Modelling and control
summaries
by Anthony Rossiter
Root-loci 14: Effects of lead compensation
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).
Lead compensator
M ( s)  K
sa
s  ra
1  r  10
How does the addition of a lead compensator affect a root-loci plot?
1. Adds a real pole and real zero, so no change to asymptote directions.
2. Asymptote centroid must change.
3. Parts of loci on real axis will change.
1. Sum (OL poles) – sum (OL zeros) = (k-m)*Centroid
2. The lead adds a zero at (-a) and a pole at (-ra), therefore the centroid moves to
the left (towards LHP):
Centroid 
 ra  a
; r  1  Centroid  0
k m
SUMMARY: As a lead moves the centroid left, one might less oscillation (better
damping) with the same value of gain and hence Lead is a high gain strategy. Also,
the real parts of the dominant poles are further into the LHP, so the loop is faster.
One expects ra to be significant compared to the initial centroid, otherwise the
centroid would not be moved much and the impact would be small.
Lead IS appropriate for open-loop unstable systems as these have a centroid
pushed right by the right half plane pole and thus a movement left is essential.
However may get a loss in steady-state gain.
Illustration of impact of lead in
pulling asymptotes to the left and
also allowing larger values of K.
G
s4
s3
53
; K ( s)  K
; C 
 1
s( s  2)(s  3)
s5
2
Without lead
Root Locus Editor for Open Loop 1 (OL1)
10
Root Locus Editor for Open Loop 1 (OL1)
With lead
8
8
6
6
4
4
2
Imag Axis
Imag Axis
2
0
0
-2
-2
-4
-4
-6
-6
-8
-10
-4
-3.5
-3
-2.5
-2
Real Axis
-1.5
-1
-0.5
Illustration of a Lead being use to
stabilise an open-loop unstable
process by moving asymptotes
into LHP.
G
-8
-5
-4.5
-4
-3.5
-3
-2.5
Real Axis
-2
-1.5
-1
-0.5
0
1
s2
42
; K ( s)  K
; C 
 1
( s  1)(s  2)
s4
2
With lead
Root Locus
1.5
6
1
4
Imaginary Axis (seconds-1)
Imaginary Axis (seconds-1)
Without lead
0
0.5
0
-0.5
Root Locus
2
0
-2
-4
-1
-1.5
-1.5
-6
-5
-1
-0.5
0
0.5
1
1.5
2
2.5
-4
-3
-2
-1
0
1
2
3
Real Axis (seconds -1)
Real Axis (seconds -1)
TRY THESE EXAMPLES FOR YOURSELF Compare and contrast the closed-loop systems with the following
compensators [loci, responses and offset].
4
s 1
; K  0.6; K lead  1.6
( s  2)( s  1)
s3
s4
s 1
G2 
; K  1.25; K lead  3.4
s( s  2)( s  3)
s3
G1 