Ch

apte

r 8

Feedback Controllers

Ch

apte

r 8

Error Signal

where

= set point (3-15 psig; 4-20mA; 1-5V)

= measured value of controlled variable

(or equivalent signal from transmitter)

(3-15 psig; 4-20mA; 1-5V)

Define

0;

e t R t B t

R t

B t

e t e t

;

0

Take Laplace Transform

R t R t R B t B t B

R B e t R t B t

E s R s B s

Proportional Control

( ) ( ) ( ) ( )

where

( ) controller output

= bias value (adjustable)

controller gain (dimensionless, adjustable)

( ) error signal

Transfer Function

0

c c

c

c c

p t p K e t p K R t B t

p t

p

K

e t

p t p t p P sG s K

e t e t E s

Ch

apte

r 8

Reverse or Direct Acting Controller

• Direct-Acting (Kc < 0): “output increases as input increases"

• Reverse-Acting (Kc > 0): “output increases as input decreases"

Ch

apte

r 8

Proportional Band (PB)

PB is the error (% of the range of controlled variable) required to move the output from its lowest to its highest value.

100%

c

PBK

• Example 2:Example 2: Flow Control Loop

Assume FT is direct-acting.

1.) Air-to-open (fail close) valve ==> ?2.) Air-to-close (fail open) valve ==> ?

• Consequences of wrong controller action??

Ch

apte

r 8

Ch

apte

r 9

Example 3:Example 3: Liquid Level Control• Control valves are air-to-open

• Level transmitters are direct acting

Ch

apte

r 8

Question: Question: Type of controller action?

9 psigp

10 psig

in order to make 170 gpmo

p

q

at steady statei oq q

where 0

1 0

c c

c

p t p K e t K

K e e R B

Ch

apte

r 8

INTEGRAL CONTROL ACTION

0

1( ) ( )

( ) 1

( )

: reset time (or integral time) - adjustable

(1) If ( ) 0, then ( ) varies continuously with time.

(2) If ( ) 0 for *, then

t

I

I

I

p t p e d

P s

E s s

e t p t

e t t t

p t

*

0

1( ) constant

t

sI

p p e d

Proportional-Integral (PI) Controller

0

1( ) ( ) ( )

where is not adjustable

Transfer Function

( ) 11

( )

t

cI

cI

p t p K e t e t dt

p

P sK

E s s

Reset TimeReset time is the time that the integral mode repeats the action of proportional mode.

It

*tt

Example: Heat Exchanger Control Loop

Reset Windup

p t

0

1( ) ( ) ( )

t

cI

v

p t p K e t e t dt

vp t vp K p t p

0e t

0 ot st

t

Reset Windup

p t

0

1( ) ( ) ( )

t

cI

v

p t p K e t e t dt

vp t vp K p t p

t

0e t

0e t

t t

Anticipatory or Derivative Control Action

( )

Used to improve dynamic response of the

controlled variable

D

dep t p

dt

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 (see Table 8.1, page 194)

• Next, we consider the three most common forms.

Parallel Form of PID Control

0

The parallel form of the PID control algorithm

(without a derivative filter) is given by

1* * τ

τ

Transfer Function

11 τ

τ

tc D

I

c DI

de tp t p K e t e t dt

dt

P sK s

E s s

Effects of Anticipatory (Derivative) Control Action

Drawbacks of Anticipatory (Derivative) Control Action

Ch

apte

r 8

Parallel-Form PID Controller with Derivative Filter

τ11

τ

where

0.05 0.2 (usually

τ

)

1

0.1

Dc

I D

P s sK

E s s s

Derivative and Proportional Kicks

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.

R t

p t

Elimination of Derivative and Proportional Kicks in Parallel-

Form Controllers

0

0

1* *

or

1* *

D

D

t

cI

t

cI

p t p K e t e t dt

p t p K e t d

dB

t

dt

dBB t

dt

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 c

ontroller

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:

τ 1 τ 1

τ ατ 1I D

cI D

P s s sK

E s s s

Elimination of Derivative Kick in Series-Form Controllers

11

versus

11

ID c

I

Ic D

I

sR s s B s K P s

s

sR s B s K s P s

s

B s

R s

B s

Expanded (Non-interacting) Form of PID Control

0

In addition to the well-known series and parallel forms,

the expanded form of PID control in Eq. 8-16 is sometimes

used:

* *t

c I Dde t

p t p K e t K e t dt Kdt

Ch

apte

r 8

Typical Response of Feedback Control SystemsConsider response of a controlled system after a sustained disturbance occurs (e.g., step change in disturbance variable)

Ch

apte

r 8

Ch

apte

r 8

Ch

apte

r 8

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-upplant shut-downemergencies

• Percentage of controllers "on manual” ?? (30% in 2001, Honeywell survey)

Ch

apte

r 8

Digital PID Controller

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)

(pd)- incremental change

1nn

D1n

1kk

Incn ee

te

teKpp

np

ne

t

Ch

apte

r 8

PID -Most complicated to tune (Kc, I, D) .-Better performance than PI-No offset-Derivative action may be affected by noise

PI -More complicated to tune (Kc, I) .-Better performance than P-No offset-Most popular FB controller

P -Simplest controller to tune (Kc).-Offset with sustained disturbance or set point change.

Controller Comparison

Ch

apte

r 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.

3. Proportional plus integral:No offset.Better dynamic response than reset alone.Possibilities exist for instability due to lag introduced.

Ch

apte

r 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 reset plus rate:

Most complex

Rapid response

No offset.

Difficult to tune.

Best control if properly tuned.

Ch

apte

r 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.

Example 1:Example 1: Temperature control of jacketed vessel.

Ch

apte

r 8

On-Off Controllers

Synonyms:“two-position” or “bang-bang” controllers.

Controller output has two possible values.

Ch

apte

r 8

Practical case (dead band)

Ch

apte

r 8