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EL 402. Xavier Neyt. 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). Regulation. How? Combine two systems the actual system S(p) the control system R(p)
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EL 402 Xavier Neyt
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
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
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
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
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
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
Example EL 402
Example EL 402
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
Bode diagram EL 402
Nyquist diagram EL 402
Stability • Aim of the Nyquist theorem • determine the stability of the closed-loop system • knowing the stability of the open-loop system EL 402
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
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
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
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
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
Gain/Phase margins Unit Gain circle Phase margin Gain margin EL 402
Gain/Phase margins Unit Gain circle Phase margin Gain margin EL 402
Gain margin Phase margin Gain/Phase margins -180° EL 402
Drive Precision EL 402