Optimization and cost-effectiveness for
estimating shock absorber states
MANI SANAEI NURMI
Master’s Degree Project
Stockholm, Sweden November 2012
XR-EE-RT 2012:032
Abstract
The role of the hydraulic shock absorber is to damp oscillations and
absorb energy which has a direct impact on the driving. Öhlins Racing has succeeded to produce a sophisticated semiactive technique; CES
(Continuously Controlled Electronic Suspension), where the damper is
controllable and therefore adaptable for various driving situations.
With the knowledge of difficulties in detecting useful information about
the damper-velocity at high frequencies, the aim of the thesis is to find a
method which can improve the estimated velocity of a damper mounted
on a motorcycle. Sensor fusion is a method which takes multiple measurements into account in an effort to reach an optimal estimate.
The results showed that a Kalman filter which estimated the dampervelocity generated useful information about the sprung and the unsprung mass of the motorcycle. By measuring the position and acceleration of the sprung and the unsprung mass, time delays that are
retarding the system could be reduced.
Contents
1 Introduction
1.1 Background . . . . .
1.2 Problem description
1.3 Objective . . . . . .
1.4 Delimitation . . . . .
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3 Theory and solution method
3.1 Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Model evaluation
4.1 Past year’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Model development . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Influence of noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Results
5.1 Kalman filter compared to a low pass filter . . . . . . . . . . . . . .
5.2 Today’s filter compared to last year’s filter . . . . . . . . . . . . . . .
5.3 Parameter adjustments . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Conclusion and future work
6.1 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Appendices
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A Kalman filter LTI
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2 System description
2.1 Motorcycle Suspension
2.2 External sensors . . .
2.2.1 Position Sensor
2.2.2 Accelerometer .
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B Models
31
C Theory
C.1 White noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
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33
Bibliography
35
Chapter 1
Introduction
1.1
Background
Öhlins Racing is a world leading manufacturer of advanced suspension systems
where the development of active suspensions is at the forefront.
There are two types of suspensions; Active and semiactive suspensions. In active
suspensions an external force can be implemented to improve the riding comfort.
Semiactive suspensions will dissipate energy and no external force needs to be implemented which is a more cost effective way of addressing the problem.
Öhlins were among the first to develop the CES (Continuously Controlled Electronic
Suspension)- technique, used in many modern vehicles such as BMW, Mercedes and
Audi.
The CES-damper comprises an electronically controlled valve and gives the shock
absorber the ability to control the pressure drop in the damper.
Figure 1.1. Illustration of Öhlins shock absorber
1
CHAPTER 1. INTRODUCTION
1.2
Problem description
A motorcycle can be equipped with sensors to get valid information of its states.
Figure 1.2 illustrates a position signal and its path through the system and can be
described as follows:
A signal is detected from the position sensor and will be sent to an electronically
controlled unit; ECU, which in turn sends the signal to an A/D converter responsible for modifying it to a discrete signal. Next, the signal is filtered allowing a
collection of valid information. After filtration and derivation the signal is sent to
a software program for evaluation. Finally a generated current will be sent to the
shock absorber valve and the flow of the viscous damper can be controlled.
From the moment the signal is detected by the sensor to which the valve receives
the current there are time delays where the main part of it is in the filtration stage.
In a master’s thesis presented by Daniel Kvalden and Andreas Johansson [1] from
the University of Linköping the results showed that by applying a Kalman filter the
delays could be reduced.
Figure 1.2. Öhlins system description
2
1.3. OBJECTIVE
1.3
Objective
The main goal of this thesis is to establish a model describing the surroundings of
the damper which includes the sprung and the unsprung mass of the motorcycle. A
previous master’s thesis[1] has aimed to locally studying the controllable behavior
of the damper and the next step is to study its surroundings.
A interpretation of the system aims to be evaluated and implemented in a state
space model describing the damping sequence of the motorcycle. When the model
has been described, the next step is to use the model in a linear Kalman filter which
will estimate the velocity of the damper. Finally, after finding the optimal filter an
economical evaluation will be performed, comparing the effects of the number of
types of sensors used.
1.4
Delimitation
A state space model will be chosen and applied in the filtering process which is very
time consuming and therefore a delimitation will be done regarding the control
current affecting the valve.
3
Chapter 2
System description
2.1
Motorcycle Suspension
The task of the hydraulic shock absorber is to damp oscillations. It works such that
the shock absorber generates a force opposite to the velocity of the shock absorber.
The velocity is defined as a vertical movement relative the road surface and can
be defined positively at compression stroke and negatively at rebound stroke. The
hydraulic shock absorber consists of a spring and a hydraulic damper with a gas
chamber where the flow in the hydraulic damper is controlled by a valve. The
velocity of the damper is the relative velocity between the sprung and unsprung
masses; vrelative = vsprung − vunsprung .
Figure 2.1 shows a motorcycle suspension (Quarter-car Suspension Model)[2].
Figure 2.1. Quarter-car Suspension Model
A motorcycle with a semiactive suspension system consists of a rigid body connected
5
CHAPTER 2. SYSTEM DESCRIPTION
to the wheels by front and rear suspensions. The sprung mass includes the chassis,
engine, steering head and rider. The unsprung masses are the masses connected to
the wheel.
Irregularities in the road surface give rise to resonance conditions when the excitation frequency is equal to the natural frequency of the motorcycle. The sprung
mass of a motorcycle has a natural frequency of 0.5 Hz to 2 Hz and the unsprung
mass of type 12 Hz to 16 Hz[2].
2.2
External sensors
The cost of electronics for a given function is decreasing and more functions can be
attached for a given component size. The cost of sensors does not decrease similarly
[3].
2.2.1
Position Sensor
The position sensor is a linear potentiometer which is mounted on the shock absorber
and measures the position of the damper. By deriving the position signal the
velocity of the damper can be captured. Position sensors are in general expensive.
2.2.2
Accelerometer
The optimal input of Quarter-car Model system is the velocity of the road. The
closest we can get the desired input is the acceleration of the unsprung mass. Accelerometers can be attached on both masses and measures the acceleration of the
vehicle. Accelerometers are considerably cheaper than position sensors.
6
Chapter 3
Theory and solution method
3.1
Execution
The execution phase focused on simulating the created model.
3.2
Kalman filter
The Kalman filter is the archetype for signal fusion. Without knowing a nonmeasurable state x, different measurement signals can be framed so that the state
x can be estimated in an optimal way. The theory behind the Kalman filter is
retrieved from [4].
A linear system can be described as
ẋ = Ax + Bu + N v1
y = Cx + Du + v2
(3.1)
Equation (3.1) describes the state space model with the input u and where v1 and
v2 are white noise with intensity R1 , respectively R2 . The cross spectrum between
v1 and v2 is constant and equal to R12 .
In the modelled system there is an estimation error
x̃(t) = x(t) − x̂(t)
(3.2)
where x̂ is the predicted state. An observer that minimizes the estimation error is
available
x̂˙ = Ax̂(t) + Bu + K(y(t) − C x̂(t) − Du(t)
(3.3)
where K is given by
K = (P C T + N R12 )R2−1
and where P is the symmetric positively semi definite solution to the matrix equation
7
CHAPTER 3. THEORY AND SOLUTION METHOD
AP + P AT − (P C T + N R12 )R2−1 (P C T + N R12 )T + N R1 N T = 0
which is called the riccati equation. The minimal variance of the estimation is given
by
E x̃(t)x̃(t)T = P
The equation (3.3) with the choise of K is called the Kalman filter. There is an
uncertainty in the system and which is formed as covariance matrices
h
T
E v1 (t)v1 (t + τ )
h
i
(
=
(
i
E v2 (t)v2 (t + τ )T =
Q if τ = 0
0, otherwise
)
R if τ = 0
0, otherwise
)
(3.4)
(3.5)
where v1 and v2 are stochastic white noise. Equation (3.4) and (3.5) performs the
functions as adjustments parameters.
The Kalman filter strikes a balance between the process and the measurement equation. The filtering process is digitalized and therefore the Kalman model must be
executed in discrete time. In this thesis the work will be done in Matlab and
Simulink by first creating a continuous model and then execute it with the help of
the commando kalman. A linear time-invariant system (LTI) is implemented for
maintaining linearity.
8
Chapter 4
Model evaluation
4.1
Past year’s model
In a master thesis from last year at Öhlins Racing Daniel Kvalden and Andreas
Johansson developed a state space model with a linear relationship between position,
velocity and acceleration1 .
4.2
Model development
The theory for building mathematical models is received from [5]. Building a mathematical model requires a work method distinguished into three phases:
The first phase is mainly focusing on structuring the problem including causations,
dependence and importance of different variables. After the analysis the system
will be divided into subsystems.
The second phase is focusing on the subsystems which the structuring at phase one
resulted in. The phase can be divided into two steps:
1. Connect the relationship between the variables and parameters in the subsystems.
2. Simplify the system by using approximations and idealizations.
In phase three the task is to organize the equations and expressions that were
developed in phase two.
The desired input of the motorcycle is the velocity of the road. Because of road
irregularities, problems with sensor placing and the absence of sensors that measures
the velocity, assumptions and idealizations must be done. There are also problems
of getting valid data of the tire because motorcycle manufacturers wont issue such
information. The developed model in Figure 4.1 gives the expression
1
See Appendix A
9
CHAPTER 4. MODEL EVALUATION
Figure 4.1. Representation of the developed model without the tire spring.
z = P osition between sprung mass and unsprung mass
d = P osition between unsprung mass and road
k = Spring constant
b = Damping coef f icient
m = W eight sprung mass
(4.1)
By studying (4.1) a force equation can be expressed as
¨
−mg − kz − bż = m(z̈ + d)
(4.2)
The damper-force of Figure 4.1 is
Fdamper = b(vsprung − vunsprung )
where b denotes the damping coefficient, vsprung is the velocity of the sprung mass
and vunsprung is the velocity of the unsprung mass.
The spring-force is
FSpring = k(Lg − Lg0 )
where k denotes the stiffness of the spring , Lg the initial length and Lg0 the length
of the deformed spring
From (4.1) and (4.2) a state space model can be obtained as
x1 = z = P osition between sprung mass and unsprung mass
x2 = ż = V elocity of the damper
x3 = Acceleration of the unsprung mass
10
(4.3)
4.2. MODEL DEVELOPMENT
The measurement signals of the model are
y1 = P osition of the damper
y2 = Acceleration sprung mass
y3 = Acceleration unsprung mass
(4.4)
The model must be observable to be able to estimate its states and therefore the
system has to be tested in the Matlab function obsv2 . The input of the Kalman
model is chosen as the gravitation g. Noteworthy is that the acceleration of the unsprung mass is still used as a measurement signal. By taking into account equations
(4.3) and (4.4) a modified state space model can be expressed as
ẋ1 = x2
1
ẋ2 = m
(−kx1 − bx2 ) − u − x3
ẋ3 = e(t)
u=g
where x1 is the position of the damper, x2 the velocity of the damper, x3 the
acceleration of the unsprung mass. Derivation of an acceleration is called jerk and
it’s seen as an perturbation modelled as white noise, e(t).
The measurement signals are
y1 = x1 + c(t)
1
y2 = m
(−kx1 − bx2 ) − u + v(t)
y3 = x3 + w(t)
The position y1 of the damper is measured by a position sensor. The acceleration of the sprung mass is y2 and y3 is acceleration of the unsprung mass. The
measurements are affected by perturbations which are modelled as white noise.
Finally, a model can be designed and expressed in state space form as

0

ẋ =  −k/m
0
1
−b/m
0

0
−1
0


0



−1
x
+


u + W
0
(4.5)

1

y =  −k/m
0
0
−b/m
0
0
0
1




x + 

0

−1  u + D
0
The covariance matrices of the model are

h
Q = WWT
2
i
σ12

= 0
0
See Appendix C
11
0
σ22
0
0
0
σ32



(4.6)
CHAPTER 4. MODEL EVALUATION

h
R = DDT
i
γ12

= 0
0
0
γ22
0
0
0
γ32



(4.7)
Matrix (4.6) and (4.7) shows that there are in total six adjustment parameters
which can be tuned. To simplify the calculations an assumption is made so cross
correlation does not exist.
4.3
Influence of noise
Simulating a real scenario requires that noise and disturbances must be taken into
account which in the tests are modeled as white noise. Because of the location
of the unsprung mass it’s more likely that its accelerometer will be more sensitive
to disturbances than the other two sensors. The Figues 4.2- 4.5 illustrates various
levels of noise generated in the simulation phase. The velocity of the damper can
be seen as a sine-wave defined positively at compression stroke and negatively at
rebound stroke.
Figure 4.2. An ideal noise free sinusoidal signal.
12
4.3. INFLUENCE OF NOISE
Figure 4.3. The shape of the sine wave can still be distinguished when a small
amount of noise is added to the ideal signal.
Figure 4.4. By adding more noise the shape of the sine wave can still be distinguished but not as clearly as in Figure 4.3.
13
CHAPTER 4. MODEL EVALUATION
Figure 4.5. The figure shows a very noisy signal were the shape of the sine wave
can’t be distinguished.
14
Chapter 5
Results
5.1
Kalman filter compared to a low pass filter
This project aims to study the damper on the rear part of the motorcycle and therefore the current mass is the half of the sprung mass. The spring constant and the
damper coefficient was received from Öhlins Mechatronics Research and Development Team.
The ideal velocity of the shock absorber is a continuous signal created in Simulink,
see Figure B.2 in Appendix B. The most essential thing is the ratio between the
ideal signal and the Kalman filtered signal, regardless the value of the velocity.
Figure 5.1 shows a test with Kalman model developed in (4.5), called "Kalman
filter LTI Phy", at 2Hz which corresponds to the natural frequency of the sprung
mass. The low pass-filtered signal is a derived and filtered position signal of order
two with a cut off frequency at 20 Hz. The Kalman filter estimate shows much less
time delay than the low pass filtered signal. However, there are still differences in
amplitude between the ideal signal and the Kalman filter which can be corrected
by adjusting the parameters in (4.6) and (4.7), see Figure 5.2. It was hard to find
a methodology tuning the parameters which had to be done manually.
The Kalman filter is not dependent on the noise affecting on the position signal
and the acceleration of the sprung mass signal, where the level of added noise could
be high as in in Figure 4.5 without affecting the results. The low pass filter is on
the other hand highly dependent on white noise acting on the position signal. The
values the of white noise were tuned in Figure B.3 in Appendix B.
The acceleration of the unsprung mass contributes with the largest noise impact
on the Kalman estimate. By adding noise corresponding to Figure 4.5 to the acceleration of the unsprung mass signal the results can be seen in Figure 5.3.
Figure 5.4 shows a test with the Kalman filter at 15 Hz which corresponds to
the natural frequency of the unsprung mass. The noise level on the unsprung mass
corresponds to Figure 4.4 and the Kalman estimate is following the ideal signal very
15
CHAPTER 5. RESULTS
Figure 5.1. Test at 2 Hz.
good while the low pass filter doesn’t. By increasing the noise level corresponding
to Figure 4.5 to the accelerometer on the unsprung mass at 15 Hz, the Kalman
estimate impairs which is illustrated in Figure 5.5.
5.2
Today’s filter compared to last year’s filter
A comparison is made between "Kalman filter LTI Phy" and the filter developed
from last years masters thesis, called "Kalman filter LTI 1". A test made at 2 Hz
is illustrated in Figure 5.6. "Kalman filter LTI Phy" shows less time delay than
"Kalman filter LTI 1".
By increasing the frequency to 15 Hz the "Kalman filter LTI 1" filter show its
strength, see Figure 5.7. The time delays are much smaller for "Kalman filter LTI
Phy" compared to "Kalman filter LTI Phy".
Figure 5.8 illustrates the sensitivity at 2 Hz on "Kalman filter LTI 1" when the
noise of the accelerometer on the unsprung mass corresponds to the noise level in
Figure 4.5. The results will impair but the amplitude of "Kalman filter LTI Phy" is
not changing that much. A difference of amplitude consisted even when adjusting
the covariance matrices Q and R in (4.6) and (4.7).
Figure 5.9 illustrates the sensitivity at 15 Hz of the "Kalman filter LTI 1" when the
noise of the accelerometer on the unsprung mass corresponds to Figure 4.5.
Even when the results impairs, the amplitude of "Kalman filter LTI Phy" does not
16
5.3. PARAMETER ADJUSTMENTS
Figure 5.2.
time delays.
Test at 2 Hz with adjusted parameters. The Kalman filter show no
change as much as for "Kalman filter LTI 1". A conclusion is drawn that "Kalman
filter LTI Phy" is more stable for higher levels of white noise.
5.3
Parameter adjustments
The tests intend to show what happens if the calculations are not precise. The
weight of the driver shifts dependent on carried equipment. The test at 2 Hz shows
in Figure 5.10 what happens when the sprung mass of the motorcycle is increased
with 25 percent. A reduction of the amplitude consists even when adjusting the
covariance matrices Q and R in (4.6) and (4.7).
A Test at 2 Hz in Figure 5.11 show what happens if the damping coefficient is
increased with 100 percent. A difference of amplitude consists even when adjusting
the covariance matrices Q and R in (4.6) and (4.7).
If the frequency increases to 15 Hz the results show that the estimation of the
damper-velocity will be more precise compared to parameter adjustments at 2 Hz
when the mass of the motorcycle is increased with 25 percent, see Figure 5.12. If
also the damping coefficient is increased with 100 percent at 15 Hz it shows that
the value of the damping coefficient is affecting the results such that there will be
a time delay, see Figure 5.13
Figure 5.14 and Figure 5.15 shows what happens if the mass is decreased with 25
17
CHAPTER 5. RESULTS
Figure 5.3. Test at 2 Hz. Illustration of the noise dependence of the accelerometer
attached on the unsprung mass.
percent at 2 Hz respectively 15 Hz.
At 15 Hz the results where much better than in the 2 Hz scenario and increasing
the weight of sprung mass showed only small time delays. The problems are in the
2 Hz scenario were it’s hard to receive any good results when increasing the weight
and damping coefficient and also when decreasing the weight of the motorcycle.
18
5.3. PARAMETER ADJUSTMENTS
Figure 5.4. Test 15 Hz. Note the large time delays on the low pass filter.
Figure 5.5. The figure shows the noise dependence of the accelerometer attached
on the unsprung mass at 15 Hz. A very noisy signal affects the estimated state of the
Kalman filter.
19
CHAPTER 5. RESULTS
Figure 5.6. "Kalman filter LTI Phy" shows less lag compared to "Kalman filter LTI
1".
Figure 5.7. Test at 15 Hz. "Kalman filter LTI 1" generates larger time delays
compared to "Kalman filter LTI Phy".
20
5.3. PARAMETER ADJUSTMENTS
Figure 5.8. Dependence of the noise acting on the accelerometer attaches on the
unsprung mass at 2 Hz. Note the amplitude difference between the filters.
21
CHAPTER 5. RESULTS
Figure 5.9. Dependence of the noise acting on the accelerometer of the unsprung
mass at 15 Hz. Both filter estimates will impair. Note the amplitude differnces
between the filters.
22
5.3. PARAMETER ADJUSTMENTS
Figure 5.10. Test at 2 Hz. The sprung mass is increased with 25 percent. A
reduction of the amplitude consists even when adjusting the covariance matrices Q
and R in (4.6) and (4.7).
Figure 5.11. Test at 2Hz. The damping coefficient is increased with 100 percent.
There are differences in amplitude between the signals.
23
CHAPTER 5. RESULTS
Figure 5.12. Test 15 Hz when the sprung mass of the motorcycle is increased with
25 percent. The Kalman filter show almost no time delays.
Figure 5.13. Test 15 Hz when the sprung mass of the motorcycle is increased with
25 percent and the damping coefficient is increased with 100 percent. The value of
the damping coefficient is affecting the results.
24
5.3. PARAMETER ADJUSTMENTS
Figure 5.14. Test at 2 Hz when the sprung mass of the motorcycle is decreased
with 25 percent. Note the amplitude differences.
Figure 5.15. Test at 15 Hz when the sprung mass of the motorcycle is decreased
with 25 percent. There are time delays in the system but it shows more stability than
in the 2 Hz scenario.
25
Chapter 6
Conclusion and future work
6.1
Conclusion and discussion
The linear time invariant Kalman filter gave less time delay than the previously
tested filters and improved the results at natural frequencies for both masses. The
more comprehensive model proved to be more reliable and less sensitive for disturbances, particularly at 15 Hz.
Multiple tests and adjustments were done without succeeding to adjust the amplitude levels at 2 Hz when the weight and damper coefficient were changed. At 15
Hz the results where much better and increasing the weight of sprung mass showed
only small time delays, while by increasing the damping coefficient time delays were
generated. Unfortunately there were no time to test the filter on a motorcycle.
There will be tests done this fall which will hopefully validate the improved results
from the simulation stage.
The filter is mostly dependent on the acceleration of the unsprung mass. By
adding large disturbances corresponding to Figure 4.5 the position sensor had only
a marginal impact which in a economical point of view means that a cheaper position sensor can be introduced, or even completely removed. An improvement of
the filtering stage can reduce time delays that are retarding the system described
in Figure 1.2. This means that the Kalman filter could both improve the estimated
states and also minimize the costs, which was the goal of the thesis.
6.2
Future work
In a future model it could be preferable to go deeper and try to create a more
comprehensive model. An object oriented modelling software program with the
advantage of supplying the user with component libraries could be of use. With
it’s ability to generate a state space model more variables and parameters could
be introduced. The assumption that disturbances are white noise can be expanded
27
CHAPTER 6. CONCLUSION AND FUTURE WORK
and an example is engine vibrations of the vehicle.
A linear time varying system can be of interest where the linear characteristics
facilitate the calculations. A more profound model can be evolved with the number
of revolutions taken into account.
Adjusting the covariance parameters were time consuming and it was hard to find
a methodology. Developing a program which speeds up the tuning phase could not
only improve the results but also be time saving.
Because of the lack of information regarding the tire the model had to be modified. The question is till how truthful the acceleration of the unsprung mass is as
an input. The limitation of the used sensors makes it hard to develop the model
any further. Therefore, a way of receiving more including information would be to
detect the road velocity by using laser position sensors which can be attached to a
short stick mounted on the front of the motorcycle.
28
Appendix A
Kalman filter LTI
In a master thesis from last year Daniel Kvalden and Andreas Johansson developed
a state space model with position, velocity and acceleration. The measurement
signals in the model are position, y1 and acceleration y2 .
x1 = P osition
x2 = V elocity
(A.0)
u = Acceleration
The state space equation of the model are
!
ẋ =
0 1
0 0
!
y=
1 0
0 0
!
x+
0
1
!
x+
0
1
u
u
and the covariance matrices are
!
Q=
σ12 0
0 σ22
!
R=
τ12 0
0 τ22
where there exist no cross correlation and therefore there are four adjustments
parameters in the system.
29
Appendix B
Models
The system is divided into subsystems. The sampling frequency is 400 Hz.
Figure B.1. Illustration of the model created in Simulink. The input of the system
is a sine wave which represents the acceleration of the unsprung mass. The system is
divided in to subsystems.
31
APPENDIX B. MODELS
Figure B.2. A representation of the motorcycle system in Figure 4.1. The ideal
signal is the velocity between the sprung mass and the unsprung mass.
Figure B.3. After usable and necessary signals are detected from the motorcycle
model disturbances will be added on each signal which will be the measurement
signals in the Kalman model,
32
Appendix C
Theory
C.1
White noise
This appendix describes the disturbances that are affecting the system. The theory
is retrieved from [6].
A signal that is generated from a stochastic process is called white noise. White
noise has it’s energy equally distributed over all frequencies and can’t be predicted
The stochastic sequence X(n) is called white noise if the mean function mX = 0
and
(
rX (k) =
2 k =0
σX
0 k = ±1, ±2, ...
)
If X(n) has a normal distribution with zero mean and standard deviation σ and no
cross correlation at any frequency it’s called white noise.
C.2
Observability
The system
ẋ = Ax + Bu
y = Cx
is observable if the observability matrix




O(A, C) = 


C
CA
.
.
CAn−1







33
APPENDIX C. THEORY
has full rank
34
Bibliography
[1] D.Kvalden and A.Johansson. Increased performance on a semiactive hydraulic
damper due to improved control. Master thesis LiTH-ISY-EX–11/4469–SE. Division of Automatic Control. Department of Electrical Engineering. Linköpings
Universitet. 2011-06-09.
[2] V.Cossalter. Motorcycle DYNAMICS. Race Dynamics, First English Edition
2002.
[3] H.Sohlström. EK1190 Mätteknik. Lab E4 Sensorer. KTH Elektro och systemteknik, 2011.
[4] L.Ljung and T.Glad. Reglerteori, Flervariabla och olinjära metoder. Studentlitteratur, 2003.
[5] L.Ljung and T.Glad. Modellbygge och Simulering. Studentlitteratur, 2004.
[6] P.Händel, R.Ottoson, H.Hjalmarsson. Signalteori, tredje upplagan, 2005.
35