredrik Strååt for his suggestions and advices concerning the Retarder control unit, and would specially thank Mr. Richard Riis for his assistance with the prototypes used for tests and experiments. I am also thankful to Mr. Sören Åberg, who guided me about the direction of my thesis from the beginning, and assisted with practical experiments. Finally, I wish to express my appreciation to the rest of the members of NEST at Scania, and friends and family that have supported me in doing this thesis. 1

4 Contents 1 Introduction Background Thesis Objectives Functional Description Actuation Requirements Notation Retarder Scania s Retarder Retarder System Today Retarder System Using On/Off Solenoid Valves Dead Volume Solenoid Valves Equipment in Experiments ECU Prototypes Retarder Pressure Sensor Software Oscilloscope Multimeter Modelling System Description ECU Model Valve Electrical Model Magnetic Model Mechanical Model Pneumatic Model Regulating Valve

5 3.5 Model Summary Model Validation Duty Cycle Limits Prototype Filling Characteristics Ventilation Characteristics Friction Validation Prototype Filling Characteristics Ventilation Characteristics Validation Model Refinements Parameter Tuning Air Gap Discharge Coefficient, C d Force Balance Valve inlet orifice Time Delay Temperature Dependent Resistance System Identification Control Design Background PID Control Control Structure Model Based Control Tuned Controller Implementation Approaches Scheme 1 - Fill valve and Empty valve activated separately for filling and venting Scheme 2 - Both valves activated simultaneously for filling and venting Results - Scheme Simulations Tests on Prototypes Control Improvements Anti-Windup

6 6.5.2 Improved control using prediction Improved control using non-linear control Comparison Between Scheme 1 and Scheme Conclusions and Future work Conclusions Future Work A Appendix 89 A.1 Linearization A.1.1 Fill valve and Ventilation valve are both activated

7 List of Figures 1.1 Functional description - schematic figure of the valve housing Valve Interior Normally closed 3/2 valve. Unaffected (left) and affected (right) Symbolic sketch of a normally closed 3/2 valve, unaffected (right) and affected (left), where Port 1 is the inlet port, Port 2 is the outlet port and Port 3 is the drain Pulse Width Modulated (PWM) Scheme System Description Modeling of the System ECU circuit when the PWM is low (left) and when the PWM is high (right) Sub models for a solenoid valve Electrical circuit when the PWM signal is set high Electrical circuit when the PWM signal is set low The outlet orifice A The regulation valve at its maximum stroke Filling the dead volume with a duty cycle of 100 %, i.e. the valves are fully open and there is a maximum flow through the valves Venting the dead volume with a duty cycle of 82 % Current in the coil using a duty cycle of 75 % Current in the coil using a duty cycle of 40 % The modeled pressure and the measured pressure when 82 % duty cycle has been applied to the fill (top) and empty (bottom) valve separately Pressure in the dead volume when filling valve and venting valve are both applied a duty cycle of 82 % Pressure in the dead volume when filling valve and venting valve are both applied a duty cycle of 50 %

8 4.8 Modeled and real pressure when a scheme of different duty cycles have been used as input to the fill and vent valve) valve separately Filling (left) and ventilating (right) the dead volume with and orifice of 1.0 mm Filling (Left) and ventilating (Right) the dead volume with an orifice of 1.3 mm Filling (Left) and ventilating (Right) the dead volume with and orifice of 1.9 mm Inserted orifice of 1.0 mm in the inlet port of Prototype Filling and Ventilation Verification with 1.0 mm orifice, 75 cm 3 volume, and 82 % applied duty cycle to fill and vent valve separately Filling and Ventilation Verification with 1.0 mm orifice, 75 cm 3 volume, and 60 % applied duty cycle to fill and vent valve as seen in the figure Pressure Change in the Dead Volume as Different Cd s are used for the Fill Valve when Fill Valve is first activated, then both valves simultaneously (Left). Zoomed plot of the Filling Characteristics (Right). Cd,vent= Current in the coil for DC = 75 % (top) and DC = 40 % (bottom) Current in the conductor at room teperature, T surround = 273 K, when a PWM signal of frequency, f P W M, and 100 % duty cycle have been used as input System identification s circular flow. The rectangles are the computer s main responsibilities, and the ovals are user s main responsibilities. [3] Error Feedback Structure Output Feedback Structure A setup with the controller and a distributor distributing the control signal to either of the two valves Traditional Pulsing Scheme (left) and Pulsing Scheme 1 (right) Resulting PID-controller in simulations, using an orifice of 1.3 mm and a volume of 75cm Control signal behavior for one specific reference Resulting PID-controller on prototype two with orifice of 1.3 mm orifice, including boosting and prediction

9 6.8 The actual error and the predicted error in the next sample (top). Control Signal when prediction is introduced (bottom) The prediction included in the control design Error in previous, actual and next sample, based on the prediction calculation Reulting PID-controller using Scheme

10 List of Tables 4.1 Duty cycle limits for valves to start to open and for valves to keep fully open when a pressure force equal the system pressure helps to open the valves Duty cycle limits for valves to start to open and for valves to keep fully open when no pressure force is acting on the valves Filling from P atm to 0.26P sup and ventilation from 0.88P sup to 0.71P sup Filling from P atm to P sup and ventilation from P sup to P atm

11 Chapter 1 Introduction This Master s Thesis was conducted at Scania CV AB in Södertälje, Sweden, at the department of Powertrain Control System Development, from November 12, 2007 to April 6, Scania is a big and global company operating in Europe, Latin America, Africa, Asia and Australia, and is one of the leading manufacturers of heavy trucks, buses, and industrial and marine engines in the world. Each year Scania offers students Master s Thesis work at the department of research and development. Modeling and control of Retarder is a work provided by Scania as a part of the development and optimization of Scania s Retarder. The work aims to investigate a new method for regulating the braking torque using two on/off valves instead of a single proportional valve used in Scania s Retarder system today. 1.1 Background The Retarder is an integrated component in Scania s trucks braking system, mounted directly on the shaft at the end of the gear box. The Retarder is an aid for reducing the speed without constant use of the regular service brake and the exhaust brake. The actual retarder system uses a proportional valve to control an air flow that determines an oil pressure in the Retarder, which results in a braking torque. The proportional valve is constructed in such way that it is very sensitive to temperature and vibrations, and has a strong influence on the internal friction and calibration. Proportional valves are quite expensive and experience has shown that they are affected by so called hysteresis. Using two on/off valves, a fill and a ventilation valve, to regulate the braking torque, the costs can be reduced and some of the disadvantages 9

12 affecting the dynamics can be eliminated. A former concept study performed at Scania [16], investigating different approaches for controlling the Retarder, confirms the advantages with the valves and concludes that a concept using on/off valves would be the most convenient and robust method. This thesis is based on the former research. 1.2 Thesis Objectives A system using two on/off valves to control the air pressure will be modeled and it will be investigated whether it is possible or not to make a controller that fulfills Scania s requirements specification. The valve unit used in both the current retarder control and this work, should be capable of applying, removing and regulating the braking torque created by the Retarder and is therefore divided into three actuating functions; an accumulating function, an emptying function and a regulating function. The main objective in this Master s Thesis is to investigate the regulating function. Elements that will be included in the work 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 control Design and implementation of controllers based on the model Recommendations on parameters in the system Verification of model and controller by measurements in real physical models and truck. Code Generation in real-time workshop 1.3 Functional Description Figure 1.1 shows the three actuating functions and how the components in the retarder system are connected. The only part that is investigated in this work is the regulating function in the lower part of the figure. 10

13 The valves convert the electrical signals provided by the Electronic Control Unit (ECU), connected to the connectors, to a pneumatic pressure in the volume between the on/off valves and the regulating valve. The pressure affects the regulating valve, which determines the braking torque in the Retarder. A pressure supply, P sup, which is the source for building pressure in the volume, is seen to be connected to the inlet port of the filling valve while the air drainage is seen to be connected to the outlet port of the ventilation valve. The pressure in the drainage equals the atmospheric pressure, P atm. Due to security reasons, the ventilation valve is thought to be designed as Figure 1.1: Functional description - schematic figure of the valve housing normally open 1. For experiments and tests on bench in this work, a valve that is normally closed 2 is used, and is due to limitations on the physical models available for use. The ventilation valve is illustrated as seen down to the right in the regulating function in the figure. 1.4 Actuation Requirements To obtain the desired control performance and robustness, there are several requirements the system has to fulfill. General requirements on the system 1 Normally Open and Normally Closed Valves are described further in Section Normally Open and Normally Closed Valves are described further in Section

14 and requirements on the regulating function will be considered, and is summarized in the following specification. General A change in the input (control signal) must result in a change in the output (air pressure) Filling and ventilation should not affect each other A pressure supply, P sup, should be used as input to the fill Included in the general requirement list are requirements of endurance, marking, deviations, resistance to oil in drainage air, ambient temperatures, available outputs from ECU, diagnostics, reliability, and testability. These will not be included in the report since it is not of importance to this work. Regulating function The function shall include one pressure sensor and two 2/2 on/off valves 3 where the filling valve shall be normally closed and the ventilation valve normally open 4. The regulating valve has a maximum actuating volume of V r,max for maximum stroke. An output volume, V ch, which includes the volume in the chamber, the volume in the valve housing, and the actuator volume in the regulating valve, has to be decided ( cm 3 ). Desired pressure, P ch, should be reached within a time of T req for all valid conditions with a tolerance of ± P tol. As a reference for the results, T req is for this work equal to 50 time units/samples. Filling from atmospheric pressure, P atm, to 0.26P sup, should be performed in less than 0.1T req. Ventilation from 0.88P sup to 0.71P sup should be performed in less than 0.05T req. 3 A 2/2 valve is a valve with two ports and two states. For more information on different kind of valves see Section A normally closed ventilation valve will be used in the modeling and in experiments 12

15 1.5 Notation Notation ECU ε Power Supply Volt U P W M Pulse Width Modulated Signal V D Duty Cycle % R M Measure Resistance Ω Electrical L c Inductance in coil H R c Resistance in coil Ω i Current in coil A N Number of turns in coil - Magnetic µ 0 Permeability in air Vs/Am A a Area of armature m 2 l g Length of air gap in valve m x p Position of armature in valve m l g,off Air gap length when the valve is closed m l g,on Air gap length when the valve is opened m Mechanical m a Mass of armature kg k s Spring coefficient N/m b Viscous friction coefficient Ns/m F prs Pressure force N F pld Preload force N F k Spring force N F b Force, viscous friction N F sf Static friction N 13

16 Thermodynamics C d Discharge coefficient - C d,fill Discharge coefficient fill valve - C d,vent Discharge coefficient vent valve - k = C p /C v Specific heat ratio in air - C v Specific heat capacity at constant volume J/KgK C p Specific heat capacity at constant pressure J/KgK R gas Gas constant - A o Area of orifice m 2 T air Temperature of supply air K M Mach number - m ch Mass of air in chamber kg m fill Mass of filling air kg m vent Mass of venting air kg V ch Chamber volume m 3 V r,max Max actuating volume in reg. valve m 3 d Inlet and outlet diameter of valves m P sup Supply pressure Bar P atm Atmospheric pressure Bar P ch Chamber pressure Bar P u Upside pressure Bar P d Downside pressure Bar 14

17 Chapter 2 Retarder In this chapter the Retarder will be described further, and the equipment used for experiments will be presented and explained. 2.1 Scania s Retarder The Retarder is used to create a braking torque for slowing down the speed of the vehicle, and with a maximum braking power up to 500 kw, the Retarder is one of the most powerful components in the truck s braking system. The Retarder is placed on the outgoing shaft on the gear box. Oil is pumped in between a fixed stator and a movable rotor, and as a result of the high oil pressure caused between the two components a braking power is created. One of the main benefits of the Retarder is the reduced requirement of the wheel brakes, which results in less brake wear. In this way, the wheel brakes remain cool and unused, and thus are more efficient and powerful in the need of additional braking. The Retarder can be used manually or in automatic mode. Using the Retarder in automatic mode allows even for maintaining a steady speed on descents [1] Retarder System Today The current retarder system uses a proportional valve to control the oil pressure between the rotor and stator. The inlet port of the proportional valve is connected to an air pressure supply and has two outlet ports, one connected to a cylinder containing a plunge and another to a drain. The valve is a solenoid valve and can be activated by applying an analogue current to its coil. Inside the valve there is a movable armature. If current is applied, a 15

18 magnetic field will appear between the armature and the iron core in the valve. This will result in a magneto motive force which will affect the armature causing it to move proportional to the current. Air will pass to the cylinder or to the drain depending on the armature s position. According to the air pressure in the cylinder the plunge will move, and an oil pressure causing a braking torque will be created Retarder System Using On/Off Solenoid Valves As described in the introduction, a new concept using on/off-valves is to be examined in this work. The cylinder is substituted by a chamber with constant volume 1. Two on/off solenoid valves are introduced, one for filling and one for venting the chamber. How they are connected can be seen in Figure 1.1. The inlet port of the filling valve is supplied with air pressure, while the outlet port is connected to the chamber and to the venting valve. To the outlet port of the ventilation valve there is a drain to the environment where there is atmospheric pressure. By activating the valves separately or simultaneously, one can fill and empty the chamber. Basically the on/off valves can either be open or closed, but with use of pulse width modulation (PWM) as input signal, it might be possible to manipulate their behaviors so that the armature in the valves can switch between the on and off position for one single PWM period, resulting in a limited air flow through the valve. A change in the air pressure due to air flow into the chamber affects a regulating valve in the Retarder. The plunge in the regulating valve will move from its initial position when the air pressure increases, and will eventually result in a higher braking torque in the Retarder that brakes down the truck. If the pressure decreases, the regulating valve will move in the opposite direction back to its initial position, and the braking torque will be reduced. Mounted on the chamber is a pressure sensor. In this way the pressure can be directly measured, and a controller based on the closed-loop principle, 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 regulation valve, the extra volume in the chamber, and the volume in the housing are 1 Due 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 chamber so that the volume can be considered constant. 16

19 connected. This volume is denoted the dead volume and will be used as a reference to the total volume in the retarder through the thesis. The little amount of air capacity in the actuating volume (21cm 3 ), makes it difficult for the controller to regulate the pressure. To increase the performance of the controller, extra chamber volume has been inserted. The total volume has yet not been decided, but is a part of the parameters that will be investigated and decided, in order to fulfill the requirements on filling time, ventilation time, and control performance. 2.3 Solenoid Valves An On/Off valve is an example of a solenoid valve and will be described in this section. A solenoid valve is an electro mechanical valve where its dynamic behavior has an influence on fluids (liquid or gas), and can be divided into two main parts: Solenoid A moveable armature The solenoid, seen as the gray boxes with a cross inside in Figure 2.1, consists of a coil of wire wounded in the form of a cylinder. The coil covers the movable armature, which is mounted on a spring that keeps the valve in its initial position. The valve is activated by applying a current to the coil, causing the armature to move away from its initial position. Pos X-dir Air out Air in Air out Figure 2.1: Valve Interior 17

20 As it appears from the figure, the interior of a solenoid valve is quite complex. It contains four subsystems that all are related to each other; electrical, magneto-dynamic, mechanical and fluid dynamical. When current is applied to the system, a magnetic field is induced around the coil contributing to a magnetic force. The magnetic force tries to overcome the counteracting forces, i.e. spring force and friction forces, resulting in opening or 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. The most common valves are 2/2 valves, 3/2 valves and 5/2 valves. As seen in Figure 2.2, a 3/2 valve has three ports (inlet port, outlet port, and a drain) and two states (on and off), thereby the name. Figure 2.2: Normally closed 3/2 valve. Unaffected (left) and affected (right) The available valves used in this work are 3/2 valves where the drain (port 3) has been sealed and operates therefore as a 2/2 valve. Observe that figure 2.2 shows a valve that opens or closes by pressing and releasing a button. This is a mechanical valve. However, the principle is the same for solenoid valves, but they are instead activated using a current. As mentioned earlier, a valve can be normally open or normally closed. In Figure 2.3 (right) a normally closed valve that is not affected by external forces is shown. The flow path 1-2 will be closed while the flow path 2-3 will be open. If the valve gets affected by external forces, see Figure 2.3 (left), the drain (port 3) will close and path 1-2 will open. Only in this case a flow from the inlet port to the outlet port can take place. For a normally open valve, path 1-2 is open while path 2-3 is closed, when the valve is not affected. Activating the normally open valve, the drain will open and path 1-2 will close. 18

21 Figure 2.3: Symbolic sketch of a normally closed 3/2 valve, unaffected (right) and affected (left), where Port 1 is the inlet port, Port 2 is the outlet port and Port 3 is the drain. 2.4 Equipment in Experiments Modeling of the complete system requires an understanding of the system s behavior and it s characteristic. To get a satisfying model which is similar to the real system, experiments have been performed on two prototypes. One containing a chamber with fixed volume and another where the volume in the chamber can be adjusted. In both prototypes an electronic control unit (ECU) is used to generate input signals to the valves. A laptop containing real-time software is used to acquire data from the pressure sensor and outputs from the ECU, such as current ECU The valves operating conditions are established by a PWM-scheme generated by the ECU. Available outputs from the ECU relevant for the work are current, pressure. The ECU can also be used to control other electrical systems for the trucks. The ECU runs in a frequency f ECU, i.e. the control unit samples only once every period, T ECU. PWM signal The PWM signal is a periodic square form signal in which the frequency and the duty cycle can be chosen, and is shown below in Figure 2.4. The duty cycle, D, is defined by the ratio, D = T on /T, where T on is the time for which the signal is high and T is the period time. T on is limited to be in the interval [0, T ]. The duty cycle is often also referred to in percentage, D 19

22 T - Period High Low D=65% T d High Low D=25% Figure 2.4: Pulse Width Modulated (PWM) Scheme [0%, 100%]. Unless the current in the coil is not at its maximum, it will continue to increase as long as 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 frequency for the filling and venting valve can be chosen separately, and is available as variables in the calibration software tool used 2. This is an advantage since the choice of the duty cycle and the frequency of the PWM signal for the two valves probably will affect the filling and venting times in different ways and will be future parameters in the control design Prototypes During the Master s Project two prototypes have been available for experiments and validation use. The first prototype is only usable for early tests and verification of the model. It has a fixed volume and can only be used 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 possible to verify the controller and its performance in the real system. Norgren Herion 3/2 valves with a sealed drain, which was described in Section 2.3, have been used in both prototypes. Prototype 1 Prototype 1 consists of a pressure supply inlet, a chamber with a fixed volume, V ch = 100cm 3, two on/off valves, each for filling and venting, and a 2 Gredi KleinKnecht [2] 20

23 pressure sensor. A given PWM signal can be used as input to the valves, which will partly or fully open the valves depending on the duty cycle and frequency of the PWM signal. Experiments can be done either with pressure supply or without. If only the electrical, magnetic and mechanical part of the valve is to be studied it is convenient to start with experiments where no pressure supply is connected. When the pressure supply is connected, the complete dynamics can be studied. The valves are equipped with a fixed orifice of 1.9 mm. Prototype 2 This prototype is equipped with the same valves as are used in Prototype 1, with the default orifice diameter of 1.9 mm. Smaller orifices can be introduced to get a smaller air flow, and are inserted into the valve housing on the inlet port to the valve. An extra orifice with diameter 1.0 mm has been inserted into the housing as default, and can be seen in figure It is not meaningful to use smaller orifices, because orifices less than 0.8 mm can cause problems with dirt. The fixed volume of the chamber is 51 cm 3. Taking into account the volume in the regulating valve (21 cm 3 ) and the connecting channels (2.8 cm 3 ) the total dead volume in the prototype is approximately 75 cm 3. However, in Prototype 2, extra volume can manually be added to the chamber if needed. The total volumes available for use are 75 cm 3, 100 cm 3 and 125 cm 3. This is convenient when the regulating requirements are to be examined. A large volume takes longer time to fill and ventilate, which is a drawback for filling and venting time constraints, but is easier to regulate than a smaller volume just because of this Retarder A full scale retarder has been available for experiments on Prototype 2. Included in the retarder is the rotor and stator, and the regulating valve. It has on the other hand not been connected to the gear box and no oil has been present in the retarder. The effect of the oil pressure on the regulating valve has therefore been neglected in the regulating valve model Pressure Sensor To measure the air pressure to be controlled, a pressure sensor has been used, manufactured by Denso Corporation with part number The operating pressure, denoted in absolute pressure, is 0.06 to 2.1 MPa, 21

24 but is represented by the ECU in relative pressure related to the atmospheric pressure, i.e 0 bar on the output corresponds to 1 bar in absolute pressure. Durability of the pressure sensor and surrounding temperature affect its precision. Operating temperature is -40 to 135 C and the sensor requires a supply voltage of 5 ± 0.1 V to work properly Software Gredi Kleinknecht is a calibration tool for use with the ECU. It includes functions to display, record and evaluate simultaneously acquired ECU internal and process data [2]. In this work Gredi Kleinknecht has been used in experiments to acquire data such as current and pressure. The data has been exported to Matlab where it easily can be examined. From Gredi, internal parameters in the ECU can be set, such as input to the valves used in the prototypes. Among the inputs that have been possible to vary are the PWM duty cycle and the frequency. Matlab and Simulink has been used in the modeling and simulation of the system Oscilloscope A Fluke 45 Dual Display Multi meter was used to examine the dynamics of the ECU and the electrical part of the valves Multimeter To examine the electrical circuit s dynamics, a basic multimeter, Fluke 75, manufactured by John Fluke has been used. 22

25 Chapter 3 Modelling A system can be seen as an object or a collection of objects which properties are to be examined [3]. Examining the system s properties can be done by doing experiments on it or by making a model of the system and perform computer simulations on the model. Experiments often require samples that can be very expensive or have to be performed under specific conditions. Often it is dangerous to perform experiments and it could be that the system to be examined still does not exist. Because of all the difficulties by doing experiments, modeling of systems is in many occasions to be preferred and is sometimes the only possibility. That the system still does not exist is very common in practice and is the case even in this work; The orifice in the valves in Prototype 2 is still unknown. Making a model of the system the system s dynamics can be examined. In this way the parameters can be varied in the model until an optimal set of parameters has been identified, without having to change the mechanical construction in the prototypes. Briefly said, a model of a system is a tool used to answer questions of the system without having to do experiments on it [3]. Verbal, mental, physical or mathematical models are all examples of different kinds of models. For the mathematical model, which is used and discussed in this Master s Thesis, observable magnitudes in the system (current, pressure, distance etc) are combined and transformed into mathematical relations. The mathematical model is basically a collection of mathematical relations and can be used to describe the behavior of the system, either by doing mathematical calculations or numerical experiments, so called simulations. If a real system such as a prototype exists, system identification can be used. A prototype is a physical model with properties as close as possible to the real system s properties. Using identification, experiments are per- 23

26 formed on the prototype and a model is built from measurements of inputs to and outputs from the system by adapting the model properties to the real system s properties. This chapter will describe the modeling of an electro-pneumatic system. First physical modeling will be described, where mathematical relations for the system are derived from known physical relations. Eventually system identification will be mentioned and handled briefly. 3.1 System Description The system consists of an ECU, two on/off solenoid valves, a chamber and a regulating valve and has the setup shown in Figure 3.1. One of the valves is used for filling air into the chamber, or more precisely into the dead volume 1, and is called the filling valve. The filling valve is connected to a system pressure, P sup. The other valve is the venting valve and is used for emptying the dead volume. 8.4 bar PWM System Pressure Fill Valve Air flow Chamber Regulating Valve PWM Vent Valve Oil Drain Figure 3.1: System Description The resulting air pressure in the dead volume and the oil pressure in the retarder balances a regulating valve which position determines the braking torque in the retarder. An overview of the complete system and its different models is shown in Figure 3.2. In this section the mathematical model for each part is separately derived and explained in details. Eventually, all parts are combined to a system and the complete mathematical model is expressed in state space equations. 1 See Section 2.2 for description of the dead volume 24

27 ECU Valve Chamber Regulating Valve 3.2 ECU Model Figure 3.2: Modeling of the System The electrical circuit in the ECU can be simplified as shown in Figure 3.3, and is divided into two cases, whether the PWM signal is high or low. + + e + U - PWM e + - U PWM - - Figure 3.3: ECU circuit when the PWM is low (left) and when the PWM is high (right) The switching behavior of the PWM can be lethal for the electronics in the ECU if a free wheel diode is not included on its output. If this is the case, when either of the valves are activated, the solenoid valve is charged with current. In periods when the PWM is low, the current discharges from the valve s coil and results in heating and worst case damaging the ECU and its components. Because of this, the retarder control unit (ECU) is constructed with free wheel diodes. The diode is connected in parallel with the electrical drives on the ECU output, as shown in Figure 3.3 (left). The diode makes the energy stored in the coil to be discharged in a closed circuit consisting of the valve s electrical components and the freewheel diode, preventing the current to be absorbed by the ECU. A drawback having the free wheel diode is the delayed current discharge in the coil. When the PWM signal is deactivated or set to zero, because of the diode, there will still be current in the circuit, resulting in a delayed closing time of the valves. The time constant for the current discharge is determined by the coil resistance and is hard to affect. Examining the system s dynamics, the delays have to be considered, and a model of the ECU is convenient. 25

28 According to Section 2.4.1, the output signal from the ECU, and input to the valves, can be expressed as { high for t < DT ; U P W M = (3.1) low for DT < t < T. where D is the duty cycle and T is the period of the PWM voltage. The period depends on the frequency of the PWM signal. Using a frequency of e.g. 100 Hz and a duty cycle of 60 %, the PWM voltage will be high for 6 ms and low for 4 ms during a total period of 10 ms. When the PWM is high, U P W M equals the voltage supply e, and opposite, when the PWM is low, U P W M equals 0 V. 3.3 Valve With U P W M as an input signal to the valves, they can be controlled by changing the duty cycle and the frequency of the PWM signal. As shown in Figure 3.4 the modeling of the valves consists of an electrical, a magnetic, a mechanical and a pneumatic part. Each part will be handled separately where the mathematical expressions are derived. U PWM Electrical i Magnetic F M Mechanical x p Pneumatic m& Figure 3.4: Sub models for a solenoid valve Electrical Model As mentioned in Section 2.3, the valves include a solenoid. The electrical part can be modeled as an RL circuit including a resistance in series with an inductor [4]. Because of the discontinuous free wheeling effect in the ECU, 26

29 two cases has to be considered when the electrical model is derived; Case 1 when the PWM is high and Case 2 when the PWM is low. Case 1 - Energizing: i R e + - V R V L L + Figure 3.5: Electrical circuit when the PWM signal is set high When the PWM is high no current will flow through the diode. The diode can be considered an open circuit. The power supply, e, works as the source and energizes the solenoid, as seen in Figure 3.5. According to Kirchhoff s Voltage Law (KVL), the sum of all voltage drops equals zero. Using KVL, the mathematical equation for the first case can be expressed as: e Ri V L = 0 (3.2) As seen in (3.2), the voltage drop over the resistance is given as V R = Ri, while the voltage drop over the inductance is more complex. As an armature exists inside the coil, an electro motive force (emf) is induced when the armature starts to move and is due to a change in the inductance. The voltage drop over the inductance can according to [6] be expressed as V L = N dφ B dt = d dt (Li) = Ldi dt + idl dt (3.3) where N is the number of turns in the coil, and Φ B is the magnetic flux density in the solenoid. Inserting (3.3) into (3.2), the electrical circuit expression becomes e Ri L di dt idl dt = 0 (3.4) 27

30 R - V D V R V L L + + i Figure 3.6: Electrical circuit when the PWM signal is set low Case 2 - Discharging: When the PWM is low, no power supply e is connected to the circuit. The charged solenoid will operate as a source and current will flow in a loop through the diode until the energy has been absorbed in the circuit. This is called the discharging case, or Case 2. Using Kirchhoff s Voltage Law, the mathematical expression can be derived: L di dt idl dt Ri V d = 0 (3.5) As the armature moves, the effective air gap and the amount of iron core in the solenoid will change, affecting the magnetic field inside the solenoid. A movement of the armature will also result in a change in the inductance. The inductance values for the on and off position of the armature, L on and L off, are provided by the valve manufacturer. Since the magnetic field is hard to measure, the inductance has been approximated as a linear function of the armature s position, x a, and is given by L(x a ) = L off + L off L on l g,on l g,off x a (3.6) where x a is the armature s position, and l g,on and l g,off is the air gap inside the solenoid in the on- respective off-position, x a,on and x a,off. It should be mentioned that the air gap is maximum in the armature s offposition, and minimum in the armature s on-position. When the armature is at rest, i.e. the armature is in the off-position, x a equals zero, but l g > 0. The air gap decreases as the armature opens, and should not be confused with the armature s position. 28

31 3.3.2 Magnetic Model Due to the current in the coil, a magnetic field will appear inside the solenoid and result in a magnetic force affecting the armature. According to [5], the change in the magnetic force can be expressed as df M dt = B2 da p 2µ 0 dt (3.7) where B is the magnetic field, µ 0 is the permeability in air, and A p = constant is the cross sectional area of the armature. To make the model as easy as possible, the solenoid has been approximated to be very long. Then, inside a long solenoid, the magnetic field is given by [7] as B = µ 0 Ni l g (3.8) where N is number of turns in the coil, and l g is the air gap inside the solenoid. The valves are constructed such that the air gap is never zero. This would then result in an infinite large magnetic field. Different air gaps for the on and off position of the armature have been provided by the valve manufacturer and can give information on how much the armature moves in total. As a function of the armature s position, the air gap is expressed as l g (x a ) = l g,off x a (3.9) Combining (3.7), (3.8), and (3.9), the magnetic force can eventually be expressed as F M = µ 0A p N 2 i 2 2(l g,off x a ) 2 (3.10) Mechanical Model As discussed in the previous section, the magnetic force affects the armature. However, other forces also have an influence on the armature. Recalling Section 2.3, a principle sketch of the valve is shown in Figure 2.1. As seen, the armature is connected to a spring that counteracts the magnetic force. The spring has the spring constant k s and is preloaded with a force F pld, both provided by the manufacturer. Due to the armature s connection to the valve house, static and viscous friction, F sf and F b respectively, could be present. If pressure supply is connected to the valves, a resulting pressure force F prs will also have an influence. In the modelling, F prs have been 29

32 assumed to be acting in the positive direction rather than counteracting the armature s motion. Newton s Second Law yields, and together with assumed and provided information, the mechanical model for the valve can be derived as Pressure force mẍ a = F M + F prs F pld F k F sf F b (3.11) According to [7] a force due to hydrostatic pressure is given by F = pa (3.12) where p is the pressure and A is the affected body area. As long as the pressure on the inlet port of the valve is bigger than the pressure on the outlet port of the valve, a pressure force will help lifting the armature in the valve. The resulting pressure force F prs for each valve, can be modeled as the difference between the upside (inlet port) pressure and the downside (outlet port) pressure of the valve, which in mathematical terms is expressed as F prs = F u F d = π( d 2 )2 (P u P d ) (3.13) where d is the diameter of the affected body area, F u is the force affecting the upside, F d is the force affecting the downside, P u is the pressure on the upside, and P d is the pressure on the downside. Viscous friction The viscous friction can be modeled according to [7] as F b = bv = bẋ a (3.14) where v is the armature s velocity and b is the viscous friction coefficient. Spring force The spring force is given by [7] as where x a is the armature s position. F k = k s x a (3.15) 30

33 Preload force The spring is preloaded a certain length, l pld, which is provided by the valve manufacturer, and corresponds to a preload force, F pld. The preload force is given by F pld = k s l pld (3.16) Pneumatic Model A net force resulting in opening the fill valve leads to an air flow into the chamber and regulating valve. Opposite, ventilation of the dead volume, i.e. an air flow out of the dead volume, will occur if a net force results in lifting the ventilation valve. If both valves are activated at the same time, there will be a pressure increase or decrease depending on the net air flow in the dead volume. The pneumatic model is used to describe the air and heat transfer, and the expansion and compression relations. Air is a compressible fluid which behavior differs from that of a perfect gas. As a pressure supply of P sup is used, which is considered as low pressure (p 1.0MP a), the deviations from an ideal gas could be neglected. This is not necessarily true since the Retarder can be subjected to extreme temperatures, where temperature has an influence on the dynamics. However, a controller based on pressure feedback can hopefully quickly compensate for errors due to temperature influence, and the temperature variations have therefore not been considered in the early work. The heat due to compression or expansion is minimal and has also been neglected. This is due to an almost constant dead volume and will be explained further in the next section. Using the common law of gas [8], the pressure in the dead volume can be expressed as P ch = m chr gas T air V ch (3.17) where P ch is the pressure [Bar], V ch is the total dead volume [m 3 /kg], R gas is the gas constant [J/(kgK)], T air is the temperature in the dead volume [K] and m ch is the mass of the air [kg]. The pressure change in the dead volume can now be derived by differen- 31

34 tiating left and right side of (3.17), and gives P ch = R gast air ṁ ch + m chr gas T air m chr gas T air V V ch V } ch V {{} ch 2 ch }{{} 0 0 (3.18) where it can be seen that the heat and the volume change have been neglected. According to [8], the mass flow for valves, pipes, couplings, filters, etc. can be calculated according to m ch = C dp u A 0 Ψ( P d )ζ (k) (3.19) Rgas T air P u where ζ (k) = Ψ( P d P u ) = 2k k + 1 k+1 k 1 1 if P d P u ( P d Pu ) k 2 ( P d Pu ) k+1 k k 1 2 ( 2 k+1 ) k+1 k 1 P cr if P cr < P d P u 1 where A o is smallest outlet area, C d the so called discharge coefficient, and k 1.4 is the specific heat ratio in air. The outlet area depends on the armature s position. If the armature is closed, no air will flow through the valve. As soon as the armature opens, the outlet area gets bigger. The smallest outlet orifice area can be expressed as A o = πd 0 x a (3.20) where d 0 is the diameter of the inlet orifice, and is illustrated in Figure 3.7. Remember that the outlet area cannot be bigger than the area of the inlet orifice. If this is the case, the outlet area has to be saturated to be A o = π which is the cross-sectional area of the inlet orifice. ( ) 2 d0 (3.21) 2 32

35 Armature Air Out A 0 Air in Figure 3.7: The outlet orifice A 0 P cr specifies the critical pressure ratio for a component, e.g. a valve, and depends on the shape of the orifice in the component. If the ratio between the upside and downside, also called the pressure ratio, is less than the critical pressure ratio, the flow is called a critical flow. For pressure ratios higher than the critical pressure ratio, the flow is called an under-critical flow. For a sharp edged orifice the value of P cr is given by the equation [8] 2 P cr = ( k + 1 ) k k (3.22) The orifice in real pneumatic components often have a different shape and it is not unusual that there are series of orifices that reduce the critical pressure ratio. Therefore the P cr -value is always less than for real pneumatic components. The net airflow to the chamber can be seen as the change in the air mass in the dead volume, and can be expressed as the mass flow into minus the mass flow out of the dead volume. ṁ = ṁ fill ṁ vent (3.23) Inserting (3.23) into (3.18) the pressure change is given by 3.4 Regulating Valve P ch = R gast air V ch (ṁ fill ṁ vent ) (3.24) The regulating valve consists of an inlet port, an outlet port, and a plunge connected to a spring, and is shown in Figure 3.8. Air from the dead volume flows through the inlet port, affecting the plunge. 33

36 air oil Figure 3.8: The regulation valve at its maximum stroke The plunge moves according to the air pressure in the dead volume and the oil pressure in the retarder, which is connected to the outlet port. Present in the valve are friction forces and spring forces that counteract the plunge. Figure 3.8 shows the regulation valve at its maximum stroke, where the positive direction is defined as a movement of the plunge from air side to the oil side. The maximum expansion when regulating has been calculated to be sufficient small compared to the total dead volume of 100 cm 3. The volume expansion due to plunge movement in the regulating valve has therefore been neglected, and the dead volume has been considered constant. If a much smaller dead volume is used, the expansion could have effects on the system s dynamics and should be considered to be included in the model. The regulating valve can be modeled using Newtons Second Law, where the sum of all the forces working on the system equals zero. m plunge ẍ plunge = F prs,ch F prs,oil F k F friction F pld = P ch A air P oil A oil k s x plunge F friction F pld (3.25) where F friction = { Fsf if ẋ plunge = 0 F df if ẋ plunge 0 A air is the affected plunge area on the air side of the valve, and A oil is the affected plunge area on the oil side of the valve. From these equations the plunge s position can be derived. Note that the friction includes two cases, static friction, F sf, which is present when the plunge is at rest, and dynamic friction, F df, when the plunge moves. 34