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Prof. Dr. Y. Samim Ünlüsoy ME 513 Vehicle Dynamics 1

Introduction Introduction to Matlabto Matlab

Prof. Dr. Y. Samim ÜNLÜSOYMech. Eng. Dept., Middle East Technical University, Ankara

Prof. Dr. Y. Samim Ünlüsoy ME 304 Control Systems 2

MATLAB MATLAB MATLAB 7.0.1.lnk

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INTRODUCTION TO MATLABINTRODUCTION TO MATLAB

A short introduction to Matlab is given in the following slides.

A well prepared, easy to follow, medium-sized (37 pages) introduction can be found on the web.

“An Introduction to Matlab” version 2.3

by David F. Griffiths

http://www.maths.dundee.ac.uk/~ftp/na-reports/MatlabNotes.pdf


INTRODUCTION TO MATLABINTRODUCTION TO MATLABMatrices

In MATLAB, a matrix is a rectangular array of numbers.

Scalars are 1-by-1 matrices.

Vectors are matrices with only one row or column.

MATLAB has other ways of storing both numeric and nonnumeric data; but in the beginning, it is usually best to think of everything as a matrix.

Where other programming languages work with numbers one at a time, MATLAB allows you to work with entire matrices quickly and easily.


HOW TO ENTER MATRICESHOW TO ENTER MATRICES

You can enter matrices into MATLAB in several different ways:

Enter an explicit list of elements.Load matrices from external data files.Generate matrices using built-in functions.Create matrices with your own functions in M-files.


Entering a MatrixEntering a Matrix

You have only to follow a few basic conventions:

Separate the elements of a row with blanks or commas.Use a semicolon, ; ,to indicate the end of each row.Surround the entire list of elements with square brackets, [ ].


Entering A MatrixEntering A MatrixStart by entering the matrix as a list of its elements. To enter, simply type in the Command Window

A = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1]

Once you have entered the matrix, it is automatically remembered in the MATLAB workspace. You can refer to it simply as A. Now that you have A in the workspace.


MatLab Command WindowMatLab Command Window>> A = [16 3 2 13; 5 10 11 8; 9 6 7 12;

4 15 14 1]

A =

16 3 2 135 10 11 89 6 7 124 15 14 1

>>


Transpose of a MatrixTranspose of a MatrixThere are two possibilities to transpose a matrix.

Use an apostrophe or single quote, '.

Type transpose(A), where A is the matrix transpose of which is sought.


MatLab MatLab Command Command WindowWindow

>> A'

ans =

16 5 9 43 10 6 152 11 7 1413 8 12 1

>> transpose(A)

ans =

16 5 9 43 10 6 152 11 7 1413 8 12 1

>>

A =

16 3 2 135 10 11 89 6 7 124 15 14 1


Inverse of a MatrixInverse of a Matrix

inv(X) is the inverse of the square matrix X.



>> C

C =

16 3 2 135 10 11 89 6 15 124 15 14 1

>> B=inv(C)

C =

0.0760 -0.1985 0.0417 0.10050.0196 0.1544 -0.1250 0.0098-0.0417 -0.1250 0.1250 0.0417-0.0147 0.2279 -0.0417 -0.1324

>>

C =

16 3 2 135 10 11 89 6 15 124 15 14 1


SubscriptsSubscriptsThe element in row i and column j of A is denoted by A(i,j).

For example, A(4,2) is the number in the fourth row and second column.

So it is possible to compute the sum of the elements in the fourth column of A by typing

A(1,4) + A(2,4) + A(3,4) + A(4,4)


Colon OperatorColon OperatorThe expression1:10

is a row vector containing the integers from 1 to 101 2 3 4 5 6 7 8 9 10

To obtain nonunit spacing, specify an increment.

100:-7:50is

100 93 86 79 72 65 58 51and

0:pi/4:piis

0 0.7854 1.5708 2.3562 3.1416



>> 1:10

ans =

1 2 3 4 5 6 7 8 9 10

>> 100:-7:50

ans =

100 93 86 79 72 65 58 51

>> 0:pi/4:pi

ans =

0 0.7854 1.5708 2.3562 3.1416

>>


Colon OperatorColon OperatorSubscript expressions involving colons refer to portions of a matrix.

A(1:k,j)is the first k elements of the jth column of A. So

sum(A(1:4,4))computes the sum of the fourth column. The colon by itself refers to all the elements in a row or column of a matrix and “end” refers to the last row or column. So

sum(A(:,end))computes the sum of the elements in the last column of A.



>> sum(A(1:4,4))

ans =

34

>> sum(A(:,end))

ans =

34

>>

A =

16 3 2 135 10 11 89 6 7 124 15 14 1


Matrix Multiplication and DivisionMatrix Multiplication and DivisionC = A*B

is the linear algebraic product of the matrices A and B. For nonscalar A and B, the number of columns of A must equal the number of rows of B. A scalar can multiply a matrix of any size.

.* Array multiplication.

A.*B is the element-by-element product of the arrays A and B. A and B must have the same size, unless one of them is a scalar.


MatLab Command MatLab Command WindowWindow

>> A*B

ans =

154 195175 70

>> A.*B

ans =

112 3670 10

>>

A =

16 35 10

B =

7 1214 1

16*7+3*14=15416*12+3*1=1955*7+10*14=1755*12+10*1=70

16*7=1123*12=365*14=7010*1=10


Matrix Multiplication and DivisionMatrix Multiplication and Division/ matrix right division.

B/A is roughly the same as B*inv(A). More precisely, B/A = (A'\B')'.

A./B is the matrix with elements A(i,j)/B(i,j). A and B must have the same size, unless one of them is a scalar.

\ matrix left division.

If A is a square matrix, A\B is roughly the same as inv(A)*B, except it is computed in a different way. If A is an n-by-n matrix and B is a column vector with n components, or a matrix with several such columns, then X = A\B is the solution to the equation AX = B computed by Gaussian elimination.



>> B/A

ans =

0.0690 1.17930.9310 -0.1793

>> B*inv(A)

ans =

0.0690 1.17930.9310 -0.1793

>>

A =

16 35 10

B =

7 1214 1



>> A./B

ans =

2.2857 0.25000.3571 10.0000

>> A\B

ans =

0.1931 0.80691.3034 -0.3034

>>

A =

16 35 10

B =

7 1214 1 >> inv(A)*B

ans =

0.1931 0.80691.3034 -0.3034

>>


Eigenvalues and EigenvectorsEigenvalues and EigenvectorsE = eig(X) is a vector containing the eigenvalues of a square matrix X.

[V,D] = eig(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that

X*V = V*D.



>> A

A =

16 3 2 135 10 11 89 6 15 124 15 14 1

>> G=eig(A)

G =

36.3739-7.10818.00004.7341

>>

A =

16 3 2 135 10 11 89 6 15 124 15 14 1


Eigenvalues and EigenvectorsEigenvalues and EigenvectorsE = eig(A,B) is a vector containing the generalized eigenvalues of square matrices A and B.

[V,D] = eig(A,B) produces a diagonal matrix D of generalized eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that

A*V = B*V*D.


VariablesVariablesMATLAB does not require any type declarations or dimension statements. A new variable name is automatically created and appropriate amount of storage is allocated. If the variable already exists, MATLAB changes its contents and, if necessary, allocates new storage.

Variable names consist of a letter, followed by any number of letters, digits, or underscores.

MATLAB uses only the first 31 characters of a variable name.

MATLAB is case sensitive; it distinguishes between uppercase and lowercase letters. A and a are not the same variable.

To view the matrix assigned to any variable, simply enter the variable name and press enter.


FunctionsFunctionsMATLAB provides a large number of standard elementary mathematical functions, such as

abs, sqrt, exp, and sin, cos, tan etc..


Advanced Mathematical FunctionsAdvanced Mathematical FunctionsTaking the square root or logarithm of a negative number is not an error; the appropriate complex result is produced automatically.

MATLAB also provides many more advanced mathematical functions, including Bessel and gamma functions. Most of these functions accept complex arguments.


Help on Available FunctionsHelp on Available FunctionsFor a list of the elementary mathematical functions, type

help elfunFor a list of more advanced mathematical and matrix functions, type

help specfunhelp elmat


BuiltBuilt--in and min and m--file Functionsfile FunctionsSome of the functions, like sqrt and sin,are built-in. They are part of the MATLAB core so they are very efficient, but the computational details are not readily accessible.

Other functions, like gamma and sinh, are implemented in m-files. You can see the code and even modify it if you want.


PLOTS >>help plotPLOTS >>help plotPLOT(X,Y) plots vector Y versus vector X.

If X or Y is a matrix, then the vector is plotted versus the rows or columns of the matrix, whichever line up.

If X is a scalar and Y is a vector, length(Y) disconnected points are plotted.

PLOT(Y) plots the columns of Y versus their index.

If Y is complex, PLOT(Y) is equivalent toPLOT(real(Y),imag(Y)).

In all other uses of PLOT, the imaginary part is ignored.


PLOTS >>help plotPLOTS >>help plotVarious line types, plot symbols and colors may be obtained withPLOT(X,Y,S) where S is a character string made from one elementfrom any or all the following 3 columns:

b blue . point - solidg green o circle : dottedr red x x-mark -. dashdot c cyan + plus -- dashed

m magenta * stary yellow s square

k black d diamond

For example, PLOT(X,Y,'c+:') plots a cyan dotted line with a plus at each data point; PLOT(X,Y,'bd') plots blue diamond at each data point but does not draw any line.


PLOTS >> help plotPLOTS >> help plotPLOT(X1,Y1,S1,X2,Y2,S2,X3,Y3,S3,...) combines the plots defined by the (X,Y,S) triples, where the X's and Y's are vectors or matrices and the S's are strings.

For example, PLOT(X,Y,'y-',X,Y,'go') plots the data twice, with a solid yellow line interpolating green circles at the data points.

The PLOT command, if no color is specified, makes automatic use of the colors specified by the axes ColorOrder property. The default ColorOrder is listed in the table above for color systems where the default is blue for one line, and for multiple lines, to cycle through the first six colors in the table. For monochrome systems, PLOT cycles over the axes LineStyleOrder property.

See also SEMILOGX, SEMILOGY, LOGLOG, PLOTYY, GRID, CLF, CLC, TITLE, XLABEL, YLABEL, AXIS, AXES, HOLD, COLORDEF, LEGEND, SUBPLOT, STEM.


Solution of Linear Ordinary Differential Solution of Linear Ordinary Differential Equations with Constant CoefficientsEquations with Constant Coefficients

>> help tf

tf : Creation of transfer functions or conversion to transfer function.

sys = tf(num, den) creates a continuous-time transfer function sys with numerator num and denominator den. The output sys is a tf object.

s = tf('s') specifies the transfer function H(s) = s (Laplace variable).

You can then specify transfer functions directly as rational expressions in s, e.g.,

s = tf('s'); H = (s+1)/(s^2+3*s+1)


Time Response for LTI SystemsTime Response for LTI Systems : lsimlsimMatlab command lsim (sys, u, t) is used to simulate time response of LTI models to arbitrary inputs described by u.

The time vector t consists of regularly spaced time samples and u is a matrix with as many columns as inputs and whose i-th row specifies the input value at time t(i).

For example, t = 0:0.01:5; u = sin(t); lsim(sys,u,t)

simulates the response of a single-input model sys to the input

u(t)=sin(t)

for 5 seconds by 0.01 second intervals.


mm--filesfilesThere are two types of m-files :

Script m-files, and

Function m-FilesM-files can be scripts that simply execute a series of MATLAB statements, or they can be functions that also accept arguments and produce output.

You create M-files using a text editor, then use them as you would any other MATLAB function or command.


mm--filesfilesScript M-Files

Do not accept input arguments or return output arguments.

Operate on data in the workspace.

Useful for automating a series of steps you need to perform many times.


Script M-FilesConsider a one degree of freedom mass-spring-damper

system representing a road vehicle travelling on a rough road surface.

Obtain the motion of the vehiclein terms of the vehicle body mass

displacement y and its derivatives.

Assume that the road surface profile z is sinusoidal with a frequency of 5 rad/s

and an amplitude of 1 cm.

m

kcz

x


Script M-Files

The equation of motion is given as :

The transfer function is then given by :

m

kc

z

x

+ + = +mx cx kx cz kz

+= =

+ +2X(s) cs kG(s)Z(s) ms cs k


Script M-Files% March 2004% Response of a vehicle travelling on a% sinusoidal road surface profile.% -------------------------------------------------% Define Parameters and Variables% m : vehicle mass [kg]% c : total damping coefficient of suspension dampers [m/s]% k : total spring constant of suspension springs [kN/m]% z : vehicle body displacement [m]% -------------------------------------------------% Enter Data Valuesm=1260; c=4000; k=60000; .....


Script M-Files.....% ---------------------------------------------------------% Generate the time array for the solutiont=0:0.01:10; % time array (from 0 to 10 seconds with 0.01

second increments)% -----------------------------------------------% Generate the road surface profileA=0.01; % road surface profile amplitude [m]omega=5; % road surface profile frequency [rad/s]z=A*sin(omega*t); % road surface profile [m]% ----------------------------------------% Plot the road surface profilefigure(1)plot(t,z)title(‘Sinusoidal Road Surface Profile - Amplitude : 1 cm, Frequency : 5

rad/s’);xlabel(‘Time[s]’);ylabel(‘y[m]’);.....


Script M-Files.....

% -------------------------------------------% Generate the transfer functionss=tf(‘s’); % Declare s as the Laplace variable% from z to yhz=(c*s+k)/(m*s^2+c*s+k);% from z to ydot (velocity)hzdot=s*hz;% from z to ydoubledot (acceleration)hzddot=s*hzdot;% -----------------------------------------------% Plot the displacement, velocity, and acceleration of % the vehicle bodyfigure(2)% Plot of displacement – note that hz is the TF from z to ylsim(hz,z,t).....


Script M-Files.....figure(3)% Plot of velocity – note that hzdot is the TF from z to

ydotlsim(hzdot,z,t).....


Script m-filesThis is what you get from the program defaults !

0 1 2 3 4 5 6 7 8 9 10-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01Sinusoidal Road Surface Profile - Amplitude : 1 cm, Frequency : 5 rad/s

Time [s]

y [m

]


Script m-files

You can modify the

plots using the edit facility on

the Matlab plot0 1 2 3 4 5 6 7 8 9 10-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Sinusoidal Road Surface Profile - Amplitude : 1 cm, Frequency : 5 rad/s

Time [s] (sec)

yd

ot

[m/

s]


Script m-files

You can modify it

using the edit facility on the

Matlab plot0 1 2 3 4 5 6 7 8 9 10

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Sinusoidal Road Surface Profile - Amplitude : 1 cm, Frequency : 5 rad/s

Time [s] (sec)

yd

ou

ble

dot

[m/

s2]


mm--filesfilesFunction m-Files

Can accept input arguments and return output arguments.

Internal variables are local to the function by default.

Useful for extending theMATLAB language for yourapplication.


Function mFunction m--filesfilesfunction [Fx,Fy]=tireforces(munom,Fx,Fz,alpha,slip,vwl)...............

Call from the main program body :

[Fxfl,Fyfl]=tireforces(munoml,Fxfl,Fzfl,alphafl,slipfl,vwlfl);


Time Response for LTI SystemsTime Response for LTI Systems : lsimlsimlsim(sys,u,t,x0) specifies the initial state vector x0 at time t(1) (for state-space models only). x0 is set to zero when omitted.

lsim(sys1, sys2,..., u, t, x0) simulates the response of multiple LTI models sys1, sys2,... on a single plot.

You can also specify a color, line style, and marker for each system, as in

lsim(sys1,'r',sys2,'y--',sys3,'gx',u,t).


Time Response for LTI SystemsTime Response for LTI Systems : lsimlsimDraw the yaw velocity response for the vehicle specified below for a 3o step steering angle of the front wheels while travelling at 60 kph.

Vehicle specifications :m=1310 [kg]

J=1760 [kg.m^2]a=1.2 [m]b=1.4 [m]

Cf=-34870 [N/rad]Cr=-31730 [N/rad]


Time Response for LTI SystemsTime Response for LTI Systems : lsimlsim% November 2003% -----------------------% Example% Plots of sideslip angle and yaw velocity% for a step fronts wheel steering angle% ---------------------------------------------------------% Enter vehicle parameters - in [kg], [kg.m^2] and [m]

m=1310;J=1760;a=1.2;b=1.4;L=a+b; % Enter axle cornering stiffnesses - in [N/rad]

Cf=-34870;Cr=-31730;% Enter forward speed of the vehicle in [kph] and convert to [m/s]

ukph=60;u=ukph/3.6;

.

.


Time Response for LTI SystemsTime Response for LTI Systems : lsimlsim..% Enter solution time range and increment - 0 to 2s by 0.01s increments

t=0:0.025:2;% Enter front wheel steering angle variation with time in [deg]

dfdeg(1:81)=3; % Express this for all t values !% Convert to [rad]

df=dfdeg*pi/180;% Plot

figure(1) plot(t,dfdeg)

xlabel('Time [s]')ylabel('Front Wheel Steering Angle [deg]')gtext('Vehicle Speed : 60 [kph]')

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


Time Response for LTI SystemsTime Response for LTI Systems : lsimlsim..% Basic System Matrices% State variables are sideslip and yaw velocities

A=[(Cf+Cr)/(m*u) (a*Cf-b*Cr)/(m*u)-u;(a*Cf-b*Cr)/(J*u) (a^2*Cf+b^2*Cr)/(J*u)];

B=[-Cf/m;-a*Cf/J];

% [Y]=[C]{x}+[D]{u}C=[1 0;0 1]; % The output will have two columns, for v and rD=[0;0];

.

.


Time Response for LTI SystemsTime Response for LTI Systems : lsimlsim..% Plot response

figure(2)lsim(A,B,C,D,df,t)

% This is going to give two plots, one for v(t) and the other for r(t).% If you prefer to produce your own plot(s), then you can store% the results in a matrix.% x=lsim(A,B,C,D,df,t)% Again, x will have the size 81x2, and the first column is going to % give variation of v(t) and the other r(t).% Then to plot variation of r% plot(t,x(:,2))% To plot variation of v% plot(t,x(:,1)

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