AIpplying Genetic Search Techniques
to Drivetrain Modeling
D. Maclay a n d R. Dorey
W
ork has been carried
eling work carried out during an
investigation into the tuning and
control of driveline torsional oscillation. The model is based on
a transfer function and state
space description that is simple,
but which allows inclusion of
nonlinear effects associated, for
example, with the operation of
the clutch and with transmission
backlash. The model has been
used to help in identifying important component characteristics to adjust the stiffness
and damping of the drive system.
Test data for the work was
acquired from a development
vehicle and this comprised
measurements of engine crank
velocity, throttle position, vehicle speed, and acceleration. Maneuvers were carried out at a
variety of engine and vehicle
speeds, and in low gear ratios, to
stimulate longitudinal oscillation.
out to identify a nonlinear model of a vehicle engine
and drivetrain. A hybrid approach ha: been adopted which
combines 30th physical modeling and parameter optimization
using gen-tic algorithm (GA)
search techniques. The resulting
models which cover a range of
operating conditions have allowed the sensitivity to variation
of key pari.meters to be assessed
and have been used to help optimize the o\.erall response of the
vehicle drivetrain. A comparison has bem made between the
GA search and a gradient based
method M hich highlights the
"intelligent" nature of the fornier appro ich.
Background
Advan:es that have been
made to iniprovc. the economy
and emis>ions of powertrains
have increased efficiencies and
lowcred frictional losses. bringing reducd powertrain daniping and a greater sensitivity to
driveline torsional disturbances.
With light, responsive engines,
rapid throi tle movements made
by the driver can excite a low
frequency torsional vibration felt by the occupants as a sudden
and undesirable longitudinal oscillation. This oscillation is most
significant in low gear ratios.
A systems approach to the synthesis of the vehicle engine and
drivetrain allows examination of the key contributors to this
important physical behavior. This article describes recent mod-
Presented at the 1992 IEEE International Symposium on Intelligent Control. D. Maclay is with Cambridge Control Ltd., The
Jeffreys Bitilding, Cowley Road, Cambridge CB4 4WS, U.K. R.
Dorey is Mith Powertrain Research, Ford Motor Company Ltd.,
Research and Engineering Centre, Laindon, Basildon. Essex
SS15 6EE U.K.
50
Model Structure
A drivetrain model structure
Stock Market: Simon
has been developed which incl u d es engine dynamics,
drivetrain dynamics, and dynamics arising from longitudinal compliance in the suspension. The equations for this model
are written in terms of physical parameter values. Some of these
parameter values were obtained by direct measurement; the
others have been identified by matching the model response to
corresponding test data using the GA search techniques described
below. The approach of developing a model structure and using
(not necessarily GA) optimization methods to identify the physical parameter values, has already been shown to work successfully - see for example [ I].
While the main objective of this work was to investigate the
acceleration response of the vehicle to changes in throttle position, it was important to model the internal behavior of the engine
and drivetrain in order that nonlinear effects, in particular backlash in the drive, could be incorporatcd. This requirement pre-
0212- 1108/93I$O3.OOO1993IEEE
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cluded the identification of a
“black box” type model andwas
the reason why the model structure described here has been
used.
Based on previous work, the
engine dynamics from throttle
- -
as:
bs + 1
____
e-sT
where u(t) is the throttle input, x ( t ) = [xix2 ~ 3 x 4 x6]’
~ 5 is the state
vector, y ( t ) = bi y2 y3] is the output vector and
-ba2+b2a2 -ba2
k
d
-
le
- 1 0
O
O
+
x5
+
X
x4
x2
In order to obtain a simplified
model of drivetrain dynamics, the drivetrain was represented as
three inertias linked by flexible couplings and dampers as shown
in Fig. 1. The inertia lerepresents the engine and flywheel inertia,
Id represents the inertia of the gearbox components, wheels and
sub-frame and Zv is an equivalent inertia representing the mass of
the vehicle. Note that in order to simplify them, all the equations
of motion have been referred to the engine crankshaft.
By transforming the engine transfer function to state space form
and combining this with a state space model of the drivetrain
dynamics,the overall systemcan be written as (a detaileddescription
of variables used here and throughout appears in the Appendix):
0
x3
+
0 1 0
0
0
0
C= 0 0 0
1
0
0
This model structure provides three outputs which are engine
crank speed, roadwheel speed, and vehicle accelerationand these
correspond to measurementsfrom tests carried out on the vehicle.
Note that at this stage, we still have a linear model structure.
However, since the internal structure of the drivetrain has been
modeled (as opposed to using a black box approach) this can be
extended to incorporate nonlinear effects such as backlash which
are embedded within the system between the engine flywheel
and the roadwheels: details of how this was done are given later.
The model structure which has been derived incorporates a
number of physical parameters: accurate estimates are available
for some of these, and for others, less accurate estimates are
available. Following some initial experimentationit was decided
to use these fixed values for each of the inertia parameters and
to apply the GA optimization techniques to search over the
parameter set
O
I
1
0
While the choice of this parameter set may appear somewhat
arbitrary, it yielded good results and this issue is given further
consideration below.
0
A=
Parameter Identification Using Genetic Algorithms
0
0
L
0
cs
- -
Ivg2
The first thorough treatment of the the use of GASwas given
in [2] and a more up to date description of the theory and
application of GA methods may be found in [3]. The use of GAS
extends across many fields including pattern recognition, structure optimization and network optimization;examples of the use
of GAS in control applications are given in [4]-[6].
A description of the basic GA is now given as well as an
explanation of how the algorithms described in [3] have been
adapted in order to deal the problem in hand. The algorithm
described here has been implemented using MATLAB [7] and is
ks -cS+cw]
lvg2
Ivg2
B = [ l b 0 0 0 0IT
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- -
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Parents
,1 0 1 0 , O1 1 1 0
Offspring
0 1 0 00 10 1 1
*
1 o 1 0 '01 o 1 1 '
Fig. 2. Crossover:
being considered for inclusion in a future release of the Optimization Toolbox [8].
The General Scheme
As suggested by the name, the inspiration for GASarose from
the study of evolutionary and biological processes which enable
a species to adapt over many generations to become better suited
to its environment. The GA owes its existence to ideas borrowed
from the study of these processes and the following description
of a basic CiA explains how these ideas have been adapted into a
form which is suitable for computer implementation:
1) Initialization: an initial population is set up consisting of
individuals whose (randomly chosen) characteristics are represented by a string of ones and zeros.
2 ) Evaluate Fitness: for each individual within the population
a fitness value is calculated which is based upon a suitable
performance criterion.
3) Selection: pairs of individuals for breeding are chosen on
a probabilistic basis, individuals with a high fitness value being
more likely to be chosen than those with a lower fitness value.
4 ) CrosJover: in this stage the binary coding of each parent is
divided into two and segments from each parent are then combined to give an offspring which has inherited part of its coding
from each parent (illustrated in Fig. 2 ) .
5 ) Mutation: inversion of bits in coding of the offspring with
probability of perhaps one in a thousand.
6) Retum to step 2.
The following sectionsconsiderhow each of these stageshas been
implemented with respect to the drivetrain identification problem.
Initialization of the Population
A population was initialized by setting up a random distribution of parameter vectors. The individual parameter values were
set up with a uniform distributionacross the allowable range. For
most of the runs, a population of size 80 was used, however, good
results were obtained with populations as small as 20.
BinaryCoding of the Parameter Set
Before proceeding, it is necessary to consider how a vector
of values from the parameters P is converted to a binary
representation. Take as an example the parameter c ; anaccurate estimate was not available for this parameter, however,
from physical considerations it was expected to lie between
0.01 and 1. By taking 16 logarithmically spaced values across
this range the parameter cv can be mapped onto a 4-bit binary
number.
It is important in this case to use a logarithmic rather than a
linear spacing since the linear spacing would give very poor
resolution at the lower end of the range. It is true that using 16
points to cover a range from 0.01 to 1 gives a fairly coarse
resolution: clearly the resolution may be increased either by
increasing the number of points or by narrowing the range. It is
questionnablehow far the resolution may profitably be increased,
in this work for example, an error of say 21% on any of the
parameter values would be considered very satisfactory.
Evaluation and Scaling of the Fitness Function
For each individual in the population it is necessary to evaluate a measure of fitnessfix) which will be used to generate a
probability with which the individual in question will be selected
for breeding.
A natural way to gauge the relative performance of different
sets of parameter values is to simulate the response of the model
with each of the sets of parameter values and calculate the least
squared error (denote this g(x)) between this simulated response
and the test data acquired under corresponding conditions.
As pointed out in [3], however, the least squared error function
g(x) cannot be used directly as a fitness function, since it does
not fit the necessary conditions that it should be i) non-negative,
and ii) increasing with increasing performance of the parameter
vector. Goldberg suggests a mapping from g(x) tofix) given by
=
{Oc,,
Cmax
- g ( x ) when
otherwise
(4)
which requires a suitable value to be chosen for Cma, A further
problem is that at the start of the GA run, it is common to have a
few individuals of extraordinary fitness and it is important that
these individuals should not dominate the population after a
single generation. To overcome this Goldberg suggests a linear
scaling method which itself may require modification to prevent
negative values of fitness.
In this work a simpler approach to mapping g(x) ontofix) has
been adopted: the cost function g(x) for each individual is evaluated and the results are used to rank each individual in the
population from highest to lowest performance; thus for a population of 10 individuals, the highest performing individual is
given a fitness of 10 and the lowest performing is given a fitness
of 1. This satisfies requirements i) and ii) above and makes the
scaling process unnecessary.
Selection,Crossover, and Mutation
These steps are performed in a standard manner: individuals
are selected for breeding with a probability proportional to their
fitness; crossover is performed with the crossover point being
chosen randomly and mutation on an individual bit is performed
with probability of one in a thousand.
Performance of the Genetic Algorithms
A discussion will be given here of the performance indicators
used during the search and consideration will be given to determining why GA search techniques worked well, whereas difficulties
were experienced with applying quadratic,gradient based methods.
Performance Indicators
For each generation, a number of performance measures are
displayed in order to indicate the progress of the search. These
performance measures are:
1) Average least squared error for all members of the population.
2 ) Least squared error for the fittest member of the population.
IEEE Control Systems
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Test 13: simulated (solid) and measured (dashed) acceleration
1-
-
o c
I
-20
J
-
02
04
Fig. 4. Dead-zonefunction used within the nonlinear equations for
modeling backlash.
___
06
08
1
12
time in seconds
Test 13. simulated (solid) and measured (dashed) crank velocity
9or-
]
/'
/
I
/'
,
~
,.,'
,'
0 -~
0.2
06
0.4
0.8
1
1.2
time in seconds
Fig. 3. Comparison of simulated and measured data
Wle I
Method Comparison
Charicteristics of M-L method
I Charicteristics of GA method
Search oker parameter T requires
interpolation routine to allow noninteger numbers of samples
Search over T is easily
accomplished by mapping to binary
coding corresponding to integer
number of samples
If it can get stuck in a local
optimum, it will often do so
Generally robust against becoming
stuck in local optima
When the global optimum is found, Good at getting close to the
the accuracy of the solution may be optimum-for high accuracy it may
be worth continuing with M-L
chosen arbitrarily
Very efficient when starting point
does lead to global optimum
~.
Always requires a large number of
function evaluations
3) Normalized variance of each of the physical parameters
across the whole population.
All of these performance indicators can be used to decide
when the optimization should be terminated. In particular, when
the average least squared error is equal or nearly equal to the least
squared error from the fittest individual,then no further improvement is likely. In practice, with the setup that was used, the least
squared error for the fittest individual reached a minimum in
about half the number of generations it took for the whole
population to achieve this value.
For each generation, the variance of each parameter across
the population was calculated and normalized such that values
between and 0 and 1 were possible. This indicator provided
considerable insight into the relative importance of the the different parameters:in general, the parameters Tand kd and cw were
the first to converge, whereas cs, ks and cv converged last.
This observation has implications regarding the choice of a
minimal parameter set, i.e., what is the largest set of parameters
for which the data is rich enough to identify a unique value for
each parameter. This is of particular importance when non-genetic search techniques, for example quadratic, gradient based
methods such as the Levenberg-Marquartd (M-L) method (see
[9]) are employed.
Comparison with a QuadraticGradient Based Method
Initial attempts to apply the Levenberg-Marquartdmethod to
this problem were unsuccessful; however, the performance of
this method depends critically upon the initial guess used as the
starting point for the search. Once a solution vector had been
obtained (using the GA search method), some tests were performed in order to gauge the relative performance of the GA and
M-L methods.
To this end a GA search was perfomed with an initial population based on randomly chosen parameter vectors as described
above. The number of function evaluations required to obtain the
best fit solution was then noted.
For the M-L search a number of runs were performed using
as initial guesses the same parameter vectors which were used in
the GA search as the initial population. The total number of
function evaluations was noted as was whether each of the searches
had approached the global optimum or become stuck in a local
optimum.
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For both search methods, the main computational burden
arose from evaluating the least squared error function so the total
number of function evalutions provides a useful performance
indicator.A difficulty remains however with the choice of termination criteria - for the GA search the process was continued
until the fitness of the fittest individual had not improved for 10
generations, and for the M-L search the process was continued
until either criteria relating to the accuracy of the objective
function and parameter vector were satisfied, or 900 function
evaluations were reached.
In order to make the comparison more meaningful, a wider
range of parameter values for the initial population was used
than for the final results; a population of size 80 was used for
the GA search with a 5-bit representation for each of the
parameters. The termination criterion was satisfied after 60
generations and the resulting parameter vector gave a good fit
for both crank velocity and vehicle acceleration. For the M-L
method ten runs were performed using the first ten parameter
vectors from the GA’s initial population as starting points. In
each case, the termination criteria were satisfied only when
900 function evaluations were reached and out of the ten
searches only one arrived at a solution which gave a good fit
for both crankshaft velocity and vehicle acceleration. Thus, in
total, the GA search required 4800 function evaluations
whereas the M-L method required 9000 function evaluations
to reach an acceptable solution.
Clearly this is only a rough comparison, and the results would
be expected to vary widely depending on the initial conditions,
termination criteria and search parameters.
Table I makes some comparisons of a more general nature for
the two methods applied to this problem. Note, in particular,that
for a gradient based method it is necessary to be able to obtain
partial denvatives of the objective function with respect to the
search parameters: this requires some customization of the algorithm to deal with the time delay parameter which is most
conveniently represented as a discrete number of samples. The
relative ease with which the GA handles this parameterhighlights
0
0.2
04
0.6
0.8
I
1.2
1.4
time in seconds
m---
Test 3, madel no. 16 crank velocity response
.
the “intelligent” nature of the algorithm: the fact that the GA is
based on discretely varying parameters means that it can even be
applied to problems where the model structure itself is included
in the search.
Results
A number of models of the engine and drivetrain system have
successfullybeen identifiedfor a range of differentengine speeds
in each of first, second and third gears. Fig. 3 shows an example
of the identifiedand measured responsesto a step input on throttle
starting from cruise at 15 km/h in first gear. This shows that a
good fit has been obtained for both the vehicle acceleration and
the engine crank velocity. Note that with the step input being
applied starting from cruise, the backlash in the drive plays no
part in the response and the system is assumed to behave linearly.
In contrast when the step input is applied from a deceleration
initial condition, the backlash in the drive must first be taken up
before the vehicle itself can start to accelerate.
As mentioned earlier, it was possible to model the effect of
backlash between the flywheel and the roadwheels. This was done
by replacing x3 (differencebetween the angular displacementof the
inertiasleand I&*) on the right hand side of the state space equations
(1) with B(x3) where B(x3) is the dead-zone function shown in Fig.
4. This alters the equations for X2 and i 4 which become
X 2 = (-ba2
+ b2a2)xl - ba2x2 + -B(x3)
kd
1,
By making this alteration the model accurately represents the
nonlinear behavior of the drivetrain which effectivelybehaves as
two decoupled systems whenever the backlash is active.
Fig. 5 shows three traces which illustrate the effect of the
backlash. The dashed trace shows measured data from a test where
the vehicle was initially decelerating at 15 km/h in first gear; the
dotted trace shows the response of the model identified at the same
operating point but without backlash; the solid trace shows the
response of the same model but with backlash included (the amount
of backlash was determined by direct measurement).The difference
between the solid and dotted traces shows that the backlash results
in a significantly greater amplitude of oscillation during acceleration. The close match between the solid and dashed responses shows
that by incorporating the nonlinear effect of backlash into the
identified model, an accuraterepresentationof the vehiclebehaviour
with backlash active is obtained.
I
Successful Application of GA Search Techniques
Fig. 5. Comparison of simulated data (with and without backlash)
and measured data. Solid-simulated response with backlash. Dotted-simulated response without backlash. Dashed-measured data
from vehicle.
GA search techniques have been successfully applied here to
the identification of vehicle drivetrain dynamics. Of particular
note regarding the GA search technique itself are the robustness
and ease of application which this method offers. This article
outlines the circumstances under which the GA search method
has been found to be most appropriate as applied to the drivetrain
identification problem and suggests how it may be used in
conjunction with a quadratic method to give a particularly powerful combination.
IEEE Control Systems
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It was shown that the modeling problem considered in this
article was in fact amenable to being solved by traditional optimization methods; however, the flexibility and robustness of the
genetic algorithm approach was noted and with regard to the
“intelligent’ nature of the algorithm, the ease with which discretely varying parameters such as a time delay or even a choice
of model structures may be incorporatedin the search was found
to be an important consideration.
Appendix: Notation
longitudinal damping between sub-frame and body referred
to engine crankshaft in “/(rads)
damping coefficient to represent wheel slip and viscous
damping within the gearbox referred to engine crankshaft
in “/(rads)
damping coeffient to represent wind resistance referred to
engine crankshaft in “/(rads)
gear box ratio
combined engine, flywheel and clutch inertia in kgm2
combined equivalent inertia of transmission,road wheels
and vehicle sub-frame referred to road wheels in kgm2
overall stiffness of components between clutch and subframe, referred to engine crankshaft in “/rad
overall stiffness of longitudinal suspension referred to
gine crankshaft, in “/rad
model state associated with engine dynamics
model state: crank angular velocity in rads
model state: difference between road wheel angular
position referred to crank and crank angular position in rad
model state: road wheel angular velocity referred to crank
in rads
model state: difference between vehicle equivalent angular
position referred to crank and road wheel angular position
referred to crank in rad
model state: vehicle equivalent angular velocity referred
to crank in rads
model output: crank angular velocity in rads
model output: road wheel angular velocity referred to crank
in rads
model output: vehicle e uivalent angular acceleration
referred to crank in rads
92
References
[ l ] R. Stanway and J. E. Mottershead, “Identification of combined viscous
and Coulomb friction - A numerical comparison of least-squares algorithms,” Trans. Znst. MC, Jan. 1986.
[21 J. H. Holland, Adaptution in Natural and Artijicial Systems. Ann Arbor,
MI: Univ. of Michigan Press, 1975.
[3] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine
Learning. Reading, MA: Addison-Wesley, 1989.
[4] D. M. Etter, M. J. Hicks, and K. H. Cho, “Recursive adaptive filter design
using an adaptive genetic algorithm,” in Proc. IEEE Inr. Conj Acoustics,
Speech, and Signal Processing, 1992, vol. 2, pp. 635-638.
[51 J.R. and M.A. Keane, “Cart centering and broom balancing by genetically
breeding populations of control strategy programs,” in Proc. Int. Joint Con5
Neural Nefworks, 1990, pp. 198-20 I .
[6] D.R. McGregor, M.O. Odetayo and D. Dasgupta, “Adaptive control of a
dynamic system using genetic-based methods,” in Proc. IEEE Int. Symp.
Intelligent Control 1992, pp. 521-525.
[7] MATLAB User Guide, The Mathworks Inc., 24 Prime Park Way, Natick,
MA, 1990.
[8] A. Grace, Optimization Toolboxfor Use with MATLAB. The Mathworks
Inc., 24 Prime Park Way, Natick, MA, 1990.
[9] D.G. Luenberger, Linear and Nonlinear Programming. Reading, MA:
Addison-Wesley, 1984.
David Maclay is a Senior Engineer with Cambndge
Control Ltd., a U.K. based company which provides
control engineering design services. He graduated
from Cambridge University with a degree in electronic and information sciences having spent two
years studying mathematics before switching to engineering. While working at Cambridge Control he
has built up experience, primanly in the automotive
and aerospace industnes, and his work has extended
to the development of a full envelope helicopter
flight control system designed using multivanable techmques. In the automotive field, his expenence includes engine modeling and development of
an idle speed control system.
Robert E. Dorey was awarded the Ph.D. degree in
1983 by the University of Bath, U.K., for work on
the design and modeling of vehicle transmission
systems. He continued this work as a Lecturer in
Mechanical Engineering at Bath, where he developed complementary interests in microprocessor
systems and digital control. He joined Ford Motor
Company in 1990 where he is a Technical Specialist
in Powertrain Control Systems. He has published
approximately thirty papers in the automotive field.
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