Lec 5 pid

Control Systems
LECT. 5 PID CONTROLLER
BEHZAD FARZANEGAN
3/1/2020 PROVIDED BY: BF(B.FARZANEGAN@AUT.AC.IR) 1

Compensation
The design of a control system is concerned with the arrangement of
the system structure and the selection of a suitable components and
parameters.
A compensator is an additional component or circuit that is inserted
into a control system to compensate for a deficient performance.
Types of Compensation
◦ Cascade compensation
◦ Feedback compensation
◦ Output compensation
◦ Input compensation

Design Specifications
often use design specifications to describe what the system should do
and how it is done.
These specifications are unique to each individual application and often
include specifications about relative stability, steady-state accuracy
(error), transient-response characteristics
In some applications there may be additional specifications on
sensitivity to parameter variations, that is, robustness, or disturbance
rejection.
PROVIDED BY: BF(B.FARZANEGAN@AUT.AC.IR) 33/1/2020

The design of linear control systems can be carried out in either the
time domain or the frequency domain.
For instance, steady-state accuracy is often specified with respect to a
step input, a ramp input, or a parabolic input, and the design to meet a
certain requirement is more conveniently carried out in the time
domain.
Other specifications such as maximum overshoot, rise time, and
settling time are all defined for a unit-step input and, therefore, are
used specifically for time-domain design

Thus, the choice of whether the design should be conducted in the
time domain or the frequency domain depends often on the preference
of the designer.
We should be quick to point out, however, that in most cases, time-
domain specifications such as maximum overshoot, rise time, and
settling time are usually used as the final measure of system
performance.

Fundamental Principles of Design
After a controller configuration is chosen, the designer must choose a
controller type that, with proper selection of its element values, will satisfy
all the design specifications.
Engineering practice usually dictates that one choose the simplest controller
that meets all the design specifications.
In most cases, the more complex a controller is, the more it costs, the less
reliable it is, and the more difficult it is to design.
Choosing a specific controller for a specific application is often based on the
designer's past experience and sometimes intuition, and it entails as much
art as it does science.

Historical Note
PID control is one of the earlier control strategies
Early feedback control devices used the ideas of proportional,
integral and derivative action in their structures.
 Its early implementation was in pneumatic devices, followed by
vacuum and solid state analog electronics, before arriving at today’s
digital implementation of microprocessors.
It has a simple control structure which was understood by plant
operators and which they found relatively easy to tune.
According to a survey for process control systems conducted in
1989, more than 95 of the control loops were of the PID type

Why PID Control
1. Simple: easy to use
2. Wide Application: Petrochemical, Pharmaceuticals, Food, Chemical,
Aerospace and Semiconductor, etc.
3. Robust: Insensitive to changes to plant parameter and disturbance.
Over 95% control loops are PID with two exceptions:
1. On/off control for those with low control requirement loops
2. Advanced control for those difficult systems and with high control
quality.

Control Modes
The two-step mode: The controller is just a switch which is activated by
the error signal and supplies just an on-off correcting signal. Example of
such mode is the bimetallic thermostat.
The proportional mode (P): This produces a control action that is
proportional to the error. The correcting signal thus becomes bigger
the bigger the error. Therefore, the error is reduced the amount of
correction is reduced and the correcting process slows down. A
summing operational amplifier with an inverter can be used as a
proportional controller.
The derivative mode: This produces a control action that is proportional
to the rate at which the error is changing. When there is a sudden
change in the error signal the controller gives a large correcting signal.
When there is a gradual change only a small correcting signal is
produced. An operational amplifier connected as a differentiator circuit
followed by another operational amplifier connected as an inverter
make an electronic derivative controller circuit.

10
Control Modes
The integral mode (I): This produces a control action that is proportional to
the integral of the error with time. Therefore, a constant error signal will
produce an increasing correcting signal. The correction continues to increase
as long as the error persists.
Combination of modes: Proportional plus derivative modes (PD),
proportional plus integral modes (PI), proportional plus integral plus
derivative modes (PID). The term three-term controller is used for PID
control.
The controller may achieve these modes by means of pneumatic circuits,
analog electronics involving operational amplifiers or by the programming of
a microprocessor or computer.
3/1/2020 PROVIDED BY: BF(B.FARZANEGAN@AUT.AC.IR)

Lead/Lag Compensation
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.

PID Controllers
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.

The Characteristics of P, I, and D
Controllers
Response Rise Time Overshoot Settling
Time
SS Error
KP Decrease Increase
Small
Change Decrease
KI Decrease Increase Increase Eliminate
KD
Small
Change Decrease Decrease
Small
Change

Improving Steady-State Error
and Transient Response
PID controller or using passive network it’s called lag-lad compensator
𝑮 𝒄(𝒔) = 𝑲 𝟏 +
𝑲 𝟐
𝒔
+ 𝑲 𝟑 𝒔 =
𝑲 𝟏 + 𝑲 𝟐 + 𝑲 𝟑 𝒔 𝟐
𝒔
=
𝑲 𝟑(𝒔 𝟐
+
𝑲 𝟏
𝑲 𝟑
𝒔 +
𝑲 𝟐
𝑲 𝟑
)
𝒔
14

Design of PID Controllers
There is a fairly standard procedure for tuning PID
controllers:
• Trial and Error Tuning
• Automatic PID Tuning (simulation)
• Ziegler and Nichols Tuning
• Cohen-Coon Tuning

Trial and Error Tuning
The trial and error tuning method is based on guess-and-check. In
this method, the proportional action is the main control, while the
integral and derivative actions refine it. The controller gain, Kp, is
adjusted with the integral and derivative actions held at a minimum,
until a desired output is obtained.

Automatic PID Tuning
MATLAB provides tools for automatically choosing optimal PID gains
which makes the trial and error process described above unnecessary.
You can access the tuning algorithm directly using pidtune or through a
nice graphical user interface (GUI) using pidtool.

PID controller design by R-Lucos
Evaluate the performance of the uncompensated system to determine
how much improvement is required in transient response
Design the PD controller to meet the transient response specifications. The
design includes the zero location and the loop gain.
Simulate the system to be sure all requirements have been met.
Redesign if the simulation shows that requirements have not been met.
Design the PI controller to yield the required steady-sate error.
Determine the gains, K1, K2, and K3 shown in previous figure.
Simulate the system to be sure all requirements have been met.
Redesign if simulation shows that requirements have not been met.

Example:
Problem: Using the system in the Figure, Design a PID controller so that
the system can operate with a peak time that is 2/3 that of the
uncompensated system at 20% overshoot and with zero steady-state
error for a step input.
Solution: The uncompensated system operating at 20% overshoot has
dominant poles at -5.415+j10.57 with gain 121.5, and a third pole at -
8.169. The complete performance is shown in next table.

Example:
To compensate the system to reduce the peak time to 2/3 of original,
we must find the compensated system dominant pole location.
𝝎 𝒅 =
𝝅
𝑻 𝒑
=
𝝅
𝟐 𝟑)(𝟎. 𝟐𝟗𝟕
= 𝟏𝟓. 𝟖𝟕
⇒ 𝝈 =
𝝎 𝒅
𝐭𝐚𝐧𝟏𝟏𝟕. 𝟏𝟑∘
= −𝟖. 𝟏𝟑
20

Calculating the PD compensator
To design the compensator, we find the sum of angles from the
uncompensated system’s poles and zeros to the desired compensated
dominant pole to be -198.37. Thus the contribution required from the
compensator zero is 198.37-180=18.37. Then we calculate the location
of the zero as:
𝟏𝟓. 𝟖𝟕
𝒛 𝒄 − 𝟖. 𝟏𝟑
= 𝐭𝐚𝐧𝟏𝟖. 𝟑𝟕∘
and 𝒛 𝒄 = 𝟓𝟓. 𝟗𝟐
⇒ 𝑮 𝑷𝑫 𝒔 = 𝒔 + 𝟓𝟓. 𝟗𝟐
⇒ 𝑲 = 𝟓. 𝟑𝟒
21

Predicted characteristics of uncompensated
and PD- compensated systems

Calculating the PI compensator
Choosing the ideal integral compensator to be
Finally to implement the compensator and find the K’s, using the PD
and PI compensators
we find K1= 259.5, K2=128.6, and K3=4.6
𝑮 𝑷𝑰(𝒔) =
𝒔 + 𝟎. 𝟓
𝒔
⇒ 𝑮 𝑷𝑰𝑫(𝒔) =
)𝑲(𝒔 + 𝟓𝟓. 𝟗𝟐)(𝒔 + 𝟎. 𝟓
𝒔
=
𝟒. 𝟔(𝒔 𝟐 + 𝟓𝟔. 𝟒𝟐𝒔 + 𝟐𝟕. 𝟗𝟔
𝒔
23

Predicted characteristics of uncompensated,
PD and PID- compensated systems

PROVIDED BY: BF(B.FARZANEGAN@AUT.AC.IR) 25
Ziegler-Nichols Tuning of PID
J. G. Ziegler and N. B. Nichols recognized that the step responses of a
large number of processes control systems exhibits a process reaction
curve
Ziegler & Nichols gave two methods for tuning the controller
• Open-Loop Tuning Method or Process Reaction Method
• Based on a stability boundary
3/1/2020

Tuning of PID Controllers
Because of their widespread use in practice, we present below several
methods for tuning PID controllers. Actually these methods are quite
old and date back to the 1950’s. Nonetheless, they remain in
widespread use today.
In particular, we will study.
• Ziegler-Nichols Oscillation Method
• Ziegler-Nichols Reaction Curve Method
• Cohen-Coon Reaction Curve Method

PID standard form
The standard form PID are:
Proportional only:
Proportional plus Integral:
Proportional plus derivative:
Proportional, integral and
derivative:

Ziegler-Nichols (Z-N) Oscillation Method
This procedure is only valid for open loop stable plants and it is
carried out through the following steps
• Set the true plant under proportional control, with a very
small gain.
• Increase the gain until the loop starts oscillating. Note that
linear oscillation is required and that it should be detected
at the controller output.
• Record the controller critical gain Kp = Kc and the oscillation
period of the controller output, Pc.
• Adjust the controller parameters according to Table below;

Numerical Example
Consider a plant with a model given by
Find the parameters of a PID controller using the Z-N oscillation method.
Obtain a graph of the response to a unit step input reference and to a
unit step input disturbance.

Solution
Applying the procedure we find:
Kc = 8 and ωc = 3.
Hence, from Table, we have
The closed loop response to a unit step in the reference at t = 0 and a unit
step disturbance at t = 10 are shown in the next figure.

Example

Example
Ku = 8 and Pu =3.62
Hence, from Table , we have
K= 0.6 ku=4.8, Ti=1/2 Pu=1.81, Td=1/8 Pu=0.45
3/1/2020

Example
Kp = 4.8, Ki = 2.64, Kd = 2.16
3/1/2020

Ziegler-Nichols Open-Loop Tuning
Method or Process Reaction Method
This method remains a popular technique for tuning controllers that use
proportional, integral, and derivative actions. The Ziegler-Nichols open-
loop method is also referred to as a process reaction method, because it
tests the open-loop reaction of the process to a change in the control
variable output. This basic test requires that the response of the system
be recorded, preferably by a plotter or computer. Once certain process
response values are found, they can be plugged into the Ziegler-Nichols
equation with specific multiplier constants for the gains of a controller
with either P, PI, or PID actions.

Reaction Curve Based Methods
A linearized quantitative version of a simple plant can be
obtained with an open loop experiment, using the following
procedure:
1. With the plant in open loop, take the plant manually to a normal
operating point. Say that the plant output settles at y(t) = y0 for
a constant plant input u(t) = u0.
2. At an initial time, t0, apply a step change to the plant input, from
u0 to u (this should be in the range of 10 to 20% of full scale).
3. Record the plant output until it settles to the new operating
point. Assume you obtain the curve shown on the next slide.
This curve is known as the process reaction curve. In Figure,
m.s.t. stands for maximum slope tangent.
4. Compute the parameter model as follows

Sample of MATLAB Implement
Consider a system with transfer function
T=2/[(S+2)*(0.18*S^2+0.6*S+1)]
With L=0.38;R=1;
1
3/1/2020

2
3
4
5
3/1/2020
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
t
y(t)
Step Response

Cohen-Coon Reaction Curve Method
Cohen and Coon carried out further studies to find controller settings
which, based on the same model, lead to a weaker dependence on the
ratio of delay to time constant. Their suggested controller settings are
shown in Table
1
)(
0
0
0 


s
eK
sG
s


.
0
x

Ziegler-Nichols closed-loop tuning method
The Ziegler-Nichols closed-loop tuning method allows you to use the
ultimate gain value, Ku, and the ultimate period of oscillation, Pu, to calculate
K . It is a simple method of tuning PID controllers and can be refined to give
better approximations of the controller. You can obtain the controller
constants K , Ti , and Td in a system with feedback. The Ziegler-Nichols closed-
loop tuning method is limited to tuning processes that cannot run in an open-
loop environment
Determining the ultimate gain value, Ku, is accomplished by finding the value
of the proportional-only gain that causes the control loop to oscillate
indefinitely at steady state

General Comments about
Controller Tuning
a) Why is the proportional gain Kp for PI control is less than the value for
P-only control?
b) Why Kp for PID control is more than that PI?

Worked out Example
Problem: You're a controls engineer working for Flawless Design company
when your optimal controller breaks down. As a backup, you figure that by
using coarse knowledge of a classical method, you may be able to sustain
development of the product. After adjusting the gain to one set of data taken
from a controller, you find that your ultimate gain is 4.3289.
From the adjusted plot below, determine the type of loop this graph
represents; then, please calculate K, Ti, and Td for all three types of
controllers.

Worked out Example
Solution:
From the fact that this graph oscillates and is not a step function, we see that
this is a closed loop. Thus, the values will be calculated accordingly.
We're given the Ultimate gain, Ku = 4.3289. From the graph below, we see that
the ultimate period at this gain is Pu = 6.28
3/1/2020

Worked out Example
From this, we can calculate the Kc, Ti,
and Td for all three types of
controllers. The results are tabulated
below. (Results were calculated from
the Ziegler-Nichols closed-loop
equations.)
3/1/2020

General comments about controller
tuning
The different methodologies of controller tuning, known as Ziegler-Nichols method have
been illustrated. It is to be remembered that the recommended settings are obtained
from extensive experimentation with number of different processes; there is no
theoretical basis behind these selections. As a result, a better combination of the P, I, D
values may always be found, that will give less oscillation and better settling time. But
with no a-priori knowledge of the system, it is always advisable to perform the
experimentation and select the controller settings, obtained from Ziegler-Nichols
method. But there is always scope for improving the performance of the controller by
fine-tuning. So, Ziegler-Nichols method provides initial settings that will give satisfactory,
result, but it is always advisable to fine-tune the controller further for the particular
process and better performance is expected to be achieved

General Comments about Controller Tuning
Nowadays digital computers are replacing the conventional analog
controllers. P-I-D control actions are generated through digital
computations. Digital outputs of the controllers are converted to analog
signals before they are fed to the actuators. In many cases, commercial
software are available for Auto tuning the process.

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