1. Feedback Controllers
Chapter 7

2. Chapter 7 On-off Controllers Examples 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?
Chapter 7

3. On-Off Controllers Chapter 7 Synonyms:
“two-position” or “bang-bang” controllers.
e = error =
set point – measured variable
Chapter 7
Controller output has two possible values.

4. Chapter 7 Practical case (dead band) δ = tolerance
system never reaches steady-state

5. Chapter 7

6. Chapter 7 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 7

7. Chapter 7 Since signals are time varying, e(t) = Ysp(t) - Ym (t)
n.b. Watch units!!
For proportional control:
where,
p(t) = controller output
= bias value (adjustable)
Kc = controller gain (dimensionless, adjustable)
Chapter 7

8. Chapter 7
Standards (ISO/ISA)
3 – 15 psi ma
0 – 10 VDC

9. Chapter 7 Proportional Band, PB Reverse or Direct Acting Controller
Kc can be made positive or negative
Recall for proportional FB control: or
Direct-Acting (Kc < 0)
“output increases as input increases"
p(t) Ym(t)
Reverse-Acting (Kc > 0)
“output increases as input decreases"
Chapter 7

10. Chapter 7 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??

11. Chapter 7 Transfer Function for Proportional Control: Let
Then controller input/output relation can written as
Take Laplace transform of each side, or
INTEGRAL CONTROL ACTION
Synonyms: "reset", "floating control"
I  reset time (or integral time) - adjustable
Chapter 7

12. Chapter 7 Proportional-Integral (PI) Control
integral provides memory of e
most popular controller
Response to unit step change in e:
Chapter 7

13. Chapter 7 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.
Transfer function for PI control
Chapter 7

14. Chapter 7 ("repeats per minute") instead of I . For PI controllers,
Some controllers are calibrated in 1/I
("repeats per minute") instead of I .
For PI controllers,
is not adjustable.
Derivative Control Action
Ideal derivative action
Used to improve dynamic response of the
controlled variable
Derivative kick (use -dym/dt )
Use alone?
Chapter 7

15. Chapter 7

16. Chapter 7 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.
Next, we consider the three most common forms.
Chapter 7
Parallel Form of PID Control
The parallel form of the PID control algorithm (without a derivative filter) is given by

17. Chapter 7 The corresponding transfer function is:
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:
Chapter 7

18. Features of PID Controllers
Expanded Form of PID Control
In addition to the well-known series and parallel forms, the expanded form of PID control in Eq is sometimes used:
Features of PID Controllers
Chapter 7
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.

19. Chapter 7

20. Chapter 7 Automatic and Manual Control Modes 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)
Chapter 7

21. Chapter 7 Digital PID Controller finite difference approximation
where,
= the sampling period (the time between
successive samples of the controlled variable)
= controller output at the nth sampling
instant, n=1,2,…
= error at the nth sampling unit
velocity form - see Equation (8-19)
(pn)- incremental change
Chapter 7

22. Chapter 7

23. Chapter 7 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.
Chapter 7

24. Chapter 7

25. Chapter 7
integral action ~

27. 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 7

28. Chapter 7 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 7

29. Chapter 7 Example 3: Liquid Level Control
Control valves are air-to-open
Level transmitters are direct acting
Chapter 7

30. Chapter 7 Question: 1. Type of controller action? Select Kc so that
air-to-open valve: sign of Kv?
sign of process gain?
Chapter 7

31. Chapter 7
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