JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

43, 3, pp. 575-591, Warsaw 2005

ONLINE OPTIMIZATION OF A PREVIEW CONTROLLER –

STRUCTURE AND ALGORITHMS 1

Joachim Lückel

Eckehard Münch

Henner Vöcking

Thorsten Hestermeyer

University of Paderborn

e-mail: Joachim.Lueckel@mlap.de; muench@mlap.de; Henner.Voecking@mlap.de;

Thorsten.Hestermeyer@mlap.de

Active suspension systems are used to increase ride comfort and safety of
vehicles. Optimal results can be achieved if disturbances from the track are
known in advance. Usually, this causes a problem as track excitations cannot
bemeasured until they take effect on the vehicle. Here, we present an appro-
ach to disturbance compensation for railway vehicles coupled in a network.
Information on stationary arising track disturbances are gathered by vehicles
and stored locally at track sections.By repeated runs over a respective section
this information is iteratively optimized and can be used for the disturbance
compensation in subsequent vehicles.
Thedevelopedoptimizationalgorithm isdescribed.Design criteria arederived
fromdigital control theory.Theprocedurewas implemented ona testbed for a
semi-vehiclewith threedegreesof freedom,where its usabilitywasproved.The
resultsarediscussedand,finally, someaspectsofgeneralizationareconsidered.

Keyword:mechatronics, learning,distributedoptimization, active suspension,
railway systems

1. Introduction

Today, active suspension systems are well established in theory and prac-
tice. This holds especially true for automotive applications. When looking at

1
This workwas developed in the course of the ”Collaborative ResearchCenter 614 – Self-

Optimizing Concepts and Structures inMechanical Engineering” –University of Paderborn,

and was published on its behalf and funded by the Deutsche Forschungsgemeinschaft.



576 J.Lückel et al.

the railway industry, vehicles with active suspensions are available, but these
systems usually focus on the tilt and centering of the coach body rather than
on ride comfort. However – even if rare – there has also been some work on
active damping in industry (Streiter et al., 2001) and public research.

Thiswork uses, as an application example, the system setup of the railway
system ”Neue Bahntechnik Paderborn”, which is described in more detail in
Section 2.
Most of the vast literature on the control of active suspension systems

focuses on single vehicles. Collaborative vehicle networks, however, offer a
promising way to improve ride comfort even further. This article shows that
it is possible to reduce bodymotion by a great extent by using the experience
gainedbyother vehicles. In order to do so, two things are necessary: Firstly, an
algorithm is required that determines information about the track excitation
and uses this in the control algorithm of the active suspension. Secondly, a
collaborative network with communication infrastructure has to be set up.
Here, we focus on the first step.

The article is structured as follows:
Section 2 gives a brief overview of an examplary suspension system and its

underlying control structure. Section 3 presents the basic idea for the overall
system setup including the optimization algorithm in the collaborative vehicle
network. With this setup in mind, Section 4 develops the learning algorithm.
In order to show the applicability of the algorithm and its benefits, the system
was implemented on a suspension test bed described in Section 5, where the
results arediscussedaswell.Thearticle concludeswithanoutlook inSection6.

2. Active suspension control

The railway system ”Neue Bahntechnik Paderborn” (NBP) (Hestermeyer,
2003) features small autonomous railway vehicles of van sizewitha fully-active
suspension system (Fig.1). The suspension system is used for both tilting of
the car body and for the adjustment of the spring and damper characteristics.
The car body and bogie are connected by air springs without any passive
dampers. This is for enhancing the comfort as passive dampers harden if their
excitation contains high frequencies.

The function of passive dampers is taken on by an active system of hy-
draulic cylinders that create damping forces by displacing spring bases. The
displacement vector xactive yields necessary cylinder displacements lcyl,i by
computing the inverse kinematics of the cylinder arrangement (Fig.2). As the



Online optimization of a preview controller... 577

Fig. 1. Structure of active suspension

input for the controller, relative positions with respect to velocities between
the car bodyand the bogie are used,which are gained fromthemeasured leng-
ths of hydraulic cylinders and position sensors in parallel to the air springs. In
this way, dynamics of the entire system can be customized arbitrarily by adju-
sting the parameters of the controller, and therefore the rules for calculating
the additional active forces.

Fig. 2. Control structure for relative damping

3. Preview systems for active suspensions

Whendesigning an active suspension, it is important to put special care on
the employed sensor concept and the control strategy as bothplay amajor role
in the success of the system. One important aspect became clear alreadywith
first realizations of active suspension systems. The disturbance compensation
using information about the ground excitation can improve the ride comfort
considerably. Jäker (1990) e.g. useda disturbance compensator as apart of the
control law for the active suspension in an off-road truck with great success2.

2The information about wheel excitation was derived from an observer based on
signals from accelerometersmounted on the axles.



578 J.Lückel et al.

The way in which the ground excitation is determined has significant in-
fluence on the compensation result. Due to actuator dynamics, it is vital to
know the ground excitation as early as possible. In the optimal case, the exci-
tation is known before it actually hits the wheel. This is known as ”preview”.
Preview information for rear wheels can be gained by using information from
front wheels (so called ”internal preview”). This is quite a convenient way for
disturbance compensation in trains, where the locomotive can collect track
information and transfer it to the carriages. In short vehicles like cars howe-
ver, the influence of the front wheels on comfort is very high and the internal
preview provides only small benefit (Rutz, 1987). The preview information
for the front wheels would therefore help to improve ride comfort even more.
Unfortunately, looking at a single vehicle, collecting the preview information
for the front wheels is an arduous and costly business. (Donahue (2001) e.g.
describes a military external preview system with an expensive radar and
optical sensors.) A much simpler way to obtain the desired information can
be found for vehicles integrated in a network. Ioannou (1998) proposed such
an infrastructure-supported network for highway vehicles.
In this work, we concentrate on the railway system ”Neue Bahntechnik

Paderborn” (NBP) (Hestermeyer, 2003), which supplies perfect infrastructure
for the new preview system presented here.

The shuttles are propelled by a double-fed asynchronous linear motor. For
the implementation of the motor, the track is divided into sectors which are
equipped with their own frequency converters and computer hardware. The
creation of propelling forces requires fast communication between the shuttles
and the track. Figure 3 shows the information and communication structure
of the NBP-system (Zanella et al., 2002). The available computation power
and communication infrastructure can be used to set up a preview system for
the active suspension (see also Hestermeyer et al., 2004; Münch et al., 2004).

Fig. 3. Communication structure for the railway-system ”Neue Bahntechnik
Paderborn” (Zanella et al., 2002)



Online optimization of a preview controller... 579

The system structure shown in Fig.3 suggests the following set-up for the
determination of the track excitation.

In the first step, the track is logically divided into different sections, and
the agent network is allocated to the track. One track agent is allocated to
each section (Fig.4)3.

Fig. 4. Determination of preview information bymulti-agent optimization

When a shuttle wants to enter a special section, it contacts the track agent
and receives in return an estimation of the track excitation it can use for
disturbance compensation4. After completing the section, the shuttle answers
with a performance rating which is used by the track agent to optimize the
trajectory. In the case of a communication error, thedisturbance compensation
is simply turned off. This results in less comfort but is otherwise uncritical.

Apart from improving the ride comfort by optimal disturbance compen-
sation, this method offers an excellent way of monitoring the track quality,
as the track information is continously updated with each shuttle. Special
measurement runs can be reduced or even excluded at all.

3Comparing Fig.3 and Fig.4, it seems obvious to select sections according to the
motor sectorsanddownload the trackagent softwareon the available sectorhardware.
However, this is not a prerequisite. The multi-agent software can also be run on
centralised hardware.
4Dynamics of the respective shuttle has to be considered when using the preview

information.Otherwise, the optimizationof thepreview information in the trackagent
might yet converge, but is now valid only for shuttles with similar dynamics.



580 J.Lückel et al.

3.1. Extended control structure

As alreadymentioned in the introduction, the work presented here focuses
on the realization of the disturbance compensation and the trajectory opti-
mization disregarding communication issues and questions arising from the
multi-agent implementation. Figure 5 shows the structure of the self-learning
control system including the learning algorithm.

Fig. 5. Control structure featuring learning algorithm

The basis of the active suspension control is a simple feedback law (block
”controller”) assuring sufficient damping of the car body as described in Sec-
tion 2.

In order tominimize the absolutemovement of the car body, an additional
relative displacement signal f is introduced, which includes the reference and
disturbance information in function of the shuttle position s (Hestermeyer et
al., 2004). The table f =(si,fi) determines f from s by interpolation.

Based on the system response evaluated by a block objective generation,
the superposed learning algorithm computes a trajectory that reduces the
influence of disturbances in the track by adding the signal f to the relative
displacement between body and bogie.

So far, this concept suggested the usage of the track excitation as di-
sturbance compensation. This requires knowledge of the vehicle and actuator
dynamics when using the excitation trajectory in the controller. In the first
step, this dynamics was not explicitly considered, so that the car body and
actuator dynamics was reflected in determined trajectories.



Online optimization of a preview controller... 581

4. Learning algorithm

During the runover a track section, different disturbances affect the chassis
of a shuttle. These disturbances can be distinguished into stochastic distur-
bances and stationary disturbances,which recur at the sameplace of the track
section. The learning algorithmpresentedhere identifies and compensates the-
se stationary disturbances on the chassis. The objective is to keep the car body
of the shuttle as still as possible in order to improve the comfort of passengers.
Asdescribed inSection 3, the learningalgorithmdetermines a trajectory as

a sequence of numbers fki , where k indicates the step number of the learning
process and thus the number of shuttles that have crossed the section.
The shuttle measures movements of the car body during the passage over

the track section. Afterwards the data is given back to the learning algorithm,
it determines the new sequence fk+1i .

4.1. Learning algorithm

As the learning algorithm, a computation instruction of the form

f
k+1
i = f

k
i −Kay

k
j (4.1)

with
j = i+h (4.2)

was chosen. The value Ka gives the learning factor of the algorithm and y
k
j

the deviation of the car body position. The value h reflects dynamics of the
car body and indicates a shift of the fki signal with respect to the associated
measuring point. This shift is chosen according to the cut-off frequency of the
car body dynamics T and the travel speed of the shuttle v.
For the passage over the regarded track section, a constant speed

v(t)= const (4.3)

is assumed.

4.2. Convergence analysis

The learning algorithm inEq. (4.1) is verymuch similar to the description
of discrete controllers. They differ in the meaning of the counting variable k,
whichdescribes the progres of timewith thediscrete controllers. In thepresen-
ted learning algorithm, the variable k indicates a new run over the respective
track section.



582 J.Lückel et al.

It is obvious to analyze the convergence characteristics of the learning
algorithm by means of well-established digital control-engineering methods
(Hanselmann, 1984). Therfore, it is necessary to describe shuttle-dynamics by
a mathematical model. In order to treat the supporting points independently
of each other, some simplificationsmust bemade for the convergence analysis
of the learning algorithm. For the feed-forward signal, a simple step-function
in place of the interpolation function is used. Furthermore, an ideal reference
reaction of the car body is assumed. The system response of the shuttle can
be described with the simple model

y
k
j =Kp(f

k
i +u

k
i ) (4.4)

in which ui describes the track disturbance at the supporting point i.

Inserting it into learning algorithm equation (4.1) yields

f
k+1
i = f

k
i −KpKa(f

k
i +u

k
i ) (4.5)

Theuseof theZ-transformationwithEq. (4.5) results ina transfer function

G(z)=
fi(z)

ui(z)
=

KpKa

1−KpKa−z
(4.6)

In order to analyze the stability of the system, poles of (4.6) can be used

z=1−KpKa (4.7)

For stable system behaviour, the pole stays within the unit circle

0<KpKa < 2 (4.8)

In the stability analysis presented above, strong simplifications were ma-
de. Dynamic behaviour of the shuttle was reduced to a simple gain factor Kp,
whereby the dependencies between neighboured supporting points were elimi-
nated. If one selects this gain Kp to be the overshoot of the car body response
to aunit step, then one receives a useful estimation for the feasible range of the
learning factor Ka. In practice, the convergence will be assured by reducing
the learning factor to a relative small value. On the one hand, this leads to
a decreased learning speed, on the other the insensitivity of the feed-forward
signal fki to the above mentioned stochastic disturbance is increased.



Online optimization of a preview controller... 583

4.3. Extended convergence analysis

In the following section, an extended convergence analysis is presented,
which considers the dependencies of neighboured supporting points by more
accurate modelling of the shuttle dynamics.

In order to do this, the response of the shuttle chassis to the supporting
points and the track excitation must be described in relation to the current
supporting point. This can be achieved by a linear difference equation. The
solution to such a difference equation has the form5

{yj}=Z
−1{Y (z)} (4.9)

The dynamics of the car body in this application scenario can bemodelled
by a second order system.

yj+2+ b1yj+1+ b0yj = gj (4.10)

By assuming of yk0,y
k
1 =0, the solution to such a system results in

yj =
j
∑

ν=2

gj−νaν−1 (4.11)

with

aj =
α
j
1−α

j
2

α1−α2
(4.12)

(see Bronstein et al., 1995) where α1 and α2 are poles of the system, and the
track excitation is denoted by gj.

By selecting the sample rate of thedifference equation equal to the clocking
of the disturbance compensation, one receives

yj =
j
∑

ν=2

(fj−ν −uj−ν)aν−1 (4.13)

With uk
−1,u

k
0 = 0 and f

k
−1,f

k
0 = 0, Eq. (4.13) yields a description of the

car body position at the supporting points 1, . . . ,nwith respect to the signal

5Here Z−1 represents the inverseZ-transformation. In this context, the inverseZ-
tranformationof thedifference equation in the timedomain ismeant (see Section4.2).



584 J.Lückel et al.

vector fk and the track excitation vector uk













y1
y2
y3
y4
...
yn













k

=













0 0 · · · 0 0 0
0 0 · · · 0 0 0
a1 0 · · · 0 0 0
a2 a1 · · · 0 0 0
...

...
... 0 0 0

an−2 an−3 · · · a1 0 0













︸ ︷︷ ︸

Ap


























f1
f2
f3
f4
...
fn













k

−













u1
u2
u3
u4
...
un













k













(4.14)

By inserting Eq. (4.14) into Eq. (4.1), one receives a mathematical de-
scription for the overall behaviour of the learning algorithm. By choosing an
appropriate shift h, it is always possible to insert the matrix Ap from (4.14)
as a lower trianglematrix into this description. Choosing h=2, results in the
following set of equations









f1
f2
...
fn









k+1

=









f1
f2
...
fn









k

−Ka









a1 0 · · · 0
a2 a1 · · · 0
...

...
...

...
an an−1 · · · a1

























f1
f2
...
fn









k

−









u1
u2
...
un









k







(4.15)

or briefly

f
k+1 =(I−KaAp,h=2)

︸ ︷︷ ︸

A

f
k+KaAp,h=2u

k (4.16)

where I is the identity matrix.

In these equations, themutual influence of neighboured supporting points
is considered by the modelling of the car body dynamics.

Inorder to examine the convergence andstability of the learningalgorithm,
the eigenvalues of the systemmatrix A are determined. These eigenvalues can
be gained from the determinant

det(I−KaAp,h=2−λI)= 0 (4.17)

i.e. ∣
∣
∣
∣
∣
∣
∣
∣
∣
∣

1−Kaa1−λ 0 · · · 0
Kaa2 1−Kaa1−λ · · · 0
...

...
...

...
Kaan Kaan−1 · · · 1−Kaa1−λ

∣
∣
∣
∣
∣
∣
∣
∣
∣
∣

=0 (4.18)



Online optimization of a preview controller... 585

Thedeterminant of the lower trinaglematrix is equal to the product of the
diagonal elements. So as the eigenvalues of A, we get the n-fold eigenvalue
λ1,...,n with

λ1,...,n =1−Kaa1 (4.19)

As one can see from Eq. (4.18),this eigenvalue is independent of the total
number of supporting points n.
The eigenvalue λ1,...,n only depends on the term a1 from the description of

the car body dynamics (see Eq. (4.12)). Comparing Eq. (4.19) and Eq. (4.7)
from the simple convergence analysis, one states that with Kp = a1 the two
results of the convergence analysis results are identical.
The presented learning algorithm is limited to systems which perform re-

peativly the same type ofmovement, i.e., with the same velocity and the same
trajectory. Under these circumstances, it is possible to design a stable and
convergent algorithm bymeans of the depicted convergence analysis. The ap-
proach is not restricted to small systems since the factor a1 can always be
determined for the linear difference equation. This applies as well to higher
order systems (Bronstein et al., 1995).

5. Realisation and results

For a practical test, the approach for the preview control presented here
was realised in a simplified environment. In the following, we describe the
incorporated testbedandpresent the implementation of the appliedprocedure.

5.1. Configuration of the testbed

The control of the damperless suspension/tilt system is described by He-
stermeyer et al. (2003) for a complete vehicle. To test the active suspension,
a testbed was built up in the context of the ”NBP”. It allowed us to design
and test the suspension control (Liu-Henke et al., 2002). Figure 6 shows the
testbed. Three lower hydraulic cylinders serve to simulate track excitations
and are able to impose forces and torques in horizontal, vertical, and rotato-
ry directions on the car body. The upper part represents the suspension/tilt
module as it might be mounted aboard the vehicle. The mass of the carriage
is supported exclusively by airsprings, as described in Section 2.

With the procedure described here, the control of the relative motion be-
tween the car bogie and body is sufficient. In order to achieve an increase in
comfort, onemight think of damping absolutemovements of the carriage body



586 J.Lückel et al.

Fig. 6. Suspension-tilt testbed

(skyhook, Hestermeyer et al., 2004), but this was not included in the analysed
control because the absolute movements were to be impacted by the preview
algorithm.

5.2. Implementation of the optimization algorithm

In order to test the self-optimization approach presented in Section 4 at
the testbed, we needed a recurring excitation for simulation of repeated runs
along a fixed section. For this purpose, a track course was defined that stret-
ched over a 100m-long track section. For determining the objective variables,
we subdivided the track section into 100 parts of 1m each; thus for any po-
ssible direction of motion an evaluation vector containing 100 elements was
recorded for each crossing. For the objective, we used the maximum of de-
viations from the middle position of the car body in the respective part; it
was measured by existing position-measuring sensors. One run over the track
section took 10s at the assumed speed v = 10m/s. After each crossing, the
evaluation vectors were transmitted to the learning algorithm which defined
new trajectories of the disturbance compensation for the next crossing. These
trajectories are parameterised over 100 supportingpoints corresponding to the
parts of the respective section. As shown in Fig.5, the signal f was interpola-
ted between the supporting points. The algorithm required to determine the



Online optimization of a preview controller... 587

optimal disturbance compensation was implemented to a real-time hardware,
in addition to the testbed control described. Here, the emphasis was put on
testing the learning algorithm. Difficulties resulting from the necessary com-
munication between the vehicle and processing of information on the track
were ignored. At the time interval where the crossing is finished, the recorded
evaluation variableswere transmitted to the optimizing procedurewhich after-
wards computed new trajectories of the disturbance compensation at exactly
the same time interval andmade them available for the next crossing to start
at the following time interval.

5.3. Results

This section describes the results of the optimization on the basis of
measurements at the testbed. The recurring excitation over the track sec-
tion in question is displayed in Fig.7 to Fig.9 in the top left-hand cor-
ner. In the translatory direction, the chassis was at first excited by a si-
nusoidal signal and subsequently by three steps in the lateral direction, re-
spectively inversely in the vertical direction. Additionally, a superposed si-
nusoidal rotation around the longitudinal axis of the chassis took effect.
The diagrams on the lower left display the disturbance trajectory acquired
by repeated runs in the course of optimization. The plot shows the cha-
racteristics as follows: at the outset of the optimization as a dotted line,
after five crossings as a broken one, and as an unbroken line after 50 cros-
sings. The part on the right displays the corresponding plots of the objective
functions w.

Fig. 7. Evaluation: lateral



588 J.Lückel et al.

Fig. 8. Evaluation: vertical

Fig. 9. Evaluation: rotatory

All in all, the optimizationmethod has proved its ability to nearly entirely
compensate movements of the car body for periodically recurring excitations.
After only five crossings, the amplitudes of the body motion fell below 10%.
After 50 crossings, the carriage bodywas nearly in the position of rest in spite
of the excitation.

Another aspect is made clear in the comparison between the excitation
behaviours and the corresponding disturbance compensation. With vertical
motion these variables converge closely to the excitation, while with torsion
and lateralmotion there remainsignificantdifferences evenafter 50 repetitions.
This is due to the coupling of motions. A lateral excitation will always bring



Online optimization of a preview controller... 589

about a torsion in the car body; andvice versa – rotation of the chassis around
the longitudinal axis will always affect the lateralmotion of the car body.This
is why the disturbance compensation has to take into account these couplings,
the result being the behaviour shown.On the other hand, the car bodymotion
in the vertical direction is decoupled from the other degrees of freedom – thus
thedisturbancecompensationwill onlyhave todealwith theexcitationportion
in this direction.

6. Conclusion

It was shown that the using of the existing data processing infrastructure
for the exchange of collected data can be effecive for the compensation of
recurring stationary disturbances. The realization at a testbed confirmed the
advantages of this approach. However, itmust not be ignored that trajectories
of the compensation do not represent the disturbances themselves. Rather
they are optimized with respect to a particular vehicle. At the testbed, this
causes no problem because dynamics of the testbed does not change during a
test. Of course in reality different vehiclesmust be considered. In this case, the
compensation adapted to a particular vehicle cannot be used. Nevertheless, in
order to be able to use the presented method, independent information on a
vehicle must be stored. For this purpose, actual track characteristics are ideal
which can be determined by observation from the respective system response
of an individual vehicle. In this way, the approach introduced here can be
generalized to different types of vehicles.

To prove the convergence of the learning algorithm, a simplified conver-
gence analysis was performed by usingmethods of digital control theory. The
disign criteria for the parameters of the algorithmwere derived from this ana-
lysis, so they can easily be chosen. Thus themethod represents a goodway to
improve dynamic behaviour for repetitive motions. It can also be transferred
to other applications which show similar characteristics.

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Online optimization of a preview controller... 591

Bieżąca optymalizacja sterowania typu ”preview”, jego struktura

i algorytmy optymalizacji

Streszczenie

Układy zawieszeń aktywnych stosowane są w celu zwiększenia komfortu i bezpie-
czeństwa ruchupojazdów.Optymalne parametry pracy takich zawieszeńuzyskuje się,
gdy zaburzenia od drogi znane są z góry. Zwykle jednak zaburzenia te mogą być do-
piero zmierzone, gdy zaczynają już oddziaływać na pojazd.Wpracy zaprezentowano
metodę kompensacji zakłóceń ruchu na przykładzie pojazdów szynowych porusza-
jących się w sieci. Informacje o nierównościach toru pojawiających się stacjonarnie
zbierane są i przechowywane dla poszczególnych sekcji toru. Poprzez kolejne przejaz-
dy pociągów wzdłuż tych sekcji informacje są iteracyjnie optymalizowane tak, aby
mogły zostać wykorzystane przez następny pociąg. W artykule opisano zapropono-
wany algorytm optymalizacji. Kryteria strukturalne procedury zaczerpnięto z teorii
sterowania cyfrowego. Algorytm wdrożono do stanowiska doświadczalnego modelu
pociągu o trzech stopniach swobody. Badania pokazały skuteczność wprowadzonej
metody sterowania. Rezultaty badań przedyskutowano także pod kątem możliwości
ich uogólnienia na inne typy pociągów.

Manuscript received May 4, 2005; accepted for print May 23, 2005