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DESCRIBING FUNCTION

DESCRIBING FUNCTION. DESCRIBING FUNCTION. Is there a way to analytically analyze oscillations in nonlinear control loops? Simulation is OK, but if control design is also required, then simulation approach is not easy

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DESCRIBING FUNCTION

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  1. DESCRIBING FUNCTION

  2. DESCRIBING FUNCTION • Is there a way to analytically analyze oscillations in nonlinear control loops? • Simulation is OK, but if control design is also required, then simulation approach is not easy • Describing function or harmonic linearization analysis provides analytical approach for both analysis and design of controllers

  3. Nonlinearity Linear part n(t) r(t) = 0 e(t) + c(t) N(e) G(jw) - A DESCRIBING FUNCTION B

  4. DESCRIBING FUNCTION • Assumptions • No input, r(t) = 0 • Linear part acts as a low-pass filter, that is, higher order harmonic components are damped • Nonlinearity does not generate subharmonics • Nonlinearity is symmetric • Nonlinearity does not depend on frequency • Assume that at point A, e(t) = E sin(wt)

  5. DESCRIBING FUNCTION • Consider the nonlinearity • At the output, point B n(t) = N(e(t)) = N(E sin(wt)) = n(wt) • Fourier series

  6. DESCRIBING FUNCTION • From assumptions • where

  7. N(jw,E) DESCRIBING FUNCTION Only fundamental frequency

  8. Tehonsäätö langattomassa tietoliikenteessä RNC = Radio Network Controller MSC =Mobile Services Switching Center PSTN = Public Switched Telephone Network

  9. Probability Outagearea SIRtarget SIR at receiver Closed-Loop Power Control for CDMA Systems Closed-loop power control Outer loop power control

  10. Tehonsäätö langattomassa tietoliikenteessä +1 dB or –1 dB

  11. DESCRIBING FUNCTION - Relay Nonlinearity Output signal This is developed into Fourier series Input signal

  12. DESCRIBING FUNCTION - Relay • Odd nonlinearity, therefore A1 = 0.

  13. DESCRIBING FUNCTION - Saturation

  14. DESCRIBING FUNCTION - Saturation

  15. DESCRIBING FUNCTION - Backlash

  16. DESCRIBING FUNCTION - Criterion

  17. DESCRIBING FUNCTION - Criterion • Criterion for predicting oscillations • Analogous to linear case • Can also be written as

  18. DESCRIBING FUNCTION – Stable limit cycle Nyquist diagram Nichols diagram

  19. DESCRIBING FUNCTION – Unstable limit cycle Nyquist diagram Nichols diagram

  20. DESCRIBING FUNCTION – Unstable limit cycle Nyquist diagram Nichols diagram

  21. DESCRIBING FUNCTION – Example of dead zone Use describing function method to investigate oscillations in the feedback system below.

  22. DESCRIBING FUNCTION – Example of dead zone, describing functions

  23. DESCRIBING FUNCTION – Dead-zone example Describing function E=1.01:0.01:100; C=1; N=(2/pi)*acos(C./E)-(C./E).*sqrt(1-(C./E).^2); hold on;Nichols(g);grid plot (-1./N, zeros(size(N))); g=zpk([],[0 -2 -10],20) ; Zero/pole/gain: 20 ------------- s (s+2) (s+10) Nichols(g);grid Multiply gain by 17 g=zpk([],[0 -2 -10],20*17) %Zero/pole/gain: 340 -------------- s (s+2) (s+10)

  24. DESCRIBING FUNCTION – Type of limit cycle

  25. DESCRIBING FUNCTION – Type of limit cycle Oscillation occurs.

  26. DESCRIBING FUNCTION – Backlash example

  27. DESCRIBING FUNCTION – Type of limit cycle

  28. M e DESCRIBING FUNCTION – Relay with Hysteresis M = relay amplitude e = relay width Constant

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