Introduction to MATLAB - MITweb.mit.edu/acmath/matlab/IAP2007/IntroMatlabLinsysCalcODE.pdf · IAP 2007 Introduction to MATLAB: Calculus, Linear Algebra, Differential Equations Topics

Introduction to MATLAB: Calculus, Linear Algebra, Differential EquationsIAP 2007

Introduction to MATLAB

Violeta Ivanova, Ph.D.Office for Educational Innovation & Technology

[email protected]://web.mit.edu/violeta/www


Topics

MATLAB Interface and Basics Calculus, Linear Algebra, ODEs Graphics and Visualization Basic Programming Programming Practice Statistics and Data Analysis


Resources

Class materialshttp://web.mit.edu/acmath/matlab/IAP2007 Previous session: InterfaceBasics <.zip, .tar> This session: LinsysCalcODE <.zip, .tar>

Mathematical Tools at MIT web sitehttp://web.mit.edu/ist/topics/math


MATLAB Toolboxes & Help MATLAB

+ Mathematics+ Matrices and Linear Algebra

+ Solving Linear Systems of Equations+ Inverses and Determinants Eigenvalues

+ Polynomials and Interpolation+ Convolution and Deconvolution

+ Differential Equations+ Initial Value Problems for ODEs and DAEs


MATLAB Calculus & ODEs

Integration & DifferentiationDifferential EquationsODE Solvers


Integration

Polynomial integration

Analytical solution Built-in function polyint

>> p = [p1 p2 p3 …];

>> P = polyint(p); assumes C=0>> P = polyint(p, k); assumes C=k

p1! x

n+ ...+ pnx + pn+1dx = P1x

n+1+ ...+ Pn+1x + C


More Polynomial Integration

Area under a curve

Built-in function polyval>> p = [p1 p2 p3 …];

>> P = polyint(p) >> area = polyval(P, b) - …

polyval(P, a)

p1

a

b

! xn+ p

2xn"1...+ pn"1x

2+ pnx + pn+1dx


Differentiation

Polynomial differentiation

Built-in function polyder>> P = [P1 P2 … Pn C]

>> p = polyder(P)

d

dx(P1xn+ ...+ Pnx + C) = p

1xn!1

+ ...+ pn


Differential Equations

Ordinary Differential Equations (ODE)

Differential-Algebraic Expressions (DAE)

MATLAB solvers for ODEs and DAEs>> ode45; ode23; ode113; ode23s …

y ' = f (t, y)

M (t, y)y ' = f (t, y)


ODE and DAE Solvers Syntax

>> [T,Y] = solver(odefun,tspan,Y0)

solver: ode45, ode23, etc.odefun: function handletspan: interval of integration vectorY0: vector of initial conditions[T, Y]: numerical solution in two vectors


ODE Example: Mars Lander Entry, descent, landing

(EDL) force equilibriumLander velocity: vDrag coefficient: k = 1.2Lander mass: m = 150 kgMars gravity: g = 3.688 m/s2

EDL velocity ODEVelocity at t0=0: v0=67.056 m/s

mdv t( )

dt= mg ! kv

2t( )

dv t( )

dt= g !

k

mv2t( )

!F = m

!a

Theoretical example courtesy of Ms. Shannon Cheng, MIT Department of Aeronautics & Astronautics.


ODE Example (continued)

Problem:

Solution: odeLV.m, MarsLander.mtspan = [0 : 0.05 : 6];V0 = 67.056;

[t, V] = ode45(@odeLV, tspan, V0)-------------------------function DV = odeLV(t, V)

K = 1.2; M = 150; G = 3.688;DV = G - K/M * V^2

dv t( )

dt= g !

k

mv2t( )


PDE Solver Partial Differential Equations (PDEs)

Elliptic and parabolic

e.g. Laplace’s equation:

Numerical PDE solver: pdepe Syntax:

>> solution = pdepe(m, pdefun,icfun, bcfun, xmesh, tspan)

Function handles: pdefun, icfun, bcfun

!2V

!x2+!2V

!y2= 0


MATLAB Linear Systems

Linear Equations & SystemsEigenvalues & EigenvectorsLinear Dynamic Networks


Useful 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)


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

!

"

###

$

%

&&&


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


State Equation Ordinary Differential Equations

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


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( )


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

!

"#

$

%&

Theoretical example courtesy of Prof. Stephen Hall, MIT Department of Aeronautics & Astronautics.


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


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)


Linear Systems Exercises

Exercise One: stateeig.m Matrices & vectors Systems of linear equations Eigenvalues & eigenvectors

Follow instructions in m-file …


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


Convolution Definition

Example: signals & systems

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

$

% u #( )d#

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


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)


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

#$%

Theoretical example courtesy of Prof. Stephen Hall, MIT Department of Aeronautics & Astronautics.


Example: RC Circuit (continued)


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-’)


Linear Systems Exercises

Exercise Two: convolution.m Matrices & vectors Convolution

Follow instructions in m-file …

Documents

Introduction to MATLAB - MITweb.mit.edu/acmath/matlab/IAP2007/IntroMatlabLinsysCalcODE.pdf · IAP 2007 Introduction to MATLAB: Calculus, Linear Algebra, Differential Equations Topics