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Me330 lecture2

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Me330 lecture2

1. 1. ME 330 Control Systems SP 2011 Lecture 2
2. 2. Complex Numbers <ul><li>Complex numbers contain both real and imaginary parts </li></ul><ul><li>Complex functions “ ” </li></ul><ul><ul><li>Example </li></ul></ul>
3. 3. Laplace Transform <ul><li>The Laplace transform is given by </li></ul><ul><li>The inverse Laplace transform is given by </li></ul><ul><ul><li>c > 0 and c > singular values of f(t) is an integration constant </li></ul></ul><ul><li>The transformation is no loss of “information” </li></ul><ul><ul><li>Can convert between the two (not like averaging) </li></ul></ul>
4. 4. Common Functions 1 <ul><li>Exponential </li></ul><ul><li>Step </li></ul>not defined at t = 0
5. 5. Common Functions 2 <ul><li>Sinusoid </li></ul><ul><li>Ramp </li></ul>Euler Formula
6. 6. General Functions <ul><li>Function Derivatives </li></ul>
7. 7. Properties of Laplace Transform <ul><li>Linearity </li></ul><ul><li>Frequency Shift </li></ul><ul><li>Convolution </li></ul>
8. 8. Laplace Tables <ul><li>In general, don’t solve Laplace integrals. </li></ul><ul><ul><li>Common functions and corresponding Laplace transforms are well documented </li></ul></ul>
9. 9. Usefulness of Laplace Transform <ul><li>Basic mathematical framework for control systems analysis and design </li></ul><ul><ul><li>differential equations  algebraic equations </li></ul></ul>f(t) Inputs Outputs
10. 10. Example Process <ul><li>Solving differential equations to get the time-domain response of the dynamical system. </li></ul>Step 2) Transform f(t) to F(s) Step 1) Represent the system differential equation in Laplace domain “transfer function” Step 3) Multiply “transfer function” with Laplace domain input F(s) Step 4) Solve for time-domain response with inverse Laplace transform f(t)
11. 11. Laplace Domain Mathematical Framework <ul><li>Solutions for time-domain response to specific input excitation </li></ul><ul><li>Analysis of system stability </li></ul><ul><ul><li>given any bounded input, the output will be bounded </li></ul></ul><ul><li>Performance of system </li></ul><ul><ul><li>speed of time-domain response, oscillatory behavior, steady-state (persistent) errors </li></ul></ul>
12. 12. Next Lecture <ul><li>Techniques for solving inverse Laplace </li></ul><ul><li>Solutions for time-domain response to specific input excitation </li></ul><ul><li>Derivation of simple dynamical system models </li></ul>