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.