Find consistent initial conditions for DAE system [MATLAB - ODE solver]

Matlab can be used to solve DAE systems. Generally speaking a DAE system is a set of differential and algebraic equations that need to be solved simoultaneosly. Matlab provides different tools able to deal with these types of equations; those solvers are named ODE solvers. Even if the name itself would suggest the capability of the solvers to solve only ODEs, also DAEs can be easily managed.

For a complete overview of the different solvers, please visit the following link where all the available builtin solvers and their properties are described in detail.

One of the most important part to get a DAE system to work is to feed it a set of consistent initial conditions. Matlab provides also a tool able to support the user in this process (the function is called decic) but here I would like to explain a more DIY method.

Using the above function (decic) you will be able to find the set of variables which satisfy the consistent initial condition, such that

f(y',y,t0 ) = 0

where f(y',y,t ) describes the set of algebraic and differential equations that you're trying to solve.

Let's suppose to have your DAE system composed of N differential equations and M algebraic equations.

A way to boost the decic function is to fix the initial values for the differential variabless and use those guesses to solve the only set of M algebraic equations.

To do this you can easily use the fsolve Matlab's function. This script will try to find the root of a set of equations; Once you get the result from the fsolve you can just use it, with the guess of the differential states, to build the array of consistent initial conditions for the ODE solver.

Please note that in order to be considered consistent, the result coming from the fsolve should have a low residual from its run.

Smart way to write "trace" command in Matlab - Boost your code, save time

Sometimes is necessary to write complex operations in Matlab involving matrices. These operations can be of different type and they operate in different ways.

Today I'll show you how write in a smart way the "trace" command of a matrix or a product/sum/difference of the using matrices notation to speed up your code.

Concatenate arrays with different sizes in Matlab

Sometimes you want to concatenate arrays with different sizes, and Matlab doesn't allow you to. This makes sense, of course, but still you just want to put all that stuff together for some reason. Then this function may help you. As any other functions I've wrote, I'm sure it an be optimized, so I'll be happy to hear suggestions.
You can also download it from my matlab file exchange and rate it :-)

function [catmat]=padconcatenation(a,b,c)

%[catmat]=padconcatenation(a,b,c)

%concatenates arrays with different sizes and pads with NaN.

%a and b are two arrays (one or two-dimensional) to be concatenated, c must be 1 for

%vertical concatenation ([a;b]) and 2 for horizontal concatenation ([a b])

%

% a=rand(3,4)

% b=rand(5,2)

% a =

%

%     0.8423    0.8809    0.7773    0.3531

%     0.2230    0.9365    0.1575    0.3072

%     0.4320    0.4889    0.1650    0.9846

% b =

%

%     0.6506    0.8854

%     0.8269    0.0527

%     0.4742    0.3516

%     0.4826    0.2625

%     0.6184    0.5161

%

% PADab=padconcatenation(a,b,1)

% PADab =

%

%     0.8423    0.8809    0.7773    0.3531

%     0.2230    0.9365    0.1575    0.3072

%     0.4320    0.4889    0.1650    0.9846

%     0.6506    0.8854       NaN       NaN

%     0.8269    0.0527       NaN       NaN

%     0.4742    0.3516       NaN       NaN

%     0.4826    0.2625       NaN       NaN

%     0.6184    0.5161       NaN       NaN

%

% PADab=padconcatenation(a,b,2)

%

% PADab =

%

%     0.8423    0.8809    0.7773    0.3531    0.6506    0.8854

%     0.2230    0.9365    0.1575    0.3072    0.8269    0.0527

%     0.4320    0.4889    0.1650    0.9846    0.4742    0.3516

%        NaN       NaN       NaN       NaN    0.4826    0.2625

%        NaN       NaN       NaN       NaN    0.6184    0.5161

sa=size(a);

sb=size(b);

switch c

    case 1

        tempmat=NaN(sa(1)+sb(1),max([sa(2) sb(2)]));

        tempmat(1:sa(1),1:sa(2))=a;

        tempmat(sa(1)+1:end,1:sb(2))=b;

       

    case 2

        tempmat=NaN(max([sa(1) sb(1)]),sa(2)+sb(2));

        tempmat(1:sa(1),1:sa(2))=a;

        tempmat(1:sb(1),sa(2)+1:end)=b;

end

catmat=tempmat;

end

Custom Colors for Matlab Plots

This is a(nother) simple function I wrote for matlab, it can also be downloaded from my file exchange at mathworks:
http://www.mathworks.com/matlabcentral/fileexchange/36514-custom-colors-for-plots

In plot graphs, often it's needed to use a several colors that can be easily differentiated on a first look on the graphic. With this function you can call easily (by name or by code) a selection of 15 colors.
It's very simple and there are more elegant ways to do it, but I think it's a very functional solution.

function ccol=CustomColors(ct1)

%This is a function to create some easy to differentiate colors, very

%useful when plotting many things in the same graphic

%The RGB coordinates and names where taken from the following page:

%http://web.njit.edu/~kevin/rgb.txt.html

%

%Examples:

%for ccc=1:15

%plot(ccc*ones(1,10),'Linewidth',4,'Color', CustomColors(ccc))

%end

%

%plot(ones(1,10),'Linewidth',4,'Color', CustomColors('Coral'))

%plot(ones(1,10),'Linewidth',4,'Color', CustomColors('DeepSkyBlue4'))

%

%Copyright: Andres Gonzalez. 2012.

CColors={

    [   [0,0.407843137254902,0.545098039215686] ];% DeepSkyBlue4

    [   [0.545098039215686,0.270588235294118,0.0745098039215686]    ];% SaddleBrown

    [   [0,0.803921568627451,0] ];% green3

    [   [0,0.498039215686275,1] ];% SlateBlue

    [   [0.956862745098039,0.643137254901961,0.376470588235294] ];% SandyBrown

    [   [0.419607843137255,0.556862745098039,0.137254901960784] ];% OliveDrab

    [   [0.800000000000000,0.196078431372549,0.600000000000000] ];% Violet,Red

    [   [0.556862745098039,0.419607843137255,0.137254901960784] ];% Sienna

    [   [0.803921568627451,0.678431372549020,0] ];% gold3

    [   [0.545098039215686,0,0.545098039215686] ];% magenta4

    [   [1,0.498039215686275,0] ];% coral

    [   [1,0.843137254901961,0] ];% gold1

    [   [0.600000000000000,0.196078431372549,0.800000000000000] ];% DarkOrchid

    [   [0.941176470588235,0.501960784313726,0.501960784313726] ];% LightCoral

    [   [0.635294117647059,0.803921568627451,0.352941176470588] ];% DarkOliveGreen3

   

    };

NColors={

    'DeepSkyBlue4';

    'SaddleBrown';

    'Green3';

    'SlateBlue';

    'SandyBrown';

    'OliveDrab';

    'VioletRed';

    'Sienna';

    'Gold3';

    'Magenta';

    'Coral';

    'Gold1';

    'DarkOrchid';

    'LightCoral';

    'DarkOliveGreen3';

    };

switch class(ct1)

    case 'double'

        if ct1<1

            ct1=1

            disp(sprintf('Color index must be between 1 and %d, taking closest value [%d]',length(CColors),ct1))

        elseif ct1>length(CColors)

            ct1=length(CColors)

            disp(sprintf('Color index must be between 1 and %d, taking closest value [%d]',length(CColors),ct1))

        end

       

       

       

    case 'char'

        ctfound=0

        for ct2=1:length(NColors)

            if (strcmp(NColors{ct2},ct1))

                ctfound=ct2

            end

        end

        if ctfound==0

            disp(sprintf('Color not found taking default value [%d]',1))

        else

            ct1=ctfound

        end

       

    otherwise

        disp(sprintf('Color not valid taking default value [%d]',1))

       

       

end

ccol=CColors{ct1}

end

Save Workspace to Struct

This was a nice function that I created today, and it allows to save all the variables from the "base" workspace to a single struct, in order to have them all nicely packed in case you want to create mat file with them and later reload it without having so much worry of overwriting the variables in the other workspace. I gratefully accept suggestions for better options.
And now it's also available at the Matlab Central (if you want to rate it, which is always welcome):  http://www.mathworks.com/matlabcentral/fileexchange/36257-save-workspace-to-struct

%This Script saves all the variables from the current workspace into a

%single structure array.

%Created by Andres Gonzalez. 2012

function WStruct=ws2struct()

WSVARS = evalin('base', 'who');

for wscon=1:size(WSVARS,1)

    thisvar=evalin('base', WSVARS{wscon});

    eval(strcat('THEWORKSPACE.(WSVARS{wscon})=thisvar;'))

end

WStruct=THEWORKSPACE;

Resample a Matrix in Matlab

This function does the same as the previous one (resample) but for a matrix. Again, maybe not the most elegant, but works

function newmatrix=resampmatrix(matrix,newlen)

%----------------

%this function uses linear interpolation to change the number of rows in a

%matrix.

%Copyright, 2011, Andrés González

%----------------

[rows cols]=size(matrix);

newmatrix=[];

for i=1:cols

len=rows;

x=1:len;x=x';y=matrix(:,i);

xx=1:(len-1)/(newlen-1):len;

xx=xx';

yy=interp1(x,y,xx,'linear');

newmatrix=[newmatrix yy];

end

Resample in Matlab

This is a simple function to resample a vector in matlab. I now there might be more elegant ways to do it, but this one works and it's simple

function newvector=resample(vector,newlen)

%%%%%%%%%%%%%%%%%%%%

%This function uses linear interpolation to change the number of samples

% Copyright 2011 Andrés Gonzalez

%%%%%%%%%%%%%%%%%%%%

len=length(vector);

x=1:len;x=x';y=vector;

xx=1:(len-1)/(newlen-1):len;

xx=xx';

yy=interp1(x,y,xx,'linear');

newvector=yy;

end