MATLAB CHAPTER 9 Simulink

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MATLAB ??CHAPTER 9 Simulink

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Simulation Diagram

• Simulation diagram (block diagram)
• Consider the equation
• ?
• ?
• Simulation diagrams for

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Introduction to Simulink

• Type Simulink
• in the MATLAB
• Command window
• to start window.

? The Simulink Library Browser
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Introduction to Simulink

• Create a new model
• Click on the icon that resembles a clean sheet of
  paper, or select New from the File menu in the
  Browser.

Click this icon
Or File ? New
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Introduction to Simulink

• ? Double-click appropriate library.
• ? See a list of blocks within that library.
• ? Click on the block name or icon.
• ? Hold the mouse button, and drag
• it to the new model window.
• ? release the button.

click, hold, drag, and release
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Introduction to Simulink

• ? Use the File menu in the model window to Open,
  Close, and Save model files.
• ? To print, File ? print
• ? Edit to copy, cut and paste blocks.
• ? you can also use mouse for these operations

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Introduction to Simulink

• E.g.
• Simulink Solution of
• Use Simulink to solve the following problem for
  0 t 13.
• The exact solution is

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Introduction to Simulink

• ? Start Simulink and open a new model window.
• ? Select and place the Sine Wave block from the
  Source library.
• Double-click to open the Block Parameters
  window.
• Make sure Amplitude 1, Frequency 1, Phase
  0, Sample time 0
• Then click OK.

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Introduction to Simulink

• ? Select and place the Gain block from the Math
  Operations library.
• the Block Parameters window the Gain value
  10.
• Note that the value 10 then appears in
  the triangle.
• ? Select and place the Integrator block from the
  Continuous library.
• Initial condition0 (because y(0)0)

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Introduction to Simulink

• ? Select and place the Scope block from the Sinks
  library.
• ? connect each input and output port.

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Introduction to Simulink

• ? click on the Simulation menu, and click the
  Configuration Parameters item.
• click on the solver tab, and enter 13 for
  the stop time.
• Make sure the Start time is 0.
• (matlab 6.5 click Simulation parameters)

matlab 6.5 Simulation parameters
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Introduction to Simulink

• ? Run the simulation by clicking on the
  Simulation menu,
• and clicking the Start item.

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Introduction to Simulink

• ? After the simulation, double-click on the Scope
  block .

Double-click!
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Linear State-Variable Models

• State-variable models can have more than one
  input and more than on output.
• Simulink has the State-space block that
  represents the linear state-variable model
• E.g.
• Simulink Model of the Two-Mass System

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Linear State-Variable Models

• m15, m23, c14, c28, k11, and k24
• The eq. of motion are
• These equations can be expressed in
  state-variable form as

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Linear State-Variable Models

• Vector matrix form
• Initinal condition

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Linear State-Variable Models

• Solution
• ?Create a new model window
• ?Select and place the Step block from the Sources
  library.
• Step time0, Initial and Final values 0 and
  1, Sample time0

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Linear State-Variable Models

• ?Select and place the State-Space lock. Enter
  A,B,C,D. Then enter initial condition.
• ?Select and place the Scope block.
• ?connect each port.

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Linear State-Variable Models

• ?experiment with different values of the Stop
  time until the Scope shows that the steady-state
  response has been reached.
• Ex) when stop time 25

steady-state
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Piecewise-Linear Models

• Closed-form solutions are not available for most
  nonlinear differential equations, We must solve
  such equations numerically.
• Ex)
• Piecewise-linear models are actually nonlinear,
  although they may appear to be linear.
• Ex) a mass attached to a spring and sliding on a
  horizontal surface with Coulomb friction.

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Piecewise-Linear Models

• These two equations can be expressed as the
  single, nonlinear equation.
• Solutions of models that contain piecewise-linear
  functions are very tedious to program. However,
  Simulink has built-in blocks that represent many
  of the commonly-found functions such as Coulomb
  friction. Therefore Simulink is especially
  useful for such applications. One such block is
  the Saturation block in the Discontinuities
  library.

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Simulink Model of a Rocket-Propelled sled

• A rocket propelled sled
• Compute the sleds velocity v for 0t6 if v(0)
  0
• The rocket thrust is 4000N and the sled mass is
  450kg.
• The sleds equation of motion is

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Simulink Model of a Rocket-Propelled sled

• To obtain ?(t)
• Thus the equation of motion becomes
• The solution is formally given by

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Simulink Model of a Rocket-Propelled sled

• (a) Create a Simulink model to solve this problem
  for 0t10s.
• (b) Now suppose that the engine angle is limited
  by a mechanical stop to 60 , which is 60p/180
  rad. Create a Simulink model to solve the
  problem.
• Solution
• (a)
• ?create ?(t) by integrating the constant
  twice.
• ?Constant block from the Sources library.
  Constant valuepi/50.
• ?Trigonometric block from the Math Operations
  library. Functioncos.

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Simulink Model of a Rocket-Propelled sled

• ? Set the Stop time 10, run the simulation, and
  examine the result in Scope.
• (b)
• ?modify the model as follows.
• ?the Saturation block from the Discontinuities
  library.
• Upper limit 60pi/180,
• Lower limit0.
• ?mux

mux
Generate the solution when the engine angle ?0.
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Simulink Model of a Rocket-Propelled sled

• Scope window

T0
T?0
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The Relay Block

• The Simulink Relay block is an example of
  something that is tedious to program in MATLAB
  but is easy to implement in Simulink.
• A graph of the logic of a relay.
• The relay switches the output between two
  specified values, named On and Off in the figure.

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Model of a Relay-Controlled Motor

• The model of an armature-controlled dc motor

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Model of a Relay-Controlled Motor

• x1i, x2?
• R0.6?, L0.002H, KT0.04Nm/A, Ke0.04Vs/rad,
  c0.01Nms/rad, and I610(-5)kgm2
• Suppose we have a sensor that measure the motor
  speed, and we use the sensors signal to activate
  a relay to switch the applied voltage v(t)
  between 0 and 100V to keep the speed between 250
  and 350rad/s.
• SwOff250, SwOn350, Off100, On0

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Model of a Relay-Controlled Motor

• Given parameter values
• To examine the speed ? as output, we choose

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Model of a Relay-Controlled Motor

• ?Create a new Simulink model.
• ?Select and place a Step block from the Sources
  library. Label it Disturbance Step.
• Step time0.05, Initial and Final time0 and 3,
  Sample time0
• ?Select and place a Relay block from the
  Discontinuities library.
• Switch-on and Switch-off points350 and 250,
• Output when on and Output when off0 and 100.

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Model of a Relay-Controlled Motor

• ?Select and place the Mux block from the signal
  Routing library.
• Display option to Signals.
• number of input2.
• ?Select and place the State-Space block from the
  Continuous library.
• enter the A,B,C,D.
• enter 00 for the initial conditoin.
• B tells 2 input, C and D tells 1 output.

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Model of a Relay-Controlled Motor

• ?Select and place the Scope block from the Sinks
  library.
• ?connect each port.
• ?Stop time0.1 and run the simulation. (the plot
  of ?(t) in the scope.)
• ?If you want examine the current i(t), change
  the matrix C to 1,0, and run the simulation
  again.

Note Connect signal1(first input) to the output
of the Relay block Connect signa2(second input)
to the output of the Disturbance Step
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Model of a Relay-Controlled Motor

• Relay logic scheme keeps the speed within the
  desired limits of 250 and 350 before the
  disturbance torque starts to act.
• Speed oscillates.
• When V0, the speed decrease because back emf,
  and viscous damping.
• The speed drops below 250 when the disturbance
  torque starts to act, because the V0.
• As soon as the speed drops for the speed to
  increase because the motor torque must now work
  against the disturbance.

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Model of a Relay-Controlled Motor

• Speed becomes constant
• V100, the system achieves a steady-state
  condition in which the motor torque equals the
  sum of the disturbance torque and the viscous
  damping torque. Thus the acceleration is zero.

Change the matrix C1,0 Current i(t)