EL 402

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EL 402
Xavier 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)

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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)
U(p)
Y(p)

S(p)
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Combination of systems

• Feedback combination
• F(p) RS/( 1 RS)
• poles of F zeros of 1RS

U(p)
Y(p)

R(p)
S(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 1RS(p)
• These zeros need to be located p lt 0
• 1RS(p) has the same poles as RS(p)
• P1RS PRS
• Principle of the argument
• T0 N - P
• T-1 N1RS - P1RS N1RS - 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
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Drive Precision