Chapter 8
Feedback Controllers
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.
Examples
•Batch process control (PLC = programmable logic controller)
•Solenoid in home heating unit
•Sprinkler systems
•Cruise control?
On-Off Controllers
Chapter 8
Synonyms:
“two-position” or “bang-bang” controllers.
e = error =
set point – measured variable
Controller output has two possible values.
Chapter 8
Practical case (dead band)
δ = tolerance
system never reaches steady-state
Chapter 8
Chapter 8
Three Mode (PID) Controller
• Proportional
• Integral
• Derivative
Proportional Control
• Define an error signal, e, by e = Ysp – Ym
where
Ysp = set point
Ym = measured value of the controlled variable
(or equivalent signal from transmitter)
Chapter 8
Since signals are time varying,
e(t) = Ysp(t) - Ym (t)
n.b. Watch units!!
p = p - p
• For proportional control: p(t) = p + Kce(t)
where,
p(t) = controller output
p = bias value (adjustable)
Kc = controller gain (dimensionless, adjustable)
Chapter 8
Figures 8.4, 8.5
in Text
Standards (ISO/ISA)
3 – 15 psi
4 - 20 ma
0 – 10 VDC
 Proportional Band, PB
100%
PB 
Kc
Chapter 8
 Reverse or Direct Acting Controller
 Kc can be made positive or negative
 Recall for proportional FB control:
p(t) = p + K ce(t)
or

p( t )  p  K c Ysp ( t )  Ym ( t )

 Direct-Acting (Kc < 0)
“output increases as input increases"
p(t)
Ym(t)
 Reverse-Acting (Kc > 0)
“output increases as input decreases"
Chapter 8
• Example 2: Flow Control Loop
Assume FT is direct-acting. Select sign of Kc so
that KcKv > 0
1.) Air-to-open (fail close) valve ==> ?
2.) Air-to-close (fail open) valve ==> ?
• Consequences of wrong controller action??
Chapter 8
 Transfer Function for Proportional Control:
Let p(t)  p(t) - p
Then controller input/output relation can written as
p(t)  K c e(t)
Take Laplace transform of each side,
P(s)  K c E(s)
or
P(s)
 Kc
E(s)
INTEGRAL CONTROL ACTION
Synonyms: "reset", "floating control"
t
1
p( t )  p   e( t)dt
I 0
P(s) 1

E(s) Is
I  reset time (or integral time) - adjustable
Proportional-Integral (PI) Control
Chapter 8
t


1
p( t )  p  K c e( t )   e( t )dt 
I 0


integral provides memory of e
most popular controller
• Response to unit step change in e:
• Integral action eliminates steady-state error
(i.e., offset) Why??? e  0  p is changing with
time until e = 0, where p reaches steady state.

P(s)
1 
• Transfer function for PI control

 K c 1 
Chapter 8
E(s)

 Is 
 Some controllers are calibrated in 1/I
("repeats per minute") instead of I .
Chapter 8
 For PI controllers, p is not adjustable.
Derivative Control Action
 Ideal derivative action
de
p( t )  p   D
dt
 Used to improve dynamic response of the
controlled variable
 Derivative kick (use -dym/dt )
 Use alone?
Chapter 8
Proportional-Integral-Derivative (PID) Control
Now we consider the combination of the proportional, integral,
and derivative control modes as a PID controller.
Chapter 8
• Many variations of PID control are used in practice.
• 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
(8-13)
The corresponding transfer function is:
P  s 
(8-14)
Chapter 8


1
 Kc 1 
 τDs
E s
 τI s

P  s 
 τ I s  1  τ D s  1 
 Kc 


E s
τ
s
ατ
s

1
 I  D

(8-15)
Expanded 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
Chapter 8
p  t   p  Kc e  t   K I  e  t * dt *  K D
(8-16)
Features of PID Controllers
Elimination of Derivative and Proportional Kick
• 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.
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
Chapter 8
∆𝑡
𝑝𝑛 = 𝑝 + 𝐾𝑐 𝑒𝑛 +
𝜏𝐼
𝑛−1
𝑘=1
𝜏𝐷
𝑒𝑘 + (𝑒𝑛 − 𝑒𝑛−1 )
∆𝑡
I
D
finite difference approximation
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,…
e n = error at the nth sampling unit
velocity form - see Equation (8-19)
(pn)- incremental change
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); y > 0 is off-spec.
No control
(Kc=0)
Increasing KC
Chapter 8
y
0
Time
Figure 8.13 Proportional control: effect of Controller gain
y
Increasing D
0
Time
Figure 8.15 PID control: effect of derivative time
Increasing I
y
Increasing KC
y
Chapter 8
0
0
Time
(b)
Time
(a)
Figure 8.14 Proportional-integral control: (a) effect of integral time, (b) effect of controller gain
integral action ~
Kc /  I
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. (OK for level
control)
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 integral plus derivative:
Most complex
Rapid response
No offset.
Best control if properly tuned.
Chapter 8
Example 3: Liquid Level Control
• Control valves are air-to-open
• Level transmitters are direct acting
Chapter 8
Question:
1. Type of controller action? Select Kc so that
Kc Kv K p  0
(a) air-to-open valve: sign of Kv?
(b) sign of process gain?
Chapter 8
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