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Waterbed Effect

Waterbed Effect. Open Loop Transfer Matrix The Law Waterbed Effect Formulas Physical Interpretation Design Implications References. Outline. L(s). Open Loop Transfer Matrix. L(s) state-space realization. Open Loop Characteristic Polynomial. The Law.

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Waterbed Effect

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  1. Waterbed Effect Nathan Sorensen Kedrick Black

  2. Open Loop Transfer Matrix The Law Waterbed Effect Formulas Physical Interpretation Design Implications References Outline Nathan Sorensen Kedrick Black

  3. L(s) Open Loop Transfer Matrix L(s) state-space realization Open Loop Characteristic Polynomial Nathan Sorensen Kedrick Black

  4. The Law • All systems must obey Bode’s Sensitivity Integrals • Increased Performance over some frequencies = Increased sensitivity in others Np -> RHP-poles at locations pi Sensitivity Function Complimentary Sensitivity Function Nathan Sorensen Kedrick Black

  5. L(s) has at least two more poles than zeros (first waterbed formula) L(s) has a RHP-zero (second waterbed formula) Waterbed Effect Causes Nathan Sorensen Kedrick Black

  6. First Waterbed Formula • At least two more poles than zeros • Ex: • Sensitivity: where the Nyquist plot is within the unit circle Np -> RHP-poles at locations pi Nathan Sorensen Kedrick Black

  7. Second Waterbed Formula • A single real RHP-zero z or a complex conjucate pair of zeros z = x jy • Ex: Nathan Sorensen Kedrick Black

  8. Physical Interpretation • Removal of sensitivity “dirt” from lower frequencies creates higher sensitivity at higher frequencies • This will lead to instability and/or non-optimal performance at higher frequencies Nathan Sorensen Kedrick Black

  9. Performance Trade-offs • All sensitivity “dirt” within the region from wi to w0 must remain within that region • Sensitivity from w0 to infinity will remain constant for any control scheme ωb is the bandwidth -> ω0 - ωi w0 is the cutoff frequency Nathan Sorensen Kedrick Black

  10. Time Domain Constraints If p is a pole of L and q is a zero of L then S(p) = 0, S(q) = 1, T(p) = 1, T(q) = 0 If the closed loop is stable then the following 2 are true: 1. error at the RHP-pole goes to 0 and the error at the RHP- zero goes to 1/q 2. output at the RHP-pole goes to 1/p and the output at the RHP-zero goes to 0 Nathan Sorensen Kedrick Black

  11. Design Interpretations Nathan Sorensen Kedrick Black

  12. Time Domain Tradeoffs For a RHP pole p or zero q For a RHP pole p and a zero q If p = q then the pole and zero cancel which has no effect on the system Nathan Sorensen Kedrick Black

  13. “Essentials of Robust Control” by Kemin Zhou “Multivariable Feedback Control” by Sigurd Skogestad and Ian Postlethwaite “1998 American Control Conference Tutorial Workshop Number 3” by Graham Goodwin, Jim Freudenberg, Rick Middleton, Julio Braslavsky, and Maria Marta Seron References Nathan Sorensen Kedrick Black

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