1) Root locations with the same percent overshoot (%OS) lie on radial lines extending from the origin. 2) Root locations with the same peak time lie on horizontal lines. 3) Root locations with the same settling time lie on vertical lines. 4) Root locations with the same natural frequency lie on circles centered at the origin.

1. Underdamped Second Order System
C (s ) ωn2
G( s) = = 2
R( s) s + 2ςω n s + ω n
2
Section 4.6
Unit Step Response (Eqn 4.28, p. 176) :
 ς 
Control Systems Engineering g (t ) = 1 − e− ςωnt  cosω n 1 − ς 2 t + sin ω n 1 − ς 2 t 
 1−ς 2 
Norman S. Nise  
Control System Performance Maximum Percent Overshoot
Common measures of control system • Must be careful 0.4
performance include: with the
0.3
• maximum % overshoot (MPO) definitions if the
steady-state 0.2
 
− ςπ / 1−ς 2  value is not 1:
%OS = 100e  
0.1 Max S.S
Max − S .S .
%OS = 100 0
S .S . 0 0.5 1 1.5 2
Time (second)
Max % Overshoot Complex Roots
+ Imaginary
• What is the damping ratio for the system
in the previous figure? r1 + jω n 1 − ς 2
 0.34 − 0. 2  − ςπ / 1−ς 2 
 
%OS = 100   = 70 = 100 e  
ωn
 0.2  ωn 1 − ς 2
70 − ςπ / 1− ς 2 
  ςπ φ = cos−1 ς
= 0. 7 =e   − = ln(0.7) =− 0. 3567 - Real φ
100 1 −ς 2
ςω n + Real
0 .1272 ς 2π 2 − ςω n
ς = 0.113 ς2 = = 0. 1272
(
π + 0. 1272
2
) 1− ς 2

2. Question #1 Same % O.S.
All roots on this radial line have same % O.S.
• What does a set of root locations with + Imaginary
the same %OS look like on the complex All roots on this
plane? radial line have
same % O.S.
- Real
+ Real
Control System Performance Estimate the Peak Time
1.6
Common measures of control system 1.4
performance include: 1.2
• peak time (time to 1st peak) 1
0.8
π
Tp = 0.6 What is the imaginary
ωn 1−ς 2 0.4 part of the root locations
for this system?
0.2
0
0 1 2 3 4 5
Time, sec
Question #2 Same Peak Time
All roots on this horizontal line have same Peak Time.
• What does a set of root locations with + Imaginary
the same peak time look like on the
complex plane?
All roots on this horizontal line
have same Peak Time.
- Real
+ Real

3. Control System Performance Question #3
Common measures of control system • What does a set of root locations with
performance include: the same settling time look like on the
• settling time (output stays within ± 2% complex plane?
of final value)
4
TS = (valid up to ζ ~ 0.7)
ςω n
Estimate the Settling Time Same Settling Time
2 All roots on this vertical line have same Settling Time.
1.8 + Imaginary
1.6
1.4
1.2
All roots on this
vertical line have
1
same Settling Time.
0.8 What is the real part of
0.6 the root locations for
this system? - Real
0.4
0.2
0 + Real
0 0.1 0.2 0.3 0.4 0.5 0.6
Time, sec
Question #4 Same Natural Frequency
• What does a set of root locations with All roots on this circle + Imaginary
the same natural frequency look like on have same natural
frequency
the complex plane?
+ Real
- Real
All roots on this
circle have same
natural frequency

4. Proportional Control System Proportional Control
Closed loop or “feedback” control system Adjustments to the Kp term often give
error conflicting results
R(s) + E(s) Va(s) 1 C(s) An increase in Kp increases ω n, which
Kp
- s (s + a ) – reduces the peak time
– increases the overshoot (MPO)
– settling time stays about the same
input output
“proportional” control