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I am Mathew K. I am a Planetary Science Assignment Expert at eduassignmenthelp.com. I hold a Masters’s Degree in Planetary Science, The University of Chicago, USA. I have been helping students with their homework for the past 8 years. I solve assignments related to Planetary Sciences.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Planetary Science Assignments.
Linear Control Hard-Disk Read/Write Controller AssignmentIsham Rashik
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In this paper, a new method for designing a reduced order observer for linear time-invariant system is
proposed. The approach is based on matrix inversion with proper dimension. The arbitrariness associated with
the method proposed by O’Reilly is presented here and has been reduced with the help of pole-placement
technique. It also helps reducing the computations regarding the observer design parameters. Illustrative
numerical examples with simulation results are also included.
Different analytical and numerical methods are commonly used to solve transient heat conduction problems. In this problem, the use of Alternating Direct Implicit scheme (ADI) was adopted to solve temperature variation within an infinitesimal long bar of a square cross-section. The bottom right quadrant of the square cross-section of the bar was selected. The surface of the bar was maintained at constant temperature and temperature variation within the bar was evaluated within a time frame. The Laplace equation governing the 2-dimesional heat conduction was solved by iterative schemes as a result of the time variation. The modelled problem using COMSOL-MULTIPHYSICS software validated the result of the ADI analysis. On comparing the Modelled results from COMSOL MULTIPHYSICS and the results from ADI iterative scheme graphically, there was an high level of agreement between both results.
The cubic root unscented kalman filter to estimate the position and orientat...IJECEIAES
In this paper we introduce a cubic root unscented kalman filter (CRUKF) compared to the unscented kalman filter (UKF) for calculating the covariance cubic matrix and covariance matrix within a sensor fusion algorithm to estimate the measurements of an omnidirectional mobile robot trajectory. We study the fusion of the data obtained by the position and orientation with a good precision to localize the robot in an external medium; we apply the techniques of kalman filter (KF) to the estimation of the trajectory. We suppose a movement of mobile robot on a plan in two dimensions. The sensor approach is based on the CRUKF and too on the standard UKF which are modified to handle measurements from the position and orientation. A real-time implementation is done on a three-wheeled omnidirectional mobile robot, using a dynamic model with trajectories. The algorithm is analyzed and validated with simulations.
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In this paper, the authors have discussed and shown
how to tune the PID controller in closed loop with time-delay
for the double integrator systems for a particular stability
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the plant is known. As a case study the authors have consid-
ered the mathematical model of the real-time beam and ball
system and analyzed the simulation and real time response.
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IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
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spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
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When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
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Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
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3. 1 Prior
All edited code was completed in MATLAB and can be found in the
accompanying Gradescope submission.
The below work references both the project brief provided for this project
and Chapter 10: The Extended Kalman Filter in the accompanying course
notes.
2
4. 2 Introduction
The problem specified required the implementation of a state estimator
that tracks the X-coordinate, Y-coordinate and heading angle of a bicycle.
Figure 1 was provided in the project brief, and illustrates the system
model. X1(t) and Y1(t) are the X- and Y- coordinates of the rear wheel, B
is the wheelbase (m), γ(t) is the steering angle relative to the bicycle
frame, θ(t) is the heading angle and ω(t) is the angular velocity of the rear
wheel.
Figure 1: Simple Bicycle Model Used
The bicycle dynamics were predefined as follows:
˙x1(t) = v(t) cos (θ(t)) (1)
˙y1(t) = v(t) sin (θ(t)) (2)
˙θ(t) =
v(t)
B
tan (γ(t)) (3)
where v(t) is the linear velocity of the bicycle.
3
5. An idealized measurement model p(k) of the discretized position
coordinates was also specified in the project brief:
p(k) =
x1(tk) + 1
2
B cos (θ(tk))
y1(tk) + 1
2
B sin (θ(tk))
(4)
It is known that measurements are obtained approximately every 0.5
seconds, however the sensors are faulty and measurements are not
guaranteed. γ(t) and ω(t) are known with certainty at each time step.
The goal is to design an estimator that can minimize the error in the
prediction of the true position and heading angle of the bicycle at the final
time step.
4
6. 3 Estimator Selection and Design
This project was completed in MATLAB. The Extended Kalman Filter
(EKF) was chosen as an acceptable estimator for this project. The EKF
functions by linearizing the system equations about the current state
estimate, and applying a standard Kalman Filter prior and measurement
update equations to the linearized equations.
In this system, three states were defined for estimation: the X-coordinate
X, the Y-coordinate Y and the heading angle θ.
As shown above in Section 2, the bicycle dynamics were given in terms of a
continuous time model. The state space was discretized at Ts = 0.25s, as
the EKF equations are applied at discrete time intervals - although
measurements are obtained at approximately every 0.5 s, I found 0.25 s to
be optimal. The MATLAB inbuilt state-space model from the control
toolbox was used for this purpose. 1
The continuous time model was specified as follows, and are obtained from
linearizing the system equations about the current state estimate (which
depends on real-time measurements):
A(k) =
1 + v(t) cos (θ(k)) 0 0
0 1 + v(t) sin (θ(k)) 0
0 0 1 + v(t)
B
tan (γ(k))
(5)
B(k) =
0 0
0 0
0 0
(6)
H(k) =
1 0 −0.5B sin (θ(k))
0 1 0.5B cos (θ(k))
(7)
1
Interestingly, I accidentally initially used the continuous model in place of the dis-
cretized model, and I actually obtained superior performance from the estimator. How-
ever, this approach is clearly incorrect, and so I sacrificed estimator performance for
modelling accuracy. I am also unable to explain and justify this behaviour!
5
7. D(k) =
0 0
0 0
(8)
The matrices L(k), M(k) are obtained from the process noise derivative of
the state equation and measurement noise derivative of the measurement
equation respectively. As the process noise
For this system, these matrices are defined as follows:
L(k) =
1 0 0
0 1 0
0 0 1
(9)
M(k) =
1 0
0 1
(10)
In this case, the position is measured and the heading angle is estimated.
However, given that the position measurements were not guaranteed at
each time step, a contingency was introduced to provide a measurement
estimate from the idealized model specified in Equation 4. When no
real-time measurements were provided, the position update was estimated
by Equation 11, in which α is a random variable in the range [0, 1] which
serves the purpose of added uniform noise.
z(k) =
0.99(x1(tk) + 1
2
B cos (θ(tk))) + 0.01α(x1(tk) + 1
2
B cos (θ(tk)))
0.99(y1(tk) + 1
2
B sin (θ(tk))) + 0.01α(y1(tk) + 1
2
B sin (θ(tk)))
(11)
There were certain other parameters that were defined to meet the
Extended Kalman Filter equation requirements:
V : approximation of the process noise variance,
W: approximation of the measurement noise variance, and
Pm: approximation of the prior conditional variance of the initial states
X,Y and θ.
The values for these parameters are shown in Equations 12, 13 and 14.
These values were converged upon by trial and error measuring the mean
6
8. and variance of the final error over all 93 data sets. However, the drastic
altering of these values did not have a huge impact on the final estimation
error.
V =
10 0 0
0 10 0
0 0 0.1
(12)
W =
0.05 0
0 0.05
(13)
Pm =
10 0 0
0 10 0
0 0 0.1
(14)
7
9. 4 Results
The output of the main() MATLAB function is illustrated in Figure 2 and
Figure 3 for two randomly selected data sets. It can be seen graphically for
both cases that the estimator appears to have a decent approximation of
both the X- and Y- coordinates appear to follow the path in both cases.
The approximation of the heading angle θ is very inaccurate in both data
sets and oscillates significantly which does not represent the true heading
angle.
Figure 2: Estimator Performance For First Data Set
Figure 3: Estimator Performance For 78th Data Set
8
10. Histograms illustrating the mean and variance of X, Y and θ are shown in
Figures 4, 5 and 6 respectively. This data was obtained by measuring the
final error over all 93 data sets. The observations of these histograms
quantify what was roughly observed from the graphs in Figure 2 and 3.
The position errors are normally distributed, however both resulting errors
have a positive bias of approximately 0.2 m. The distribution of the Final
X-coordinate error has a tighter variance than the Y-coordinate equivalent.
Figure 4: Histogram of Heading Angle Error at Final Position
9
11. Figure 5: Histogram of Heading Angle Error at Final Position
The distribution of the final heading angle error, however, is uniform and
random. Given that the range of possible values of θ is [−π, π], the
estimator does not estimate the heading angle with any precision. This
may be because no measurements of heading angle are received, and the
value of θ is initialized as π
2
, as it is stated in the project brief that the
cyclist is headed ”approximately North-East” for each of the test cases.
10
13. 5 Conclusion
As stated in the course notes, if the actual state and noise values are close
to the values that we linearize about, then the linearization is a good
approximation of the actual nonlinear dynamics. The underlying
assumption we make, when using the EKF, is that we can push a random
variable through a linear function and use the linearization to compute the
resulting random variable’s statistics.
In the case of Gaussian noise, these quantities are not guaranteed to be
small since the noise is actually unbounded.
The EKF provides only an approximation of the conditional mean and
variance, and not the true value. In the case of this bicycle system,
significant effort was made to tune the parameters V , W, Pm(0) along
with the initial conditions of the state. This tuning resulted in little
reduction of the variance of the final error of the state estimation. The
EKF can be applied in this system, as the simplified bicycle dynamics are
not extremely nonlinear and the distribution is unimodal. The use of a
simplified model and the lack of measurement data for each time step
result in added uncertainty of estimation.
However, in hindsight the Unscented Kalman Filter or the Particle Filter
would probably provide a more accurate state estimation of this system’s
true final position.
12