PID controller in control systems

Assistant Professor Dr. Khalaf S Gaeid
Electrical Engineering Department/Tikrit University
khalafgaeid@tu.edu.iq
gaeidkhalaf@gmail.com
+9647703057076
April 2018

1.Introduction
A controller is a device that generates an output signal based on the input
signal it receives.
The input signal is actually an error signal, which is the difference
between the measured variable and the desired value as can be shown in
feedback control system ( Fig.1).
Fig.1.Feedback control system

A sensor measures and transmits the current value of the process
variable(PV) back to the controller.
▪ Controller error(e(t)) at current time t is computed as set-point(SP)
minus measured process variable as in (1).
e(t) = SP – PV (1)
▪ The controller uses this e(t) in a control algorithm to compute a new
controller output signal.
▪ The controller output signal is sent to the final control element (e.g.
valve, pump, heater, fan) causing it to change.
▪ The change in the final control element causes a change in a
manipulated variable
▪ The change in the manipulated variable (e.g. flow rate of liquid or gas)
causes a change in the PV

1.Introduction
A proportional-integral-derivative controller (PID controller or
three term controller) is a control loop feedback mechanism
widely used in industrial control systems and a variety of other
applications requiring continuously modulated control.
A PID controller continuously calculates an error value e(t) as
the difference between a desired set point (SP) and a measured
process variable (PV) and applies a correction based on
proportional, integral, and derivative terms (denoted P, I, and
D respectively) which give the controller its name.

The first theoretical analysis and practical application was in
the field of automatic steering systems for ships, developed
from the early 1920s onwards. It was then used for automatic
process control in manufacturing industry, where it was widely
implemented in pneumatic, and then electronic, controllers.
Today there is universal use of the PID concept in applications
requiring accurate and optimized automatic control.
PID control is widely used in all areas where control is
applied(solves(90%of all control problems).

2. Requirements of a Good Control System
The essential requirements of a good Control System can be listed as
follows:
1) Accuracy: Accuracy must be very high as error arising should be
corrected. Accuracy can be improved by the use of feedback element.
2) Sensitivity: A good control system senses quick changes in the
output due to an environment, parametric changes, internal and external
disturbances.
3) Noise: Noise is a unwanted signal and a good control system should
be sensitive to these type of disturbances.

2. Requirements of a Good Control System
4) Stability: The stable systems has bounded input and bounded output.
A good control system should response to the undesirable changes in the
stability.
5) Bandwidth: To obtain a good frequency response, bandwidth of a
system should be large.
6) Speed: A good control system should have high speed that is the
output of the system should be fast as possible.
7) Oscillation: For a good control system oscillation in the output
should be constant or at least has small oscillation.

3.Controller Modes
In industry there are many control modes as follows:
1. ON-OFF controller/two position controller as temperature controller
used for domestic heating system.
2. Three-position controller
3. Proportional Action Control
4. Integral/Reset Action Control
5. Derivative/Rate Action Control
6. P+I Control
7. P+D Control
8. P+I+D Control

• P depends on the present error
• I on the accumulation of past errors
•D is a prediction of future errors, based on
current rate of change
So the importance of PID comes from the above
P_Controller
I _Controller
D_Controller

4. The Characteristics of P, I, and D Controllers
A proportional controller (Kp ) will have the effect of reducing the rise
time and will reduce but never eliminate the steady state error.
An integral control (Ki ) will have the effect of eliminating the steady-
state error for a constant or step input, but it may make the transient
response slower.
A derivative control (Kd ) will have the effect of increasing the stability
of the system, reducing the overshoot, and improving the transient
response.

In fact, changing one of these variables can change the effect of the other
two.
With the PID controller we can set the P+I+D values so that we will not
have any Over or undershoot and reach set point directly.
PID controller has all the necessary dynamics: fast reaction on change
of the controller input (D mode), increase in control signal to lead error
towards zero (I mode) and suitable action inside control error area to
eliminate oscillations (P mode).
This combination of{Present + Past + Future} makes it possible to
control the application very well.

Proportional Controller
In a proportional controller the output (also called the actuating/control
signal) is directly proportional to the error signal. The position of Kp
can be as shown in Fig.2.
Fig.2. Proportional controller
Control signal = Kp * e(t) (2)
If the error signal is a voltage, and the control signal is also a voltage,
then a proportional controller is just an amplifier.

Properties of proportional controller:
In a proportional controller, steady state error tends to depend inversely
upon the proportional gain, so if the gain is made larger the error goes
down as in (3).
(3)
Where SSE is the steady state error
Proportional controller helps in reducing the steady state error, thus
makes the system more stable.
Slow response of the over damped system can be made faster with the
help of these controllers.
P controller has the advantage of reducing down the steady state error of
the system , but along with that it also has some serious disadvantages.
These properties can be shown in Fig.3.
1 / (1 (0))p
SSE K G 

Fig.3.Response of PV to step change of SP vs time, for three values of Kp (Ki and Kd held
constant)

Disadvantages of P Controller
1. Due to presence of these controllers we have some offsets in the
system.
2. Proportional controllers also increase the maximum overshoot of the
system.
3. It directly amplifies process noise.
To avoid these errors and to make the controller more accurate and
practical, we use the advanced and modified version of it known as the
Proportional Integral Controllers (PI) and Proportional Derivative
Controllers (PD).

Integral
An integral term increases action in relation not only to the error but
also the time for which it has persisted. So, if applied force is not
enough to bring the error to zero, this force will be increased as time
passes. A pure "I" controller could bring the error to zero, however, it
would be both slow reacting at the start (because action would be small
at the beginning, needing time to get significant), brutal (the action
increases as long as the error is positive, even if the error has started to
approach zero), and slow to end (when the error switches sides, this for
some time will only reduce the strength of the action from "I", not make
it switch sides as well), prompting overshoot and oscillations (see Fig.4).
Moreover, it could even move the system out of zero error: remembering
that the system had been in error, it could prompt an action when not
needed. An alternative formulation of integral action is to change the
electric current in small persistent steps that are proportional to the
current error. Over time the steps accumulate and add up dependent on
past errors; this is the discrete-time equivalent to integration.

Fig.4. Response of PV to step change of SP vs time, for three values of Ki (Kp and Kd held
constant)

Derivative
A derivative term does not consider the error (meaning it cannot bring
it to zero: a pure D controller cannot bring the system to its set-point),
but the rate of change of error, trying to bring this rate to zero. It aims at
flattening the error trajectory into a horizontal line, damping the force
applied, and so reduces overshoot (error on the other side because too
great applied force). Applying too much impetus when the error is small
and is reducing will lead to overshoot. After overshooting, if the
controller were to apply a large correction in the opposite direction and
repeatedly overshoot the desired position, the output would oscillate
around the set-point in either a constant, growing, or decaying sinusoid.
If the amplitude of the oscillations increase with time, the system is
unstable. If they decrease, the system is stable. If the oscillations remain
at a constant magnitude, the system is marginally stable. This can be
illustrated in Fig.5.

Fig.5. Response of PV to step change of SP vs time, for three values of Kd (Kp and Ki held
constant)

5. Mathematical form
The overall control function can be expressed mathematically as in(4)
(4)
where Kp , Ki , and Kd , all non-negative, denote the coefficients for the
proportional, integral, and derivative terms respectively (sometimes
denoted P, I, and D).
In the standard form of the equation (see later in article), Ki and Kd
are respectively replaced by Kp/Ti and Kd*Td ; the advantage of this
being that Ti and Td have some understandable physical meaning, as
they represent the integration time and the derivative time respectively
the above relationship is obtained from parallel configuration of PID
controller as illustrated in Fig.6.
0
( )
( ) ( ) ( )
t
p i d
d e t
u t k e t k e d k
d t
   

Fig.6. Parallel configuration of PID controller
Although a PID controller has three control terms, some applications
use only one or two terms to provide the appropriate control. This is
achieved by setting the unused parameters to zero and is called a PI, PD,
P or I controller in the absence of the other control actions. PI controllers
are fairly common, since derivative action is sensitive to measurement
noise, whereas the absence of an integral term may prevent the system
from reaching its target value.

6. Electronic analogue controllers
Electronic analog PID control loops were often found within more
complex electronic systems, for example, the head positioning of a disk
drive, the power conditioning of a power supply, or even the
movement-detection circuit of a modern seismometer. Discrete
electronic analogue controllers have been largely replaced by digital
controllers using microcontrollers or FPGAs, to implement PID
algorithms. However, discrete analog PID controllers are still used in
niche applications requiring high-bandwidth and low-noise
performance, such as laser-diode controllers.

7.Limitations of PID control
While PID controllers are applicable to many control problems, and
often perform satisfactorily without any improvements or only coarse
tuning, they can perform poorly in some applications, and do not in
general provide optimal control. The fundamental difficulty with PID
control is that it is a feedback control system, with constant parameters,
and no direct knowledge of the process, and thus overall performance is
reactive and a compromise. While PID control is the best controller in an
observer without a model of the process, better performance can be
obtained by overtly modeling the actor of the process without resorting
to an observer.
PID controllers, when used alone, can give poor performance when the
PID loop gains must be reduced so that the control system does not
overshoot, oscillate or hunt about the control set point value.

7.Limitations of PID control
They also have difficulties in the presence of non-linearities, may
trade-off regulation versus response time, do not react to changing
process behavior (say, the process changes after it has warmed up),
and have lag in responding to large disturbances.
The most significant improvement is to incorporate feed-forward
control with knowledge about the system, and using the PID only to
control error. Alternatively, PIDs can be modified in more minor ways,
such as by changing the parameters (either gain scheduling in
different use cases or adaptively modifying them based on
performance), improving measurement (higher sampling rate, precision,
and accuracy, and low-pass filtering if necessary), or cascading multiple
PID controllers.

Closed-loop Response performance depends on the effects of PID
parameters as can be shown in the following table when the parameters
are increased as can be shown in table1.
Table1. Increasing PID controller parameters with the performance of the closed loop
system
• Note that these correlations may not be exactly accurate, because P, I
and D gains are dependent of each other.
StabilitySSESettling timeOvershootRise timeparameter
degradedecreaseSmall changeincreasedecreaseKp
degradeEliminateincreaseincreasedecreaseKi
Improve if
Kd small
No effectdecreasedecreaseMinor changeKd

8. The tuning parameters essentially determine:
How much correction should be made? The magnitude of the
correction (change in controller output) is determined by the
proportional mode of the controller.
How long the correction should be applied? The duration of the
adjustment to the controller output is determined by the integral mode
of the controller.
How fast should the correction be applied? The speed at which a
correction is made is determined by the derivative mode of the
controller.

8. PID Tuning Algorithm
Typical PID tuning objectives include:
Closed-loop stability: The closed-loop system output remains bounded
for bounded input.
Adequate performance: The closed-loop system tracks reference
changes and suppresses disturbances as rapidly as possible. The larger
the loop bandwidth (the frequency of unity open-loop gain), the faster
the controller responds to changes in the reference or disturbances in the
loop.
Adequate robustness: The loop design has enough gain margin and
phase margin to allow for modeling errors or variations in system
dynamics.

9.Electronic PID Controllers
Electronic PID controllers can be obtained using operational amplifiers
and passive components like resistors and capacitors. A typical scheme is
shown in Fig.7.
Fig. 7. Electronic PID controller

It is evident from Fig. 5, the proportional gain Kp is decided by the ratio
R2/R1 of the first amplifier; the integral action is decided by R3 and C1
and the derivative action by R5 and C2.
The final output however comes out with a negative sign, compared to
eqn.(1) (though the positive sign can also be obtained by using a
noninverting amplifier at the input stage, instead of the inverting
amplifier).
The op. amps. shown in the circuits are assumed to be ideal.

10. PID Tuning
Users of control systems are frequently faced with the task of adjusting
the controller parameters to obtain a desired behavior. There are many
different ways to do this. One approach is to go through the
conventional steps of modeling and control design as described in the
previous section. Since the PID controller has so few parameters, a
number of special empirical methods have also been developed for direct
adjustment of the controller parameters. The first tuning rules were
developed by Ziegler and Nichols. Their idea was to perform a simple
experiment, extract some features of process dynamics from the
experiment and determine the controller parameters from the features

Ziegler–Nichols tuning rules (Table2). (a) The step response methods
give the parameters in terms of the intercept a and the apparent time
delay τ. (b) The frequency response method gives controller parameters
in terms of critical gain kc and critical period Tc.
Table2. Ziegler–Nichols tuning rules
by extensive simulation of a range of representative processes. A
controller was tuned manually for each process, and an attempt was then
made to correlate the controller parameters with a and τ
TdTiKpType
1/aP
3τ0.9/aPI
0.5τ2τ1.2/aPID
TdTiKpType
0.5kcP
0.8Tc0.4kcPI
0.125Tc0.5Tc0.6kcPID
(a) Step response method (b) Frequency response method

33
The series configuration of PID control consists of a proportional plus
derivative (PD) compensator cascaded with a proportional plus integral
(PI) compensator.
The purpose of the PD compensator is to improve the transient
response while maintaining the stability.
The purpose of the PI compensator is to improve the steady state
accuracy of the system without degrading the stability.
Since speed of response, accuracy, and stability are what is needed for
satisfactory response, cascading PD and PI will suffice.

34
Lead/Lag compensation is very similar to PD/PI, or PID control.
The lead compensator plays the same role as the PD controller,
reshaping the root locus to improve the transient response.
Lag and PI compensation are similar and have the same response: to
improve the steady state accuracy of the closed-loop system.
Both PID and lead/lag compensation can be used successfully, and
can be combined.

12.Conclusions
• Proportional action gives an output signal proportional to the size of
the error. Increasing the proportional feedback gain reduces steady-state
errors, but high gains almost always destabilize the system.
• Integral action gives a signal which magnitude depends on the time
the error has been there. Integral control provides robust reduction in
steady-state errors, but often makes the system less stable.
• Derivative action gives a signal proportional to the change in the Error.
It gives sort of “anticipatory” control .Derivative control usually
increases damping and improves stability, but has almost no effect on the
steady state error
• These three kinds of control combined from the classical PID
controller.
• PID can be implemented in Hardware and software.
• The PI controller can be considered as Lag compensator, The PD
controller can be considered as lead compensator and PID same as
Lag-Lead compensator works to improve transient and steady state
region.

12.Conclusions
• The tuning of the controller is one of the limitations of PID controller.
• Proportional and integral control modes are essential for most control
loops, while derivative is useful only in some cases.
• Designing and tuning a PID controller appears to be conceptually
intuitive, but can be hard in practice, if multiple (and often
conflicting) objectives such as short transient and high stability are to
be achieved.
• Control engineers usually prefer P-I controllers to control first order
plants. On the other hand, P-I-D control is vastly used to control two
or higher order plants.
• The major reasons behind the popularity of P-I-D controller are its
simplicity in structure and the appilicability to variety of processes.
Moreover the controller can be tuned for a process, even without
detailed mathematical model of the process.
•

12.Conclusions
• The choice of P-D, P-I or P-I-D structure de pends on the type of the
process we intend to control.
• There are few more issues those need to be addressed while using P-I
controller. The most important among them is the anti-windup control.

PID controller in control systems

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