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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