S.W.H.Simons. DCT Report Technische Universiteit Eindhoven

Transcription

1 Shift dynamics modelling for optimization of variator slip control in a continuously variable transmission S.W.H.Simons DCT Report Technische Universiteit Eindhoven Master of Science Thesis Committee: ir. T. W. G. L. Klaassen (Coach) dr. P. A. Veenhuizen dr. ir. A. A. H. Damen prof. dr. ir. M. Steinbuch Eindhoven University of Technology Department of Mechanical Engineering Section Control Systems Technology, Automotive Engineering Science Eindhoven University of Technology Department of Electrical Engineering Section Measurement and Control Systems, Control Systems Eindhoven, 3rd July 26

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3 Samenvatting Steeds meer duwband Continu Variabele Transmissies (CVT) worden geïmplementeerd, vanwege hun ongeëvenaarde schakel gemak. Een hoge overdrive ratio zorgt voor lagere motor snelheid, verbeterd rijcomfort op snelwegen en een gereduceerd brandstof gebruik. Doordat de 6 of 7 traps automatische versnellingsbakken echter steeds meer concurrerend worden, is het erg belangrijk de prestaties van de CVT te verbeteren. Met name op het gebied van efficiëntie, robuustheid en koppel capaciteit. Efficiëntie verbetering, door middel van het reduceren van knijpkrachten in de variator tot een minimal niveau, heeft zich bewezen. Door het reduceren van de knijpkrachten kan de variator in zijn meest efficiënte werkpunt opereren. Ook wordt de mechanische belasting op de variator geminimaliseerd en worden de hydraulische actuatie verliezen teruggebracht. De CVT slip regeling techniek, zorgt voor de best mogelijk efficiëntie, in combinatie met verbeterde robuustheid ten opzichte van schade als gevolg van slip. Terwijl voorafgaande onderzoeken zich voornamelijk bezig hielden met de dynamica tijdens stationair rijgedrag, bleek dat tijdens snel schakelen andere dynamische effecten optreden. Hiervoor zijn theoretische schakel modellen gevalideerd en met elkaar vergeleken. Het beste model is gebruikt voor de optimalisatie van de huidige slip regeling. Hiervoor is een slip regeling ontworpen op basis van Linear Quadratic Gaussian (LQG) control. Tevens is een PI regeling voor de ratio van de CVT variator ontworpen. Experimenten in een test voertuig zijn gebruikt om de regeling te valideren en te vergelijken met het ontwerp voor de slip regeling uit vorige onderzoeken. i

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5 Abstract V-belt type Continuously Variable Transmissions (CVT) are applied in an increasing number of vehicles as a result of their unparalleled shift comfort. Large ratio coverage allows for reduced engine speed, improved highway driving comfort and reduced fuel consumption. With the advent of the competing automatic transmissions with 6 or even 7 steps, it becomes increasingly important to further improve the performance in terms of efficiency, robustness and torque capacity of the CVT. Improvements on the efficiency of the pushbelt CVT by reducing variator clamping forces to minimum values are well established. By reducing clamping forces such that the variator operates in its most efficient point, the mechanical load on this variator is minimized and hydraulic actuation losses are reduced. The CVT slip control technique allows for best possible transmission efficiency, combined with improved robustness for slip damage. However, CVT slip dynamics during transient behaviour could not be neglected compared to steady state behaviour. Theoretical CVT shifting models are validated and compared to best agreement with experimental results. The results are used for model optimization. A Linear Quadratic Gaussian (LQG) controller is designed for the CVT slip control, as well as a Proportional Integral (PI) controller for CVT ratio control. Experimental results in a test vehicle prove validity and compare the design with the previous work on this approach. iii

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7 Contents Samenvatting Abstract Nomenclature and acronyms i iii vii 1 Introduction 1 2 Variator slip control Variator slip and functional properties Vehicle implementation previous slip controller Experimental vehicle Analysis previous slip controller Transient pushbelt variator models Literature on shifting mechanisms Transient models Ide s model CMM model Pulley thrust ratio Validation and comparison of transient models Modeling system dynamics Variator dynamics Actuation system dynamics Clamping force actuation Shifting force actuation Linear CVT model Interaction analysis Control design and strategy Ratio control Slip control LQG feedback control Feedforward control Slip control strategy Control implementation in test-vehicle Torque converter control Safety measures Implementation results Simulation results Vehicle implementation results v

8 vi CONTENTS 7 Conclusions and recommendations Conclusions Recommendations Bibliography 42 A Jatco CKkai-CVT 45 B Nissan Primera test vehicle 47 C Shift speed experiments 51 D Pulley thrust ratio 55 E Dimensional analysis 59 F CMM model 61 G Linearized model 63 H Electro-hydraulic system 65 I Linearizing shift valve model 67 J Ratio control analysis 69 K Non-linear model 71 L Chassis dyno 75

9 Nomenclature and acronyms Nomenclature Symbol Description Value [Unit] A car frontal area of the vehicle 1.8 [m 2 ] A p Primary cylinder surface [m 2 ] A s Secondary cylinder surface [m 2 ] A v Shift valve orifice [m 2 ] F air Air drag [N] F grad Gradient resistance [N] F p Primary clamping force [N] Fp Primary clamping force at steady state conditions [N] F rol Rolling resistance [N] F s Secondary clamping force [N] F spr, Preload spring secondary moveable pulley [N] I r Integral gain for ratio control [-] J e Engine side or primary inertia.356 [kgm 2 ] J s Vehicle side or secondary inertia 4.96 [kgm 2 ] K step Axial movement spindle pro step of the stepper motor [m] L Belt length.73 [m] P r Proportional gain for ratio control [-] R p Primary running radius [m] R s Secondary running radius [m] S f Safety factor [-] T d Road load torque [Nm] T e Engine torque [Nm] T p Primary pulley belt torque [Nm] T pump,loss Torque losses at the oil pump [Nm] T rl Road load torque [Nm] T s Secondary pulley belt torque [Nm] T var,loss Variator torque losses [Nm] V p Initial primary cylinder volume 1 4 [m 3 ] a Axial pulleys distance.168 [m] vii

10 viii CONTENTS c f Discharge coefficient.6 [-] c pl Primary leak coefficient.6 [-] c w Air drag coefficient.34 [-] f cp Centrifugal coefficient primary pulley [Ns 2 /rad] f cs Centrifugal coefficient secondary pulley [Ns 2 /rad] f r Rolling resistance coefficient.12 [-] g Gravitational constant 9.81 [m/s 2 ] k c Constant in the CMM model for rate of ratio changing [-] k c,x Constant in the CMM model for axial primary pulley speed [-] k i Constant in Ide s model [-] k oil Oil compressibility [m 2 /N] k spr Spring constant secondary pulley [N/m] m i Constants in pulley thrust ratio approximation [-] m car Simulated vehicle mass 13 [kg] n p Primary pulley rotational speed [rpm] p d Drain pressure 1 4 [Pa] p p Primary pulley pressure [Pa] p p Primary pulley pressure at steady state conditions [Pa] p p Minimal primary pulley pressure [Pa] p ph Primary pulley pressure at turning point between slip and creep [Pa] mode p p,ss Primary pulley pressure at steady state [Pa] p p,val Primary pulley pressure due to shift valve position operation [Pa] p s Secondary pulley pressure [Pa] r fd Final drive.1827 [-] r g Geometric ratio [-] r s Speed ratio [-] r s Speed ratio at no load conditions [-] u sol Solenoid input [-] u step Stepper motor input [-] x p Axial position primary pulley [m] x p,min Minimal axial position primary pulley [m] x p,ref Axial position reference [m] x s Axial position secondary pulley [m] x s,max Maximal axial position secondary pulley [m] x step Stepper motor position [m] x v Valve position [m] F Absolute shifting force [N] ln F Logarithm relative shifting force [-] p i Pressure drop over the shift valve [bar] Λ Relative gain array [-] β Maximum amplitude of the wedge half-angle variations along the contact arc [-]

11 CONTENTS ix Ψ Pulley thrust ratio [-] α Gradient of the road [ ] β Pulley wedge angle 11 [ ] β Pulley wedge angle at loaded conditions [ ] γ Throttle position [%] η Efficiency [%] κ Interaction measure [-] λ Relative gain array element [-] µ Traction coefficient.9 [-] µ eff Effective traction coefficient [-] ν Relative belt slip [-] ν ref Relative belt slip reference [-] ρ air Air density 1.29 [kg/m 3 ] ρ oil ATF oil density (at 8 C) [kg/m 3 ] τ Torque ratio [-] ω e Engine rotational speed [rad/s] ω p Primary pulley rotational speed [rad/s] ω s Secondary pulley rotational speed [rad/s] Acronyms Symbol Description AT F Automatic Transmission Fluid CAN Controller Area Network CM M Carbone Mangialardi Mantriota CV T Continuously Variable Transmission ECM Engine Control Module LOW Low ratio (r g =.43) LQG Linear Quadratic Gaussian LV DT Linear Variable Displacement Transducer M IM O Multiple Input Multiple Output MED Medium ratio (r g = 1) OD Overdrive ratio (r g = 2.15) P I Proportional Integral P LT R Power Loop Test Rig P W M Pulse Width Modulation RGA Relative Gain Array SISO Single Input Single Output T CM Transmission Control Module V DT Van Doorne s Transmissie

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13 Chapter 1 Introduction The low efficiency in a present production CVT is to a large extent caused by high clamping force levels. To prevent major slip events between pulleys and belt, the clamping forces are much higher than necessary for proper operation. These higher forces result in higher torque losses in the variator. Additionally, higher clamping forces require higher hydraulic pressures, thereby leading to increased pumping losses. Previous studies have shown that reducing these clamping forces result in a remarkable increase in efficiency [1]. However, the risk of belt slip is increased, using these low clamping forces. Present work by Van Drogen and Van der Laan [23] has shown that belt slip is allowed to a certain extent. Bonsen et al. [1] have demonstrated a possible efficiency gain by using slip control instead of the conventional control strategy. In earlier studies, conditions for optimum performance regarding efficiency and robustness were identified and validated for steady state conditions. However, stability robustness during ratio changes proved to be insufficient during tests in an experimental vehicle [1]. To deal with CVT slip dynamics during transient behaviour, a theoretical model is necessary. In this report, a novel CVT shifting model, recently proposed by Carbone et al. [7], is tested experimentally and compared with the model of Ide et al. [9]. The relationship between the clamping forces acting on the moveable pulley sheave, the rate of change of speed ratio, the loading conditions and the belt velocity are investigated both from a theoretical and an experimental point of view. Experiments are carried out on a test rig and in a test vehicle. The main goal of this report is to propose a variator control strategy, which achieves increased robustness and drivability compared to the control strategy proposed and implemented by Bonsen et al. [1]. To achieve this goal, first the basic principles of slip behaviour in the pushbelt variator are described. Also in Chapter 2, the previous designed slip controller is analysed. Especially the fast shifting events, where failure of the previous slip controller occurred, are investigated to improve slip control robustness. To get more insight in shifting behaviour of the CVT, a short literature study is given in Chapter 3. Subsequently, the obtained models are validated and compared. This leads to the model, which gives the best prediction of the CVT s dynamical behaviour during shifting manoeuvres for given values of the applied clamping forces, torque load and pulley angular velocity. In Chapter 4 the dynamic model of the variator, including actuation system, is derived. The model makes use of the CVT slip and shifting dynamics, which are obtained in the previous chapters. The interaction of the derived plant is analysed for control design possibilities. The designed controllers, for both slip and ratio control are described in Chapter 5. For slip control, a design based on LQG control is proposed. The obtained controllers are tested in a experimental vehicle and compared with the previous proposed gain scheduled PI controller in Chapter 6. Finally, in Chapter 7 conclusions and recommendations are given. 1

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15 Chapter 2 Variator slip control Bonsen et al. [1] presents the possible efficiency gain by using slip control instead of the conventional strategy. This controller is based on the slip between pulleys and pushbelt in a CVT. This relative slip and its functional properties are defined in the next section. Afterwards the previous slip controller, proposed by Bonsen et al. [1], is discussed and analysed. 2.1 Variator slip and functional properties The theory of the slip controller is based on the power transfer in the CVT, which is due to the friction between pulleys and pushbelt. For a pushbelt variator, the relation between secondary clamping force F s and input torque T p can be represented by the effective friction coefficient µ eff (ν, r g ) = T p cos β 2F s R p (2.1) where R p denotes the running radius of belt on the primary pulley and β denotes the pulley wedge angle. Experiments have shown that this µ eff depends strongly on the CVT ratio r g and the amount of slip ν between belt and pulleys, but only weakly on clamping force and shaft speed. The dependency between slip and µ eff at fixed ratio and secondary clamping force is shown in the upper part of Figure 2.1. The traction increases linearly with slip until a maximum is reached at 1 [%] to 3 [%]. At higher slip levels, the traction decreases slowly with slip. A distinction between a stable micro-slip area and a unstable macro-slip area can be made. The turning point between these regimes is at the highest possible effective friction coefficient. Close to this turning point the maximum variator efficiency can be reached, as shown in the lower part of Figure 2.1. In this study the relative slip ν is defined as ν = 1 r s r s (2.2) where r s is the speed ratio and r s the speed ratio at no load conditions, assuming that there is no slip when the secondary shaft is unloaded, i.e. T s =. The speed ratio r s is defined as r s = ω s ω p (2.3) where ω p and ω s represent the primary and secondary rotational pulley speed, respectively. Assuming the belt runs on perfectly circular paths on the wrapped angles of both pulleys, the geometric ratio r g can be defined by r g = R p R s (2.4) 3

16 4 CHAPTER 2. VARIATOR SLIP CONTROL µ eff [ ] Overdrive Medium Low ν [%] 1 η [ ] ν [%] Figure 2.1: Effective friction coefficient µ eff and efficiency η versus slip ν in the variator measured at input speed of 3 [rad/s] for variator ratios low (.43), medium (1.) and overdrive (2.25). The slip values at which variator efficiency reaches its maximum are indicated. with R s the secondary running radius. Since the variator is assumed to be slip free when T s =, the continuity relation R p ω p = R s ω s leads to r s = r g. As shown in Figure 2.1 there is an optimum of the variator efficiency depending on the amount of slip. A safety factor can be defined that gives an indication of excessive clamping forces, that is equal to 1 where the efficiency coefficient reaches its maximum. S f (ν, r g ) = max(µ eff ) µ eff (ν, r g ) (2.5) where max(µ eff ) represents the maximum value of µ eff over the complete range of positive values of ν in a certain variator ratio r g. Parallel to the safety factor, a torque ratio τ can be defined. This equals the transmitted torque over the torque that maximally can be transmitted by the given clamping force level. i.e. τ is the inverse of the safety factor defined by Eq. (2.5), given by τ = 1 S f (2.6) If the clamping force is much larger than needed to transfer the torque, µ eff is smaller and therefore S f increases. In a normal production CVT, an overclamping factor S f of 3 up to 5 is not uncommon. In order to increase efficiency, it is preferred to control the amount of slip, such that S f 1. As shown in the micro slip area in Figure 2.1, an efficiency increase in the variator is achieved when S f is decreased. Disadvantage of this slip control strategy can also be seen in Figure 2.1. At slip values beyond the point where the slope of µ eff is zero, i.e. S f = 1, the micro slip regime changes to the macro slip

17 2.2. VEHICLE IMPLEMENTATION PREVIOUS SLIP CONTROLLER 5 regime as also indicated in [23]. This macro slip area is unstable and slip could increase rapidly due to driveline disturbances. Limited excursions into this area may however be allowed for the pushbelt variator. At steady state operation points with small disturbances, the present slip control achieved good results as indicated in [1]. Furthermore, disturbances by shock loads due to full throttle and emergency braking were overcome by a predictive feedforward, guaranteeing robustness in most operating points. However during fast shifting actions, in combination with load peaks due to engine torque and inertias, major slip events occurred. Therefore shifting behaviour is taken into account in the slip model as described in Chapter Vehicle implementation previous slip controller Experiments on test rigs by Pulles [16] showed that the strategy to control slip had great potential for efficiency improvement, while remaining robustness in a production CVT. For more realistic experiments, especially with regard to robustness due to high and unpredictable loads disturbances, the previous slip controller proposed by Bonsen et al. [1] is implemented in a production car Experimental vehicle Figure 2.2: Nissan Primera test vehicle The test vehicle used in this study is a Nissan Primera 2.5i with Jatco CKkai-CVT. This type of transmission is very similar to the Jatco CK2-CVT, used during experiments on test rigs, but has an increased torque capacity of 25 [Nm], made possible by a small decrease of the ratio coverage. The engine of this vehicle delivers a maximum power of 115 [kw] and a maximum torque of 25 [Nm]. By making a small adaptation in this production vehicle, slip control could be implemented. Therefore a linear displacement sensor is attached to the mechanical sensor following the axial position of the primary pulley. With this sensor the ratio r s under no load conditions can be estimated, while the two original Hall sensors provide the conventional ratio signal r s. With these two quantities the relative slip number is calculated with Eq. (2.2). The system is accurate enough to detect slip with a resolution of.1 [%]. The slip controller is implemented using a dspace Autobox system. This system together with

18 6 CHAPTER 2. VARIATOR SLIP CONTROL a signal conditioning box provides the measurement data of the transmission during experiments. The setup of the system can be seen in Appendix B. Furthermore, the three actuation signals of the hydraulic system of the CVT can be controlled. These are the Pulse Width Modulation (PWM) signals for the line pressure and torque converter lockup clutch and the stepper motor control signal. In the previous slip controller torque converter lockup and ratio control were preformed by the original TCM (Transmission Control Module) Analysis previous slip controller The previous slip controller consists of a gain scheduled proportional integral (PI) controller and feedforward. The parameters of the PI controller vary, depending on the CVT ratio and between the micro- and macro regime, as obtained by Pulles [16]. The feedforward is needed because the bandwidth of the slip controller is not sufficient to compensate for the fast dynamics of a combustion engine, due to the delay in the hydraulic circuits. This feedforward is based on the minimal clamping force necessary to transmit the engine torque T e, as shown in Eq. (2.7). F s,f F W = T e(ω e, γ) cos β 2R p max(µ eff ) The engine torque T e is estimated using a engine map, dependent on throttle position γ and engine speed ω e. Moreover, fast increase of the throttle position requires fast downshifts. This in combination with torque peaks induced by engine and accelerations in the variator, can trigger slip peaks. Therefore, an additional compensation in the feedforward is necessary. This compensation increases the minimal clamping force with a safety factor, if the derivative of the throttle position exceeds 8 [%/s]. This increase is maintained for 2.5 [s]. The amount of this safety depends on the maximum amount of the derivative during a throttle position increase. On the Bosch test track at Boxberg, Germany, over 5 people drove the test vehicle on a handling circuit with different driving styles. When driving in a more aggressive way, the slip controller was not able to attenuate slip peaks fast enough. Therefore over 5 major slip events exceeding 1 [%] slip occurred. These unnecessary large slip peaks are summarized and analysed next. (2.7) [ ] [%] [ ] [bar] Speed Ratio No Load Ratio Slip Duty cycle Line Pressure Time [s] Figure 2.3: Response of a sudden downshift

19 2.2. VEHICLE IMPLEMENTATION PREVIOUS SLIP CONTROLLER 7 In spite of the precautions during fast throttle increase and sudden downshifts, not all slip peaks caused by this event could be prevented. An example is shown in Figure 2.3. During downshift the primary pressure drops and therefore torque capacity of the variator decreases, while due to inertias, transferable torque increases. This causes an increase in slip. Bad implementation of the safety increase for fast throttle position increase, gave sudden drops in the duty cycle during downshifting. This triggered slip peaks instead of preventing them. [ ] [%] [ ] Speed Ratio No Load Ratio Slip Duty cycle [bar] Line Pressure Time [s] Figure 2.4: Response of a sudden upshift Also fast upshifting events gave problems. During sudden upshifting, feedforward as described in Eq. (2.7) decreases, since primary running radius increases. This implies a lower necessary clamping force. The decrease in feedforward triggers a slip events which cannot be followed accurately by the PI controller. This causes the amount of slip to increase to unacceptable levels, as shown in Figure 2.4. The slip peaks will be experienced as shocks during driving and are therefore undesirable. Regarding damage to the belt and pulleys, the method described by Van Drogen and Van der Laan [23] is used. In this research it is shown that adhesive wear between belt and pulleys is caused by a combination of high slip speeds and normal forces on the elements. Results can be depicted in a force versus speed (F/v) diagram. The ratio dependent estimation of the failure limit in this diagram is based on measurements from this research. For both up- and downshifting events, normal force and slip speed are depicted in the F/v-diagrams in Figure 2.5. During both slip events slip peaks arise when the CVT ratio r g is near 1. Therefore the failure limit at that ratio is depicted. As shown in the F/v-diagrams, both events marginally exceed the failure limit, causing adhesive wear in the contact point between belt and pulley. Most of the occurred slip events at the test track, gave the same result. This led to severe damage as shown in Figure 2.6. This causes a decrease in the torque capacity and can lead to complete failure of the CVT. Analysis shows that transient behaviour cannot be neglected. Therefore the PI controller based on quasi-static variator dynamics gives insufficient robustness at that point. Furthermore the bandwidth of the slip controller should be increased, for better attenuation of disturbances. However,

20 8 CHAPTER 2. VARIATOR SLIP CONTROL 1.8 Normal Force [kn] Normal Force [kn] Slip speed [m/s] (a) Sudden upshifting Slip speed [m/s] (b) Sudden downshifting Figure 2.5: F/v-failure diagrams for the up- and downshifting events described in Figures 2.3 and 2.4, with failure limit for rg = 1 (a) Left side (b) Right side Figure 2.6: Belt flank pictures after the test drives at the Bosch test track in Boxberg, Germany the delay in the hydraulic actuation system will still bound this latter point of improvement.

21 Chapter 3 Transient pushbelt variator models As shown in the previous chapter, shifting dynamics are not negligible for the slip controller. In contrast with steady state pushbelt variator models, the number of transient models is exceptionally small. Different shift models are compared in order to find out which model gives the best representation of characteristics during shifting. The pulley thrust ratio Ψ, important for all transient model is discussed afterwards. Finally validation and comparison of the transient models, which are of interest for this research, are described. 3.1 Literature on shifting mechanisms One of the first studies considering shifting mechanisms of the pushbelt CVT was by Ide et al. [8]. A dynamic response analysis was carried out, showing that downshifting has a negative influence on the vehicle acceleration. Here also a primary balance force F p was introduced to keep the CVT ratio constant, proportional to the secondary clamping force and depending on speed ratio and torque ratio. Ide also discovered that the movement of the belt on the pulleys in radial direction per revolution is somewhat independent of the input speed and clamping force and only dependent on the absolute shifting force (F p F p ), with F p defined as the actual primary clamping force. Combining this, a simple formula was obtained revealing the relation between shifting force and shifting speed. The research from Ide was expanded [9]. This work shows that the maximal shifting speed in the pushbelt CVT is limited by the orifice of the primary valve, but not depended on the input speed. Therefore the equation given in the previous work [8] only holds for relative small shifting speeds. Two shifting modes can be defined, creep mode, with a maximal shifting speed dependent on input speed and slip mode where radial slip between the belt and pulleys occurs. In Figure 3.1 a simple visualization of these two modes is represented. In slip mode less force is needed to achieve increasing shifting speeds, compared to creep mode. Shafai et al. [2] introduced a simple belt model based on kinematics and the steady state ratio of both clamping pressures. It was shown that when the pressure ratio differs from the steady state situation, acceleration in axial direction of the primary pulley sheave is possible. The second order equation of motion derived, contains a representing mass of the sheave and a representing friction or damping coefficient. Fujimura et al. [13] stated that during shifting radial slip occurs over the whole contact arc of the pulley where the running radius decreases. At the pulley where the running radius increases elastic deformation of the elements occurs, pushing each element outwards. Since the next undeformed element enters the pulley at the increased running radius, the running radius of the total belt increases by each element that enters the pulley during the shifting cycles. Shifting 9

22 1 CHAPTER 3. TRANSIENT PUSHBELT VARIATOR MODELS dx p dt Creep mode Slip mode dx p dt dx p dt h p p, p p,h p p * p p Figure 3.1: Relation between primary pressure and axial pulley speed for both creep and slip mode speed is only dependent on the plastic deformation. In this work also the friction coefficient between elements and both pulleys is investigated during shifting but no direct relations were derived. Comparable to this latter work, Kataoke et al. [17] also carried out a study regarding the friction between blocks and pulleys in relation to the shifting speed. In this work the shift speed is dependent on the shift force while the elastic deformation increase with increasing normal force. A numerical model is obtained for the proposed relation regarding elastic deformation. Bonsen et al. [3] did research on the behaviour of both tangential and radial friction coefficient related to the creep and slip mode as introduced by Ide. The friction coefficient is not directly measured, but modelled by a Coulombs friction model. Also a relation between the shifting speed and shifting force is proposed based on the Ide model [9] in creep mode and the Shafai model [2] in slip mode. Finally Carbone et al. [6] proposed a theoretical model of the pushbelt to determine the transient behaviour of the belt in the CMM (Carbone Mangialardi Mantriota) model. The model can represent dynamics in both creep and slip mode and the turning point between both modes can be evaluated. With this model, a simple relation between shifting speed and shifting force in creep mode is obtained. In succeeding work of Carbone [7], this relation is further analysed. In this latter research a chain is modelled for torque transfer. The model for the pushbelt differs slightly from the chain model. In the model for the pushbelt the clearance between the elements at the entrance of the driving pulley and the shock section are included. Summarizing, some principle work regarding the friction behaviour of the system has been done. For control purposes in this research, a simple relation between shifting force and shifting speed is necessary for better slip and ratio control models. Further investigation on this topic is not of main importance in this research. Due to low line pressure level when using slip control, shift force levels are limited, as shown in Appendix C. This leads to small shifting speeds, which are assumed to occur in creep mode as shown by Ide et al. [9]. Therefore, the models that handle shifting in creep mode, Ide and CMM, are further evaluated.

23 3.2. TRANSIENT MODELS Transient models Transient variator models describe the relation between the CVT shift speed ṙ g and clamping forces in the variator. The models to be compared are further explained in the next section Ide s model The model of Ide is based on experimental results. During his study a number of ratio change experiments were carried out with various settings of pressures, primary speed, load and speed ratio. The resulting model can be expressed as ṙ g = k i (r g )ω p [F p F s Ψ(r g, τ)] (3.1) where k i (r g ) is an experimentally obtained constant, which depends on the geometric ratio r g. Ψ represents the pulley thrust ratio necessary for steady state behaviour respectively, as defined in the next section. Eq. (3.1) relates the shift speed to the shift force (F p F s Ψ) and input speed ω p. Analysing k i (r g ) gives a parameter, which can be divided in two parts, i.e. a ratio independent part and a dependent part, transforming axial pulley speed to the rate of ratio change. k i (r g ) = k i, ṙ g ẋ p where ẋ p denotes the axial speed of the primary pulley. (3.2) Eq. (3.1) holds in creep mode, while in slip mode the rate of change of the radial position of the belt on the pulley is independent of input speed. It is thought that the shifting speed also depends on the bending of the pulleys, proportional to the absolute shifting force. This should be compensated in the constant k i (r g ). In this model little physical explanation is given and Eq. (3.1) is estimated on experimental results. Also no difference between up- and downshifting is distinguished CMM model In contrast to Ide s model, the CMM model ([5], [7]) originates from a theoretical investigation preformed on a one-dimensional model, while friction forces are modelled on the basis of Coulomb friction hypothesis. Also in this model a creep mode and slip mode can be distinguished. In the creep mode, close to the steady state operation point, a linear dependency between shift speed and shift force can be estimated by [ ṙ g = ω p β (F s )k c (r g ) ln F ] p ln Ψ(r g, τ) (3.3) F s where β = max β β is the maximum amplitude of the wedge half-angle variations along the contact arc. The groove angle is, indeed, not constant along the wrapped arc because of the pulley bending due to elastic deformation and clearance in the bearings. β is the non-uniform wedge half-angle of the deformed pulley, whereas β is the wedge half-angle of the undeformed pulley. the difference (β β) is described on the basis of the Sattler s model [19], by means of simple trigonometric functions. The quantity β depends on the clamping forces acting on the two pulleys always being of the order 1 3 [rad]. Eq. (3.3) shows that creep mode shifting takes place only due to bending of the pulley sheaves. k c (r g ) is a known function calculated using the theoretical model, in contrast to k i (r g ) from the Ide model. The derivation of this latter term is given in Appendix F Besides the physical explanation of the shifting behaviour in creep mode, the CMM model has another big difference with the model of Ide. By taking the logarithm of the shifting force, a distinction between up- and downshifting can be made, based on dimensional analysis, explained in Appendix E. This results in a symmetrical up- and downshifting around r g = 1, dependent on the logarithmic of the shift force.

24 12 CHAPTER 3. TRANSIENT PUSHBELT VARIATOR MODELS 3.3 Pulley thrust ratio All models for transient behaviour use the pulley thrust ratio Ψ, as defined by Vroemen [24]. Ψ(r g, τ) = F p F s where Fp denotes the primary clamping force needed to maintain stationary CVT speed ratio. This pulley thrust ratio is dependent on the geometric ratio r g and torque ratio τ. Measurements of Ψ are carried out at a test rig, where two electrical motors are both driving and braking the CK2-CVT, as shown in Appendix C. During the experiments to obtain this thrust ratio for different operating point, it became clear that it also was depending on the amount of secondary pressure p s. Due to limitations of the test rig this could not be measured for all working point as shown in Appendix D. In this research on slip control, the results from measurements at minimum pressure levels are sufficient and the influence of increased secondary pressure will be neglected. Measurement results of Ψ at p s = 7 [bar] are shown in Figure 3.2. In order to reveal symmetry around r g = 1 and Ψ = 1 as proposed by Carbone et al. [5], it is chosen to depict ln(ψ) versus ln(r g ). From the CMM model an approximation for the pulley thrust ratio is derived. This approximation, which is dependent on geometric ratio r g and torque ratio τ, is defined by (3.4) ln(ψ(r g, τ)) = m + m 1 ln(r g ) + m 2 τ + m 3 τ 2 (3.5) The values for parameters m i are chosen to match the results of the measurements. This relation is visualized by the solid lines in Figure τ =.2.4 τ =.6 τ = ln(ψ).2.4 ln(ψ).2 ln(ψ) ln(r g ).8.6 τ = ln(r g ).8.6 τ = ln(r g ).8.6 τ =.8 ln(ψ).4.2 ln(ψ).4.2 ln(ψ) ln(r g ) ln(r g ) ln(r g ) Figure 3.2: Ψ measurements (points) on the CK2-CVT at F s = 1 [kn], compared to the theoretical values for Ψ derived from the CMM model (solid line) It is clearly shown that when the torque ratio τ is small, there is significant disagreement between the theoretical predictions and the experimental outcomes. Nonetheless, it has been found

25 3.4. VALIDATION AND COMPARISON OF TRANSIENT MODELS 13 that in the case of used belts a very good agreement occurs also at low torque levels [5]. This may support the idea that band-elements interaction, not considered by the CMM model, may have a key role at small torque ratios. Thus, further investigation should be carried out. However, since the slip control strategy needs relatively high torque ratio values, this allows the use of the CMM model in the subsequent sections. 3.4 Validation and comparison of transient models In the previous sections two models were introduced. The Ide model, based on experimental results and the CMM model based on theoretical calculations. The latter model is investigated and validated by spin-loss measurements at a variator preformed by Carbone et al. [5]. This variator is preferred since, at the Jatco CK2-CVT side effects of the other parts of the transmission should be taken into account. In Figure 3.3(a) some results of the validations of the CMM model are shown. The shifting speed is depicted as a function of the logarithm of the thrust ratio F p /F s for zero torque load and different values of instantaneous speed ratio r g. A very good agreement between theory (solid lines) and experiment (points) is shown. At some points there is a small difference between theory and experiment observable. This is mainly due to a different value of the pulley thrust ratio Ψ, i.e. at ṙ g =, rather than a difference in slope. In Figure 3.3(b) it is shown that by matching the pulley bending parameter β depending on the clamping force F s as proposed in Appendix F, also at higher clamping force levels, the slope of the curves fits the measurement data. At a geometrical ratio equal to 1, an unexpected antisymmetric behaviour is visible in all of the diagrams of Figure 3.4. This may be caused by the fact that the assumption that the friction coefficient in radial direction is constant as made by Carbone et al. [5] does not hold. At ratios r g < 1 and ṙ g >.3 measurements start to deviate from the model in positive direction. This could denote the change from creep to slip mode shift regime. All other differences between experiments and theory can be accorded to scattering of the test results and are therefore negligible. At Figure 3.3(c) the experimental results at double input speed are shown. Again a good agreement of the CMM model is visible. This also confirms the prediction of a direct proportional relation between input speed n p and shifting speed ṙ g. Ide gives a linear relation between the shifting speed ṙ g and the shift force ratio F p /F s, instead of the ln(f p /F s ). Thus, if the Ide relation is really accurate, it is expected that the experimental results should follow a straight line when plotted against the shift force ratio F p /F s. However, Figure 3.4 shows this is not the case. Mainly at small ratios r g, a nonlinear behaviour is clearly visible. This is in agreement with early experiments, where up- and downshifting gave different results regarding the absolute shifting forces needed to achieve equal shifting speeds in both directions. Downshifting during these tests with the same shifting speed was reached at lower shift forces. Furthermore, the difference in relation between both models can be shown by the quantity [ln(f p /F s ) ln Ψ] as proposed in the CMM model. If this relation is expanded in a Taylor series in the neighbour of the pulley thrust ratio Ψ the following is obtained according to ln F p F s ln Ψ = 1 Ψ [ Fp F s Ψ ] Ψ 2 [ ] 2 Fp Ψ +... (3.6) F s After rewriting the relation of the Ide model, it equals the first order term of the Taylor expansion and neglecting higher order terms. The experimental Ide model can therefore be defined as an approximation of the theoretical CMM model. The difference between the linear relation of Ide and the logarithmic relation of the CMM model increases if Ψ is decreased below 1. This occurs at geometric ratios below 1, which is in agreement with Figure 3.4. For r g > 1 the correction is less important.

26 14 CHAPTER 3. TRANSIENT PUSHBELT VARIATOR MODELS dr g /dt r g =.6 r g =.8 r g = 1. r g = 1.4 r g = ln (F p / F s ) (a) Initial validation at F s = 2 [kn]and n p = 1 [rpm] dr g /dt dr g /dt ln (F p / F s ) ln (F p / F s ) (b) Validation at higher clamping force with more elastic pulleys bending, F s = 3 [kn[ (c) Validation of the linear dependency of the input speed, n p = 2 [rpm] Figure 3.3: Validation of the CMM model; points depict the measurements, the lines depict the CMM model at the corresponding operation point From this evaluation of shift models in creep mode it is shown that the CMM model gives a good agreement. Also the logarithmic CMM model fits the experimental results better than the linear relation of Ide, which was in agreement with early tests on the CK2-CVT. Therefore this latter model is used for modelling variator dynamics.

27 3.4. VALIDATION AND COMPARISON OF TRANSIENT MODELS dr g /dt r g =.6 r g =.8 r g = 1. r g = 1.4 r g = F p / F s Figure 3.4: Measured ṙ g as function of the shifting force F p /F s at F s = 2 [kn]and n p = 1 [rpm]

28 16 CHAPTER 3. TRANSIENT PUSHBELT VARIATOR MODELS

29 Chapter 4 Modeling system dynamics In contrast to the model in the previous work from Bonsen et al. [1], both geometric ratio r g and relative belt slip ν are taken into account here. Therefore, slip dynamics will differ from the previous models as will be shown in the next section. Subsequent, the variator and actuation dynamics are derived and interaction of the complete plant is analysed. 4.1 Variator dynamics In contrast to the previous work, the geometric ratio is not regarded as quasi-stationary. From analysis in Chapter 2 it became clear that the assumption of geometric dynamics being much slower than slip dynamics does not hold for fast shifting events. Therefore derivation of Eq. (2.2) gives ν = r sṙ g r g ṙ s r 2 g (4.1) where ṙ g is given by the CMM model (Eq. (3.3)) and the derivative of the speed ratio r s is given by ṙ s = ω p ω s ω s ω p ω 2 p (4.2) A A M F 6 F H C 6 I M I I Figure 4.1: CVT dynamics A simple representation of the CVT variator dynamics is shown in Figure 4.1. On the input side of the variator T e represents the engine torque and J e describes the equivalent engine and CVT inertia on the primary shaft. At the output side T d represents the road load torque, defined 17

30 18 CHAPTER 4. MODELING SYSTEM DYNAMICS by road load conditions, and J s describes the equivalent vehicle inertia on the secondary shaft. The dynamics of the primary and secondary shaft of the CVT variator are given by ω p = T e T p J e ω s = T s T d J s with T p and T s denoting the torque on the primary and secondary shaft respectively. These torques generated on both shafts of the variator are described, based on Eq. (2.1), by (4.3) (4.4) T p,s = 2F sµ eff (ν, r g )R p,s cos β (4.5) In this description torque losses are neglected. It is assumed that these losses are not significant for the modelling of variator dynamics. Substituting Eqs. (3.3), (4.2), (4.3), (4.4) and (4.5) in Eq. (4.1) leads to ν = 1 ( 2F sr s µ eff (ν, r g ) + T ) d + 1 ν ω p J s r g cos β J s r g ω p ( 2F sr s r g µ eff (ν, r g ) + T e J e cos β J e ) + 1 ν r g ṙ g (4.6) Compared to the model with neglecting geometric ratio dynamics described by Bonsen et al. [2], the last term in Eq. (4.6) is added, since ratio changing dynamics are not negligible, i.e. ṙ g. The amount of friction between pulley and belt is related to the slip in the variator. The friction coefficient used in this model is therefore described by µ eff (ν, r g ) = k 1i (r g )ν + k 2i (r g ) (4.7) which is a piecewise linear approximation of the results shown in Figure 1. The ratio dependency is taken into account by the choice of k 1i and k 2i. The micro slip regime is denoted by i = 1, whereas i = 2 denotes the macro slip regime. Besides the amount of slip ν, also the ratio r g is a control variable in the controller proposed in Chapter 5. Since the primary axial pulley position x p is measured and to avoid the nonlinear calculation from x p to r g, this linear position is used as control variable. Considering this, the dynamic equations are rewritten to ν = 1 ( F sx p µ eff (ν, r g ) ω p sin βj s rg 2 + T ) d + 1 ν ( F sx p µ eff (ν, r g ) + T ) e + (1 J s r g ω p sin βj e J ν)ẋp [1 + r g h(r g )] e x p (4.8) and ẋ p = ω p β k c,x (r g )x 2 p [ ln F ] p ln Ψ(r g, τ) F s (4.9) where both h(r g ) and k c,x (r g ) are defined in Appendix F. Defining the state vector x = [ ν x p ] T, input vector u = [ F s ln F T e T d ] T and output vector y = x, the dynamics can be described, when linearized around a certain working point x = [ ν x p ] T, by ẋ = A var x + B var u (4.1) y = C var x (4.11) Note that the terms dependent on ratio are also related to the axial primary pulley position. In Appendix G a more complete derivation of the linearized model is given. It is assumed that ν

31 4.2. ACTUATION SYSTEM DYNAMICS 19 1 [%] and d 1 1 and higher order terms of ν can be neglected. This results in [ ] [ ] a11 a A var = 12 b11 b and B a 21 a var = 12 b 13 b b 21 b 22 with ( ) ) (k2i k 1i) J e k1i Te rg 2 Js J ( e 2[1+rgh] 1 ( a 11 = 1 Fsx p ω p sin β ( a 12 = 1 ω p T d(1+r gh) x pj sr g + Fsk2i sin β +ω p β k c,x x p [1 + r g h)] ( ln F c a 21 = ω p β k c,x x 2 p ln F a 22 = ω p β x 2 p and ( ν 2kc,x ln F x p b 11 = xpk2i ω p sin β b 12 = ω p β k c,x x p [1 + r g h] b 13 = 1 J eω p b 14 = 1 J sω pr g ( b 21 = 1 ω p β k c,x F s b 22 = ω p β k c,x 1 J e + 1 r 2 g Js ) J sr 2 g + ln F k c,x x p + ω p β k c,x x p [1 + r g h)] ( ) ln F ν ln F )) 1 J e ) c x p + ln F r g + ln F x p ) + k ln F c,x x p ω p β k c,x x p [1 + r g h] 1 F s The partial derivatives can be found in Appendix G. In case of the CMM model, the input ln F represents the logarithm of the shift force ratio F p /F s minus the logarithm of the pulley thrust ratio Ψ. ln F = ln F p F s ln Ψ (4.12) Furthermore, the terms T e and T d are calculated to match the maximum torque that can be transmitted in the chosen working point, leading to T e = x pf s µ eff sin β T d = T e r g This results in the transfer function H var for the variator dynamics. (4.13) (4.14) 4.2 Actuation system dynamics In the Jatco CK2-CVT the clamping force F s is applied by pressurizing the two cylinders attached to the moveable pulley sheaves. The hydraulic scheme concerning the pressurizing of the two cylinders behind the moveable pulleys is shown in Figure 4.2. The secondary pressure cylinder is directly connected to the line pressure and is directly related to the clamping force. The shifting quantity ln F is regulated by a valve between the secondary and primary pressure circuit. All pressures are regulated by a complex electro-hydraulic actuation system Clamping force actuation The line pressure is determined by the PWM signal duty cycle used to control the solenoid. In this system the pressure is limited between 6.6 and 42 [bar] and can be assumed linear proportional to the duty cycle. The transfer function from solenoid input signal to clamping force is shown in

32 2 CHAPTER 4. MODELING SYSTEM DYNAMICS Figure 4.2: Hydraulic scheme for primary and secondary pressure Figure 4.3: Subsystem clamping force actuation Figure 4.3. To avoid the complex and time consuming dynamic modelling of the combination of solenoid and valve, a Frequency Response Function (FRF) measurement is preformed by Bonsen et al. [1]. The FRF-estimation H LP V from duty cycle u sol to secondary pressure p s resulted in a third order low pass filter with a cut-off frequency of 6 [Hz]. This estimation holds for all operation

33 4.2. ACTUATION SYSTEM DYNAMICS 21 points. To rewrite the pressure to an actual clamping force, both the centrifugal force and spring pre-load force should be added, resulting in F s = A s p s + f cs ω 2 s + k spr [x s,max x s ] + F spr, (4.15) Here A s and f cs represent the pressure cylinder area and centrifugal coefficient of the secondary pulley respectively. k spr is the spring constant of the spring, preloading the moveable secondary pulley sheave and F spr, the preload by the spring at its initial position. x s represents the axial position of this sheave. The linearized transfer function H LP from duty cycle u sol to secondary clamping force F s is obtained by H LP = F s u sol = H LP V H p2f (4.16) where H p2f is equal to the secondary cylinder surface A s. Since variables ω s and x s are lost due to linearizing of Eq. (4.15), transfer function H p2f is dependent on the chosen working point, i.e. speed and CVT ratio dependent Shifting force actuation The shifting in the CK2 is controlled by the stepper motor, regulating a primary clamping force differing from the steady state clamping force. The mechanical feedback, attached to the moveable primary pulley sheave, moves the valve back to its equilibrium position when the desired position is reached. The dynamic model concerning the primary pressure is based on the model given by Figure 4.4: Dynamic model for the shifting valve and primary pressure the turbulent orifice flow through the shifting valve derived from the work by Vroemen [24] and is visualized in Figure 4.4. The total primary pressure consists of the pressure change due to valve operation and an external force on the cylinder originated from the secondary pressure passed via the pushbelt. The total pressure in the primary circuit is therefore determined by p p = p p,val + p p,ss = ṗ p,val dt + F sψ f cp ω 2 p A p (4.17) Here p p,val and p p,ss denote the primary pressure change obtained by changing the valve position and steady state primary pressure respectively. The dynamic representation of the first term, assuming p s > p p, is denoted by ( ) 1 2 p i 2(p p p d ) ṗ p,val = c f A v (x v ) c f A pl A p ẋ p (4.18) k oil (A p [x p x p,min ] + V p ) ρ oil ρ oil

34 22 CHAPTER 4. MODELING SYSTEM DYNAMICS Here k oil and ρ oil are the compressibility and density properties of the ATF-oil respectively; A p the primary cylinder surface, V p the initial primary cylinder volume, A pl the primary leakage orifice and c f represents the orifice resistance coefficient. Furthermore, p i is the pressure drop over the valve, depending on the position of the valve x v. Also the orifice surface A v depends on this position. The shifting valve position is dependent on stepper motor and axial pulley position and it is assumed that x v = when the valve is closing both hydraulic circuits. Finally, the axial pulley speed is determined by the CMM model as given in Eq. (4.9). This results in p i (x v > ) = p s p p (4.19) p i (x v > ) = p p p d (4.2) x v = x step [x p x p,min ] (4.21) 2 The orifice surface is discontinuously function of the position of the valve. Measuring the geometry of the valve and its housing gives the mapping depicted in Figure A v [mm 2 ] x v [mm] Figure 4.5: Orifice surface as function of the shift valve position For linearizing purposes a first order relation is defined of this mapping, i.e. A v (x v ) = k v1i x v + k v2i (4.22) where k v1i denotes the slope and k v2i the offset of i-th part of the curve represented in Figure 4.5. By separating these parts, a distinction of the valve orifice near steady state, up- and downshifting can be made. The model of the primary valve is validated using measurements on the CK2-CVT. The primary pressure measured in the CVT, is compared to the value obtained from Eq. (4.17). Experiments have been carried out at different shift speeds, over the whole ratio range. A typical result is shown in Figure 4.6. The valve model shows good agreement with the measurements over the whole primary pulley axial position range. An offset of.5 [mm] is added to the valve equilibrium position, as shown in the upper part of Figure 4.6. This gives a small flow towards the primary pulley, to overcome the leakage at the primary cylinder during steady state operating points. Small errors can be assigned to the measured primary pressure, which is not measured directly at the primary cylinder and estimated leakage flows. Eq. (4.18) results in a non-linear system. Linearizing this model to obtain the transfer function H P P V for control purposes is shown in Appendix I. The input of the stepper motor u step is manipulatable. Inputs x p and ẋ p are fed back from the resulting shifting force by usage of the CMM model as shown in Figure 4.7 and p s acts as a disturbance on the system.

35 4.2. ACTUATION SYSTEM DYNAMICS 23 x pos [mm] stepper motor primary pulley shift valve measurement model p p [bar] time [s] Figure 4.6: Validation of the primary valve model. This example is carried out at ω p = 15 [rad/s], F s = 15 [kn] and axial speed of the spindle attached to the stepper motor ẋ step = 4 [mm/s] For determining the shifting quantity ln F, Eq. (4.17) should be considered. Implementation in the CMM model leads to ( ) pp,val A p f cp ωp 2 ln F = ln + 1 (4.23) F s Ψ Linearizing the relation leads to static gain transfer function H ln F with pressures p p,val and p s and position x p as inputs and the shifting quantity ln F as output. The derivations are shown in Appendix I. The linearized model for the whole subsystem is visualized in Figure 4.7. The transfer func- Figure 4.7: Subsystem shifting force actuation, based on the CMM model tion from stepper motor input u step to the shifting quantity ln F can be denoted as H P P = ln F H ln F,pp H P P V,ustep = ( ) (4.24) u step 1 H ln F,pp HP P V,xp H var,22 + H P P V,ẋp H shift H ln F,xp H var,22

36 24 CHAPTER 4. MODELING SYSTEM DYNAMICS with H shift denoting the transfer function from ln F to ẋ p, equal to the b 22 term of the model described in Eq. (4.1) and H var,22 the corresponding term of the variator transfer function. In Magnitude H PP [db] Phase H PP [deg] ω p = 1 [rad/s]; r g =.45 ω p = 1 [rad/s]; r g = 2.15 ω p = 3 [rad/s]; r g =.45 ω p = 3 [rad/s]; r g = Frequency [Hz] Figure 4.8: Transfer function from stepper motor u step input to shifting quantity ln F including the mechanical feedback Figure 4.8 the transfer function as obtained in Equation 4.24 is shown. The high frequent pole is due to the primary pressure model H P P V and is mainly dependent on ratio r g. The low frequent pole is provided by the feedback with the H var,22 term and is mainly input speed ω p dependent. 4.3 Linear CVT model With the derived transfer functions in the previous sections the complete plant H visualized in Figure 4.9 is obtained. Figure 4.9: Complete plant with variator dynamics H var and actuation dynamics H LP and H P P Here, H var defines the transfer function from clamping force F s and shifting force ln F to relative slip ν and axial primary pulley position x p. H LP defines the transfer function from solenoid input u sol to clamping force F s and H P P defines the transfer function from stepper motor input u step to shifting quantity ln F. For the latter plant the mechanical feedback is included, as explained in section 4.2. Here the input vector is defined u = [ u sol u step T e T d ] T and output vector y = [ ν xp ] T.

37 4.4. INTERACTION ANALYSIS 25 The derived linearized system will be used for controller design. The model has 4 inputs, where only u sol and u step can be manipulated. In the present setup of the hydraulic system, the solenoid input u sol controls the slip and the input of the stepper motor u step controls the geometric ratio that is related to the primary axial pulley position. The input torque T e is controlled by the driver via the throttle pedal, where output torque T d is determined by the road conditions. The latter two can therefore be regarded as disturbances on the system. Phase H 21 [deg] Magnitude H 21 [db] Phase H 11 [deg] Magnitude H 11 [db] Frequency [Hz] Phase H 22 [deg] Magnitude H 22 [db] Phase H 12 [deg] Magnitude H 12 [db] Frequency [Hz] Figure 4.1: Bode plots for the complete plant H at overdrive (r g = 2.15) and an input speed ω p of 1 [rad/s] (solid line) and 3 [rad/s] (dashed line) The derived linearized system will be used for controller design. The plant is shown for two operating points in Figure 4.1. As shown, the H 11 term is inverse dependent on input speed ω p, while the magnitude of the other transfer functions enlarges with the increase of this speed. This can be assigned to the definition of relative slip, which decreases if input speed increases, while in creep mode the rate of ratio changing is proportional to the input speed. 4.4 Interaction analysis To analyse the degree of mutual influence for 2 x 2 systems, the interaction measure proposed by Rijnsdorp [18] is most widely used. This interaction measure is defined as κ = H 12H 21 H 11 H 22 (4.25) For κ < this leads to diagonal dominance of the plant. The Relative Gain Array (RGA) Λ, a measure of interaction for decentralized control, is defined by [ Λ(H) = H (H 1 ) T = ] λ 11 1 λ 11 1 λ 11 λ 11 (4.26)

38 26 CHAPTER 4. MODELING SYSTEM DYNAMICS where denotes element-by-element multiplication and λ 11 = 1 1 κ (4.27) For the modelled linearized system, the values of λ 11 and λ 12 = 1 λ 11 are depicted in Figure Relative gain λ ω p = 1 [rad/s]; r g =.45 Relative gain λ ω p = 1 [rad/s]; r g = 2.15 ω p = 3 [rad/s]; r g =.45 ω p = 3 [rad/s]; r g = Frequency [Hz] Frequency [Hz] Figure 4.11: Relative gain at different operation points As can be seen, at low input speeds, Λ is near the identity matrix I. This implies small interaction between ratio and slip plant and a good choice of pairings. At higher speeds diagonal dominance of the whole plant is still present, however more interaction is visible. This can be assigned to the proportional dependency of the input speed on the shift speed, while slip dynamics depend on the inverse of the primary speed, as mentioned in Section 4.3. This leads to more influence of the off diagonal terms and therefore more interaction. The inverse dependency of the primary speed on the slip dynamics is caused by the definition of relative slip used in this research. Defining the absolute slip (ω p R p ω s R s ) could be of interest in future work. Since diagonal dominance of the complete plant is present, the design of a decentralized controller is possible. This results in the design of two separate Single Input Single Output (SISO) controllers instead of a more complicated Multiple Input Multiple Output (MIMO) controller, since off diagonal influence is negligible. The design procedure of these controllers is described in Chapter 5.

39 Chapter 5 Control design and strategy Based on the results of the interaction analysis, i.e. limited influence of the off-diagonal terms, a decentralized controller for both ratio and slip is proposed, as shown in Figure 5.1. Figure 5.1: Block diagram representation of the control design To take the interaction between the plant in- and outputs into account, sequential loop closing is applied. Since the ratio controller should always work during driving and the clamping force can also be applied in open-loop, i.e. the slip controller can be switched, the ratio control loop is closed first. This ensures stability of the ratio controller as shown by Klaassen and Steinbuch [1] 5.1 Ratio control For optimal performance, a PI controller is used for controlling the stepper motor. ( u step = P r + I ) r (x p,ref x p ) (5.1) s The shifting process already exhibits a sufficient amount of damping. Therefore a differential action is not necessary, but a high controller gain can be applied. By manually tuning the controller, an optimum in driveability and performance was found. In Figure 5.2 the open loop H OL,22 of the ratio loop is shown. The controller has a low bandwidth between.1 and.4 [Hz]. At operation points with higher input speeds, bandwidth is also higher. In Figure 5.3 the open loop of the ratio control is depicted for different shift force levels F. The depicted shift force levels give a shifting speed of approximately 4 [mm/s] in both up- and downshifting, as depicted in Appendix C. Bandwidth with the proposed controller is decreased 27

40 28 CHAPTER 5. CONTROL DESIGN AND STRATEGY Magnitude H OL,22 [db] ω = 1 [rad/s]; r =.45 p g 5 ω p = 1 [rad/s]; r g = 2.15 ω p = 3 [rad/s]; r g =.45 1 ω = 3 [rad/s]; r = 2.15 p g Phase H OL,22 [deg] Frequency [Hz] Figure 5.2: Open loop ratio control at steady state operation points ( F = ) Magnitude H OL,22 [db] 5 5 F = 3 [kn] F = [kn] F = 6 [kn] Phase H OL,22 [deg] Frequency [Hz] Figure 5.3: Open loop of the ratio control at ω p = 2 [rad/s], r g = 1 and at different shift force levels F during upshifting operation points. For the setpoint x p,ref of this controller the variogram for the CK2-CVT is used [12]. This variogram is constrained by a low drive ratio of.44 [-], an overdrive ratio of 2.17 [-], maximum engine speed of 6 [rpm] and maximum vehicle speed of 255 [km/h] as shown in Figure 5.4. In this research the optimal strategy concerning driveline efficiency, fuel consumption and driveability were not of main importance. An optimization as proposed by Bonsen et al. [4] could be of interest in future research. Also the addition of a stepwise ratio mode, common in present CVT s, is recommended.

41 5.2. SLIP CONTROL Low 4 n p [rpm] 3 2 Throttle 1 Overdrive v car [km/h] 5.2 Slip control Figure 5.4: Variogram for ratio controller setpoint With the interaction coefficient κ depicted in Eq. (4.25), the equivalent for the slip plant H 11 can be described by H 11,eq = H 11 (1 κh CL,22 ) (5.2) With H CL,22 defined as the closed loop transfer function of the ratio control. After closing the ratio control loop, the slip control loop can be closed using H 11,eq. Magnitude [db] Phase [deg] 9 18 H 11 ; ω p = 1 [rad/s] H 11,eq ; ω p = 1 [rad/s] H ; ω = 3 [rad/s] 11 p H ; ω = 3 [rad/s] 11,eq p Frequency [Hz] Figure 5.5: Slip plant H 11 and equivalent slip plant H 11,eq at r g = 2.15 [-] for two different input speeds ω p In Figure 5.5 only low frequent a small difference between H 11 and H 11,eq is visible, indicating small interaction. Only low frequent small difference between H 11 and H 11,eq is visible. At higher input speed more deviation is visible, as expected considering the results in Section 4.4.

42 3 CHAPTER 5. CONTROL DESIGN AND STRATEGY LQG feedback control In order to find a controller with stability of the closed loop system, good gain and phase margins, robustness with respect to unmodelled dynamics and optimal performance, a LQG (Linear Quadratic Gaussian) control design [21] is proposed. The slip plant is both observable and controllable in all operating points and therefore LQG can be applied. The state space notation of the slip plant should be extended with process noise ξ and measurement noise θ, that are uncorrelated Gaussian stochastic processes. ẋ = Ax + Bu + ξ (5.3) y = Cx + θ LQG control is a combination of optimal state feedback and optimal state estimation. Therefore the state feedback matrix K r and the Kalman gain K f are obtained by minimizing the criteria (5.4) K r = arg min E{x T Qx + u T Ru} (5.5) K f = arg min E{(x ˆx) T (x ˆx)} (5.6) Where Q and R are the weighting matrices for states and input respectively and ˆx represents the optimal estimate for state x. In standard LQG design an integral action is not included, therefore the slip plant is augmented with an integrator. Therefore in the slip plant the output equals [ ν s ν ]. For Q = qc T C, q = [ 1 1 ] and R = 1 are chosen. By using this term for Q, the output y is weighted rather than the state x. For the noise weighting matrices, covariances equal Ξ = E{ξξ T } = ξi, with ξ = 1 and Θ = E{θ 2 } = 1 4. This latter term is chosen this high, in order to keep control gains at high frequencies low, such that the influence of the large amount of measurement noise, typical in automotive environment, is reduced. The slip plant is depending on many parameters, mainly on the difference between the micro and macro slip regime, but also on ratio r g and input speed ω p. Since the setpoint for the slip controller is on the turning point of both slip regimes and in the micro slip area the plant is stable, the macro slip regime is of main concern for robust stability. Assuming a friction coefficient µ eff independent of the amount of slip in the macro slip area, the plant is marginally stable at that point. Magnitude C [db] Phase C [deg] 9 LQG PI Frequency [Hz] Figure 5.6: The controller based on LQG designed compared to the PI controller developed by Bonsen et al. [1] for the nominal plant

43 5.2. SLIP CONTROL 31 The worst case plant regarding robustness stability, disturbance and noise rejection was found at variator ratio r g = 2.15 and input speed ω p = 1 [rad/s]. This is chosen to be the nominal plant. The LQG control design was performed on this plant. The designed controller is shown in Figure 5.6 and is compared to the previous PI controller proposed by Bonsen et al. [1]. Since the PI controller makes use of gain scheduling, the working point of the nominal plant is chosen, to get a good comparison. As shown in Figure 5.6, the gain of the LQG controller is increased with respect to the PI controller. At the crossover frequency controller a double differential action is visible to get more phase margin at that point. After the crossover frequency this action is cut off to reduce the influence of the large amount of measurement noise at high frequencies. The open loop including the designed controller at the operation point of the nominal plant has the smallest phase and gain margins (45.1 [deg] and 5.6 [db], respectively), largest maximum sensitivity (7.2 [db]) and largest bandwidth (4.3 [Hz]) compared to other operating points. However, bandwidth in other operating points decreases with respect to their optimal control design. Phase S 11 [deg] Magnitude S 11 [db] Phase S 12 [deg] Magnitude S 12 [db] Magnitude S 21 [db] Magnitude S 22 [db] Phase S 21 [deg] Frequency [Hz] Phase S 22 [deg] Frequency [Hz] Figure 5.7: Input sensitivity of the complete system at ω p = 1 [rad/s] (solid line) or ω p = 3 [rad/s] (dashed line) and r g =.45 [-] (dark line) or r g = 2.15 [-] (bright line) The input sensitivity of the system is shown in Figure 5.7. At higher input speeds, the non diagonal sensitivities terms increase, but always remain below [db]. Bandwidth of the slip controller lays between 1.9 and 4.3 [Hz], depending on operation point. In Figure 5.8 the relative slip ν is plotted in the time domain, during a disturbance of the engine torque T e. As shown LQG control gives a better disturbance attenuation than the previous PI controller, especially at higher CVT ratios. Compared to the gain scheduled PI controller proposed by Bonsen et al. [1], the overall plant performance and robustness of the slip control is increased. Especially when using the modelled plant including ratio dynamics at an input speed of 3 [rad/s], performance and robustness drops to poor values. The gain scheduling in the previous design is based on a maximum sensitivity of 5 [db] in all operating points. If this design is

44 32 CHAPTER 5. CONTROL DESIGN AND STRATEGY 16 T e [Nm] 12 8 r g =.45 r g = ν [%] time [s] Figure 5.8: Response of the system with LQG (solid line) or PI (dashed line) control at ω p = 2 [rad/s] during a step of 1/4T e at 11 < t < 13 [s] implemented in the model proposed in this research this requirement is not met Feedforward control For optimal control with the limited bandwidth of the hydraulic actuation system the addition of a feedforward could be of great benefit. The engine torque T e can be estimated by using engine speed ω e, throttle position γ and the engine map of the ICE, supplied by Jatco [12]. During transient behaviour, the shifting speed is calculated by means of the axial speed of the spindle attached to the stepper motor. The axial position of this spindle can be calculated by x step = K step u step (5.7) with K step the axial movement per step of the spindle. This axial speed in combination with the CMM model gives the shifting force. Adding these two terms gives a secondary force equal to F s,f F W = T e(ω e, γ) cos β 2R p max(µ eff ) + F (ẋ step ) (5.8) where max(µ eff ) is dependent on CVT ratio r g as depicted in Figure 2.1. Because of the linear relation between clamping force and the PWM signal duty cycle, the amount of clamping force can be mapped to this duty cycle and added to the controller output Slip control strategy The setpoint for the slip controller is set near the turning point between the micro and macro slip regime, where the variator efficiency is close to its optimum. From Figure 2.1 it is shown that this turning point strongly dependents on ratio. Therefore, the setpoint is chosen to be dependent only on this parameter as shown in Table 5.1. Between the three given setpoints for the amount of slip, the setpoint is linear interpolated.

45 5.3. CONTROL IMPLEMENTATION IN TEST-VEHICLE 33 Table 5.1: Slip setpoint dependent on CVT ratio r g [-] ν ref [%] Control implementation in test-vehicle For vehicle implementation some additional features have to be added. The actuation of the torque converter lock-up clutch should be controlled. Furthermore some measures must be taken for fail-safe conditions Torque converter control The lock-up of the torque converter is achieved by controlling the lock-up control valve in the hydraulic circuit, as shown in Appendix H. This lock-up is determined by the PWM signal duty cycle used to control the lock-up solenoid. For good driveability and minimal vibrations in the driveline, the torque converter is gradually locked in approximately 6 [s]. Unlocking is done instantly, since this gives less vibrations in the driveline, due to the damping by the oil in unlocked situation. The strategy of locking the torque converter, is based on engine speed ω e, CVT ratio r g and difference (ω e ω p ) between in- and outgoing speed of the torque converter. However, vibration in the driveline can still be noticed when locking the torque converter at minimum clamping force level when using slip control, as shown by Bonsen et al. [1]. Therefore, clamping force level should be increased during the lock-up cycle by feedforward implementation to prevent uncomfortable driving experience Safety measures Since the controller is implemented in a test vehicle some safety measures should be taken. The relative slip is calculated based on the geometric ratio r g and speed ratio r s. These signals should be monitored to notice failure. If failure occurs and no slip is detected afterwards, i.e. ν =, this could lead to minimum clamping force level even if slip is increased. Therefore, if failure of one of these signals occurs, an increased clamping force level should be maintained to prevent damage due to slip peaks. Due to fast shifting from drive to reverse and vice versa, shift peaks occur. Since dynamic behaviour due to reversing speeds in the variator are much faster than the dynamics in the hydraulic circuit, this slip peak cannot be solved by the slip control. In normal use this does not occur and the TCM also does not react on this feature, but to prevent the transmission from any large excursions in the macro slip area, clamping force is increased if this event is detected.

46 34 CHAPTER 5. CONTROL DESIGN AND STRATEGY

47 Chapter 6 Implementation results First the ratio and slip controller are tested on a non-linear vehicle model. After that, both controllers are implemented in a test vehicle with a more realistic disturbance environment. 6.1 Simulation results Before implementation in a test vehicle, simulations on the nonlinear vehicle model are carried out. This nonlinear model is described in Appendix K. v car [km/h] 8 4 ref meas r g [ ] ref meas x p,ref x p [mm] Time [s] Figure 6.1: Ratio control simulation In Figure 6.1 the results regarding the ratio controller are shown. The cruise control implemented in the model, gives poor results, but this is not of interest in this research. Nevertheless, ratio control gives good results, except near overdrive ratio. Regarding the slip control proposed in Section 5.2, results are shown in Figure 6.2. The implemented slip controller is without feedforward. Small peaks are visible, but they remain within acceptable margins. These simulations show stable results for both controllers. Implementation in a test vehicle is the next step. 35

48 36 CHAPTER 6. IMPLEMENTATION RESULTS ν [%] Duty Cycle [ ] ref meas F s [kn] time [s] Figure 6.2: Slip control simulation 6.2 Vehicle implementation results For further evaluation of the closed loop system, especially with regard to robustness due to high and unpredictable load disturbances, the proposed controller is implemented in a production car. The test vehicle used in this study is a Nissan Primera 2.5i with Jatco CKkai-CVT with a torque capacity of 25 [Nm], as already introduced in Section The slip is measured as proposed in that section, while the axial primary pulley position is directly measured and used for ratio control. The vehicle is tested on a chassis dyno, including a flywheel and eddy current brake to simulate road load conditions. Details about this test rig and its control are found in Appendix L. The results on the chassis dyno are described next. v car [km/h] 8 4 ref meas r g [ ] ref meas x p,ref x p [mm] time [s] Figure 6.3: Ratio tracking

49 6.2. VEHICLE IMPLEMENTATION RESULTS 37 A vehicle speed trajectory is followed to study the performance of the ratio controller, as shown in Figure 6.3. The trajectory was chosen to get insight during a common acceleration while driving. It shows that the steady state error is smaller than.1 [mm]. During fast shifting the error peaks to 1.5 [mm], due to limited closed loop bandwidth. However, the overall performance of the ratio controller is satisfying for the present actuation system. More results of the ratio controller performance are shown in Appendix J. The same trajectory is used to check disturbance rejection of the slip controller. To get a good comparison between the gain-scheduled PI controller and the developed LQG controller, a feedforward was not used in this experiment. The slip setpoint was reduced between 1 and 1.5 [%] depending on the ratio. This setpoint was used in the previous controller. v car [km/h] ν [%] 8 4 ref meas ref 1 LQG 5 PI Duty Cycle [ ].7.5 LQG PI F s [kn] 3 2 LQG PI time [s] Figure 6.4: Slip control comparison during a vehicle speed trajectory As shown in Figure 6.4 the LQG controller gave similar results at steady state behaviour as the gain scheduled PI controller. Since the controller maintains the secondary clamping force at minimum level during steady state driving, increased efficiency level compared to the conventional Jatco controller is achieved, as established by Bonsen et al. [1]. This efficiency gain is still bounded by the mechanically limited minimum clamping force level in the CK2-CVT. However, due to the higher gains of the LQG control, the clamping forces are increased to a much higher level during slip events. Since the gain of the integrator is comparable to the one of the PI controller, during this cycle the overall level of clamping force is increased, reducing the efficiency gain. Increasing the integrator gain by means of increasing the first term of the weighting matrix Q in the LQG design, should decrease the steady state error. An acceleration from 1 to 3 [km/h], as shown in Figure 6.4 and enlarged in Figure 6.5 gives the largest slip peak induced by the engine torque. The throttle position during this event is approximately 25 [%]. The LQG controller gives significant better disturbance attenuation. The other two slip events, i.e. at t = 2 [s] and t = 6 [s], are enlarged in Figures 6.6 and 6.7. During the acceleration from 3 to 6 [km/h] a throttle position of 4 [%] is maintained, while during the acceleration from 2 to 8 [km/h] it reached a level of 5 [%]. Both slip events did not cause much trouble for both controllers. However in both cases the steady state error is

50 38 CHAPTER 6. IMPLEMENTATION RESULTS F Duty Cycle [ ] ν [%] r [ ] s [kn] g ref meas ref LQG PI LQG PI LQG PI time [s] Figure 6.5: Response to a sudden accelerations from 1 to 3 [km/h] 1.5 F Duty Cycle [ ] ν [%] r g [ ] s [kn] ref meas ref LQG PI LQG PI LQG PI time [s] Figure 6.6: Response to a sudden accelerations from 3 to 6 [km/h] less for the LQG controller compared to the gain scheduled PI controller. Although slip still exceeds 5 [%] using the LQG-controller, this can be reduced using feedforward control by estimating the required clamping force from the engine torque and stepper motor speed as proposed in Eq. (5.8). This shows feedforward is still necessary, due to the delay in the hydraulic system. Nevertheless, LQG control has increased bandwidth compared to the previous controller. Also feedforward control is extended with shift forces,. As shown in Figure 6.8 for both slip controllers the combination of slip speed and normal force does not exceed the limit in the F/v-diagram as described by Van Drogen and Van der Laan [23].

51 6.2. VEHICLE IMPLEMENTATION RESULTS 39 1 F s [kn] Duty Cycle [ ] ν [%] r g [ ] ref meas ref LQG PI LQG PI LQG PI time [s] Figure 6.7: Response to a sudden accelerations from 2 to 8 [km/h].8 Limit LQG PI.6 Nromal Force [kn] Slip speed [m/s] Figure 6.8: F/v-failure diagrams for the complete trajectory shown in Figure 6.4 This also implies that the torque peak during these test were insufficient to damage the variator.

52 4 CHAPTER 6. IMPLEMENTATION RESULTS

53 Chapter 7 Conclusions and recommendations 7.1 Conclusions Different shift models in the literature are investigated. The best suitable models considering this pushbelt type of CVT and slip control technique are validated and compared. As a result the CMM model gave the best agreement to reality and is therefore used in the rest of the variator modelling. A more complete model of the variator considering slip and ratio dynamics, as well as the hydraulic actuation dynamics of both line and primary pressure, are derived. Also interaction between both slip and ratio dynamics is analysed, leading to the possibility of decentralized control design. The newly developed slip controller based on LQG control design, gives better disturbance attenuation and increased bandwidth compared to the previous designed PI controller, while the improved efficiency level at steady state behaviour is maintained. 7.2 Recommendations More principle work on the friction behaviour between elements of the pushbelt and between element and bands could be of interest to improve the shift model. Especially at low torque ratios and low clamping force levels the CMM model does not give good agreement to reality. Also the additional dynamics in the production CVT compared to a single variator can be of interest to declare the anti-symmetry between up- and downshifting. Since slip control technique is applied, leading to slip in tangential direction, the influence of slip in radial direction should be further analysed. This should lead to a better identification of slip behaviour during shifting, improving the modelling of the variator regarding slip between pulleys and belt. With the present modelling, the optimal performance of the slip control is achieved with the present hydraulic actuation system. Adapting the hydraulic system for best possible control performance and robustness, instead of adapting the control to the hydraulics at the CK2-CVT, is preferred. Research on the long-term effects with respect to variator damage while using slip control is still necessary, to validate the use of this control technique in pushbelt variators. 41

54 42 CHAPTER 7. CONCLUSIONS AND RECOMMENDATIONS

55 Bibliography [1] B. Bonsen, T.W.G.L. Klaassen, R.J. Pulles, S.W.H. Simons, M. Steinbuch, and P.A. Veenhuizen. Performance optimization of the push-belt CVT by variator slip control. Journal of vehicle design, 25. [2] B. Bonsen, T.W.G.L. Klaassen, K.G.O. van de Meerakker, P.A. Veenhuizen, and M. Steinbuch. Measurement and control of slip in a continuously variable transmission. In Australia IFAC, Sydney, editor, Mechatronics 24, pages 43 48, 24. [3] B. Bonsen, T.W.G.L. Klaassen, K.G.O. van de Meerakker, P.A. Veenhuizen, and M. Steinbuch. Modelling slip- and creepmode shift speed characteristics of a push-belt type continuously variable transmission. In International Continuously Variable and Hybrid Transmission Congress, Frank, Davis, CA, United States, 24. [4] B. Bonsen, P.A. Veenhuizen, and M. Steinbuch. CVT ratio control strategy optimization. In United States IEEE, Chicago, editor, VPPC, 25. [5] G. Carbone, L. Mangialardi, B. Bonsen, C. Tursi, and P.A. Veenhuizen. CVT dynamics: Theory and experiments. Mechanism and Machine Theory. in press. [6] G. Carbone, L. Mangialardi, and G. Mantriota. EHL visco-plastic friction model in CVT shifting behaviour. Journal of vehicle design, 32(3/4): , 23. [7] G. Carbone, L. Mangialardi, and G. Mantriota. The influence of pulley deformations on the shifting mechanism of metal belt CVT. Journal of mechanical design, 25. [8] T. Ide, H. Uchiyama, and R. Kataoka. Simulation approach to the effect of the ratio changing speed of a metal V-belt CVT on the vehicle response. Vehicle System Dynamics, 24: , [9] T. Ide, H. Uchiyama, and R. Kataoka. Experimental investigation on shift speed characteristics of a metal V-belt CVT. JSAE no , [1] T.W.G.L. Klaassen and M. Steinbuch. Identification and control of the Empact CVT. Submitted to IEEE Transactions on Control Systems Technology, May 26. [11] D. Kobayashi, Y. Mabuchi, and Y. Katoh. A study on the torque capacity of a metal pushing V-belt for CVTs. SAE technical paper series, no 98822, [12] K. Michael. Jatco Europe GmbH. Personal contact. [13] T. Fujii S. Kanehara O. Fujimura, S. Kuwabara. Study on shifting mechanisms of metal pushing V-belt type CVT - dealing with shifting rate and mean coefficient of friction. JSAE no , [14] M.F. Oudijk. Application of a TNO-MACS system on a Nissan CK2-CVT. Dct-nr , TU/e, Eindhoven,

56 44 BIBLIOGRAPHY [15] H. Peeters. Design and realization of a spin loss transmission test rig. Master s thesis, TU/e, Eindhoven, 21. [16] R.J. Pulles. Slip controller design and implementation in a continuously variable transmission. Dct-nr , TU/e, Eindhoven, 24. [17] T. Fujii S. Kanehara R. Kataoke, K. Okubo. A study on a metal pushing V-belt type CVT - a novel approach to characterize the friction between blocks and a pulley, and shifting mechanisms. SAE no , 22. [18] J. Rijnsdorp. Interaction in two variable control systems for distillation columns. Automatica, 1:15 28, [19] H. Sattler. Efficiency of metal chain and V-belt CVT. In Proceedings CVT 99 Congress, Eindhoven, The Netherlands, pages 99 14, [2] E. Shafai, M. Simons, U. Neff, and P. Geering. Model of a continuously variable transmission. Vehicle System Dynamics, [21] S. Skogestad and I. Postlethwaite. Multivariable feedback control. Wiley, [22] P. Tenberge. Efficiency of chain-cvts at constant and variable ratio. In 24 International Continuously Variable and Hybrid Transmission Congress, UC Davis, United States, 24. [23] M. van Drogen and M. van der Laan. Determination of variator robustness under macro slip conditions for a push belt CVT. SAE world congress 24, 24. [24] B. Vroemen. Component control for the zero inertia powertrain. Technische Universiteit Eindhoven, 21.

57 Appendix A Jatco CKkai-CVT Cross-section view The transmission used in the Nissan Primera is the CKkai-CVT of Jatco. It is nearly the same as the CK2-CVT transmission, which is installed on the test rigs used in this research. Mechanically it is the same, therefore the cross section (Figure A.1) of the CK2 can be used in this case. Figure A.1: Cross-section of the Jatco CK2-CVT 45

58 46 APPENDIX A. JATCO CKKAI-CVT Only the ratio coverage is somewhat narrowed to be able to increase the maximal transmittable torque in the CVT. With respect to the 2. liter engine, where the CK2 is used, the maximal torque in the 2.5 liter engine is increased from 18 till 25 [Nm]. Detailed information about the mechanical and hydraulic part of the transmission are obtained by Peeters [15]. The Transmission Control Module (TCM) in this case, is different from the one used at the CK2. This is mainly due to the different ECM (Engine Control Module) and the communication with the TCM. Also more signals between both control modules are transmitted by CAN-signals. However, signals necessary for controlling the CVT remained the same and also with the same connection points. Signal overview In order to implement the slip controller on the Nissan Primera first of all the necessary main controllers of the transmission in the TCM should be replaced by one in a dspace system. Therefore all sensors and actuator signals of the CKkai-CVT must be determined. A short overview of these signals is represented in the tables below. Table A.1: Overview of all sensor signals from the CKkai-CVT, ECM and added sensors Sensor Signal Range PNP Switch N Binary or 12 [V] PNP Switch R Binary or 12 [V] PNP Switch D Binary or 12 [V] PNP Switch L Binary or 12 [V] Line pressure Analog [V] Oil temperature Analog [V] Primary pulley speed Frequency - 24 [Hz] Secondary speed Frequency - 77 [Hz] Engine speed Frequency - 3 [Hz] Throttle position Analog.5-4. [V] LVDT Analog - 1 [V] All the sensors are present in the CKkai-CVT, except the latter three in Table A.1. The engine speed ω e is measured at the crankshaft, but conditioned at the ECM and than forwarded to the TCM with a low frequency. The throttle position is directly measured at the engine. The LVDT is added and connected to the lever connected to the shift control valve. It measures the axial position of the primary pulley x p, as explained in [16]. Table A.2: Overview of all actuator signals to the CKkai-CVT Actuator Signal Range Line pressure solenoid 5 [Hz] PWM - 1 [%] duty cycle Lock up solenoid 5 [Hz] PWM - 1 [%] duty cycle Stepper motor coil A Binary or 12 [V] Stepper motor coil B Binary or 12 [V] Stepper motor coil C Binary or 12 [V] Stepper motor coil D Binary or 12 [V] A detailed description of all sensors, actuators and controllable signals available at the CK2, are distinguished by Oudijk [14].

59 Appendix B Nissan Primera test vehicle To make the CKkai-CVT transmission in the Nissan Primera available for controlling, some adaptation in the electrical circuit of the car should be made. The signals referred in Appendix A should be connected to a dspace Autobox to operate them by a controller made in MATLAB/Simulink. This Autobox and extra power source are located in the trunk of the car. Thereby, an interface unit is necessary for signal conditioning, amplification, buffering and switching. This unit is mounted instead of the locker in front of the passenger seat. The overall layout is shown in Figure B.1. TCM Cut off CK-kai CVT Amp Old power supply Buffer/switch Analog Digital Engine speed Primary speed Secondary speed PNP switch (4x) Closed throttle switch Open throttle switch Line pressure solenoid Lock-up clutch solenoid 5/12V Signal conditioning PWM AMP Engine speed Primary speed Secondary speed PNP switch (4x) Closed throttle switch Open throttle switch 12V Power supply Stepper motor (4x) 12V supply buffer 5V supply sensor Oil temperature Line pressure Throttle position Line pressure solenoid Lock-up clutch solenoid Stepper motor (4x) Pulley position Line pressure (manual) Laptop Timing IN Digital IN DS 42 (Timing I/O, 8 channels + Digital I/O, 32 channels) Timing OUT Digital OUT DS23 (A/D, 32 channels) DS213 (D/A, 32 channels) DS13 (Processor) SRAM PC card Ethernet network adapter dspace Autobox Figure B.1: Layout for adapting Primera for measuring and controlling the CVT With the presented layout an electrical scheme is setup for measuring and controlling the CKkai- 47

60 48 APPENDIX B. NISSAN PRIMERA TEST VEHICLE CVT with dspace, TCM or a combination of both, shown in Figure B.2. dspace Autobox SIGNAL 25 POLIGE SUBDM X12 CONDITIONING brake switch U2 wide open throttle position switch closed throttle position switch engine speed prim. speed sec. speed PNP switch L pos PNP switch D pos PNP switch R pos PNP switch N pos lock-up clutch solenoid stepper motor A stepper motor B stepper motor C stepper motor D Laptop brake switch TCM 45 wide open throttle position switch nc closed throttle position switch nc throttle position sensor 41 engine speed 39 prim. speed 38 sec. speed 29 oil temperature 47 line pressure 37 PNP switch L pos 27 PNP switch D pos 34 PNP switch R pos 35 PNP switch N pos 36 line pressure solenoid 1 lock-up clutch solenoid 3 line pressure solenoid dropping resistor 2 stepper motor A 11 stepper motor B 12 stepper motor C 2 stepper motor D 21 +5V supply throttle sensor 32 +5V supply pressure sensor 46 PNP switch L pos PNP switch D pos PNP switch R pos PNP switch N pos line pressure solenoid lock-up clutch solenoid stepper motor A stepper motor B stepper motor C stepper motor D LAST REVISION (for History see Revision Log ) REV SUBJECT 1125 COMPANY 13 DEPARTMENT Gemeenschappelijke Technische Dienst DESCRIPTION TITLE ORDER NR. FILENAME PRIMERA.VSD BLOKSCHEMA PRIMERA SUBJECT LOCATION FULL FILENAME DATE --2 DRAWN BY DATE CTDKRUYS M:\PROJEKTEN\PRIMERA\VISIO\PRIMERA.VSD APROVED BY A 25 POLIGE SUBDM XC12 25 POLIGE SUBDF XC1 25 POLIGE SUBDM XC11 Buffered Signals +5V -5V PWM-AMP 15 POLIGE SUBDM X13 U3 line pressure solenoid Twisted Pair Switched Signals 2x8x,22 +5V +12V 25 POLIGE SUBDM X12 pulley position sensor oil temperature line pressure throttle position sensor Buffered Signals line pressure manual Switched Signal DC-DC +12V U4 X1 Q1 DC-DC +5V Accu 12VDC U5 FUSE 3A DC-DC -5V U6 INTERFACEKAST ONDER DASHBOARD PRIMERA BUFFER/SWITCH U1 S1 S2 S3 1X 4X 7X VOEDING: +5V +12V 25 POLIGE SUBDM X11 25 POLIGE SUBDF X1 brake switch red/green CK-kai wide open throttle position switch color? closed throttle position switch color? throttle position sensor grey engine speed bleu/orange prim. speed green/yellow sec. speed green/red oil temperature brown line pressure white blue white/green green/white green red/white grey/red line pressure solenoid dropping resistor pink/black purple bleu/white bleu/yellow pink/bleu +5V supply throttle sensor red/black +5V supply pressure sensor red/bleu pulley position sensor +12V supply red/brown pulley position sensor color? ground pin 25! DS213 DS23 DS42 5 POLIGE SUBDF 5 POLIGE SUBDM 5 POLIGE SUBDF Figure B.2: Schematic overview of electronic circuit and signals to control the CKkai-CVT with dspace or the TCM As mentioned the control of the transmission can be switched between the original TCM or a

61 49 slip controller using the dspace system. To choose or combine both options, three switches are available at the interface unit. The configuration of these switches is represented in Figure B.3. To switch between TCM and dspace mode both switches S2 and S3 are used. In the proposed slip controller by Pulles [16] only the line pressure is controlled by the developed slip controller, but ratio control and torque converter lock-up still is realized by the TCM. Therefore the switch between TCM and dspace is split between two switches. If this latter configuration is used, the power resistor is needed to dissipate the power that is send by the TCM to the line pressure solenoid, otherwise the TCM switches into a fail-safe mode and the transmission is not controllable anymore. Switch S1 was added to manipulate the measured line pressure to prevent the TCM from going in to fail-safe mode, but for all configurations this is not necessary. Line pressure (dspace) Line pressure (CK-kai, after buffer) S1 Line pressure (TCM) Power resistor (ground) 17 Ω Line pressure solenoid (CK-kai) Line pressure solenoid dropping resistor (CK-kai) S2 Line pressure solenoid (TCM) Line pressure solenoid (dspace) Line pressure solenoid dropping resistor (TCM) Lock up clutch solenoid (CK-kai) Stepper motor A (CK -kai) Stepper motor B (CK-kai) Stepper motor C (CK-kai) Stepper motor D (CK-kai) 5V supply throttle sensor (CK-kai) 5V supply pressure sensor (CK-kai) S3 Lock up clutch solenoid (TCM) Lock up clutch solenoid (dspace) Stepper motor A (TCM) Stepper motor A (dspace) Stepper motor B (TCM) Stepper motor B (dspace) Stepper motor C (TCM) Stepper motor C (dspace) Stepper motor D (TCM) Stepper motor D (dspace) +5V supply throttle sensor (TCM) +5V supply pressure sensor (TCM) 5VDC Figure B.3: Signal routing for switches at the interface unit in the Primera

62 5 APPENDIX B. NISSAN PRIMERA TEST VEHICLE

63 Appendix C Shift speed experiments To get more familiar with the shifting behaviour in the CK2-CVT, several experiments are carried out in different operation points. The available test rig and the results are described next. BTS Test rig The test rig shown in Figure C.1(a) and schematically represented in Figure C.1(b) is used for these experiments. This test facility includes two identical asynchronous electric motors with the following specifications: a maximum power of 78 [kw], a maximum torque of 298 [Nm] and a maximum speed of 525 [rad/s]. Between the two motors a Jatco CK2-CVT is present. Behind the CVT a manual transmission is mounted in reverse direction, used as a gearbox-case to adapt the maximum load or speed. (a) Test setup (b) Layout of the test setup Figure C.1: Test rig used for measuring pulley thrust ratio For measuring the thrust ratio first the torque ratio is defined by increasing the load at the 51

64 52 APPENDIX C. SHIFT SPEED EXPERIMENTS desired constant input speed ω p, secondary pressure p s and ratio r g, until the belts start to slip in the macro slip regime. Than the relative load τ is increased from -.6 till 1. Shift behavior results r g [ ] dx step /dt = 1 [mm/s] dx step /dt = 2 [mm/s] dx step /dt = 4 [mm/s] dx step /dt = 6 [mm/s] dx step /dt = 1 [mm/s] dx p /dt [mm/s] ν [%] time [s] dx step /dt = 1 [mm/s] dx step /dt = 2 [mm/s] dx step /dt = 4 [mm/s] dx step /dt = 6 [mm/s] dx step /dt = 1 [mm/s] dx step /dt = 1 [mm/s] dx /dt = 2 [mm/s] step dx step /dt = 4 [mm/s] dx step /dt = 6 [mm/s] dx step /dt = 1 [mm/s] p p [bar] dx step /dt = 1 [mm/s] dx step /dt = 2 [mm/s] dx step /dt = 4 [mm/s] dx step /dt = 6 [mm/s] dx step /dt = 1 [mm/s] (F p ) [kn] dx step /dt = 1 [mm/s] dx step /dt = 2 [mm/s] dx step /dt = 4 [mm/s] dx step /dt = 6 [mm/s] dx step /dt = 1 [mm/s] lnf [ ] time [s] dx step /dt = 1 [mm/s] dx step /dt = 2 [mm/s] dx step /dt = 4 [mm/s] dx step /dt = 6 [mm/s] dx step /dt = 1 [mm/s] Figure C.2: Shift speed experiment on the CK2-CVT at p s = 1 [bar] and ω p = 15 [rad/s] Some of the results of the shift behavior experiments are shown in Figure C.2. During these ex-

65 53 periments, CVT ratio r g is changed from lowdrive to overdrive and back. By controlling the speed of the stepper motor, different levels of shifting speed are achieved. These experiments are also carried out at higher secondary pressure p s, different input speeds ω p and higher stepper motor speeds ẋ step, but the results shown are the most representable for shifting while using slip control. At this pressure level (p s = 1 [bar]), shifting speeds are bounded due to the limited shifting force. Axial primary pulley speed ẋ p is equal to the stepping motor speed ẋ step till 4 [mm/s] during upshifting and till 6 [mm/s] during downshifting. Also more symmetry between up- and downshifting regarding shifting force is visible, when denoting it as a logarithmic relation instead of a linear relation. Slip in tangential direction is hardly influenced by shifting at these operation points.

66 54 APPENDIX C. SHIFT SPEED EXPERIMENTS

67 Appendix D Pulley thrust ratio For measuring the thrust ratio as defined by Eq. (3.4) the test rig shown in Appendix C is available. Procedure of these experiments are described. Furthermore results are discussed and the approximation derived from the CMM model is analysed. Pulley thrust ratio results For measuring the pulley thrust ratio first the maximal transmittable torque at a certain operation point should be obtained. This is defined as the point where the slip behaviour enters the macro slip regime. When this maximal transmittable torque is known, the pulley thrust is obtained while increasing torque ratio τ, as defined by Eq. (2.6), from -.6 to 1. By measuring this cycles for each ratio between low- and overdrive, the result as shown in Figure D.1 is obtained, at a secondary pressure of 7 [bar]. Pulley thrust ratio Ψ r g =.5 r g =.8 r g = 1. r g = 1.3 r g = Torque ratio τ Figure D.1: Typical result of Ψ measurements at 7 [bar] secondary pressure At ratios below 1, at a certain positive torque ratio, a leap in Ψ can be seen. To be able to clarify this, first the torque transfer principle of the belt must be described. In the pushbelt CVT, torque is transmitted by compressive forces between the elements and by tensile forces in the bands 55

68 56 APPENDIX D. PULLEY THRUST RATIO that keep the elements in place, as described by Kobayashi [11]. The leap in Ψ can be assigned to the change in segment where the torque is transmitted by compressive force in the belt. If the input torque is larger than the torque transmitted by the tensile force in the bands, the compressive force acts on the elements of the upper straight segment, otherwise on the lower segment. Likewise at ratios higher than 1, this phenomena happens at the same conditions. However, since at overdrive the segment where the torque is transmitted by tensile force is changed, the torque at which the compressive side changes from segment is negative. However, during preliminary tests it became clear that the amount of secondary pressure influences the results of Ψ. To find out the exact dependency, further experiments were carried out at higher secondary pressures. Problem was the maximum torque of the electric motor on the test rig. Due to this limitation the maximum torque cannot be applied at high secondary pressures p s and ratios r g. Increase of the latter parameter gives an increase in primary running radius R p. Both parameters increase the maximal transmittable torque. Results can be extrapolated leading to some errors in the estimation of the pulley thrust ratio. Figure D.2 shows the typical trends at increasing clamping force p s = 7 [bar] p s = 15 [bar] r g = r g = Ψ Ψ Ψ τ r g = 1. Ψ τ r g = τ τ Figure D.2: Results for Ψ at p s = 7 [bar] and 15 [bar] From the extra measurements it becomes clear that due to increasing clamping force, Ψ increases before the leap. After the leap Ψ decreases with increasing secondary pressure. This could be declared by the increase in transmittable torque by compressive force between the elements of the pushbelt, due to the increase in clamping force. In the situation of high torque ratios (τ >.4) and low CVT ratios (r g < 1), compressive force works in the lower part of the belt and tensile force in the upper part. Due to the increase of the transferred torque, this results in higher primary clamping forces to maintain a constant CVT ratio. For the other case of low torque ratios (τ <.3) and low CVT ratios (r g < 1), where compressive and tensile force both work in the lower part of the belt, total torque transfer is decreased. Hereby, primary clamping force to maintain stationary behaviour is decreased.

69 57 Pulley thrust ratio model In the CMM model introduced by Carbone et al. [7] and further explained in section an approximation (Eq. (3.5)) for the pulley thrust ratio Ψ is given. Since the measurements are performed on a unloaded variator, model and experimental results are given at τ =. Experiments ln(ψ) F = 1 [kn] s F s = 2 [kn].4 F = 3 [kn] s CMM model ln(r g ) Figure D.3: Logarithm of the thrust force ratio Ψ as a function of the logarithm of the geometrical ratio r g. Compared are the theoretical model and experimental results at different clamping force levels F s at the belt box were performed at different input speeds (n p = 1, 2, 3 [rpm]) and at different clamping force levels (F s = 2, 3 [kn]). Here, both primary speed and clamping force levels have significant influence, as shown in Figure D.3. The fit through all the experiment shows good agreement with the theoretical prediction. A more steeper part around r g = 1 at F s = 1 [kn] is visible. This is in agreement with the nonlinear behaviour, shown in Figure 3.2 at τ = and F s = 1 [kn]. The measurements show that this nonlinear behaviour seems to vanish at higher clamping force levels. For accurate modelling this phenomenon should be further investigated. Hereby, also the clearance between elements as described by Carbone et al. [6] could be of interest.

70 58 APPENDIX D. PULLEY THRUST RATIO

71 Appendix E Dimensional analysis From dimensional analysis it can be seen that the CVT dynamical response will depend not separately on the clamping forces acting on primary and secondary pulley, but on their ratio F p /F s. If it is assumed for simplicity that the variator is unloaded, CVT design is fixed, and ratio is equal to 1, than the rate of ratio changing is a function of ṙ g = f(f p, F s, ω p ) (E.1) With the assumptions that ω p is constant and if the dependent parameters are divided by F s by means of dimensional analysis the shift speeds only depends on the ratio F p /F s, namely ṙ g = ω p f(f p /F s ) (E.2) Not considering actual distribution along the belt, there is a perfect symmetry and primary and secondary pulley are physically indistinguishable. This yields to the following result f(f p, F s ) = f(f s, F p ) (E.3) Beside this property, when both clamping forces are equal shift speed must be zero, independent on the quantity of this force f(f, F ) = With the given properties the following relation is proposed by Carbone et al. [6] ( ) ( ) Fp Fs ln = ln F s F p Substituting this last relation into Eq. (E.3) gives ( f ln F ) ( p = f ln F ) s F s F p (E.4) (E.5) (E.6) The previous equation is an odd function of ln(f p /F s ). Thus, the Taylor expansion of f must contain only odd terms i.e. ( f ln F ) p c 1 ln F ( p + c 2 ln F ) 3 ( p + c 3 ln F ) 5 p +... (E.7) F s F s F s F s The CMM model satisfies Eq. (E.7) under the given hypothesis of no load and ratio equal to 1. Since F p /F s is always near 1 in creep mode, higher order terms can be neglected and a second order approximation can be given by f ( ln F p F s ) c 1 ln F p F s 59 (E.8)

72 6 APPENDIX E. DIMENSIONAL ANALYSIS This result is obtained by interpreting some physical consideration in a mathematical form. Considering load conditions, influence of friction forces and different values of ratio, the relation will become more complicated. However dependency of the logarithm is expected, when F s is replaced by F p obtained by rewriting Eq. (E.1) in the form of Eq. (3.3) 1 + cos 2 [ β ṙ g = 2ω p β sin(2β ) k c(r g ) ln ( Fp F s ) ( F )] p ln F s (E.9) Rewriting will give Eq. (3.3).

73 Appendix F CMM model When the friction coefficient µ at the pulley-belt interface and the geometry of the system is known, the CVT shifting dynamics in creep may simply be described by 1 + cos 2 [ β ṙ g = ω p β sin 2β k c(r g ) ln F ] p ln Ψ(r g, τ, p s ) (F.1) F s Here, the bending of the pulley can be described as function of the clamping force, retained by experiments β = d + d 1 F s (F.2) with d = and d 1 = The bending of the pulley is only affected by the clamping force, as expected due to linear elastic response of the system. Dependence on running radius of the belt is negligible due to symmetry of the pulley pair. If the belt runs on a high radius on the one pulley it runs on a low radius on the opposite pulley. The term d is equal to the bending due to the clearance between the pulley and its shaft. For no-load conditions β = 6 1 4, which is in good agreement with some data on guidance clearance of the moveable pulley by Tenberge [22]. In the pulley model of Sattler [19] β = is assumed constantly, which is in agreement with measurements at F s 2 [kn]. The ratio dependent term is defined as k c (r g ) = R p a r gc(r g ) with a defined as the axial pulleys distance and c(r g ) = c + c 1 (ln r g ) 2 (F.3) (F.4) The latter relation is a symmetric relation dependent on the logarithm of the ratio, with c = 5 and c 1 = To convert the shifting speed to the axial speed of the primary pulley, only this ratio dependent constant will change. Taking the time derivative of r g = R p /R s gives ( ) ṙ g = r g Ṙ p R p 1 Ṙs Ṙ p r g (F.5) To determine the term Ṙs/Ṙp the length of the belt L can be considered as function of R p and R s by ( ) Rp R s L = π(r p + R s ) + 2(R p R s ) arcsin + 2 a a 2 (R p R s ) 2 (F.6) 61

74 62 APPENDIX F. CMM MODEL Thus neglecting the belt longitudinal deformation and taking the time derivative of Eq. (F.6), the following relation is obtained Ṙs = π 2 arcsin[r s R p )/a] Ṙ p π + 2 arcsin[(r s R p )/a] = π 2 arcsin[r p(1 r g )/r g a] π + 2 arcsin[r p (1 r g )/r g a] (F.7) With the relations h(r g ) = Ṙs Ṙ p and R p = x p /(2 tan β) Eq. (F.5) can be rewritten to x p ẋ p = ṙ g r g (1 + r g h(r g )) (F.8) With this the relation between the axial primary pulley speed ẋ p and the logarithm of the shifting force becomes [ ẋ p = ω p β k c,x (r g )x 2 p ln F ] p ln Ψ(r g, τ, p s ) (F.9) F s with k c,x (r g ) = cos2 β + 1 c(r g ) 4a (cos 2 β 1) (1 + r g h(r g )) (F.1)

75 Appendix G Linearized model In Section 4.1 the dynamic model for the slip and ratio dynamics are presented. In Eq. (F.6) it is proven that x p is a function of r g only. For deriving all the relations (G) dependent on the ratio r g to state x p the following relation is necessary. with dg dx p = G (r g ) dr g dx p dr g dx p = r g x p [1 + r g h(r g )] Deriving the relations h(r g ) and c(r g ) with ρ = (2 tan β) 1 gives h (r g ) = dh dx p = 4πρx p 1 + h(r g ) (π + 2 arcsin(ρx p (1 r g )/r g )) r 2 g 2 ρ 2 x 2 p(1 r g ) 2 ) 1 + r g h(r g ) 4π(1 + h(r g ))ρr g (π + 2 arcsin(ρx p (1 r g )/r g )) 2 r 2 g ρ 2 x 2 p(1 r g ) 2 ) (G.1) (G.2) (G.3) (G.4) c (r g ) = 2c 1 ln r g r g (G.5) dc dx p = 2c 1 ln r g x p [1 + r g h(r g )] (G.6) Deriving the equation of k c,x (r g ) and combining it with the previous equations gives k c,x(r g ) = cos2 β + 1 c (r g )[1 + r g h(r g )] c(r g )(h(r g ) + r g h (r g )) 4a (cos 2 β 1) [1 + r g h(r g )] 2 (G.7) dk c = cos2 β + 1 r g c (r g )[1 + r g h(r g )] r g c(r g )(h(r g ) + r g h (r g )) dx p 4a (cos 2 β 1) x p [1 + r g h(r g )] For the derivative for the logarithm of the shift force (G.8) ln F (r g, ν) = ln F p F s ln Ψ(r g, ν) k 1i ν + k 2i ln Ψ(r g, ν) = m 1 ln(r g ) + 2m 2 cos β d ln F = m 1 F p [1 + r g h(r g )] dx p x p F p + F s Ψ ( d ln F 2m2 k 1i = dν cos β + 8m 3k 1i (k 1i ν + k 2i ) cos 2 β + 4m 3 (k 1i ν + k 2i ) 2 cos 2 β ) F p F p + F s Ψ (G.9) (G.1) (G.11) (G.12) 63

76 64 APPENDIX G. LINEARIZED MODEL

77 Appendix H Electro-hydraulic system Figure H.1: Schematic overview of the electro-hydraulic system 65

78 66 APPENDIX H. ELECTRO-HYDRAULIC SYSTEM

79 Appendix I Linearizing shift valve model As shown in Figure 4.7 this dynamic model is based on the primary pulley pressure p p,val. The inputs of the model are the controllable input of the stepper motor u step, the secondary pressure p s, pulley position x p and the axial speed of this pulley ẋ p, which is determined with the CMM model. Defining the state space as x ppv = [p p,val ] and u ppv = [ u step x p ẋ p p s ] T the system can be linearized around a certain working point x ppv = [p p,val ] resulting in the linear system ˆx ppv = A ppv ˆx ppv + B ppv û ppv where ˆx ppv = x ppv x ppv, and û ppv = u ppv u ppv,. With assuming that p d p p the linearized matrices A ppv and B ppv can be derived [ A ppv = 1 c f A v ] c f A pl σ 2ρoil (p s p p) 2ρoil p p (I.1) and B ppv = 1 σ 1 2 c f k vi1 2(ps p p) ρ oil 1 2 K 2(ps p stepc f k p) ( vi1 ρ oil 2(ps p c f A p) v ρ oil Apk oil σ A p c f A v 2ρoil (p s p p) c f A pl 2pp ρ oil A p ẋ p ) T for x v > While A ppv = 1 σ [ cf ] (A v A pl ) 2ρoil p p and B ppv = 1 σ for x v 1 2 c f k vi1 2pp ρ oil 1 Apk oil σ 2 K 2pp stepc f k vi1 ρ ( oil ) 2pp 2pp c f A v ρ oil c f A pl ρ oil A p ẋ p A p T with σ = k oil (A p [x p x p,min ] + V ) (I.2) 67

80 68 APPENDIX I. LINEARIZING SHIFT VALVE MODEL In this model the position of the spindle attached to the stepper motor is defined by x step = K step u step (I.3) It is assumed the dynamics of the stepper motor are much faster compared to the dynamics of the hydraulic valve. Therefore, the output of the stepper motor is proportional with its input with no significant delay. Here K step has a value of [m] per step. When using the CMM model is used the static gain transfer function H ln F must also be linearized resulting in the following three relations. ln F p p,val = ln F x p = m 1 ln F p s = A s F s A p A p p p,val f cp ωp 2 + F sψ ( ) A p p p,val [1 + r g h] x p A p p p,val f cp ωp 2 + F sψ ( ) A p p p,val A p p p,val f cp ωp 2 + F sψ (I.4) (I.5) (I.6)

81 Appendix J Ratio control analysis The ratio controller was tuned manually. In Figure J.1 ratio is changed stepwise in four situations. Upshifting from low drive to mid drive and from mid drive to overdrive. Downshifting is preformed from overdrive to mid drive and from mid drive to low drive. Results are shown with an integral gain I r of 2 [mm 1 s 1 ] and two different proportional gains P r of 15 and 3 [mm 1 ]. 1.2 Ratio [ ] ref P r = 3; I r = 2 P r = 15; I r = Time [s] Ratio [ ] Time [s] 2 1 Ratio [ ] Time [s] Ratio [ ] Time [s] Figure J.1: Step response of the ratio control, for different control parameters Experiments are preformed with a steady throttle position of 1 [%] and active slip control. As shown with the increased proportional gain performance is increased, while robustness stability remains. It also can be seen that downshifting gives less overshoot than upshifting. Using these results the control parameters P r = 3 [mm 1 ] and I r = 2 [mm 1 s 1 ] are used for vehicle implementation. 69

82 7 APPENDIX J. RATIO CONTROL ANALYSIS

83 Appendix K Non-linear model Figure K.1: Schematic representation of the non-linear CK2-CVT model in Matlab/Simulink To safely experiment with different control strategies on the CK2-CVT, a simulation model of the CVT in Matlab/Simulink is created. The schematic representation of this non-linear model is shown in Figure K.1. The stepper motor position and the duty cycle of the line pressure solenoid are the inputs of the model. The control parameters slip and axial primary pulley position are 71

84 72 APPENDIX K. NON-LINEAR MODEL the outputs. Also the primary and secondary speed is used as input, while primary and secondary torque are outputs, to connect the model with the overall driveline model. The line pressure control circuit is described by a third order butterworth filter with 6 [Hz] cut-off frequency, as derived in Section The ratio control circuit is described by Eqs. (4.15) to (4.22) from Section 4.2.2, whereas the pulley thrust ratio is obtained from measurements on the CK2-CVT in Appendix D. For the shift model, the CMM model as described by Eq. (4.9) is used, where the forces are obtained by Eqs. (4.15) and (4.23) and the pulley thrust ratio as mentioned above. The axial pulley position is obtained by integrating Eq. (4.9). The belt model is derived from Eq. (4.5) and rewritten to T p = 2R pf s µ eff (ν, r g ) cos β T s = T p r g T var,loss (p s, r g ) (K.1) (K.2) with T var,loss representing the variator torque losses depending on CVT ratio r g and line pressure p s obtained from measurements shown in Table K.1. Table K.1: Variator torque losses in [Nm] r g [-] p s [bar] The torque losses T pump,loss in the oil pump at the CK2-CVT are measured and shown in Table K.2. The complete non-linear CK2-CVT model is implemented in an overall test rig simulation Table K.2: Pump torque losses in [Nm] ω p [rad/s] p s [bar] model, as shown in Figure K.2. In this complete model the test rig model is derived by Eqs. (4.3) and (4.4) and described by Te T p T pump,loss ω p = (K.3) J e Ts r fd T rl ω s = (K.4) J s with T rl and r fd representing the road load torque and final drive reduction of the CK2-CVT respectively. The road load torque is depending on vehicle speed and the gradient of the road, as obtained in Appendix L. The engine torque T e is calculated by means of throttle position γ, engine

85 73 Figure K.2: Schematic representation of the complete simulation model speed ω e and the engine map of the 2.5 liter gasoline engine of the Nissan Primera, supplied by Jatco [12]. The throttle position in this case is obtained by a PI cruise controller, described by ( γ = P cruise + I ) cruise (v car,ref v car ) (K.5) s This controller is used for a vehicle speed cycle input, to safely test the designed CVT ratio and slip controller.

86 74 APPENDIX K. NON-LINEAR MODEL

87 Appendix L Chassis dyno To carry out experiments with the test vehicle in a safe environment a chassis dyno is available at the TU/e. In this setup road load conditions are simulated by an eddy current brake and a flywheel. An overview of the setup is shown in Figure L.1. Figure L.1: Schematic overview of the chassis dyno present at the TU/e In this setup the flywheel represents the inertia resistance. While testing Nissan Primera the representing mass is chosen 136 [kg]. The eddy current brake simulates the air drag F air, rolling resistance F rol and gradient resistance F grad. The available brake has a maximum brake power of 23 [kw] at its maximum equivalent vehicle speed of 2 [km/h]. The brake force reference applied to the eddy current brake can be calculated by F brake,ref = F air + F rol + F grad (L.1) F air =.5ρ air A car c w v 2 car (L.2) F rol = f r m car g cos α (L.3) F grad = m car g sin α (L.4) where ρ air represents the density of the air, A car the frontal area of the vehicle, c w the air drag coefficient and v car the vehicle speed. f r equals the rolling resistance coefficient, m car the vehicle 75