Modelling and control
summaries
by Anthony Rossiter
Root-loci 17: Angles of arrival/departure
The focus is on the simplest form of block diagram, a process G(s) and a compensator M(s) which is
expressed as a gain K multiplied by a transfer function. The closed-loop transfer function is Gc(s).
Conditions for closed-loop poles
~
M (s)  KM (s) G 
c




 Kn  d  0 
n


pc  0   K  1 
d


n


o
arg   180 
  d 

~
GKM
Kn

~
1  GKM Kn  d
Complex poles or zeros
At what angle does
loci approach this
zero?
Root Locus
1
Imaginary Axis (seconds-1)
When a system has complex
conjugate pairs of poles or zeros,
these serve as departure or arrival
points for the loci.
A key question linked to design is to
determine from what direction the
loci approach these positions.
0.5
0
-0.5
-1
-3
-2
-1
0
1
-1
Real Axis (seconds )
How to determine angles of arrival and departure
Use the angle criteria
n  ( s  z1 )( s  z2 ) ( s  zm )  n  ( s  zi )
for a candidate value
‘s’ near the relevant
d  ( s  p1 )( s  p2 ) ( s  pk )  d  ( s  pi )
pole/zero, that is:
o
( s  zi )   ( s  pi )  180
n  d  180o 


Using approximation
If s (on loci) is very close to zi, then one can approximate most of the angles by
writing s=zi.
( z  z )  s  z  ( z  p )  180o

j i
i
j
i

i
j
Rearranging the above equation we find the angle or arrival/departure directly.
s  zi   ( zi  z j )   ( zi  p j )  180o
j i
( s  1  j )( s  1  j )
s ( s  2)( s  1)
z1  1  j , z 2  1  j
Example of using approximation
Using formulae above, substitute in zi,pi:
G
p1  0, p 2  1, p3  2
Find angle of arrival at z1.
s  z1  [(1  j )  (1  j )]  [(1  j )  (0)]
 [(1  j )  (1)]  [(1  j )  (2)]  180o
Hence:
s  z1    2 j  (1  j )    j  (1  j )  180o
s  z1  90  (135)  90  45  180o  0o
Root Locus
s-z1.
Imaginary Axis (seconds-1)
A blow up of the root locus
near z1 shows that the loci
does indeed approach in
the direction given, that is 0
degrees!
1.1
1
0.9
0.8
-1.1
-1
-0.9
-0.8
-1
Real Axis (seconds )
REMARKS: Calculating angles of arrival and departure is quite tedious by
hand and in the days of modern computing it would be rare to do this.
However, understanding the procedure can give some useful insight that
helps with design.