MATLAB Tutorials

Violeta Ivanova, Ph.D.Educational Technology ConsultantMIT Academic Computing

violeta@mit.edu

16.01 & 16.02 Unified Engineering - Fall 2006

Unified Engineering: MATLAB Tutorials

This Tutorial

Class materialsweb.mit.edu/acmath/matlab/unified/linsys

Topics Basics Review Linear Systems

Unified Engineering: MATLAB Tutorials

Other References

Mathematical Tools at MITweb.mit.edu/ist/topics/math

Unified: Intro to MATLAB Tutorialweb.mit.edu/acmath/matlab/unified/intro

Other Course 16 MATLAB Tutorialsweb.mit.edu/acmath/matlab/course16/

MATLAB Basics Review

Toolboxes & HelpVectors & MatricesOperators & Functions

Unified Engineering: MATLAB Tutorials

Help in MATLAB

Command line help>> help <command>

e.g. help eig>> lookfor <keyword>

e.g. lookfor eigenvalue Help Browser

Help->Help MATLAB

Unified Engineering: MATLAB Tutorials

MATLAB Toolboxes & Help MATLAB

+ Getting Started+ Mathematics

+ Matrices and Linear Algebra+ Solving Linear Systems of Equations Eigenvalues

+ Polynomials and Interpolation+ Convolution and Deconvolution

+ Programming+ Graphics

Unified Engineering: MATLAB Tutorials

Variables & Data Types Data type detection

>> a = 5; a =‘ ok’; a = 1.3

Numeric data types>> x = 5>> y = 5.34; x = round(y)>> w = 5 + 6i>> wr = real(w); wi = imag(w)

Built-in constants: pi i j Inf Special characters

[] () {} ; % : = . … @

Unified Engineering: MATLAB Tutorials

Vectors

Row vector>> R1 = [1 6 3 8 5]>> R2 = [1 : 5]>> R3 = [-pi : pi/3 : pi]

Column vector>> C1 = [1; 2; 3; 4; 5]>> C2 = R2'

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Matrices

Creating a matrix>> A = [1 2.5 5 0; 1 1.3 pi 4]

>> A = [R1; R2]

Accessing elements>> A(1,1)

>> A(1:2, 2:4)

>> A(:,2)

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Matrix Operations Operators + and -

>> X = [x1 x2]; Y = [y1 y2]; A = X+Y A =

x1+y1 x2+y2

Operators *, /, and ^>> Ainv = A^-1 Matrix math is default!

Operators .*, ./, and .^>> Z = [z1 z2]; B = [Z.^2 Z.^0]B =

z12 z2

2 1 1

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Built-In Functions Matrices & vectors

>> [n, m]= size(A)>> n = length(X)>> M1 = ones(n, m)>> M0 = zeros(n, m)>> En = eye(n)>> N1 = diag(En)>> [evecs, evals] = eig(A)

And many others …>> y = exp(sin(t)+cos(t))

Unified Engineering: MATLAB Tutorials

Graphics 2D linear plots: plot

>> plot (t, z, ‘r-’) Colors: b, r, g, y, m, c, k, w Markers: o, *, ., +, x, d Line styles: -, --, -., :

Annotating graphs>> legend (‘z = f(t)’)>> title (‘Position vs. Time’)>> xlabel (‘Time’)>> ylabel (‘Position’)

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Multiple Plots Multiple datasets on a plot

>> p1 = plot(xcurve, ycurve)>> hold on>> p2 = plot(Xpoints, Ypoints, ‘ro’)>> hold off

Subplots on a figure>> s1 = subplot(1, 2, 1)>> p1 = plot(time, velocity)>> s2 = subplot(1, 2, 2)>> p2 = plot(time, acceleration)

MATLAB Linear Systems

Linear Equations & SystemsEigenvalues & EigenvectorsConvolution

Unified Engineering: MATLAB Tutorials

Systems of Linear Equations Definition:

Matrix form: Ax = b

a11x1+ a

12x2+ ...+ a

1mxm= b

1

a21x1+ a

22x2+ ...+ a

2mxm= b

2

...

an1x1+ a

n2x2+ ...+ a

nmxm= b

m

a11

... a1n

... ... ...

an1

... amn

!

"

###

$

%

&&&

x1

...

xm

!

"

###

$

%

&&&

=

b1

...

bm

!

"

###

$

%

&&&

Unified Engineering: MATLAB Tutorials

Solving Linear Equations Define matrix A

>> A = [a11 a12 … a1m a21 a22 … a2m …

an1 an2 … anm]

Define vector b>> b = [b1; b2; … bm]

Compute vector x>> x = A \ b

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Differential Equations Ordinary Differential Equation (ODE)

Linear systems -> State Equation

Eigenvalues, λi, and eigenvectors, vi,of a square matrix A

A vi = λi vi

y ' = f (t, y)

!x1

...

!xn

"

#

$$$

%

&

'''

=

a11

... a1n

... ... ...

an1

... ann

"

#

$$$

%

&

'''

x1

...

xn

"

#

$$$

%

&

'''

x

•

= Ax

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Linear Dynamic Networks Solution using eigenvalue method

Identify the network’s states ………………………………..

Identify initial conditions …………………………………..

Find the state equation …………………………….

Find eigenvalues & eigenvectors …………

The general solution is ……….

Solve for ai …………….

x

•

= Ax

x

x t( ) = ai

i

! vie

"it

x 0( )

!i,"

i

a = v1

v2... v

n[ ]!1

x 0( )

Unified Engineering: MATLAB Tutorials

Example: RCL Circuitdv

1

dt= !

1

2v1! 2i

4

di4

dt=1

2v1! 3i

4

v40( ) = 2

i40( ) = 1

R3

6Ω

+_

R24Ω

C10.5F

L42Hv1

i4

State equation

x

•

=x1'

x2'

!

"#

$

%& =

dv1

dt

di4

dt

!

"

####

$

%

&&&&

=

'1

2'2

1

2'3

!

"

####

$

%

&&&&

v1

i4

!

"#

$

%& = Ax x 0( ) =

2

1

!

"#

$

%&

Unified Engineering: MATLAB Tutorials

Example: RCL Circuit (continued)

R3

6Ω

+_

R24Ω

C10.5F

L42Hv1

i4

Analytical solution

x t( ) = a1v

1e

!1t+ a

2v

2e

!2t"

v1

t( ) =4

3e

#t+

2

3e

#2.5t

u4

t( ) =1

3e

#t+

2

3e

#2.5t

x

•

= Ax

Avi= !

iv

i

a = v1

v2

"# $%&1

x 0( )x t( ) = a

i

i

' vie

!it

Unified Engineering: MATLAB Tutorials

RCL Circuit: MATLAB Solution

x

•

= Ax

x 0( )Av

i= !

i"

i

a = v1

v2

#$ %&'1

x 0( )x t( ) = a

i

i

( vie

!it

>> A = [a11 a12; a21 a22]

>> x0 = [v1,0; i4,0]

>> [V, L] = eig(A)

>> λ = diag(L)>> a = V \ x0>> v1 = a1*V11*exp(λ1*t)…

+ a2*V12*exp(λ2*t) i4 = a1*V21*exp(λ1*t)…

+ a2*V22*exp(λ2*t)

Unified Engineering: MATLAB Tutorials

Linear Systems Exercises Exercise 1: stateeig.m

Matrices & vectors Systems of linear equations Eigenvalues & eigenvectors

Follow instructions in m-file …

Questions?

Unified Engineering: MATLAB Tutorials

Complex Eigenvalues Example: complex eigenvalues and

eigenvectors from state equation

Euler’s Formula

Solution in terms of real variables

>> x1 =(cos(3*t)-sin(3*t))*exp(-t)

x t( ) ==0.5 + 0.5 j

0.5 ! j

"

#$

%

&'e

!1+3 j( )t+0.5 ! 0.5 j

0.5 + j

"

#$

%

&'e

!1!3 j( )t

e! +" j

= e! cos" + j sin"( )

x

1t( ) = ... = cos3t ! sin3t( )e!t

Unified Engineering: MATLAB Tutorials

State Space State-space model of a linear system

X, u & Y: state, input & output vectorsA, B & C: state, input & output matricesD: usually zero (feedthrough) matrix

Transfer functions

>> [Num, Den] = ss2tf(A, B, C, D)

X

•

= AX + Bu

Y = CX +Du

H s( ) =Num s( )Den s( )

= C sI ! A( )!1

B = D

Unified Engineering: MATLAB Tutorials

Convolution Definition

Example: signals & systems

g t( )! u t( ) = g t " #( )"$

$

% u #( )d#

Gu(t) y(t) = u(t)*g(t)

Unified Engineering: MATLAB Tutorials

Polynomial Convolution Polynomials:

x(s) = s2 + 2s + 5; y(s) = 2s2 + s + 3>> x = [1 2 5]; y = [2 1 3]

Convolution and polynomial multiplication>> w = conv(x, y)>> w =

2 5 15 11 15

Deconvolution and polynomial division>> [y, r] = deconv(w, x)

Unified Engineering: MATLAB Tutorials

Example: RC Circuit

u(t) =e! t, t " 0

0, t < 0

Impulse response:

System response:

g(t) =e!1.5t, t " 0

0, t < 0

#$%

1Ω

u(t)+

_2Ω 1F y(t)

+

_

Input:

y(t) =2e

! t ! 2e!1.5t , t " 0

0, t < 0

#$%

Unified Engineering: MATLAB Tutorials

Example: RC Circuit (continued)

Unified Engineering: MATLAB Tutorials

RC Circuit: MATLAB Solution

0 ! t ! tmax

g(t) = e"t

u(t) = e#t

y (t) = g(t)$u(t)

>> t = [0 : dt : tmax]>> nt = length(t)>> g = exp(α*t)>> u = exp(β*t)>> y = conv(g, u)*dt>> y = y(1 : nt)>> plot(t, u, ‘b-’, … t, g, ‘g-’, … t, y, ‘r-’)

Unified Engineering: MATLAB Tutorials

Linear Systems Exercises Exercise 2: convolution.m

Matrices & vectors Linear systems of equations Convolution

Follow instructions in m-file …

Questions?