Multivariable Control. Root Locus . By Kirsten Mølgaard Nielsen. Outline. Root Locus Design Method Root locus definition The phase condition Test points, and sketching a root locus Dynamic Compensation Lead compensation Lag compensation. Introduction.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
(1st order)
(2nd order)
Root locus
The characteristic equation can be written in various ways
breakaway point
Some root loci examples (K from zero to infinity)
fl
a
For example
3 poles:
q = 3
f1,dep = 60 deg.
f2,dep = 180 deg.
f3,dep = 300 deg.
Characteristic equation
Root locus for double integrator with Pcontrol
Rule 1: The locus has two branches that starts in s=0. There are no zeros. Thus, the branches do not end at zeros.
Rule 3: Two branches have asymptotes for s going to infinity. We get
Rule 2: No branches that the real axis.
Rule 4: Same argument and conclusion as rule 3.
Rule 5: The loci remain on the imaginary axis. Thus, no crossings of the jwaxis.
Rule 6: Easy to see, no further multiple poles. Verification:
Root locus for satellite attitude control with PDcontrol
r(t)
e(t)
controller
D(s)
u(t)
plant
G(s)
y(t)
+

+
+
noise
(neglecting the add. req.)
Possible
pole location
Matlab – design 1
sysG = tf([1],[1 1 0])
sysD = tf([1 7],[1 5*7])
rlocus(sysD*sysG)
No possible
pole location !
Matlab – design 2
sysG = tf([1],[1 1 0])
sysD = tf([1 7],[1 20])
rlocus(sysD*sysG)
Possible
location
New zero
location
Matlab – design 3
sysG = tf([1],[1 1 0])
sysD = tf([1 4],[1 20])
rlocus(sysD*sysG)
Pole location
Matlab – design B
sysG = tf([1],[1 1 0])
sysD = tf([1 5.4],[1 20])
rlocus(sysD*sysG)
You can use, triangle
a/sin(A) = b/sin(B)
r(t)
e(t)
controller
D(s)
u(t)
plant
G(s)
y(t)
+

Step
Ramp
Step response using the leadlag controller
If zn < 1, complex zeros
and double pole at w0
Cancellation
Root Locus
The leadlag controller with notch compensation
w0 = 50
z = 0.01
Too much overshoot !!
Also, exact cancellation might cause problems due to modelling errors
Thus, we need different parameters
The overshoot has been lowered to an acceptable level !
The oscillations have almost been removed !
So, we have reached a final design !
No delay
Padé (1,1)
Exact
Root locus for some example system (heat exchanger)
Padé (2,2)
(p,p) Approx. in Matlab :
[num,den]=pade(T,p)
However, Matlab does not support this approach...
In discrete systems the sampling time introduce a time delay