CHAPTER 12: Controller Design, Tuning,
& Troubleshooting
Anis Atikah Ahmad
anisatikah@unimap.edu.my
Steps in Process Control
Formulate the control
objectives.
Select controlled, manipulated,
and measured variables.
Choose the control strategy
and the control structure
Specify controller settings.
Outline


Performance Criteria for Closed-Loop Systems
PID Controller Design:

Model-Based Design Method



Controller Tuning Relations






IMC Tuning Relations
Tuning Relations Based on Integral Error Criteria
Miscellaneous Tuning Method
Controllers with Two Degrees o Freedom
On-Line Controller Tuning


Direct Synthesis Method
Internal Model Control (IMC)
Continuous Cycling Method
Guidelines for Common Control Loops
Troubleshooting Control Loops
Performance Criteria for Closed-Loop Systems
Which of
the
following
provides the
best
response????
Unit-step disturbance responses of FOPTD model & PI controller
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
steady-state response characteristics.
Ideally, we would like the closed-loop system to satisfy
the following performance criteria:
The closed-loop system must be stable.
The effects of disturbances are minimized,
providing good disturbance rejection
Rapid, smooth responses to set-point changes are
obtained, that is, good set-point tracking.
Performance Criteria for Closed-Loop
Systems
Steady-state error (offset) is eliminated.
Excessive control action is avoided
The control system is robust, that is, insensitive to
changes in process conditions and to inaccuracies
in the process model.
PID Controller Settings
PID controller settings can be determined by a number of
alternative techniques:
Direct
Synthesis
(DS)
method
Internal
Model
Control
(IMC)
method
Controller
tuning
relations
Frequency
response
techniques
Computer
simulation
On-line
tuning after
the control
system is
installed.
1. Direct Synthesis (DS) method
In the Direct Synthesis (DS) method, the controller design is based on
a process model and a desired closed-loop transfer function.
Consider the block diagram of a feedback control system in Figure
12.2.
The closed-loop transfer function for set-point changes is:
K mGc Gv G p
Y

Ysp 1  Gc Gv G p Gm
Eq. 12-1
1. Direct Synthesis (DS) method [cont.]

For simplicity, let G  GvG pGm and assume that Gm = Km.
Then Eq. 12-1 reduces to:
GcG
Y

Ysp 1  GcG
Eq. 12-2
Rearranging and solving for Gc gives an expression for the feedback
controller:
Gc 
Y Ysp
G 1  Y Ysp 
Eq. 12-3a
1. Direct Synthesis (DS) method [cont.]
Equation 12-3a cannot be used for controller design because the
closed-loop transfer function Y/Ysp is not known.
Also, it is useful to distinguish between the actual process G and the
model, that provides an approximation of the process behavior.
A practical design equation can be derived by replacing the
unknown G by , and Y/Ysp by a desired closed-loop transfer function,
~
(Y/Ysp)d:
G
Gc
Y Y 
 ~
G 1  Y Y 
sp d
sp d
Eq. 12-3b
1. Direct Synthesis (DS) method [cont.]
Note that the controller transfer function in (12-3b) contains the
inverse of the process model.
Gc
Y Y 
 ~
G 1  Y Y 
sp d
Eq. 12-3b
sp d
For processes without time delays, the first-order model in Eq. 12-4
is a reasonable choice,
Y 
   1
 Y   s 1
c
 sp  d
Eq. 12-4
Where τc is the desired closed loop time constant.
Because the steady-state gain is one, no offset occurs for set-point
changes.
1. Direct Synthesis (DS) method [cont.]
By substituting (12-4) into (12-3b) :
Y 
   1
 Y   s 1
c
 sp  d
Gc
Y Y 
 ~
G 1  Y Y 
sp d
sp d
Solving for Gc , the controller design equation becomes:
1 1
Gc  ~
G  cs
Eq. 12-5
The term 1  c s provides integral control action and thus eliminates
offset.
Design parameter  c provides a convenient controller tuning parameter
that can be used to make the controller more aggressive (small c) or less
aggressive (large  c ).
Direct Synthesis (DS) method [cont.]
If the process transfer function contains a known time delay θ, a
reasonable choice for the desired closed-loop transfer function is:
s
Y 
e
  
 Y   s 1
c
 sp  d
Eq. 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.
Combining Eqs. 12-6 and 12-3b :
s
Y 
e
  
 Y   s 1
c
 sp  d
Gc
Y Y 
 ~
G 1  Y Y 
sp d
sp d
Direct Synthesis (DS) method [cont.]
Combining Eqs. 12-6 and 12-3b
s
Y 
e
  
 Y   s 1
c
 sp  d
Gc
Y Y 
 ~
G 1  Y Y 
sp d
sp d
Gives:
1
e s
Gc  ~
s

s

1

e
G c
Eq. 12-7
The following derivation is based on approximating the time-delay
term in the denominator of (12-7) with a truncated Taylor series
expansion:
e s  1  s
Eq. 12-8
1. Direct Synthesis (DS) method [cont.]
Substituting (12-8) into the denominator of Eq. 12-7
e s  1  s
1
e s
Gc  ~
G  c s  1  e s
and rearranging gives
1 e s
Gc  ~
G  c   s
Eq. 12-9
Note that this controller also contains integral control action.
1. Direct Synthesis (DS) method [cont.]
1. First-Order-plus-Time-Delay (FOPTD) Model
Consider the standard FOPTD model,
~ Ke s
G
s  1
Eq. 12-10
Substituting Eq. 12-10 into Eq. 12-9 and rearranging gives a PI
controller;
s
~ Ke
G
s  1
1 e s
Gc  ~
G  c   s

1 

Gc  K c 1 
 Is 
with the following controller settings:
1 
Kc 
K   c
I 
Eq. 12-11
Direct Synthesis (DS) method [cont.]
2. Second-Order-plus-Time-Delay (FOPTD) Model
Consider a second-order-plus-time-delay model,
~
G
Ke s
 1s  1 2 s  1
Eq. 12-12
Substitution into Eq. 12-9 and rearranging gives a PID controller;
~
G
Ke s
 1s  1 2 s  1
1 e s
Gc  ~
G  c   s


1
Gc  K c 1 
  D s 
 Is

Eq. 12-13
with the following controller settings:
1 1   2
Kc 
K c 
 I  1   2
 1 2
D 
1   2
Eq. 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
Consider three values of the desired closed-loop time constant: .
 c  1,3,10
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 incorrect, K = 0.9, instead of the actual
value, K = 2.
Example 12.1- Solution
Use the DS design method to calculate PID controller settings for
the for two cases:
~
G
2e s
10s  15s  1
Gc
s
1 e
Gc  ~
G  c   s

10s  15s  1

2e s
e s
 c   s
Comparing with standard PID controller;

10 s  15s  1
Gc 
2 c   s
Thus;
50s 2  15s  1

2 c   s
 I  15


1
Gc  K c 1 
  D s 
 Is

  I D s 2   I s  1 
comparing

 K c 
Is


 D  50 15  3.33
K c  15 2 c  1
Example 12.1- Solution
The controller settings are as follows:
 I  15
 D  50 15  3.33
K c  15 2 c  1
(a)For K =2
K c  15 0.9 c  1
(b)For K =0.9
The values of Kc decrease as τc increases, but the values of τI and τD and do not
change
Example 12.1- Solution
~
~
K
2
Simulation results for (a) (G = G),
Increasing τc
As τc increases, the
responses become
more sluggish
Example 12.1- Solution
~
Simulation results for (b) ( K = 0.9).
Increasing τc
As τc increases, the
responses become
more sluggish
Example 12.1- Solution
Which one is better???WHY?
~
K 2
1 1   2
Kc 
K c 
~
K  0.9
2. Internal Model Control (IMC)
The Internal Model Control (IMC), similar to DS method, is based on an assumed
process model and leads to analytical expressions for the controller settings.
The IMC method is based on the simplified block diagram shown in Fig. 12.6b.
~
The model response Y is subtracted
from the actual response Y, and the
~
difference, Y  Y is used as the
input signal to the IMC controller,
Fig. 12.6
Internal Model Control (IMC)
The two block diagrams are identical if controllers Gc and Gc* satisfy the relation:
Gc*
Gc 
*~
1  Gc G
Eq. 12.16
Any IMC controller is
equivalent to a standard
feedback controller Gc,
and vice versa.
Fig. 12.6
2. Internal Model Control (IMC)
The following closed-loop relation for IMC can be derived as follows:
~
GG
1 G G
Y
~ Ysp 
~ D
*
*
1  Gc G  G
1  Gc G  G
*
c



For the special case of a perfect model,


~
Y  G GYsp  1  G G D
*
c
*
c
*
c
~
G G

Eq. 12.17
Eq. 12.17 reduces to
Eq. 12.18
2. Internal Model Control (IMC)
The IMC controller is designed in two steps:
1.
The process model is factored as
~ ~ ~
G  G G
where
2.
~
G
Eq. 12.19
contains any time delay and right half plane zeros.
The controller is specified as:
1
G  ~ f
G
*
c
Eq. 12.20
Where f is a low-pass filter with steady state gain of one and:
1
f 
 c s  1r
Eq. 12.21
2. Internal Model Control (IMC)
For an ideal case where the process is perfect
~
(G  G ), substituting the IMC
1
controller transfer function G  ~ f in closed loop relation,
G
~
*
*
*
c


Y  Gc GYsp  1  Gc G D


~
~
Y  G fYsp  1  fG D
gives
Eq. 12.22
Thus, the closed-loop transfer function for set-point changes (D=0) is:
Y
~
 G f
Ysp
Eq. 12.23
Example 12.2
Use the IMC design method to design a FOPTD model. Assume that f is
specified by eq 12.21 with r=1, and consider 1/1 Padé approximation for the
time delay term
FOPTD Model:
~ Ke s
G
s  1
s
According to 1/1 Padé approximation, e 
Substituting into FOPTD Model;
  
K 1  s 
~
2 

G
  
1  s s  1
 2 
1
1

2

2
s
Eq. 12.24a
s
Eq. 12.25
Example 12.2 cont.
Using 2 steps in IMC Controller design;
STEP 1:
~ ~ ~
Factor this model as G  G G
  
K 1  s 
~
 2 
Model: G 
  
1  s s  1
 2 

~
G  1  s
2
~
G
K
~
Thus, G
  ~ 
G   
1  s s  1
 2 
contains any time
delay and right half
plane zeros
Eq. 12.26
Eq. 12.27
Example 12.2 cont.
Using 2 steps in IMC Controller design;
STEP 2: Specify the controller into
Thus, substituting
Thus,
~
G 
1
G  ~ f
G
*
c
K
  
1  s s  1
 2 
  
1  s s  1
2 

*
Gc 
K  c s  1
and
1
f 
r
 c s  1
,
Eq. 12.28
Example 12.2 cont.
Using 2 steps in IMC Controller design;
STEP 2: Specify the controller into
Thus, substituting
Thus,
~
G 
1
G  ~ f
G
*
c
K
  
1  s s  1
 2 
  
1  s s  1
2 

*
Gc 
K  c s  1
and
1
f 
r
 c s  1
,
Eq. 12.28
Example 12.2 cont.
The equivalent controller Gc can be obtained from eq 12.16;
  
1  s s  1
2 
Gc  


K  c   s
2

Gc*
Gc 
*~
1  Gc G
Eq. 12.29
And rearranged into PID controller;
 
2   1
1  
Kc 
K c 
2   1
 
I 

2


D 
 
2   1
 
Eq. 12.30
*Type of controller (PI or PID) depends on time-delay approximation. Repeating this
derivation for Taylor series approximation gives a standard PI controller.
3. Controller Tuning Relations
3.1. IMC Tuning Relations
Table 12.1
3. Controller Tuning Relations
3.1. IMC Tuning Relations
Lag-dominant models (θ/τ<<1)
•
•
First- or second-order models with relatively small time delays (θ/τ<<1 )
are referred to as lag-dominant models.
The IMC and DS methods provide satisfactory set-point responses, but
very slow disturbance responses, because the value of τI is very large.
Fortunately, this problem can be solved in three different ways.
1. Approximate the lag-dominant model by integrator-plus-time delay model.
K *e s
G s  
s

K K 
*
Then apply IMC tuning relation in Table 12.1. (Case M or N)
3. Controller Tuning Relations
3.1. IMC Tuning Relations
Lag-dominant models (θ/τ<<1)
2. Limit the value of τI
For lag-dominant models, the standard IMC controllers for first-order and
second-order models provide sluggish disturbance responses because τI is
very large.
As a remedy, Skogestad (2003) has proposed limiting the value of :
 I  min  1 ,4 c   
3. Controller Tuning Relations
3.1. IMC Tuning Relations
Lag-dominant models (θ/τ<<1)
3. Design the controller for disturbance rejection, rather than set-point tracking.
For example, develop an extension of the DS approach based on closed-loop
transfer function for disturbance.
(Y/D)d, rather than (Y/Ysp)d
Example 12.4
Consider a lag-dominant model withθ/τ =0.01:
~
G s  
100
es
100 s  1
Design four PI controller;
(a)IMC (τc=1)
(b)IMC (τc=2) based on the integrator approximation
(c)IMC (τc=1) with Skogestad’s modification
(d)Direct synthesis method for disturbance rejection. The
controller settings are Kc=0.551 and τI=4.91
Evaluate the four controllers by comparing their performance for unit step
changes in both set point and disturbance.
Example 12.4-solution
The PI controller settings are:
IMC
Integrator
approximation
Skogestad
DS (disturbance)
Kc
τI
0.5
100
0.556
5
0.5
8
0.551
4.91
Example 12.4-solution
Set-point response
IMC controller provides an excellent
set-point response, while the other
three controllers have significant
overshoots and longer settling
times.
However, the IMC controller produces
an unacceptably slow disturbance
response owing to its large τI value.
Disturbance
response
3. Controller Tuning Relations
3.2 Tuning Relations Based on Integral Error Criteria
Controller tuning relations have been develop that optimize the closedloop response for a simple process model & a specified disturbance or
set-point change. The optimum settings minimize an integral error criterion.
Three popular integral error criteria are:
a. Integral of the absolute value of the error (IAE)

IAE   et dt
0
a. Integral of squared error (ISE)

IAE   et  dt
2
0
a. Integral of the time-weighted absolute error (ITAE)
IAE   t et dt
0
3. Controller Tuning Relations
3.2 Tuning Relations Based on Integral Error Criteria
IAE
value
Graphical interpretation of IAE
3. Controller Tuning Relations
3.2 Tuning Relations Based on Integral Error Criteria
3. Controller Tuning Relations
3. 3 Miscellaneous Tuning Relations
•
•
Hägglund and Åström tuning relations
G(s)
Kc
Ke s
s
0.35
K
Ke s
s  1
0.14 0.28

K
K
τI
7
0.33 
6.8
100  
Skogestad tuning relations
Condition
1  8
1  8
Kc
τI
τD
0.5 1   2 
K
1   2
 1 2
 1   2 
0.5 1  8   2 


K  8 
8   2
8 2
8   2
3. Controller Tuning Relations

Important Notes on Controller Design & Tuning
Relations
1. Kc  1/K where K = KvKpKm .
2.  / ↑  Kc ↓ (more time delay → poorer performance)
3.  / ↑  I and D ↑ ( D / I = 0.1 -0.3 )
4. When integral control action is added to a proportional-only controller,
 Kc ↓
The addition of derivative action  Kc ↑ (stability margin increased)
Controllers with two degree of freedom
The specification of controller settings for standard PID controller typically requires
a tradeoff between set-point tracking and disturbance rejection.
Fortunately, two simple strategies can be used to adjust the set-point and
disturbance responses independently. These strategies referred to as controllers
with two degrees of freedom.
First strategy: set point changes are introduced gradually rather than as abrupt
step changes . Eg: the set-point can be ramped or “filtered” by passing it through first
order transfer function:.
Ysp*
1

Ysp  f s  1
Controllers with two degree of freedom
Second strategy: adjusting the set-point response, based on a simple
modification of the PID control law:
1 t * *
dym 
pt   p  K c ysp t   ym t   K c   e t dt   D


dt
 I 0



 
Set-point weighting factor, 0<β<1
As β increases, the set point response become faster but exhibits more
overshoot. When β =1, the modified control law reduces to standard PID
control law.
Example 12.6

For the first-orderplus delay model of
Example 12.4, the PI
controller with DSd settings provided
the best
disturbance
response. Can setpoint weighting
significantly reduce
the overshoot
without adversely
affecting the settling
time?

Example 12.6
Set-point weighting with ß =0.5
provides a significant improvement,
because the overshoot is greatly
reduced and the settling time is
significantly decreased.
4. On-line Controller Tuning
4.1 Continuous Cycling Method
Step 1. After the process has reached steady state (at least approximately),
eliminate the integral and derivative control action by setting τD to zero and τI 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 continuous cycling (for proportionalonly control) is called the ultimate gain, Kcu.
- The period of the corresponding sustained oscillation is referred to as the
ultimate period, Pu.
4. On-line Controller Tuning
4.1 Continuous Cycling Method
Step 4. Calculate the PID controller settings using the Ziegler-Nichols (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.
4. On-line Controller Tuning
4.1 Continuous Cycling Method
Figure 12.12 Experimental determination of the ultimate gain Kcu.
4. On-line Controller Tuning
4.1 Continuous Cycling Method
Table 12.6. Controller settings based on the continuous cycling method
Guidelines for Common Control Loops
1. Flow Rate
- Flow control loops are characterized by fast response with essentially no time delay
- For flow control loops, PI control is generally used.
- The presence of recurring high-frequency noise discourages the use of
derivative action because it amplifies the noise.
2. Gas Pressure
- PI controllers are normally used with only a small amount of integral control
action (тI is large)
- Derivative action is normally not needed because the process response
times are usually quite small compared to those of other process operations.
Guidelines for Common Control Loops
3.Temperature
- General guidelines are difficult to state because of the wide variety of processes and
equipment involving heat transfer and different time scales.
-PID controller are commonly employed to provide more rapid responses than can be
obtained with PI controllers.
4. Liquid Level
- Standard P or PI controllers are commonly used for level control.
- Because offset is not important in averaging level control, it is reasonable to use a
proportional-only controller.
Troubleshooting Control Loops
If a control loop is not performing satisfactorily, then troubleshooting is
necessary to identify the source of the problem.
Based on experience in the chemical industry, Buckley (1973) has observed
that a control loop that once operated satisfactorily can become either
unstable or excessively sluggish for a variety of reasons that include:
a. Changing process conditions, usually changes in throughput rate.
b. Sticking control valve stem
c. Plugged line in a pressure or differential pressure transmitter.
d. Fouled heat exchangers, especially reboilers for distillation columns.
e. Cavitating pumps (usually caused by a suction pressure that is too low).
Troubleshooting Control Loops
The starting point for troubleshooting is to obtain enough background information
to clearly define the problem. Many questions need to be answered:
1.What is the process being controlled?
2.What is the controlled variable?
3.What are the control objectives?
4.Are closed-loop response data available?
5.Is the controller in the manual or automatic mode? Is it reverse or direct acting?
6.If the process is cycling, what is the cycling frequency?
7. What control algorithm is used? What are the controller settings?
8. Is the process open-loop stable?
9. What additional documentation is available, such as control loop summary
sheets, piping and instrumentation diagrams, etc.?
After acquiring this background information, the next step is to check out
each component in the control loop.