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Initial GitHub commit. Modifications listed below based on version 1.0 on the MATLAB File Exchange.
Fixes - Fixed control continuity constraints (should not have been present) Changes - PS_Fmatrix requires the derivative function to be vectorized now Features - Added capability to solve variable final time problems - Added moon lander example (variable horizon problem) - Added defaults options function (see the function for the options)
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‎INSTALL_Basic_Pseudospectral.m

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%--------------------------------------------------------------------------
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% INSTALL_Basic_Pseudospectral
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% This scripts helps you get this code up and running
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%--------------------------------------------------------------------------
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% Automatically adds project files to your MATLAB path, downloads the
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% required files, and opens an example.
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%--------------------------------------------------------------------------
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% Install script based on MFX Submission Install Utilities
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% https://github.com/danielrherber/mfx-submission-install-utilities
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% https://www.mathworks.com/matlabcentral/fileexchange/62651
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%--------------------------------------------------------------------------
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% Primary Contributor: Daniel R. Herber, Graduate Student, University of
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% Illinois at Urbana-Champaign
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% Link: https://github.com/danielrherber/basic-multiple-interval-pseudospectral
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%--------------------------------------------------------------------------
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function INSTALL_Basic_Pseudospectral
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% add contents to path
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AddSubmissionContents(mfilename)
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% download required web files
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RequiredWebFiles
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% download required web zips
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RequiredWebZips
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% add contents to path
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AddSubmissionContents(mfilename)
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% open examples
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OpenThisFile('BD_main')
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% close this file
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CloseThisFile(mfilename) % this will close this file
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end
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%--------------------------------------------------------------------------
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function RequiredWebFiles
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disp('--- Obtaining required web files')
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% initialize index
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ind = 0;
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% initialize structure
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files = struct('url','','folder','');
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% file 1
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ind = ind + 1; % increment
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files(ind).url = 'http://dmpeli.math.mcmaster.ca/Matlab/Math4Q3/Lecture2-1/LagrangeInter.m';
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files(ind).folder = 'LagrangeInter';
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% file 2
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ind = ind + 1; % increment
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files(ind).url = 'http://www1.spms.ntu.edu.sg/~lilian/bookcodes/legen/lepoly.m';
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files(ind).folder = 'lepoly';
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% obtain full function path
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full_fun_path = which(mfilename('fullpath'));
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outputdir = fullfile(fileparts(full_fun_path),'include');
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% download
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DownloadWebFiles(files,outputdir)
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disp(' ')
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end
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%--------------------------------------------------------------------------
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function RequiredWebZips
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disp('--- Obtaining required web zips')
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% initialize index
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ind = 0;
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% initialize structure
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zips = struct('url','','folder','','test','');
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% zip 1
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ind = ind + 1; % increment
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zips(ind).url = 'https://github.com/altmany/export_fig/archive/master.zip';
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zips(ind).folder = 'MFX 23629';
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zips(ind).test = 'export_fig';
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% zip 2
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ind = ind + 1; % increment
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zips(ind).url = 'http://www.mathworks.com/matlabcentral/mlc-downloads/downloads/submissions/40397/versions/7/download/zip/mfoldername_v2.zip';
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zips(ind).folder = 'MFX 40397';
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zips(ind).test = 'mfoldername';
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% obtain full function path
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full_fun_path = which(mfilename('fullpath'));
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outputdir = fullfile(fileparts(full_fun_path),'include');
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% download and unzip
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DownloadWebZips(zips,outputdir)
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disp(' ')
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end
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%--------------------------------------------------------------------------
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function AddSubmissionContents(name)
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disp('--- Adding submission contents to path')
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disp(' ')
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% current file
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fullfuncdir = which(name);
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% current folder
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submissiondir = fullfile(fileparts(fullfuncdir));
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% add folders and subfolders to path
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addpath(genpath(submissiondir))
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end
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%--------------------------------------------------------------------------
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function CloseThisFile(name)
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disp(['--- Closing ', name])
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disp(' ')
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% get editor information
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h = matlab.desktop.editor.getAll;
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% go through all open files in the editor
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for k = 1:numel(h)
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% check if this is the file
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if ~isempty(strfind(h(k).Filename,name))
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% close this file
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h(k).close
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end
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end
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end
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%--------------------------------------------------------------------------
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function OpenThisFile(name)
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disp(['--- Opening ', name])
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try
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% open the file
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open(name);
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catch % error
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disp(['Could not open ', name])
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end
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disp(' ')
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end
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%--------------------------------------------------------------------------
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function DownloadWebFiles(files,outputdir)
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% store the current directory
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olddir = pwd;
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% create a folder for outputdir
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if ~exist(outputdir, 'dir')
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mkdir(outputdir); % create the folder
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else
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addpath(genpath(outputdir)); % add folders and subfolders to path
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end
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% change to the output directory
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cd(outputdir)
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% go through each file
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for k = 1:length(files)
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% get data
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url = files(k).url;
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folder = files(k).folder;
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[~,nameurl,exturl] = fileparts(url);
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name = [nameurl,exturl];
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% first check if the test file is in the path
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if exist(name,'file') == 0
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try
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% download file
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outfilename = websave(name,url);
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% create a folder utilizing name as the foldername name
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if ~exist(fullfile(outputdir,folder), 'dir')
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mkdir(fullfile(outputdir,folder));
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end
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% move the file
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movefile(outfilename,fullfile(outputdir,folder))
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% output to the command window
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disp(['Downloaded ',folder,'/',name])
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catch % failed to download
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% output to the command window
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disp(['Failed to download ',folder,'/',name])
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% remove the html file
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delete([name,'.html'])
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end
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else
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% output to the command window
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disp(['Already available ',name])
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end
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end
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% change back to the original directory
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cd(olddir)
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end
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%--------------------------------------------------------------------------
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function DownloadWebZips(zips,outputdir)
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% store the current directory
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olddir = pwd;
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% create a folder for outputdir
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if ~exist(outputdir, 'dir')
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mkdir(outputdir); % create the folder
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else
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addpath(genpath(outputdir)); % add folders and subfolders to path
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end
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% change to the output directory
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cd(outputdir)
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% go through each zip
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for k = 1:length(zips)
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% get data
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url = zips(k).url;
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folder = zips(k).folder;
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test = zips(k).test;
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% first check if the test file is in the path
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if exist(test,'file') == 0
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try
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% download zip file
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zipname = websave(folder,url);
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% save location
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outputdirname = fullfile(outputdir,folder);
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% create a folder utilizing name as the foldername name
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if ~exist(outputdirname, 'dir')
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mkdir(outputdirname);
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end
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% unzip the zip
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unzip(zipname,outputdirname);
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% delete the zip file
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delete([folder,'.zip'])
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% output to the command window
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disp(['Downloaded and unzipped ',folder])
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catch % failed to download
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% output to the command window
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disp(['Failed to download ',folder])
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% remove the html file
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delete([folder,'.html'])
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end
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else
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% output to the command window
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disp(['Already available ',folder])
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end
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end
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% change back to the original directory
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cd(olddir)
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end

‎License

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BSD 3-Clause License
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Copyright (c) 2015, Daniel Herber
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All rights reserved.
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions are met:
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* Redistributions of source code must retain the above copyright notice, this
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list of conditions and the following disclaimer.
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* Redistributions in binary form must reproduce the above copyright notice,
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this list of conditions and the following disclaimer in the documentation
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and/or other materials provided with the distribution.
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* Neither the name of the copyright holder nor the names of its
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contributors may be used to endorse or promote products derived from
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this software without specific prior written permission.
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THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
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DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
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FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
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SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
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CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
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OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

‎README.md

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## README (basic-multiple-interval-pseudospectral)
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[![GitHub release](https://img.shields.io/github/release/danielrherber/basic-multiple-interval-pseudospectral.svg)](https://github.com/danielrherber/basic-multiple-interval-pseudospectral/releases/latest)
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[![](https://img.shields.io/badge/language-matlab-EF963C.svg)](https://www.mathworks.com/products/matlab.html)
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[![](https://img.shields.io/github/issues-raw/danielrherber/basic-multiple-interval-pseudospectral.svg)](https://github.com/danielrherber/basic-multiple-interval-pseudospectral/issues)
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[![GitHub contributors](https://img.shields.io/github/contributors/danielrherber/basic-multiple-interval-pseudospectral.svg)](https://github.com/danielrherber/basic-multiple-interval-pseudospectral/graphs/contributors)
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[![license](https://img.shields.io/github/license/danielrherber/basic-multiple-interval-pseudospectral.svg)](https://github.com/danielrherber/basic-multiple-interval-pseudospectral/blob/master/License)
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This project implements multiple-interval pseudospectral methods to solve optimal control problems.
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![readme_image.svg](http://www.danielherber.com/img/projects/basic-multiple-interval-pseudospectral/readme_image.svg)
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---
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### Install
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- Download the [project files](https://github.com/danielrherber/basic-multiple-interval-pseudospectral/archive/master.zip)
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- Run [INSTALL_Basic_Pseudospectral.m](https://github.com/danielrherber/basic-multiple-interval-pseudospectral/blob/master/INSTALL_Basic_Pseudospectral.m) in the MATLAB Command Window *(automatically adds project files to your MATLAB path, downloads the required files, and opens an example)*
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```matlab
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INSTALL_Basic_Pseudospectral
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```
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- See [BD_main.m](https://github.com/danielrherber/basic-multiple-interval-pseudospectral/blob/master/examples/BD_main.m) to run the Bryson-Denham example
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```matlab
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open BD_main
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```
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- See the technical report [[PDF]](http://systemdesign.illinois.edu/publications/Her15a.pdf) for the theory and case study results
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### Citation
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The code is complementary material for the following publication:
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- DR Herber. **Basic Implementation of Multiple-Interval Pseudospectral Methods to Solve Optimal Control Problems.** Technical report, Engineering System Design Lab, UIUC-ESDL-2015-01, Urbana, IL, USA, Jun 2015. [[PDF]](http://systemdesign.illinois.edu/publications/Her15a.pdf)
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### Description
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The two numerical schemes are used: the Legendre pseudospectral method with LGL nodes and the Chebyshev pseudospectral method with CGL nodes. The results from the case studies using the Bryson-Denham problem demonstrate the effect of user's choice in mesh parameters and little difference between the two numerical pseudospectral schemes. The solution procedure is independent of Bryson-Denham problem the test so other optimal control problems can be solved with the accompanying code.
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The main purpose of this submission is to provide a reference for the basic implementation of multiple-interval pseudospectral methods. Paired with the technical report of the same name, I hope to help bring this advanced method for solving optimal control problems to a broader audience (especially in the classroom).
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### External Includes
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See [INSTALL_Basic_Pseudospectral.m](https://github.com/danielrherber/basic-multiple-interval-pseudospectral/blob/master/INSTALL_Basic_Pseudospectral.m) for more information
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- MATLAB File Exchange Submission IDs (**23629**, **40397**)
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- Code from Lloyd N. Trefethen. **Spectral Methods in MATLAB**, SIAM, 2000. [[URL]](https://people.maths.ox.ac.uk/trefethen/spectral.html)
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- Code from J. Shen, T. Tang, and L. Wang. **Spectral Methods: Algorithms, Analysis and Applications**, Springer, 2011. [[URL]](http://www.ntu.edu.sg/home/lilian/book.htm)
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---
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### General Information
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#### Contributors
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- [Daniel R. Herber](https://github.com/danielrherber)
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#### Project Links
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- [https://github.com/danielrherber/basic-multiple-interval-pseudospectral](https://github.com/danielrherber/basic-multiple-interval-pseudospectral)
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- [http://www.mathworks.com/matlabcentral/fileexchange/51104](http://www.mathworks.com/matlabcentral/fileexchange/51104)

‎examples/.gitignore

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!.gitignore
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_private/

‎examples/bryson-denham/.gitignore

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!.gitignore
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saved/

‎examples/bryson-denham/BD_boundary.m

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%--------------------------------------------------------------------------
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% BD_boundary.m
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% Bryson-Denham boundary constraint function
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%--------------------------------------------------------------------------
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% b = BD_boundary(X,U,p)
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% X: state
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% U: control
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% p: parameter structure
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% b: boundary constraint vector
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%--------------------------------------------------------------------------
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% Primary Contributor: Daniel R. Herber, Graduate Student, University of
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% Illinois at Urbana-Champaign
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% Link: https://github.com/danielrherber/basic-multiple-interval-pseudospectral
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%--------------------------------------------------------------------------
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function b = BD_boundary(X,U,p)
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% initialize boundary constraint
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b = [];
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% position boundary constraints
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b = [b;X(1,1)-p.prob.p0;X(end,1)-p.prob.pf];
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% velocity boundary constraints
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b = [b;X(1,2)-p.prob.v0;X(end,2)-p.prob.vf];
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end

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