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## Chapter 6 Design of Control System

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### NUAA-Control System Engineering Chapter 6 Design of Control System What we have learned How to build the mathematical model of a control system Write differential ... – PowerPoint PPT presentation

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Title: Chapter 6 Design of Control System

1
Chapter 6 Design of Control System
NUAA-Control System Engineering
2
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 (Masons formula)

3
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)
• 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

4
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

5
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 N0
• 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)

6
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 root-locus analysis
• Can be applied to high-order system
• Can use the data from the measurements of a
physical system without deriving its mathematical
model

7
What we have learned
• To change the performance of a control system
• Vary the gain K

8
PID Control
9
Proportional-Integral-Derivative (PID) Control
10
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.
11
Ziegler-Nichols Rules for Tuning PID Controllers
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.
12
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.
The S-shape curve may be characterized by two
constants Delay time L and Time constant T
13
First method.
Table I
Type of Controller
P
PI
PID
14
Second method.
--Set and use the proportional
control only --Increase
from 0 to 8 to a critical value at
which the output first exhibits sustained
oscillations --If the output does not exhibit
sustained oscillations for whatever value
may take, then this method does not apply.
15
Second method.
16
Second method.
Table II
Type of Controller
P
PI
PID
17
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
Ziegler-Nichols tuning rules
18
Example
Consider the following control system
PID controller
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.
19
The PID controller has the transfer function
Since the plant has a integrator, we use the
second method.
20
The value of Kp that makes the system marginally
stable so that sustained oscillation occurs can
be obtained by use of Rouths stability
criterion.
The CE of the closed-loop system is
The Rouths array
When Kp30, the closed-loop system is marginally
stable. Thus the critical gain
21
With Kp set to Kcr(30), the CE becomes
To find the frequency of the sustained
oscillation, we substitute sjw into the CE as
follows
or
Hence the period of the sustained oscillation is
22
With Table II, we determine the parameters of the
PID controller as follows
The transfer function of the PID controller is
thus
23
The closed-loop transfer function with the PID
controller is
Now let us examine the unit-step response of the
closed-loop system to see if it exhibits
approximately 25 maximum overshoot.
gtgtnum 6.3223 18 12.811 gtgtden 1 6 11.3223
18 12.811 gtgtstep(num,den) gtgtgrid
24
The maximum overshoot is about 62 and is
excessive with respect to the requirement of 25.
25
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.
The maximum overshoot is reduced to 18.
The speed of response is increased and the
maximum overshoot is also increased to 28.
26
The parameters of the PID controller after fine
tuning
Compared with the parameters obtained by
Ziegler-Nichols second method
The new ones are approximately twice the values
suggested by Ziegler-Nicholss method.
Note that the Ziegler-Nichols tuning rule has
provided a starting point for fine tuning.
27
END OF THE COURSE