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# Me330 lecture8

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### Me330 lecture8

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