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Available online at ScienceDirectIFAC-PapersOnLine 49-11 (2016) 742748Estimation of the Clutch CharacteristicMap for Wet Clutch TransmissionsConsidering Actuator Signal and ClutchSlipT. Arndt A. Tarasow C.Bohn G.WachsmuthR. SerwayInstitute for Electrical Information Technology, Clausthal Universityof Technology, Clausthal, Germany, (e-mail: arndt, bohniei.tu-clausthal.de).IAV Automotive Engineering, Berlin, Germany, (e-mail: alex.tarasow, guido.wachsmuth, roland.serwayiav.de).Abstract: The wet dual clutch transmission is an essential part of the powertrain in modern high-power engine vehicles. This clutch is a complex mechatronic system which has high requirements on control to provide comfort, sportiness and economy. Many control concepts use clutch characteristics in control. To fulfil the demands of the costumer a high quality of the estimated clutch behaviour is necessary. This paper introduced a new approach to estimate the clutch characteristic map of an automated wet clutch as a function of the actuator signal and the clutch slip. The identification of the characteristic map is based on the clutch characteristic of the relation between the actuator signal and the transferred clutch torque. The information of this characteristic is used to calculate the slope of the characteristic map. This slope can be used to identify the slip influence on the clutch torque. The identification results in two separate characteristics, which can be combined to a characteristic map to describe the clutch behaviour with a static characteristic map. The advantage of this approach is the usability of the clutch characteristic with and without the slip influence. This paper shows the results of this identification and the improvement of using the three-dimensional characteristic map instead of the two-dimensional clutch characteristic. The results of the introduced approach are shown in an example with data of a wet dual clutch. 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.Keywords: clutch characteristic, friction, dual clutch transmission, clutch identification, clutch control1. INTRODUCTIONThe demands of customers increase continuously. Thus, the components and software have to be improved to fulfil the customer requirements. Sportiness, comfort and econ-omy are important factors for the purchase of a vehicle. One very important component in modern vehicles with high-power engines is the wet dual clutch. It is used in the powertrain to transfer the torque of the engine to the wheels. Especially during the launch and shift process, the quality of the clutch is evaluated. Because of the complex-ity of the clutch system, advanced control algorithms are necessary to achieve a high quality of the shifting process. Many approaches to control the clutch are proposed in the literature. In Arndt et al. (2013) an overview of dierent control designs is given. Possible control designs are e.g. the optimal control (Garofalo et al. 2002), model predic-tive control (Heijden et al. 2007) or decoupled PI control (Vasca et al. 2008). In many control concepts the clutch behaviour is considered as an inverse clutch characteristic to estimate the desired actuator signal of the clutch, e.g. see Vasca et al. (2008), or it is used as feed forward control. The clutch characteristic describes the relationbetween the actuator signal and the clutch torque. In Tarasow et al. (2013) an overview of dierent approaches for the description of the clutch behaviour is given. A new approach to represent the clutch torque characteristic as a function of the actuator signal is proposed in Tarasow (2015). Here, the parameters correspond to the shape of the characteristic and physical properties. Dierent iden-tification methods are used to estimate the parameters of the clutch characteristic. Because of the demands on the quality of the control process, the clutch characteristic has to be in good accordance with the real behaviour. However, in Tarasow (2015) only the actuator signal is considered whereas other influences are neglected. In wet Dual Clutch Transmissions (DCT), the clutch slip has an important influence on the friction. To get a better description of the behaviour, the clutch characteristic can be extended to a characteristic map which considers the clutch slip.The friction is a well-researched eect in mechanical sys-tems. A overview is given in Nouailletas et al. (2010). To describe the friction, dierent approaches exist which can be categorized in static and dynamic models. Static models are e.g. the Stribeck friction and Karnopp-model in Olsson2405-8963 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control.10.1016/j.ifacol.2016.08.108T. Arndt et al. / IFAC-PapersOnLine 49-11 (2016) 742748743et al. (1998). Alternatively, friction can be described with dynamic models, e.g. the LuGre-model in De Wit et al. (1995). To parametrize these models, identification meth-ods are used to estimate the parameters. In the literature dierent approaches are presented. Chu et al. (2014) use a recursive least-squares algorithm and Verge (2005) pro-poses genetic algorithms to identify the friction coecient. Parlitz et al. (2004) and Rizos and Fassois (2009) Rizos et al. (2009) identify the friction with linear and non-linear regression approaches for dynamic friction models.In Tarasow (2015) the clutch is described with a quasi-stationary clutch characteristic and identified with recur-sive and non-recursive least-squares estimation algorithms. But in dynamic situations and during the shift process this clutch description is not exact. Therefore, in Arndt et al. (2016) the clutch characteristic is extended to a charac-teristic map to consider the influence of the clutch slip on the friction coecient. The complete characteristic map with its dependencies is identified with recursive and non-recursive least-squares estimation algorithms. It has been shown that the extension increases the quality of the clutch model, but the clutch characteristic in Tarasow (2015) can dier from the actuator influence of the characteristic map.In this paper a further identification approach is intro-duced to present the clutch behaviour. Here, the clutch characteristic is extended to a characteristic map based on the actuator signal and the clutch slip. The new approach is based on the clutch characteristic in Tarasow (2015). The slope of the characteristic is used for identification to include the clutch slip influence. The behaviour of the clutch characteristic is preserved in this approach. The remainder of this paper is organised as follows. In Section 2 the clutch system is introduced and the mathematical descriptions of the system are given. Additionally the clutch characteristic is extended to the characteristic map. Section 3 introduces the applied identification method and the identification process is presented. In Section 4 the results of the identification on real measurement data are given. At the end of the paper a conclusion and a short outlook are given.2. WET CLUTCH TRANSMISSION2.1 Wet Single and Dual Clutch SystemsThe wet clutch transmission is an essential component in modern vehicles. The clutch is used in high-power engine vehicles to transfer the torque from the engine to the tires. The wet clutch is able to transfer a torque up to 500 Nm, but this causes a temperature increase. Because of theSteelDiscClutch PistonPressure ChamberFrictionClutch PackDiscPressure SupplyReturn SpringTclTeTorsionalDisc Carrier Gearbox SideDamperDisc Carrier Engine SideFig. 1. Scheme of a wet clutch (based on Tarasow 2015).side and the gearbox side. The clutch pack consists of steel discs which are connected to the chassis of the clutch, and friction discs which are connected via a torsional damper to the input shaft of the gearbox. Additionally a return spring is integrated in the clutch design to release the clutch pack if the clutch is disengaged.The design of a wet dual clutch is similar to a wet single clutch. Here a combination of two alternately used clutches is employed. This concept allows a continuous transfer of the engine torque Te without interrupting the torque flow. At any time only one clutch is completely engaged while the other clutch is disengaged. During the launch and shift process the states of the clutches change. One clutch engages and the other clutch disengages, but in sum both clutches transfer the torque of the engine to the gearbox. Therefore, a high control quality is necessary to reach this desired behaviour.2.2 Mathematical DescriptionThe torque transfer is aected by two dierent friction influences. Both influences can be described physically. The torque transfer is caused by liquid friction and solid friction (Tarasow et al. 2013). Liquid friction mainly occurs during the engagement and disengagement process and is caused by the liquid between the steel and friction discs. This torque transfer is a function of the distance between the discs. It disappears if the discs are far away from each other or if the liquid is pushed out. In the second case the liquid friction changes to solid friction. It is assumed that the torque transfer is mainly caused by solid friction and that liquid friction can be neglected. In Tarasow et al. (2013) the clutch torque caused by solid friction is described asliquid-bedded clutch the heat can be removed from the system. Another advantage of wet clutches is the compact design that saves construction space.Tcl = (, fp, top) rg (FN, fp, top) z F (ucl, ucl, cl, cl, top)(1)Fig. 1 shows a scheme of a single wet clutch. The clutch connects the engine with the gearbox. The engine torque Te is transferred to the input shaft of the gear-box with the torque Tcl. This torque transfer is done by engaging the clutch. To engage the clutch a pressure is built up in the pressure chamber via a pressure supply. The pressure is the actuator signal ucl. Caused by the pressure the clutch piston is moved to engage the clutch pack to create a physical connection between the enginewith the friction coecient , a parameter rg that depends on geometrical properties, the number of friction contacts z and the normal force F . The aforementioned parame-ters are also dependent on many other influences. Main influences are the temperature of the friction plate fp and the temperature of the clutch cl, the clutch slip , the normal force FN, the rotational clutch speed cl, the supplied pressure ucl and its gradient ucl and the operating time of the clutch top. To represent the whole physics ofHere, the parameters of the clutch characteristic describe physical properties and the shape of the characteristic. In this equation, the parameter pT1 is the slope of the clutch characteristic and the parameter pT2 is the kiss point, where the friction areas have their first contact. The parameters are combined in the parameter vector744T. Arndt et al. / IFAC-PapersOnLine 49-11 (2016) 742748the clutch a complex non-linear mathematical description is necessary.2.3 Clutch CharacteristicIn Tarasow (2015) a two-dimensional clutch torque charac-teristic is presented which describes the clutch behaviour as a function of the actuator signal ucl. There, the clutch torque in (1) is simplified toTcl = rg z F (ucl)(2)Instead of the varying friction coecient (), a con-stant friction coecient is used. This friction coecient can be interpreted as the mean friction coecient of the clutch behaviour. The approach assumes that the influence of the clutch clip and the change of the friction coecient over time are negligible. Thus, the clutch characteristic of a wet clutch can be defined as a function of the actuator signal by using the following mathematical description(ucl pT2) .(3)Tcl(ucl) = pT1F () = FC + (FS FC) e|v/vS|S + FVv.(6)Here FC denotes the Coulomb friction force, FS the Stribeck friction, S a geometric parameter, vS the Stribeck speed and FV the viscous friction force. In Arndt et al. (2016) the Stribeck function in (6) is modified to describe a normalized friction coecient. To describe the clutch slip influence more generally, in this paper, the description is modified to () = p1 + p2ep23 + p4 .(7)with four unknown parameters p1, . . . , p4. Equation (7) can be interpreted as friction coecient. The parameters are combined in the parameter vector= p1, p2, p3, p4T.(8)Equation (7) can be split in two parts. The first part is a linear equation and the second part an exponential equation which is similar to an impulse response of a first-order lag element.2.5 Characteristic Map = pT1, pT2T.T ,clThe characteristic map in Arndt et al. (2016) describes the clutch behaviour as a function of the actuator signal ucl and the clutch slip . By combining the clutch torquecharacteristic () described in Section 2.3 and theTcl ucl(4) friction coecient () introduced in Section 2.4, the characteristic map becomesDue to the utilization of the mean friction coecient a more dynamical behaviour based on the clutch slip is not modelled. In practice this missing information about the slip dependence results in an inaccurate clutch torque. Thus, a slightly dierent torque is estimated from this map. To overcome this problem the friction coecient and the influence of the clutch slip have to be considered. Instead of the simplification in (2) the clutch slip is considered in the clutch torque. Main influences on the clutch torque are the actuating signal ucl and the present clutch slip . Thus, Equation (1) can be simplified to().(9)Tcl,map(ucl, ) = Tcl(ucl) With (3) and (7), this yields) = (pT1 (ucl pT2) Tcl,map(ucl, .(10) p1 + p2ep23 + p4In (10) the parameters of (4) and (8) appear. They result in the parameter vector of (10) toTcl = () rg z F (ucl) .(5) = , T.(11),clT ,mapTIt is assumed that the neglected parameters have only a small influence or have only a very slow time-varying eect on the clutch torque Tcl.2.4 Friction CharacteristicTo take the behaviour of the friction coecient into account the friction has to be modelled. The friction can be described with dynamic and static modelling approaches. Dynamic models describe the friction with dierential equations, e.g. with the LuGre-model in De Wit et al. (1995). Descriptions which use a static friction model can be found in Tarasow (2015) and Maki (2005). Here, the Stribeck friction described in Olsson et al. (1998) is used to model the friction influence. The Stribeck friction function is a static non-linear model, which describes the friction force F as a function of the velocity , given byThe parameter pT1 of the clutch characteristic and the parameters of the friction coecient are redundant and can be combined. This leads to() = ()Tcl,map ucl,ucl pT2 2(12)p1 + p2ep3 + p4.with p1 = pT 1p1, p2 = pT 1p2 and p4 = pT 1p4. The result is a three-dimensional map of the clutch torquebased on the two-dimensional clutch characteristic and the friction coecient (). The calculation of the absolute friction coecient is not intended, in the following it is considered as a clutch slip based influence. Thus, the description in (12) is sucient to represent the clutch behaviour even if the parameters lose their physical corre-spondence.T. Arndt et al. / IFAC-PapersOnLine 49-11 (2016) 742748The behaviour of the clutch can be used in control toTcl(k)calculate the desired actuator signal ucl,des. In Vasca etal. (2008) an inverse clutch characteristic is used. Theinvertibility of the clutch characteristic in (3) is shownucl(k)in Tarasow (2015). Here, (9) can be inverted using theseparate inversion of the clutch characteristic toucl,des(Tcl, ) = ucl ()(13)TclThe inversion of the characteristic map is possible as long(k)as the clutch characteristic is invertible and the identifiedfriction coecient does not become zero.745 T ,clLMTcl,map(16)(9)p1,ref (k)LM3. IDENTIFICATION METHODThe identified clutch characteristic according to (3) leads to a mean characteristic without the influence of clutch slip. To consider the influence of the clutch slip, it was proposed in Arndt et al. (2016) to split up the character-istic map into the clutch characteristic and a normalizedfriction coecient. The first part () of the character-T uclistic map is similar to the characteristic in (3). Because of the good shift results of the characteristic map in Tarasow (2015) the clutch characteristic has to be preserved, be-cause the identification of the whole characteristic map can lead to an implausible behaviour of the friction coecient or a deviation between the clutch characteristic in (3) and the clutch characteristic part in (9). Possible reasons are the considered number of reference values or the allocation of the reference values along the clutch slip. Here, the slope of the clutch characteristic is used to describe the clutch slip influence. The result is based on the clutch characteristic in Tarasow (2015) and the behaviour of the clutch characteristic is preserved in a fallback solution. To estimate the behaviour of the clutch, identification meth-ods are used. In the following, the identification methods and the approach are presented.3.1 Levenberg-MarquardtThe Levenberg-Marquardt method (LM) is a non-recursive identification approach. This method is a damped Gauss-Newton method which was introduced by Levenberg (1944) and Marquardt (1963). It is used to solve non-linear least-squares problems with a cost functionalT=Ff f =1xref (k) xident k, (14)2k,wheretheerror vector. Here, xref denotes thef() isreference variables and xident the variables calculated withthe identified parameter vectorfor the discrete timeindex k.During the identification process the parameter vector is iteratively corrected with the correction step width.This step is described by = J TJ+ I 1J Tf (15)Fig. 2. Scheme of the identification of the characteristic map.where J is the Jacobian of f and is a weighting factorof the parameter correction. So the parameter vector iscorrected iteratively until the cost function in (14) reaches a local minimum.3.2 Characteristic Map IdentificationIn this section, the introduced identification method is used to estimate the characteristic map given by (12). In Fig. 2, the scheme of the identification is shown. The inputs of the identification algorithm are the clutch torque Tcl(k), the actuator signal ucl(k) and the clutch slip (k). This approach is based on the estimation of the clutch characteristic map in Tarasow (2015). In the first stepthe parameters of the clutch characteristic are identifiedwith the LM method. The LM block includes the data collection and pre-processing, because the LM algorithm needs a certain amount of valid data to estimate the parameter vector. The identification result represents the clutch characteristic as a function of the actuator signal ucl. This can be interpreted as a regression curve through the reference data.It is assumed that mostly the slope is aected by the friction coecient, especially for varying the clutch slip. Other parameters, e.g. the kiss point pT2, are assumed to be time invariant. This assumption is clearly a limitation of this approach. However, the eects of this limitation do not appear to be too severe. In future work, appraoches that include a varying kiss-point will be investigated. Here, the shape parameters of the clutch characteristic are used directly and the slope is calculated analytically withp1,ref (k) =Tcl(k)(16)ucl(k) pT2for each discrete time instant k where valid measurement data exist. The vector p1,ref is a vector of the calculated parameters p1,ref (k). These values are the reference data to calculate the clutch slip dependency on the clutch char-acteristic slope. With the slope values p1,ref , the clutch slip influence is estimated with a LM algorithm. The identifi-cation process results in the estimated parameter vectorgiven in (8). The final result of both LM algorithmsis the parameter vector, which describes the,cl,mapTcharacteristic map in (12).746T. Arndt et al. / IFAC-PapersOnLine 49-11 (2016) 742748250(Nm/bar)80200Reference Values60Identified Characteristic150Initial CharacteristicReference Values40Identified Characteristic100Initial CharacteristicClutchTorque(Nm)ClutchCharacteristic2050of0Slope020010001234560400600800Actuator Signal (bar)Clutch Slip (rpm)Fig. 3. Clutch torque characteristic of the first clutch in the DCT.4. IDENTIFICATION RESULTS4.1 Experimental SetupTo evaluate the introduced approach in Section 3, it was applied on measured data of test runs. Dierent vehicles with wet dual clutch technology were examined. Most of vehicles were equipped with DCT380 or DCT500. Because of the usage and a planned implementation of the algorithm in test vehicles, the necessary measured data are reduced to existing signals from the transmission control unit. The processed signals are downloaded via a CAN interface. The apprach is used oine.4.2 Identification ResultsIn the following, the results of the identification approach are shown exemplarily for one test run. To identify the characteristic map, the measured data are reduced to the launch and upshift situations in order to get reasonable results. Both clutches of the DCT are identified separately.The identification of the characteristic map is started with the evaluation