el 402
DESCRIPTION
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) - PowerPoint PPT PresentationTRANSCRIPT
EL 402Xavier Neyt
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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)
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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
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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)
R(p) S(p)U(p) Y(p)
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Combination of systems
Parallel combination–: F(p) = R(p) + S(p)–does not move the poles of S(p)
R(p)
S(p)
U(p) Y(p)+
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Combination of systems
Feedback combination–: F(p) = RS/( 1 + RS)– poles of F = zeros of 1+RS
R(p) S(p)U(p) Y(p)
+-
+
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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
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Example
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Example
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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...
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Bode diagram
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Nyquist diagram
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Stability Aim of the Nyquist theorem
– determine the stability of the closed-loop system
– knowing the stability of the open-loop system
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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
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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.
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Example: unstable 1st order sys.
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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)
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Example: stable 4th order sys.
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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
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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
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Gain/Phase margins
Unit Gain circle
Phase margin
Gain margin
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Gain/Phase margins
Unit Gain circle
Phase margin
Gain margin
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Gain/Phase margins
-180°
Gain margin
Phase margin
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Drive Precision