بسم الله الرحمن الرحيم

PID Controllers

1- Action, types and Modifications

2- Offline and online tuning

Ref 1: Smith & Corripio “Principles and Practice of Automatic Process Control”, 3 rd Ed., Wiley, 2006, Chapter 5 & 7.

Ref 2: C. C. Yu, Autotuning of PID controllers, 2nd ed., springer, 2006, Chapters 2 &3.

Ref 3: K. J. Astrom and T. Hagglund, Advanced PID Control, ISA, 2006, Chapter 3.

Lecturer: M. A. Fanaei Ferdowsi University of Mashhad

1 -Action of PID Controllers

If the action is not correctly selected, the controller will not control

• Reverse action (increase/decrease)

In feedback control loop, the

multiplication of Process gain

(Kp), Control valve gain (Kv),

Sensor gain (Km) and Controller

gain (Kc) must be positive.

Reverse action : If Kp Kv Km > 0 → Kc > 0 2

1 -Action of PID Controllers

• Direct action (increase/increase)

To determine the action of a controller, the engineer must know:

1. The process characteristics

2. The fail-safe action of the control valve

Direct action : If Kp Kv Km < 0 → Kc < 0

3

2 -Types of PID Controllers

• Classic PID:

• Parallel PID:

• Series PID:

dt

tdeKdtte

KteKmtm Dc

I

cc

)()()()(

s

sK

sE

sMsG D

Icc

1

1)(

)()(

1

11

)(

)()(

s

s

sK

sE

sMsG

D

D

Icc

1

111

)(

)()(

s

s

sK

sE

sMsG

D

D

Icc

Range :0.05 to 0.2(0.1)

4

2 -Types of PID Controllers

5

3 -Problems and Modifications of PID Controllers

Reset Windup

dt

tdeKdtte

KteKmtm Dc

I

cc

)()()()(

6

3 -Problems and Modifications of PID Controllers

Reset Feedback (RFB)

Internal Reset Feedback

Mmin

Mmax

Mmax

Mmin

mM

mMmmM

100max

min

s

sK

s

K

sE

sM

I

Ic

I

c

1

11

1)(

)(

7

3 -Problems and Modifications of PID Controllers

Reset Feedback (RFB)

8

3 -Problems and Modifications of PID Controllers

Reset Feedback (RFB)

External Reset Feedback 9

3 -Problems and Modifications of PID Controllers

Proportional and Derivative Kick

dt

tdeKdtte

KteKmtm Dc

I

cc

)()()()(

yyeycyeybyeWheredt

tdeKdtte

KteKmtm

spspdspp

dDc

I

cpc

,,

)()()()(

Proportional Kick Derivative Kick

Range: 0-1 Range: 0-1, Commonly zero

Two Degrees of Freedom or ISA - PID

10

3 -Problems and Modifications of PID Controllers

11

4 -Off-Line Tuning of PID Controllers

Ziegler-Nichols(1942): Recommended for 0.1< D/ t <0.5 ( )1

s

eK Dsp

More than 250 tuning rules are exist for PI and PID Controllers

What is the suitable tuning rule?

It really depends on your process (Type, Order, Parameters,

Nonlinearity, Uncertainty, etc)

12

4 -Off-Line Tuning of PID Controllers

Tyreus-Luyben(1992): Recommended for time-constant dominant

processes ( D/ t <0. 1 )

Ciancone-Marlin(1992): Recommended for dead-time dominant

processes ( D/ > t 2.0 )

13

4 -Off-Line Tuning of PID Controllers

PID tuning based on IMC (Rivera et al., 1986)

14

Final Control Element

ProcessSensor/

Transmitter

Step Change

Record

m(t), % c(t) , %

1)(

)(

:

s

eK

sM

sC

timedeadplusorderfirst

Ds

Process Gain:

m

cK s

15

4 -Off-Line Tuning of PID Controllers

Fit 3: 212 ,)(2

3tDtt

16

4 -Off-Line Tuning of PID Controllers

5 -On-Line Tuning of PID Controllers

Ziegler-Nichols Test (1942)

1. Set the controller gain Kc at a low value, perhaps 0.2.

2. Put the controller in the automatic mode.

3. Make a small change in the set point or load variable and observe the

response. If the gain is low, then the response will be sluggish.

4. Increase the gain by a factor of two and make another set point or

load change.

5. Repeat step 4 until the loop becomes oscillatory and continuous

cycling is observed. The gain at which this occurs is the ultimate gain

Ku , and the period of oscillation is the ultimate period Pu. 17

Relay Feedback Test (Astrom & Hagglund, 1984)Luyben popularized relay feedback method and called this method “ATV” (autotune variation).

18

5 -On-Line Tuning of PID Controllers

Relay Feedback Test

1. Bring the system to steady state.

2. Make a small (e.g. 5%) increase in the manipulated input. The magnitude of change depends on the process sensitivities and allowable deviations in the controlled output. Typical values are between 3 and 10%.

3. As soon as the output crosses the SP, the manipulated input is switched to the opposite position (e.g. –5% change from the original value).

4. Repeat step 3 until sustained oscillation is observed .

5. Read off ultimate period Pu from the cycling and compute Ku from the following Equation: Ku = 4h/(πa) , ωu = 2π/Pu

19

5 -On-Line Tuning of PID Controllers

Advantages of Relay Feedback Test

1. It identifies process information around the important frequency, the ultimate frequency (where the phase angle is -π).

2. It is a closed-loop test; therefore, the process will not drift away from the nominal operating point.

3. The amplitude of oscillation is under control (by adjusting h ).

4. The time required for a relay feedback test is roughly equal to two to four times the ultimate period.

5. If the normalized dead time D /t is less than 0.28, the ultimate period is smaller than the process time constant. Therefore the relay feedback test is more time efficient than the step test. Since the dead time can not be too large, the temperature and composition loops in process industries seem to fall into this category. 20

5 -On-Line Tuning of PID Controllers

Advantages of Relay Feedback Test

1

s

eK Dsp

21

5 -On-Line Tuning of PID Controllers

Relay feedback responses of FOPDT processes

Assume an integrator plus dead time (Time constant dominant processes)

Assume a FOPDT (Most slow processes)

Assume a pure dead time (Dead time dominant processes)

22

5 -On-Line Tuning of PID Controllers

6 -Discrete Form of PID Controllers

Position Form

dt

tdeKdtte

KteKutu Dc

t

I

ccs

)()()()(

0

Sampling Time : Ts , Number of Sampling : k , Time : t = kTs

k

iss

tTiTedtte

10

)()(:iomapproximatr rectangulaUpper

s

ssk

T

TkekTe

dt

tde ))1(()()( :Difference Finite Backward

Position Form

Velocity Form

dt

tdeKdtte

KteKutu Dc

t

I

ccs

)()()()(

0

)1()()()()(1

kekeT

Kie

TKkeKuku

s

DcK

iI

sccs

)2()1(2)()()1()()1()( kekekeT

Kke

TKkekeKkuku

s

Dc

I

scc

6 -Discrete Form of PID Controllers

Velocity Form

Where:

)2()1()()1()( 210 kegkegkegkuku

s

Dc

s

Dc

s

D

I

sc

T

Kg

TKg

T

TKg

2

1

0

21

1

6 -Discrete Form of PID Controllers

Backward Shift Operator (q -1) : y(k-n)=q-ny(k)

Tuning of Digital PID : Moore et al. (1969)

Use the continuous tuning formula of PID controller with corrected dead time

1

22

110

1)(

)(

q

qgqgg

ke

ku

2s

c

TDD

6 -Discrete Form of PID Controllers