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A framework for modeling and optimal control of automatic transmission systems
4th IFAC Workshop on 4th Workshop on 4th IFAC IFACand Workshop on Control, Simulation and Modeling Engine Powertrain 4th IFAC Workshop on Control, Simulation and Modeling Engine Powertrain Engine and and Powertrain Control, OH, Simulation and online Modeling August 23-26, 2015. Columbus, USA Available at www.sciencedirect.com Engine and Powertrain Control, Simulation and Modeling August 23-26, 2015. August 23-26, 2015. Columbus, Columbus, OH, OH, USA USA August 23-26, 2015. Columbus, OH, USA
ScienceDirect
IFAC-PapersOnLine 48-15 (2015) 285–291
A A framework framework for for modeling modeling and and optimal optimal A framework for modeling and optimal control of automatic transmission control of automatic transmission systems systems control of automatic ∗transmission systems ∗
V. Nezhadali ∗ L. Eriksson ∗ V. Nezhadali ∗∗ L. Eriksson ∗∗ V. V. Nezhadali Nezhadali L. L. Eriksson Eriksson ∗ Electrical Engineering Department, Link¨ oping University, SE-581 83 ∗ ∗ Electrical Engineering Department, Link¨ o ping lars.eriksson}@liu.se) University, SE-581 83 Electrical Engineering Department, Link¨ o University, ∗Link¨ o ping, Sweden (e-mail: {vaheed.nezhadali, Electrical Engineering Department, Link¨ oping ping lars.eriksson}@liu.se) University, SE-581 SE-581 83 83 Link¨ o ping, Sweden (e-mail: {vaheed.nezhadali, Link¨ o ping, Sweden (e-mail: {vaheed.nezhadali, lars.eriksson}@liu.se) Link¨ oping, Sweden (e-mail: {vaheed.nezhadali, lars.eriksson}@liu.se) Abstract: Development of efficient control algorithms for the control of automatic transmission Abstract: of efficient control algorithms for the control oflife automatic Abstract: Development of control for control automatic transmission systems is Development crucial to maintain passenger comfort and operational of the transmission Abstract: Development of efficient efficient control algorithms algorithms for the the control of oflife automatic transmission systems is crucial to maintain passenger comfort and operational of the transmission systems is crucial to maintain passenger comfort and operational life of the transmission components. An optimization framework is comfort developedand by state space modeling of atransmission powertrain systems is crucial to maintain passengeris operational life of the components. An optimization framework developed by state space modeling of a components. An optimization framework is developed state space of aa powertrain powertrain including a nine speed automatic transmission, dieselby engine, torquemodeling converter and a model components. An optimization framework is developed by state space modeling of powertrain including a nine speed automatic transmission, diesel engine, torque converter and aa model including a nine speed automatic transmission, diesel engine, torque converter and for longitudinal vehicle dynamics considering drive shaft as the only flexibility of the driveline. including a ninevehicle speed dynamics automaticconsidering transmission, diesel engine, torque converter and driveline. a model model for longitudinal drive shaft as the only flexibility of the for longitudinal vehicle dynamics considering drive shaft as the only flexibility of the driveline. Emphasis is set on the kinematics of the automatic transmission with the aim of modeling for for longitudinal vehicle dynamics considering drive shaft as the only flexibility of the driveline. Emphasis is set on the kinematics of the automatic transmission with the aim of modeling for Emphasis is set on the kinematics of the automatic transmission with the aim of modeling for gearshift optimal control during the inertia phase. Considering the interacting forces between Emphasis is set on the kinematics of the automatic transmission with the aim of modeling for gearshift during the inertia phase. Considering the interacting between gearshift optimal control during the phase. Considering the forces between planetaryoptimal gearsets,control clutches and brakes in the transmission, kinematic equationsforces of motion are gearshift optimal control during the inertia inertia phase. Considering the interacting interacting forces between planetary gearsets, clutches and brakes in the transmission, kinematic equations of motion are planetary gearsets, clutches and brakes in the transmission, kinematic equations of motion are derived for rotating transmission components enabling to calculate both transmission dynamics planetary gearsets, clutches and brakes in the transmission, kinematic equations of motion are derived for rotating to calculate both transmission dynamics derived for transmission components enabling to calculate both dynamics and internal forces. transmission The model iscomponents then used enabling in optimal control problem formulations for the derived for rotating rotating transmission components enabling to control calculateproblem both transmission transmission dynamics and internal forces. The model is then used in optimal formulations for the and internal forces. The is used in control analysis of optimal control transients in two up-shift cases. and internal forces.control The model model is then then usedup-shift in optimal optimal control problem problem formulations formulations for for the the analysis of optimal transients in two cases. analysis of optimal control transients in two up-shift cases. analysis of optimal control transients in two up-shift cases. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Automatic transmission, optimal control, modeling and simulation framework Keywords: Keywords: Automatic Automatic transmission, transmission, optimal optimal control, control, modeling modeling and and simulation simulation framework framework Keywords: Automatic transmission, optimal control, modeling and simulation framework 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION Various driveline configurations are utilized in todays cars Various driveline utilized cars Various driveline configurations configurations are utilized inoftodays todays cars while Automatic Transmission are (AT) is onein the comVarious driveline configurations are utilized in todays cars while Automatic Transmission (AT) is one of the comwhile Automatic Transmission (AT) is one of the commonly used systems, Wagner (2001). Such transmission while Automatic Transmission (AT) isSuch one of the commonly systems, Wagner (2001). monly used systems, Wagnergearsets (2001). where Such transmission transmission systemsused consist of planetary various gear monly used systems, Wagner (2001). Such transmission systems consist of planetary gearsets where various gear gear systems consist of planetary gearsets where various ratios are achieved by coupling different components of systems consist of planetary gearsets where various gear ratios are achieved by coupling different components of ratios are achieved by coupling different components of planetary gearsets with each other. The coupling takes ratios are gearsets achieved by coupling different components of planetary each other. The takes planetary gearsets with with each other. clutches The coupling coupling takes place via engagement of oil immersed and brakes, planetary gearsets with each other. The coupling takes place via engagement of oil immersed clutches and brakes, place viatoengagement engagement of oil oil immersed immersed clutches and brakes, brakes, referred as shift elements, at the time of gearshift. Using place via of clutches and referred to as shift elements, at the time of gearshift. Using referred to as shift elements, at the time of gearshift. Using combinations of planetary gearsets in the transmission, referred to as shift elements, gearsets at the time of gearshift. Using combinations of in transmission, combinations of planetary planetary gearsets in the thewhile transmission, wide range of gear ratios become available the comcombinations of planetary gearsets in the transmission, wide range of gear ratios become available while the wide range of of gear gear ratios become available available while the comcompact structure of such transmission makes it favorable for wide range ratios become while the compact structure of such transmission makes it favorable for pact structure of such transmission makes it favorable for application in both normal passenger cars, Greiner et al. pact structure of such transmission makes itGreiner favorable for application in both normal passenger cars, et al. application in both normal passenger cars, Greiner et al. (2004), and heavy duty machinery, A40G (2015). application in bothduty normal passenger cars, Greiner et al. (2004), (2004), and and heavy heavy duty duty machinery, machinery, A40G A40G (2015). (2015). (2004), and machinery, A40G (2015). However, asheavy power is transferred through transmission However, as power is transferred through transmission However, as power is transferred through transmission components during gearshifts, the difference between inerHowever, asduring powergearshifts, is transferred through between transmission components difference inercomponents duringspeed gearshifts, the difference between inertia and rotational of thethe coupled bodies introduces components during gearshifts, the difference between inertia and speed of bodies introduces tia and rotational rotational speed of the the coupled coupled bodiesApart introduces discontinuities in the powertrain dynamics. from tia and rotational speed of the coupled bodies introduces discontinuities in the powertrain dynamics. Apart from discontinuities in the powertrain dynamics. Apart from the long-term destructive impacts of the induced oscilladiscontinuities in the powertrain dynamics. Apart from the destructive impacts of the long-term destructive impacts of the the induced induced oscillationslong-term on powertrain components, passenger comfortoscillais also the long-term destructive impacts of the induced oscillations on powertrain components, passenger comfort is tions on powertrain components, passenger comfort is also also greatly dependent oncomponents, the gearshiftpassenger smoothness, see Huang tions on powertrain comfort is also greatly dependent on the gearshift smoothness, see Huang greatly dependent on the gearshift smoothness, see Huang and Wang (2004) and Horste (1995). greatly dependent and on the gearshift smoothness, see Huang and and Wang Wang (2004) (2004) and and Horste Horste (1995). (1995). and Wang (2004) Horste (1995). Simultaneous control of off-going and on-going shift eleSimultaneous of off-going shift Simultaneous control of determines off-going and and on-going shift eleelements during control gearshifts theon-going shift characterisSimultaneous control of off-going and on-going shift elements during gearshifts determines the shift characterisments during gearshifts determines the shift characteristic which in its turn has direct effects on other vehicle ments during its gearshifts determines the shift characteristic which turn has direct on tic which in in its as turn has comfort, direct effects effects on other other vehicle vehicle properties such drive fuel efficiency, tic which in its turn has direct effects on other vehicle properties such as drive comfort, fuel efficiency, vehicle properties such as drive comfort, fuel efficiency, vehicle operability and durability of gearbox components. In addiproperties such as drive of comfort, fuel efficiency, Invehicle operability and durability gearbox components. addioperability and durability of gearbox components. In addition to the improvement of vehicle properties, proper AT operability and durabilityof of vehicle gearboxproperties, components. In addition to the improvement proper AT tion to the improvement of vehicle properties, proper AT control strategies also open up the design space for new tion to the improvement of vehicle properties, proper AT control strategies also open up design for control strategies also with openlesser up the the design space space for new new transmission concepts components resulting in control strategies also open up the design space for new transmission concepts with lesser components resulting in transmission concepts with lesser components resulting in lower total vehicle mass and consequently fuelresulting consumptransmission concepts with lesser components in lower total lower total vehicle vehicle mass mass and and consequently consequently fuel fuel consumpconsumption reduction. lower total vehicle mass and consequently fuel consumption reduction. tion reduction. tion reduction.
Fig. 1. Cutaway drawing of a nine speed automatic gearbox Fig. Cutaway of Fig. 1. 1. Cutaway drawing of aaa nine nine speed speed automatic automatic gearbox gearbox and torque drawing converter. Fig. 1. Cutaway drawing of nine speed automatic gearbox and torque converter. and torque converter. and torque converter. Extensive research has been carried out on the topic of AT Extensive has been carried topic of Extensive research has and beenoperating carried out out onofthe the topic of AT AT control forresearch ride quality lifeon shift elements, Extensive research has been carried out on the topic of AT control for ride quality and operating life of shift elements, control for ride quality and operating life of shift elements, see Goetz et al. (2005), Sun and Hebbale (2005), Han and control for ride quality and operating life of shift elements, see Goetz al. and Hebbale (2005), and see (2003), Goetz et et al. (2005), (2005), Sun and and Hebbale (2005), Han and Yi Minowa et al.Sun (1999) Haj-Fraj andHan Pfeiffer see Goetz et al. (2005), Sun and Hebbale (2005), Han and Yi (2003), Minowa et al. (1999) and Haj-Fraj and Pfeiffer Yi (2003), Minowa et al. (1999) and Haj-Fraj and Pfeiffer (2002), concluding that the main criteria for the evaluation Yi (2003), Minowa that et al.the (1999) and Haj-Fraj and Pfeiffer (2002), main the evaluation (2002), concluding that the main criteria criteria for theprocess evaluation of shift concluding quality arethat the the duration of gear for shift and (2002), concluding main criteria for the evaluation of shift quality are the duration of gear shift process and of shift quality are the duration of gear shift process and emerging oscillations in the powertrain. An efficient of shift quality are theinduration of gear shift process and and emerging oscillations the powertrain. An efficient emerging oscillations in the powertrain. An efficient and cost effective approach obtain suitable strategies for emerging oscillations in to theobtain powertrain. Anstrategies efficient and cost effective suitable for cost control effectiveis approach approach tomodels obtainwith suitable strategies for AT to utilize to a time dependent cost effective approach to obtain suitable strategies for AT control is to utilize models with aa time dependent AT control is to utilize models with time dependent structure where the dynamics of different components are AT control is tothe utilize models with a time dependent structure where of are structure where during the dynamics dynamics of different different components are well described gearshift. Previouscomponents AT modeling structure where the dynamics of different components are well described during gearshift. Previous AT modeling well described during gearshift. Previous AT modeling effortsdescribed can be classified two different categories. In the well during into gearshift. Previous AT modeling efforts can be two categories. In efforts can be classified classified into two different different categories. In the the first the focus is on theinto dynamics of hydraulic actuation efforts can be classified into two different categories. In the first the focus is on the dynamics of hydraulic actuation first the focus is on the dynamics of hydraulic actuation system initiated by the change of actuator lever position first the focus is by on the change dynamics actuator of hydraulic actuation system lever system initiated initiated by the the change of of pressure actuator at lever position ending in the rise of hydraulic theposition brakes system initiated by the change of actuator lever position ending in the rise of hydraulic pressure at the brakes ending in the rise of hydraulic pressure at the brakes and clutches, see Minowa et al. (1999), Thornton et al. ending in the rise of hydraulic pressure at the brakes and clutches, see Minowa et al. (1999), Thornton et and clutches, see Minowa et al. (1999), Thornton et al. al. (2013) and Gao et al. (2010). In the second category, and clutches, see et Minowa et al. In (1999), Thornton et al. (2013) and Gao al. (2010). the second category, (2013) and Gao et al. (2010). In the second category, the kinematics of planetary gearsets before, during and (2013) and Gaoof etplanetary al. (2010). In thebefore, secondduring category, the gearsets and the kinematics kinematics of of planetary gearsets before, during and after gearshifts are interest while the before, hydraulic pressures the kinematics of planetary gearsets during and after gearshifts are of interest while the hydraulic pressures after gearshifts are of interest while the hydraulic pressures are considered as control input signals, see Haj-Fraj and after gearshifts are of interest while the hydraulic pressures are as signals, are considered considered as control control input signals, see see Haj-Fraj Haj-Fraj and and Pfeiffer (2002) and Kim etinput al. (2003). are considered as control input signals, see Haj-Fraj and Pfeiffer (2002) and Kim et al. (2003). Pfeiffer (2002) and Kim et al. (2003). Pfeiffer (2002) and Kim et al. (2003).
Copyright © 2015, 2015 IFAC 285 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 285 Copyright ©under 2015 responsibility IFAC 285Control. Peer review of International Federation of Automatic Copyright © 2015 IFAC 285 10.1016/j.ifacol.2015.10.041
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uf Engine
u1-2 = {uf , TB5 , TK3 } (1) where uf , fuel mass injected per combustion cycle is the control input to the diesel engine model, and TB5 and TK3 are the torques from brake 5 and clutch 3.
TKi TBi Te Tp
Tt
Torque Converter
Tf Transmission
Tw Final drive Drive shaft
B1
Fig. 2. Schematic view of the powertrain and driveline components with focus on transmission inputs and outputs. In this work, the aim is to develop a framework enabling optimal control analysis of desirable AT gearshifts during the inertia phase. A nine speed AT as shown in Figure 1 comprised of a torque converter, five planetary gearsets and eight shift elements will be considered. Analyzing the kinematics of rotating bodies and calculating the internal forces/torques between various components, transmission dynamics at different gears are obtained as a state space system which is utilized for optimal control problem formulation. Then the AT transients during the inertia phase are calculated and analyzed for two example gearshifts to illustrate the framework applicability in different gearshifting scenarios. 2. DRIVELINE MODEL Figure 2 shows a schematic overview of the developed powertrain model where a diesel engine generates the required power transferred to the wheels via torque converter, transmission, final drive and a drive shaft. Considering the large number of AT components and the corresponding state variables, which will be later introduced, in order to reduce the number of required state variables for the complete powertrain kinematics and for simplicity, the following assumptions and simplification are made during powertrain modeling: (1) The turbocharger lag and engine pumping losses are neglected in diesel engine model assuming that the engine has already reached high speed when a gearshift starts. (2) All components except the drive shaft are considered without torsional damping or flexibility. (3) Friction and viscous losses in the planetary gearsets are neglected. (4) Component properties are chosen according to the information available in A40G (2015). (5) Perfect clutch fill and coordination is assumed. Only the inertia phase of the gearshift is considered and therefore, the transmission actuation system is modeled such that the shift element torques are directly applied as control inputs. Engine speed ωe , rotational speed of the transmission components ω1,...,8 , torsion of drive shaft x and wheel speed ωw are the state variables of the model which are described by equations (3), (9)-(16),(21) and (22). Considering actuation torques from all shift elements, TB1,2,3,4,5 and TK1,2,3 as depicted in Figure 2, there are eight choices for the transmission model control inputs depending on the selected gear. The kinematic analysis is described for 1-2 up-shift only but similar methodology can be followed for other gearshifts. Therefore the control inputs (u1-2 ) for 1-2 up-shift modeling will be: 286
K1
B2
K2
In
B3
B4
B5
R
R
R
R
R
C
C
C
C
C
S
S
S
S
S
1
2
3
4
5
K3
Out
Fig. 3. Schematic view of a nine speed automatic gearbox and mechanical links between planetary gearsets components, Sun(S)-Carrier(C)-Ring(R), with respect to clutches and brakes. Gear 1 2 3 4 5 6 7 8 9
K1 × × × × × × ×
K2
K3
B1
B2
×
× × ×
× × × × ×
×
B3
× ×
B4 × ×
B5 × ×
× × ×
×
Table 1. The shift elements (clutches and brakes) that are active at each gear.
2.1 Diesel engine A simplified version of the mean value engine model (MVEM) in Walstr¨om and Eriksson (2011) with properties according to A40G (2015) is used since continuity and differentiability of MVEM makes it more appropriate for optimal control applications. Indicated torque Ti for the six cylinder diesel engine and engine friction torque Tf r are calculated as: 6 × 10−6 ηig qlhv uf (2a) Ti = 4π 2 Tf r = cf 1 ω e + c f 2 ωe + c f 3 (2b) where ηig is combustion efficiency, qlhv is specific heating value for diesel fuel and cf i are friction model coefficients. The diesel engine model has only one state variable, engine speed ωe , with the dynamics calculated as follows: 1 dωe = (Ti − Tf r − Tp ) (3) dt Je + Jtc,p where the inertia of the pumping side of the torque converter Jtc,p is added to the engine inertia Je , and Tp is the required torque on the pumping side of torque converter described in the following section. 2.2 Torque converter In torque converter modeling, the focus is more on the continuity and differentiability of the model than the dynamics of the torque converter components. Therefore the typical approach as in Haj-Fraj and Pfeiffer (2002) using torque converter characteristic curves, ξ and M P1000 , is implemented.
400
2.4
300
1.8
200
1.2
100
0.6
0 0
0.2
0.4
0.6
0.8
R
R
F1R1
F1R1
ξ (−)
MP1000 (Nm)
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C
S
F1S1
1
Fig. 4. Torque converter characteristic curves. Using the characteristic curves, illustrated in Figure 4, the required torque on pumping side and generated torque on turbine side of the torque converter are calculated as follows: ωe 2 ) (4) Tp = M P1000 (φ) ( 1000 (5) Tt = M P1000 (φ) ξ(φ) where φ=ωt /ωe and ωt is the turbine shaft speed connected to the transmission input and ωe in equation (4) is in (rpm).
C
2
F2S2
F3S3
F4S4
S 3
Tt
F 5R 5 F 4R 4
R
Tf C
F4S4
F3S3
S In
F 3R 3
R
S
F5S5
4
S
F5R5 TK3 F5S5
TK3 TB5
5
Out
where ωin = ωt = ωS3S4 and ωout = ωC5 . Performing similar calculations for all nine gears, the gear ratios γi as function of αi are obtained as follows: γ2 = γ3 = γ4 =
The transmission unit is comprised of five planetary gearsets, three clutches and five brakes illustrated in Figure 3. The brakes are used to fix a component to the casing (zeros speed) while clutches connect two rotating components with each other. Different gear ratios can be obtained by activating shift elements according to Table 1. There are three widely used methods for analysis of automatic transmissions, Shushan and Peng (2013), namely algebraic equation method, lever analogy and matrix methods. While the first two are handy for transmissions with fewer (one or two) number of planetary gearsets, the matrix methods is prioritized for larger transmissions and is also utilized here. In the next two sections it is first described how complying dimensions of sun and ring components, required in matrix method, are calculated and then using the matrix method, dynamics of the transmission components are calculated.
γ5 =
287
C
F 4R 4
Fig. 5. Graphical representation of exerted torques on transmission components in a 1-2 gearshift.
2.3 Transmission
Gearbox model parametrization Dimensions of planetary gearsets are required for calculation of transmission dynamics, however, only the total gear ratio of the transmission at each gear γi , individual planetary gearset properties such as ratios and dimensions are in many cases not publicly available. Therefore it is necessary to calculate the complying gearset properties such that correct total gear ratio can be obtained after solving transmission dynamics. The kinematic constraint due to the mechanical structure of each planetary gearset is: (6) ωC (1 + αi ) = ωS αi + ωR where αi is the ratio between sun and ring radius in gearset i. Considering equation (6) and the mechanical links between the gearset components illustrated in Figure 3, the total gear ratios can be calculated as the ratio between input and output speeds of the transmission. For example in case of the first gear the gear ratio (γ) calculations is: α4 ωin (7a) ωC4 = ωR5 , ωR4 = 0 ⇒ ωR5 = 1 + α4 (7b) ωS5 = 0 ⇒ ωout (α5 + 1) = ωR5 ωin (1 + α4 )(1 + α5 ) ⇒ = = γ1 (7c) ωout α4
R TB4
F2S2
F1S1
φ (−)
F2R2 F3R3
F 2R 2
C
0 1
287
γ6 =
(1 + α4 ) α4 (1 + α3 )(1 + α4 )(1 + α5 ) α4 + α3 α4 + α3 (1 + α3 )(1 + α4 ) α4 + α3 α4 + α3 α4 + 1 α3 + α 2 α 3 , g1 = α4 + g1 α2 + α2 α3 + α3 α4 + 1 α3 , g2 = 1 α4 + g2 1 + α3 − 1+α1+α 2 +α1 α2
γ7 = 1
α3 (1 + α4 )(1 + α5 ) γ8 = (α3 + α4 + α3 α4 )g3 − α4 (1 + α1 ) γ8 γ9 = , g 3 = 1 + α1 + α 1 α 2 1 + α5
(8a) (8b) (8c) (8d) (8e) (8f) (8g) (8h)
Using the gear ratios, γi and equation (7c)-(8h) the unknown αi can be calculated. Knowing that the transmission is almost one meter long and using this as a scale, the radii of sun gears (Si ) are read from Figure 1 and then the radii of the ring gears (Ri ) are calculated from Ri = Si /αi . Applying the matrix method Mechanical links between gearset components are shown in Figure 3. In order to reduce the required number of state variables for describing all component dynamics, the linked components of different gearsets namely sun(S)-ring(R)-carrier(C) are considered as a single body resulting in the following inertias: J1 : S1 J 5 : C4 R5
J2 : R1 C2 R3 J 3 : C1 S2 J6 : S3 S4 + JT C,t J7 : C5
J4 : R2 C3 R4 J8 : S5
where Jtc,t is the inertia of torque converter on the turbine side, the output shaft inertia is lumped into J7 and the inertia of clutch and brake components are neglected. A schematic view of the transmission components including applied torques from active shift elements during first and second gear operation together with the input and output torque of the transmission is depicted in Figure 5 where Si and Ri are the sun and ring radii and Fi represents the internal force in each planetary gearset. Considering Newton’s second law for all eight rotating bodies of the transmission, the dynamics are obtained as follows:
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J1 J2 J3 J4 J5 J6 J7 J8
dω1 dt dω2 dt dω3 dt dω4 dt dω5 dt dω6 dt dω7 dt dω8 dt
− F1 S1 = 0
(9)
− F1 R1 + F2 (S2 + R2 ) + F3 R3 = 0
(10)
+ F1 (S1 + R1 ) − F2 S2 = 0
(11)
− F2 R2 − F3 (S3 + R3 ) + F4 R4 = TB4
(12)
− F4 (S4 + R4 ) + F5 R5 = 0
(13)
+ F3 S3 + F4 S4 = Tt
(14)
− F5 (S5 + R5 ) = −TK3 − Tf
(15)
+ F5 S5 = TK3 + TB5
(16)
The dynamic equations can be written in the following matrix form:
I K κ Ω= T 0 KT 0
The required TB4 can then be calculated from equation (12) after obtaining F1,2,3 from equation (18) which can be useful for estimation of required torque bandwidth in brake 4. Therefore TB4 is not used as a control inputs during the 1-2 up-shift and with same reasoning the control inputs during 4-5 up-shift are reduced to {uf , TB5 , TB3 } by omitting TK3 . 2.4 Drive shaft flexibility and Longitudinal Dynamics According to Eriksson and Nielsen (2014)-chapter 14, the main flexibility of driveline is located in the drive shaft connecting the final drive to the wheels. Therefore the drive shaft is modeled as a damped flexibility with stiffness k and damping coefficient of c with the output and input torques Tw and Tf as follows: Tw = k xf lex + c ( Tf =
(17)
Tw γf d
ω7 − ωw ) γf d
(19) (20)
where γf d is the final drive ratio and xf lex is the state variable describing the drive shaft torsion which can be T with I = diag(J1 , J2 , · · · , J8 ), Ω = [ω˙ 1 , . . . , ω˙ 8 , F1 , . . . , F5 ] , obtained as follows: ω7 dxf lex = − ωw (21) −S1 0 0 0 0 dt γ fd R3 0 0 −R1 R2 + S2 R + S The wheel speed ωw is the last state variable in the com−S2 0 0 0 1 1 0 plete driveline model calculated from the vehicle longitu−R −R − S R 0 2 3 3 4 , K= dinal dynamics for level road condition according to: 0 0 0 −R − S R 4 4 5 1 1 dωw 0 0 S3 S4 0 2 3 = T w − ρ c d A ωw rw − m g cr rw (22) 2 0 0 0 0 −S5 − R5 dt m rw + Jw 2 0 0 0 0 S5 where rw , m and A are wheel radius,, vehicle mass, and vehicle frontal area, cr and cd are rolling resistance and 00 0 0 0 aerodynamic drag coefficients and g represent the earth 0 0 0 0 0 Tt gravity. 0 0 0 0 0 T B4 0 1 0 0 0 3. OPTIMAL CONTROL PROBLEM FORMULATION TK3 . κ= 0 0 0 0 0 and T = T B5 1 0 0 0 0 Using the developed state space model, optimal conTf 0 0 −1 0 −1 trol problems are formulated and solved for optimization 00 1 1 0 of gearshift transients. There are several properties of where Tf is the transferred torque from the transmission gearshift which can be considered as the optimization to the final drive and is calculated in the next section. objective, see Haj-Fraj and Pfeiffer (2002) and Haj-Fraj Rearranging equation (17) into (18), the speed dynamics and Pfeiffer (2001) for a discussion. Here shift duration (T ) of every rotating component and the internal forces in each corresponding to the operating life of shift elements and changes in the vehicle acceleration (jerk) corresponding planetary gearset can be calculated from the following: to the passenger comfort, are focused. The shift duration as one of the objectives in the optimal control analysis is −1 I K κ T (18) represented by: Ω= t 0 KT 0 T = dt (23) 0 The dynamic equations of the transmission components where t denote the duration of the gearshift. The jerk, A, i can be simplified during stationary operations ( dω dt = 0) is represented by integrating the squared derivatives of the and also when a component is fixed by means of shift vehicle accelerations during the gearshift as follows: elements to another component or to the transmission t da housing such that it has same dynamics as another com( )2 dt (24) A= ponent or is stationary. For example during 1-2 up-shift 0 dt equation (12) can be removed from dynamic equations where a = dωw dt . since TB4 fixes R2 C2 R1 to the transmission housing and 4 To obtain the trade-off between min T and min A soluω4 = dω dt = 0. The transmission dynamics can then be calculated by removing J4 from I, ω˙ 4 from Ω, fourth row tions, the optimal control problem objective is often forof matrix K, second column of κ and TB4 from T vector. mulated as the weighted sum of the time and jerk terms 288
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Considering the 1-2 up-shift, state vector x can be summarized as: x = {ω1,2,3,5,6,7,8 , ωe , ωw , xf lex }
(25)
while the state dynamics f (x(t), u(t)) are obtained by the differential equations presented in the previous sections. The complete optimal control problem formulation for different scenarios stated earlier can be summarized as follows: min
x(t),u(t),γ(t)
T or A
4. OPTIMAL CONTROL RESULTS Gearshifting transients are optimized for 1-2 and 4-5 upshifts where in both cases there are one off-going and one in-coming shift element. The major phenomena observed in the calculated optimal transients are described in the following sections. 4.1 Trade-off between jerk and time Figure 6 shows the trade-off between min T and min A solutions for both gearshift cases where the dashed-lines are obtained by interpolation between the calculated points. The trade-offs show that small increase of gearshift duration from the time optimal solution can largely reduce the jerk. For example, at C12 and C45 a good compromise is achieved between the two objectives as the gearshift duration in the 1-2 and 4-5 up-shifts has increased only 0.22 % and 4 % while the jerk has reduced as much as 64.7 % and 55.9 % respectively. However, the gain in the jerk reduction reduces when the gearshift time increases further. 1−2 up−shift 3
Another approach is used here where after solving the min T and min A solutions as the start and end points on the trade-off, a time grid is selected on the interval between these two solutions. Then, the gearshift duration t is repeatedly set equal to the grid times and equation (24) is solved to obtain the A values. The first point on the time grid is obtained by solving the min T problem, but there is no unique solution in time when solving the min A problem (v1 = 0) since after a certain gearshift duration, zero jerk can be obtained for various gearshift lengths. Therefore, to obtain the shortest time where zero jerk gearshift is possible, equation (23) is chosen as the objective and the problem is solved including an additional a = 0 constraint in the optimal control problem formulation. The calculated gearshift duration is then used as the end point of the time grid.
Implementing direct multiple shooting method, the formulated optimal control problem is transformed into a nonlinear program (NLP). The NLP is then solved by CasADi, Andersson (2013), which is an open source optimal control framework using Ipopt, W¨achter and Biegler (2006), and the powertrain transients are obtained. The results of this optimization are presented and analyzed in the following section.
A (rad/s )
(v1 T + v2 A, v1 + v2 = 1) where the points on the tradeoff are calculated by solving with different weights (v1,2 ). However, considering the large difference in the order of magnitude between the two objectives, when formulating the weighted sum, A and J have to be normalized with respect to their maximums. This makes the solution of the weighted sum sensitive to the normalization and weight values, which makes it difficult to calculate the trade-off with an acceptable spread of points.
10
C12
5 0 1.32
(26)
x(t) ˙ = f (x(t), u(t)) umin ≤ ui (t) ≤ umax xmin ≤ xi (t) ≤ xmax Te (ωe , uf (t)) ≤ Te,max x(0) = x0 x(T ) = xT x(0) ˙ =0 x(T ˙ )=0 a = 0, when solving for min T |A=0
289
1.33
1.34
s.t.
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A (rad/s3)
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20
45
10 0 0.55
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T (s)
0.7
0.75
0.8
Fig. 6. The trade-off between time and jerk for 1-2 and 4-5 up-shifts. At C12,45 there is a compromise between the two objectives.
where the min and max are the upper and lower limits for states and controls and x0 and xT correspond to the state values at the end of the first gear and beginning of the second gear operation, respectively. These are calculated in similar manner as described by equations (7a)-(7c) for rw ωw =6 km/h and rw ωw =15 km/h in 1-2 and 4-5 upshifts respectively. The requirements on x˙ at the beginning and end of the gearshift are applied to guaranty that the gearshift starts and ends at stationary condition. It should be noted that the kinematic constraint mentioned in equation (6) does not need to be added to problem constraints as their effect is already present in the dynamic equations of states via internal forces Fi . 289
4.2 Optimal transients for 1-2 and 4-5 up-shifts The min T , min A and C12,45 state and control input transients for 1-2 and 4-5 up-shifts are illustrated in Figures 7, 8, 9 and 10. According to Figures 7 and 9 it is both time and jerk optimal to release the offgoing shift element, ub5 and ub3 for 1-2 and 4-5 cases respectively, as soon as the gearshifts starts. Also in the min A case, it is optimal to reach the maximum torque capacity of the in-coming shift element, uk3 and ub1 for 12 and 4-5 up-shifts, immediately after gearshift starts such that the initial torque phase is completely avoided. This requirement for fast actuation of in-coming shift elements
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45
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150
ω (rad/s)
150 0 300 200 100 0 0 0.4
0.6
ice
200
0.5
Fig. 9. Optimal control transients for min T , min A and C45 cases in a 4-5 up-shift .
T (s)
Fig. 8. Optimal state transients for min T , min A and C12 cases in a 1-2 up-shift where ωout =ω7 and ωin =ω6 . can be considered when designing the hydraulic actuation system. Considering C12,45 transients, the magnitude of disturbances in the vehicle speed, ωw , is reduced compared to the min T solution especially in case of the 1-2 up-shift which can also be verified comparing the dωw /dt values in Figure 11. The required power in b1 and k3 shift elements T T calculated as 0 ub1 ω2 dt and 0 |uk3 ω8 | dt is also 8.8 % and 12.11 %, respectively, lower than the min T case. Figure 11-top shows the input torque to the transmission. The transients are similar for both up-shift cases in the min A solutions in a sense that the fuel injection umf is cut off so that the absence of an input torque on the engine side of the torque converter together with the constant torque from the in-coming shift element, smoothly reduce the kinetic energy of the transmission components down to the level required for the engagement of the next gear. However, in the min T case of both up-shifts, in addition to the fuel cut-off, the bang-bang type dynamics 290
xflex (rad)
xflex (rad)
ω8 (rad/s) ωice (rad/s) ωin (rad/s) ω (rad/s) ωout (rad/s) w
Fig. 7. Optimal control transients for min T , min A and C12 cases in a 1-2 up-shift.
0.4 T (s)
0 100 50 0 −50 0.40 0.2 0 0
Fig. 10. Optimal state transients for min T , min A and C12 cases in a 4-5 up-shift where ωout =ω7 and ωin =ω6 . of the in-coming shift elements introduce disturbances into the driveline such that the induced excitation is transferred to the vehicle mass at wheels via the the drive shaft. These excitations get partially damped by the drive shaft flexibility, vehicle mass and wheel inertia which increases the rate of reduction in the kinetic energy of the transmission components and shortens the time before the next gear can be engaged. Subtracting the total kinetic energy of all transmission components (0.5 Ii ωi2 ) at the beginning and end of the gearshifts, the difference is 25.17 % larger in case of the 1-2 up-shift which could be the main reason for the slower transients compared to the 4-5 case as more reduction in the kinetic energy is required before the next gear can be engaged.
0.05
t
T (Nm)
600 400 200
dωw/dt (rad/s )
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Fig. 11. Transmission input torque and vehicle acceleration/decceleration in min T , min A and C45 solutions of 1-2 and 4-5 up-shifts. 5. CONCLUSION A framework is developed for gearshift transient optimization during inertia phase via state space modeling of a nine speed heavy duty automatic transmission. Using the developed model and in order to analyzed the minimum time/jerk transients and the trade-off between these, optimal control problems are formulated and solved. Two example up-shifts are considered and in order to calculate the trade-off between time and jerk objectives while avoiding objective function normalizations difficulties, the minimum jerk problem is iteratively solved for various preselected gearshift durations. The results show that the developed framework is applicable for efficient optimization of inertia phase gearshift transients. As future model developments, hydraulic actuation dynamics can be included and sensitivity of the gearshift transients with respect to the parameters such as clutch fill dynamics, components’ inertia and driveline flexibilities can be analyzed. REFERENCES A40G (2015). Volvo A40G hauler, ”http://www. volvoce.com/SiteCollectionDocuments/VCE/ Documents%20Global/articulated%20haulers/ Brochure_A35GFS_A40GFS_SV_12_20040745_B_2015. 02.pdf”, Accessed July. 2015. Andersson, J. (2013). A General-Purpose Software Framework for Dynamic Optimization. PhD thesis, Arenberg Doctoral School, KU Leuven, Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center, Kasteelpark Arenberg 10, 3001Heverlee, Belgium. Eriksson, L. and Nielsen, L. (2014). Modeling and Control of Engines and Drivelines. John Wiley & Sons. Gao, B., Chen, H., Zhao, H., and Sanada, K. (2010). A reduced-order nonlinear clutch pressure observer for automatic transmission. Control Systems Technology, IEEE Transactions on, 18(2), 446–453. Goetz, M., Levesley, M., and Crolla, D. (2005). Dynamics and control of gearshifts on twin-clutch transmissions. 291
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Proceedings of the institution of mechanical engineers, Part D: Journal of Automobile Engineering, 219(8), 951–963. Greiner, J., Doerr, C., Nauerz, H., and Graeve, M. (2004). The new 7g-tronic of mercedes-benz: Innovative transmission technology for better driving performance, comfort and fuel economy. SAE Technical Paper 2004-010649. Haj-Fraj, A. and Pfeiffer, F. (2001). Optimal control of gear shift operations in automatic transmissions. Journal of the Franklin Institute, 338(2), 371–390. Haj-Fraj, A. and Pfeiffer, F. (2002). A model based approach for the optimisation of gearshifting in automatic transmissions. International journal of vehicle design, 28(1), 171–188. Han, W. and Yi, S.J. (2003). A study of shift control using the clutch pressure pattern in automatic transmission. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 217(4), 289–298. Horste, K. (1995). Objective measurement of automatic transmission shift feel using vibration dose value. SAE Technical Paper 951373. Huang, Q. and Wang, H. (2004). Fundamental study of jerk: evaluation of shift quality and ride comfort. SAE Technical Paper 2004-01-2065. Kim, D.H., Yang, K.J., Hong, K.S., Hahn, J.O., and Lee, K.I. (2003). Smooth shift control of automatic transmissions using a robust adaptive scheme with intelligent supervision. International journal of vehicle design, 32(3), 250–272. Minowa, T., Ochi, T., Kuroiwa, H., and Liu, K.Z. (1999). Smooth gear shift control technology for clutch-to-clutch shifting. SAE Technical Paper 1999-01-1054. Shushan, Bai, M.J. and Peng, H. (eds.) (2013). Dynamic Analysis and Control System Design of Automatic Transmissions. SAE International. Sun, Z. and Hebbale, K. (2005). Challenges and opportunities in automotive transmission control. In American Control Conference, 2005. Proceedings of the 2005, 3284–3289. IEEE. Thornton, S., Pietron, G.M., Yanakiev, D., McCallum, J., and Annaswamy, A. (2013). Hydraulic clutch modeling for automotive control. In Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, 2828–2833. IEEE. W¨achter, A. and Biegler, L.T. (2006). On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical programming, 106(1), 25–57. Wagner, G. (2001). Application of transmission systems for different driveline configurations in passenger cars. SAE Technical Paper 2001-01-0882. Walstr¨om, J. and Eriksson, L. (2011). Modeling engines with a variable-geometry turbocharger and exhaust gas recirculation by optimization of model parameters for capturing non-linear system dynamics. Proceedings of the Institution of Mechanical Engineers, Part D, Journal of Automobile Engineering, 960–986.
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IFAC-PapersOnLine 48-15 (2015) 285–291
A A framework framework for for modeling modeling and and optimal optimal A framework for modeling and optimal control of automatic transmission control of automatic transmission systems systems control of automatic ∗transmission systems ∗
V. Nezhadali ∗ L. Eriksson ∗ V. Nezhadali ∗∗ L. Eriksson ∗∗ V. V. Nezhadali Nezhadali L. L. Eriksson Eriksson ∗ Electrical Engineering Department, Link¨ oping University, SE-581 83 ∗ ∗ Electrical Engineering Department, Link¨ o ping lars.eriksson}@liu.se) University, SE-581 83 Electrical Engineering Department, Link¨ o University, ∗Link¨ o ping, Sweden (e-mail: {vaheed.nezhadali, Electrical Engineering Department, Link¨ oping ping lars.eriksson}@liu.se) University, SE-581 SE-581 83 83 Link¨ o ping, Sweden (e-mail: {vaheed.nezhadali, Link¨ o ping, Sweden (e-mail: {vaheed.nezhadali, lars.eriksson}@liu.se) Link¨ oping, Sweden (e-mail: {vaheed.nezhadali, lars.eriksson}@liu.se) Abstract: Development of efficient control algorithms for the control of automatic transmission Abstract: of efficient control algorithms for the control oflife automatic Abstract: Development of control for control automatic transmission systems is Development crucial to maintain passenger comfort and operational of the transmission Abstract: Development of efficient efficient control algorithms algorithms for the the control of oflife automatic transmission systems is crucial to maintain passenger comfort and operational of the transmission systems is crucial to maintain passenger comfort and operational life of the transmission components. An optimization framework is comfort developedand by state space modeling of atransmission powertrain systems is crucial to maintain passengeris operational life of the components. An optimization framework developed by state space modeling of a components. An optimization framework is developed state space of aa powertrain powertrain including a nine speed automatic transmission, dieselby engine, torquemodeling converter and a model components. An optimization framework is developed by state space modeling of powertrain including a nine speed automatic transmission, diesel engine, torque converter and aa model including a nine speed automatic transmission, diesel engine, torque converter and for longitudinal vehicle dynamics considering drive shaft as the only flexibility of the driveline. including a ninevehicle speed dynamics automaticconsidering transmission, diesel engine, torque converter and driveline. a model model for longitudinal drive shaft as the only flexibility of the for longitudinal vehicle dynamics considering drive shaft as the only flexibility of the driveline. Emphasis is set on the kinematics of the automatic transmission with the aim of modeling for for longitudinal vehicle dynamics considering drive shaft as the only flexibility of the driveline. Emphasis is set on the kinematics of the automatic transmission with the aim of modeling for Emphasis is set on the kinematics of the automatic transmission with the aim of modeling for gearshift optimal control during the inertia phase. Considering the interacting forces between Emphasis is set on the kinematics of the automatic transmission with the aim of modeling for gearshift during the inertia phase. Considering the interacting between gearshift optimal control during the phase. Considering the forces between planetaryoptimal gearsets,control clutches and brakes in the transmission, kinematic equationsforces of motion are gearshift optimal control during the inertia inertia phase. Considering the interacting interacting forces between planetary gearsets, clutches and brakes in the transmission, kinematic equations of motion are planetary gearsets, clutches and brakes in the transmission, kinematic equations of motion are derived for rotating transmission components enabling to calculate both transmission dynamics planetary gearsets, clutches and brakes in the transmission, kinematic equations of motion are derived for rotating to calculate both transmission dynamics derived for transmission components enabling to calculate both dynamics and internal forces. transmission The model iscomponents then used enabling in optimal control problem formulations for the derived for rotating rotating transmission components enabling to control calculateproblem both transmission transmission dynamics and internal forces. The model is then used in optimal formulations for the and internal forces. The is used in control analysis of optimal control transients in two up-shift cases. and internal forces.control The model model is then then usedup-shift in optimal optimal control problem problem formulations formulations for for the the analysis of optimal transients in two cases. analysis of optimal control transients in two up-shift cases. analysis of optimal control transients in two up-shift cases. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Automatic transmission, optimal control, modeling and simulation framework Keywords: Keywords: Automatic Automatic transmission, transmission, optimal optimal control, control, modeling modeling and and simulation simulation framework framework Keywords: Automatic transmission, optimal control, modeling and simulation framework 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION Various driveline configurations are utilized in todays cars Various driveline utilized cars Various driveline configurations configurations are utilized inoftodays todays cars while Automatic Transmission are (AT) is onein the comVarious driveline configurations are utilized in todays cars while Automatic Transmission (AT) is one of the comwhile Automatic Transmission (AT) is one of the commonly used systems, Wagner (2001). Such transmission while Automatic Transmission (AT) isSuch one of the commonly systems, Wagner (2001). monly used systems, Wagnergearsets (2001). where Such transmission transmission systemsused consist of planetary various gear monly used systems, Wagner (2001). Such transmission systems consist of planetary gearsets where various gear gear systems consist of planetary gearsets where various ratios are achieved by coupling different components of systems consist of planetary gearsets where various gear ratios are achieved by coupling different components of ratios are achieved by coupling different components of planetary gearsets with each other. The coupling takes ratios are gearsets achieved by coupling different components of planetary each other. The takes planetary gearsets with with each other. clutches The coupling coupling takes place via engagement of oil immersed and brakes, planetary gearsets with each other. The coupling takes place via engagement of oil immersed clutches and brakes, place viatoengagement engagement of oil oil immersed immersed clutches and brakes, brakes, referred as shift elements, at the time of gearshift. Using place via of clutches and referred to as shift elements, at the time of gearshift. Using referred to as shift elements, at the time of gearshift. Using combinations of planetary gearsets in the transmission, referred to as shift elements, gearsets at the time of gearshift. Using combinations of in transmission, combinations of planetary planetary gearsets in the thewhile transmission, wide range of gear ratios become available the comcombinations of planetary gearsets in the transmission, wide range of gear ratios become available while the wide range of of gear gear ratios become available available while the comcompact structure of such transmission makes it favorable for wide range ratios become while the compact structure of such transmission makes it favorable for pact structure of such transmission makes it favorable for application in both normal passenger cars, Greiner et al. pact structure of such transmission makes itGreiner favorable for application in both normal passenger cars, et al. application in both normal passenger cars, Greiner et al. (2004), and heavy duty machinery, A40G (2015). application in bothduty normal passenger cars, Greiner et al. (2004), (2004), and and heavy heavy duty duty machinery, machinery, A40G A40G (2015). (2015). (2004), and machinery, A40G (2015). However, asheavy power is transferred through transmission However, as power is transferred through transmission However, as power is transferred through transmission components during gearshifts, the difference between inerHowever, asduring powergearshifts, is transferred through between transmission components difference inercomponents duringspeed gearshifts, the difference between inertia and rotational of thethe coupled bodies introduces components during gearshifts, the difference between inertia and speed of bodies introduces tia and rotational rotational speed of the the coupled coupled bodiesApart introduces discontinuities in the powertrain dynamics. from tia and rotational speed of the coupled bodies introduces discontinuities in the powertrain dynamics. Apart from discontinuities in the powertrain dynamics. Apart from the long-term destructive impacts of the induced oscilladiscontinuities in the powertrain dynamics. Apart from the destructive impacts of the long-term destructive impacts of the the induced induced oscillationslong-term on powertrain components, passenger comfortoscillais also the long-term destructive impacts of the induced oscillations on powertrain components, passenger comfort is tions on powertrain components, passenger comfort is also also greatly dependent oncomponents, the gearshiftpassenger smoothness, see Huang tions on powertrain comfort is also greatly dependent on the gearshift smoothness, see Huang greatly dependent on the gearshift smoothness, see Huang and Wang (2004) and Horste (1995). greatly dependent and on the gearshift smoothness, see Huang and and Wang Wang (2004) (2004) and and Horste Horste (1995). (1995). and Wang (2004) Horste (1995). Simultaneous control of off-going and on-going shift eleSimultaneous of off-going shift Simultaneous control of determines off-going and and on-going shift eleelements during control gearshifts theon-going shift characterisSimultaneous control of off-going and on-going shift elements during gearshifts determines the shift characterisments during gearshifts determines the shift characteristic which in its turn has direct effects on other vehicle ments during its gearshifts determines the shift characteristic which turn has direct on tic which in in its as turn has comfort, direct effects effects on other other vehicle vehicle properties such drive fuel efficiency, tic which in its turn has direct effects on other vehicle properties such as drive comfort, fuel efficiency, vehicle properties such as drive comfort, fuel efficiency, vehicle operability and durability of gearbox components. In addiproperties such as drive of comfort, fuel efficiency, Invehicle operability and durability gearbox components. addioperability and durability of gearbox components. In addition to the improvement of vehicle properties, proper AT operability and durabilityof of vehicle gearboxproperties, components. In addition to the improvement proper AT tion to the improvement of vehicle properties, proper AT control strategies also open up the design space for new tion to the improvement of vehicle properties, proper AT control strategies also open up design for control strategies also with openlesser up the the design space space for new new transmission concepts components resulting in control strategies also open up the design space for new transmission concepts with lesser components resulting in transmission concepts with lesser components resulting in lower total vehicle mass and consequently fuelresulting consumptransmission concepts with lesser components in lower total lower total vehicle vehicle mass mass and and consequently consequently fuel fuel consumpconsumption reduction. lower total vehicle mass and consequently fuel consumption reduction. tion reduction. tion reduction.
Fig. 1. Cutaway drawing of a nine speed automatic gearbox Fig. Cutaway of Fig. 1. 1. Cutaway drawing of aaa nine nine speed speed automatic automatic gearbox gearbox and torque drawing converter. Fig. 1. Cutaway drawing of nine speed automatic gearbox and torque converter. and torque converter. and torque converter. Extensive research has been carried out on the topic of AT Extensive has been carried topic of Extensive research has and beenoperating carried out out onofthe the topic of AT AT control forresearch ride quality lifeon shift elements, Extensive research has been carried out on the topic of AT control for ride quality and operating life of shift elements, control for ride quality and operating life of shift elements, see Goetz et al. (2005), Sun and Hebbale (2005), Han and control for ride quality and operating life of shift elements, see Goetz al. and Hebbale (2005), and see (2003), Goetz et et al. (2005), (2005), Sun and and Hebbale (2005), Han and Yi Minowa et al.Sun (1999) Haj-Fraj andHan Pfeiffer see Goetz et al. (2005), Sun and Hebbale (2005), Han and Yi (2003), Minowa et al. (1999) and Haj-Fraj and Pfeiffer Yi (2003), Minowa et al. (1999) and Haj-Fraj and Pfeiffer (2002), concluding that the main criteria for the evaluation Yi (2003), Minowa that et al.the (1999) and Haj-Fraj and Pfeiffer (2002), main the evaluation (2002), concluding that the main criteria criteria for theprocess evaluation of shift concluding quality arethat the the duration of gear for shift and (2002), concluding main criteria for the evaluation of shift quality are the duration of gear shift process and of shift quality are the duration of gear shift process and emerging oscillations in the powertrain. An efficient of shift quality are theinduration of gear shift process and and emerging oscillations the powertrain. An efficient emerging oscillations in the powertrain. An efficient and cost effective approach obtain suitable strategies for emerging oscillations in to theobtain powertrain. Anstrategies efficient and cost effective suitable for cost control effectiveis approach approach tomodels obtainwith suitable strategies for AT to utilize to a time dependent cost effective approach to obtain suitable strategies for AT control is to utilize models with aa time dependent AT control is to utilize models with time dependent structure where the dynamics of different components are AT control is tothe utilize models with a time dependent structure where of are structure where during the dynamics dynamics of different different components are well described gearshift. Previouscomponents AT modeling structure where the dynamics of different components are well described during gearshift. Previous AT modeling well described during gearshift. Previous AT modeling effortsdescribed can be classified two different categories. In the well during into gearshift. Previous AT modeling efforts can be two categories. In efforts can be classified classified into two different different categories. In the the first the focus is on theinto dynamics of hydraulic actuation efforts can be classified into two different categories. In the first the focus is on the dynamics of hydraulic actuation first the focus is on the dynamics of hydraulic actuation system initiated by the change of actuator lever position first the focus is by on the change dynamics actuator of hydraulic actuation system lever system initiated initiated by the the change of of pressure actuator at lever position ending in the rise of hydraulic theposition brakes system initiated by the change of actuator lever position ending in the rise of hydraulic pressure at the brakes ending in the rise of hydraulic pressure at the brakes and clutches, see Minowa et al. (1999), Thornton et al. ending in the rise of hydraulic pressure at the brakes and clutches, see Minowa et al. (1999), Thornton et and clutches, see Minowa et al. (1999), Thornton et al. al. (2013) and Gao et al. (2010). In the second category, and clutches, see et Minowa et al. In (1999), Thornton et al. (2013) and Gao al. (2010). the second category, (2013) and Gao et al. (2010). In the second category, the kinematics of planetary gearsets before, during and (2013) and Gaoof etplanetary al. (2010). In thebefore, secondduring category, the gearsets and the kinematics kinematics of of planetary gearsets before, during and after gearshifts are interest while the before, hydraulic pressures the kinematics of planetary gearsets during and after gearshifts are of interest while the hydraulic pressures after gearshifts are of interest while the hydraulic pressures are considered as control input signals, see Haj-Fraj and after gearshifts are of interest while the hydraulic pressures are as signals, are considered considered as control control input signals, see see Haj-Fraj Haj-Fraj and and Pfeiffer (2002) and Kim etinput al. (2003). are considered as control input signals, see Haj-Fraj and Pfeiffer (2002) and Kim et al. (2003). Pfeiffer (2002) and Kim et al. (2003). Pfeiffer (2002) and Kim et al. (2003).
Copyright © 2015, 2015 IFAC 285 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 285 Copyright ©under 2015 responsibility IFAC 285Control. Peer review of International Federation of Automatic Copyright © 2015 IFAC 285 10.1016/j.ifacol.2015.10.041
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uf Engine
u1-2 = {uf , TB5 , TK3 } (1) where uf , fuel mass injected per combustion cycle is the control input to the diesel engine model, and TB5 and TK3 are the torques from brake 5 and clutch 3.
TKi TBi Te Tp
Tt
Torque Converter
Tf Transmission
Tw Final drive Drive shaft
B1
Fig. 2. Schematic view of the powertrain and driveline components with focus on transmission inputs and outputs. In this work, the aim is to develop a framework enabling optimal control analysis of desirable AT gearshifts during the inertia phase. A nine speed AT as shown in Figure 1 comprised of a torque converter, five planetary gearsets and eight shift elements will be considered. Analyzing the kinematics of rotating bodies and calculating the internal forces/torques between various components, transmission dynamics at different gears are obtained as a state space system which is utilized for optimal control problem formulation. Then the AT transients during the inertia phase are calculated and analyzed for two example gearshifts to illustrate the framework applicability in different gearshifting scenarios. 2. DRIVELINE MODEL Figure 2 shows a schematic overview of the developed powertrain model where a diesel engine generates the required power transferred to the wheels via torque converter, transmission, final drive and a drive shaft. Considering the large number of AT components and the corresponding state variables, which will be later introduced, in order to reduce the number of required state variables for the complete powertrain kinematics and for simplicity, the following assumptions and simplification are made during powertrain modeling: (1) The turbocharger lag and engine pumping losses are neglected in diesel engine model assuming that the engine has already reached high speed when a gearshift starts. (2) All components except the drive shaft are considered without torsional damping or flexibility. (3) Friction and viscous losses in the planetary gearsets are neglected. (4) Component properties are chosen according to the information available in A40G (2015). (5) Perfect clutch fill and coordination is assumed. Only the inertia phase of the gearshift is considered and therefore, the transmission actuation system is modeled such that the shift element torques are directly applied as control inputs. Engine speed ωe , rotational speed of the transmission components ω1,...,8 , torsion of drive shaft x and wheel speed ωw are the state variables of the model which are described by equations (3), (9)-(16),(21) and (22). Considering actuation torques from all shift elements, TB1,2,3,4,5 and TK1,2,3 as depicted in Figure 2, there are eight choices for the transmission model control inputs depending on the selected gear. The kinematic analysis is described for 1-2 up-shift only but similar methodology can be followed for other gearshifts. Therefore the control inputs (u1-2 ) for 1-2 up-shift modeling will be: 286
K1
B2
K2
In
B3
B4
B5
R
R
R
R
R
C
C
C
C
C
S
S
S
S
S
1
2
3
4
5
K3
Out
Fig. 3. Schematic view of a nine speed automatic gearbox and mechanical links between planetary gearsets components, Sun(S)-Carrier(C)-Ring(R), with respect to clutches and brakes. Gear 1 2 3 4 5 6 7 8 9
K1 × × × × × × ×
K2
K3
B1
B2
×
× × ×
× × × × ×
×
B3
× ×
B4 × ×
B5 × ×
× × ×
×
Table 1. The shift elements (clutches and brakes) that are active at each gear.
2.1 Diesel engine A simplified version of the mean value engine model (MVEM) in Walstr¨om and Eriksson (2011) with properties according to A40G (2015) is used since continuity and differentiability of MVEM makes it more appropriate for optimal control applications. Indicated torque Ti for the six cylinder diesel engine and engine friction torque Tf r are calculated as: 6 × 10−6 ηig qlhv uf (2a) Ti = 4π 2 Tf r = cf 1 ω e + c f 2 ωe + c f 3 (2b) where ηig is combustion efficiency, qlhv is specific heating value for diesel fuel and cf i are friction model coefficients. The diesel engine model has only one state variable, engine speed ωe , with the dynamics calculated as follows: 1 dωe = (Ti − Tf r − Tp ) (3) dt Je + Jtc,p where the inertia of the pumping side of the torque converter Jtc,p is added to the engine inertia Je , and Tp is the required torque on the pumping side of torque converter described in the following section. 2.2 Torque converter In torque converter modeling, the focus is more on the continuity and differentiability of the model than the dynamics of the torque converter components. Therefore the typical approach as in Haj-Fraj and Pfeiffer (2002) using torque converter characteristic curves, ξ and M P1000 , is implemented.
400
2.4
300
1.8
200
1.2
100
0.6
0 0
0.2
0.4
0.6
0.8
R
R
F1R1
F1R1
ξ (−)
MP1000 (Nm)
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C
S
F1S1
1
Fig. 4. Torque converter characteristic curves. Using the characteristic curves, illustrated in Figure 4, the required torque on pumping side and generated torque on turbine side of the torque converter are calculated as follows: ωe 2 ) (4) Tp = M P1000 (φ) ( 1000 (5) Tt = M P1000 (φ) ξ(φ) where φ=ωt /ωe and ωt is the turbine shaft speed connected to the transmission input and ωe in equation (4) is in (rpm).
C
2
F2S2
F3S3
F4S4
S 3
Tt
F 5R 5 F 4R 4
R
Tf C
F4S4
F3S3
S In
F 3R 3
R
S
F5S5
4
S
F5R5 TK3 F5S5
TK3 TB5
5
Out
where ωin = ωt = ωS3S4 and ωout = ωC5 . Performing similar calculations for all nine gears, the gear ratios γi as function of αi are obtained as follows: γ2 = γ3 = γ4 =
The transmission unit is comprised of five planetary gearsets, three clutches and five brakes illustrated in Figure 3. The brakes are used to fix a component to the casing (zeros speed) while clutches connect two rotating components with each other. Different gear ratios can be obtained by activating shift elements according to Table 1. There are three widely used methods for analysis of automatic transmissions, Shushan and Peng (2013), namely algebraic equation method, lever analogy and matrix methods. While the first two are handy for transmissions with fewer (one or two) number of planetary gearsets, the matrix methods is prioritized for larger transmissions and is also utilized here. In the next two sections it is first described how complying dimensions of sun and ring components, required in matrix method, are calculated and then using the matrix method, dynamics of the transmission components are calculated.
γ5 =
287
C
F 4R 4
Fig. 5. Graphical representation of exerted torques on transmission components in a 1-2 gearshift.
2.3 Transmission
Gearbox model parametrization Dimensions of planetary gearsets are required for calculation of transmission dynamics, however, only the total gear ratio of the transmission at each gear γi , individual planetary gearset properties such as ratios and dimensions are in many cases not publicly available. Therefore it is necessary to calculate the complying gearset properties such that correct total gear ratio can be obtained after solving transmission dynamics. The kinematic constraint due to the mechanical structure of each planetary gearset is: (6) ωC (1 + αi ) = ωS αi + ωR where αi is the ratio between sun and ring radius in gearset i. Considering equation (6) and the mechanical links between the gearset components illustrated in Figure 3, the total gear ratios can be calculated as the ratio between input and output speeds of the transmission. For example in case of the first gear the gear ratio (γ) calculations is: α4 ωin (7a) ωC4 = ωR5 , ωR4 = 0 ⇒ ωR5 = 1 + α4 (7b) ωS5 = 0 ⇒ ωout (α5 + 1) = ωR5 ωin (1 + α4 )(1 + α5 ) ⇒ = = γ1 (7c) ωout α4
R TB4
F2S2
F1S1
φ (−)
F2R2 F3R3
F 2R 2
C
0 1
287
γ6 =
(1 + α4 ) α4 (1 + α3 )(1 + α4 )(1 + α5 ) α4 + α3 α4 + α3 (1 + α3 )(1 + α4 ) α4 + α3 α4 + α3 α4 + 1 α3 + α 2 α 3 , g1 = α4 + g1 α2 + α2 α3 + α3 α4 + 1 α3 , g2 = 1 α4 + g2 1 + α3 − 1+α1+α 2 +α1 α2
γ7 = 1
α3 (1 + α4 )(1 + α5 ) γ8 = (α3 + α4 + α3 α4 )g3 − α4 (1 + α1 ) γ8 γ9 = , g 3 = 1 + α1 + α 1 α 2 1 + α5
(8a) (8b) (8c) (8d) (8e) (8f) (8g) (8h)
Using the gear ratios, γi and equation (7c)-(8h) the unknown αi can be calculated. Knowing that the transmission is almost one meter long and using this as a scale, the radii of sun gears (Si ) are read from Figure 1 and then the radii of the ring gears (Ri ) are calculated from Ri = Si /αi . Applying the matrix method Mechanical links between gearset components are shown in Figure 3. In order to reduce the required number of state variables for describing all component dynamics, the linked components of different gearsets namely sun(S)-ring(R)-carrier(C) are considered as a single body resulting in the following inertias: J1 : S1 J 5 : C4 R5
J2 : R1 C2 R3 J 3 : C1 S2 J6 : S3 S4 + JT C,t J7 : C5
J4 : R2 C3 R4 J8 : S5
where Jtc,t is the inertia of torque converter on the turbine side, the output shaft inertia is lumped into J7 and the inertia of clutch and brake components are neglected. A schematic view of the transmission components including applied torques from active shift elements during first and second gear operation together with the input and output torque of the transmission is depicted in Figure 5 where Si and Ri are the sun and ring radii and Fi represents the internal force in each planetary gearset. Considering Newton’s second law for all eight rotating bodies of the transmission, the dynamics are obtained as follows:
IFAC E-COSM 2015 288 August 23-26, 2015. Columbus, OH, USA V. Nezhadali et al. / IFAC-PapersOnLine 48-15 (2015) 285–291
J1 J2 J3 J4 J5 J6 J7 J8
dω1 dt dω2 dt dω3 dt dω4 dt dω5 dt dω6 dt dω7 dt dω8 dt
− F1 S1 = 0
(9)
− F1 R1 + F2 (S2 + R2 ) + F3 R3 = 0
(10)
+ F1 (S1 + R1 ) − F2 S2 = 0
(11)
− F2 R2 − F3 (S3 + R3 ) + F4 R4 = TB4
(12)
− F4 (S4 + R4 ) + F5 R5 = 0
(13)
+ F3 S3 + F4 S4 = Tt
(14)
− F5 (S5 + R5 ) = −TK3 − Tf
(15)
+ F5 S5 = TK3 + TB5
(16)
The dynamic equations can be written in the following matrix form:
I K κ Ω= T 0 KT 0
The required TB4 can then be calculated from equation (12) after obtaining F1,2,3 from equation (18) which can be useful for estimation of required torque bandwidth in brake 4. Therefore TB4 is not used as a control inputs during the 1-2 up-shift and with same reasoning the control inputs during 4-5 up-shift are reduced to {uf , TB5 , TB3 } by omitting TK3 . 2.4 Drive shaft flexibility and Longitudinal Dynamics According to Eriksson and Nielsen (2014)-chapter 14, the main flexibility of driveline is located in the drive shaft connecting the final drive to the wheels. Therefore the drive shaft is modeled as a damped flexibility with stiffness k and damping coefficient of c with the output and input torques Tw and Tf as follows: Tw = k xf lex + c ( Tf =
(17)
Tw γf d
ω7 − ωw ) γf d
(19) (20)
where γf d is the final drive ratio and xf lex is the state variable describing the drive shaft torsion which can be T with I = diag(J1 , J2 , · · · , J8 ), Ω = [ω˙ 1 , . . . , ω˙ 8 , F1 , . . . , F5 ] , obtained as follows: ω7 dxf lex = − ωw (21) −S1 0 0 0 0 dt γ fd R3 0 0 −R1 R2 + S2 R + S The wheel speed ωw is the last state variable in the com−S2 0 0 0 1 1 0 plete driveline model calculated from the vehicle longitu−R −R − S R 0 2 3 3 4 , K= dinal dynamics for level road condition according to: 0 0 0 −R − S R 4 4 5 1 1 dωw 0 0 S3 S4 0 2 3 = T w − ρ c d A ωw rw − m g cr rw (22) 2 0 0 0 0 −S5 − R5 dt m rw + Jw 2 0 0 0 0 S5 where rw , m and A are wheel radius,, vehicle mass, and vehicle frontal area, cr and cd are rolling resistance and 00 0 0 0 aerodynamic drag coefficients and g represent the earth 0 0 0 0 0 Tt gravity. 0 0 0 0 0 T B4 0 1 0 0 0 3. OPTIMAL CONTROL PROBLEM FORMULATION TK3 . κ= 0 0 0 0 0 and T = T B5 1 0 0 0 0 Using the developed state space model, optimal conTf 0 0 −1 0 −1 trol problems are formulated and solved for optimization 00 1 1 0 of gearshift transients. There are several properties of where Tf is the transferred torque from the transmission gearshift which can be considered as the optimization to the final drive and is calculated in the next section. objective, see Haj-Fraj and Pfeiffer (2002) and Haj-Fraj Rearranging equation (17) into (18), the speed dynamics and Pfeiffer (2001) for a discussion. Here shift duration (T ) of every rotating component and the internal forces in each corresponding to the operating life of shift elements and changes in the vehicle acceleration (jerk) corresponding planetary gearset can be calculated from the following: to the passenger comfort, are focused. The shift duration as one of the objectives in the optimal control analysis is −1 I K κ T (18) represented by: Ω= t 0 KT 0 T = dt (23) 0 The dynamic equations of the transmission components where t denote the duration of the gearshift. The jerk, A, i can be simplified during stationary operations ( dω dt = 0) is represented by integrating the squared derivatives of the and also when a component is fixed by means of shift vehicle accelerations during the gearshift as follows: elements to another component or to the transmission t da housing such that it has same dynamics as another com( )2 dt (24) A= ponent or is stationary. For example during 1-2 up-shift 0 dt equation (12) can be removed from dynamic equations where a = dωw dt . since TB4 fixes R2 C2 R1 to the transmission housing and 4 To obtain the trade-off between min T and min A soluω4 = dω dt = 0. The transmission dynamics can then be calculated by removing J4 from I, ω˙ 4 from Ω, fourth row tions, the optimal control problem objective is often forof matrix K, second column of κ and TB4 from T vector. mulated as the weighted sum of the time and jerk terms 288
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Considering the 1-2 up-shift, state vector x can be summarized as: x = {ω1,2,3,5,6,7,8 , ωe , ωw , xf lex }
(25)
while the state dynamics f (x(t), u(t)) are obtained by the differential equations presented in the previous sections. The complete optimal control problem formulation for different scenarios stated earlier can be summarized as follows: min
x(t),u(t),γ(t)
T or A
4. OPTIMAL CONTROL RESULTS Gearshifting transients are optimized for 1-2 and 4-5 upshifts where in both cases there are one off-going and one in-coming shift element. The major phenomena observed in the calculated optimal transients are described in the following sections. 4.1 Trade-off between jerk and time Figure 6 shows the trade-off between min T and min A solutions for both gearshift cases where the dashed-lines are obtained by interpolation between the calculated points. The trade-offs show that small increase of gearshift duration from the time optimal solution can largely reduce the jerk. For example, at C12 and C45 a good compromise is achieved between the two objectives as the gearshift duration in the 1-2 and 4-5 up-shifts has increased only 0.22 % and 4 % while the jerk has reduced as much as 64.7 % and 55.9 % respectively. However, the gain in the jerk reduction reduces when the gearshift time increases further. 1−2 up−shift 3
Another approach is used here where after solving the min T and min A solutions as the start and end points on the trade-off, a time grid is selected on the interval between these two solutions. Then, the gearshift duration t is repeatedly set equal to the grid times and equation (24) is solved to obtain the A values. The first point on the time grid is obtained by solving the min T problem, but there is no unique solution in time when solving the min A problem (v1 = 0) since after a certain gearshift duration, zero jerk can be obtained for various gearshift lengths. Therefore, to obtain the shortest time where zero jerk gearshift is possible, equation (23) is chosen as the objective and the problem is solved including an additional a = 0 constraint in the optimal control problem formulation. The calculated gearshift duration is then used as the end point of the time grid.
Implementing direct multiple shooting method, the formulated optimal control problem is transformed into a nonlinear program (NLP). The NLP is then solved by CasADi, Andersson (2013), which is an open source optimal control framework using Ipopt, W¨achter and Biegler (2006), and the powertrain transients are obtained. The results of this optimization are presented and analyzed in the following section.
A (rad/s )
(v1 T + v2 A, v1 + v2 = 1) where the points on the tradeoff are calculated by solving with different weights (v1,2 ). However, considering the large difference in the order of magnitude between the two objectives, when formulating the weighted sum, A and J have to be normalized with respect to their maximums. This makes the solution of the weighted sum sensitive to the normalization and weight values, which makes it difficult to calculate the trade-off with an acceptable spread of points.
10
C12
5 0 1.32
(26)
x(t) ˙ = f (x(t), u(t)) umin ≤ ui (t) ≤ umax xmin ≤ xi (t) ≤ xmax Te (ωe , uf (t)) ≤ Te,max x(0) = x0 x(T ) = xT x(0) ˙ =0 x(T ˙ )=0 a = 0, when solving for min T |A=0
289
1.33
1.34
s.t.
1.35
1.36 T (s)
1.37
1.38
1.39
1.4
A (rad/s3)
4−5 up−shift C
20
45
10 0 0.55
0.6
0.65
T (s)
0.7
0.75
0.8
Fig. 6. The trade-off between time and jerk for 1-2 and 4-5 up-shifts. At C12,45 there is a compromise between the two objectives.
where the min and max are the upper and lower limits for states and controls and x0 and xT correspond to the state values at the end of the first gear and beginning of the second gear operation, respectively. These are calculated in similar manner as described by equations (7a)-(7c) for rw ωw =6 km/h and rw ωw =15 km/h in 1-2 and 4-5 upshifts respectively. The requirements on x˙ at the beginning and end of the gearshift are applied to guaranty that the gearshift starts and ends at stationary condition. It should be noted that the kinematic constraint mentioned in equation (6) does not need to be added to problem constraints as their effect is already present in the dynamic equations of states via internal forces Fi . 289
4.2 Optimal transients for 1-2 and 4-5 up-shifts The min T , min A and C12,45 state and control input transients for 1-2 and 4-5 up-shifts are illustrated in Figures 7, 8, 9 and 10. According to Figures 7 and 9 it is both time and jerk optimal to release the offgoing shift element, ub5 and ub3 for 1-2 and 4-5 cases respectively, as soon as the gearshifts starts. Also in the min A case, it is optimal to reach the maximum torque capacity of the in-coming shift element, uk3 and ub1 for 12 and 4-5 up-shifts, immediately after gearshift starts such that the initial torque phase is completely avoided. This requirement for fast actuation of in-coming shift elements
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100
0.4
0.6
0.8
1
1.2
1.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 ub1 (Nm)
min T min A C
−200
−2000 −4000
12
0
0.2
0.4
0.6
0.8
1
1.2
−6000
1.4
0 −200 −400 −600
(Nm)
2000 1000 0
0
u 0.2
u
uk3 (Nm)
0
0
−400
200
mf
100 0
300
(gr/cycle)
200
b3
ub5 (Nm)
umf (gr/cycle)
300
0
0.2
0.4
0.6
0.8
1
1.2
1.4
min A min T C45 0
0.1
0.2
0.3
T (s)
(rad/s) ω (rad/s) ω (rad/s) ω (rad/s) in out
100 50 0 1.9
0.2
0.4
0.6
0.8
1
1.2
1.4
w
1.8 1.7 0 250 200 150 0 250
0.2
0.4
0.6
0.8
1
1.2
1.4
0.2
0.4
0.6
0.8
1
1.2
1.4
0.8
1
1.2
1.4
0.2
0.4
0.6
0.8
1
1.2
1.4
0.2
0.4
0.6
0.8
1
1.2
1.4
ω
0.6
ω (rad/s)
0.2 min0.4 A min T C
0.8
100 50 4.650
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.8
4.6 4.55 0 200
0.1
0.2
0.3
0.4
0.5
0.6
min A min T C 0.7
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4 T (s)
0.5
0.6
0.7
0.8
45
150 3000 250
200 1000 0 −100
3
12
2
0.2 0 0
0.7
150
ω (rad/s)
150 0 300 200 100 0 0 0.4
0.6
ice
200
0.5
Fig. 9. Optimal control transients for min T , min A and C45 cases in a 4-5 up-shift .
T (s)
Fig. 8. Optimal state transients for min T , min A and C12 cases in a 1-2 up-shift where ωout =ω7 and ωin =ω6 . can be considered when designing the hydraulic actuation system. Considering C12,45 transients, the magnitude of disturbances in the vehicle speed, ωw , is reduced compared to the min T solution especially in case of the 1-2 up-shift which can also be verified comparing the dωw /dt values in Figure 11. The required power in b1 and k3 shift elements T T calculated as 0 ub1 ω2 dt and 0 |uk3 ω8 | dt is also 8.8 % and 12.11 %, respectively, lower than the min T case. Figure 11-top shows the input torque to the transmission. The transients are similar for both up-shift cases in the min A solutions in a sense that the fuel injection umf is cut off so that the absence of an input torque on the engine side of the torque converter together with the constant torque from the in-coming shift element, smoothly reduce the kinetic energy of the transmission components down to the level required for the engagement of the next gear. However, in the min T case of both up-shifts, in addition to the fuel cut-off, the bang-bang type dynamics 290
xflex (rad)
xflex (rad)
ω8 (rad/s) ωice (rad/s) ωin (rad/s) ω (rad/s) ωout (rad/s) w
Fig. 7. Optimal control transients for min T , min A and C12 cases in a 1-2 up-shift.
0.4 T (s)
0 100 50 0 −50 0.40 0.2 0 0
Fig. 10. Optimal state transients for min T , min A and C12 cases in a 4-5 up-shift where ωout =ω7 and ωin =ω6 . of the in-coming shift elements introduce disturbances into the driveline such that the induced excitation is transferred to the vehicle mass at wheels via the the drive shaft. These excitations get partially damped by the drive shaft flexibility, vehicle mass and wheel inertia which increases the rate of reduction in the kinetic energy of the transmission components and shortens the time before the next gear can be engaged. Subtracting the total kinetic energy of all transmission components (0.5 Ii ωi2 ) at the beginning and end of the gearshifts, the difference is 25.17 % larger in case of the 1-2 up-shift which could be the main reason for the slower transients compared to the 4-5 case as more reduction in the kinetic energy is required before the next gear can be engaged.
0.05
t
T (Nm)
600 400 200
dωw/dt (rad/s )
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0
0.4
0.6
T (s)
0.8
1
0
−0.05 0
0.1
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0.4 T (s)
0.5
0.6
1.2 1.4 min A, 4−5 min T, 4−5 min A, 1−2 min T, 1−2 C45 C0.7
0.8
1.2
1.4
12
0.05 0
w
dω /dt (rad/s2)
0.2
−0.05 0
0.2
0.4
0.6
T (s)
0.8
1
Fig. 11. Transmission input torque and vehicle acceleration/decceleration in min T , min A and C45 solutions of 1-2 and 4-5 up-shifts. 5. CONCLUSION A framework is developed for gearshift transient optimization during inertia phase via state space modeling of a nine speed heavy duty automatic transmission. Using the developed model and in order to analyzed the minimum time/jerk transients and the trade-off between these, optimal control problems are formulated and solved. Two example up-shifts are considered and in order to calculate the trade-off between time and jerk objectives while avoiding objective function normalizations difficulties, the minimum jerk problem is iteratively solved for various preselected gearshift durations. The results show that the developed framework is applicable for efficient optimization of inertia phase gearshift transients. As future model developments, hydraulic actuation dynamics can be included and sensitivity of the gearshift transients with respect to the parameters such as clutch fill dynamics, components’ inertia and driveline flexibilities can be analyzed. REFERENCES A40G (2015). Volvo A40G hauler, ”http://www. volvoce.com/SiteCollectionDocuments/VCE/ Documents%20Global/articulated%20haulers/ Brochure_A35GFS_A40GFS_SV_12_20040745_B_2015. 02.pdf”, Accessed July. 2015. Andersson, J. (2013). A General-Purpose Software Framework for Dynamic Optimization. PhD thesis, Arenberg Doctoral School, KU Leuven, Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center, Kasteelpark Arenberg 10, 3001Heverlee, Belgium. Eriksson, L. and Nielsen, L. (2014). Modeling and Control of Engines and Drivelines. John Wiley & Sons. Gao, B., Chen, H., Zhao, H., and Sanada, K. (2010). A reduced-order nonlinear clutch pressure observer for automatic transmission. Control Systems Technology, IEEE Transactions on, 18(2), 446–453. Goetz, M., Levesley, M., and Crolla, D. (2005). Dynamics and control of gearshifts on twin-clutch transmissions. 291
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