EL 402

1. EL 402 Xavier Neyt

2. Regulation • Why? • Stabilize unstable systems • e.g. inverted pendulum • Modify the dynamic behaviour • e.g. car suspension, B747 • Increase the “drive precision” • e.g. static error (lift) EL 402

3. Regulation • How? • Combine two systems • the actual system S(p) • the control system R(p) • Such that the new system has the desired behaviour • poles at a convenient position EL 402

4. U(p) Y(p) R(p) S(p) Combination of systems • Serial combination • : F(p) = R(p) S(p) • does not move the poles of S(p) • ! Zeroes of R(p) should NOT cover unstable poles of S(p) EL 402

5. Combination of systems • Parallel combination • : F(p) = R(p) + S(p) • does not move the poles of S(p) R(p) U(p) Y(p) + S(p) EL 402

6. Combination of systems • Feedback combination • : F(p) = RS/( 1 + RS) • poles of F = zeros of 1+RS U(p) + Y(p) + R(p) S(p) - EL 402

7. Example • S(p): First order system: • : S(p) = 1/( pT - 1) • pole in p = 1/T  unstable • R(p): Proportional (constant) • : R(p) = K • F(p) = K/(pT -1 + K) • pole in p = (K-1)/T EL 402

8. Example EL 402

9. Example EL 402

10. Nyquist diagram • Plot of RS(p) in parametric form • : x = Re( RS(p) ) • : y = Im( RS(p) ) • for p  Nyquist contour • Can be deduced from the Bode plot • in the simple cases... EL 402

11. Bode diagram EL 402

12. Nyquist diagram EL 402

13. Stability • Aim of the Nyquist theorem • determine the stability of the closed-loop system • knowing the stability of the open-loop system EL 402

14. Stability • How does it work? • Need to know the zeros of 1+RS(p) • These zeros need to be located p < 0 • 1+RS(p) has the same poles as RS(p) • P1+RS = PRS • Principle of the argument: • T0 = N - P • T-1 = N1+RS - P1+RS = N1+RS - PRS = PF - PRS EL 402

15. Stability • Nyquist theorem • :T-1 = PF - PRS • La boucle fermee sera stable ssi le contour de Nyquist enlace (ds le sens negatif) autant de fois le point (-1,0) que le systeme en boucle ouverte possede de poles instables • De gesloten lus zal stabiel zijn als en slechts als het aantal toeren (in negatieve zin) die de Nyquist kromme rond het punt (-1,0) doet gelijk is aan het aantal onstabiele polen van de open lus. EL 402

16. Example: unstable 1st order sys. EL 402

17. Stability • Nyquist theorem • particular case: the open-loop system is stable • PRS = 0  T-1 = 0 • If the open-loop system is stable, the closed-loop system will be stable iff the Nyquist curve does not go round the point (-1,0) EL 402

18. Example: stable 4th order sys. EL 402

19. Robustness • Introduces the notion of stability margins • define some kind of distance between the point (-1,0) and the Nyquist curve. • Most often used distances • gain margin • phase margin EL 402

20. Robustness • Most often used distances • gain margin • Distance to the point having a phase = -180º • Maximum gain allowed in R without compromising the system stability • maximum & minimum gain • phase margin • Angle to the first point having unit gain (0dB gain) • How much phase rotation is R allowed to introduce without compromising the system stability • max phase lag & max phase lead EL 402

21. Gain/Phase margins Unit Gain circle Phase margin Gain margin EL 402

22. Gain/Phase margins Unit Gain circle Phase margin Gain margin EL 402

23. Gain margin Phase margin Gain/Phase margins -180° EL 402

24. Drive Precision EL 402