Elements of Feedback Control

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Name_________________
In order to have transient die out, all
closed-loop poles of the system must __in the
LHP, or stable__ Use z, s, wn , and wd to fill
in the spaces below. Settling time is inversely
proportional to ____ s_____ Rise time is
inversely proportional to ____ wn
_____ Percentage overshoot is most directly
determined by ___ z ____ Oscillation frequency
determined by ____ wd _____
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Prototype 2nd order system
target
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Settling time
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Effects of additional zeros
Suppose we originally have
i.e. step response
Now introduce a zero at s -z
The new step response
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Effects

• Increased speed,
• Larger overshoot,
• Might increase ts

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When z lt 0, the zero s -z is gt 0, is in the
right half plane. Such a zero is called a
nonminimum phase zero. A system with nonminimum
phase zeros is called a nonminimum phase system.
Nonminimum phase zero should be avoided in
design. i.e. Do not introduce such a zero in
your controller.
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Effects of additional pole
Suppose, instead of a zero, we introduce a pole
at s -p, i.e.
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L.P.F. has smoothing effect, or averaging effect
Effects

• Slower,
• Reduced overshoot,
• May increase or decrease ts

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Matlab program template
enter plant transfer function Gp(s) nump .
denp. enter desired closed loop step
response specification you may allow both
uppper and lower limits convert from specs
to zeta, omegan, sigma, omegad plot
allowable region for pole location obtain
controller transfer function for now make C(s)
1 numc1 denc1 obtain closed loop transfer
function from Gp(s) and C(s) numcl
dencl obtain closed-loop step response
compute actual step response specs, using
your program from last week are they
good? compute the actual closed-loop poles,
place x at those locations are they in
the allowable region?
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Root locus

• A technique enabling you to see how close-loop
  poles would vary if a parameter in the system is
  varied
• Can be used to design controllers or tuning
  controller parameters so as to move the dominant
  poles into the desired region

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• Recall step response specs are directly related
  to pole locations
• Let p-sjwd
• ts proportional to 1/s
• Mp determined by exp(-ps/wd)
• tr proportional to 1/p
• It would be really nice if we can
• Predict how the poles move when we tweak a system
  parameter
• Systematically drive the poles to the desired
  region corresponding to desired step response
  specs

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Root Locus
k s(sa)
y
e
r
Example

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Two parameters k and a. would like to know how
they affect poles
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The root locus technique

• Obtain closed-loop TF and char eq d(s) 0
• Rearrange terms in d(s) by collecting those
  proportional to parameter of interest, and those
  not then divide eq by terms not proportional to
  para. to get
• this is called the root locus equation
• Roots of n1(s) are called open-loop zeros, mark
  them with o in s-plane roots of d1(s) are
  called open-loop poles, mark them with x in
  s-plane

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1. The o and x marks falling on the real axis
   divide the real axis into several segments. If a
   segment has an odd total number of o and/or x
   marks to its right, then n1(s)/d1(s) evaluated on
   this segment will be negative real, and there is
   possible k to make the root locus equation hold.
   So this segment is part of the root locus. High
   light it. If a segment has an even total number
   of marks, then its not part of root locus. For
   the high lighted segments, mark out going arrows
   near a pole, and incoming arrow near a zero.

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• Let npolesorder of system, mzeros. One root
  locus branch comes out of each pole, so there are
  a total of n branches. M branches goes to the m
  finite zeros, leaving n-m branches going to
  infinity along some asymptotes. The asymptotes
  have angles (p 2lp)/(n-m). The asymptotes
  intersect on the real axis at

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• Imaginary axis crossing
• Go back to original char eq d(s)0
• Use Routh criteria special case 1
• Find k value to make a whole row 0
• The roots of the auxiliary equation are on jw
  axis, give oscillation frequency, are the jw axis
  crossing points of the root locus
• When two branches meet and split, you have
  breakaway points. They are double roots. d(s)0
  and d(s) 0 also. Use this to solve for s and k.
• Use matlab command to get additional details of
  root locus
• Let num n1(s)s coeff vector
• Let den d1(s)s coeef vector
• rlocus(num,den) draws locus for the root locus
  equation
• Should be able to do first 7 steps by hand.