by Frank Owen PhD PE polyXengineering Inc San Luis Obispo California general aims for any controller Reduce percent overshoot Shorten time to reach peak value after a step change minimize T ID: 713719
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Slide1
Designing pid controllers using root locus
by Frank Owen, PhD, PE
polyXengineering, Inc.
San Luis Obispo, CaliforniaSlide2
general aims for any controller
Reduce percent overshoot
Shorten time to reach peak value after a step change (minimize T
P
)
Shorten time it takes to reach the new final value in a step change (minimize T
S
)
Reduce or eliminate steady-state errorSlide3
Conflicting aims
These aims often work against each other:
For example: You get quicker response at the expense of greater overshoot
…or, you get a gentler, less vigorous, less “wiggly” response by making it slowerSlide4
More specifically
Increase
z
to reduce percent overshoot
Let’s use this system:
, with
z
= 0.2 and = 1 rad/secCompare that with the response when z is increased to 0.7Enter the following commands in Matlab to compare the two systems:
K = 1
z = 0.2wn = 1.0s = tf(‘s’)gpt2 = K*wn^2/(s^2+2*z*wn*s+wn^2)step(gpt2)hold onz = 0.7gpt7 = K*wn^2/(s^2+2*z*wn*s+wn^2)grid on
Try this yourself before going on…Slide5
Results
Longer
T
p
with higher
z
z
= 0.2z = 0.7But the more heavily damped system settles out faster
It might seem like the roughly 1-second difference in time to arrive at the peak is trivial, but in a repetitive industrial process like machining, welding, or assembly, those seconds add up.
Of course the ideal response would be to have the speed without the overshoot. Is this possible? We shall see…Slide6
context
How much overshoot you can stand depends on the application of the system
For example, in an automated milling operation, where you move a cutting head to a location to begin a milling operation, probably no overshoot is tolerable.
This will be accomplished at the expense of speed.
But so far we have not said a word about steady-state error.
This will require putting a pole at the origin
…and that often slows a system down too and makes it less stableSlide7
The root locus will aid in the design of a PID controller
We are able to see with the root locus how we are “sculpting” the system’s dynamicsSlide8
Important diagram
This diagram from my textbook is very important in making decisions about how to place the poles when installing a controller from the PID family.
It applies strictly to 2
nd
-order systems, but it can be used for higher-order systems if they have a pair of dominant
2
nd
-order
polesSlide9
Effects of moving in various directions
Re
Im
w
d
(
T
p = p/w
d)
T
p decreasesz increases, so% overshootdecreases-z·wnTs decreases
(
T
s
=
4
/
z
·
w
n
)
q
Constant %OS
cos
q = z = -
ln %OS/100
(p
2
+ ln
2
%OS/100)
1/2Slide10
An example: halve Ts
while keeping
t
p
the same
Re
Im
Tp = p/4 =0.785 sec …for both cases
4
-2
q11
T
s1
= 4/2 = 2.0 sec
-4
q
2
2
T
s2
= 4/4 = 1.0 sec
But
q
2
<
q
4
, so
z
is less and thus %OS will be greater for the second case
Let’s check this out with some simple Matlab commandsSlide11
Try this yourself
Start out with poles at s = -2
±
j
·
2 and halve
T
p while keeping Ts the same Demonstrate this with two step responses shown together on the same plot
Start out with poles at s = -2
±
j·2 and halve Ts while keeping the %OS the same Demonstrate this with two step responses shown together on the same plot