NUAA-Control System Engineering
Chapter 6
Design of Control System
Control System Engineering-2008
What we have learned…
 How to build the mathematical model of a control
system
 Write differential equations based on physical laws
 Transform the differential equations to the s-
domain by Laplace transform
 Obtain the transfer function
 Depict the control system with
 Block diagram
 Signal-flow graph (Mason’s formula)
Control System Engineering-2008
What we have learned…
 Time-domain analysis of control systems
 Time-domain responses and time-domain
specifications
 Transient response (rise time, peak time, maximum
overshoot, settling time)
 Steady-state response (steady-state error)
 Time-domain response of
 First-order system
 Prototype second-order system
 Adding poles or zeros to loop or closed-loop
transfer functions
 Has varying effects on the transient response of the
closed-loop system
Control System Engineering-2008
What we have learned…
 Stability condition of a linear system
 The roots of its characteristic equation (CE) must
all be located in the left-half s-plane (LHP).
 Methods of determining stability
 Routh-Hurwitz criterion
 Test whether any of the roots of CE lie in RHP
 Indicate the number of roots that lie on the jw-axis and in
RHP
Control System Engineering-2008
What we have learned…
 Methods of determining stability
 Nyquist criterion
 Semi-graphical method -- Nyquist plot
 Analyze closed-loop stability based on the loop transfer
function G(s)H(s)
 Closed-loop stability criterion: N=-P
 For minimum-phase loop function: N=0
 Bode diagram
 Plot the magnitue of the loop transfer function G(jw)H(jw)
in dB and the phase in degrees versus frequency w
 Closed-loop stability can be determined by observing the
behavior of the plots (gain margin, phase margin)
Control System Engineering-2008
What we have learned…
 Control system design technique
 Root-locus design
 A graphical method
 Investigate how the roots of the CE of a LTI system move
when one or more parameters vary
 Frequency-response design
 A graphical method
 Provide information different from what we get from rootlocus analysis
 Can be applied to high-order system
 Can use the data from the measurements of a physical
system without deriving its mathematical model
Control System Engineering-2008
What we have learned…
 To change the performance of a control system
 Vary the gain K
 Add poles
 Add zeros
Control System Engineering-2008
PID Control
Control System Engineering-2008
Proportional-Integral-Derivative (PID) Control
R( s)
E ( s)
KI
KP 
 KD s
s
U c ( s)
Plant
Y (s)
PID controller
t
KI


U c ( s)   K P 
 K D s  E ( s )  uc (t )  K P e(t )  K I  e( )d  K D e(t )
0
s


Proportional (P) control: K P
Proportional-Integral (PI) control:
Proportional-Derivative (PD) control:
KI
s
KP  KD s
KP 
Control System Engineering-2008
More than half of the industrial controllers in
use today utilize PID or modified PID control
schemes.
When the mathematical model of the plant is
unknown and therefore analytical design cannot
be used, PID control proves to be most useful.
Many different types of tuning rules have been
developed to adjust the parameters of PID
controllers on-site.
Control System Engineering-2008
Ziegler-Nichols Rules for Tuning PID
Controllers


KI
1
Gc ( s )  K P 
 K D s  K P 1 
 TD s 
s
 TI s

Ziegler and Nichols proposed rules for determining
values of the proportional gain Kp, integral time Ti and
derivative time Td based on the transient response of
a given plant.
There are two methods called Ziegler-Nichols tuning
rules: the first method and the second method.
Control System Engineering-2008
First method.
--Obtain the response of the plant to a unit-step input
--If the plant involves neither integrator nor dominant
complex-conjugate poles, then it step response exhibits an
S-shape curve , the first method can be applied.
unit-step input
Plant
S-shape response
The S-shape curve
may be
characterized by
two constants:
Delay time L and
Time constant T
y (t )
0
L
T
t
Control System Engineering-2008
First method.
Type of
Controller
P
PI
Table I
KP
TI
TD
T
L

0
T
0.9
L
L
0.3
0
T
L
2L
0.5L
PID
1.2
Control System Engineering-2008
Second method.
--Set K I  K D  0 and use the proportional control only
--Increase K P from 0 to ∞ to a critical value K cr at which
the output first exhibits sustained oscillations
--If the output does not exhibit sustained oscillations for
whatever value K P may take, then this method does not
apply.
R( s)
E ( s)
KP
U c ( s)
Plant
Y (s)
Control System Engineering-2008
Second method.
y (t )
Pcr
0
t
Control System Engineering-2008
Second method.
Type of
Controller
P
PI
PID
Table II
KP
TI
TD
0.5K cr

0
0.45K cr
1
Pcr
1.2
0
0.6 K cr
0.5Pcr
0.125Pcr
Control System Engineering-2008
Comments
Ziegler-Nichols tuning rules have been widely
used to tune PID controllers in process control
systems where the plant dynamics are not
precisely known.
If the plant dynamics are known, many analytical
and graphical approaches to the design of PID
controllers are available, in addition to ZieglerNichols tuning rules
Control System Engineering-2008
Example
R( s)
Consider the following control system
E ( s)
Gc ( s )
U c ( s)
PID controller
1
s( s  1)( s  5)
Y (s)
Plant
Design a PID controller to make the maximum overshoot
of the system to be approximately 25% or less.
Solution. We start design the PID controller by applying
Ziegler-Nichols rules.
Here the transfer function of the plant is known, we can
use analytical method instead of experimental method.
Control System Engineering-2008
R( s)
E ( s)
Gc ( s )
U c ( s)
PID controller
1
s( s  1)( s  5)
Y (s)
Plant
The PID controller has the transfer function


KI
1
Gc ( s )  K P 
 K D s  K P 1 
 TD s 
s
 TI s

Since the plant has a integrator, we use the second method.
By setting K I  K D  0, we obtain the closed-loop TF:
Y ( s)
KP

R( s ) s( s  1)( s  5)  K P
Control System Engineering-2008
R( s)
E ( s)
Gc ( s )
U c ( s)
Y (s)
1
s( s  1)( s  5)
PID controller
Plant
The value of Kp that makes the system marginally stable so
that sustained oscillation occurs can be obtained by use of
Routh’s stability criterion.
The CE of the closed-loop
system is
s  6s  5s  K P  0
3
2
When Kp=30, the closed-loop
system is marginally stable. Thus
the critical gain K  30
cr
The Routh’s array
s3
1
5
s2
6
KP
s
1
s0
30  K P
6
Kp
Control System Engineering-2008
R( s)
E ( s)
Gc ( s )
U c ( s)
PID controller
1
s( s  1)( s  5)
Y (s)
Plant
With Kp set to Kcr(=30), the CE becomes
s 3  6s 2  5s  30  0
To find the frequency of the sustained oscillation, we
substitute s=jw into the CE as follows:
( j)3  6( j)2  5( j)  30  0
or
6(5   2 )  j(5   2 )  0
 5
Hence the period of the sustained oscillation is
2 2
Pcr 

 2.8099

5
Control System Engineering-2008
R( s)
E ( s)
Gc ( s )
U c ( s)
PID controller
1
s( s  1)( s  5)
Y (s)
Plant
With Table II, we determine the parameters of the PID
controller as follows:
K P  0.6 K cr  18
TI  0.5Pcr  1.405
TD  0.125Pcr  0.35124
The transfer function of the PID controller is thus
1

 6.3223  s  1.4235
Gc ( s )  18  1 
 0.35124 s  
s
 1.405s

2
Control System Engineering-2008
R( s)
E ( s)
Gc ( s )
U c ( s)
PID controller
1
s( s  1)( s  5)
Y (s)
Plant
The closed-loop transfer function with the PID controller is
Y ( s)
6.3223s 2  18s  12.811
 4
R( s ) s  6s 3  11.3223s 2  18s  12.811
Now let us examine the unit-step response of the closedloop system to see if it exhibits approximately 25%
maximum overshoot.
>>num = [6.3223 18 12.811];
>>den = [1 6 11.3223 18 12.811];
>>step(num,den)
>>grid
Control System Engineering-2008
Unit-Step Response
1.8
1.6
1.4
Amplitude
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
Time (sec)
The maximum overshoot is about 62% and is excessive
with respect to the requirement of 25%.
Control System Engineering-2008
The amount of maximum overshoot can be reduced by fine
tuning the parameters of the PID controller.
Such fine tuning can be made on the computer.
1

 6.3223  s  1.4235
Gc ( s )  18  1 
 0.35124 s  
s
 1.405s

2
Move the pole locations
1

 13.836  s  0.65
Gc ( s )  18  1 
 0.7692 s  
s
 0.3077 s

2
The maximum overshoot is reduced to 18%.
Increase the gain
1

 13.836  s  0.65
Gc ( s )  39.42  1 
 0.7692 s  
s
 0.3077 s

The speed of response is increased and the maximum
overshoot is also increased to 28%.
2
Control System Engineering-2008
The parameters of the PID controller after fine tuning
K P  39.42
TI  3.077
TD  0.7692
Compared with the parameters obtained by ZieglerNichols’ second method
K P  0.6 K cr  18
TI  0.5Pcr  1.405
TD  0.125Pcr  0.35124
The new ones are approximately twice the values suggested
by Ziegler-Nichols’s method.
Note that the Ziegler-Nichols’ tuning rule has provided a
starting point for fine tuning.
END OF THE COURSE