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Signal Generator for Prediction of Transient Control Signals of an Automotive Transmission Control Unit Depending on Scalar Calibration Parameters.

INTRODUCTION

Due to rising demands on driving comfort, the complexity of vehicle powertrain components is increasing continuously. Especially the number of functions and parameters of the transmission control unit (TCU) rises strongly. To achieve a high customer acceptance, these TCU parameters must be calibrated with respect to high gearshift comfort. Because of growing variant diversity and shortened product development time, the calibration task constitutes a challenge but also a key aspect in further vehicle development. In future, tools and methods are required for an efficient adjustment of calibration parameters. For combustion engines, model based calibration methods are established and represent a promising approach to calibrate the engine control unit (ECU) parameters effectively [1]. Model based methods for TCU parameter calibration are in development and are not common in practice so far. For computer-aided calibration a powertrain model may be used to simulate the behavior of the system. Furthermore the functionality of the TCU software is needed to calculate certain signals for engine and clutch control during the gearshift operation. The transient progresses of these control signals depend on the TCU calibration parameters and influence the gearshift process and the gearshift comfort respectively. The TCU software is linked with the powertrain model to accomplish a complete software-in-the-loop (SIL) simulation environment. This software requires certain state quantities of the powertrain, which have to be simulated explicitly and accurately by the model. Furthermore a residual bus simulation for all input signals of the TCU software has to be implemented, which normally are not provided by a simplified powertrain model. The implementation of a TCU model, which provides the original TCU software functionality, is very extensive and time-consuming. For the modeling task, detailed knowledge of the TCU software is required. This knowledge may be obtained from the manufacturer. Alternatively, using information from general literature merely allows creating an universal model of TCU software, which may not be compared with a certain manufacturer's TCU functionality. Moreover a large number of measured data is required to adjust the parameters of a TCU model [24].

In further progress of this work a signal generator will be introduced, which offers the same functionality of the TCU software regarding the generation of transient control signals during a gearshift prozess. The signal generator enables a time efficient implementation of a simulation environment and represents a promising tool for model based TCU calibration.

GEARSHIFT PROCESS OF AN AUTOMATIC TRANSMISSION

During vehicle acceleration the engine generates the torque [T.sub.E] to drive the downstream components. This torque is converted by the torque converter and the planetary gearbox. The planetary gearbox contains multiple clutches or brakes, which lock the gear sets to each other or to the gearbox housing. Depending on the clutches states various gear ratios can be realized. Before the command for a gearshift is triggered, the torque is completely transmitted by clutch C1 and the gear ratio of the current gear is [i.sub.1]. During the gearshift operation, in general two clutches are acting at the same time. Within the process one clutch is disengaging (C1) and the other clutch (C2) is engaging simultaneously, whereas the torque is transmitted via both actuating clutches. After the second clutch is fully engaged the complete torque is transmitted via clutch C2 and the gear ratio of the following gear is [i.sub.2].

Furthermore the powertrain contains different control units, which set control signals on certain powertrain components depending on the driver demands and the current system states. During a gearshift process the actuated clutches are controlled by the TCU. The transmission control unit calculates a current for each clutch. This current actuates directly an electrohydraulic valve, which sets a pressure for clutch actuation. Additionally the TCU also requests a desired engine torque to increase the gearshift comfort or the gearshift spottiness. A schematic overview of the characteristic signals which exemplary describe the traction upshift process is shown in Figure 1. Both the target pressures [p.sub.C1.T] and [p.sub.C2.T] of the operating clutches and the engine torque [T.sub.E] influence the turbine speed [n.sub.T] and the vehicle acceleration [a.sub.veh] directly. The course of these signals is determined by the TCU calibration parameters P(1),..., P(L), which may be calibrated according to individual requirements.

The whole gearshift process can be subdivided into four characteristic phases. The filling phase begins after the gearshift command is initialized. To provide a defined initial state for the engaging clutch, the target pressure increases rapidly to prefill the electrohydraulic unit with fluid. Concurrently the target pressure of the disengaging clutch C1 drops to a pressure level, where the clutch is still in stick state. During the overlapping phase, the target pressure of the engaging clutch begins to increase continuously while the target pressure of the disengaging clutch decreases simultaneously. Thus, the transmitted torque is transferred from clutch C1 to clutch C2. In the synchronizing phase, after the transmitted torque is completely transferred to clutch C2, the disengaging clutch passes into the slip state and the turbine speed begins to synchronize to the target speed of the following gear. During the speed synchronization the engine reduces its torque to support the synchronization process. After the turbine reaches the target speed the engaging clutch passes into the stick state. In the final connecting phase the target pressure of clutch C2 increases abruptly to ensure that this clutch remains in the stick state.

Depending on the operating point and the underlying control strategy, the TCU also calculates a target pressure signal for the actuation of the torque converter clutch. The torque converter clutch allows a desired slip between pump and turbine of the torque converter and increases primary the fuel consumption efficiency.

SIGNAL PREPROCESSING

The signal preprocessing is required to prepare a dataset for parameter estimation of the signal generator. In the first step, single upshift events are extracted from the entire measured signal, which contains differently calibrated gearshifts. For this purpose, the start point of each upshift is identified via the clutch control state (STAT), which is used to define a reference point RP for each gearshift signal. The reference point allows extracting the signals in order that each signal begins from the same sampling point. Both all extracted signals and all predicted signals using the signal generator are related to this particular reference point. Thus, each value of a sampling point characterizes a certain discrete property of the extracted signal during the gearshift operation.

From top to bottom, the first and the second diagram in Figure 2 show 15 exemplary target pressure signals of the engaging clutch within the same operating point (constant turbine speed and load at gearshift request) for different calibrated traction upshifts. These signals were extracted and subsequently superposed. Additionally a second signal is plotted, which describes the current discrete clutch control state of the disengaging clutch STAT during the gearshift operation. Each extracted signal is put into a data set and has the same time length or the same number of sampling points respectively. In this example the total number of sampling points [N.sub.totel] of each extracted signal is 701. This quantity is defined as [N.sub.total] = [N.sub.a] + [N.sub.b] and consists a range of sampling points before ([N.sub.a]) and after ([N.sub.b]) the reference point RP (see Figure 2). Both ranges may be predefined and are valid for each extracted signal, whereas [N.sub.b] should be large to capture also prolonged gearshifts.

SIGNAL GENERATOR

The task of the signal generator is to predict transient TCU signals for gearshift control depending on scalar TCU calibration parameters. The relevant control signals, which are directly influenced by the TCU calibration parameters, are the desired engine torque and the control currents for both active clutches inside the planetary gearbox during the gearshift process. To simplify the powertrain model, the effective engine Torque [T.sub.E] and the target pressures [p.sub.C1.T] and [p.sub.C2.T] are used for prediction using the signal generator. These signals are provided directly by the engine control unit (ECU) and the TCU respectively.

Owing to the fact that all preprocessed signals are referring to the same reference point RP, the value of each individual sampling point depends clearly on the corresponding calibration parameter set P. This correlation between the calibration parameters and the value of each sampling point constitutes the key element of the introduced signal generator.

For each sampling point N, the signal generator contains an individual function [f.sub.EM,N] (P), which exclusively predicts the assigned signal value [Q.sub.N] on the sampling point N depending on the TCU calibration parameter set P:

[Q.sub.N] = [f.sub.EM,N] (P). (1)

In this work, the nonlinear functions [f.sub.EM,N] (P) are represented by experimental models (EMs), which will be explained in the further course.

Once all values Q = [Q.sub.1], ..., [Q.sub.N total] were predicted, they are processed by the function [f.sub.SG] (Q). The function reconstructs the desired signal passing through the signal nodes Q. Additionally, this function also resamples the outcome signal S(t) by linear interpolation if needed and assigns all sampling points a corresponding timestamp. In Figure 2, an exemplary node [Q.sub.360](P) on sampling point N=390 is presented, which may adopt different values along the ordinate depending on P.

The output of the processing function is the signal S(t):

S(t) = [f.sub.SG] (Q). (2)

Before the measured signals were used to train the behavior of the experimental models within the signal generator, they are scaled by the following equation [22]:

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

with the means [??] and [??] and the standard deviations [??] and [??]. The quantity W describes the index within the input parameter vector P and R is the number of measured data. Both quantities P and Q are referred to all measured signals. The scale of the data is required to eliminate the different orders of magnitude between each input and the output. Therefore all inputs have the same weight and thus the same impact on the process within the experimental model structure. The predicted output signal of the signal generator S*(t) is also scaled and has to be scaled back to the signal S(t), which represents the input of the powertrain model (see Figure 3).

In general the introduced approach enables to predict a transient signal with any course. A schematic overview of parameter and signal processing of the signal generator is shown in Figure 3.

ALLOCATION OF EXPERIMENTAL MODELS

As mentioned before each experimental model predicts the value of the signal on one certain sampling point. Therefore the maximum number of experimental models which may be used for signal prediction equals to the number of all sampling points within the extracted signal. In this case the maximum quality of the predicted signal can be archived. On the other hand the number of experimental models increases the computational effort. The total computational runtime [t.sub.r] increases linear to the number of experimental models [n.sub.EM].

[t.sub.r] [member of] O([n.sub.EM]). (5)

Hence, in terms of computational effort it is more efficient to reject experimental models on sampling points, which has no significant impact on the course of the signal respectively. For a computational resource efficient allocation of experimental models a criteria-based, multi-stage approach is proposed. First, a number of experimental models has to be specified which depends on computational resources that are available for utilization. Subsequently the specified experimental models will then be allocated in four stages:

Within the first stage, overall two experimental models are allocated on the first and on the last sampling point of the preprocessed signal (see triangles in Figure 4).

In the second stage the standard deviation on each sampling point N of all superposed signals [[sigma].sub.S,N] is used to allocate a predefined number of EMs [[xi].sub.[sigma]] The consideration of the standard deviation is important in order to include the areas on the sampling point coordinate, where corresponding signal values have a wide valuation. The standard deviations on all sampling points are summed up to [[SIGMA].sub.[sigma]s].

[mathematical expression not reproducible] (6)

The sum of all standard deviations is divided by the predefined number of EMs [[xi].sub.[sigma]]:

[mathematical expression not reproducible] (7)

where [[zeta].sub.[sigma]] represents the imaginary boundary, which decides, where the EMs within the second phase have to be placed (see squares in Figure 4). This procedure is shown schematically in the code below:

Algorithm: 'Allocation procedure'

1. Set c = 0 (c is an auxiliary variable)

2. Set N = 0 (first sampling point)

3. Set c = c + [[sigma].sub.s,N]

a. If c is larger than [[zeta].sub.[sigma]]

b. Place EM at actual sampling point N

4. c = c - [[zeta].sub.[sigma]]

5. If N == 700

a. Brake 'Allocation procedure'

6. SetN = N+1

7. Go to step 3.

To avoid an accumulation of EMs on particular locations, a minimum distance between neighboring EMs is introduced. If the distance is below the minimum distance, the mentioned EM is rejected on this particular sampling point [N.sub.rej]. To ensure that all [[xi].sub.[sigma]] EMs are allocated by this procedure, the allocation process is performed iteratively. To avoid that EMs in further iterations are allocated at the same location the according values of [??] are set to zero. The rejected EMs will be placed by a further iteration, until there are exactly [[xi].sub.[sigma]] EMs allocated. A simplified schematic code of the iterative algorithm is shown below:

Algorithm: 'Minimum-distance-based reallocation'

1. 'Allocation procedure'

2. If neighboring EM's distance is smaller than the minimum predefined distance (min. distance)

a. Reject these specified EMs on [N.sub.rej]. = [N.sub.rej].(1),..., [N.sub.rej.](end)

b. Set [[sigma].sub.S,N]([N.sub.rej] - [min. distance/2]:[N.sub.rej] + [min. distance/2]) = 0 for each [N.sub.rej].

c. Go to step 1 and reallocate the rejected EMs

3. 'Minimum-distance-based reallocation' finished

In the third stage, a further number of EMs are allocated using the absolute second derivation of the signal's arithmetic mean [[[??].S].sub.N] (see circles in Figure 4). The arithmetic mean is calculated on each N and considers all measured data. The process within the third stage equals the introduced process of the second stage, whereas the standard deviation has to be substituted by |[[[??].S].sub.N]| Because of linear interpolation between neighbored nodes of the predicted signal, it is required to allocate more EMs in sections with a varying gradient, where the first derivation of the signal is not constant (e. g. steps or distinctive dynamics). The use of the signal's second derivation is appropriate to identify these sections within a signal.

In the fourth and last stage a predefined number of EMs is allocated between gaps with the largest distance. This step ensures that sections with poor standard deviation or low gradient of the mean signal will also be allocated with EMs (see diamonds in Figure 4).

The fifth and the sixth diagram in Figure 4 show each an exemplary plotted signal with 40 equidistant and 40 criteria-based sampling points respectively. Regarding the signals within the fifth and sixth diagram, it is noticeable, that the criteria-based sampling point allocation leads to a better fit compared to the signal with equidistant sampling points using the same number of EMs. Hence, the criteria-based method is a suitable approach to allocate sampling points by a given number of sampling points.

MODELING APPROACHES

For the experimental models different modeling approaches can be used. The target of each model is to predict a scalar value [Q.sub.N] on a certain sampling point N, depending on a scalar parameter set P (see Eq. (1)). For the modeling task, the formulation in Eq. (1) is expanded by the term [[epsilon].sub.N]:

[Q.sub.N] = [f.sub.EM,N] (P) + [[EPSILON].sub.N]. (8)

This equation contains an error [[epsilon].sub.N] that considers uncertainties, e.g. sensor noise and system uncertainties. In this study two different and frequently applied modeling approaches are presented, which are used to express the nonlinear function [f.sub.EM,N](P).

Artificial Neural Network

In past, artificial neural networks (ANN) were used in different applications in science and engineering. An important field of application is system identification, where ANN are applied for modeling of complex, usually nonlinear systems [2], [3], [4]. Mathematically an ANN can be understood as an unknown function, which maps input data to corresponding output data. There are different types of ANN which differ in network topologies [5].

In this investigation, a feedforward neural network (FNN) is used to predict the output of the signal generator. The FNN is a comparatively simple artificial neural network approach, where the connections between neurons are in one direction. Hence, there are no feedback connections between layers. Every node within a layer may be connected to every node in its adjacent forward layer [5].

The optimal structure of a FNN for the signal generator has been estimated iteratively by numerical investigations. A suitable and simple structure of a FNN contains one hidden layer and three neurons:

[mathematical expression not reproducible] (9)

In Eq. (9). the vectors [w.sup.(1).sub.r,N] represent the weights of the hidden neurons and [b.sup.(1).sub.r,N] are the related biases. Similarly, [w.sup.(2).sub.r,N] and [b.sup.(2).sub.r,N] are the parameters of the output layer. All parameters of the presented FNN are trained by the Levenberg-Marquardt backpropagation algorithm [6] using measured data.

Gaussian Process

The Gaussian process (GP) is a stochastic process and is presented briefly in this work. The GP represents a comparatively new approach, which can be used for regression. Within the GP, the function [f.sub.EM,N] (P) is modelled as a multivariate Gaussian distribution with zero mean:

[f.sub.EM,N](P1), ... , [f.sub.EM,N](PZ)~N(0, [K.sub.N]), (10)

where the index of P describes any constellation of calibration parameters. The model error may be modelled as white noise with the variance [[sigma].sup.2.sub.[epsilon],N]:

[mathematical expression not reproducible] (11)

The matrix [K.sub.N] represents the covariance matrix, which describes the covariances between values of the function [f.sub.EM,N](P). The covariance matrix may be generated using an appropriate covariance function, e.g. rational quadratic (RQ) covariance function, which has been successfully used in previous work [6]:

[mathematical expression not reproducible] (12)

where D is the dimension of the vector P and [delta] is the Kronecker delta and i,j = 1,...,Z. The length scale [[lambda].sub.d,N], the steepness [[alpha].sub.N], the scaling parameter [[sigma].sup.2.sub.f,N] and the white noise variance [[sigma].sup.2.sub.[epsilon],N] represent the hyperparameters [[theta].sub.N] = [[gamma].sub.1,N], ... , [[gamma].sub.D,N], [[alpha].sub.N], [[sigma].sup.2.sub.f,N], [[sigma].sup.2.sub.[epsilon],N] of the function. In general, two outputs [f.sub.EM,N]([P.sub.i]) and [f.sub.EM,N]([P.sub.j]) have higher covariance, when the corresponding inputs are closer to each other. The estimation of these hyperparameters is carried out by maximization of the log-likelihood [8]:

[mathematical expression not reproducible] (13)

where [K.sub.N] is the M x M training covariance matrix and [Q.sub.N] the training data vector with the length M relating to the sampling point N. The maximization of Eq. (13) is executed iteratively by applying a conjugate gradient based method provided by gpml software [9]. The estimation of the hyperparameters depends strongly on the initial parameter set within the maximization process [6].

The GP is fully described by a mean [[mu].sub.N]([P.sub.n]) and a variance [[sigma].sup.2.sub.N] ([P.sub.n]) [21]:

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

where [k.sub.N] represents an entry within the covariance matrix [K.sub.N] and M + 1 describes the index which does not belong to the known data set 1,...,M with the index n= 1,...,M Equations (14) and (15) are used to predict the model output corresponding to a new given input [P.sub.M+1] Further information about the GP are given in [8], [20].

Generation of Measuring Data

Before the signal generator is capable to predict the course of the desired signals, the experimental models have to be trained using measured data. In this work a powertrain with standard drive and automatic transmission is utilized to generate measuring data. The powertrain is mounted on a powertrain test rig. Two electric machines are attached on the side shafts rigidly to simulate the driving resistances of the vehicle (see Figure 5). The test rig environment allows to adjust the TCU calibration parameters and to perform different gearshifts under constant ambient conditions. The engine torque including the required torque reduction within the synchronizing phase is captured directly from the ECU. Furthermore the currents and the target pressures of the actuating clutches of the planetary gearbox and of the clutch inside the torque converter are captured directly from the TCU. The engine and the turbine speed are measured via the CAN-Bus signals. Additionally separate sensors measure the torque on both driven side shafts. A schematic setup of the powertrain test rig is shown in Figure 5. All measured signals were captured with 100 Hz.

For training of the experimental models, it is necessary to consider a wide spectrum of different training data [10]. Usually a design of experiments approach is used to plan experiments efficiently. Space-filling based designs consider the whole design space and they are suitable if no a priori system knowledge is known. [11]. Thus the space-filling based S-optimal design is used for variation of certain TCU calibration parameters within one operating point throughout [A.sub.D] = 1000 single experiments [phi]. The S-optimal design [D.sub.S] represents a subset of a full factorial design [D.sub.0]. S-optimality aims to maximize the geometric mean of the distances between nearest neighbor points [12]:

[mathematical expression not reproducible] (16)

The expression d([phi], D - [phi]) is the distance measure to the nearest neighbor of [phi].

SIGNAL GENERATOR FOR PREDICTION OF TRANSIENT GEARSHIFT-COMFORT RELEVANT SIGNALS

Due to the flexible structure the signal generator is capable to predict signals with any course. Hence, for model based gearshift calibration, the signal generator may also be used to predict gearshift comfort relevant signals directly. This approach is particularly useful, if merely one or more transient signals should be assessed in respect to gearshift comfort criteria without simulation of all system states. To output the comfort relevant vehicle acceleration signal, it is necessary to measure the toque of both side shafts additionally. The measured side shaft torque is used to train the behavior of the signal generator. Subsequently the vehicle acceleration may be calculated by applying Eq. (18) (see chapter 'Powertrain model') using the predicted side shaft torque.

EVALUATION OF EXPERIMENTAL MODELS

The accuracy of the signal generator depends on one hand on the modeling approach of the experimental models and their parameters and on the other hand on the number of measured data, which are considered for the training process. In this example the predicted signal accuracy is evaluated using the mean root-mean-square error (RMSE) between 25 different measured and predicted signals. As can be seen in Figure 6, the RMSE declines in general, if more measurements within the training process are considered. Nevertheless the GP approach has a slightly lower RMSE investigated for all signals compared to the ANN approach.

A further aspect represents the mean calculation time (1) [t.sub.calc] for each EM of the different modeling approaches. For an iterative model based calibration task, it is desirable to predict the signals with low calculation effort. As shown in Figure 6, within the training process the GP approach requires significantly more calculation time compared to the ANN approach. The calculation time rises exponentially when using more measurement data. In practice, the calculation time of the training process has no major importance, because the training process is performed merely once for each signal. For an efficient and iterative model based calibration process the calculation effort for signal prediction is more crucial. For signal prediction, the number of considered measured training data has no relevant impact on the mean calculation time [t.sub.Calc], which is comparably low for both modeling approaches regarding the investigated number of measurements. Related to the signal generator, the GP approach offers a better overall performance with respect to accuracy and calculation effort. Hence, in further course of this work the GP approach with a training dataset of 700 measurements is used for signal prediction.

POWERTRAIN MODEL

The following powertrain model is used to simulate the gearshift process. It contains a combustion engine, a torque converter including a torque converter lockup clutch, a planetary gear box, a gear box output shaft, a differential gear, a combined side shaft and an equivalent vehicle load side (see Figure 7).

The powertrain model is based on the approach in [13] and was already presented in similar form in [14], [15]. The model has up to four rotational degrees of freedom depending on the state of the clutches inside the torque converter and the gear box. The combined side shaft includes both driven side shafts and has elastic and damping properties. The quantity [T.sub.SS] combines the torque of both driven side shafts and is calculated with:

[mathematical expression not reproducible] (17)

where [i.sub.Diff] is the differential gear ratio. The vehicle acceleration [a.sub.veh] is an appropriate signal to assess the perceived shift comfort [16]. The characteristics of the tires are neglected, thus the vehicle acceleration is proportional to the side shaft torque [T.sub.SS] and is calculated depending on the vehicle mass [m.sub.veh], and the dynamic tire radius [r.sub.dyn]:

[mathematical expression not reproducible] (18)

where [F.sub.res] represents the overall force of the driving resistances:

[mathematical expression not reproducible] (19)

The constants [k.sub.0], [k.sub.1] and [k.sub.2] express the coefficients of the constant, linear and quadratic terms of the overall driving resistance force.

The predicted signals [T.sub.E], [p.sub.C1.T] and [p.sub.C2.T] from the signal generator represent the inputs of the powertrain model. The engine torque [T.sub.E] is already given. Hence, no extensive engine model for torque calculation is needed. Given the mass inertia of the engine-pump unit [[THETA].sub.E/P], the equation of motion for the relating degree of freedom is:

[[THETA].sub.E/P] X [[[??].[phi]].sub.E/P] = [T.sub.E] - [T.sub.P] - [T.sub.CC]. (20)

The torque of the pump [T.sub.P] and the torque of the slipping converter clutch [T.sub.CC] are given with:

[mathematical expression not reproducible] (21)

[T.sub.CC] = [A.sub.CC] x [r.sub.m,CC] x [N.sub.CC] x [[mu].sub.CC] x [p.sub.CC]. (22)

According to Eq. (71) [T.sub.P] depends on the power coefficient [kappa], the density of the operating fluid [[rho].sub.Oil] and on the profile diameter of the pump impeller [D.sub.C]. Moreover, [A.sub.CC] represents the surface of the clutch disk, [r.sub.m,CC] is the mean radius of the clutch disks and [N.sub.CC] is the number of the grinding disk surfaces. The friction coefficient [[mu].sub.CC] is a function of difference speed at the converter clutch [DELTA][n.sub.CC] = [n.sub.P] - [n.sub.T] The pressure [p.sub.CC] acts directly on the torque converter clutch disks. For simplification, a PI control is implemented, which calculates a target pressure [p.sub.CCT] depending on the controller error e = [DELTA][n.sub.CCT] - [DELTA][n.sub.CC]. For the considered constant operating point, the target slip speed [DELTA][n.sub.CC.T] is set to 20 rpm and equals the observed measured data. Alternatively, the clutch converter pressure may also be predicted by the signal generator.

During a gear shift operation usually two clutches are active simultaneously. Thus, within the planetary gear box overall two clutches are considered. The underlying gear box model is sufficient to simulate the torque transfer from the disengaging clutch to the engaging clutch and thus also to simulate the crossover from the gear ratio of the prior gear to the gear ratio of the next gear [13]. During the gearshift operation, both clutches pass through different states. The transmittable clutch torque depends strongly on, whether the clutch is in stick or slip state. Following equation represents the condition for the clutch stick state:

|TC|[less than or equal to] [T.sub.C,max] = [A.sub.C] x [r.sub.m,c] x [N.sub.c] x [[mu].sub.0,C] x [p.sub.c]. (23)

Index C symbolizes any clutch, whereas [[mu].sub.0,C] represents the static friction coefficient. While the clutch C1 is in stick state, the clutch torque is calculated with:

[mathematical expression not reproducible] (24)

The quantities [[theta].sup.+.sub.OS] and [[theta].sup.+.sub.T] represent reduced mass inertias of the transmission output and turbine including the masses of the transmission input on clutch C1. The index n represents the path and the index m represents the position in relation to the corresponding clutch of the gear ratio [i.sub.nm]. The torque of the turbine [T.sub.T] is proportional to [T.sub.P] and depends on the torque factor [[mu].sub.TF] = f(v):

[T.sub.T] - [T.sub.P] x [[mu].sub.TF], (25)

where the angular speed ratio v is defined with [??]. Equivalent to Eq. (24), the torque of the sticking clutch C2 is given by:

[mathematical expression not reproducible] (26)

The parameters [[theta].sup.++.sub.OS] and [[theta].sup.++.sub.T] are the mass inertias reduced on clutch C1. As illustrated in Eq. (24) and Eq. (26) the torque of a sticking clutch depends also on the transmitted torque of the other clutch. The torques [T.sub.C1] and [T.sub.C2] represent the torques of clutch C1 and C2 in slipping state, which are calculated equivalent to Eq. (22). A clutch passes into the slip state once the condition in Eq. (23) is not fulfilled anymore. Otherwise, a clutch passes into the stick state, if condition in Eq. (23) is fulfilled and the differential speed on the clutch is approximately zero.

The effective pressure [p.sub.C] acting on the clutch disks is simulated using an electrohydraulic model, which is schematically shown in Figure 8.

An electrically actuated hydraulic solenoid valve provides a target pressure [p.sub.C.T], which is affected by the control current [i.sub.C] The relationship between the control current and the target pressure is described via characteristic map. Both signals are provided by the TCU. A transfer function of second order and the coefficients [z.sub.1] and [z.sub.2] describe the dynamic behavior of the electrohydraulic valve [23]:

[mathematical expression not reproducible] (27)

The calculation of the volume flow V(t) through the orifice is based on the principle of linear momentum and is given by:

[mathematical expression not reproducible] (28)

whereas [A.sub.Or] is the cross section surface of the orifice and [zeta] specifies the resistance coefficient of the hydraulic route. The quantity [DELTA]p describes the pressure difference between the target pressure [p.sub.C.T] and the cylinder pressure [p.sub.CY], which is yielded by solving the differential equation of motion of the piston:

[m.sub.Pi] x x = [p.sub.Cy] x [A.sub.cy] - x x [d.sub.pi] - [F.sub.S] (x). (29)

The parameters [m.sub.Pi] and [d.sub.Pi] are the mass and the damping coefficient of the piston and [A.sub.CY] describes the cross-section surface of the cylinder. The spring force [FS](x) contains the return spring [F.sub.R](x) and also the reaction force of the clutch [F.sub.C](x):

[mathematical expression not reproducible] (30)

The quantity [F.sub.C](x) also covers the force resulting from a limit stop spring, which is connected in series with the clutch disks. As shown in Eq. (30), the clutch force does not act before the piston overcomes the idle stroke [x.sub.C]. Given the force acting on the clutch and the surface area of the clutch disks [A.sub.C], the effective clutch pressure is calculated with:

[p.sub.C] = [F.sub.C](x)/[A.sub.C] (31)

The stroke of the piston x is affected by the volume flow:

[mathematical expression not reproducible] (32)

whereas x([t.sub.0]) is the initial stroke of the piston. Within the model of the engaging clutch, the piston stroke begins to rise after the empty volume of the cylinder [V.sub.0] is prefilled with fluid and the following condition is satisfied:

[mathematical expression not reproducible] (33)

In this study, some model parameters were provided by the manufacturer of the powertrain components. The remaining parameters were estimated based on literature [17], [18], [19].

SIMULATION RESULTS

To analyze the potential of the signal generator, the predicted transient control signals were used for input signals of the powertrain model to simulate the turbine speed [n.sub.T] and the comfort relevant side shaft torque [T.sub.SS]. In Figure 9, two different calibrated gearshifts (A and B) are shown, which are used for model validation and which were not considered within the training process. The results demonstrate the high potential of the signal generator. The control signals for the powertrain model are predicted precisely. Especially the steps within the pressure signals are predicted accurately. Furthermore the simulation results of the powertrain model match well with measured data. The presented powertrain model is suitable to simulate all relevant dynamic effects of the gearshift process using predicted input signals. Also the explicitly via signal generator predicted transient side shaft torque has a high accuracy, which is comparable to the simulated side shaft torque.

The main objective of the model based calibration task is the optimization of the calibration parameters according to different objective gearshift comfort criteria, which depends strongly on the calibration parameters. To demonstrate the capability of the signal generator for the TCU parameter calibration process, three different objective evaluation criteria are applied. Two of these criteria describe the slump of the vehicle acceleration signal within the overlapping phase (AS-OP) and on the end of the synchronizing phase (AS-SP) [16] (see Figure 9). The third exemplary criterion (Waviness) represents the summed deviation between the 9 Hz low pass filtered and the 2 Hz low pass filtered vehicle acceleration signal and considers the oscillations induced by the gearshift process. The vehicle acceleration is calculated according to Eq.(18) using the side shaft torque. Figure 10 illustrates the results of the evaluation criteria applied on the measured, the simulated and the explicitly predicted signals. These results show two aspects. First of all, the values of the evaluation criteria differ strongly for both demonstrated gearshifts A and B. The height of the acceleration slumps but also the oscillations of gearshift A are higher compared to gearshift B and can also be observed in Figure 9. These characteristics lead to a poor gearshift comfort, which may be improved by optimization of calibration parameters. A further essential result represents the high prediction quality of the evaluation criteria. The evaluation criteria values of the measured signal, of the simulated signal using predicted model inputs and of the explicitly predicted signal are close to each other. Thus both approaches may be used for a precise evaluation of the gearshift comfort.

The overall calculation time for prediction of the side shaft torque signal depends linear on the number of the EMs (see Eq. (5)).

Considering the results in Figure 5, the calculation for a side shaft torque signal prediction with 80 EMs takes approximately (2) 2.9 s or 70% less compared to the same gearshift simulation of the powertrain model.

SUMMARY/CONCLUSIONS

In this investigation an innovative signal generator was introduced in order to predict transient control signals of a transmission control unit (TCU) depending on scalar TCU calibration parameters. The signal generator contains multiple experimental models, which calculate the value on certain sampling points of the predicted signal. The number of the experimental models increases the computational effort. Thus an algorithm was applied, which effectively allocates a given number of experimental models depending on several criteria, in order to decrease the computing time and simultaneously to increase the accuracy of the signal generator. For the experimental models, two different modeling approaches based on artificial neural network and on Gaussian process (GP) were investigated. The parameters of the experimental models were estimated using measured data from a real powertrain mounted on a powertrain test rig. Additionally a comparatively simplified powertrain model was presented, which is suitable to simulate all dynamic effects of a gearshift process. The signal generator was connected with the powertrain model to provide an overall model for the application in a model based calibration task. This overall model enables to simulate gearshift comfort relevant signals depending on scalar TCU calibration parameters, which may be optimized according to different objective evaluation criteria. Because of high adaptability, the introduced signal generator also enables to predict various dynamic signals with any course. Thus, the signal generator was additionally used to predict a gearshift comfort relevant signal explicitly.

Considering both investigated modelling approaches for the experimental models within the signal generator, the GP approach provides a good trade-off between prediction accuracy and prediction calculation effort. Furthermore the results in this work demonstrate the high potential of the signal generator in several applications. On the one hand, the signal generator predicts all control signals very accurately. In contrast to an original TCU software model, the signal generator has no specific requirements to the powertrain model relating to modeling scope. Hence, simplified powertrain models can be used. In this work, the dynamic engine torque was predicted explicitly by the signal generator. Thus, no comprehensive engine model for engine torque simulation is required. The presented overall model, containing the signal generator and the simplified powertrain model, is suitable to simulate different calibrated transient gearshifts. Hence, it represents a powerful tool for the model based calibration task of TCU parameters.

Another potential of the signal generator is the ability to predict dynamic signals of any course. Thus the signal generator may also be used to predict gearshift comfort relevant signals explicitly. For purpose of demonstration, the side shaft torque was predicted explicitly and was also simulated via the powertrain model. Both side shaft torque signals were used to calculate the corresponding vehicle acceleration in order to evaluate the gearshift comfort. The values of evaluation criteria applied on both signals were close to each other and comparable to values based on the measured side shaft torque signal. In terms of computational effort, the calculation of the predicted side shaft torque signal via the signal generator is significantly less time consuming compared to a gearshift simulation via the powertrain model. Nevertheless, if multiple signals or specific non measurable system states (e. g. friction power at the clutches within the gearbox) have to be taking into account, a simulation including the powertrain model is required.

In future work, the signal generator may be applied in a model based calibration task for optimization of TCU calibration parameters. Because of the flexibility of the presented signal generator, it is conceivable to adopt the introduced approach in further subject areas, e.g. for combustion engines.

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DEFINITIONS/ABBREVIATIONS

ANN - Artificial neural network

AS-OP - Acceleration slump - overlapping phase

AS-SP - Acceleration slump - synchronizing phase

CAN - Controller area network

CPU - Central processing unit

DDR - Double data rate

ECU - Engine control unit

EM - Experimental model

FNN - Feedforward neural network

GP - Gaussian process

RAM - Random access memory

RMSE - Root mean square error

TCU - Transmission control unit

Ivan Rot and Stephan Rinderknecht

TU Darmstadt

CONTACT INFORMATION

Ivan Rot, M. Sc.

TECHNISCHE UNIVERSITAT DARMSTADT

Institute for Mechatronic Systems in Mechanical Engineering (IMS)

Otto-Berndt-Strasse 2

64287 Darmstadt

Germany

Phone:+49 6151 16-23 272

Fax:+49 6151 16-23 264

rot@ims.tu-darmstadt.de

www.ims.tu-darmstadt.de

(1.) The calculation time in this work is referred to an Intel[R] Core[TM] i5 3570k CPU and 12 GB of DDR3 RAM. The GP (ANN) algorithm performed on 4(1-2) CPU cores.

(2.) The signal generator with the GP approach runs on four CPUs, whereas the powertrain model runs on one CPU.
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Author:Rot, Ivan; Rinderknecht, Stephan
Publication:SAE International Journal of Passenger Cars - Mechanical Systems
Article Type:Report
Date:Apr 1, 2017
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