ERT 210 Process Control & dynamics Anis Atikah binti Ahmad CHAPTER 8 Feedback Controllers.

ERT 210Process Control & dynamics

Anis Atikah binti Ahmad

CHAPTER 8Feedback Controllers

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Control objective: to keep the tank exit composition, x, at the desired value (set point) by adjusting the flow rate, w2, via the control valve

Figure 8.1: Schematic diagram for a stirred-tank blending system

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Figure 8.2: Flow control system

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In feedback control, the objective is to reduce the error signal to zero where

(8-1)sp me t y t y t

and

error signal

set point

measured value of the controlled variable

(or equivalent signal from the sensor/transmitter)

sp

m

e t

y t

y t

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Basic Control ModesThree basic control modes:

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requires very intensive energy

costly

highly endothermic reaction

Derivative control

Proportional control

Integral control

Proportional ControlC

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For proportional control, the controller output is proportional to the error signal,

(8-2)cp t p K e t

where:

controller output

bias (steady-state) value

controller gain (usually dimensionless)c

p t

p

K

•At time t = 25 min, e(25) = 60–56 = 4•At time t = 40 min, e(40) = 60–62 = –2

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The key concepts behind proportional control are the following:

1. the controller gain (Kc) can be adjusted to make the controller output changes as sensitive as desired to deviations between set point and controlled variable;

2. the sign of Kc can be chosen to make the controller output increase (or decrease) as the error signal increases.

Some controllers have a proportional band setting instead of a controller gain. The proportional band PB (in %) is defined as

Large PB correspond to a small value of Kc and vice versa

%1001

cK

PB ≜

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In order to derive the transfer function for an ideal proportional controller (without saturation limits), define a deviation variable

as p t

Then Eq. 8-2 can be written as

(8-5)cp t K e t

The transfer function for proportional-only control:

(8-6)cP s

KE s

An inherent disadvantage of proportional-only control is that a steady-state error (or offset) occurs after a set-point change or a sustained disturbance.

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)48( )( )( -ptptp ≜

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For integral control action, the controller output depends on the integral of the error signal over time,

0

1* * (8-7)

τ

t

I

p t p e t dt where ,= integral time/reset timeτI

• 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.

Integral Control

The corresponding transfer function for the PI controller in Eq. 8-8 is given by

τ 111 (8-9)

τ τI

c cI I

P s sK K

E s s s

Integral control action is normally used in conjunction with proportional control as the proportional-integral (PI) controller:

0

1* * (8-8)

τ

tc

I

p t p K e t e t dt

Proportional-Integral (PI) Control

The integral mode causes the controller output to change as long as e(t*) ≠ 0 in Eq. 8-8

•Can produce p(t) that causes the final control element (FCE) to saturate.

•That is, the controller drives the FCE (e.g. valve, pump, compressor) to its physical limit of fully open/on/maximum or fully closed/off/minimum.

If an error is large enough and/or persists long enough

•the integral term continue growing, •the controller command the FCE to move to 110%, then 120% and

more•this command has no physical meaning (no impact on the process)

If this extreme value is still not sufficient to eliminate the error

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*Antireset windup reduce the windup by temporarily halting the integral control action whenever the controller output saturates and resumes when the output is no longer saturates.

Disadvantage of Integral Action: Reset Windup

Can grow very largeCan grow very large

0

1* * (8-8)

τ

tc

I

p t p K e t e t dt

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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

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The function of derivative control action is to anticipate the future behavior of the error signal by considering its rate of change.

Controller output is proportional to the rate of change of the error signal or the controlled variable.

Thus, for ideal derivative action,

τ (8-10)D

de tp t p

dt

τDwhere , the derivative time, has units of time.

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Derivative Control

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Advantages of derivative action

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For example, an “ideal” PD controller has the transfer function:

1 τ (8-11)c D

P sK s

E s

• Derivative action always used in conjunction with proportional or proportional-integral control.

• Unfortunately, the ideal proportional-derivative control algorithm in Eq. 8-11 is physically unrealizable because it cannot be implemented exactly.

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Physically unrealizable

Physically unrealizable

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• For “real” PD controller, the transfer function in (8-11) can be approximated by

τ1 (8-12)

ατ 1D

cD

P s sK

E s s

where the constant α typically has a value between 0.05 and 0.2, with 0.1 being a common choice.

• In Eq. 8-12 the derivative term includes a derivative mode filter (also called a derivative filter) that reduces the sensitivity of the control calculations to high-frequency noise in the measurement.

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Proportional-Integral-Derivative (PID) ControlCh

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requires very intensive energy

costly

highly endothermic reaction

Expanded form

Parallel form

Series form

3 common PID control forms are:

Parallel Form of PID Control

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

0

1* * τ (8-13)

τ

tc D

I

de tp t p K e t e t dt

dt

The corresponding transfer function for “Ideal” PID control is:

11 τ (8-14)

τc DI

P sK s

E s s

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The corresponding transfer function for “Real” PID control (parallel), with derivative filter is:

1

11

)(

)(

1 S

S

S D

DcK

sE

sP

Figure 8.8 Block diagram of the parallel form of PID control (without a derivative filter)

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Series Form of PID ControlConstructed by having PI element and a PD element operated in series.Commercial versions of the series-form controller have a derivative filter as indicated 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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Figure 8.9 Block diagram of the series form of PID control (without a derivative filter)

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PIPI PDPD

sI1

1

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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

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Used in MATLABUsed in MATLAB

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 setpoint change.

Controller Comparison

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• This sudden change is undesirable and can be avoided by basing the derivative action on the measurement, ym, rather than on the error signal, e.

• Replacing de/dt by –dym/dt gives

0

1* * τ (8-17)

τ

t mc D

I

dy tp t p K e t e t dt

dt

• One disadvantage of the previous PID controllers :

Derivative and Proportional KickFeatures of PID Controllers

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derivative kick : when there is a sudden change in set point (and hence the error, e) that will cause the derivative term to become very large.

From a parallel form of PID control in Eq. 8-13From a parallel form of PID control in Eq. 8-13

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(8-22)c sp mp t p K y t y t

• The controller gain can be made either negative or positive.

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Direct (Kc < 0) Reverse (Kc > 0)

Controller output p(t) increases as the input signal

ym(t) increases

Controller output p(t) increases as its input signal

ym(t) decreases

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Reverse acting (Kc > 0)e(t)↑, p(t) ↑ym(t)↓, p(t) ↑

Reverse acting (Kc > 0)e(t)↑, p(t) ↑ym(t)↓, p(t) ↑

(8-22)c sp mp t p K y t y t

Direct acting (Kc < 0)e(t) ↓, p(t) ↑ym(t)↑, p(t) ↑

Direct acting (Kc < 0)e(t) ↓, p(t) ↑ym(t)↑, p(t) ↑

On-Off Controllers

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

Controller output has two possible values.

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ym(t)↓ym(t)↓

ym(t)↑ym(t)↑

eg: thermostat in home heating system.-if the temperature is too high, the thermostat turns the heater OFF. -If the temperature is too low, the thermostat turns the heater ON.

eg: thermostat in home heating system.-if the temperature is too high, the thermostat turns the heater OFF. -If the temperature is too low, the thermostat turns the heater ON.

On-Off Controllers (continued)C

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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 the disturbance variable)

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Figure 8.12. Typical process responses with feedback control.

No control: the process slowly reaches a new steady state

P – speed up the process response & reduces the offset

PI – eliminate offset & the response more oscillatory

PID – reduces degree of oscillation and the response time

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Figure 8.13. Proportional control: effect of controller gain.Ch

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-Increasing Kc tends to make the process response less sluggish (faster)

-Too large of Kc, results in undesirable degree of oscillation or even become unstable

-Intermediate value of Kc usually results in the best control.

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-Increasing τI tends to make the process response more sluggish (slower)

-Too large of τI, the controlled variable will return to the set point very slowly after a disturbance change @ set-point change occurs.

Figure 8.14. PI control: (a) effect of reset time (b) effect of controller gain.Chap

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Figure 8.15. PID control: effect of derivative time.

-Increasing τD tends to improve the process response by reducing the maximum deviation, response time and degree of oscillation.

-Too large of τD: measurement noise is amplified and process response more oscillatory.

-The intermediate value of τD is desirable.

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