1

ProportionalIntegral

Differential (PID) Control

Proportional Control–Equation–Example

• PI Control–Equation–Numerical Integration–Example

• PID Control–Web Simulation–Equation–Numerical Differentiation–A 3-mode opamp controller–Industrial PID tuning

• Summary

2

Proportional Control-1

opp pekp

Input signal

FeedbackSignal

OutputController

Actuator(Plant)

Transducer

+

-

Error

er

b

minmax bb

bre

bre

p

3

Proportional Control-2

OutputK G

H

+

-

Error

er

b

p

)(

)(

HprGKp

Hpb

brGKGKep

bre

GKH

GKrp

GKrGKHpp

GKHpGKrp

1

4

Proportional Control-3

OutputK G=8

H=1/8

+

-

Error

er

b

p

K

Krp

K

Krp

GKH

GKrp

1

8

81

81

81

rKHKzG

zGzP

rKHKsG

sGsP

)(1

)()(

)(1

)()(

5

Proportional Control-4

OutputK=1/4 G=4

H=1

+

-

Error

er

b

p

rp

rp

GKH

GKrp

21

41

41

141

41

K=4 G=4

H=1

rp

rp

GKH

GKrp

1716

1441

441

6PC-5

opp pekp kp

+

+

ep p

minmax

0

pp

ppe p

p0

Proportional controller

pp k

ppe 0

Output

Setpoint

In units of the output

In terms of % of span

7

PC-6

9.8 For a proportional controller, the controlled variable is a process temperature with a range of 500 to 1300C and a setpoint of 73.50C. Under nominal conditions, the setpoint is maintained with an output of 50%. Find the proportional offset that results from a load change which requires a 55% output if the proportional gain is (a) .1 (b) .7 (c) 2.0 and (d) 5.0

opp pekp

minmax

0

pp

ppe p

pp k

ppe 0

In units of the output

In terms of % of span

ppp kke

5%50%55

pp k

Te

5%100)

50130

5.73(

pkCT

45.73 0 Output

Temperature=

(a) T=33.50C(d) T=72.70C

8PC-7http://www.fourmilab.ch/hackdiet/www/subsection1_2_3_0_5.html

Red-line is On-Off Control (furnace on at 60)(air conditioner on at 80)

High heat load: air conditioner takesa while to bring temp down.

Cold wind: heater takes a while to bring temp up.

Time of day

Green-line is ProportionalControl: above 70 air-conditioner fan speed increases proportional tothe T-700

Green-line is Proportional Control: below 70amount of furnace gas increases proportional to the 700 – T.

9

PC-8

http://newton.ex.ac.uk/cgi-bin/metaform?http://newton.ex.ac.uk/teaching/CDHW/Feedback/OvSimForm-gen.html

On-Off Control

Proportional Control

10

Proportional-Integral (PI) Control

http://newton.ex.ac.uk/cgi-bin/metaform?http://newton.ex.ac.uk/teaching/CDHW/Feedback/OvSimForm-gen.html

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Proportional-Integral-Derivative(PID) Control

http://newton.ex.ac.uk/cgi-bin/metaform?http://newton.ex.ac.uk/teaching/CDHW/Feedback/OvSimForm-gen.html

12

PI, PD, and PID Composite Mode Equations

)0()(

)0()(

0

0

0

Ip

Dp

t

pIppp

pDppp

I

t

pIppp

pdt

dekkdttekkekp

pdt

dekkekp

pdttekkekp

PI

PD

PID

http://members.aol.com/pidcontrol/booklet.html

CONTROLLER TUNING AND CONTROL LOOP PERFORMANCE, A PRIMER, SECOND EDITION Subtitled PID WITHOUT THE MATH by David W. St. Clair

13

Example 9.8 in text

20)(550

t

pp dtteepEquation:

percent

14Ex 9.8

20)(550

t

pp dtteep

+20% would just raise the PI curve.

15Simple Numerical

Integration

Calculations:.1*1 +0=.1.1*2+.1=.3.1*3+.3=.6.1*3+.6=.9.1*3+.9=1.2etc.

0 0.1 0.2 0.3 0.4

0

3

2

1

seconds

volts

16

PID Example

dt

dedtteep p

t

pp 5)(550

17

Numerical Derivative

Calculations:(1-0)/(1-0)=1(2-1)/(2-1)=1(3-2)/(3-2)=1(3-3)/(4-3)=0(3-3)/(5-4)=0etc.

0 1 2 3 4

0

3

2

1

seconds

volts

5

18

Opamp PID Circuit

Set kp

Set kD

Set kI

dt

dekdttekekp pD

t

pIpp 0

)(

19

Industrial PID Tuning

http://www.expertune.com/ • DCS: Digital Control System. DCS refers to larger analog

control systems like Fisher, Foxboro, Honeywell, and Bailey systems. DCS's were traditionally used for PID control in the process industries, whereas PLC's were used for discrete or logic processing. However, PLC's are gaining capability and acceptance in doing PID control. Most utilities, refineries and larger chemical plants use DCS's. These systems cost from 20 thousand to millions of dollars.

• DCS systems by: ABB (http://www.abb.com/ ), Bailey, Foxboro (http://www.foxboro.com/industries/ ), Fisher, Moore, Honeywell, Yokogawa, etc.

20

Matlab PID Tuning-1

http://www.mathworks.com/

21

• Introduction• Proportional Control

– Equation– Example

• PI Control– Equation– Numerical Integration– Example

• PID Control– Web Simulation– Equation– Numerical Differentiation– A 3-mode opamp controller– Industrial PID tuning

Summary