Control Optimo de Engranes

Journal of the Franklin Institute 338 (2001) 371–390

Optimal control of gear shift operationsin automatic transmissions

A. Haj-Fraj*, F. Pfeiffer

Lehrstuhl B fur Mechanik, TU Munchen, 85747 Garching, Germany

Abstract

An optimal control approach for gear shift operations in automatic transmissions isproposed in this paper. Starting from a verified model of a typical power train with anautomatic transmission a performance measure for evaluating the gear shift process in terms

of passengers comfort and control expenditure is developed. The gear shift operation is statedas a multistage decision process by making use of the dynamic programming method.Thereby, the synchronization of the gear box is formulated as a constraint at the end of the

process. A control law is derived analytically in an explicit form by minimizing theperformance measure over each process stage. Simulation results show a significantimprovement in terms of gear shift comfort by different driving load cases. Furthermore,the shift time and the frictional losses in the shift elements can be reduced by applying

the proposed control. # 2001 The Franklin Institute. Published by Elsevier Science Ltd.All rights reserved.

Keywords: Power train dynamics; Automatic transmission; Optimal control; Gear shift operations;

Passenger’s comfort

1. Introduction

The increasing requirements for more comfort, less fuel consumption and higherpower performance lead more and more to the use of electronics in combination withoptimal control applications. Therefore, passenger cars have become verycomplicated mechatronic systems, where many power train components have tobe controlled simultaneously. Fig. 1 shows a part of the electronic car managementwith a controller area network (CAN), which is responsible for data exchange

*Corresponding author. Tel.: +49-89-289-15201; fax:+49-89-289-15213.

E-mail address: [email protected] (A. Haj-Fraj).

0016-0032/01/$20.00 # 2001 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved.

PII: S 0 0 1 6 - 0 0 3 2 ( 0 0 ) 0 0 0 9 1 - 0

between the different electronic devices in the car. In this work, we will concentrateon the control of the gear shift process, which is generally carried out by controllingoil-immersed clutches and brakes. The main criteria for evaluating the shift qualityare the duration of the gear shift process and some measures for the emergingacceleration change.During a gear shift operation, two clutches are operating simultaneously, one

clutch engaging and the other one disengaging. The clutch pressure is determined bythe electronic control unit (ECU) and applied by the hydraulic device of the gearbox. At the same time, the engine torque is reduced by the digital motor electronics(DME) by retarding the ignition timing or adjusting the throttle opening. Theengagement and disengagement of the shift elements lead to a system with time-varying structure and require the development of a suitable control approach whichis able to deal with these transitions.In [1,2] the fundamentals of powershifting in automatic transmissions are

illustrated using a simple model. An efficient method for detailed modeling ofpower trains with automatic transmissions as multibody systems with time-varyingstructure is presented in [3,4].An approach based on the sliding mode control was used in [5] for developing a

clutch-to-clutch shift controller. In [6], the Ricatti equation was applied to minimizethe jerk and the dissipative losses during the gear shift process. Though thesimulation results show a reduction of the dissipative losses, no significantimprovement could be observed in terms of the acceleration smoothness. This ismainly due to the used performance measure, which evaluates the difference betweenthe actual and the reference state trajectories of the system resulting from a non-optimal reference feedforward control. In [7], an adaptive control approach ispresented. The control variables for the load reduction and the clutch pressure arethereby interpolated from characteristic maps depending on the torques and speeds

Fig. 1. Electronic control of the automatic gear box in the car.

A. Haj-Fraj, F. Pfeiffer / Journal of the Franklin Institute 338 (2001) 371–390372

of the engine and the turbine of the hydrodynamic torque converter. This approachwas improved by the superposition of another feedback controller for the turbinespeed. The characteristic maps mentioned above are defined by a set of samplingpoints. A method to optimize these maps off-line using a model-based optimizationis presented in [8].In this paper, we propose a new model-based control approach based on the

method of dynamic programming developed by Bellman [9–11]. The aim of thisapproach is to determine an explicit optimal control law for the shift time whichminimizes the jerk during the gear shift process in automatic transmissions.

2. Mechanical model

The numerical implementation of a simulation program can be performed veryefficiently by decomposing a technical system into subsystems, showing particularadvantages in terms of a modular structure of the mechanical model. A drive trainwith automatic transmission in general consists of five main components: engine,torque converter, gearbox, output train and vehicle. Each component of the drivetrain can be considered as a rigid multibody subsystem. The rigid bodies areconnected to each other by ideal rigid joints, clutches and force elements (Fig. 2).

2.1. Engine

For investigations on gear shift comfort the high-frequency vibrations of theengine can be neglected. Therefore, the engine can be modelled as a rotating rigidbody. The drive train excitation caused by the engine is described by its torque

MM ¼ MMðaTH; _jMÞ; ð1Þ

which can be interpolated from a measured two-dimensional characteristic map(Fig. 3) as a function of the throttle opening aTH and the engine speed _jM . The

Fig. 2. Power train model.

A. Haj-Fraj, F. Pfeiffer / Journal of the Franklin Institute 338 (2001) 371–390 373

equation of motion of the engine can be written as

JM �jM ¼ ðð1� bÞMM � MPÞ: ð2ÞThe factor b represents the load-reduction applied to the engine during the gear shiftprocess. Outside of the process b is set to zero. The torque MP is explained in thenext section.

2.2. Torque converter model

The hydrodynamic torque converter consists of a pump (P) which is connected tothe input shaft, a turbine (T) which is connected to the output shaft, and an impeller(I) which is pivoted on the housing through a one-way clutch (Fig. 4).Since only the transmission behaviour of this component is of interest, it is

described by a force law. With the definition of the velocity ratio n and torque ratio m(Fig. 4)

n ¼ _jT_jP; mðnÞ ¼ MT

MPð3Þ

and for given input and output speed _jP and _jT the turbine torque can be calculatedas

MT ¼ mðnÞMP: ð4ÞThe pump torque itself can be calculated using a torque characteristic MPCðnÞmeasured for a constant pump speed _jPC

MP ¼ MPCðnÞ_jP_jPC

� �2: ð5Þ

2.3. Gear box model

The gear box investigated here is a five-speed gear box. For the simulation of thegear shift process from the first to the second gear, we use the reduced model shownin Fig. 5. Thereby, only gearwheels and shift elements that are in the power flow

Fig. 3. Characteristic map of the engine.


during the first (index 1) and the second gear (index 2) are considered. JE and JArepresent the moment of inertia of the driving side (input) and driven side (output),respectively.The transmission ratio i1 ¼ i1Ei1A of the first gear results from the gear

transmission ratios i1E before and i1A after the corresponding shift element. Thesame applies to the transmission ratio i2 ¼ i2Ei2A of the second gear.The shift element which is active during the first gear is a one-way clutch. The

transition to the second gear is achieved by engaging a wet clutch. Once the clutchpressure is high enough so that the wet clutch is able to transmit the whole inputdrive torque, the one-way clutch releases and its relative speed becomes positive. Theclutch pressure is increased continuously until the relative speed of the wet clutchdisappears. This indicates that the clutch sticks and the second gear is fully engaged.The equations of motion of the gear box are

JE 0

0 JA

" #�jT�jA

!¼

MT þ 1i1E

M1 þ 1i2E

M2

�MW � i1AM1 � i2AM2

!: ð6Þ

Fig. 4. Hydrodynamic torque converter and its characteristics.

Fig. 5. Power train model.


The determination of the torques M1 and M2 depends on the operating state of theshift elements. The relative velocity of the one-way clutch can be defined as

D _j1 ¼ _jAi1A �_jTi1E

: ð7Þ

The one-way clutch allows a relative motion in one direction and blocks it up in theother direction. By defining this free direction as positive, the two operating statescan be described as follows:

* D _j1 ¼ 0Ð one-way clutch blocks, and* D _j1 > 0Ð one-way clutch unblocks.

During the first gear, we obtain D�j1 ¼ 0 for the one-way clutch. Substituting Eq. (6)in this constraint for its relative acceleration, we get

M1 ¼1

JA=i21E þ i21AJE�JA

i1EMT � i1AJEMW � JA

i1E i2Eþ i1A i2AJE

� �M2

� �50: ð8Þ

As long as M1 > 0, the one-way clutch remains blocked. Once M1 reaches zero, theone-way clutch starts rotating and it cannot transmit any torque as long as D _j1 > 0.This transmission behaviour is described by the complementary transmission law

D _j150; M150; D _j1M1 ¼ 0: ð9Þ

The transition to the second gear is carried out by engaging the second shift element,which is a wet clutch. Depending on its relative speed

D _j2 ¼ _jAi2A �_jTi2E

: ð10Þ

Two operating phases are distinguished

* D _j2 6¼ 0Ð wet clutch slips, and* D _j2 ¼ 0Ð wet clutch sticks.

The torque transmitted by the wet clutch during the sliding phase is calculated as

M2 ¼ �signðD _j2ÞmC Apz rmp; ð11Þ

with the clutch pressure p, the dynamic friction coefficient mC, the piston surface Ap,the number of friction surfaces z and the mean friction radius rm.Once the control pressure is high enough for the wet clutch to transmit the whole

driving torque, the one-way clutch releases. This transition occurs when the torqueM1 becomes zero and the relative acceleration D�j1 becomes positive.The second gear is engaged when the wet clutch sticks. The sticking point is

characterized by the condition D _j2 ¼ 0. During the sticking phase, the clutch torquerepresents a static friction torque and is limited by the inequality

�mC0Apzrmp4M24mC0Apzrmp; ð12Þ

where mC0 is the static friction coefficient of the clutch.


Substituting Eq. (6) in the constraint for sticking D�j2 ¼ 0, we obtain

M2 ¼1

ðJA=i22EÞ þ i22AJE�JA

i2EMT � i2AJEMW

� �: ð13Þ

One should note that, as mentioned above, the one-way clutch releases before thewet clutch sticks. Therefore, M1 is set to zero while evaluating Eq. (13).

2.4. Output train model

This submodel represents a simplified description of the output train from thecardan shaft to the wheels and chassis of the vehicle. The elasticities of the shafts andtires are modeled as a torsional force element with the spring coefficient cW and thedamping factor dW (Fig. 2). The torque of the force element can be calculated as

MW ¼ cW ðjA � jRÞ þ dW ð _jA � _jRÞ: ð14Þ

The load torque ML results from the wind force, rolling force and the inclinationforce

ML ¼ ðFwind þ Fincl þ FrollÞrRiRD

; ð15Þ

where

Fwind ¼ 12 rcwA _jR

rRiRD

� �2; ð16Þ

Fincl ¼ mg sin a; ð17Þ

Froll ¼ msmg cos a; ð18Þ

with the wheel radius rR, the transmission ratio of the rear differential iRD, the airdensity r, the air resistance coefficient cw, the vehicle front surface A, the mass of thevehicle m, the gravitation g, the road inclination a and the rolling coefficient ms.The equation of motion of the output train with the vehicle mass can be given as

JW �jR ¼ ðMW � MLÞ: ð19Þ

3. Equations of motion

3.1. Nonlinear state equations

The mechanical model for the power train can be described by the five states_jM ; _jT; _jA; _jR, and jA � jR. For the comfort evaluation the jerk is necessary.Therefore, we add the acceleration as an additional component to the state vectorwhich can later be used to calculate the jerk according to (34) and (35). Then the


state vector is defined as

x ¼

x1

x2

x3

x4

x5

x6

0BBBBBBBBB@

1CCCCCCCCCA

¼

_jM

_jT_jA_jR

jA � jRa

0BBBBBBBBB@

1CCCCCCCCCA

ð20Þ

with the vehicle acceleration

a ¼ rRiRD

�jR: ð21Þ

Substituting �jR by (19) in (21), the differentiation with respect to the time yields

_a ¼ rRiRD

1

JW

d

dtðMW � MLÞ: ð22Þ

The equations of motion of the whole mechanical model (Fig. 2) can be formulatedas

_x1

_x2

_x3

_x4

_x5

_x6

0BBBBBBBBB@

1CCCCCCCCCA

¼

1JMðð1� bÞMM � MPÞ

1JEðMT þ

1

i1EM1 þ

1

i2EM2Þ

1JAð�MW � i1AM1 � i2AM2Þ

x6iRDrR

x3 � x4

rRJW iRD

cwðx3 � x4Þ þ dwð _x3 � x6iRDrRÞ � rcwA

r2R

iRD2

x4x6

h i

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA: ð23Þ

Introducing the control vector consisting of the reduction factor for the enginetorque and the clutch pressure

u ¼b

p

!ð24Þ

and eliminating _x3 in the sixth equation by the third equation in (23), we obtain thecompact form

_x ¼ fðx; uÞ: ð25Þ

While the torques MM ; MP; MT; MW and ML can be calculated at each instant oftime during the simulation by Eqs. (1), (4), (5), (14) and (15), the evaluation of thetorques M1 and M2 depends on the operating state of the shift elements

* one-way clutch

D�j1 ¼ 0) M150 ðM1 from Eq: 8ÞD�j1 > 0) M1 ¼ 0


* wet clutch

D _j2 6¼ 0) M2 ¼ �signðD _j2ÞmCApzrmpD _j2 ¼ 0) �mC0Apzrmp4M24mC0Apzrmp ðM2 from Eq: 13Þ:

The comparison between simulations performed with the presented model andmeasurements carried out during a gear shift process from the first to the second gearshows a very good agreement (Fig. 6). In the simulation, the control strategy used inthe car, was implemented and used as reference in the following sections. Therefore,the mechanical model of the power train can be used to develop a model-basedoptimal control for the gear shift process in automatic transmissions.To solve the dynamic programming problem of the gear shift process analytically,

the equations of motion (25) must be available in a discrete linear form. This can beachieved in two steps: the nonlinear system equations have to be first linearized andthen discretized.

3.2. Linear state equations

The gear shift operation is a highly transient process coupled with a change of thegear box configuration. This results in a nonsmooth dynamical behaviour which isdescribed by a set of equations with time-varying structure. Because of the largerange of change in some state variables like the engine and turbine speed during theprocess, the linearization must be performed with respect to a reference trajectoryðx0ðtÞ; u0ðtÞÞ rather than to a constant reference operating point ðx0; u0Þ.Therefore, we consider for a given load case aTH a reference control u0ðtÞ which

yields the state trajectory x0ðtÞ according to the equations of motion (25). Thisreference control is the same one used to verify the model in Section 3.1.

Fig. 6. Comparision of simulation and measurements.


The linear equations of motion with respect to the reference state vector x0ðtÞ andreference control u0ðtÞ can then be obtained using the series expansion

_xðtÞ ¼ Aðx0ðtÞ; u0ðtÞÞ xðtÞ þ Bðx0ðtÞ; u0ðtÞÞ uðtÞ þ eðtÞ; ð26Þ

where

Aðx0ðtÞ; u0ðtÞÞ ¼@f

@x

��0

; ð27Þ

Bðx0ðtÞ; u0ðtÞÞ ¼@f

@u

��0

; ð28Þ

eðtÞ ¼ _x0ðtÞ � ½Aðx0ðtÞ; u0ðtÞÞ x0ðtÞ þ Bðx0ðtÞ; u0ðtÞÞ u0ðtÞ: ð29Þ

3.3. Time-varying discrete state equations

Considering the gear shift process as a sequence of equal time increments T , thelinear equations of motion (26) can be transformed into a discrete form

xkþ1 ¼ Akxk þ Bkuk þ ek ð30Þ

with the definitions [12,13]

Ak ¼ Eþ AðkTÞT þ 1

2A2ðkTÞT2 þ � � � þ 1

m!AmðkTÞTm

� �; ð31Þ

Bk ¼ TEþ 1

2AðkTÞT2 þ � � � þ 1

ðm þ 1Þ! AmðkTÞTmþ1

� �BðkTÞ; ð32Þ

ek ¼ x0kþ1 � ðAkx0k þ Bku0kÞ: ð33Þ

For the formulation of the performance function in Section 4.2, we need the jerk _a ofthe vehicle during the gear shift process. Therefore, the discrete equations of motion(30) must be extended such that the vehicle jerk can be calculated. This can beachieved by adding a new component to the discrete state space vector

x7kþ1 ¼ x6k ¼ ak: ð34Þ

With x6k and x7kþ1 the sixth components of the vector xk and the seventhcomponents of the vector xkþ1. The jerk can be calculated as an output variable

yk ¼ _ak ¼ x6k � x7kT

: ð35Þ

With the extended state space vector xk 2 R7 the discrete description of the systemcan be written as

xkþ1 ¼ Akxk þ Bkuk þ ek; ð36Þ

yk ¼ cTxk ð37Þ


with

cT ¼ ½0 0 0 0 0 1T � 1

T ð38Þ

the dimension of the matrices Ak;Bk and ek must be extended according to Eq. (34).

4. Optimal control

4.1. Control strategy

In this paper, we will consider the gear upshifting from the first to the second gear.In order to develop a control strategy we split up the gear shift process into threephases as shown in Fig. 7. These phases can be found in every gear upshifting anddownshifting process.

* The first phase starts with the shifting signal which indicates the beginning of thegear change into the next gear. After a delay due to the dead time of the electric

Fig. 7. Control strategy of the gear shifting.


and hydraulic actuators of the system, the pressure of the upcoming clutch israised continuously. During this phase the off-going clutch, in this case the one-way clutch, remains blocked. Although the gear shift process has already begun,the transmission ratio corresponds to the previous gear, in this case the first gear.

* The second phase begins when the torque of the one-way clutch becomes zero.The one-way clutch releases and the gear box has no fixed transmission ratiobecause there is no determined kinematic relationship between the gear box inputand output speed. The beginning of this phase is accompanied by a turnaround inthe turbine speed.

* The third phase starts when the upcoming clutch sticks. This point is called thesynchronous point and is reached when the relative speed of the wet clutchbecomes zero and the actual clutch torque is less than the maximal transmittabletorque.

A reasonable optimal control can only be applied when the gear box has twodegrees of freedom, one for the input shaft and one for the output shaft, which arekinematically independent. This state is only given for the second phase, when therelative speeds of both shift elements are inequal zero and the gear box has no fixedtransmission ratio. The first phase must be used as a pre-control phase to put thegear box in a controllable state. This can be achieved by applying a feedforwardcontrol.

uðtÞ ¼0

pðtÞ

!: ð39Þ

The pressure of the upcoming clutch is raised smoothly after a time delay. Since, apart of the driving input torque can now be transmitted by the wet clutch, the torqueof the one-way clutch decreases. When its torque disappears, the one-way clutchreleases. Now an optimal control law in the form

uðxðtÞ; tÞ ¼bðxðtÞ; tÞpðxðtÞ; tÞ

!ð40Þ

can be applied to the power train in order to achieve the gear shift in a given shifttime. The control law is optimal in terms of minimizing some performance criteria,which will be specified in the following section. It should be noted that once thetarget gear is engaged, the wet clutch cannot be controlled anymore because theclutch torque cannot be influenced by the clutch pressure in the sticking phase (13).Furthermore, the load reduction must be finished at the synchronous point. Thisavoids the undesirable excitation of the power train by the change of the enginetorque after the new gear is engaged.The dynamic torque (11) of the clutch, just before the synchronous point, is

generally greater than the static torque (13) just after it. An abrupt drop in the clutchtorque at the synchronous point leads to an excitation of the output train and causesan undesired vehicle jerk during the third phase. Since no control intervention can beapplied after the synchronization of the clutch, the control law during the second


phase must be designed such that the jerk after the gear change is suppressed. Thisrequires a predictive control approach.

4.2. Problem formulation

It is desired to determine the control law which drives the power train systemgoverned by Eq. (36) from the initial state x0 ¼ xðt0Þ to the end state xK ¼ xðteÞ,which obeys the constraint

gðxðteÞÞ ¼ D _j2 ¼ _jAi2A �_jTi2E

¼ 0 ð41Þ

¼ wTxK ð42Þ

with wT ¼ ð0 � 1

i2Ei2A 0 0 0 0Þ, by minimizing the performance measure

J ¼ lyðxðteÞ; teÞ þZ te

t0

fðxðtÞ; uðtÞ; tÞ dt: ð43Þ

The cost function fðxðtÞ; uk; tÞ evaluates the cost of the process in the second phase,whereas the function yðxðteÞÞ evaluates the cost resulting from the end state xðteÞ.The weighting factor l is included to permit the adjustment of the relativeimportance of the terms in J.Making use of the dynamic programming method based on the principle of

optimally formulated by Bellman [9–11], the second phase of the gear shift can beconsidered as a multistage decision process. This is accomplished by considering theprocess during the second phase as K equal time increments, T , in the interval ½t0; te.Using the discrete form of the equations of motion (36) the performance measure isformulated as

J ¼ lyðxKÞ þXK�1

k¼0Fðxk; uk; kÞ: ð44Þ

Definition of the cost function F: The acceleration change is the most critical issuewhich affects the passengers comfort during the gear shift process [2,7,8,14].Thereby, the smoother the acceleration the more comfortable is the gear shift. Weformulate the process cost function as

Fðxk; uk; kÞ ¼ _a2kþ1ðxkþ1Þ þ uTkRkuk ð45Þ

¼ xTkþ1ccTxkþ1 þ uTkRkuk: ð46Þ

This cost function reflects the desire to keep the jerk close to zero without excessiveexpenditure of the control effort.

Definition of the end cost function y: Although this term in the performancemeasure depends only on the end state, it can describe a behaviour which extendsover a time interval after the synchronous point. Therefore, we can choose a criterionwhich evaluates the jerk after the end of the gear shift over a period of time. For the


formulation of this criterion, we need a prediction function which estimates the jerkat each discrete step considered after the synchronous point as a function of the endstate xK . This can be accomplished by making use of the discrete linear form of theequation of motion of the system. Since no control is applied to the power train theequations of motion yield during the third phase

xKþ1 ¼ Ak xK þ eK ; ð47Þ

xKþ2 ¼ AKþ1xKþ1 þ eKþ1; ð48Þ

..

.

xKþN ¼ AKþN�1xKþN�1 þ eKþN�1: ð49Þ

Because of the recursive form of Eqs. (47)–(49), we can formulate each state xKþi as afunction of xK by beginning with xKþ1 and substituting xKþi by xKþiþ1. Since thestate xK relates to xK�1 and uK�1 through the state equation

xK ¼ AK�1xK�1 þ BK�1uK�1 þ eK�1 ð50Þ

the state xKþ1 can be written as a function of xK�1; uK�1 and K � 1. Continuing inthis manner, we obtain for i50

xKþ1þiðxK�1; uK�1Þ ¼Yi

l¼�1AKþl

!xK�1 þ

Yi

l¼0AKþl

!BK�1uK�1

þXi

j¼0

Yi

l¼i�j

AKþl

!eK�1þi�j þ eKþi: ð51Þ

Assuming that the interval considered for the evaluation of the jerk in the thirdphase has N equal increments, T , the cost function can be written as

yðxKÞ ¼ yðxK�1; uK�1Þ ¼XKþN

i¼K

xTi ccTxi: ð52Þ

Note that the performance measure y permits a prediction of the jerk over the nextN þ 1 discrete time increments, which results from applying a given control uK�1 to agiven state xK�1 at the discrete ðK � 1Þth stage.

4.3. Optimal control law

Now the method of dynamic programming can be applied to determine an optimalcontrol law which can lead the system given by Eq. (36) from the initial state x0 tothe end state xK which satisfies the constraint gðxKÞ ¼ 0 by minimizing theperformance measure

J ¼ lyðxK�1; uK�1Þ þXK�1

k¼0ð _a2kþ1ðxkþ1Þ þ uTkRkukÞ: ð53Þ


In the dynamic programming algorithm, the first step in the computationalprocedure is to find the optimal control for the last stage of the process [9–11].This is essentially a matter of finding a law which satisfies the end constraintgðxKÞ ¼ 0. In other words, we have to find the control vector uK�1, which forces theclutch to stick at the end of the last stage.By taking into account that the engine torque reduction must be finished at the

end of the gear shift operation, the first control variable yields

uK�1½1 ¼ bK�1 ¼ 0: ð54Þ

Since the end state xK is related to xK�1 and uK�1 through the state equation, weobtain by solving the end constraint gðxKÞ with respect to uK�1

uK�1 ¼b

p

!K�1

¼0

Iðe½2 � i2E i2Ae½3Þ

!K�1

þ0

IðA½2 � i2E i2AA½3Þ

!K�1

xK�1

¼ uK�1ðxK�1Þ; ð55Þ

where A½i and e½i are, respectively, the ith row of the matrix A and the ith element ofthe vector e and the definition

IK�1 ¼ ði2E i2ABK�1½3; 2 � BK�1½2; 2Þ�1: ð56Þ

The cost resulting from driving the system during the last stage to the required endof the process is then

JK�1 ¼ðcTxKðxK�1; uK�1ðxK�1ÞÞÞ2 þ uK�1ðxK�1ÞRK�1uK�1ðxK�1Þþ lyðxK�1; uK�1ðxK�1ÞÞ: ð57Þ

It should be emphasized here that JK�1 which depends only on xK�1 and K � 1, isthe only possible cost to satisfy the end constraint and reach the synchronous pointat the end of the process. Because of the quadratic form of the performance measure,we can transform Eq. (57) as follows

JK�1ðxK�1Þ ¼ qK�1 þ �qTK�1xK�1 þ xTK�1QK�1xK�1 ð58Þ

with qK�1 as a scalar, �qK�1 2 R7;QK�1 2 R7 7 which all depend on AK�1;BK�1; eK�1and RK�1 [15].In the next step, we consider the ðK � 2Þth stage. The cost of operation over the

last two stages is given as

JK�2 ¼ JK�1 þ FðxK�2; uK�2Þ: ð59Þ

Observe that JK�2 is the cost of a two-stage process with the initial state xK�2.Based on the Bellman recursion, the optimal performance during the last two


intervals is found from

minuK�2

JK�2 ¼ J *K�2 ¼ minuK�2

fðcTðeK�2 þ AK�2xK�2 þ BK�2uK�2ÞÞ2

þ uTK�2RK�2uK�2 þ ðeK�2 þ AK�2xK�2 þ BK�2uK�2ÞTQK�1

ðeK�2 þ AK�2xK�2 þ BK�2uK�2Þ þ qK�1

þ �qTK�1ðeK�2 þ AK�2xK�2 þ BK�2uK�2Þg ð60Þ

where again we have used the dependence of xK�1 on xK�2; uK�2 and K � 2. Tominimize JK�2 with respect to uK�2 we need to consider the control vectors for which

@JK�2@uK�2

¼ 0 ð61Þ

is satisfied.Since JK�2 is quadratic in uK�2, Eq. (61) is linear in uK�2 and therefore we obtain a

unique solution

uK�2ðxK�2Þ ¼ IK�2½�BTK�2ðQTK�1 þQK�1ÞeK�2 � BTK�2�qK�1

� 2BTK�2ccTeK�2 þ IK�2½�BTK�2ðQTK�1 þQK�1ÞAK�2

� 2BTK�2ccTAK�2xK�2: ð62Þ

with

IK�2 ¼ ½BTK�2ðQK�1 þQTK�1ÞBK�2 þ 2BTK�2ccTBK�2

þ RK�2 þ RTK�2�1: ð63Þ

It can be shown that the matrix of the second partials @2JK�2=@u2K�2 ¼ 0 is positive

definite [16] so that solution (62) yields the absolute, or global, minimum J *K�2 ofJK�2 in the form

J *K�2ðxK�2Þ ¼ qK�2 þ �qTK�2xK�2 þ xTK�2QK�2xK�2: ð64Þ

It is important to state that this solution is based on the assumption that the controlvalues are not bounded or at least that they do not violate the boundary constraints,if there are any. Actually the control variables are constrained by

04b4bmax;

04p4pmax ð65Þ

because of design constraints of the actuators. Since the control expenditure isincluded in the performance measure J as a penalty function, the control values canbe forced to lie in the admissible intervals by choosing a suitable matrix Rk.Continuing backwards in this manner, we obtain for the first stage the optimalcontrol

u0ðx0Þ ¼ I0½�BT0 ðQT1 þQ1Þe0 � BT0 �q1 � 2BT0 c cTe0þ I0½�BT0 ðQT1 þQ1ÞA0 � 2BT0 c cTA0x0 ð66Þ


with I0 defined like IK�2 by substituting ðK � 2Þ by 0. The resulting whole minimalcost for driving the system optimally from the initial state x0 to the end constraintgðxKÞ ¼ 0 over K stages can then be written analogously to (64) as

J *0 ðx0Þ ¼ q0 þ �qT0 x0 þ xT0Q0x0: ð67Þ

Let us now summarize the results provided by the applied dynamic programmingalgorithm. We have obtained an optimal control law for each interval ½kT ; ðk þ 1ÞT in the explicit, analytical form

uk ¼ rðxk; kÞ ð68Þ

¼ �uðkÞ þ �rðxk; kÞ; ð69Þ

which drives the system from the initial state xk to the next state xkþ1 with minimaljerk and control expenditure. It is important to note here that the optimal control(68) consists of both a feedforward �uðkÞ and a feedback �rðxk; kÞ portion.Now the control law, which was developed in a backward algorithm, can be

applied forward. Beginning with any given initial state x0, the control vector (68)drives the power train stage by stage to the synchronous point. In the last stage thecontrol uK�1 forces the wet clutch to stick at the end of the operation and drives thesystem to the synchronous point whatever the initial state x0 was.

5. Results

In the following, some results achieved by applying the developed control law tothe verified nonlinear simulation model are presented.Fig. 8a and b show the acceleration and jerk for the load case (100% throttle

opening) resulting from considering only the function F in the performance measure(44) by setting l to zero. As expected, the jerk is kept close to zero during thesecond phase. The acceleration change becomes consequently very smooth. Since thecost function y is not considered the jerk after the synchronization remainsunimproved. This changes as soon as the function y is considered in the performancemeasure (44) by using an adequate weighting factor l (Fig. 8c and d). The appliedcontrol drives the system to the synchronous point in the given shift time byminimizing the jerk during the second and third phases. During the first phase, theapplied pre-control is the same as the reference. Therefore, the acceleration remainunchanged.The proposed approach can be applied for any load case of the car. Fig. 9 shows

further results for the 80 and 40% throttle openings. In both cases, the jerk andacceleration smoothness are apparently improved by keeping the desired shifttime.Moreover, the control approach is found to be robust with respect to varying the

desired shift time. In Fig. 10, some results are presented for the full-load case wherethe desired shift time was reduced by 20% with respect to the reference. The plot ofthe relative speed of the wet clutch (Fig. 10c) shows that the synchronization is


achieved at the desired time. The acceleration and jerk are still smooth (Fig. 10aand b). Furthermore, the frictional losses in the wet clutch (Fig. 10d) are reduced,which improves the life expectancy of the friction discs.

Fig. 8. Results for 100% throttle opening: (a) and (b) without the end cost function y ðl ¼ 0Þ, (c) and (d)with the end cost function y ðl 6¼ 0Þ.

Fig. 9. Results for 80% throttle opening: (a) and (b), and 40% throttle opening: (c) and (d).


6. Conclusions

In this article, an optimal control approach for gear shift operations in vehicleautomatic transmissions has been presented. First, a mechanical model is developedfor the whole power train and verified by measurements. After the discretization ofthe equations of motion and making use of the dynamic programming method, thegear shift operation is considered as a multistage process with constraints.Furthermore, a suitable performance measure for evaluating the gear shift comfortduring the process is formulated. The analytical solution of the dynamicprogramming problem leads to an explicit discrete optimal control law for the gearshift process. The application of the derived optimal control to the verified nonlinearmodel in computer simulations shows major improvements in terms of thepassengers comfort for different throttle openings. Moreover, the shift time andthe frictional losses during the process are reduced.

References

[1] H.-J. Forster, Getriebeschaltung ohne Zugkraftunterbrechung, Automob. Ind. (1962) 60–76.

[2] F.J. Winchell, W.D. Route, Ratio changing the passenger car automatic transmission, Design

Practices } Passenger Car Automatic Transmissions, Part 1 SAE AE-1 (1962) 57–80.

[3] A. Haj-Fraj, F. Pfeiffer, Dynamics of gear shift operations in automatic transmissions, Proceedings of

the First International Conference on the Integration of Dynamics, Monitoring and Control for the

21st Century, Manchester, UK, 1–3 September 1999, pp. 29–35.

Fig. 10. Results for 100% throttle opening with a reduction of the shift time of 20% with respect to the

reference.


[4] F. Pfeiffer, Ch. Glocker, Multibody Dynamics with Unilateral Contacts, Wiley, New York, 1996.

[5] D. Cho, Nonlinear control methods for automotive powertrain systems, Ph.D. Thesis, Department of

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Elektronik, ATZ Automob. Z. 96 (4) (1994) 228–235.

[8] A. Haj-Fraj, F. Pfeiffer, Optimization of gear shift operations in automatic transmissions,

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[9] R.-E. Bellman, Mathematical Theory of Control Processes, Academic Press, New York, 1967.

[10] R.-E. Bellman, Some Vistas of Modern Mathematics, University of Kentucky Press, Kentucky, 1968.

[11] R.-E. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.

[12] W.S. Levine et al., The Control Handbook, CRC Press, Boca Raton, FL, 1996.

[13] K. Ogata, Discrete-Time Control Systems, Prentice-Hall International Inc., Englewood Cliffs, NJ,

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[14] Y. Hojo et al., Toyota Five-Speed Automatic Transmission with Application of Modern Control

Theory, SAE 920610.

[15] M. Feiler, Optimale Regelung der Schaltvorgange in Automatikgetrieben, Diploma Thesis at the

Lehrstuhl B fur Mechanik, TU Munchen, 1999.

[16] S.E. Dreyfus, Dynamic Programming and the Calculus of Variations, Academic Press, New York,

1965.


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