Feedback Controllers

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

e = error = set point – measured variable Controllers output has two possible values

 Practical case (dead band)

δ = tolerance

system never reaches steady-state

Three Mode (PID) Controller  Proportional  Integral  Derivative

Proportional Control

 Define an error signal, e, by e= Ysp - B 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

p(t) = p + K ce(t) p = p - p where p(t)  controller output p  bias value (adjustable)

K c  controller gain (dimensionless, adjustable) Standards (ISO/ISA)

3 – 15 psi 4 - 20 ma 0 – 10 VDC

 Proportional Band, PB 100% PB  K c  Reverse or Direct Acting Controller

 K c can be made positive or negative  Recall for proportional FB control:

p(t) = p + K ce(t) or p(t)=p+K轾 Y (t)-B(t) c臌 sp

 Direct-Acting (K c < 0) “output increases as input increases" p(t) B(t)

 Reverse-Acting (K c > 0) “output increases as input decreases"

Example: 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 (fall open) valve ?  Consequences of wrong controller action??  Transfer Function for Proportional Control: Let p(t)  p(t) - p Then controller input/output relation can written as

p(t)  K ce(t) Take Laplace transform of each side,

P(s)  K cE(s) or P(s)  Kc E(s)

Integral Control Synonyms: "reset", "floating control"

1 t P(s) 1 p(t)  p   e(t)dt  I 0 E(s) Is

I  reset time (or integral time) - adjustable

Proportional-integral (PI) Control  1 t  p(t)  p  K c e(t)   e(t)dt integral provides memory of e  I 0  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  Transfer function for PI control P(s)  1   K c 1   E(s)  Is 1  Some controllers are calibrated in ("repeats per minute") Instead of I I  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 db  Derivative kick (use ) dt  Use alone?

PID CONTROLLER

 Ideal controller  1 t de p(t)  p  K c e(t)   e(t)dt  D   I 0 dt   Transfer function (ideal) P(s)  1   K c 1   D s E(s)  Is   Transfer function (actual) lead /lag units

P(s)  Is  1 Ds  1   K c    E(s)  Is  Ds  1  = 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 轾犏 Dt n-1 t D Finite difference approximation pn= p + K c犏 e n + e k +( e n - e n-1 ) t Dt 犏 142I k =1 43 1 442 4 43 臌 I D where t = the sampling period (the time between successive samples of the controlled variable)

pn = controller output at the nth sampling instant, n=1,2,…

en = error at the nth sampling unit velocity form - see Equation (8-19)

(pn ) - incremental change

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. Price Comparison (1979 Prices)

Controller Pneumatic Electronic P $ 840 $1470 PI $ 900 $1350 PID $1000 $1470

Digital PID (1994) < $1000. integral action ~ Kc/t I 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

Questions: 1. Type of controller action? Select Kc so that Kc K v K p > 0

(a) air-to-open valve: sign of Kv? (b) sign of process gain?