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System Response Characteristics

System Response Characteristics. ISAT 412 -Dynamic Control of Energy Systems (Fall 2005). Review. We have overed several O.D.E. solution techniques Direct integration Exponential solutions (classical) Laplace transforms

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System Response Characteristics

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  1. System Response Characteristics ISAT 412 -Dynamic Control of Energy Systems (Fall 2005)

  2. Review • We have overed several O.D.E. solution techniques • Direct integration • Exponential solutions (classical) • Laplace transforms • Such techniques allow us to find the time response of systems described by differential equations

  3. Generic 1st order model • Solution in Laplace domain • Solution comprised of • Free Response (homogeneous solution) • Forced Response (non-homogeneous solution)

  4. Free response of 1st order model • Free response means: • Converting back to the time domain:

  5. Time constant • Define the system time constant as • Rewriting the free response or

  6. Free response behavior Unstable Stable Unstable

  7. Meaning of the time constant • When t = t • When t = 2t, t = 3t, and t = 2t,

  8. Transfer Functions and Common Forcing Functions ISAT 412 -Dynamic Control of Energy Systems (Fall 2005)

  9. Forced response of 1st order system • The forced response corresponds to the case where x(0) = 0 • In the Laplace domain, the forced response of a 1st order system is

  10. Transfer functions • Solve for the ratio X(s)/F(s) • T(s) is the transfer function • Can be used as a multiplier in the Laplace domain to obtain the forced response to any input

  11. Using the transfer function • Now that we know the transfer function for a 1st order system, we can obtain the forced response to any input if we can express that input in the Laplace domain

  12. Step input • Used to model an abrupt change in input from one constant level to another constant level • Example: turning on a light switch

  13. Heaviside (unit) step function • Used to model step inputs

  14. Time shifted unit step function • For a unit step shifted in time, • Using the shifting property of the Laplace transform (property 6)

  15. Step input model • For a step of magnitude b at time D

  16. Pulse input

  17. Pulse input model • Use two step functions

  18. Pulse input model • For a pulse input of magnitude M, starting at time A and ending at time B

  19. Impulse input • Examples: explosion, camera flash, hammer blow

  20. Impulse input model • Unit impulse function • For an impulse input of magnitude M at time A

  21. Ramp input

  22. Ramp input model • For a ramp input beginning at time A with a slope of m

  23. Other input functions • Sinusoidal inputs • Combinations of step, pulse, impulse, and ramp functions

  24. Modeling periodic inputs

  25. Square wave input model • Addition of an infinite number of step functions with amplitudes A and -A

  26. Laplace transform of square wave

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