Feedback Controllers
Feedback Controllers
[pic]
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 system
- Sprinkler systems
- Cruise control?
Synonyms: "Two position" or "bang-bang" controllers
[pic]e = error = set point – measured variable
Controllers output has two possible values
Practical case (dead band)
δ = tolerance
system never reaches steady-state
[pic]
Three Mode (PID) Controller
Proportional
Integral
Derivative
Proportional Control
Define an error signal, e, by [pic]
where
Ysp = set point
B = measured value of the controlled variable (or equivalent signal from transmitter)
Since signals are time varying,
e(t) = Ysp(t) - B(t)
n.b. Watch units!!
For proportional control
[pic]
where
[pic]controller output
[pic]bias value (adjustable)
[pic]controller gain (dimensionless, adjustable)
[pic]
Standards (ISO/ISA)
3 – 15 psi
4 - 20 ma
0 – 10 VDC
Proportional Band, PB
[pic]
Reverse or Direct Acting Controller
13. [pic]can be made positive or negative
14. Recall for proportional FB control:
[pic]
or
[pic]
Direct-Acting [pic]
“output increases as input increases"
p(t) B(t)
Reverse-Acting [pic]
“output increases as input decreases"
Example: Flow Control Loop
[pic]
Assume FT is direct-acting. Select sign of Kc so that KcKv > 0.
1) Air-to-open (fail close) valve (?
2) Air-to-close (fall open) valve (?
[pic]
Consequences of wrong controller action??
Transfer Function for Proportional Control:
Let [pic]
Then controller input/output relation can written as
[pic]
Take Laplace transform of each side,
[pic]
or
[pic]
Integral Control
Synonyms: "reset", "floating control"
[pic]
[pic]reset time (or integral time) - adjustable
Proportional-integral (PI) Control
[pic] integral provides memory of e
most popular controller
Response to unit step change in e:
[pic]
Integral action eliminates steady-state error (i.e., offset)
Why?? [pic] is changing with time
Transfer function for PI control
[pic]
[pic]
Some controllers are calibrated in [pic] ("repeats per minute") Instead of [pic]
For PI controllers, [pic] is not adjustable.
Derivative Control Action
Ideal derivative action
[pic]
Used to improve dynamic response of the controlled variable
Derivative kick (use [pic])
Use alone?
PID CONTROLLER
Ideal controller
[pic]
Transfer function (ideal)
[pic]
Transfer function (actual)
[pic]
α = small number (0.05 to 0.20)
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” ??
Digital PID Controller
Finite difference approximation
where
[pic]= the sampling period (the time between successive samples of the controlled variable)
[pic]= controller output at the nth sampling instant, n=1,2,…
[pic]= error at the nth sampling unit
velocity form - see Equation (8-19)
[pic]- incremental change
[pic]
Typical Responses 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.
[pic]
Price Comparison (1979 Prices)
|Controller |Pneumatic |Electronic |
|P |$ 840 |$1470 |
|PI |$ 900 |$1350 |
|PID |$1000 |$1470 |
Digital PID (1994) < $1000.
[pic]
[pic]
integral action ~
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 reset:
No offset.
Better dynamic response than reset alone.
Possibilities exist for instability due to lag introduced.
4. Proportional plus rate:
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.
Example: Liquid Level Control
Control valves are air-to-open
Level transmitters are direct acting
[pic]
Questions:
1. Type of controller action?
Select Kc so that
a) air-to-open valve: sign of Kv?
b) sign of process gain?
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[pic]
[pic]
[pic]
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