ee4314 lab 1 session matlab and simulink spring 2014 ta: joe sanford joe.sanford@mavs.uta.edu...

EE4314 Lab 1 SessionMatlab and Simulink

Spring 2014

TA: Joe Sanford joe.sanford@mavs.uta.edu

(with special thanks to Suresh Sampathkumar and Md Ahsan Habib)

MATLAB Overview• When MATLAB launched, Command Window Appears• Command prompt(>>) in Command Windows to accept instruction or input• Objects → Data• Objects are placed in MATLAB workspace

>> a = 4; b = 3+2j; c = a * b;• whos → Another way to view workspace

>> whos• who → short version of whos -> reports only the names of workspace objects

>> who• clear a → remove specific variables (a) from the workspace

>> clear a• clc → clears the command window

>> clc• help -> most important and useful command for new users

>> help whos• exit -> terminates MATLAB

Algebra of Complex Numbers• Complex number: z = a + jb

• Re z = a; Im z = b• Complex number expressed in polar coordinates (r,θ)

• a = rcos θ, b = rsin θ, z = r(cos θ+ jsin θ)• Euler Formulae:

• ejθ=cos θ + jsin θ, z = r ejθ

• z = |z|ej∠z

• |z| = r = √(a2+ b2)• ∠z = θ = tan-1 (b/a), π≥ θ ≥-π

• Conjugate of z, z* = a – jb = r e-jθ= |z|e-j∠z

• zz* = (a+jb)(a-jb) = a2 + b2 = |z|2

• Useful Identities (7)• e∓jnπ = -1, n → odd integer e∓j(2n+1)π = -1, n → integer• e∓j2nπ = 1, n → integer

Complex Numbers – A common mistakez1 = a +jb

z2=-a -jb

θ1

θ2

∠ z1 = tan-1(b/a) = θ1

∠ z2 = tan-1(-b/-a) = θ2

∠ z2 ≠ ∠ z1

∠ z2 = θ2 = θ1 - 180

z1 = -a +jb

z2=a -jb

θ1

θ2

∠ z1 = tan-1(b/-a) = θ1

∠ z2 = tan-1(-b/a) = θ2

∠ z1 ≠ ∠ z2

∠ z1 = θ1 = 180 + θ2

Example B.1 (9)

Complex Numbers - MATLAB• Matlab predefines i = j =

>> z = -3-j4• real and imag operators extract real and imaginary components of z.

>> z_real = real(z)>> z_imag = imag(z)

• Modulus or Magnitude of a complex number•

>> z_mag = sqrt(z_real^2+z_imag^2)• |z|2 = zz*

>> z_mag = sqrt(z*conj(z))>> z_mag = abs(z)

• Angle of a complex number>> z_rad = atan2(z_mag, z_real)

• atan2 -> two-argument arc-tangent function; ensures the angle reflects in the proper quadrant.>> z_rad = angle(z)

• MATLAB function pol2cart number polar form to Cartesian form• z = 4 e-j(3π/4)

>> [z_real, z_imag] = pol2cart(-3*pi/4,4)

Complex Numbers - Exercise

• Determine z1z2 and z1/z2 if z1 = 3+j4 and z2 = 2+3j>> Verify your results using MATLAB• Convert your results from Cartesian coordinates to Polar coordinates>> Verify your results using MATLAB function pol2cart

(13)

MATLAB - Vector Operation• Vectors of even valued integers

>> k = 0:2:11• Negative and non-integer step sizes

>> k = 11:-10/3:0• If step size not specified, value of one assumed

>> k = 0:11• In MATLAB, ascending positive integer indices specify particular vector elements.

>> k(5), k(1:4),• Vector representation to create signals

• 10 Hz sinusoid described by f(t) = sin(2π10t+π/6) when 0≤t<0.2>> t = 0:0.0004:0.2-0.0004; f = sin(2*pi*10*t+pi/6); f(1)

• Find the three cube roots of minus 1, • →

>> k = 0:2;>> w = exp(j*(pi/3 + 2*pi*k/3))

Exercise (56)

• Find the 50th cube root of minus 1?

Simple Plotting• MATLAB’s plot command

>> plot(t,f);• Axis labels are added using xlabel and ylabel

>> xlabel(‘t’); ylabel(‘f(t)’)• Plotting discrete points, 100 unique roots of w^100=-1

>> plot(real(w), imag(w), ‘o’);>> xlabel(‘Re(w)’); ylabel(‘Im(w)’);>> axis equal

Element by Element Operations• Multiplication, Division and Power• x = [5 4 6]; y = [1 2 3];

>> x_mul_y = x * y >> x_elem_mul_y = x.*y>> x_div_y = x/y>> x_elem_div_y = x./y>> x_elem_pow_y = x.^y

•Suppose h(t) = f(t)g(t) where g(t) = exp(-10*t)>> g = exp(-10*t);>> h = f.*g;>> plot (t,f,’-k’,t,h,’-b’);>> xlabel(‘t’); ylabel(‘Amplitude’);>> legend (‘f(t)’,’h(t));

h g(t)

Damped Sinusoid

Matrix Operations• Common Useful function

• eye(m) creates the m×m identity matrix>> eye(3)

• ones(m,n) creates the m×n matrix of all ones>> ones(2,3)

• zeros(m,n) creates the m×n matrix of all zeros>> zeros(3,2)

•Row vector>> r = [1 3 2];

•A 2×3 matrix>> A = [2 3; 4 5; 0 6];

Matrix Operations• Transpose

>> c= r’;• Concatenation

>> B = [c A];• Matrix inverse

>> D = inv(B);

• Matrix indices>> B(1,2)>> B(1:2,2:3)

• Colon can be used to specify all elements along a specified dimension

>> B(2,:)>> B(:,2)

Matrix Operations

Solve

• Ax = y; • x = A-1Ax = A-1y >> A = [1 -2 -3; -sqrt(3) 1 –sqrt(5); 3 –sqrt(7) 1]; >> y = [1; pi; exp(1)]; >> x = inv(A)*y

,

Control Systems Toolbox

• Useful functions (example code to follow)

– tf(num, den) - creates a continuous-time transfer function with numerator(s) num and denominator(s) den.

– ss(a,b,c,d) - creates the continuous-time state-space model– impulse() - calculates the unit impulse response of a linear system.– step() - calculates the unit step response of a linear system.– lsim() - simulates the time response of continuous or discrete linear systems

to arbitrary inputs. – residue(b,a) - finds the residues, poles, and direct term of a partial fraction

expansion of the ratio of two polynomials, b(s) and a(s).– ode23() - solves initial value problems for ordinary differential equations.– Dsolve () – solves differential equations symbolically.

Control Systems Toolbox (exa)

• How do we solve this in Matlab?– Define variables for numerator and denominator– Build a system

• Create a transfer function

– Show response to input

Control Systems Toolbox (exa)>> Num = [1 1]Num = 1 1>> Den = [1 3 12]Den = 1 3 12>> sys = tf(Num, Den)

sys = s + 1 -------------- s^2 + 3 s + 12 Continuous-time transfer function.

>> impulse(sys)

What about the step response?

Control Systems Toolbox (exa)

• This can also be done in a program!– This allows you to create complex systems with

individual blocks, each with their own dynamics!– Allows for multiple responses all at once (less

typing in command prompt)• For example:

– Time Response to arbitrary input>> t = 0 : 0.1 : 10;>> u = sin(2.*t);>> lsim(sys,u,t)

Control Systems Toolbox (cont’d)

• What about Partial Fraction Expansions?– After defining system using Num, Den formulation

• Use “residue()” function>> [r,p,k] = residue(Num, Den)r = 0.5000 + 0.0801i 0.5000 - 0.0801ip = -1.5000 + 3.1225i -1.5000 - 3.1225ik = []

Can now be written as

Control Systems Toolbox - Differential Equations

Matlab can only solve first order differential equations

So, what do you do then?

example

Control Systems Toolbox - Differential Equations

• Convert Second Order system – Introduce intermediate variable

• In this case, the first derivative of the velocity

– Divide through by the mass, m

– Then introduce intermediate variable…

Control Systems Toolbox - Differential Equations

= y(0) = 600z(0) = 0

Control Systems Toolbox - Differential Equations

• Can use either ode45 or ode23 function• Must give system using only first order

equations• Let y be variable 1 and z be variable 2• Values will be returned by ODE function in

vector form– dy/dt first and dz/dt second

Control Systems Toolbox - Differential Equations

Function rk = f(t,y)mass = 80;g = 9.81;rk = [y(2); -g + 4/15*y(2).^2/mass];

Control Systems Toolbox - Differential Equations

clear alltimerange = [0 15]; %secondsinitialvalues = [600 0];[t, y]=ode45(@f, timerange, initialvalues)plot(t, y(:,1))ylabel(‘height (m)’)xlabel(‘time (s)’)

Control Systems Toolbox

• Other Useful Functions to Review– det(sI-A)– poles(eig A)– ss(A,B,C,D)– lsim(sys,u,t,x0)

Control Systems Toolbox – Stability Example

• Define Matrices– A = [ 0 1 0; 980 0 -2.8; 0 0 -100 ]; B = [ 0; 0; 100 ]; C = [ 1 0 0 ];

• Type in program or command window– poles = eig(A);– t = 0:0.01:2; – u = zeros(size(t)); – x0 = [0.01 0 0]; – sys = ss(A,B,C,0); – [y,t,x] = lsim(sys,u,t,x0); – plot(t,y)– title('Open-Loop Response to Non-Zero Initial Condition') – xlabel('Time (sec)') – ylabel(‘System Position(m)')

• What does the resultant plot show us about the system?• What about the output of the “poles = eig(A)” command?

Control Systems Toolbox – Stability Example

• Stability of the System– The eigenvalues of the A matrix are the values of s

where det(sI - A) = 0. – equivalent to the poles of the transfer fucntion

• What about the graph?

Simulink!

(very exciting stuff)(no, really. It’s actually really, really powerful)

Simulink

• Simulink is a tool for simulating dynamic systems with a graphical interface specially developed for this purpose

• Dynamic systems are described by differential equations.

• Simulink is a numerical differential equation solver.

• Launch Simulink by typing in Matlab command prompt>> simulink

Create a new model

• Click the new-model icon in the upper left corner to start a new Simulink file

• Select the Simulink icon to obtain elements of the model

Your workspace

Library of elements Model is created in this window

Save your model

• You might create a new folder, like the one shown below, called simulink_files

• Use the .mdl suffix when saving

Example 1: a simple model

• Build a Simulink model that solves the differential equation

• Initial condition• First, sketch a simulation diagram of this

mathematical model (equation)

tx 2sin3.1)0( x

Simulation diagram

• Input is the forcing function 3sin(2t)• Output is the solution of the differential

equation x(t)

• Now build this model in Simulink

xxs1

3sin(2t)(input)

x(t)(output)

1)0( x

integrator

Select an input block

Drag a Sine Wave block from the Sources library to the model window

Select an operator block

Drag an Integrator block from the Continuous library to the model window

Select an output block

Drag a Scope block from the Sinks library to the model window

Connect blocks with signals

• Place your cursor on the output port (>) of the Sine Wave block

• Drag from the Sine Wave output to the Integrator input

• Drag from the Integrator output to the Scope input Arrows indicate the direction of the

signal flow.

Select simulation parameters

Double-click on the Sine Wave block to set amplitude = 3 and freq = 2.

This produces the desired input of 3sin(2t)

Select simulation parameters

Double-click on the Integrator block to set initial condition = -1.

This sets our IC x(0) = -1.

Select simulation parameters

Double-click on the Scope to view the simulation results

Run the simulation

In the model window, from the Simulation pull-down menu, select Start

View the output x(t) in the Scope window.

Simulation results

To verify that this plot represents the solution to the problem, solve the equation analytically.

The analytical result,

matches the plot (the simulation result) exactly.

ttx 2cos)( 23

21

Example 2• Build a Simulink model that solves the

following differential equation– 2nd-order mass-spring-damper system– zero ICs– input u(t) is a step with magnitude 1– parameters: m = 1, c = 0.5, k = 2

(Continued)

(Continued)

• You can flip the orientation of the block by right click > Format > Flip Block (under “Format” or hitting “Ctrl-I”

(Continued)

• The last thing we need to do is to add all the signals together using ‘Sum’ block.

(Continued)

• At this point, the model accurately solves the ordinary differential equation.

(Continued)

• At this point, the model accurately solves the ordinary differential equation.

(Continued)

u(t) Signal z(t) Signal

Introduction of Labview and myDAQ

By: Suresh Sampathkumar

Graphical programming language& Data flow

• LabVIEW relies on graphical symbols rather than textual language to describe programming actions

• The principle of dataflow, in which functions execute only after receiving the necessary data, governs execution in a straightforward manner

How does LabVIEW work?

• LabVIEW programs are called:– Virtual Instruments (VIs) – because their appearance and operation imitate actual instruments.

• However, they are analogous to main programs, functions and subroutines from popular language like C, Fortran, Pascal, …

LabVIEW Programs Are Called VirtualInstruments (VIs)• Front Panel

• Controls = Inputs• Indicators = Outputs

• Block Diagram• Accompanying “program”

for front panel• Components “wired”

together

LabVIEW Introduction

• Two “sets” for development– Front Panel– Block Diagram

• Wiring connections

• LabVIEW Conventions

• Running LabVIEW programs

Controls Palette

• The Controls palette contains the controls and indicators you use to create the front panel.

LabVIEW Front Panel

• All user interfaces goes here!

• Used to display Controls or Indicators

• Highly customizable

LabVIEW Block Diagram

• Actual program

• Invisible to user

• Read left to right, like a book

• Where the MAGIC happens!

TerminalsWhen you place a control

(or indicator) on the

FRONT PANEL

LabVIEW automatically

creates a corresponding

control (or indicator) terminal on the BLOCKDIAGRAM

Control? or Indicator?

Controls = Inputs from the user = Source Terminals

Indicators = Outputs to the user = Destinations

Manipulating Controls and Indicators

• Right click on an indicator to– Change to control– Change format or precision

• Right click on a control to– Change to indicator– Change mechanical action (whether to latch open

or closed, and what to use as default…)

Wiring Connections• Wires transport data through the block diagram

• Wire color indicates variable type

• A red “X” means something is wrong!

Wires

A LabVIEW VI is held together by wires connecting nodes and terminals; they deliver data from one source terminal to one or more destination terminals.

Basic wires used in block diagramsand corresponding types

Each wire has different style or color, depending on the data type that flows through the wire:

Scalar 1D array 2D array Color

Floating-point number

orange

Integer number blue

Boolean green

String pink

Dataflow Programming

• In this case, the block diagram executes from left to right, not because the objects are placed in that order, but because the Subtract function cannot execute until the Add function finishes executing and passes the data to the Subtract function. Remember that a node executes only when data are available at all of its input terminals and supplies data to the output terminals only when the node finishes execution.

Data Flow programming (Contd)

• consider which code segment would execute first—the Add, Random Number, or Divide function. You cannot know because inputs to the Add and Divide functions are available at the same time, and the Random Number function has no inputs.

Running LabVIEW Programs

• ALMOST ALWAYS put your program in some sort of loop that can be stopped with a control

• AVOID using the red “x” to stop your program

Add/edit text

Wire features together tocontrol flow of data

Select a feature toedit or move

Operate a control

Probe Data(troubleshoot)

Insert a digital indicator or control

Insert a boolean control (button or switch)

Add a structure such as for, while, and case statements

Add a numericoperator (+,-,…)

File I/O

Add a booleanoperator (and, or…)

Data Acquisition

Signal analysis

Comparison

Mathematical Functions

Timing/dialog

Reorder objects

Run

Continuous run

PauseStop

Font ring

Alignment ring

Distribution ring

Debugging featuresmore on this later…

Examples

• Area of a triangle.• Print “Hello World”.• Connecting different data types (if time

permits)

What is MyDAQ• General Purpose Data Acquisition Device• Multimeter, Oscillocope, Function Generator, • Spectrometer and much more

=

+

+

$50-$200

$300-$10,000

$300-$5,000

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+ much more

• MyDAQ - Designed for hands-on experimentation. NI MyDAQ combines portability with a comprehensive set of features. NI MyDAQ allows for real engineering and, when combined with NI LabView and Multisim, gives students the power to prototype systems and analyze circuits in or outside of the classroom.

• NI MyDAQ hardware integrates with NI Labview graphical development software, giving the students hands-on interaction with real analogue circuits, sensor measurements, and signal processing. It bridges the gap between theory and real-world practice by providing students with eight Labview software-based instruments including a digital multimeter (DMM), oscilloscope, function generator, bode analyzer, dynamic signal analyzer, arbitrary waveform generator, digital reader and digital writer.

• NI MyDAQ is compact enough to fit in a student's pocket, and is powered by a USB connection.

Digital Multimeter (DMM)

Measuring Resistance Across 1000Ω

Examples

• Read 5 volts from analog input using a wire.

Questions?