1 Eindhoven University of Technology MASTER Stop-Go with the ZI-powertrain a first glance Verhagen, T.C.P. Award date: 2000 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain
2 Eindhoven University of Technology Department of Mechanical Engineering System and Control Stop-Go with the a first glance T.C.P Verhagen WFW report Committee Prof. dr. ir. J.J. Kok (chair) Prof. ir. N.J.J. Liebrand dr. ir. F.E. Veldpaus dr. S. Shen ir. A.F.A. Serrarens ir. B.G. Vroemen 11 Juli 2000
3 Summary Early 1997, Van Doorne's Transmissie b.v. (VDT), in co-operation with the Technical University Eindhoven (TUE) and the Netherlands Organisation for Applied Scientific Research (TNO), launched the Eco Drive project to improve the driveability, fuel economy and exhaust emissions of passenger cars with a CVT. Part of this project, is the optimisation of the mechanical design and the control strategy of a CVT-based driveline with an integrated flywheel. The flywheel in combine with the CVT offers the possibility to use the engine in a most efficient way while a comfortable vehicle driveability is guaranteed. An additional facility of this so called 21'-driveline, is the Stop-Go process. The idea of Stop-Go is to shut-off the engine during full vehicle stops. The fuel that is normally combusted during idling (e.g waiting for red traffic lights) is saved with this type of operation. When launches are requested by the driver the energy of the flywheel is used to accelerate the engine until fuel ignition can be resumed. At the same time the vehicle is accelerated. During the Stop-Go process several clutches of the driveline are opened and closed in order to achieve a more or less comfortable vehicle launch while guaranteeing the engine crankshaft to be accelerated up to a minimum ignition speed. A mathematical method is developed to handle the change in number of degrees of freedom (DOF's) after (dis)engagement of a clutch. The method, based on the use of system transformation matrices, is used for developing system models describing the closing of one clutch, a part of the Stop-Go process. This manner of approach can be used to eventually describe the closing and opening of all the clutches during Stop-Go. With the usage of the system model and the feedback linearization theory a control model is also developed. Both the system model and the control model have been incorporated in a Matlab simulation program. The simulation results show that the mathematical method of using transformation matrices is an useful and effective way to take a loss of degrees of freedom into account. They also give a strong indication that the required minimum engine speed can be achieved while a comfortable vehicle launch is possible. Further research must determine if during the whole Stop-Go process a comfortable vehicle launch can be established.
4 Samenvat t ing In het begin van 1997 startte Van Doorne's Transmissie b.v. (VDT) samen met de Technische Universiteit Eindhoven (TUE) en de Nederlandse Organisatie voor Toegepast-wetenschappelijk onderzoek (TNO) een project op om de rijeigenschappen, brandstofverbruik en uitlaat-emissies van personenauto's met een CVT te verbeteren. Een deel van dit EcoDrive project beslaat het ontwerpen en regelen van een aandrijflijn met CVT waaraan een vliegwiel is toegevoegd. Dit vliegwiel maakt het mogelijk om de motor zo efficient mogelijk te laten werken terwijl aan het rijgedrag geen concessies worden gedaan. Een extra faciliteit van deze aandrijflijn, genaamd, is het zogenaamde Stop- Go proces. Het idee van Stop-Go is het stopzetten van de motor wanneer de auto volledig stil staat. Hierdoor wordt de brandstof bespaard die anders nodig zou zijn om de motor stationair te laten draaien, bijv. bij het wachten voor een rood stoplicht. Wanneer de bestuurder weer wenst te rijden wordt de energie van het vliegwiel gebruikt om zowel de motor als de auto tegelijkertijd te versnellen totdat de motor een minimale toerental heeft bereikt. Vanaf dat moment wordt de brandstoftoevoer naar en verbranding in de motor weer gestart. Tijdens het Stop-Go proces moeten een aantal koppelingen in de zodanig geopend en gesloten worden dat de bestuurder het wegrijden als comfortabel ervaart terwijl tegelijkertijd het minimale toerental van de motor bereikt wordt waarop deze kan ontsteken. Voor het beschrijven van de werking van de koppelingen is een mathematische methode opgesteld die het veranderen van het aantal graden van vrijheid (wat gebeurt bij het openen en sluiten van koppelingen) beschrijft. De methode, gebaseerd op het gebruik van systeem transformatie matrices, is toegepast bij het ontwikkelen van een systeemmodel dat de eerste stap van het Stop-Go process, waarbij een koppeling gesloten wordt, beschrijft. Deze methode kan gebruikt worden om uiteindelijk het gebruik van alle koppelingen tijdens Stop-Go te beschrijven. Met behulp van dit systeemmodel en met de theorie van feedback linearisatie is ook een regelmodel ontwikkeld. Zowel het systeemmodel als het regelmodel zijn verwerkt in een simulatieprogramma. De simulatieresulaten tonen aan dat het mathematisch model zeer bruikbaar is. Tevens blijkt uit de resultaten dat in de eerste fase van Stop-Go, waarbij 66n koppeling gesloten wordt, dat het benodigde minimale toerental kan worden bereikt en dat de auto zodanig versneld kan worden dat een gewenst rijgedrag ontstaat. Verder onderzoek moet aantonen of dit tijdens het sluiten van de andere koppelingen van Stop-Go ook het geval is.
5 Contents Summary 1 Samenvatting 2 1 Introduction The outline of this report The ZI-driveline and Stop-Go External torques Itoad-load torque Engine torque Planetary gearset DNRse t Multiplate wet clutch Continuously Variable Transmission Torque converter Stop-Go conditions Description of the Stop-Go process Simulations for step 1 of the Stop-Go process System configuration for Step one The three phases of clutch Cf Differentiai equations for phase two Differential equations of phase three Differential equations for phase one Simulation model Simulation results Control Feedback linearization Control model Desired output yd(t) Derivation of control input u Simulation results Internal dynamics... 31
6 5 Conclusions and Future Recommendations Conclusions Recommendations Bibliography A Equations for step 1 A.1 Matrix representztion B Determination timepoint of transitions B.l Transition of phase one into phase two B.2 Transition of phase two into phase three..... B.3 Transition of phase three into phase two..... C Outline Simulation program D Derivation of F and I?. E System Parameters
7 Chapter 1 Introduction Early 1997, Van Doorne's Transmissie b.v. (VDT), in co-operation with the Technical University Eindhoven (TUE) and the Netherlands Organisation for Applied Scientific Research (TNO), launched a project to improve the driveability, fuel economy and exhaust emissions of passenger cars with a CVT. This EcoDrive project, is subsidised by the Dutch Ministry of Economic affairs. Part of this research is the optimisation of the mechanical design development of a control strategy of a CVT-based driveline with an integrated flywheel. The flywheel is used as a torque assist unit for the internal combustion engine whenever the driver desires fast transients of the vehicle. In this way it is possible to operate the engine at points of high torque and low speed, which is desirable for fuel economy. Normally, such operating points are not acceptable from a driveability point of view because at high torques and low speeds, the engine can not generate enough power to quickly accelerate its own inertia and the vehicle's inertia at the driver's request. The acceleration of the engine inertia is now achieved with the flywheel such that the engine can fully deploy its combustion power for acceleration of the vehicle. The flywheel completely or partly compensates the engine inertia in some transients situations. Therefore the new driveline is called Zero Inertia Driveline or for short. [Serrarens and Veldpaus, The Stop-Go operation of the driveline is an effective way to save a considerable amount of fuel. In short, the principle of Stop-Go comprises engine shut-08 during full vehicle stops. When launches are requested the flywheel - which was still rotating - accelerates the engine until fuel ignition can be initiated again. At the same time the vehicle is accelerated. The fuel that is normally cornbusted during idling (e.g waiting for the red traffic lights) is saved with this type of operation. In this report the relevant components of the for Stop-Go are described including a description of wet plate clutches. During the Stop-Go process several of these clutches are opened and closed. The main concern is closing these various clutches in order to achieve a more or less comfortable vehicle launch while guaranteeing that the engine crankshaft is accelerated up to a minimum ignition speed. The difficulty in developing a controller for the clutches is the alteration of degrees of freedom (DOF's) after (dis)engagement of the clutches. A mathematical method to handle this modelling challenge will be explained in this report. The mathematical approach has been used to derive a model of the Using this model a control model has been developed. The results of simulations with these models will be given and discussed. This manner of approach could serve as a starting point for further investigations for the whole Stop-Go process.
8 1.1 The outline of this report The remainder of this report is organised as follows. In Chapter 2 the relevant components of the are introduced and the conditions and steps of the Stop-Go process are given. In Chapter 3 the mathematical method of handling the alteration of degrees of freedom is explzined on the basis of step one of Stop-Go. Subsequently the results of a Matlab simulation r~utir,e, used fa testing the methed, xe given md discussed. The developed system iilode!s, dong with the theory of feedback linearization, is the basis of the three control models in Chapter 4. The results and evaluation of these models are presented in Chapter 4. Finally, in chapter 5, the main conclusions are summarised and a list of recommendations for future research modelling and for control of the Stop-Go process are given.
9 Chapter 2 The ZI-driveline and Stop-Go In Figure (Fl) on the last an outline of the with its relevant components for the Stop-Go process is depicted. This driveline is a standard driveline with a metal pushbelt CVT plus a power assist unit, embodied as a compact flywheel and a planetary gearset. The CVT consists of a primary and secondary pulley connected by a metal V pushbelt. The Drive/Neutral/Reverse set (DNR set) is a special planetary gearset with two clutches Cd and C,. The hydraulic torque converter is modelled by two inertias representing the impeller and turbine part and the engine is modelled as one inertia (representing engine flywheel, crank shaft, cam-shaft, pistons etc.). The inertias of the engine, torque converter and DNR-set are lumped into one inertia J,. Jv is the total inertia of the vehicle and the rotating parts as seen at the driven wheels (final drive assembly, drive shafts and wheels). The total gear ratio of the final drive assembly is denoted by i,. The weakest elements in the driveline are taken into account representing respectively the driveshaft and the torque converter spring. Two additional multiplate wet clutches, denoted by Cf and C,,,, are required for Stop-Go. The modification of the standard driveline consists of a flywheel and a planetary gearset with additional gears to the primary and secondary pulley. These components are positioned in parallel to the standard CVT driveline. Here, i, is the fixed gear ratio between the ring gear of the planetary set and the primary pulley whereas i, is the fixed gear ratio between the carrier and the secondary pulley. Jf represents the flywheel inertia. The ratio of the annulus radius and the sun radius in the planetary gearset is denoted by z. In the next sections more background information will be given on the planetary gearset, the ciutches, the C'JT, the torque converter and the DPu'R-set. Also the modeis for these components and the models for the engine torque T, on the engine and the road-load torque Tload on the vehicle will be given. The Stop-Go facility to be designed is an add-on to the Its goal is to fastly accelerate the vehicle and the engine, using the stored flywheel energy. In section 2.7 relevant conditions to be met in Stop-Go are described. Subsequently, the Stop-Go process is described in more detail. 2.1 External torques Two external torques act on the being the road-load torque zoad and the engine torque T,. In the following two sections these external load torques are described. l~his figure can be found at the last, to be unfolded, page
10 2.1.1 Road-load torque The load torque on a vehicle consists of three parts: 0 Troll sign(+,), rolling resistance 0 Tad = + Cur +: - sign(+,), air drag and 0 Tdist, disturbance torque caused by up-hill driving, road irregularities, gusts of wind etc. During Stop-Go the vehicle velocity +, is very small and therefore the air drag is negligible. No information is available regarding Tdist. This is left out of consideration in the sequel. Hence, Tload is equal to the rolling resistance. It is assumed that zoad = Troll sign(+,) if +, j: 0 and that I qoad I< Troll if +, = 0. Therefore the complete model for Tload is: In the simulations Troll = 53.9 Nm is used. This value is obtained from manufacturer's data Engine torque During the first step of Stop-Go the angular speed of the engine +, is increased from zero until the speed is high enough to resume ignition and fuel injection. So the engine torque T, in this step is the so-called drag torque. The experimentally obtained relation between Te and +, is given in Figure (2.1). It is approximated in a least square sense e.g.: For very small engine speeds this approximation is not very accurate. Especially p(0) $: 0 whereas in practice the drag torque is very small for +, = 0. For this reason and to avoid numerical problems, the relation for Te is adjusted by multiplication of p(+,) with a factor (1 - e t), so +e[?l + Figure 2.1: Engine torque 8
11 Here 0 < R << 0, where R, is the engine speed at which fuel injection and ignitation is resumed. In the simulation 0, = 50 [radlsec] and R = 1 is used. 2.2 Planetary gearset A planetary set consists of a sun wheel, a number of planet wheels borne on a carrierframe, and an annulus wheel, see Figure (2.2). With a planetary gearset it is possible to connect the engine, flywheel and vehicle and split or combine their power flows. In the 1. shaft of sun 2. hollow shaft of carrier 3. hollow shaft of annulus 4. carrier 5. annulus 6. carrier frame 7. sun 8. shaft of carrier Figure 2.2: planetary set Figure 2.3: speed diagram planetary set the flywheel is connected to the sun (s), the vehicle to the carrier (c) and the engine to the annulus (a). The choice for this configuration is explained in Druten et al. . The carrier circumferential speed Vc = +, is the mean of the speed V, = rsgs of the sun, and the speed Va = ra+, of the annulus, see Figure (2.3), e.g.: Rearranging (2.4) and using the definition z := ra/rs yields The sign conventions for the torques on sun, carrier ar,d annulus are depicted in Figure (2.4). If power losses and inertias in the planetary set are neglected then the torques satisfy: 2.3 DNR set A standard CVT-driveline contains a Drive/Neutral/Reverse set (DNR set). A DNR set is a planetary gearset with three double planet wheels borne on the carrierframe. In Figure (2.5) a schematic of the DNR set can be seen. The left shaft is connected by the torque converter to the engine. The right shaft is connected to the primary pulley of the CVT. The left shaft is directly locked to the right shaft when clutch Cd (drive) is closed and clutch C, (reverse) is open. Consequently, the annulus rotates along with the other two elements. Then
12 Figure 2.4: Sign conventions for the planetary gearset Figure 2.5: DNR set the gear ratio between the left and the right shaft idnr = is equal to 1. When clutch C, is closed and Cd is open the rotation direction of the the left shaft is opposite to that of the right shaft. The gear ratio then depends on the sizes of the two carrier wheels and their spatial arrangements. For the DNR-set at hand the gear ratio in reverse is id, = The planetary gearset is completely locked and the speeds are linked to the housing by C, when Cd and C, are both closed. Then id, = 0. The gear ratio id, is not kinematic determined when C, and/or Cd is partially closed. Therefore the used model for the DNR set is: 1 Cd closed, C, open -1. C, closed, Cd open 2.4 Multiplate wet clutch The purpose of clutches is to permit gradual connection and disconnection of two shafts and to transmit torque when gradually connected. A multiplate clutch has intermediate plates that are connected to one shaft and friction plates that are connected to the other shaft, see Figure (2.6) [TRi!!iarr,s, When the c!utch is dise~gaged the plates are free to ratate ad, Tc1 w Tic i Jcll +c12 +dl Figure 2.6: clutch Figure 2.7: clutchmodel apart from an oil drag loss torque, no torque can be transmitted. When the clutch is engaged
13 the discs are clamped together by a normal force, caused by a controllable pressure in the hydraulic cylinder of the clutch. The plates rotate with different velocities causing friction and consequently a torque Tcl can be transmitted. The clutch is modelled as two rotating inertias (Jcll and Jc12 see Figure (2.7)). The inertias are not crucial from a physical point of view but they provide a way to circumvent algebraic loops in Simulink, used in the EcoDrive project. The model for Tcl is based on the principle of Coulomb friction. It consists of two parts: the clutchplates rotate with different velocities (dip) or with zero speed difference (stick). Tne model is given by: with +,l1 the velocities of the input and output shaft, N the normal force, pstat the static coefficient of friction, pdyn the dynamic coefficient of friction and Reff the so called 'effective radius' at which the friction force pn acts. 2.5 Continuously Variable Transmission The CVT consists of a metal belt composed of thin metal segments and strings, mounted between two V-shaped pulleys, see Figure (2.8). One of the conical sheave of each pulley can Figure 2.8: CVT hydraulically be moved in axial direction, established by hydraulically increasing or decreasing the pressure on the sheaves. This changes the radius of the metal belt on the pulleys, therefore adjusting the transmission ratio between the pulleys. The tensile force in the strings, combined with a pushing force between the V-elements, allows the CVT to transmit a torque. The relevant relations for the CVT are given by, e.g [Serrarens and Veldpaus, Torque converter The torque converter (Figure 2.9) [W.H.Crouse and D.L.Anglin, essentially is a fluid coupling. It uses a fluid and vaned rotors to transmit power between shafts. Inside the fluidfilled circular housing are three members. These are the impeller, (a vaned rotor connected to the engine crankshaft), the turbine, (a vaned rotor connected to the transmission input shaft) and a stator (connected to the transmission housing). The rotating impeller blades increase the velocity of the fluid. The fluid is then thrown into the turbine blades, and the turbine is driven. The discharge from the turbine is opposite to the impeller rotation, caused
14 cover engine cranks1 Figure 2.9: Torque converter by the curvature of the turbine blades. This is corrected by the stator, rn :h has a series of blades curved in the reverse direction of the turbine. This causes the fluid to pass through the impeller and then push on the turbine vanes again. This result in torque multiplication under certain conditions. During normal vehicle operation, the impeller turns faster than the turbine. This difference in speed (slip) causes a power loss in the torque converter. Most automatic transmissions have a torque-converter clutch (Ctc) that can mechanically lock the torque converter to prevent slippage. This improves the efficiency and lowers the temperature of the fluid. The torque converter clutch is attached to the turbine hub and when it is closed it locks the impeller to the turbine. The torque converter turns as a unit with no slippage or power loss. Then the gear ratio between the engine shaft and turbine shaft it, = 1. The relations describing the torque converter when Ctc is disengaged are given in [Mussaeus et al., Isolator springs with stiffness kt, help dampen the shock that can occur when the torque converter locks. The value kt, = 1182 [Nm/rad] for the torque converter at hand in is obtained from manufacturer's data. 2.7 Stop-Go conditions The following conditions apply for Stop-Go: 1. No extra components, except for the necessary clutches and their pressure control units may be added to the 2. The DNR set can be applied in Drive, Neutral and Reverse mode. 3. The initial launch that must be achievable is set to 3.5 [$I. 4. The clutches must be able to transmit a torque of at least 300 [Nm]. 5. The minimum engine speed a, to resume fuel injection and ignition is 50 [radlsec]. 6. The initial flywheel speed df is set at 1900 [RPM] E 200 [radlsec]. 7. The Stop-Go process is completed when the normal configuration is obt ained. 8. The parameters Jf, i,, ic and z of the are used [Druten et al.,
15 2.8 Description of the Stop-Go process Figure (Fl), on the last page, shows a simplified configuration of the with the relevant components for Stop-Go. During the Stop-Go process the stored flywheel energy is used for simultaneously launching the vehicle and accelerating the engine crankshaft up to at least the minimum speed that is required to resume ignition. This implies that nearly no time delay occurs during launch because the vehicle is accelerated immediately. Of course other options are possible. For instance, if an extra clutch is placed between the CVT and the venicie it is possibie to only accelerate the engine. Then the minimum ignition speed can be reached in a shorter time but it will take some time before the vehicle starts moving. The main concern is controlling the various clutches in order to achieve a more or less comfortable vehicle launch while guaranteeing the acceleration of the engine crankshaft up to at least R,. The available clutches are the torque converter lockup clutch Ct,, the DNR clutches Cd and C,, the flywheel clutch Cf and the clutch C,,, between the secondary pulley and the driveshaft. The vehicle and engine are halted (+, = +, = 0) at the starting point of Stop-Go. The Stop-Go process is completed when the normal configuration is obtained. At that moment the stored flywheel energy is (almost) completely used, Cf, C,,, and Cd are closed and Ct, and C, are open. At a first glance of Stop-Go functionality, the following steps are distinguished during a vehicle launch: Step one: at the start of step one the clutches C,,,, Cd and Cf are open whereas the clutches Ct, and C, are closed. During this step the flywheel is decelerated (Cf < 0) by closing clutch Cf. Then the flywheel torque Ts = - Jf cij is positive. According to Eq. (2.6) this implies that Tc = (x + l)ts is positive and that T, = -zt, is negative. The positive torque Tc results in a positive acceleration +, of the vehicle. To obtain a positive engine with the negative torque T, it is necessary that the DNR-set is in reverse mode. Therefore the DNR set is in reverse meaning that Cd is open and C, is closed during this step. This step lasts until +, becomes equal to R, = 50 [radlsec]. Simulations with an initial flywheel speed of 200 [radlsec] show that Cf is not fully closed at this moment. Step two: in the normal the DNR-set is in drive mode, i.e. C, is open and Cd is ched. 2% change from the reverse mnde at the end of step one tc? the drive mode it is necessary to control the primary shaft speed edn, to zero by closing Cd without opening C,. Because +, for a locked torque converter it is necessary to unlock the torque converter by opening Ctc at the very beginning of step two in order to prevent the engine be shut off again. Next, fuel injection and ignition are started and the engine speed is controlled to the operating speed. Simultaneously clutch Cd is closed gradually. It is assumed that the time, needed to open Ct,, is negligible compared to the time needed to close Cd. Furthermore, it is assumed that the shock, caused by suddenly opening Ct, is "dampened" by the torque converter spring. Consequently the opening of Ctc is not gradually but logically closed. During step 2 the closing of Cf proceeds. Step 3: this step starts when Cd is fully closed. At that moment Cf can be fully closed or the closure of this clutch can still be in progress. In step three C, is gradually or
16 suddenly opened and the engine starts to deliver power towards the drive shaft via the torque converter and the planetary gear set. Step 4: at the end of step three the DNR-set is in drive mode, the flywheel clutch is fully closed and the engine is accelerating the vehicle. It is demanded that Cf is fully closed before step four starts. In step four the secondary clutch C,,, is closed suddenly or pretty fast as soon as the speeds of the secondary pulley and the secondary shaft are synchmnised. (+secp,ii,, - +sec,ilaft Remark: it seems to be possible to perform step two and step three simultaneously, i.e. to open C, while closing Cd. This alternative has the advantage that the time to switch from reverse mode to drive mode can be reduced because it is unnecessary that +dnr becomes equal to zero. will decrease and therefore also with this strategy it is required to open the torque converter clutch Ct, first. The disadvantage of combing these steps is that several clutches, i.e. Cr, Cd and Cf are partially closed. This requires more advanced control of these clutches. Further research is recommended. Figure (2.10) gives a schematic representation of the disengaging/engaging of the clutches. A part of the curve of Cf is represented by a dotted line because this part is uncertain at this moment. The extra possible paths of Cr and Cd (step two and three simultaneously) have been represented by dashed lines. ;or , , i i I I I - bme start step l SLcp2 Stcp3 SCp4 Stur Step 1 Step2 Step3 Stqr4 Figure 2.10: diagram clutches
17 Chapter Simulations for step 1 of the Stop-Go process The main difficulty of Stop-Go is to control the various clutches and at the same time comply with the Stop-Go conditions. The development of a controller is hampered by the alteration of degrees of freedom (DOF's) after (dis)engagement of a clutch. In this chapter a mathematical method will be expounded to handle this modelling challenge. The method is based on the use of system transformation matrices. It allows formal transitions between models with a different number of DOF's. The mathematical approach is explained for the basis of step one of the Stop-Go process and could serve as a starting point for further investigations of the next steps. Simulations show that during step one the engine speed +, exceeds the minimum speed R, = 50 rad/sec to resume ignition before Cf is fully closed. Therefore step two of Stop-Go can be initiated during this step. However, in order to test the mathematical method for handling the change of number of DOF's it is essential to test the whole process of closing clutch Cf. Therefore clutch Cf is fully closed in this chapter. Subsequently the Matlab simulation routine for testing the method will be explained. Finally the results of the simulation will be given and discussed. 3.1 System configuration for Step one In Figure (F2) on the last, to be unfolded page, the configuration of the driveline in step one is depicted. The gray elements around the clutch inertias represent power losses in the driveline and have been modelled as two dampers. During step one the CVT is decoupled from the driveline (C,,, is open), the torque converter is locked (Ct, closed, it, = 1) and the DNR set is in reverse (Cd open, Cr closed, id,, = -1.1). At the beginning of step one the vehicle and engine are at rest (& = +, = 0) and the flywheel speed Gf = 199 [rad/sec] (=I900 [RPM]). In step one Cf is closing in order to accelerate the engine and vehicle. At a first glance the closing of clutch Cf can be divided into three phases. These phases are distinguished because the number of degrees of freedom (DOF's) in the phases alters. In section (3.2) these phases will be presented and the differential equations for the model for each of these phases are derived using the earlier mentioned mathematical method.
18 Figure 3.1: driveshaft torque 3.2 The three phases of clutch Cf Phase one Step 1 starts at t = 0. It is assumed that $, = Ge = 0 and that Cf is fully opened (Tcl = 0) for t < 0. Cf is closing for t > 0. At first the driveshaft torque Tkiv (see Figure (3.1)) acting on the vehicle is not large enough to compensate the rolling resistance TTOll. Therefore only the engine will be accelerated. So Tki, and Tload are at balance = 0 ((Goad( 5 Troll, see Eq. (2.1)). By closing Cf, Tcl and thus Tkiu increases. Phase one is finished when the vehicle starts moving # 0) i.e. when Tkiv > Troll (Goad = Trollsign((Pv), see Eq. (2.1)). During phase one Cf will not be fully closed. and (C7e increase and clutch Cj is slipping ($cll -ecl2# Phase two During phase two 0). There is one extra degree of freedom compared to phase one: the velocity $, of the vehicle. Simulations show that during this phase the engine exceeds the minimum speed S1, of 50 rad/sec. Therefore step two of Stop-Go can be initiated during this phase. However, in order to test the mathematical method for handling the change of the number of DOF's it is essential to test an additional phase three: Cf is closed. When phase one, two and three are correctly handled then this manner of approach could be used as a basis for further investigation of the next steps of Stop-Go. Therefore an extra phase is introduced to test the mathematical method. Phase three This phase starts when clutch Cf is completely closed i.e. when $cll - $c12 = 0. Then a degree of freedom is lost, because $cll = dc12 = $cl (see Figure (F2) last page). The clutch remains locked unless the magnitude of Tcl exceeds the static friction capacity, TStat. TStat is set to be 1.1 times the value of Tcl at time t = t, when the clutch started to stick so: When ITc1 I exceeds the value of Tstat then the situation of phase two applies. In Figure (3.2) an outline of these three phases is given. In the following subsections the differential equations of these phases are given. Phase two has the greatest amounts of degrees of freedom, therefore the differential equations of this phase is given first. The mathematical method of taking the loss of a degree of freedom into account is explained for phase three and repeated for phase one. This type of approach can be used in similar circumstances which a changing number of DOF's.
19 Figure 3.2: The three phases in step one Differential equations for phase two When the clutch is partially open, (+,11 - +,12 # O), and the vehicle is moving (+, # 0), then the five differential equations that have to be solved to obtain the vector - q of degrees of freedom are described by: Where - q is given by (see also Figure (Fl) last page) Furthermore, the vector h is equal to g2 - y, where vector gj (j = 1,2,..., 5) is the jth unit vector, so & = [I , ec = [ , etc. Expressions for the mass matrix M, the damping matrix S and the stiffness matrix K are given in Appendix A Differential equations of phase three If clutch Ct is closed then +ell - +,12 -- etq - = 0, whereas Tcl is unknown a priori. The number of degrees of freedom is four instead of five because htq = 0 represents a = ctq - = -pq kinematic restriction. In order to take this loss of freedom into account a vector -r q with four components is introduced: The condition htq = 0 is satisfied for every -r = 0. The difficulty of this transformation from - q to -T q lies in the determinationof the initial conditions. Assume because ht4 = ht%q
20 that the clutch is partially open for t < t, and closed for t 2 t,. Then = %q (t,). Let Q,&5x4 satisfy the condition: - -r - = 0 and Then it is easily seen that:. [+ - nt;(+ C.\v) - wr 2tv-J -, (3.6) With the choice for matrix Q, the value for q (t,) is set. Integration of q(t) = &q (t) for t 2 t, gives: - 4(t) = %grw + 2, r -7 - = q(tr)-rrg,(tr) (3.7) In principle the choice for q (t,) is free. A logical choice, keeping (3.6) in mind is: -r which means that Substitution in Eq. (3.2) gives the equations whom have to be solved: Here Tcl is unknown. Pre-multiplication of (3.9) with RF and substitution of RY~ = 0 gives: with the so-called reduced matrices M, = $%I%, S, = K ~ T S ~ ~ and, K, = RFKR,. M is positive definite (and thus regular) and R, is a matrix of maximum rank. Therefore M, is &c pnsiti~~e definite and regular. This implies that 6 can be calculated from Eq. (3.10). -r Pre-multiplication of Eq. (3.9) with ht, Tcl yields a relation for Tcl: The actual value for Tcl is needed to check the slip condition: the clutch remains locked until ITcl 1 exceeds Tstat. The matrices R, and Q, used in the simulations are:
21 3.2.3 Differential equations for phase one During phase one of step = 0 and last until qoad = Troll a at time t = t,. This phase can be analysed in the same way as phase by considering the as a kinematic constraint. This constraint is given by: = el q(t) = 0 for t 5 t, - (3.13) The procedure can be the same as for phase three. Therefore a vector q -21 components is introduced; (see (3.4)): with the four Furthermore, a (5x4) matrix Qw is introduced such that In the same way as for phase three (3.8) this results in: Substitution of (3.17) into Eq. (3.2) gives: with unknown qoad. Pre-multiplication of (3.18) with RT and substitution of R T = ~ 0 ~ gives: and Kw = RTKR,. Here M, is also positive definite and regular which implies that q, can be calculated with Eq. (3.19). The vehicle remains at rest = 0), until the magnitude of qoad is equal to the rolling resistance TTOll. For the torque qoad applies (pre-multiplication of (3.18) with &: with the M, = RTMG, S, = RTSR, The matrices &, Q,, used in the simulations, are:
22 3.3 Simulation model For a better understanding of the dynamic behaviour of the model, a simulation program has been written in Matlab. In this section important features of this program will be explained. An outline of the program can be found in Appendix C. Equations (3.2), (3.10), and (3.19) describe the distinguished phases. In summary: The initial conditions at time t = 0 are given by: Yv = 'Pdz = Yczl = ~f = Pe = 0 no angular displacements cpv = cpe = 0 vehicle and engine at rest(3.22) = 199[rad/sec] flywheel speed at t=o cpf = All The simulation starts at t = 0 and is stopped at t = tend. The used numerical integration method is a forward (explicit) Euler scheme with time step At = The relevant input for the considered system is the pressure p in the hydraulic cylinder of clutch Cf. This pressure determines the normal force N between the clutchplate and therefore, according to Eq. (2.?), also the maximum torque that can be transmitted by the clutch. In the simulation this normal force N is seen as the controllable input. In a practical implementation of Stop-Go in the the controllaw for this input will have a feedforward and a feedback component. Since the main issue in this section is to check the simulation problem here only a priori known function of time for the input N is used. The chosen function is depicted in Figure 3.3. First, N is a linear function of t, starting with N = 0 for t = 0. At time t = t, the vehicle starts moving > 0). This is the end of phase one. It is assumed that the clutch is not closed at this this time. For t, 5 t < tr the vehicle is moving and the input N increases linearly with time t (see Figure 3.3). At time t = tt the clutch is closed and the clutch plates stick together. This is the end of phase two and the start of phase three. For t 5 t, the input is constant. The torque that can be transmitted in this case is equal to (no slip over the clutch): Tstat = l.l,unref f. where 1.1~ is the static coefficient of friction whereas,u itself is the dynamic coefficient of friction. For the torque Tcl transmitted by the clutch holds Eq. (3.11). To test a transition from phase three to phase two it is assumed that at time t = t, the input N decreases. This decrease in N wiii cause a decrease in Tstat and will be large enough to cause slip over the clutch (TC1 > TStat), i.e. to cause a transition of phase three to phase two. A problem with the simulations is that the points of time where a phase transition takes place are not known a priori. A possible strategy to determine these transition points (usually between two integration points) is outlined in B. The results of the simulations with the earlier specified function for the input N are presented and discussed in the next section. 3.4 Simulation results In this section the simulation program for closing the clutch is validated, amongs others, by comparing the frequencies in the simulation results with the natural frequencies of the system. Also the results of the transitions are presented. Subsequently the kinetic energy of several
23 Figure 3.3: set N components and other parameters are given. All the results are validated and important features emphasized. The natural frequencies of a system are properties of the system model and depend on its constant parameters only and can be found by solving the eigenvalue problem of the undamped system: det (-w K) = 0 (3.23) If M and K are (nxn) matrices then Eq. (3.23) is of degree n in w2 and has n roots w:. The natural eigenfrequencies f, (r = i, 2,..., n) follow from: Figure (3.4) depicts the calculated values for &ll and &12 around the timepoint t, at which clutch Cf is closed. These two quantities are obtained from a simulation with the Matlab program for the model without damping (S = 0). In Figure (3.4) several frequencies are perceptible. These frequencies represent several frequencies of the model. In order to validate the simulation program these frequencies are compared with the natural frequencies as determined from Eq. (3.23). Before Cj is closed, the natural frequencies of the systemmodel are calculated by solving
24 Figure 3.4: Visible frequencies After Cf is closed they are calculated by solving the equation det(-w2~, + K,) = 0 (3.26) In the table below the obtained from Eq. (3.25) or Eq.(3.26) and the frequencies obtained from the simulations are given: Cf open Cf closed - natural frequency model [Hz] simulation [Hz] model [Hz] simulation [Hz] not visible Both systemmodels (Cf open and closed) contain only two natural frequencies consequently rigid body modes are possible. From this table follows that visible and calculated frequencies within reading accuracy are similar. Also the sampling time of the simulation At = is small enough, using the Euler method, to obtain accurate results. In Figure (3.5(a)) the kinetic energy of the flywheel, engine and vehicle and the energy dissipated by the clutch Cf are presented. Eused represents the energy decrease of the flywheel. These result are obtained from a simulation until timepoint t = t, when Cf is completely closed. At that moment 57.1 % of the used energy is dissipated by the clutch, 23.8 % is used to accelerate the engine, 4.6 % is used for accelerating the vehicle and 14.5 % is used to increase the kinetic energy of the rest of the driveline (dampers, springs, etc.). Notice that, in proportion to the vehicle and engine, a large amount of energy is absorbed by Cf. In Figures (3.5(c) and 3.5(b)) the angular velocities of the engine +,, flywheel +f and vehicle +, are given. The angular acceleration of the that: can be seen in Figure (3.5(d)). It is noticed The minimum engine speed R, = 50 [rad/sec] to resume ignition is obtained before Cf is closed.
25 (a) Energy distribution (b) velocity of the vehicle +, (c) velocity of the engine +, and flywheel +f (d) acceleration of the vehicle (tj, Figure 3.5: Various results simulation The maximum value of yj, almost meets the launch acceleration condition of 3.5 [m/s2]. 0 The vehicle speed +, represents a comfortable vehicle launch. 0 All the curves are smooth and show no irregularities, indicating that the mathematical method is correctly incorporated in the Matlab program. In Figure (3.6(a)) the relevant variables for the transition of phase 1 into phase two, Tload and Troll are depicted. In Figure (3.6(b)) the variable +,lip = Pcll - cpclz can be seen. At timepoint tv yields zoad(tv) = Troll and the condition +,rip = 0 holds at t = f. Therefore the strategy of determination of the transition points have been correctly incorporated in the Matlab program.
26 (a) transition phase one into two (b) transition phase two into three Figure 3.6: validation transitions
27 Chapter 4 Control Although the system model that has been derived in chapter 3 only describes a part of the Stop-Go process a control model is used to design a control law for the input variable N in order to obtain a more or less comfortable vehicle launch. This is done mainly to obtain more information about the feasibility of the Stop-Go facility in the The control law is based on feedback linearization. First, feedback linearization is explained. Subsequently three control laws are derived. In the last part of this chapter the results of these three laws are given and discussed. 4.1 Feedback linearization Feedback linearization, or more precisely input-output linearization, can be seen as a method to derive an explicit relation between the output y of a non-linear system (with n dimensional state g) and the input u. Let the system be given by: - 7J = Lf h(:) + Lgh(:)u w(:) + Pl(d~ (4.3) ) : f 0 then this relation gives the desired explicit relation between y and u. If Pl (z) = 0 Differentiation of the output with respect to time results in: If,01( then the relation y = a1 ) : ( is diiferentiated again, resulting in: If,B2(:) = 0 then the process of differentiating is repeated until finally a relation of the form Y' = a, ) : ( + P, (:)u (4.5) is obtained with,o,(:) # 0. The number T is called the relative degree of the considered system. Suppose that the objective is to make the output y(t) track a desired output yd(t), i.e. to make the tracking error ~ (t) = y(t) - yd(t) equal to zero. Then, introducing a new input p according to
29 is set at the constant value Troll. Therefore the control model for the derivation of control laws is given by (see Eq. (3.2)): - qt = [ PV ~c12 ~cll ' ~ j Pe. ] With the state vector, : defined by and the engine speed +, as the output y and the clutch torque TCl the control input u the model is given by: with: T y=c : Our goal is to make the output y(t) track the desired trajectory yd(t) From Eq. (4.14) and (4.15) it follows that: Because g5 b = 0, y is differentiated: It turns out that ctb = 0 so - y is differentiated once more: = C ~(A+F)~ = c~(a+f)(az+ - f(~) +bu) Where the gradient matrix F is defined by (see also Appendix D):
30 SF with: F = -bi i=l 6% For the derivation of F see Appendix D. In this case the multiplying factor of u is unequal to zero, so a direct relation between and u can be found: with: a(g) = c~[(a + I?)')' + (A: f f (:))G~F~o](A: + (GTB)C~F~~(A: P(.> = c K A + ~ )'b - control law is: + f(d) and + f (g)]. For F10 see Appendix D. The proposed with kl > 0, k2 > 0 and k3 > 0 and where E = y(t) - yd(t) is the tracking error and leads to... exponentially convergent tracking. Unfortunately it is very hard to measure i' = (Fe - (P,,) in reality. Therefore ks = 0 and This however implies that the error i is not dampened. Simplified type In order to circumvent algebraic loops in Simulink the two inertias JCl1 and Jc12 have been introduced. However when assuming Jell x 0 and the damping is not incorporated S = 0 a simple relation between y and u can be found: The proposed control law is: with kl > 0 and E = y(t) - yd(t). 4.3 Simulation results In this section the data are presented, obtained with simulation program of Appendix C. The simulation stops at t = 0.55 s, because then the = 50 [rad/sec] has been satisfied and step two of the Stop-Go process can be initiated. The input u(= Tcl) is now calculated using the control law (4.25) or (4.27). For the elaborate type kl = 5 and k2 = 5 is used and for the simplified type also kl = 5 is used. Thus these two control laws have been used: 1 ul = -(-a(.) + 'jid - 5e - 5i) elaborate type (4.29) P(:>
31 In the discussion of the results in this section a subindex d stands for the desired value (e.g. the desired engine speed +,), index s is used for the calculated value using input u, (e.g the clutch torque TCls), and index 1 means the calculated value using ul. In Figure (4.1) output y (the engine speed) is depicted. In Figure (4.l(b)) a detail of (4.l(a)) is given. It is noted that the tracking error is very small for both laws. Also a slight vibration is perceptible in +,. This is dued to the fact that ks is set to zero and therefore the error is not dampened tbl (a) Engine (b) Selection of (L7e Figure 4.1: Results engine speed In Figure (4.2(a)) two controlled input variables u(= TC1) are depicted obtained with the two controllaws. In the same way in Figure (4.2(b)) the vehicle speed is given and in Figure (4.2(c)) the vehicle acceleration. The following features are noticeable in Figure (4.2): The input variable u(= T,.) needed to obtain the demanded output variable doesn't exceed 300 [Nm]. 0 The launch acceleration condition of 3.5 [m/s2] is almost met by &. 0 The condition +, = 50 [rad/sec] is met and therefore step 2 of Stop-Go can be initiated. The path of +, represents a comfortable vehicle launch When step 2 can be initiated the flywheel is still rotating and contains energy. This is noticeable in Figure (4.3(a)) and Figure (4.3(b)). The energy of the flywheel can be used in the next steps of Stop-Go. In order to test both controllaws a disturbance has been introduced and an initial condition is changed. The disturbance is moddeled by increasing Tload for a short period of time, for instance due to uphill climbing, see Figure (4.4). The initial condition +, = 0 [rad/sec] is changed into +, = 5 [radlsec]. In reality this means that the engine is still running when the Stop-Go process is initiated. In Figure (4.4(a)) output y (=+,) and in Figure (4.4(b)) the acceleration of the vehicle. These are a result from a
32 (a) Clutch Torque T,t (b) Velocity of the vehicle gv (c) Acceleration of the vehicle Gv (d) Tload disturbance Figure 4.2: Various results simulation simulation with the disturbance Tload. It is noted that when using either of the controllaws the effect of this disturbance is clearly visible in the curves but has no great effect on the output variable +,. In Figure (4.5) output y is depicted resulting from simulations with the initial condition +, = 5 [rad/sec]. It is noted that for using either of the controllaws yields that the desired value yd is tracked after a short period of time. Hoewever, the tracking error using the elaborate law decelerates much faster than for the simplified controllaw.
33 (a) Energy of the flywheel Ef (b) Velocity of the flywheel df Figure 4.3: Results flywheel 4.4 Internal dynamics When the relative degree r is smaller then the order n of the system then control of the tracking error only accounts for part of the closed loop dynamics. Therefore, a part of the system dynamics (described by (n-r) state components) is not directly influenced by the control law. This part of the dynamics is called the internal dynamics. If this internal dynamics is stable the tracking control design problem has indeed been solved. Otherwise, the above tracking controller is practically meaningless, because the instability of the internal dynamics would imply undesirable phenomena such as violent vibration of mechanical members. Therefore the effectiveness hinges upon the stability of the internal dynamics. For the current case the choice of the output function y is such that internal dynamics is present. After all the elaborate model has relative degree 2 and for the simplified model applies r = 0 while the system model is of order 6. In this case it is very difficult to directly determine the stability of the internal dynamics. An indication that the internal dynamics are stable is the fact that the simuiations the behaview ef the shte variables are smooth and all stat variables remain within realistic bounds even though the system model is heavily excited.