Introduction to
       Presentation Outline

   What is Simulink?
   Basic operations with Simulink
   Examples
   Exercise
                 What is Simulink?
   Simulink is an interactive tool for modeling,
    simulating and analyzing dynamic systems.
   Simulink integrates seamlessly with MATLAB,
    providing immediate access to an extensive range of
    analysis and design tools.
   Simulating a dynamic system is a two-step process
    with Simulink:
     create a model of the system to be simulated using
      Simulink’s model editor (BLOCK DIAGRAM)
     use Simulink to simulate the behavior of the system for
      a specified time span
            Launch Simulink
• First launch MATLAB.
• To open Simulink, type simulink at the MATLAB
  command window or click on the Simulink icon
  on the MATLAB toolbar.
           Simulink Block Libraries
Simulink provides a library
  browser that allows you to
  select blocks from libraries
  of standard blocks:
 Continuous - blocks that
  describe linear functions
 Discrete - blocks that
  describe        discrete-time
 Functions & Tables -
  general functions and table
  look-up operations
 Math - blocks that describe
  general mathematics
         Simulink Block Libraries
   Nonlinear - blocks that describe nonlinear functions
   Signal & systems - blocks that allow multiplexing,
    de-multiplexing, implement external input/output, pass
    data to other parts of the model, create subsystems
    and perform other functions
   Sinks - blocks that display or write block output
   Sources - blocks that generate signals
   Blocksets and toolboxes - the extras block library
    of specialized blocks
         Creating 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

                 Block Diagram
A Simulink block diagram is a pictorial model of a
  dynamic    system.    It  consists    of blocks
  interconnected by lines.
Blocks represent elementary dynamical systems
  that Simulink knows how to simulate. A block
  comprises one or more of the following:
  A   set of inputs.
  A   set of states.
  A   set of outputs.
To introduce blocks in your model, choose the block
  from the library, click on it and drag it in your
  model. Double clicking on the block will allow you
  to change the block parameters.
          Model Execution Phase
In this phase Simulink successively computes the
  states and the outputs of the system at intervals
  from the simulation start time to the stop time,
  using information provided by the model.
Time steps - successive time points at which the
  states and the outputs are computed.
Step size - the length of time between steps. It
  depends on the type of solver:
   Fixed-step - a smaller step size produces a more
    accurate simulation but results in a longer
    execution time.
          Model Execution Phase
   Variable step - depending on the application, it
    can produce more accurate results without
    sacrificing execution speed.

Parameters set up:
       Simulation > Simulation parameters …

Simulink simulates a system when you choose start
  from the model editor’s simulation menu.
      Example 1: a Simple Model

   Build a Simulink model that solves the
    differential equation

                 x  3sin 2t 
   Initial condition x (0)  1.

   First, sketch a simulation diagram of this
    mathematical model (equation) (3 min.)
         Simulation Diagram
   Input is the forcing function 3sin(2t)
   Output is the solution of the differential
    equation x(t)

                        x (0) = -1

                    x       1        x
         3sin(2t)                          x(t)
         (input)            s            (output)

   Now build this model in Simulink
Select in 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
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
   Drag from the sine
    wave output to the
    integrator input
   Drag from the
    integrator output to       Arrows indicate the
    the scope input        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
   Select Simulation Parameters

on the
block to set
condition = -1

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

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

Double-click on
the Scope to
view the
              Simulation Results

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

The analytical

x(t )  1  3 cos2t 
        2   2

matches the plot
(the simulation
result) exactly.
               Example 2

 Build a Simulink model that solves the
  following differential equation

           m  cx  kx  f (t )
            x 
  –   2nd-order mass-spring-damper system
  –   Zero ICs
  –   Input f(t) is a step with magnitude 3
  –   Parameters: m = 0.25, c = 0.5, k = 1
  –   m->mass; c->damping factor; k->spring
    Example 2

      m         f (t)

            x           c
 Create the Simulink Diagram

 On the following slides:
  – The simulation diagram for solving the
    ODE is created step by step.
  – After each step, elements are added to
    the Simulink model.
 Optional exercise: first,    sketch   the
  complete diagram (5 min.).
      Create the Block Diagram
   First, solve for the term with highest-
    order derivative

           m  f (t )  cx  kx
            x              
   Make the left-hand side of this equation
    the output of a summing block


Drag a Sum block from the Math library

Double-click to change
the block parameters
to rectangular and + - -
      Create the Block Diagram

   Add a gain (multiplier) block to
    eliminate the coefficient and produce
    the highest-derivative alone

                       ..       ..
                      mx    1   x

   Drag a Gain block from the Math library

                         The gain is 4 since 1/m = 4.

Double-click to change
the block parameters.
Add a title.
      Create the Block Diagram

   Add integrators to obtain             the
    desired output variable

            x    1   
                     x    1   
                              x   1   x
                 m        s       s
                       Drag Integrator blocks from
                       the Continuous library

                                      ICs on the
                                      integrators are

Add a scope from the Sinks library.
Connect output ports to input ports.
Label the signals by double-clicking on the leader line.
      Create the Block Diagram

   Connect to the integrated signals with
    gain blocks to create the terms on the
    right-hand side of the equation

             x    1    
                       x        1   
                                    x   1   x
                  m             s       s
      summing          
                      cx        c
                           kx       k
                                Drag new Gain blocks
                                from the Math library
                              To flip the gain block, select it and
                              choose Flip Block in the Format pull-
                              down menu or double-clock on it.

                                                          c = 0.5

 Double-click on gain
  blocks to set parameters
 Connect from the gain
  block input backwards up
                                                          K = 1.0
  to the branch point.
 Re-title the gain blocks.
               Complete the Model

   Bring all the signals and inputs to the
    summing block.
   Check signs on the summer.

  f(t)   +
input          m
                x    1   
                         x    1       
                                      x       1       x    x(t)
                     m        s               s           output
                         c       c       
                         kx           k           x
Drag the Step function from the
Source library
Double-click on Step block
to set parameters. For a
step input of magnitude 3,
set Final value to 3
Final Simulink Model
Run the Simulation

     Overshoot of 0.5.
     Final value of 3.
     Is this expected?
             Checking Results
                            x     c       1
   Standard form                 x  x  f (t )
                        k         k       k

    Natural frequency        k
                       n     2.0

                        2  c
                                0.5
   Damping ratio       n k

   Static gain         K  1
          Checking Results

Damping ratio of 0.5 is less than 1:
  – Expect the system to be underdamped.
  – Expect to see overshoot.
Static gain is 1:
  – Expect output magnitude to equal input
  – Input has magnitude 3, so does output.
     Simulation results conform to
Saving to Workspace
      Drag the To Workspace
      block from the Sink library
            Saving to Workspace
                         Double click on the To
                         Workspace     block  to
                         change the parameters.

Check on MATLAB workspace if the variable is there.
Example: plot (tout, x); y = sqrt (x )
Inserting a S-Function
          Drag a S-Function block
          from the Functions &
          Tables library
  Inserting a S-Function
                Double click on the S-
                Function block to
                change the S-Function
                name and include
                additional parameters.

Use the template that comes with
 Change the template based on
          your project.
          Inserting a S-Function
•   Type sfundemos at the MATLAB
    command line.
•   Double click on M-files
•   Double click on M-file S-Function
•   Save the file in another folder and with
    another name
•   Change the function name:

    function [sys,x0,str,ts]=sfungains(t,x,u,flag)
            Inserting a S-Function
•   Change the S-Function size parameters:

sizes = simsizes;

sizes.NumContStates = 0;
sizes.NumDiscStates = 0;
sizes.NumOutputs    = 2;
sizes.NumInputs    = 0;
sizes.DirFeedthrough = 0;
sizes.NumSampleTimes = 1; % at least one sample time is

sys = simsizes(sizes);
      Inserting a S-Function

•   Edit the mdlOutputs m-function in
    accordance to your project:

function sys=mdlOutputs(t,x,u)

K1 = 50;
K2 = 20;
sys = [K1 K2];
Inserting a S-Function

           Drag a Demux block from
           the Signals & Systems
Inserting a S-Function

            Drag Display blocks from
            the Sink library

    Run your project

Given the following block diagram:



 u          +                        +       y
      K1                        K4
                _                        +


1) Show the correspondence of this block diagram
   with the RC circuit simulated in Assignment #1
2) Find K1, K2, K3, K4 and K5 in accordance to the
   parameters of Assignment #1.
3) Implement the system in Simulink. Use
   MATLAB to enter your parameters through a M-
4) Simulate your system in Simulink and compare
   the response with your expected results.
5) K3 is related to the initial conditions.
   Incorporate I.C. in your integrator block.

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