Improvement of Traction Through mmpc in Linear Vehicle Dynamics Based on Electrohydraulic Dry-Clutch

Intell Ind Syst (2015) 1:153 161 DOI 10.1007/s40903-015-0017-6 ORIGINAL PAPER Improvement of Traction Through mmpc in Linear Vehicle Dynamics Based on Electrohydraulic Dry-Clutch Mario Pisaturo 1 Adolfo Senatore 1 Received: 14 April 2015 / Revised: 9 June 2015 / Accepted: 12 June 2015 / Published online: 14 July 2015 Springer Science+Business Media Singapore 2015 Abstract Today s several algorithms are currently managed by the control units in passenger cars to deliver higher comfort level to vehicle driver and passengers. One route to smoother longitudinal dynamics as well as deliver higher safety is provided by the engine torque limiters and traction control. The drawback of such filters is that they bring in time delays which may give to the driver the feeling that the vehicle engine does not respond in a prompt way to her/his demand [1]. On the other hand, recent highly-supercharged engines suffer for inadequate engine torque rise when setting the vehicle in motion from stationary with fast accelerator request: this behaviour is related to fueling response, fuel-air ratio control, soot emissions regulation, etc. The goal of this paper is to design a feedback controller based on multiple model predictive controller (mmpc) to manage the clutch opening/closing operations in a dry-clutch automated manual transmission architecture. The controller aims at ensuring a comfortable vehicle launch by assuring a reduced engagement time and by maintaining the wheel slip at a desired value to have the maximum traction force. The simulations have been carried out by considering the engine dynamics as well as the delay ensuing engine torque build-up together to an imposed wheel slip in fast launch manoeuvre to prove the effectiveness of the proposed clutch control strategy and encourage the development of real-time routines for the testing in real-time environment. B Mario Pisaturo mpisaturo@unisa.it Adolfo Senatore a.senatore@unisa.it 1 Department of Industrial Engineering, University of Salerno, 84084 Fisciano, SA, Italy Keywords Model predictive control Dry clutch Engine torque build-up Engagement speed-tracking Introduction The ever increasing use of automatic transmissions in modern passenger cars along with the development of advanced sensors, actuators and control unit functions could provide a more effective electronically-managed solution through the design of proper control strategies [2 4]. For market sectors such as large-series and ecological cars, the automated manual transmissions (AMTs) show all the advantages ensuing lower weight and higher efficiency with respect to other typologies of automatic transmissions [5]. In both the shifting conditions semi-automatic, based on the driver s command, or fully automatic the transmission control unit manages the shifting steps, through suitable signals to the engine, the clutch assembly and the gearbox, according to current engine regime, driving conditions and setup program (economy, sport, winter profiles, etc.). In addition, an AMT is directly derived from a manual one through the integration of actuators; then, development and production costs are generally lower than other automatic transmissions, while the reliability and durability are at highest level. For high class sport cars, vehicle dynamic performances and driving quality can be strongly improved with respect to automatic transmissions [6]. Unfortunately, the higher is the power potential of a highly-supercharged engine the higher is the discrepancy between driver s expectations and vehicle response in quick start-up manoeuvres. In this paper a high order dynamic model of the powertrain system which encompass the electro-hydraulic actuator behaviour has been analysed to design a feedback controller. It is based on multiple model predictive controller (mmpc)

154 Intell Ind Syst (2015) 1:153 161 ϑ gw = gw = g w (9) Fig. 1 Driveline scheme: 5 DoF to manage the clutch opening/closing and gearbox shifting operations in a dry-clutch AMT architecture [2,7,8]. The mmpc is developed to comply with constraints both on the inputs and outputs. The controller aims at ensuring a comfortable vehicle launch by assuring a reduced engagement time and by maintaining the wheel slip at a desired value to have the maximum traction force. The engine dynamics as well as the delay ensuing engine torque build-up in fast launch manoeuvre has been modelled to prove the effectiveness of the proposed clutch control strategy and encourage the development of real-time routines for the testing in realtime environment. Vehicle Longitudinal Dynamics This section describes a model for simulating the driveline and the vehicle longitudinal dynamics. In Fig. 1 the driveline scheme is shown, where the subscripts e, f, c, g, w indicate engine, flywheel, clutch disc, (primary shaft of) gearbox, and wheels, respectively. A dynamic model of the driveline can be obtained by applying the torque equilibrium at the different nodes of the driveline scheme, where T indicates the torques, J the inertias and ϑ the angles. The equations which describe the driveline model are: J e e = T e ( e ) b e e T ef ϑef, ef (1) J f f = T ef ϑef, ef T fc (y) (2) J c c = T fc (y) T cg ϑcg, cg (3) J g (r) g = T cg ϑcg, cg bg g 1 r T gw ϑgw, gw (4) J w w = T gw ϑgw, gw Tw (5) where T e is the engine torque (assumed to be a control input of the model), T fc is the torque transmitted by the clutch (the second control input), y is the throwout bearing position, and T w is the equivalent load torque at the wheels (a measured disturbance). The gear ratio is r (which here includes also the final conversion ratio), and J c is an equivalent inertia, which includes the masses of the clutch disc, friction pads and the cushion spring. Furthermore the following equations also hold: J g (r) = J g1 + J g2 r 2 (10) T ef ϑef, ef = kef ϑ ef + b ef ef (11) T cg ϑcg, cg = kcg ϑ cg + b cg cg (12) T gw ϑgw, gw = kgw ϑ gw + b gw gw (13) T w = F x R l + F zf x (14) where k are torsional stiffness coefficients, b viscous dampings, F x is the traction force in longitudinal direction, R l is the rolling radius which could be assumed as 92 % of the wheel radius for radial tires [9], F zf is the normal load applied on the front wheels and x is the offset shown in Fig. 1. These equations represent the driveline system during the slipping phase. The only non-linearities in these equations are the plant inputs: the engine torque T e ( e ), clutch torque T fc (y) and the wheel load torque T w (measured disturbance), [10,11]. During the engaged phase, the flywheel angular speed f and the clutch angular speed c are the same: thus the Eqs. (2) and (3) can be summed each other, which yields: Jc + J f c = T ef ϑef, ef Tcg ϑcg, cg (15) The driveline parameters used for the simulation are typical for a mid-size car and can be found in literature. In the scheme of Fig. 2 is depicted the vehicle longitudinal motion by considering a front-wheel drive car. The vehicle has been considered as a rigid body and the difference between the left and the right tires have been ignored (the so called single-track model). The lateral, yaw, pitch and roll dynamics have also been neglected. The equations which describe the vehicle longitudinal motion are: and the angular speeds dynamics are: ϑ e = e (6) ϑ ef = ef = e f (7) ϑ cg = cg = c g (8) Fig. 2 Vehicle longitudinal dynamics in acceleration

Intell Ind Syst (2015) 1:153 161 155 m v x = F x F R (16) F x = μ (s) F zf (17) 1 F R = mgf roll + c d 2 ρ a Avx 2 (18) where m is the vehicle mass, v x is the longitudinal speed of the vehicle center gravity, μ (s) is the Pacejka s tire model coefficient, F R is the force resistance which includes both the roll resistance and the air resistance as explained in (18), g is the acceleration gravity, f roll is the rolling resistance coefficient, ρ a the air density, A the front surface vehicle area, c d the air resistance coefficient. The traction force has been assumed as function of the normal load and the slip ratio. The normal force has been assumed as a static force by ignoring the influence of the suspension. This method provides an accurate estimation of the normal force, particularly when the road surface is not bumpy [12]. The longitudinal slip s for a wheel is defined as the relative difference between a driven wheel angular speed and the vehicle absolute speed: s = R w w v x R w w R w w >v x, w = 0 acceleration R w w v x v x R w w <v x, v x = 0 braking (19) The traction force F x in the longitudinal direction is a non-linear function of the longitudinal slip s, of the normal force applied on the drive wheels F zf and of the peak road adhesion coefficient μ p which depends from the road conditions. Smaller values of μ p correspond to more slippery road conditions. The tire-road friction model used in this paper is the so-called Magic-Formula developed by Pacejka and co-workers [13]. μ (s) = D sin (C arctan (Bs E (Bs arctan (Bs)))) (20) where BCD is the slope of the curve at zero-slip, B is the stiffness factor, C is the curve factor, E is the shape factor and D is the peak value. If the shape factor E < 1 and the curve factor 1 < C 2 the curve has one relative maximum and if E = 0 its value is: s max = tan π 2C (21) B It is worth noting that at s max corresponds μ p i.e., the peak road adhesion coefficient and beyond of this value the road adhesion is lost. In Fig. 3 the relationship between the traction longitudinal force F x and the slip s for a given normal load F z is depicted. Fig. 3 Traction longitudinal force versus slip The driveline parameters used for the simulation are typical for a mid-size passenger car and can be found in literature [14]. A mathematical representation of the driveline useful for the model predictive control (MPC) approach is the State- Space representation. In the continuous domain the driveline model can be written as follows: ẋ (t) = [ A sl d + A eng (1 d) ] x (t) + [ B sl d + B eng (1 d) ] u (t) (22) y (t) = Cx (t) where the state, input and output vectors are respectively: x = { } T e ϑ e f ϑ f c ϑ c g ϑ g w ϑ w u = { T e T fc T w } T y = { e c } T and d is a switching variable equal to 1 when the system is in the slipping phase and 0 otherwise. The subscripts sl and eng indicate the slipping and the engaged system matrices, respectively, and the matrices can be simply deduced from the Eqs. (1) (15). Instead, the longitudinal vehicle motion represented by Eqs. (16) (18) has been implemented regardless of the State-Space model i.e., it has not been taken into account in the MPC design. The MPC has been designed with the discrete time version of the driveline model (22) obtained by using the zero-order hold method with a sampling time of 0.01 s. This value is compatible for automotive applications. In fact, as reported in [15] the computational cycle adopted for these applications is set as 5 10 ms. Engine Static Model This section deals with the description of the engine dynamics implemented in the Simulink model used for the simulations. In particular the engine behaviour has been simulated

156 Intell Ind Syst (2015) 1:153 161 Fig. 4 Engine static map by using a static map with the engine torque depending on the accelerator pedal (first MPC output) and the measured engine speed (plant output), see Fig. 4. As stated before, the engine torque build-up has been considered for the simulations. In details, the torque build-up phenomenon is related, in the modern automotive and truck diesel engine, to the match between turbocharger and engine usually optimized for steady-state conditions and for low specific consumption. Thus, the turbocharger size is determined for high torque output, which usually leads to high moment of inertia and consequently slow air-charge response. This slow air-flow response cannot match the required fast fueling commands during transients and so it causes poor response and black smoke emissions. The key reason is the turbocharger inertia, which is responsible for the biggest part of the delay concerned. This turbocharger lag is more pronounced with the increase in engine rating or degree of turbocharging, which explains why it has become more prominent during recent years [1]. The engine torque build-up has been introduced in the simulation algorithm by using a rate limiter (with a limitation only on the rising rate of 200 Nm/s). radius of the contact surface and F fc is the cushion spring reaction. From the Eq. (23) it is clear that the cushion spring compression δ f and the corresponding force F fc (δ f ) determine the torque transmissibility as function of the throwout bearing position y [16]. Details on more complex model of the torque transmitted by a dry clutch can be found in literature [8,16,18]. Moreover, as previously explained the aim of the actuator is to control the throwout bearing position and consequently, the torque transmitted from the engine to the wheels. The actuator is mainly composed by a hydraulic piston connected to the clutch diaphragm spring; for a passively closed architecture, the clutch is engaged when no pressure is applied to the piston. The piston chamber is connected to a servovalve by means of pipeline. The other two-ways of the servovalve are connected to a supply circuit and to a discharge circuit, see Fig. 5 for details. The position of the spool valve, which is controlled by an electromagnetic circuit, switches the states of the hydraulic circuit in filling phase or in dumping phase. The servovalve connects the piston chamber to the discharging circuit in order to disengage the clutch. The springs push the piston back and the oil flows to the tank. Conversely, to engage the clutch the servovalve connects the piston chamber to the supply circuit in this way the piston force overcomes the springs reactions. The servovalve displacement is controlled in current and to keep the clutch at a certain position an offset current is needed to hold the spool in its neutral point, that corresponds to no oil flowing in the circuit. For currents greater than this offset value, the actuator is connected to the high pressure power supply, while for currents smaller than the offset value, the actuator is connected to the low-pressure circuit [17]. A regulator PI has been designed on a linearised model of the actuator. For the sake of brevity Clutch and Actuator Models In this section the model of the hydraulic actuator system coupled to the clutch is described. The torque transmitted by a dry clutch is generated by the friction phenomenon between the friction pads on the two sides of the clutch disk and the flywheel on one side and pressure plate on the other side. In [16] a model of dry clutch torque transmissibility has been proposed. It explains the role of clutch springs and their influence on the torque transmissibility. The connection between torque transmissibility and cushion spring force obtained in [16]isgivenbyEq.(23). T fc (y) = nμr eq F fc (δ f (y)) (23) where n (n = 2 in this case) is the number of contact pairs, μ is the dynamic friction coefficient, R eq is the equivalent Fig. 5 Clutch actuator and its position at clutch disengaged or open

Intell Ind Syst (2015) 1:153 161 157 Fig. 6 Control scheme details about the mathematical model and the design of the PI regulator have been omitted but they can be found in [3]. Controller Design Two closed loops have been designed in order to manage the engagement phase. The outer closed loop manages the driveline dynamics and a mmpc strategy has been implemented with the aim of generating the reference signals for the engine and the clutch subsystems. The inner loop manages the actuator dynamics and a PI controller has been designed by neglecting the sensor dynamics. In Fig. 6 the closed loop scheme is reported. The clutch torque characteristic as function of throwout bearing position and cushion spring compression is implemented in the Clutch torque map block [18,19]. In the latter block the map of the dynamic friction coefficient has been included as from [20] in which the Authors explored the tribological behaviour of the clutch facings and listed the experimental results together with predictions of the performance through artificial neural-network algorithm. Multiple Model Predictive Control The MPC approach has been developed to manage the engine and clutch operations during vehicle launch since it provides numerous advantages over the conventional control algorithms. Indeed, it naturally handles multivariable control problem, it can take account of actuator limitations, it allows the system to operate closer to constraints than conventional control, and control update rates are relatively low in these applications; so that there is plenty of time for the necessary on-line computations [21]. As explained above, the clutch operates in two different working conditions: the slipping phase and the engaged phase. Two different controllers have been designed for each one of the latter phases as explained in detail in [8]. A switching parameter selects the controller by considering the absolute value of the difference between the engine and the clutch angular speed. In particular, the switching condition from the first controller to the second one is attained when sl = e c 1 rad/s. It is important to emphasize that in no way the two controllers can work simultaneously and so any conflict between them is avoided a priori. The MPC has been designed with the discrete time version of the driveline model (22) obtained by using the zero-order hold method with a sampling time of 0.01 s. As explained above, this value is compatible with automotive applications. Constraints The solution here proposed is based on the design of a multiple controller working in sequence according to the powertrain operating conditions. These controllers are designed to comply with some constraints, both on the plant inputs and on the plant outputs, which allow the comfort to be improved during the engagement process and increase the safety of the system. In particular, the plant inputs saturation constraints have been imposed on both the manipulated variables, the accelerator pedal and the clutch torque: [ α α min,α max] (24) [ ] T fc T fc min, T fc max (25) where α min = 5 % is the minimum accelerator pedal value to avoid the engine stall, α max = 100 % is the maximum admissible accelerator pedal value, T fc min = 0 Nm is the minimum torque value transmitted by the clutch, T fc max = 400 Nm is the maximum torque value that the clutch can transmit. Instead, on the plant outputs, engine and clutch angular speeds, the following constraints hold: [ ] e e kill,e max (26) c c min (27) where e kill = 60 rad/s represents the so-called no-kill condition [14], e max = 600 rad/s is the maximum value of the engine speed before attaining critical conditions and = 0 rad/s is the minimum value of clutch speed during min c

158 Intell Ind Syst (2015) 1:153 161 Table 1 MPC1 parameters, slipping phase Symbol Description Value 1 2 W u Input weight 0.6 0.1 W u Input rate weight 0.3 1.0 W y Output weight 0.3 0.5 P Prediction horizon 20 m Control horizon 5 ρ ɛ Overall penalty weight 0.8 Table 2 MPC2 parameters, engaged phase Symbol Description Value 1 2 W u Input weight 0.9 0.0 W u Input rate weight 0.5 0.0 W y Output weight 0.5 0.0 P Prediction horizon 40 m Control horizon 15 ρ ɛ Overall penalty weight 0.8 the vehicle launch. It is worth noting that it is not necessary to constrain the clutch angular speed upper bound, because it is equal to the engine angular speed during the engaged phase and it can only decrease for passive resistance during the opening phase. Tuning The parameters have been tuned by trial and error procedure by using the MPC Toolbox implemented in SIMULINK. The driving criteria adopted to select these parameters have been a trade-off between fast engagement and comfortable lock-up. For the sake of brevity, details on their meanings and influence on the plant response are deeply described in [8]. The parameters used during the simulations are listed in Tables 1 and 2. During the engaged phase, i.e., when the engine is synchronized with the transmission, the clutch torque value (second input) does not influence the plant [8]. Indeed, in these conditions the driveline has one degree of freedom less than the slipping state and the plant model is represented by Eq. (15) which has been obtained by adding the Eqs. (2) and (3). In particular, when the clutch is engaged the engine angular speed (first plant output) and the clutch angular speed (second plant output) are the same and the system has one degree of freedom less than the slipping phase. In Table 2 opportune values have been used to avoid singularity problem on Hessian matrix. A lower value of the weight means that the first plant input accounts for weaker influence on the behaviour of the overall performance. Simulation Results This section describes the result of the simulations of two different fast launch manoeuvres by using MATLAB/ SIMULINK environment. In order to have the maximum traction force it has been selected a tire-road slip s near its peak s max on the linear segment of the curve shown in Fig. 3. This means that the tire-road adhesion conditions never exceeds the value beyond which the traction control acts. The role of the MPC is to manage the clutch during the engagement phase by avoiding to exceed the tire-road adhesion coefficient μ p (s max ). The switch between the slipping phase and the lock-up phase is managed by a Finite-State Machine. In particular, the clutch is considered to be engaged when the value of the slip speed is lower than 1 rad/s. Once the clutch is engaged, the throwout bearing position is rapidly increased to its maximum value by the control algorithm. Manoeuvre 1 The following figures show the results for a fast launch manoeuvre. In Fig. 7 are depicted the plots of the engine and the clutch angular speeds, respectively. The solid lines represent the set point trajectories e sp and c sp, instead the dashed lines represent the output of the plant e and c. The engagement condition is achieved earlier than the same occurrence of the reference signals. This is due to the faster increase of the clutch angular speed respect to its reference trajectory and this leads to a shorter engagement time. After that the engagement is attained, the MPC switches to the second controller which manages the driveline with one degree of freedom less. The response delay of the clutch actuator (Fig. 8) is due to the actuator dynamics and to the differences between the simpler friction map implemented in the transmission control unit and the real system. This results also in difference between the reference clutch torque (output of the mmpc) and the actual clutch torque (output of the clutch torque map), Fig. 9. It is worth highlighting that a good choice of the parameters of the PI regulator gives a fast actuator response and the wind up does not arise. This means that the throwout bearing position never exceeds its limit making the saturation unnecessary [3]. In Fig. 10, the reference engine torque (output of the engine static map) together with the engine torque response which presents a lag due to the torque build-up phenomenon are displayed. The torque build-up weakly appears in the first second. Indeed, the mmpc in order to track the reference engine speed produces an accelerator pedal signal (Fig. 11) which results in a reference engine torque which cannot

Intell Ind Syst (2015) 1:153 161 159 Fig. 7 Engine and clutch speeds Fig. 10 Engine torque Fig. 8 Bearing position Fig. 11 Accelerator pedal signal be adequately tracked because of the delayed response of the turbocharger occurs. Manoeuvre 2 Fig. 9 Clutch Torque The following figures show the results for a second fast launch manoeuvre. Such a manoeuvre is faster than previous one. In Fig. 12 the plots of the engine and the clutch angular speeds are depicted, respectively. Also in this case the engagement time is slightly shorter than the set point trajectories. Once more time this is due to the faster increase of the clutch speed with respect to its reference trajectory. After that the engagement is attained the MPC switches on the second controller. The effect of the response delay of the clutch actuator (Fig. 13) is added to the differences between the simplified clutch frictional torque characteristic implemented in the transmission control unit and the real system. This results in sensible difference between the reference clutch torque

160 Intell Ind Syst (2015) 1:153 161 Fig. 12 Engine and clutch speeds Fig. 14 Clutch Torque Fig. 13 Bearing position Fig. 15 Engine torque (output of the mmpc) and the actual clutch torque (output of the clutch torque map), Fig. 14. In Fig. 15, the reference engine torque (output of the engine static map) and the response delay of the engine torque due to the build-up phenomenon are displayed. This phenomenon heavily appears in the first half second because of the abrupt torque request. In fact, the engine torque request is higher than the previous manoeuvre; thus, the mmpc produces a high accelerator pedal signal (Fig. 16). In this case the lag response of the engine is more evident, Fig. 15. Moreover, in Figs. 10 and 15 it is possible to note that after the engagement the engine torque request is higher because in this condition all the road load weighs on the engine. The result is a marked build-up phenomenon. Finally in Fig. 17, the vehicle speed for both the manoeuvres is plotted. The graphs show that the vehicle in the second Fig. 16 Accelerator pedal signal

Intell Ind Syst (2015) 1:153 161 161 Fig. 17 Vehicle speed for the first v x,1 and the second v x,2 manoeuvre manoeuvre, labelled with v x,2, is faster than the first one, v x,1, for the reasons explained above. Concluding Remarks A multiple model predictive controller for dry clutch engagement to manage passenger car launch delay even in presence of engine torque build-up has been proposed. The controller aims at ensuring a comfortable vehicle launch by assuring a reduced engagement time with wheel slip at desired value to achieve the maximum traction force. Two controllers, the first for the slipping phase and the second for the engaged phase, have been designed to deliver a good trade-off between fast engagements and comfortable manoeuvres by complying with the constraints both on the input and the output variables. Two fast launch manoeuvres have been simulated to prove the effectiveness of the proposed control method. 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