Lecture -10
PID CONTROL SCHEMES
10.1 Introduction
• So far, we have touched on the PID controllers and their design
based on models of plants.
• However, in many practical situations, we deal with very
complicated plants whose mathematical models are difficult to
obtain. Hence, the analytical approach of designing PID
controllers is not possible.
• In this chapter, we shall introduce tuning rules for basic PID
controllers.
• Note that more than half of industrial plants today still employ
PID control schemes.
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3.2 PID Controller Tuning
Consider the system
The standard form of PID controller is
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Gc ( s ) K P 1 Td s
Ti s KI
KP KDs
s
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P(Proportional) action K P e(t )
P-action provides an action which depends on the instantaneous
value of the control error. It can be used to improve speed of
response and steady-state error but with limited performance.
KP t
Ti 0
I (Integral) action e( s )ds
I-action gives a controller output that is proportional to the
accumulated error, which implies that it is a slow reaction control
mode. It is used to achieve zero-steady error in the presence of a
step reference and/or disturbance.
The shortcomings are: its pole at the origin is detrimental to
loop stability and undesirable effect in the presence of actuator
saturation, known as wind-up.
D (Derivative) action K PTd e(t )
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It acts on the changing rate of the control error, sometimes
referred to as a predictive mode. Its limitation is the tendency to
yield large control signals in response to high frequency
control errors such as error induced by set-point changes or
measurement noises.
Tuning Rules for PID Controllers
The process of selecting the controller parameters KP , Ti and Td
is known as Controller Tuning.
• Ziegler-Nichols Rules
Two classical methods for determining the parameters of PID
controllers were presented by Ziegler-Nichols in 1942.
The Ziegler-Nichols rules determine K P , Ti and Td based on the
transient response characteristics of a given plant. They often
form the basis for tuning procedures used by manufacturers
and process industry. 4
1) The Step Response Method
Many plants, particularly those in the process industries
(stable plant involving neither integrators nor dominant
complex poles) has step response of S-shape (reaction curve).
An open-loop experiment for deriving step response can be
• With the plant in open-loop, let the plant run at normal
operating condition with constant input u (t ) u0 and steady
state output y (t ) y0 .
• At the initial time t0 , apply a step change to the plant, from u0
to u (in the range of 10 to 20% of full scale).
• Record the output response to get the reaction curve. In the
figure, the m.s.t. stands for the maximum slope tangent.
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L T
• Compute
y y0
K , L t1 t0 , T t2 t1
u u0
where K is the system gain, L and T are often called the
apparent dead time and the apparent time constant.
Note that an approximate transfer function for the plant can be
Ke Ls
G( s)
Ts 1 6
The PID parameters are then chosen according to the table.
Controller KP Ti Td
P T
KL
PI 0.9 T
KL 3L
PID 1.2 T 2L 0.5L
KL
Example 10.1
Consider a process with transfer function
1
G( s)
( s 1)3 7
From the unit step response of the process, it is known that T 3.7
and L 0.8 .
1 0
K 1
1 0
Then, the PID parameters given by the Ziegler-Nichols tuning
rule is
K P 5.55, Ti 1.60, Td 0.40
The closed-loop response is shown in the figure.
PID P
-
8
Pcr
2) Frequency Response Method
• Set the plant under P-control, with a very small gain.
• Increase the gain until the loop starts oscillation. Note
that it should be detected at the controller output.
• Record the critical gain K cr K P and the oscillation
period of controller output, Pcr .
• Set the PID controller according to
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Controller KP Ti Td
P 0.5K cr
PI Pcr
0.45K cr
1.2
PID 0.6 K cr 0.5Pcr 0.125 Pcr
Example 10.2 Consider the example discussed in Example
10.1. In the frequency response method, we first apply P-
controller and determine K cr and Pcr .
1
KP
- ( s 1)3
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The closed-loop characteristic equation is
s 3 3s 2 3s 1 K P 0
Routh array is given by
s3 1 3
s2 3 1 KP
8 KP
s1
3
s0 1 KP
Clearly, the critical gain is K cr 8 . The oscillation frequency
can be calculated from
3s 2 9 0 s j 3
as 3.
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Hence, 1 2 2
Pcr
2 f 2 f
Pcr 3.63
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The PID setting from the table is then
K P 5.184, Ti 1.681, Td 0.420
The closed-loop response is
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Remarks:
• The response shown in the examples exhibits significant
overshoot that might be unacceptable in some applications.
However, Ziegler-Nichols tuning provides a starting point
for finer tuning. For example, by increasing Ti and Td , it
can be expected that the overshoot is reduced. Certainly, for
applications where measurement noise is significant, care
needs to be taken in increasing Td .
• Ziegler-Nichols tuning rules have been widely used in
process control where the plant dynamics are not known.
For processes with known dynamics, other analytical or
graphical methods can be applied.
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• Generally, to apply the step response method, one needs
to obtain the `S'-shape responses. Plants with complicated
dynamics but no integrators are usually the cases.
• The frequency response method requires that the plant be
forced to oscillate; this can be dangerous and expensive.
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