PROCESS INSTRUMENTATION AND
CONTROL
CHAPTER 6. Designing of Process Control
System - Feedback Controller Design
Chapter 6 – Part 1
Designing of Process
Control System
 Disturbance variables: variables that are varied uncontrollably
 Controlled variables: process variables that need to be regulated
/ maintained steadily at predetermined values (called setpoints)
 Manipulated variables: variables that are varied by controllers
(within certain ranges) so as to compensate for the changes in
controlled variables caused by disturbance variables
The objective of Process Control System is to “move” process
variations from disturbance variables to manipulated variables
such that controlled variables are kept steadily at setpoint values.
The following figure illustrates that “action” in a “single inputsingle output” control system
Degrees-of-Freedom Analysis
 The number of degrees of freedom is given by:
ND = NV - NE
where ND is the number of degrees of freedom, NV is the
number of process variables, and NE is the number of independent
equations that describe the process.
 The number of manipulated variables NM is given by:
NM = ND – NED = NV - NE - NED
where NED is the number of disturbances (they are externally
defined in the process model)
 The number of manipulated variables equals the number of
controlled variables that can be regulated.
Selection of Controlled Variables
1. Select output variables that are either non self-regulating or
unstable: values of self-regulating variables do not "runaway": they automatically (without a controller) transit to new
steady values in the presence of disturbances
2. Choose output variables that must be kept within equipment
and operating constraints (e.g., temperatures, pressures, and
liquid level).
3. Select output variables that are a direct measure of product
quality (e.g., composition, refractive index) or that strongly
affect it (e.g., reaction temperature)
Selection of Controlled Variables
4. Choose output variables that exhibit significant interactions
with other output variables: the pressure in a steam header
that supplies steam to downstream units is a good example. If
this supply pressure is not well regulated, it will act as a
significant disturbance to downstream units.
5. Choose output variables that have favorable dynamic and
static characteristics. Output variables that have large
measurement time delays, or are insensitive to the
manipulated variables are poor choices.
To be more specific, usually the controlled variables:
 Directly affect the operational stability and safety, and
performance of a certain process equipment
 Directly affect the quantity and quality of products
Controlled variables:
 Pressure and liquid level in
reactor (for steady operation
of the reactor)
 Reactor temperature (affecting
the product quality)
 Flowrate of all feedstock
streams entering the reactor
(affecting the product quantity
and quality)
 SIC (affecting the product
quality)
Selection of Manipulated Variables
To select a manipulated variable for regulating a controlled
variable:
1. Ideally, change in the selected manipulated variable should
cause a significant change in controlled variable in a direct
and rapid manner
2. Avoid recycling disturbances: it is preferable not to
manipulate an inlet stream or a recycle stream. It is usually
better to eliminate the effect of disturbances by allowing them
to leave the process in an effluent stream rather than having
them propagate through the process by the manipulation of a
feed or recycle stream
Selection of Manipulated Variables
Controlling reactor
temperature: option A
Controlling reactor
temperature: option B
The hot shot / cold shot (option A) as manipulated variable has
more rapid / direct effect on the controlled variable (reactor
temperature) than the cooling / heating fluid (option B)
To be more specific, usually manipulated variables are:
• Flowrates of utilities in the process (steam, hot oil, fuel oil,
fuel gas, cooling water,…): the most popular choice
• Flowrate of a process stream that is not further processed in a
downstream unit; examples: gas stream goes to flare or gas
gathering unit, product stream goes to storage tank
• Other types of manipulated variables: liquid level in heat
exchanger with mixed-phase stream (affecting the heat
transfer area and the amount of heat transferred), flowrate of
bypass stream in a heat exchanger, reflux ratio in distillation
column,..
Manipulated variables: liquid level in heat exchanger that uses
steam as hot utility (affecting the heat transfer area and the
amount of heat transferred)
Controlled variables:
 Liquid level: for stable operation and safety of CSTR
 Reactor temperature: for stable operation and safety of CSTR,
affecting the product composition (affecting reaction rate)
 Product composition / product quality is controlled by adjusting
feed flowrate (this variable determines the reactor residence
time)
Controlled variables:
 Liquid level: for stable operation and safety of the flash drum
 Temperature and pressure: for stable operation and safety of the
flash drum, affecting the product composition
 Feed flowrate (optional): determine the product flowrates
Control scheme for binary distillation tower:
Configuration (a) has better dynamic performance than the config.
(b): easier to control the composition of the overhead product
Configuration (b) is used when reflux ratio is large, the flowrate of
the overhead product is small when compared to the reflux flowrate
 The design of the Process Control System requires technical
expertise of the designers, usually performed by engineers
specializing in instrumentation and control
 The approach to design Process Control System based on
mathematical models of process control systems becomes more
and more popular
 A step-by-step procedure, not involving mathematical models,
is used in this course
Process Control System design procedure
A step-by-step procedure to design the Process Control Systems is
described as follows:
1. Establish the Control Objectives. Common control objectives
are:
•
Maintain product flowrate at required value
•
Achieve maximum feedstock-to-product conversion
•
Satisfy specifications on product quality
2. Determine the Control Degrees of Freedom: the number of
control valves in the flowsheet equals the degrees of freedom
Process Control System design procedure
3. Establish the Energy-management System
In this step, control loops are positioned to regulate
exothermic and endothermic reactors at desired temperatures.
In addition, temperature controllers are positioned to ensure
that disturbances are removed from the process through utility
streams rather than recycled by heat-integrated process units.”
4. Set the Production Rate. This is accomplished by placing a
flow control loop on the principal feed stream (referred to as
fxed feed or fresh feed) or on the principal product stream
(referred to as on-demand product)
Process Control System design procedure
5. Control the Product Quality and Handle Safety,
Environmental, and Operational Constraints: set the control
loops for composition of the product streams or the process
variables that strongly affect the product composition (e.g.
reaction temperature)
6. Fix a Flow Rate in Every Recycle Loop and Control
Vapor and Liquid Inventories (i.e. pressures and liquid
levels in vessels)
7. Check Component Balances. In this step, control loops are
installed to prevent the accumulation of individual chemical
species in the process
Process Control System design procedure
8. Control the Individual Process Units. At this point, the
remaining degrees of freedom are assigned to ensure that
adequate local control is provided in each process unit.
Usually, the Process Control System has been fully
configured in steps 1-7, this steps often requires no additions
to the control system.
9. Optimize Economics and Improve Dynamic Controllability.
Usually, this step involves the use of advanced control
techniques such as cascade control, combined feedforward /
feedback control to improve the dynamic and economic
performance of the process.
Use the following guidelines to design the Process Control
System:
 There can only be a single control valve on any given stream
between unit operations.
 A level controller is needed anywhere where a vapour-liquid or
liquid-liquid interface is maintained.
 Always control the pressure for unit operations and process
units whose function depends on pressure. Examples are flash
drums, hydrocyclones…
 Pressure control is more responsive when the pressure
controller actuates a control valve on a vapour
stream
Use the following guidelines to design the Process Control
System (tt):
 Two operations cannot be controlled at different pressures
unless there is a valve or other restriction (or a compressor or
pump) between them.
 Temperature control is usually achieved by controlling the
flow of a utility stream (such as steam or cooling water) or a
bypass around an exchanger.
 The overall plant material balance is usually set by flow
controllers or flow ratio controllers on the process feeds
Feedback control loop
Single-input/single-output feedback control loop
Corrective actions: Taking corrective action after upset
propagated, to eliminate the error after occurrence
Feedback control loop
Advantages:
 Simple: easy to design, install, and operate
 The most popular choice, the first candidate to be considered
when designing the process control system
 Adequately satisfy the job requirement for “not-toocomplicated” applications: pressure control, level control,…
Disadvantages:
 Not a good choice for “demanding” applications, for example:
when disturbances frequently occur; or the time lapse since
the appearance of disturbances to the appearance of induced
change in controlled variable is long (a few minutes)
Cascade control loops
Cascade control loops
Single-input/single-output
feedback control loop
Flowrate / pressure of steam
input to the heat exchanger
frequently changes
Cascade control loops
Cascade control loops
Single-input/single-output
feedback control loop
Flowrate of hot oil input to
the heat exchanger frequently
changes
Cascade control loops
Cascade control loops
Cascade control loops
When to use cascade control architecture?
- When the conventional feedback control loop has a poor
performance that makes it not suitably used for the intended
application
- When flowrate / pressure of the manipulated stream (usually
an utility stream) frequently changes
- When it is possible / advisable to control a process variable
(for example, composition) via controlling another process
variable (for example, reaction temperature)
Cascade control loops
Remote setpoint
Cascade control loops
Feedforward control
"Proactive mode": Taking corrective action before upset
propagated. It “predicts” the disturbance and proactively takes
action to prevent it. It prevents error before occurrence
Feedback vs. Feedforward control
Feedback
Feedforward
Combined Feedback / Feedforward control
Combined Feedback / Feedforward control
Combined Feedback / Feedforward control
Ratio control
Ratio control can be
used where it is
desired to maintain
two flows at a
constant ratio; for
example, it is usually
required to maintain a
constant ratio
between two feed
stream flowrates of a
reactor or a mixing
tank
Ratio control
Method 1
Method 2
Common configurations for level control
Level control
Common configurations for pressure control
Common configurations for flowrate control
Common configurations for temperature control
Some examples of process control systems
Some examples of process control systems
Some examples of process control systems
Some examples of process control systems
Distillation column – Preheat train
Some examples of process control systems
Comment: outlet
temperature of the
overhead stream is
usually controlled
via the temperatureto-flow cascade
control loop
Distillation column
– Overhead system
Some examples of process control systems
Distillation column
– Bottom section
Kettle reboiler,
Ex-705, utilizes a
natural circulation
feed system
Designing the Process Control System – Example 1
Designing the Process Control System – Example 1
1. Establish the Control Objectives: the primary goal is to meet
the required production rate. There are two approaches:
•
Control / regulate the flowrate of the product stream (set
up a flow control that uses the control valve V-7): the
“on-demand product” option
•
Control / regulate the flowrate of the principal feed
stream (set up a flow control that uses the control valve
V-1): the “fixed feed” option
2. Determine the Control Degrees of Freedom: the number of
controlled variables = the number of control valves = 7
Designing the Process Control System – Example 1
3. Establish the Energy-management System:

Temperature in the reactor R-100 is controlled by adjusting
the flowrate of cooling water (use valve V-2)

Temperature of the feed stream entering the reactor R-100 is
controlled by adjusting the flowrate of heating steam (use
valve V-3)
4. Set the Production Rate: already established in step 1: use
valve V-7 (the “on-demand product” option) or valve V-1
(the “fixed feed” option)
Designing the Process Control System – Example 1
5. Control the Product Quality:

Composition of the effluent stream of R-100 (stream that has
valve V-4) is to be controlled. The process variable that strongly
affects the composition of the effluent stream is the temperature
of R-100 => it is needed to control temperature of R-100
(already established in step 3)

Composition of the product stream B is determined by the
composition of the effluent stream of R-100, as well as the
pressure and temperature in V-100: use valve V-6 to control
temperature and valve V-5 to control pressure in flash drum V100
Designing the Process Control System – Example 1
6. Fix a Flow Rate in Every Recycle Loop and Control Vapor and
Liquid Inventories: there is no recycle stream

Pressure control: applicable for V-100 (use valve V-5). R-100
has liquid phase only

Level control: liquid level in a vessel is controlled by
manipulating either the liquid feed stream or the liquid effluent
stream of the vessel:
 The “on-demand product” option: liquid level in V-100 is
controlled via valve V-4 => liquid level in R-100 is
controlled via valve V-4
 The “fixed feed” option: liquid level in R-100 is controlled
via valve V-4 => liquid level in V-100 is controlled via
valve V-7
Designing the Process Control System – Example 1
7. Check Component Balances: N/A
8. Control the Individual Process Units: N/A
9. Optimize Economics and Improve Dynamic Controllability:
assuming that it is possible to measure stream composition
(with fast response time), we will establish a “composition-totemperature” cascade controllers with primary controller
being the composition controller, secondary controller being
the temperature controller of R-100
Designing the Process Control System – Example 1
Designing the Process Control System – Example 1
Designing the Process Control System – Example 2
Designing the Process Control System – Example 2
1. Establish the Control Objectives: the primary goal is to meet
the required production rate. Use only the “fixed feed” option
because it is technically not recommended to control flowrate
of a vapor / gas stream coming out of a vessel:
•
Control / regulate the flowrate of the principal feed
stream (set up a flow control that uses the control valve
V-1): the “fixed feed” option
2. Determine the Control Degrees of Freedom: the number of
controlled variables = the number of control valves = 6
Designing the Process Control System – Example 2
3. Establish the Energy-management System:

Temperature in the reactor R-100 is controlled by adjusting
the flowrate of cooling water (use valve V-2)
4. Set the Production Rate: already established in step 1: use
valve V-1 (the “fixed feed” option)
Designing the Process Control System – Example 2
5. Control the Product Quality:

Composition of the effluent stream of R-100 (stream that has
valve V-3) is to be controlled. The process variable that strongly
affects the composition of the effluent stream is the temperature
of R-100 => it is needed to control temperature of R-100
(already established in step 3)

Composition of the product stream B is determined by the
composition of the effluent stream of R-100, as well as the
pressure and temperature in V-100: use valve V-5 to control
temperature and valve V-4 to control pressure in flash drum V100
Designing the Process Control System – Example 2
6. Fix a Flow Rate in Every Recycle Loop…:

Fix / control the flowrate of the recycle stream (use valve V-6)

Pressure control: applicable for V-100 (use valve V-4). R-100
has liquid phase only

Level control:
 Liquid level in V-100 is controlled via valve V-3 because the
flow of liquid effluent stream is already regulated
 For R-100: manipulation of the effluent stream via V-3 (used
by the LC in V-100) and regulation of recycle stream via
valve V-6 is needed for stable process operation. Whereas
regulation of input stream via valve V-1 has been established
to meet the required production rate => flowrate of input
stream can be adjusted to control the liquid level in R-100
Designing the Process Control System – Example 2
7. Check Component Balances: N/A
8. Control the Individual Process Units: N/A
9. Optimize Economics and Improve Dynamic Controllability:
“To maximize conversion, a cascade controller is installed as
in the previous example in which the setpoint of the reactor
temperature controller (TC on V-2) is adjusted to control the
concentration of B in the reactor effluent. Again, for an
irreversible reaction, it is enough to operate the reactor at the
highest possible temperature”
Designing the Process Control System – Example 2
Thiết kế hệ thống điều khiển – ví dụ minh họa 3
Một thiết bị bay hơi (evaporator) được dùng để cô đặc một dung dịch
(của chất tan + dung môi D) đến nồng độ mong muốn của chất tan xB.
Nhiệt cho quá trình hóa hơi được cung cấp bởi hơi nước. Các biến có
thể được điều chỉnh là lưu lượng dòng hơi, lưu lượng hơi nước, lưu
lượng dòng sản phẩm. Các yếu tố gây nhiễu (yếu tố thay đổi) là lưu
lượng và thành phần dòng nhập liệu. Giả sử thành phần dòng sản phẩm
có thể được đo lường với thời gian có kết quả nhanh. Thiết kế hệ thống
điều khiển cho thiết bị này.
Thiết kế hệ thống điều khiển – ví dụ minh họa 3
1. Thiết lập mục tiêu điều khiển: sản phẩm đạt tiêu chuẩn chất
lượng về thành phần sản phẩm:
2. Xác định bậc tự do điều khiển = 3
3. Kết quả của các bước 3, 4, 5 ở slide sau
Thiết kế hệ thống điều khiển – ví dụ minh họa 3
 Vì dung môi D bay hơi ở nhiệt độ xem như không đổi (khi áp
suất được giữ cố định), vòng điều khiển nhiệt độ không cần
thiết
 Vì thành phần dòng sản phẩm có thể được đo lường với thời
gian có kết quả nhanh, chúng ta có thể thiết lập một vòng điều
khiển thành phần dòng sản phẩm.
 Để đạt được tiêu chuẩn về thành phần sản phẩm (xB theo yêu
cầu), 1 lượng dung môi tương ứng với thành phần và lưu
lượng dòng nhập liệu phải được hóa hơi
 Ở điều kiện áp suất (và nhiệt độ) được giữ cố định, lượng dung
môi bay hơi (trong khoảng thời gian bằng thời gian lưu của lưu
chất) phụ thuộc vào lượng nhiệt cấp vào thiết bị
 Phương án được áp dụng là sử dụng lưu lượng hơi nước như
biến điều chỉnh của vòng điều khiển thành phần sản phẩm
Thiết kế hệ thống điều khiển – ví dụ minh họa 3
6. Điều khiển lưu lượng các dòng hồi lưu, điều khiển áp suất và
mực chất lỏng:

Điều khiển áp suất: gắn và sử dụng van điều khiển trên dòng
hơi ra khỏi thiết bị

Điều khiển mực chất lỏng: mực chất lỏng được điều khiển
bằng cách điều chỉnh lưu lượng dòng sản phẩm lỏng ra khỏi
thiết bị
7. Kiểm tra cân bằng vật chất của các cấu tử: N/A
8. Điều khiển từng thiết bị cụ thể trong quy trình: N/A
Kết quả đến bước 8 được trình này ở slide sau
Thiết kế hệ thống điều khiển – ví dụ minh họa 3
Thiết kế hệ thống điều khiển – ví dụ minh họa 3
9. Tối ưu hóa tính kinh tế, cải thiện đặc tính điều khiển động học
của quy trình (nếu có thể): phương án thiết kế vừa trình bày
có thể được cải thiện thêm bằng cách thêm feedforward
control, trong đó thông tin về lưu lượng (hoặc thành phần)
dòng nhập liệu sẽ được dùng để điều chỉnh lưu lượng hơi
nước, ví dụ: khi lưu lượng dòng nhập liệu tăng lên thì lưu
lượng hơi nước được điều chỉnh tăng lên. Như vậy ta sử dụng
combined feed forward/feedback control để điều khiển thành
phần dòng sản phẩm với biến được điều chỉnh là lưu lượng
dòng hơi nước.
Thiết kế hệ thống điều khiển – ví dụ minh họa 3
Signal selector
Feedforward controller
FFC
FT
Chapter 6 – Part 2
Feedback Controller
CHAP 6 - Part 2: THE FEEDBACK LOOP
v1
4-20 mA
T
A
4-20 mA
v2
3-15 psi
CHAP 6 - Part 2: THE FEEDBACK LOOP
Music: “I cannot define good music, but I know what I like.”
Control Performance: We must be able to define what we desire, so that we can design
equipment and controls to achieve our objectives.
Controlled Variable
Set point
1.5
entered by
person 1
0.5
Controlled variable, value from a sensor
0
0
5
10
15
Manipulated Variable
2
20
25
Time
30
35
40
45
50
Manipulated variable, usually a valve
1.5
1
0.5
0
0
5
10
15
20
25
Time
30
35
40
45
50
CHAP 6 - Part 2: THE FEEDBACK LOOP
1.5
Controlled Variable
Let’s be sure we understand the
variables in the plot. We will see this
plot over and over and over …!
1
0.5
0
0
Manipulated Variable
2
1.5
1
0.5
0
0
5
10
15
20
25 30 35
Time
40 45 50
5
10
15
20 25 30
Time
35
40
45 50
CHAP 6 - Part 2: THE FEEDBACK LOOP
Set point Change

= IAE = |SP(t)-CV(t)| dt

0
1.5
IAE = Integral of absolute
value of the error
A
B
1
Return to set point,
“zero offset
0.5
B/A = Decay ratio
0
0
5
10
15
20
Rise time
25
Time
30
35
40
45
50
2
C/D = Maximum overshoot of manipulated variable
1.5
C
1
D
0.5
0
0
5
10
15
20
25
Time
30
35
40
45
50
CHAP 6 - Part 2: THE FEEDBACK LOOP
Disturbance Response

= IAE = |SP(t)-CV(t)| dt

0
0.8
Maximum CV deviation from set point
0.6
0.4
0.2
0
-0.2
0
5
10
15
20
25
Time
30
35
40
45
50
0
5
10
15
20
25
Time
30
35
40
45
50
0
-0.5
-1
-1.5
CHAP 6 - Part 2: THE FEEDBACK LOOP
Disturbance Response
Often, the process is subject to many large and small
disturbances and sensor noise. The performance
measure characterizes the variability.
S-LOOP plots deviation variables (IAE = 5499.9786)
Controlled Variable
20
10
Variance or
standard
deviation of CV
0
-10
-20
0
100
200
300
400
500
Time
600
700
800
900
1000
Manipulated Variable
20
10
Variance or
standard
deviation of MV
0
-10
-20
0
100
200
300
400
500
Time
600
700
800
900
1000
CHAP 6 - Part 2: THE FEEDBACK LOOP
20
Controlled Variable
• To reduce the variability in the CV,
we increase the variability in the MV.
• We must design plant with MV’s
that can be adjusted at low cost.
10
0
-10
-20
0
Manipulated Variable
20
10
0
-10
-20
0
100 200 300 400 500 600 700 800 900 1000
Time
100 200 300 400 500 600 700 800 900 1000
Time
Class exercise: Comment on the quality of control for the
four responses below.
S-LOOP plots deviation variables (IAE = 43.9891)
3
1
2
0.5
Controlled Variable
Controlled Variable
S-LOOP plots deviation variables (IAE = 17.5417)
1.5
A
0
-0.5
0
20
40
60
Time
80
100
B
0
-1
120
2
0
20
0
20
40
60
Time
80
100
120
40
60
80
100
120
4
1.5
Manipulated Variable
Manipulated Variable
1
1
0.5
0
3
2
1
0
-0.5
0
20
40
60
Time
80
100
120
-1
Time
S-LOOP plots deviation variables (IAE = 24.0376)
1.5
1.5
1
1
Controlled Variable
Controlled Variable
S-LOOP plots deviation variables (IAE = 34.2753)
0.5
C
0
-0.5
0
20
40
60
Time
80
100
0
20
40
60
80
100
120
80
100
120
Time
1.5
Manipulated Variable
Manipulated Variable
D
0
-0.5
120
1
0.5
0
-0.5
0
0.5
20
40
60
Time
80
100
120
1
0.5
0
-0.5
0
20
40
60
Time
Class exercise: Comment on the quality of control for the
four responses below.
S-LOOP plots deviation variables (IAE = 43.9891)
3
1
2
0.5
Controlled Variable
Controlled Variable
S-LOOP plots deviation variables (IAE = 17.5417)
1.5
A
0
-0.5
0
20
40
60
Time
80
100
0
2
20
1.5
1
0.5
0
40
60
Time
80
60
80
100
120
100
120
Too oscillatory
4
Manipulated Variable
Manipulated Variable
B
0
-1
120
Generally acceptable
1
3
2
1
0
-0.5
0
20
40
60
Time
80
100
120
-1
0
20
1.5
1
1
Controlled Variable
Controlled Variable
1.5
0.5
C
0
-0.5
0
20
40
60
Time
Too slow
80
100
D
0
0
1.5
0.5
0
20
40
60
Time
Gets close quickly;
Gets to set point slowly
20
40
60
80
100
120
80
100
120
Time
Manipulated Variable
Manipulated Variable
0.5
-0.5
120
1
-0.5
0
40
Time
S-LOOP plots deviation variables (IAE = 24.0376)
S-LOOP plots deviation variables (IAE = 34.2753)
80
100
120
1
0.5
0
-0.5
0
20
40
60
Time
CHAP 6 - Part 2: THE FEEDBACK LOOP
We can apply feedback via many approaches
1, No control - The variable responds to all inputs, it
“drifts”.
2. Manual - A person observes measurements and
introduces changes to compensate, adjustment
depends upon the person.
3. On-Off - The manipulated variable has only two
states, this results in oscillations in the system.
4. Continuous, automated - This is a modulating control
that has corrections related to the error from desired.
5. Emergency - This approach takes extreme action
(shutdown) when a dangerous situation occurs.
Chapter 6 – Part 3
The PID Controller
CH 6 - Part 3: THE PID CONTROLLER
PROPERTIES THAT WE SEEK IN A CONTROLLER
• Good Performance - feedback
measures (e.g. IAE)
• Wide applicability - adjustable
parameters
• Timely calculations - avoid
convergence loops
• Switch to/from manual bumplessly
• Extensible - enhanced easily
v1
TC
v2
CH 6 - Part 3: THE PID CONTROLLER
SOME BACKGROUND IN THE CONTROLLER
• Developed in the 1930-40’s,
remains workhorse of practice
• Not “optimal”, based on good
properties of each mode
• Programmed in digital control
equipment
• ONE controlled variable (CV) and
ONE manipulated variable (MV).
Many PID’s used in a plant.
v1
TC
v2
CH 6 - Part 3: THE PID CONTROLLER
MV =
controller
output
Proportional
E
+
Integral
-
Derivative
Note: Error = E  SP - CV
Final
element
SP = Set
point
+
CV =
Controlled
variable
sensor
Process
variable
PROCESS
Three “modes”: Three ways of using the time-varying
behavior of the measured variable
CH 6 - Part 3: THE PID CONTROLLER
Closed-Loop Model: Before we learn about each
calculation, we need to develop a general dynamic model
for a closed-loop system - that is the process and the
controller working as an integrated system.
v1
TC
v2
This is an example; how
can we generalize?
• What if we measured
pressure, or flow, or …?
• What if the process
were different?
• What if the valve were
different?
CH 6 - Part 3: THE PID CONTROLLER
GENERAL CLOSED-LOOP MODEL BASED ON BLOCK DIAGRAM
D(s)
SP(s)
+
-
E(s)
Gd(s)
MV(s)
GC(s)
Gv(s)
CVm(s)
GP(s)
+
+
CV(s)
GS(s)
Transfer functions
Variables
GC(s) = controller
Gv(s) = valve
GP(s) = feedback process
GS(s) = sensor
Gd(s) = disturbance process
CV(s) = controlled variable
CVm(s) = measured value of CV(s)
D(s) = disturbance
E(s) = error
MV(s) = manipulated variable
SP(s) = set point
CH 6 - Part 3: THE PID CONTROLLER
D(s)
SP(s)
+
-
E(s)
MV(s)
GC(s)
Gv(s)
CVm(s)
Let’s audit
our
understanding
Gd(s)
GP(s)
+
+
CV(s)
GS(s)
• Where are the models for the transmission, and signal
conversion?
• What is the difference between CV(s) and CVm(s)?
• What is the difference between GP(s) and Gd(s)?
• How do we measure the variable whose line is
indicated by the red circle?
• Which variables are determined by a person, which by
computer?
CH 6 - Part 3: THE PID CONTROLLER
D(s)
SP(s)
E(s)
+
Gd(s)
CV(s)
MV(s)
GC(s)
Gv(s)
GP(s)
+
+
CVm(s)
GS(s)
Set point response
G p ( s )Gv ( s )Gc ( s )
CV ( s )

SP( s ) 1  G p ( s )Gv ( s )Gc ( s )GS ( s )
Disturbance Response
Gd ( s )
CV ( s )

D( s ) 1  G p ( s )Gv ( s )Gc ( s )GS ( s )
• Which elements in the control system affect
system stability?
• Which elements affect dynamic response?
Proportional
CH 6 - Part 3:
MV
E
SP
-
+CV
Integral
+
Derivative
THE PID CONTROLLER,
Note: Error = E  SP - CV
The Proportional Mode
PROCESS
“correction proportional to
error.”
Time domain : MV (t )  K c E(t )  I p
MV ( s )
Transfer function : GC ( s ) 
 KC
E( s )
KC = controller gain
How does this differ from
the process gain, Kp?
Proportional
MV
CH 6 - Part 3:
E
SP
-
+CV
Integral
+
Derivative
THE PID CONTROLLER,
Note: Error = E  SP - CV
The Proportional Mode
“correction proportional to
error.”
PROCESS
Time domain : MV (t )  K c E(t )  I p
Kp depends upon the process (e.g.,
reactor volume, flows, temperatures,
etc.)
KC = controller gain
How does this differ from
the process gain, Kp?
KC is a number we select; it is used
in the computer each time the
controller equation is calculated
Proportional
CH 6 - Part 3:
THE PID CONTROLLER,
MV
+
E
SP
-
+CV
Integral
Derivative
Note: Error = E  SP - CV
The Proportional Mode
PROCESS
Time domain : MV (t )  K c E(t )  I p
Proportional
CH 6 - Part 3:
THE PID CONTROLLER,
MV
+
E
SP
-
+CV
Integral
Derivative
Note: Error = E  SP - CV
The Proportional Mode
PROCESS
Key feature of closed-loop performance with P-only
Final value
Kd
D K d
D
CV ' (t ) t   lim s

0
after
s 0
s 1  Kc K p 1  Kc K p
disturbance:
• We do not achieve zero offset; don’t return to set point!
• How can we get very close by changing a controller
parameter?
• Any possible problems with suggestion?
Proportional
MV
CH 6 - Part 3:
+
THE PID CONTROLLER,
E
SP
-
+CV
Integral
Derivative
Note: Error = E  SP - CV
The Proportional Mode
PROCESS
Disturbance in concentration of A in the solvent
CV = concentration of A in effluent
MV = valve % open of pure A stream
FS
solvent
FA
CV
pure A
From person
AC
MV
SP
E (t )  SP(t )  CV (t )
MV (t )  K c E (t )  I
THE PID CONTROLLER,The Proportional Mode
0.8
Controlled Variable
Controlled Variable
0.8
0.6
No control
0.4
0.2
0.6
0.4
0
0
20
40
60
80
100
Time
120
140
160
180
0
200
0
0.5
valve
20
40
60
80
100
Time
120
140
160
180
200
0
Manipulated Variable
Manipulated Variable
1
Kc = 0
0
-0.5
-1
0
20
40
60
80
100
Time
120
140
160
180
Kc =10
-2
-4
-6
200
0
Note change of scale!
20
40
20
40
60
80
100
Time
120
140
160
180
200
60
80
100
Time
120
140
160
180
200
80
100
Time
120
140
160
180
200
0.3
Controlled Variable
0.25
Controlled Variable
Offset (bad)
0.2
0.2
0.15
Less Offset, better
(but not good)
0.1
0.05
0.2
0.1
0
-0.1
0
0
0
20
40
60
80
100
Time
120
140
160
180
200
Unstable, very bad!
20
-5
Manipulated Variable
valve
Manipulated Variable
0
Kc = 100
-10
-15
-20
0
-20
-40
-60
-25
0
0
20
40
60
80
100
Time
120
140
160
180
200
Kc = 220
20
40
60
Proportional
CH 6 - Part 3:
MV
+
SP
-
+CV
Integral
Derivative
THE PID CONTROLLER,
E
Note: Error = E  SP - CV
The Integral Mode
PROCESS
“The persistent mode”
t
Kc
Time domain : MV (t ) 
E (t ' )dt '  I I

TI 0
MV ( s ) KC 1
Transfer function : GC ( s ) 

E( s )
TI s
TI = controller integral time (in denominator)
Proportional
MV
CH 6 - Part 3:
E
SP
-
+CV
Integral
+
Derivative
THE PID CONTROLLER,
Note: Error = E  SP - CV
The Integral Mode
PROCESS
t
Kc
Time domain : MV (t ) 
E (t ' )dt '  I I

TI 0
MV(t)
Slope = KC E/TI
time
Behavior when E(t) = constant
Proportional
MV
CH 6 - Part 3:
+
THE PID CONTROLLER,
E
SP
-
+CV
Integral
Derivative
Note: Error = E  SP - CV
The Integral Mode
PROCESS
Key feature of closed-loop performance with I mode
Final value
after
disturbance:
CV ' (t ) t 
Kd
D
 lim s
s 0
s 1  Kc K p
0
sTI
• We achieve zero offset for a step disturbance;
return to set point!
• Are there other scenarios where we do not?
Proportional
CH 6 - Part 3:
MV
E
SP
-
+CV
Integral
+
Derivative
THE PID CONTROLLER,
Note: Error = E  SP - CV
The Derivative Mode
PROCESS
“The predictive mode”
dE(t )
Time domain : MV (t )  K cTD
 ID
dt
MV ( s )
Transfer function : GC ( s ) 
 K cTd s
E( s )
TD = controller derivative time
Proportional
MV
CH 6 - Part 3:
+
THE PID CONTROLLER,
E
SP
-
+CV
Integral
Derivative
Note: Error = E  SP - CV
The Derivative Mode
PROCESS
Key features using closed-loop dynamic model
Final value
after
disturbance:
CV ' (t ) t 
Kd
D
 lim s
 D K d
s 0
s 1  K cTd s
We do not achieve zero offset; do not return to
set point!
Proportional
CH 6 - Part 3:
THE PID CONTROLLER,
MV
+
E
SP
-
+CV
Integral
Derivative
Note: Error = E  SP - CV
The Derivative Mode
PROCESS
dE(t )
Time domain : MV (t )  K cTD
 ID
dt
• What would be the behavior of the manipulated
variable when we enter a step change to the set point?
• How can we modify the algorithm to improve the
performance?
Proportional
MV
CH 6 - Part 3:
+
THE PID CONTROLLER,
E
SP
-
+CV
Integral
Derivative
Note: Error = E  SP - CV
The Derivative Mode
PROCESS
dE(t )
Time domain : MV (t )  K cTD
 ID
dt
X
We do not want to take the derivative of the set
point; therefore, we use only the CV when
calculating the derivative mode.
d CV (t )
Time domain : MV (t )   K cTD
 ID
dt
Proportional
MV
CH 6 - Part 3:
+
SP
-
+CV
Integral
Derivative
THE PID CONTROLLER
E
Note: Error = E  SP - CV
PROCESS
Let’s combine the modes to formulate the PID Controller!
E (t )  SP(t )  CV (t )

1
MV (t )  K c  E (t ) 
TI

d CV 
0 E (t ' )dt 'Td dt   I
t
Please explain every term and symbol.
Proportional
MV
CH 6 - Part 3:
E
SP
-
+CV
Integral
+
Derivative
THE PID CONTROLLER
Note: Error = E  SP - CV
PROCESS
Let’s combine the modes to formulate the PID Controller!
E (t )  SP(t )  CV (t )

1
MV (t )  K c  E (t ) 
TI

proportional
Error from set point
d CV 
0 E (t ' )dt 'Td dt   I
t
integral
derivative
Constant (bias) for bumpless transfer
Reverse or Direct Acting
Controller
• Direct-Acting (Kc < 0): MV (manipulated
variable) increases if CV (controlled
variable) increases (Error = Setpoint –
Measured value of CV < 0)
• Reverse-Acting (Kc > 0): MV decreases if
CV increases
When the liquid level (the CV) falls you want to increase
opening of valve A (MV: inlet flowrate) so the inlet flow will
increase and raise the tank level. This is a reverse response.
When the liquid level falls the valve B will close, reducing the
outlet flow rate and raising the tank level. This is a direct
response.
PID Controller
CH 6 - Part 3
 Ideal controller
• Transfer function (ideal)
t

1
de 
p( t )  p  K c e( t )   e( t )dt    D 
I 0
dt 



P(s)
1
 K c 1 
 Ds 
E(s)
 Is

 Transfer function (actual)
 Is  1  Ds  1 
P(s)


 K c 
E(s)
 Is  Ds  1 
α = small number (0.05 to 0.20)
lead / lag units
The parallel form
& Expended form
of the PID controller
CH 6 - Part 3
Typical Response of Feedback Control Systems
Consider response of a controlled system after a
sustained disturbance occurs (e.g., step change in
the disturbance variable)
y
Figure 8.12. Typical process responses with feedback control.
111
CH 6 - Part 3
y
Figure 8.13.
Proportional control:
effect of controller
gain.
Figure 8.15. PID
control: effect of
derivative time.
112
CH 6 - Part 3
y
y
Figure 8.14. PI control: (a) effect of reset time (b) effect of
controller gain.
113
CH 6 - Part 3
Position and Velocity Algorithms for Digital PID
Control
A straight forward way of deriving a digital version of the parallel
form of the PID controller (Eq. 8-13) is to replace the integral and
derivative terms by finite difference approximations,
k
0 e  t * dt   e j t
(8-24)
de ek  ek 1

dt
t
(8-25)
t
j 1
where:
t = the sampling period (the time between successive
measurements of the controlled variable)
ek = error at the kth sampling instant for k = 1, 2, …
114
CH 6 - Part 3
There are two alternative forms of the digital PID control
equation, the position form and the velocity form. Substituting (824) and (8-25) into (8-13), gives the position form,


D
t k
pk  p  K c ek   e j 
 ek  ek 1 
1 j 1
t


(8-26)
Where pk is the controller output at the kth sampling instant. The
other symbols in Eq. 8-26 have the same meaning as in Eq. 8-13.
Equation 8-26 is referred to as the position form of the PID
control algorithm because the actual value of the controller output
is calculated.
115
CH 6 - Part 3
In the velocity form, the change in controller output is
calculated. The velocity form can be derived by writing the
position form of (8-26) for the (k-1) sampling instant:


D
t k
pk  p  K c ek   e j 
 ek  ek 1 
1 j 1
t


(8-26)
Note that the summation still begins at j = 1 because it is assumed
that the process is at the desired steady state for
j  0 and thus ej = 0 for j  0. Subtracting (8-27) from (8-26)
gives the velocity form of the digital PID algorithm:


D
t
pk  pk  pk 1  K c  ek  ek 1   ek 
 ek  2ek 1  ek 2 
I
t


(8-28)
116
CH 6 - Part 3
The velocity form has three advantages over the position form:
1. It inherently contains anti-reset windup because the
summation of errors is not explicitly calculated.
2. This output is expressed in a form, pk, that can be utilized
directly by some final control elements, such as a control
valve driven by a pulsed stepping motor.
3. For the velocity algorithm, transferring the controller from
manual to automatic mode does not require any initialization
of the output ( p in Eq. 8-26). However, the control valve (or
other final control element) should be placed in the
appropriate position prior to the transfer.
117
Automatic and Manual Control Modes
CH 6 - Part 3
•
Automatic Mode
Controller output, p(t), depends on e(t), controller
constants, and type of controller used.
( PI vs. PID etc.)
 Manual Mode
Controller output, p(t), is adjusted manually.
 Manual Mode is very useful when unusual
conditions exist:
plant start-up
plant shut-down
emergencies
• Percentage of controllers "on manual” ??
(30% in 2001, Honeywell survey)
CH 6 - Part 3
Controller Comparison
P
- Simplest controller to tune (Kc).
- Offset with sustained disturbance or setpoint
change.
PI
-
More complicated to tune (Kc, I) .
Better performance than P
No offset
Most popular FB controller
PID
-
Most complicated to tune (Kc, I, D) .
Better performance than PI
No offset
Derivative action may be affected by noise
Controller Comparison
 If a steady state error (offset) in the controlled variable is
CH 6 - Part 3
acceptable, the use of a proportional (P) controller is
advised
 If the system has signal noise or dead times, the use of a
proportional-integral (PI) controller is recommended => PI
controller is the most commonly used type of PID
controller
 If the signal noise is negligible, a proportional-integral-
derivative (PID) controller is recommended
CH 6 - Part 3
Controller Comparison
Controller Comparison
Effect of PID parameters (Kc, TI and Td)
Effect of PID parameters (Kc, TI and Td)
Effect of PID parameters (Kc, TI and Td)
S-LOOP plots deviation variables (IAE = 12.2869)
Controlled Variable
1.5
1
0.5
0
0
20
40
60
Time
80
• Is this good
performance?
• How do we
determine:
100
120
Kc, TI and Td?
Manipulated Variable
40
30
20
10
Kc = 30, TI = 11, Td = 0.8
0
0
20
40
60
Time
80
100
120
Effect of PID parameters (Kc, TI and Td)
S-LOOP plots deviation variables (IAE = 20.5246)
Controlled Variable
2
1.5
1
0.5
0
0
20
40
60
Time
80
• Is this good
performance?
• How do we
determine:
100
120
Kc, TI and Td?
Manipulated Variable
150
100
50
0
Kc = 120, TI = 11, Td = 0.8
-50
0
20
40
60
Time
80
100
120
CH 6 - Part 3: THE PID CONTROLLER
HOW DO WE EVALUATE THE DYNAMIC
RESPONSE OF THE CLOSED-LOOP SYSTEM?
• In a few cases, we can do this analytically
• In most cases, we must solve the equations
numerically. At each time step, we integrate
- The differential equations for the process
- The differential equation for the controller
- Any associated algebraic equations
• Many numerical methods are available
CH 6 - Part 3: THE PID CONTROLLER
CH 6 - Part 3: THE PID CONTROLLER
CH 6 - Part 3: THE PID CONTROLLER
CH 6 - Part 3: THE PID CONTROLLER
CH 6 - Part 3: THE PID CONTROLLER
CH 6 - Part 3: THE PID CONTROLLER
More details and more examples can be found in
Lecture “11. Dynamic Behavior and Stability of
Closed-Loop Control Systems” in the main
textbook “Process Dynamics and Control 4th Ed
(2017, Wiley)
Chapter 6 – Part 4
PID Tuning
CH6-Part 4
Controller Tuning: A Motivational Example
Fig. 12.1. Unit-step disturbance responses for the candidate controllers
(FOPTD Model: K = 1, θ  4, τ  20).
138
PID Controller Design, Tuning, and
Troubleshooting
CH6-Part 4
Performance Criteria For Closed-Loop Systems
• The function of a feedback control system is to ensure that
the closed loop system has desirable dynamic and steadystate response characteristics.
• Ideally, we would like the closed-loop system to satisfy the
following performance criteria:
1. The closed-loop system must be stable.
2. The effects of disturbances are minimized, providing
good disturbance rejection.
3. Rapid, smooth responses to set-point changes are
obtained, that is, good set-point tracking.
139
4. Steady-state error (offset) is eliminated.
CH6-Part 4
5. Excessive control action is avoided.
6. The control system is robust, that is, insensitive to
changes in process conditions and to inaccuracies in the
process model.
PID controller settings can be determined by a number
of alternative techniques:
1. Direct Synthesis (DS) method
2. Internal Model Control (IMC) method
3. Controller tuning relations
4. Frequency response techniques
5. Computer simulation
6. On-line tuning after the control system is installed.
140
Direct Synthesis Method
CH6-Part 4
• In the Direct Synthesis (DS) method, the controller design is
based on a process model and a desired closed-loop transfer
function.
• The latter is usually specified for set-point changes, but
responses to disturbances can also be utilized (Chen and
Seborg, 2002).
• Although these feedback controllers do not always have a PID
structure, the DS method does produce PI or PID controllers
for common process models.
• As a starting point for the analysis, consider the block diagram
of a feedback control system in Figure 12.2. The closed-loop
transfer function for set-point changes was derived in Section
11.2:
K mGcGvG p
Y

(12-1)
Ysp 1  GcGvG pGm
141
CH6-Part 4
Fig. 12.2. Block diagram for a standard feedback control system.
142
For simplicity, let G
Eq. 12-1 reduces to
GvG pGm and assume that Gm = Km. Then
CH6-Part 4
GcG
Y

Ysp 1  GcG
(12-2)
Rearranging and solving for Gc gives an expression for the
feedback controller:
1  Y / Ysp 
Gc  
(12-3a)


G  1  Y / Ysp 
• Equation 12-3a cannot be used for controller design because the
closed-loop transfer function Y/Ysp is not known a priori.
• Also, it is useful to distinguish between the actual process G
and the model, G , that provides an approximation of the
process behavior.
• A practical design equation can be derived by replacing the
unknown G by G, and Y/Ysp by a desired closed-loop transfer
function, (Y/Ysp)d:
143
CH6-Part 4




1  Y / Ysp d 
Gc 
(12-3b)
G 1  Y / Ysp 
d

• The specification of (Y/Ysp)d is the key design decision and will
be considered later in this section.


• Note that the controller transfer function in (12-3b) contains
the inverse of the process model owing to the 1/ G term.
• This feature is a distinguishing characteristic of model-based
control.
Desired Closed-Loop Transfer Function
For processes without time delays, the first-order model in
Eq. 12-4 is a reasonable choice,
 Y 
1
(12-4)

 
 Ysp d  c s  1
144
• The model has a settling time of ~ 4τc, as shown in
Section 5. 2.
CH6-Part 4
• Because the steady-state gain is one, no offset occurs for setpoint changes.
• By substituting (12-4) into (12-3b) and solving for Gc, the
controller design equation becomes:
Gc 
1 1
G τc s
(12-5)
• The 1/ τc s term provides integral control action and thus
eliminates offset.
• Design parameter τc provides a convenient controller tuning
parameter that can be used to make the controller more
aggressive (small τc ) or less aggressive (large τc).
145
CH6-Part 4
• If the process transfer function contains a known time delay θ ,
a reasonable choice for the desired closed-loop transfer
function is:
 Y

 Ysp

e  θs
 
d τc s  1
(12-6)
• The time-delay term in (12-6) is essential because it is
physically impossible for the controlled variable to respond to
a set-point change at t = 0, before t = θ .
• If the time delay is unknown, θ must be replaced by an
estimate.
• Combining Eqs. 12-6 and 12-3b gives:
1
e  θs
Gc 
G τ c s  1  e  θs
(12-7)
146
• Although this controller is not in a standard PID form, it is
physically realizable.
CH6-Part 4
• Next, we show that the design equation in Eq. 12-7 can be used
to derive PID controllers for simple process models.
• The following derivation is based on approximating the timedelay term in the denominator of (12-7) with a truncated Taylor
series expansion:
eθs  1  θs
(12-8)
Substituting (12-8) into the denominator of Eq. 12-7 and
rearranging gives
Gc 
1
eθs
G  τc  θ 
s
(12-9)
Note that this controller also contains integral control action.
147
First-Order-plus-Time-Delay (FOPTD) Model
Consider the standard FOPTD model,
CH6-Part 4
Keθs
G s 
τs  1
(12-10)
Substituting Eq. 12-10 into Eq. 12-9 and rearranging gives a PI
controller, Gc  K c 1  1/ τ I s  ,with the following controller
settings:
1 τ
Kc 
,
τI  τ
(12-11)
K θ  τc
Second-Order-plus-Time-Delay (SOPTD) Model
Consider a SOPTD model,
Keθs
G s 
 τ1s  1 τ2 s  1
(12-12)
148
Substitution into Eq. 12-9 and rearrangement gives a PID
controller in parallel form,
CH6-Part 4


1
Gc  K c 1 
 τDs 
 τI s

(12-13)
where:
1 τ1  τ 2
Kc 
,
K τc  
τ I  τ1  τ 2 ,
τ1τ 2
τD 
τ1  τ 2
(12-14)
Example 12.1
Use the DS design method to calculate PID controller settings for
the process:
2e s
G
10s  1 5s  1
149
CH6-Part 4
Consider three values of the desired closed-loop time constant:
 c  1, 3, and 10. Evaluate the controllers for unit step changes in
both the set point and the disturbance, assuming that Gd = G.
Repeat the evaluation for two cases:
a. The process model is perfect ( G = G).
b. The model gain is K = 0.9, instead of the actual value, K = 2.
Thus,
0.9e s
G
10s  1 5s  1
The controller settings for this example are:


K c  K  0.9 
Kc K  2
τI
τD
τc  1
3.75
8.33
15
3.33
τc  3
1.88
4.17
15
3.33
 c  10
0.682
1.51
15
3.33
150
CH6-Part 4
The values of Kc decrease as τc increases, but the values of τ I
and τ D do not change, as indicated by Eq. 12-14.
Figure 12.3 Simulation results for Example 12.1 (a): correct
model gain.
151
CH6-Part 4
Fig. 12.4 Simulation results for Example 12.1 (b): incorrect
model gain.
152
Internal Model Control (IMC)
CH6-Part 4
• A more comprehensive model-based design method, Internal
Model Control (IMC), was developed by Morari and
coworkers (Garcia and Morari, 1982; Rivera et al., 1986).
• The IMC method, like the DS method, is based on an assumed
process model and leads to analytical expressions for the
controller settings.
• These two design methods are closely related and produce
identical controllers if the design parameters are specified in a
consistent manner.
153
Controller Tuning Relations
CH6-Part 4
In the last section, we have seen that model-based design
methods such as DS and IMC produce PI or PID controllers for
certain classes of process models.
IMC Tuning Relations
The IMC method can be used to derive PID controller settings
for a variety of transfer function models.
154
CH6-Part 4
C is the desired closed-loop time constant
See the text for the rest of this table.
155
156
CH6-Part 4
CH6-Part 4
Method 1: Integrator Approximation
Kes
K * es
Approximate G ( s ) 
by G ( s ) 
s  1
s
where K * K / .
• Then can use the IMC tuning rules (Rule M or N)
to specify the controller settings.
157
Method 2. Limit the Value of I
CH6-Part 4
• For lag-dominant models, the standard IMC controllers for firstorder and second-order models provide sluggish disturbance
responses because τ I is very large.
• For example, controller G in Table 12.1 has τ I  τ where τ is
very large.
• As a remedy, Skogestad (2003) has proposed limiting the value
of τ I :
τ I  min τ1,4  τc  θ 
(12-34)
where 1 is the largest time constant (if there are two).
158
Example 12.4
CH6-Part 4
Consider a lag-dominant model with θ / τ  0.01:
100  s
G s 
e
100 s  1
Design four PI controllers:
a) IMC  τ c  1
b) IMC  τc  2  based on the integrator approximation
c) IMC  τ c  1 with Skogestad’s modification (Eq. 12-34)
d) Direct Synthesis method for disturbance rejection (Chen and
Seborg, 2002): The controller settings are Kc = 0.551 and
τ I  4.91.
159
Evaluate the four controllers by comparing their performance for
unit step changes in both set point and disturbance. Assume that
the model is perfect and that Gd(s) = G(s).
CH6-Part 4
Solution
The PI controller settings are:
Controller
Kc
(a) IMC
(b) Integrator approximation
0.5
0.556
(c) Skogestad
(d) DS-d
0.5
0.551
I
100
5
8
4.91
160
CH6-Part 4
Figure 12.8. Comparison
of set-point responses
(top) and disturbance
responses (bottom) for
Example 12.4. The
responses for the Chen
and Seborg and integrator
approximation methods
are essentially identical.
161
CH6-Part 4
Tuning Relations Based on Integral
Error Criteria
• Controller tuning relations have been developed that optimize
the closed-loop response for a simple process model and a
specified disturbance or set-point change.
• The optimum settings minimize an integral error criterion.
• Three popular integral error criteria are:
1. Integral of the absolute value of the error (IAE)

IAE   e  t  dt
(12-35)
0
where the error signal e(t) is the difference between the set
point and the measurement.
162
CH6-Part 4
4a
Figure 12.9. Graphical
interpretation of IAE.
The shaded area is the
IAE value.
163
2. Integral of the squared error (ISE)

ISE   e  t   dt
2
(12-36)
0
CH6-Part 4
3. Integral of the time-weighted absolute error (ITAE)

ITAE   t e  t  dt
(12-37)
0
164
165
CH6-Part 4
CH6-Part 4
Comparison of Controller Design and
Tuning Relations
Although the design and tuning relations of the previous sections
are based on different performance criteria, several general
conclusions can be drawn:
166
CH6-Part 4
1. The controller gain Kc should be inversely proportional to the
product of the other gains in the feedback loop (i.e., Kc  1/K
where K = KvKpKm).
2. Kc should decrease as θ / τ , the ratio of the time delay to the
dominant time constant, increases. In general, the quality of
control decreases as θ / τ increases owing to longer settling
times and larger maximum deviations from the set point.
3. Both τ I and τ D should increase as θ / τ increases. For many
controller tuning relations, the ratio, τ D / τ I, is between 0.1 and
0.3. As a rule of thumb, use τ D / τ I = 0.25 as a first guess.
4. When integral control action is added to a proportional-only
controller, Kc should be reduced. The further addition of
derivative action allows Kc to be increased to a value greater
than that for proportional-only control.
167
CH6-Part 4
Controllers With Two Degrees
of Freedom
• The specification of controller settings for a standard PID
controller typically requires a tradeoff between set-point
tracking and disturbance rejection.
• These strategies are referred to as controllers with two-degreesof-freedom.
• The first strategy is very simple. Set-point changes are
introduced gradually rather than as abrupt step changes.
• For example, the set point can be ramped as shown in Fig.
12.10 or “filtered” by passing it through a first-order transfer
function,
*
Ysp
1

(12-38)
Ysp τ f s  1
168
*
where Ysp
denotes the filtered set point that is used in the control
calculations.
CH6-Part 4
• The filter time constant, τ f determines how quickly the filtered
set point will attain the new value, as shown in Fig. 12.10.
Figure 12.10 Implementation of set-point changes.
169
• A second strategy for independently adjusting the set-point
response is based on a simple modification of the PID control
law in Chapter 8,
t

dym 
1
*
*
p  t   p  K c e  t    e t dt   D
(8-7)

I 0
dt 

where ym is the measured value of y and e is the error signal.
e ysp  y. m
CH6-Part 4
 
• The control law modification consists of multiplying the set
point in the proportional term by a set-point weighting factor, β :
p  t   p  K c βysp  t   ym  t  
1 t * *
dym 
 K c   e t dt  τ D

dt 
 τ I 0
 
(12-39)
The set-point weighting factor is bounded, 0 < ß < 1, and serves as
a convenient tuning factor.
170
CH6-Part 4
Figure 12.11 Influence of set-point weighting on closed-loop
responses for Example 12.6.
171
On-Line Controller Tuning
CH6-Part 4
1. Controller tuning inevitably involves a tradeoff between
performance and robustness.
2. Controller settings do not have to be precisely determined. In
general, a small change in a controller setting from its best
value (for example, ±10%) has little effect on closed-loop
responses.
3. For most plants, it is not feasible to manually tune each
controller. Tuning is usually done by a control specialist
(engineer or technician) or by a plant operator. Because each
person is typically responsible for 300 to 1000 control loops, it
is not feasible to tune every controller.
4. Diagnostic techniques for monitoring control system
performance are available.
172
CH6-Part 4
Continuous Cycling Method
Over 60 years ago, Ziegler and Nichols (1942) published a
classic paper that introduced the continuous cycling method for
controller tuning. It is based on the following trial-and-error
procedure:
Step 1. After the process has reached steady state (at least
approximately), eliminate the integral and derivative control
action by setting τ D to zero and τ I to the largest possible value.
Step 2. Set Kc equal to a small value (e.g., 0.5) and place the
controller in the automatic mode.
Step 3. Introduce a small, momentary set-point change so that the
controlled variable moves away from the set point. Gradually
increase Kc in small increments until continuous cycling occurs.
The term continuous cycling refers to a sustained oscillation with
a constant amplitude. The numerical value of Kc that produces
173
continuous cycling (for proportional-only control) is called the
ultimate gain, Kcu. The period of the corresponding sustained
oscillation is referred to as the ultimate period, Pu.
CH6-Part 4
Step 4. Calculate the PID controller settings using the ZieglerNichols (Z-N) tuning relations in Table 12.6.
Step 5. Evaluate the Z-N controller settings by introducing a
small set-point change and observing the closed-loop response.
Fine-tune the settings, if necessary.
The continuous cycling method, or a modified version of it, is
frequently recommended by control system vendors. Even so, the
continuous cycling method has several major disadvantages:
1. It can be quite time-consuming if several trials are required and
the process dynamics are slow. The long experimental tests
may result in reduced production or poor product quality.
174
CH6-Part 4
Pu
Figure 12.12 Experimental determination of the ultimate gain
Kcu.
175
176
CH6-Part 4
CH6-Part 4
2. In many applications, continuous cycling is objectionable
because the process is pushed to the stability limits.
3. This tuning procedure is not applicable to integrating or
open-loop unstable processes because their control loops
typically are unstable at both high and low values of Kc,
while being stable for intermediate values.
4. For first-order and second-order models without time delays,
the ultimate gain does not exist because the closed-loop
system is stable for all values of Kc, providing that its sign is
correct. However, in practice, it is unusual for a control loop
not to have an ultimate gain.
177
Relay Auto-Tuning
CH6-Part 4
• Åström and Hägglund (1984) have developed an attractive
alternative to the continuous cycling method.
• In the relay auto-tuning method, a simple experimental test is
used to determine Kcu and Pu.
• For this test, the feedback controller is temporarily replaced by
an on-off controller (or relay).
• After the control loop is closed, the controlled variable exhibits
a sustained oscillation that is characteristic of on-off control
(cf. Section 8.4). The operation of the relay auto-tuner includes
a dead band as shown in Fig. 12.14.
• The dead band is used to avoid frequent switching caused by
measurement noise.
178
CH6-Part 4
Figure 12.14 Auto-tuning using a relay controller.
179
• The relay auto-tuning method has several important advantages
compared to the continuous cycling method:
CH6-Part 4
1. Only a single experiment test is required instead of a
trial-and-error procedure.
2. The amplitude of the process output a can be restricted
by adjusting relay amplitude d.
3. The process is not forced to a stability limit.
4. The experimental test is easily automated using
commercial products.
180
Step Test Method
CH6-Part 4
• In their classic paper, Ziegler and Nichols (1942) proposed a
second on-line tuning technique based on a single step test.
The experimental procedure is quite simple.
• After the process has reached steady state (at least
approximately), the controller is placed in the manual mode.
• Then a small step change in the controller output (e.g., 3 to
5%) is introduced.
• The controller settings are based on the process reaction curve
(Section 7.2), the open-loop step response.
• Consequently, this on-line tuning technique is referred to as the
step test method or the process reaction curve method.
181
CH6-Part 4
Figure 12.15 Typical process reaction curves: (a) non-selfregulating process, (b) self-regulating process.
182
CH6-Part 4
An appropriate transfer function model can be obtained from the
step response by using the parameter estimation methods of
Chapter 7.
The chief advantage of the step test method is that only a single
experimental test is necessary. But the method does have four
disadvantages:
1. The experimental test is performed under open-loop conditions.
Thus, if a significant disturbance occurs during the test, no
corrective action is taken. Consequently, the process can be
upset, and the test results may be misleading.
2. For a nonlinear process, the test results can be sensitive to the
magnitude and direction of the step change. If the magnitude of
the step change is too large, process nonlinearities can
influence the result. But if the step magnitude is too small, the
step response may be difficult to distinguish from the usual
fluctuations due to noise and disturbances. The direction of the
step change (positive or negative) should be chosen so that 183
the controlled variable will not violate a constraint.
CH6-Part 4
3. The method is not applicable to open-loop unstable processes.
4. For analog controllers, the method tends to be sensitive to
controller calibration errors. By contrast, the continuous
cycling method is less sensitive to calibration errors in Kc
because it is adjusted during the experimental test.
Example 12.8
Consider the feedback control system for the stirred-tank blending
process shown in Fig. 11.1 and the following step test. The
controller was placed in manual, and then its output was suddenly
changed from 30% to 43%. The resulting process reaction curve is
shown in Fig. 12.16. Thus, after the step change occurred at t = 0,
the measured exit composition changed from 35% to 55%
(expressed as a percentage of the measurement span), which is
equivalent to the mole fraction changing from 0.10 to 0.30.
Determine an appropriate process model for G GIPGvG pGm .
184
CH6-Part 4
Figure 11.1 Composition control system for a stirred-tank
blending process.
185
CH6-Part 4
Figure 12.16 Process reaction curve for Example 12.8.
186
CH6-Part 4
Figure 12.17 Block diagram for Example 12.8.
187
CH6-Part 4
Solution
A block diagram for the closed-loop system is shown in Fig.
12.17. This block diagram is similar to Fig. 11.7, but the feedback
loop has been broken between the controller and the current-topressure (I/P) transducer. A first-order-plus-time-delay model can
be developed from the process reaction curve in Fig. 12.16 using
the graphical method of Section 7.2. The tangent line through the
inflection point intersects the horizontal lines for the initial and
final composition values at 1.07 min and 7.00 min, respectively.
The slope of the line is
 55  35% 
S 
 3.37% / min

 7.00  1.07 min 
and the normalized slope is
S 3.37% / min
R

 0.259min 1
p 43%  30%
188
The model parameters can be calculated as
CH6-Part 4
xm 55%  35%
K

 1.54  dimensionless 
p 43%  30%
θ  1.07 min
τ  7.00  1.07 min  5.93 min
The apparent time delay of 1.07 min is subtracted from the
intercept value of 7.00 min for the τ calculation.
The resulting empirical process model can be expressed as
X m  s 
1.54e1.07 s
 G s 
P  s 
5.93s  1
Example 12.5 in Section 12.3 provided a comparison of PI
controller settings for this model that were calculated using
different tuning relations.
189
CH6-Part 4