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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.

ExamplesExamples•Batch process control (PLC = programmable logic controller)•Solenoid in home heating unit•Sprinkler systems•Cruise control?

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On-Off Controllers

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

Controller output has two possible values.

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e = error = set point – measured variable

Practical case (dead band)

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system never reaches steady-state

δ = tolerance

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

Proportional Control• Define an error signal, e, by e = Ysp – Ym

whereYsp = set pointYm = measured value of the controlled variable (or equivalent signal from transmitter)

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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)

p-p=p e(t)K+p=p(t) c

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p

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Standards (ISO/ISA)

3 – 15 psi4 - 20 ma0 – 10 VDC

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"

cK

%100PB

e(t)K+p=p(t) c

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)t(Y)t(YKp)t(p mspc

• Example 2: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??

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Transfer Function for Proportional Control:Let

Then controller input/output relation can written as

Take Laplace transform of each side,

or

INTEGRAL CONTROL ACTIONSynonyms: "reset", "floating control"

I reset time (or integral time) - adjustable

p-p(t)(t)p

e(t)K(t)p c

E(s)K(s)P c

cKE(s)

(s)P

s

1

E(s)

(s)P td)t(e

1p)t(p

I

t

0I

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t

0Ic td)t(e

1)t(eKp)t(p

Proportional-Integral (PI) Control

• Response to unit step change in e:

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integral provides memory of emost popular controller

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• 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.

s

11K

E(s)

(s)P

Ic

• Transfer function for PI control

Derivative Control Action Ideal derivative action

Used to improve dynamic response of the controlled variable Derivative kick (use -dym/dt ) Use alone?

Some controllers are calibrated in 1/I

("repeats per minute") instead of I .

p

dt

dep)t(p D

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For PI controllers, is not adjustable.

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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.

Parallel Form of PID Control

The parallel form of the PID control algorithm (without a derivative filter) is given by

0

1* * τ (8-13)

τ

tc D

I

de tp t p K e t e t dt

dt

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The corresponding transfer function is:

11 τ (8-14)

τc DI

P sK s

E s s

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:

τ 1 τ 1(8-15)

τ ατ 1I D

cI D

P s s sK

E s s s

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Expanded Form of PID Control

In addition to the well-known series and parallel forms, the expanded form of PID control in Eq. 8-16 is sometimes used:

0

* * (8-16)t

c I Dde t

p t p K e t K e t dt Kdt

Features of PID Controllers

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.

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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)

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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

1

11

nD

n c n k n nkI

DI

tp p K e e e e

t

np

ne

t

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Typical Response of Feedback Control SystemsConsider response of a controlled system after a sustained disturbance occurs (e.g., step change in disturbance variable); y > 0 is off-spec.

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integral action ~ /c IK

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.

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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.

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

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Question:Question:1. Type of controller action? Select Kc so that

0c v pK K K

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

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