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