Me330 lecture8
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Me330 lecture8

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Me330 lecture8 Me330 lecture8 Presentation Transcript

  • ME 330 Control Systems SP 2011 Lecture 8
  • Nonlinear Systems
    • Every real system is truly nonlinear
    • Linear models often approximate nonlinear systems within acceptable operating range
    spring damper
  • Linearization of Nonlinear Systems
    • Pendulum model
     l m
    • Nonlinear differential equation
    • Free-body diagram
    l mg T inertia for simple pendulum
    • Linear differential equation approximation (for small angles  )
  • Linear Model Response
    • Pendulum response
    from the model Laplace domain representation
    • Assume and
    • Inverse Laplace transform
  • Inverted Pendulum
    • One of the most studied control problems (in the nonlinear form)
    Define the center of gravity coordinates
    • Open-loop unstable (will fall unless suitable force is applied)
    • Force f(t) applied to cart
    • Assume center of gravity is at the geometric center
     l mg x f M
  • Inverted Pendulum
    • Nonlinear equations of motion
     l mg V H horizontal vertical rotational horizontal f V H M
  • Linearization of Inverted Pendulum
    • Assume  to be small
    horizontal vertical
  • Transfer Function Inverted Pendulum
    • Solve for X(s)
    • Plug into
  • Next Lectures
    • Direct transfer function manipulation.