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### What is Modelling?

This blog is all about system dynamics modelling and simulation applied in the engineering field, especially mechanical, electrical, and ...

### Discussion of the Secant Method to Solve an Equation

Let's consider the following equation:

eˣ = 2x² + 1

We may write the above expression as,

f(x) = eˣ - 2x² - 1

To find the value of x or roots of the equation, we may apply the Secant method. In this method, the derivative may be approximated by a backward finite divided difference approach. So, the function may be expressed as,
This method is convenient when evaluating derivative for some function is difficult in Newton-Raphson method. Now, the above equation may be substituted in Newton-Raphson’s formula to get the following algorithm.
In contrast to Newton-Raphson method, two initial assumptions are made for the two unknowns’ xi-1 and xi.
For the proper initial guess for x, graphical method is applied to visualize the characteristics of the function. The MATLAB commands generate the plot of the function f(x). This function is plotted with respect to the values of x from -10 to +10. The following figure shows the function plot. From the plot, the function changes sign when x is between 2 to 3. So, the root lies in this interval. And, in the program developed for Secant method, two initial guesses for xi-1 and xi are 2 and 3 respectively.

However, more close observation reveals two roots, which are shown in the following figure. In this time, the range of x is decreased significantly for precise investigation between -2 to +2 as this interval seems unclear in the previous figure for the presence of any roots.

So, the above figure shows the existence of other two roots, which lie in between 0.1 to 1 and at the origin. The origin is obvious for this case, and there is no need for any further estimation by numerical methods. But, for the interval 0.1 to 1, the accurate and precise approximation is necessary. For this, the two initial guesses are set to 0.1 and 1 for xi-1 and xi accordingly.
The following Secant formula is implemented to approximate the two roots lie in the intervals 2 to 3 and 0.1 to 1. The results are shown after the program.

Secant Formula Implementation by a MATLAB Program

% Definition of a function "secant" to solve equation by Secant Method

function [x,ea] = secant(X,X0,etol)

format long;

% Input: X,X0, etol=Tolerance definition for error
% Output: x, ea=Calculated error in loop
% Iterative Calculations

while (1)

% Implementation of Secant Algorithm

solution = X-((exp(X)-(2*X^2)-1)*(X0-X))/((exp(X0)-(2*X0^2)-1)-(exp(X)-(2*X^2)-1));
solutionprevious=X;
X0=X;
X=solution;

if (solution-solutionprevious)~=0        % Approximate percent relative error

ea=abs((solution - solutionprevious)/solution)*100;

end

if ea<=etol,

break,
end

x = solution;

disp(x);

end

% Display of output parameters

disp(x);

disp(ea);

%disp(solution-(0.1*abs(ea)*solution));
end

Outputs:

>> secant (3, 2, 1e-5)
% Iterations

2.597424571529533
2.796706966294471
2.858723109017894
2.841817367110900
2.842657964680533
2.842673428005184
2.842673428005184

>> secant(1, 0.1, 1e-5)
% Iterations

0.308929151717719
0.570046187158243
1.157486960694727
0.682985699401547
0.723807551583747
0.742065590882227
0.740827136083267
0.740850398734311
0.740850398734311

So, the solutions for the equation are 2.842673428005184, 0.740850398734311 and 0.

#SecantMethod #RootsofEquations #Matlab #NumericalMethod #Blog #Blogger

### Modelling and Simulation of an Electric Vehicle in SIMULINK

In this tutorial, I am going to present you a very simple way to design and control your very first electric vehicle in SIMULINK. You need to have MATLAB version 2019/2020 with full features. Most of the features that I will be using are from Simulink and Simscape libraries. Simscape is a very powerful tool that Matlab has integrated lately. This tutorial will be straightforward - showing you the diagrams and explaining afterwards. At first, let's begin with a simple question, how an overall electric vehicle system looks like? Below is the image of the electric vehicle system.

As it appears, this is a Simulink block diagram, which contains many blocks from the library browser. Let's see what are the fundamental components of an electric vehicle system. From the diagram, there are five main blocks:

1. Vehicle Body
2. DC Motor
3. DC Motor Driver
4. Battery
5. Controller

## 1. Vehicle Body

In this modelling, we can consider some basic properties of a vehicle. Below, we see the model of the vehicle body, which has four tires, a simple gear, and vehicle property block. When you click on each block, you will find the respective properties that you need to choose or enter independently. Just remember, these blocks represent the governing dynamics that Matlab has already prepared for us in the Simulink blocks. You do not need to spend hours for deriving system equations. Rather, here, we ought to be paying attention while connecting the blocks and providing appropriate parameters.

When you click on the vehicle body block, it will pop-up a new window as below. For this tutorial, I just chose the default parameters, but you can certainly change these for your specific problem.

The following is the tire property block.

This is how a simple gear block appears when you double click the block.

## 2. DC Motor

The purpose of the DC motor in electric vehicle is to drive the tires/wheels. It should be noted that from the Simscape library, you can choose different types of motors. The one side of the DC motor block is connected to the vehicle body blocks, and the other side is connected to the DC motor driver, which controls and rotates the motor. The following image shows the DC motor block in Simulink.

## 3. DC Motor Driver

To precisely control a DC motor, we are required to design a driver for that. There are numerous ways you may develop the motor controller. I used an H-Bridge block from Simscape library along with the PWM voltage controller block. In the following image, you see that there are some other blocks, such as, current and voltage sensors, electrical reference etc.

## 4. Battery

For any electric vehicle, battery is a very important part. Still, in industries, people are struggling to find a better solution for the battery that can run an electric vehicle with more mileage before it drains fully. Here, I have chosen a simple battery model from Simscape library where you can obviously tune the parameters for your design requirement. To implement the battery and connect it to the rest of the system, a controlled current source block is added. As before, we need to have an electrical reference for the current source block, which is depicted in the subsequent figure.

## 5. Controller

In newer versions Matlab with full features, you will find a block named 'Longitudinal Driver' that may act as a controller for the electric vehicle system. Below, we see the block parameters. I am using all the default properties for this block. You can decide the type of the controllers at first, then shift type, units, output gear signal, and so forth. There is a whole lot option to tune this block parameter for your specific goals.

We begin this tutorial with the final version of the Simulink system model of an electric vehicle. There were some blocks that were masked by some images, below you can see the original version of the complete system diagram for your first virtual electric vehicle model. As you can see, in this model, most of the blocks are from the Simscape library, and you cannot connect them directly to any other Simulink blocks. But, with a converter, you can easily make the connections. In Simscape, there are two blocks - PS-Simulink and Simulink-PS converter - which are used to convert physical signals from Simscape block into Simulink compatible signal and vice versa.

## Some Results

Everything is done. Now, if you would like to see the responses of different parts of the electric vehicle, you can do so. You need to add the oscilloscope blocks from the Simulink library to those parts.

Here is the source signal that is fed into the system to run the vehicle.

This is the signal that is an output from the longitudinal driver block.

This oscilloscope shows the current from a current source block.

That's all. We have designed a virtual model for an electric car. Certainly, this is not a Tesla as the picture of the vehicle body block shows in the first image. Nevertheless, we can at least have a little sense of accomplishment that a simple model may even be designed in our laptops.

#ElectricVehicle #Simulink #Simscape #Modelling #Simulation #DCmotor #Blog #Blogger

### Medical Image Segmentation with 3D Slicer: A Beginner's Guide

3D Slicer is an open source software that is widely used for image processing, visualization, filtering, and so forth. In this article, I would like to discuss its effectiveness and application in medical image segmentation. This software is completely FREE, and there is a continuous development of many add-ins/extensions for this platform. You may download the software from 3D Slicer Webpage. After downloading, the installation procedure is straightforward, you may also follow the instructions mentioned in their webpage.

Medical image segmentation to extract the size or volume of an organ or complex airways/channels from computed tomography (CT) or micro-computed tomography (𝛍CT) is very interesting and has been playing a crucial part in biomedical engineering. For example, human nasal cavities or airways have such a complex formation that from the CT scans, we are unable to extract the volume. However, with the power of modern image processing tools, we can manually trace the nasal airways path slice by slice and finally obtain the volume of the cavities.

In this tutorial, I am going to highlight the 3D Slicer's interface, and how we can do the manual/automatic segmentation for the purpose of extracting the volume or shape information. When you launch the software, you will see the following welcome window. The window has four sectional views. The three black windows show the transverse (or, axial), sagittal (or, longitudinal), and coronal (or, frontal) views of the CT images. The other blue colored section highlights the three dimensional reconstruction of the segmented images.

Medical images are often processed as .DICOM file, which can be easily opened in 3D Slicer. In the welcome window, there are options to load different format of data, but Digital Imaging and Communications in Medicine (DICOM) images are smoothly handled by the 3D Slicer. In the following, I have opened a CT scan of a human brain with a tumor, which is available in 3D Slicer's sample data for demonstration. Our task here is to find the volume of the tumor quantitatively from the CT scan images so that physicians can have more in-depth information regarding the size and growth of the tumor. They can make their next decisions confidently based on these data.

To do so, at first, we need to open the segment editor, which is located at the top of the window where you will see an icon says 'All Module'. Click on it and then, select the segment editor as highlighted below.

Once you select the segment editor, a new small window will pop-up just like the screenshot shown beneath. You will find some features in that new little window, such as, 'ADD', 'Show 3D' etc.

You need to choose the option 'Add (+)' and then, a new segment will be created named as 'Segment_1' (see below). From the screenshot, you see that a new segment is ready to carry on for further operation.

Now, if you double click on the 'Segment_1', you will see the following options will appear to select the segmentation nature. For example, if we segment a bone, then choose 'bone' in that window; or if the segmentation is related to the brain or tooth, select the respective 'brain' and 'tooth' segmentation features. In our example, it is a tumor that we are going to segment. A tumor is a mass of abnormally growth of tissues, so we can choose the segmentation media as 'tissue'.

So, we have decided how we are going to segment the tumor, now it's time to move forward using the tools to do the segmentation. There are two types of segmentations: automatic and manual. To understand the difference between the automatic and manual segmentation, let me first explain how the CT scans are formatted or processed for image analysis. There are many single images that form the CT scan of an organ, which alternatively implies that a CT scan itself an image that has subsets of many images. So, when we consider automatic segmentation, it is a process of tracing or narrowing a particular section from the CT scan that is done by the software, and manual segmentation is that you need to paint or trace that particular region by hand for all the individual images that eventually form the CT scan. As you see below, the red circled area is the tumor that we need to separate out and get the volume of it. We also see that there are some options, like 'Paint', 'Draw', 'Level Tracing' to trace the tumor. Here, I used the 'Level Tracing' option for that, traced few images or slices and then used the 'Fill between slices' option as an automatic segmentation approach.

Once the segmentation is done, then we can estimate the volume of the tumor. For that, select again 'All Modules', and from the dropdown menu, click on the 'Quantification', and then hit 'Segment statistics'.

After that, at the bottom of your window, you will find the volume of the segmented tumor. From the screenshot below, we see the volume of the tumor is approximately 346.99 mm³. Now, remember, this is not an accurate or quantitatively perfect representation of the tumor. However, it gives a reasonable view and idea about the shape and size of the tumor. Below, we can see the looks of the tumor in a three dimensional space.

#ComputedTomography #3DSlicer #ImageSegmentation #ImageProcessing #3DModelling #Blog #Blogger

### A Taylor Series Expansion to Approximate a Function near a Point

The Taylor series expansion up to seventh terms for approximation of near point x =1 is,

Letting starting point xₒ is at 0 and the step size (xi - xₒ) is 1. Now, the value of eˣ at x = 1 is to be calculated by Taylor series expansion on the basis of the starting point and its derivatives at x = 0 with step size 1. A MATLAB program is written to evaluate this value from zero order approximation to sixth order approximation and show the improvement of accuracy and minimization of truncation error as well.
The true value of is 2.718281828459046. The truncation error is the difference between true and approximate values. The following MATLAB program describes each steps of the calculation.

Taylor Series Implementation by a MATLAB Program

% Taylor Series expansion to approximate exp(x) near point x=1

function e = taylorseries(x0,x1,h)

e0=exp(x0);              % calculates exp(x) value when x=0

derivative1=1;           % all the six derivatives calculated at x=0
derivative2=1;
derivative3=1;
derivative4=1;
derivative5=1;
derivative6=1;

% zero order approximation

e=e0;
error=exp(x1)-e;
disp (e);        % display of exp(x) and error at command prompt
disp(error);

% first order approximation

e=e0+(derivative1*h);
error=exp(x1)-e;
disp (e);
disp(error);

% second order approximation

e=e0+(derivative1*h)+(derivative2*(h^2/factorial(2)));
error=exp(x1)-e;
disp (e);
disp(error);

% third order approximation
e=e0+(derivative1*h)+(derivative2*(h^2/factorial(2)))+(derivative3*(h^3/factorial(3)));
error=exp(x1)-e;
disp (e);
disp(error);

% fourth order approximation
e=e0+(derivative1*h)+(derivative2*(h^2/factorial(2)))+(derivative3*(h^3/factorial(3)))+(derivative4*(h^4/factorial(4)));
error=exp(x1)-e;
disp (e);
disp(error);

% fifth order approximation
e=e0+(derivative1*h)+(derivative2*(h^2/factorial(2)))+(derivative3*(h^3/factorial(3)))+(derivative4*(h^4/factorial(4)))+(derivative5*(h^5/factorial(5)));
error=exp(x1)-e;
disp (e);
disp(error);

% sixth order approximation
e=e0+(derivative1*h)+(derivative2*(h^2/factorial(2)))+(derivative3*(h^3/factorial(3)))+(derivative4*(h^4/factorial(4)))+(derivative5*(h^5/factorial(5)))+(derivative6*(h^6/factorial(6)));
error=exp(x1)-e;
disp (e);
disp(error);
end

Table 1: Outputs from the Above Program to Approximate at x=1

The above table tells us as the order of the Taylor series increases the truncation error also decreases. However, after sixth and higher order approximations, improvement of the accuracy is not much significant. So, we can stop the approximation to first seven terms of the Taylor series and the approximate value of ex near point x = 1 is 2.718.

### Finding Roots of a Equation by Newton-Raphson Method

In this tutorial, we are interested to the roots of the following function.

f(x) = sin x + 2 - 0.1 x

To find the value of x or roots of the equation, Newton-Raphson method is applied. For the proper initial guess for x, graphical method is applied to visualize the characteristics of this nonlinear function. This graphical technique also helps to find multiple roots for the unknown function. The following MATLAB commands generate the plot of the function, f(x). This function is plotted with respect to the values of x from -50 to +50.

>> X=-100:100;
>> plot(X,(sin(X)+2-(0.1*X)))

From the following figure 1, it is clear that the equation has seven roots and plotting the function helps to get the roots. The function changes its sign from, x = 10 - 11, 11 -12, 16 -17, 18 - 19, 21 - 22, 25 - 26, and 27 - 28. And, the roots of the equation must lie between these intervals. Now, to get good approximations of the roots, a Newton-Raphson algorithm is written in MATLAB.

Figure 1: Plot of the function f(x) to identify the intervals where roots of the function lie.

The program is developed by MATLAB user defined function called “sinusoidal”. The initial assumptions for approximating seven roots of x are 10.5, 11.5, 16.5, 18.5, 21.5, 25.5 and 27.5 respectively. The codes are given below. It is straightforward to run the program seven times as we have seven initial guesses for the corresponding roots.

## Newton-Raphson Algorithm by MATLAB

% Definition of a function "sinusoidal" to solve equation by Newton-Raphson
function [x,ea] = sinusoidal(X, etol)
format long;

% Input: X=21.5, etol=Tolerance definition for error
% Output: x=Solution of equation, ea=Calculated error in loop

% Program Initialization

solution = 21.5;

% Iterative Calculations

while (1)
solutionprevious = solution;

% Implementation of Newton-Raphson Method

solution = X - (sin(X)+2-(0.1*X))/(cos(X)-0.1);
X=solution;

if solution~=0                         % Approximate percent relative error

ea=abs((solution - solutionprevious)/solution)*100;
end

if ea<=etol,                  % Condition to meet specified error tolerance

break,
end

x = solution;

disp(x);

end

% Display of output parameters

disp(x);
disp(ea);

end

## Program Outputs:

>> sinusoidal (10.5,1e-10)
ans =
10.636782424281707
>> sinusoidal(11.5, 1e-10)
ans =
11.562055865585652
>> sinusoidal (16.5,1e-10)
ans =
16.107752970689450
>> sinusoidal (18.5,1e-10)
ans =
18.721338781142521
>> sinusoidal (21.5,1e-10)
ans =
21.809224297055675
>> sinusoidal (25.5,1e-10)
ans =
25.744697403517950
>> sinusoidal (27.5,1e-10)
ans =
27.435908953388580

So, the seven roots from the program are 10.637, 11.562, 16.108, 18.721, 21.809, 25.745 and 27.436. In all the cases, the error tolerance limit is kept extremely small (0.0000000001) to get more accurate and precise results.

#Blog #Blogger #NewtonRaphson #Matlab #Sinusoidal