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Chapter 8
Feedback Controllers
Chapter 8
Error Signal
e t   R t   B t 
where
R  t  = set point (3-15 psig; 4-20mA; 1-5V)
B  t  = measured value of controlled variable
(or equivalent signal from transmitter)
(3-15 psig; 4-20mA; 1-5V)
Define
e  t   e  t   0; R  t   R  t   R ; B  t   B  t   B
0  R  B  e  t   R  t   B  t 
Take Laplace Transform
E   s   R  s   B  s 
Proportional Control
p (t )  p  K c e(t )  p  K c  R (t )  B (t ) 
where
p (t )  controller output
p = bias value (adjustable)
K c  controller gain (dimensionless, adjustable)
e(t )  error signal
Transfer Function
p  t   p  t   p 
P  s 
 Kc
  Gc  s  
e  t   e  t   0 
E  s 
Chapter 8
Reverse or Direct Acting
Controller
• Direct-Acting (Kc < 0): “output increases
as input increases"
• Reverse-Acting (Kc > 0): “output increases
as input decreases"
Chapter 8
Proportional Band (PB)
PB is the error (% of the range of controlled
variable) required to move the output from
its lowest to its highest value.
100%
PB 
Kc
Chapter 8
• Example 2: Flow Control Loop
Assume FT is direct-acting.
1.) Air-to-open (fail close) valve ==> ?
2.) Air-to-close (fail open) valve ==> ?
• Consequences of wrong controller action??
Chapter 9
Chapter 8
Example 3: Liquid Level Control
• Control valves are air-to-open
• Level transmitters are direct acting
Question:
Type of controller action?
p  10 psig
in order to make qo  170 gpm
p  9 psig
qi  qo at steady state
p t   p  Kce t 
Kce     1
where K c  0
e   R  B   0
INTEGRAL CONTROL ACTION
Chapter 8
p (t )  p 
1
I
t
 e( )d
0
P( s )
1

E ( s )  I s
 I : reset time (or integral time) - adjustable
(1) If e(t )  0, then p(t ) varies continuously with time.
(2) If e(t )  0 for t   t*, then
p  t    ps  p 
1
I
t*
 e( )d  constant
0
Proportional-Integral (PI)
Controller
t


1
p (t )  p  K c e(t )   e(t ) dt 
I 0


where p is not adjustable
Transfer Function

P( s )
1 
 K c 1 


E ( s)
 Is 
Reset Time
Reset time is the time that the integral mode repeats the
action of proportional mode.
t I
t
t*
Example:
Heat Exchanger Control Loop
Reset Windup
e t   0
t


1
p(t )  p  K c e(t )   e(t )dt 
I 0


vp  t   vp  K v  p  t   p 
p t 
t
0
to
ts
Reset Windup
e t   0
e t   0
t


1
p(t )  p  K c e(t )   e(t )dt 
I 0


vp  t   vp  K v  p  t   p 
p t 
t
t 
t 
Anticipatory or Derivative Control
Action
de
p (t )  p   D
dt
Used to improve dynamic response of the
controlled variable
Proportional-Integral-Derivative
(PID) Control
Now we consider the combination of the
proportional, integral, and derivative control
modes as a PID controller.
• Many variations of PID control are used in
practice (see Table 8.1, page 194)
• Next, we consider the three most common
forms.
Parallel Form of PID Control
The parallel form of the PID control algorithm
(without a derivative filter) is given by

1
p  t   p  K c e  t  
τI

de  t  
0 e t * dt *  τ D dt 
t
Transfer Function
P  s 


1
 K c 1 
 τDs
E  s 
 τI s

Effects of Anticipatory
(Derivative) Control Action
Drawbacks of Anticipatory
(Derivative) Control Action
Chapter 8
Parallel-Form PID Controller with
Derivative Filter
P  s 

τDs 
1
 K c 1 


E  s 
 τ I s  τ D s  1
where
0.05    0.2 (usually 0.1)
Derivative and Proportional Kicks
One disadvantage of the previous PID
controllers is that a sudden change in set
point (and hence the error, e) will cause the
derivative term momentarily to become
very large and thus provide a derivative kick
to the final control element.
R t 
p t 
Elimination of Derivative and
Proportional Kicks in ParallelForm Controllers

1
p  t   p  K c e  t  
I

or
dB  

0 e t * dt *    D dt 

1
p t   p  Kc  B t  
I

t
dB  

0 e  t * dt *    D dt 
t
Series Form of PID Control
Historically, it was convenient to construct early analog controllers
(both electronic and pneumatic) so that a PI element and a PD element
operated in series. Commercial versions of the series-form controller
have a derivative filter that is applied to either the derivative term,
as in Eq. 8-12, or to the PD term, as in Eq. 8-15:
P  s 
 τ I s  1  τ D s  1 
 Kc 


E  s 
τ
s
ατ
s

1
 I  D

Elimination of Derivative Kick in
Series-Form Controllers
R  s 
B s
B  s 
  s 1 
 R  s    D s  1 B  s   K c  I
  P  s 

s
 I 
versus
  s 1 
 R  s   B  s   K c  D s  1  I
  P  s 
 Is 
Expanded (Non-interacting) Form
of PID Control
In addition to the well-known series and parallel forms,
the expanded form of PID control in Eq. 8-16 is sometimes
used:
t
de  t 
0
dt
p  t   p  K c e  t   K I  e  t * dt *  K D
Chapter 8
Chapter 8
Typical Response of Feedback Control Systems
Consider response of a controlled system after a
sustained disturbance occurs (e.g., step change in
disturbance variable)
Chapter 8
Chapter 8
Automatic and Manual Control Modes
Chapter 8
•
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)
Digital PID Controller


t n 1
D
p n  p  K c e n   e k  e n  e n 1 
I k 1
t


Chapter 8
where,
t = the sampling period (the time between
successive samples of the controlled variable)
p n = controller output at the nth sampling
instant, n=1,2,…
en = error at the nth sampling unit
velocity form - see Equation (8-19)
(pd)- incremental change
Chapter 8
Controller Comparison
P
-Simplest controller to tune (Kc).
-Offset with sustained disturbance or set point
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
Chapter 8
Summary of the Characteristics of the Most
Commonly Used Controller Modes
1. Two Position:
Inexpensive.
Extremely simple.
2. Proportional:
Simple.
Inherently stable when properly tuned.
Easy to tune.
Experiences offset at steady state.
3. Proportional plus integral:
No offset.
Better dynamic response than reset alone.
Possibilities exist for instability due to lag
introduced.
Chapter 8
4. Proportional plus derivative:
Stable.
Less offset than proportional alone (use of
higher gain possible).
Reduces lags, i.e., more rapid response.
5. Proportional plus reset plus rate:
Most complex
Rapid response
No offset.
Difficult to tune.
Best control if properly tuned.
Chapter 8
On-off Controllers
•
•
•
•
Simple
Cheap
Used In residential heating and domestic refrigerators
Limited use in process control due to continuous
cycling of controlled variable  excessive wear
on control valve.
Example 1: Temperature control of jacketed vessel.
On-Off Controllers
Chapter 8
Synonyms:
“two-position” or “bang-bang” controllers.
Controller output has two possible values.
Chapter 8
Practical case (dead band)