by Frank Owen PhD PE polyXengineering Inc San Luis Obispo California Ponly controller simple gain adjustment K P 0 K I 0 K D 0 Controller does not change root locus of existing G ID: 725629
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Slide1
P-only controllervia root locus
by Frank Owen, PhD, PE
polyXengineering, Inc.
San Luis Obispo, CaliforniaSlide2
P-only controller: simple gain adjustment
K
P
≠ 0, K
I
= 0, K
D
= 0
Controller does not change root locus of existing G
OL
K
P
-adjustment moves poles along existing root locus to a point that gives desired performanceSlide3
Example 1: 1
st
-order system
A 1
st
-order system has a steady-state gain of 1 and a time constant of 2 seconds. Install a P-only controller to double the speed of the system.
Re
Im
-1/2
Pole at -1/T
-1/1
Need to move pole to hereSlide4
Example 1: 1
st
-order system, cont.
Re
Im
-1/2
-1/1
M
p1
= 0.5
Draw vector from OL pole to desired pole location
Use graphical method to solve:
q
p1
= 180
°
There is only one such vector, so angle criterion met
To meet the magnitude criterion, K
P
= 0.5Slide5
Example 2: 2
nd
-order system
Re
Im
-3
A 2
nd-order system has open-loop poles at s = -6 and s = -3. KOL without KP = 1. A controller is added with the goal of minimizing the time to peak value but without exceeding 10% overshoot.
Design this controller.
-6
10% overshoot means:
z
= 0.591 means:
53.8
°
Put closed-loop pole hereSlide6
Example 2: 2
nd
-order system, cont.
Re
Im
-3
Again, the graphical solution:
-6
53.8
°
M
p2
M
p1
q
p1
q
p2
Draw vectors from the OL poles to the desired location of the CL pole.
q
p2
Since
s
a
is halfway between the two OL poles, the red vectors form an isosceles triangle. Note that the two bottom interior angles are
q
p2
.
So
q
p1
+
q
p2
= 180
°
…and the angle criterion is satisfied.Slide7
Example 2: 2
nd
-order system, cont.
Re
Im
-3
Now use K
P to satisfy the magnitude criterion:
-6
53.8
°
M
p2
M
p1
q
p1
q
p2
q
p2
To meet the magnitude criterion, K
P
= M
p1
·M
p2
=39.9Slide8
That’s all folks!
Fin
©
polyXengineering, Inc.
San Luis Obispo, California
www.polyxengineering.com