el 402 xavier neyt. 24/1/01el 402 2 regulation f why? –stabilize unstable systems u e.g. inverted...
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EL 402Xavier Neyt
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24/1/01 EL 402 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)
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24/1/01 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
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24/1/01 EL 402 4
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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24/1/01 EL 402 5
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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24/1/01 EL 402 6
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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24/1/01 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
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24/1/01 EL 402 8
Example
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24/1/01 EL 402 9
Example
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24/1/01 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...
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24/1/01 EL 402 11
Bode diagram
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24/1/01 EL 402 12
Nyquist diagram
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24/1/01 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
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24/1/01 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
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24/1/01 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.
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24/1/01 EL 402 16
Example: unstable 1st order sys.
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24/1/01 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)
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24/1/01 EL 402 18
Example: stable 4th order sys.
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24/1/01 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
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24/1/01 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
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24/1/01 EL 402 21
Gain/Phase margins
Unit Gain circle
Phase margin
Gain margin
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24/1/01 EL 402 22
Gain/Phase margins
Unit Gain circle
Phase margin
Gain margin
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24/1/01 EL 402 23
Gain/Phase margins
-180°
Gain margin
Phase margin
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24/1/01 EL 402 24
Drive Precision