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Matlab and Ode Solvers

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Lab 0: MATLAB and ODE Solvers | ME 4173 Robot Kinematics

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The following report will display the results and conclusions of an experiment to simulate the output of an inverse pendulum system in MATLAB. The objectives of this experiment were to review MATLAB programming and using MATLAB to simulate ODEs and systems.

* Examine the basics of MATLAB

* Use MATLAB to simulate a system

* Use ODE solvers to numerically integrate the system over a set time period
The apparatus used in this experiment was MATLAB. It was used to provide a simulation environment to analyze the inverse pendulum’s motion.
Experiments and Results
There were six components of this experiment. This experiment was mostly familiarizing with MATLAB. All code used is illustrated in the Appendix – Code.

The first part consisted of learning commands within the MATLAB environment. It was a brief overview of how commands work in MATLAB. There was no code used in the part of the experiment.

The second part of the experiment examined how arrays were created and used in MATLAB. The first step was to create a matrix. This matrix was then subjected to various commands including eye( ), zeroes( ), and ones( ). Indexing was also used to access various parts of the matrix. Matrix operations such as transpose, inverse, size and length were also shown.

Part three of the experiment explained how a script was created and what it was used for. A script file was created and used for other parts of the lab.
Part four illustrated what a function was and how they are created. It explained how syntax was important in the naming and calling of the function. This allowed the user to write a function and save in within a directory. This function could be called by the function name and input variables and return an answer depending on what the function was coded to perform. An addTwo(a,b) function was created as an example.

Part five contained the method for utilizing MATLAB`s ordinary differential equation solver (ODE). It uses this solver to perform numerical integration which allows systems to be solved within the MATLAB environment. MATLAB contains a number of ODE solvers: ode23, ode113, ode15s, ode23s, ode23t, ode23tb, ode45. An example of an ODE solver was used to illustrate the process of utilizing the solver.

Part six of the experiment was the simulation of an inverted pendulum. The pendulum was simulated using the ODE solver. Before the ODE solver could be used, the ordinary differential equations for the inverted pendulum needed to be developed and placed into state space variables. This was done by creating a function called xdot( ). The function has two element: xdot(1) and xdot(2). The two were plotted using a plot( ) function in MATLAB. The result was two sinusoidal waves which were cyclic in nature.
The first problem which arose concerned the trigonometric functions within MATLAB. The code was written in degrees but the answer was expected to be in radians. It was giving a response which looked like an analog signal which was growing exponentially. Once the code was changed to display in radians, there was another issue. The code was checked by the TA and verified to be correct but the response was not what was expected. The issue was actually within MATLAB. Another MATLAB application was used with the same code and the issue was resolved.
The experiment was much more difficult than anticipated but that was due to small errors in syntax and in the software itself. The expected results were achieved in all questions of the experiment. The analysis of the inverted pendulum was completed using the ODE solver within MATLAB. The plot revealed two undamped sinusoidal waves which represented the angular velocity and angle of the system with a specific input.

Appendix- MATLAB Code * Part 2
A = [1 2 3 4 5; 6 7 8 9 10];


x = [1 2 3 4 5] x =
1 2 3 4 5 x(1) x(end)

size(A) length(x) * Part 4 function var = addTwo(a,b) var = a + b;


* Part 5
function ydot = rates(t,y) ydot(1,1) = y(3); ydot(2,1) = y(4); ydot(3,1) = -0.1; ydot(4,1) = 0.2;

Plotting y0 = [0;0;0;0]; %initial conditions tInitial = 0; %initial time tFinal = 10; %final time
[tout, yout] = ode23(’rates’,[tInitial tFinal],y0); % call to the ode solver

* Part 6
% Function xdot with u = 0

function xdot = func(t,x) m = 2;
M = 8;
L = 1; g = 9.8; xdot = zeros(2,1); xdot(1) = x(2); xdot(2) = 2*g*sind(x(1))-m*L/(2*(m+M))*x(2)^2*sind(2*x(1))/(4*L/3-m*L*cosd(x(1))^2/(M+m));

Plotting Results close all clear all clc x = [5,0];

[tOut , xOut] = ode45(@func, [0 60], x);

plot (tOut, xOut)…...

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...Lab 2: Linear Time-Invariant Systems In this experiment, you will study the output response of linear time-invariant (LTI) systems using MATLAB, and learn how to use MATLAB to implement the convolution sum. You will also investigate the properties of LTI systems. The objective of this experiment is: (1) to study how to compute the output of LTI systems, and (2) to study the properties of discrete-time LTI systems. 1. Introduction to Linear Time-Invariant (LTI) Systems In discrete time, linearity provides the ability to completely characterize a system in terms of its response [pic] to a signal of the form [pic] for all [pic]. If a linear system is also time-invariant, then the responses [pic] will become [pic]. The combination of linearity and time-invariance therefore allows a system to be completely described by its impulse response [pic]. The output of the system [pic] is related to the input [pic] through the convolution sum as follows: [pic] Similarly, the output [pic] of a continuous-time LTI system is related to the input [pic] and the impulse response [pic] through the following convolution integral: [pic] The convolution of discrete-time sequences [pic] and [pic] represented mathematically by the expression given in [pic] can be viewed pictorially as the operation of flipping the time axis of the sequence [pic] and shifting it by [pic] samples, then multiplying [pic] by[pic] and summing the resulting product......

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...Memulai Menggunakan Matlab Matlab merupakan bahasa canggih untuk komputansi teknik. Matlab merupakan integrasi dari komputansi, visualisasi dan pemograman dalam suatu lingkungan yang mudah digunakan, karena permasalahan dan pemecahannya dinyatakan dalam notasi matematika biasa. Kegunaan Matlab secara umum adalah untuk : • • • • • Matematika dan Komputansi Pengembangan dan Algoritma Pemodelan,simulasi dan pembuatan prototype Analisa Data,eksplorasi dan visualisasi Pembuatan apilikasi termasuk pembuatan graphical user interface Matlab adalah sistem interaktif dengan elemen dasar array yang merupakan basis datanya. Array tersebut tidak perlu dinyatakan khusus seperti di bahasa pemograman yang ada sekarang. Hal ini memungkinkan anda untuk memecahkan banyak masalah perhitungan teknik, khususnya yang melibatkan matriks dan vektor dengan waktu yang lebih singkat dari waktu yang dibutuhkan untuk menulis program dalam bahasa C atau Fortran. Untuk memahami matlab, terlebih dahulu anda harus sudah paham mengenai matematika terutama operasi vektor dan matriks, karena operasi matriks merupakan inti utama dari matlab. Pada intinya matlab merupakan sekumpulan fungsi-fungsi yang dapat dipanggil dan dieksekusi. Fungsi-fungsi tersebut dibagi-bagi berdasarkan kegunaannya Kuliah Berseri IlmuKomputer.Com Copyright © 2004 IlmuKomputer.Com yang dikelompokan didalam toolbox yang ada pada matlab. Untuk mengetahui lebih jauh mengenai toolbox yang ada di matlab dan......

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...An Introduction to Matlab for Econometrics John C. Frain TEP Working Paper No. 0110 February 2010 Trinity Economics Papers Department of Economics Trinity College Dublin An Introduction to MATLAB for Econometrics John C. Frain. February 2010 ∗ Abstract This paper is an introduction to MATLAB for econometrics. It describes the MATLAB Desktop, contains a sample MATLAB session showing elementary MATLAB operations, gives details of data input/output, decision and loop structures, elementary plots, describes the LeSage econometrics toolbox and maximum likelihood using the LeSage toolbox. Various worked examples of the use of MATLAB in econometrics are also given. After reading this document the reader should be able to make better use of the MATLAB on-line help and manuals. Contents 1 Introduction 1.1 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The MATLAB Desktop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7 1.2.8 1.2.9 ∗ Comments 4 4 6 6 7 8 8 9 9 9 The Command Window . . . . . . . . . . . . . . . . . . . . . . . . The Command History Window . . . . . . . . . . . . . . . . . . . The Start Button . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Edit Debug window . . . . . . . . . . . . . . . . . . . . . . . . The Figure Windows . . . . . . . . . . . . . . . . . . . . . . . . . . The Workspace Browser . . . . . . . . . . . . . . . . . . . . . . . .......

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Ode to Autumn

...Conspiring with him how to load and bless With fruit the vines that round the thatch-eves run; * Ah, so now the sun and autumn are "conspiring," eh? Looks like we might have to separate the two of them. What are they whispering about over there? * OK, so not quite as thrilling as we thought. They are planning how to make fruit grow on the vines that curl around the roofs ("eves") of thatched cottages. * The image highlights the weight of the fruit as it "loads" down the vines. * Thatched cottages suggest a pastoral setting, characterized by shepherds, sheep, maidens, and agriculture. The "pastoral" as a literary genre was thought to originate in Ancient Greece, and the ode is a Greek form, so it is appropriate for this ode to include pastoral themes. Keats's other Great Odes, especially "Ode on a Grecian Urn," include similar imagery. Lines 5-6 To bend with apples the moss'd cottage-trees, And fill all fruit with ripeness to the core; * Keats is going nuts (pun!) with images of weight and ripeness. The richness here is like Willy Wonka's Chocolate Factory set in an orchard. * The apples "bend" down the branches of mossy trees with their weight. The trees belong not to some big farming cooperative, but to the simple cottages of country folk. * The ripeness penetrates deep to the very center of the fruit. They're not like those apples that look delicious until you take a bite and realize that the fruit is hard and sour. No, these babies are......

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