digital controllers_anjan raksit

8/3/2019 Digital Controllers_anjan raksit

1/49

Digital Controller in a Process Control Loop


2/49


In a processor based digital controller, rapid switching from onealgorithm to another (e.g. a P controller to a PID controller) andautomatic tuning of controller parameters are possible.

No drift problem is encountered in a digital system.

All the desirable features of a process controller (e.g. anti-integralwind-up, auto/manual modes of operation with bump-less transfer

etc.) may be easily incorporated while maintaining the highaccuracy and precision of digital systems.


3/49


+ ADCDigital

ControllerDACr

+

_

e en mn

Set pointError

Errorsequence controlleroutputsequence

m FinalControlElement

VProcess

Controlleroutput

C

ADC equivalent

e en

Sampler

m

n

m

C

Sample & hold

(holdcapacitor)

DAC equivalent

e

time0

Error

en 2

n

Errorsequence

0 1 2 3

Controlleroutput

sequence

DACoutput

m

mn

0 1 2 3n

time0 2

m

Sampling interval (ADC conversion time + computation time)


4/49


The error computation may be performed digitally if the set-point is available indigital form (say, from digital keyboard) and the measured variable is digitized withthe ADC.

+rn+

DigitalController

DACen

Errorsequence

_Set pointsequence

(digital)

mn m Final

ControlElement

Process

ADCCn

outputsequence

CV

Digital System

Proper selection of the sampling interval is necessary for satisfactory operation ofthe process control loop.

A large may lead to unstable operation of the loop (because of the extra lagintroduced in the loop), whereas a very small requires a high speed digital hardware(hence high cost) to implement the controller.


5/49

Realization of Digital Controllers(through discrete approximation of analog controllers)

Proportional (P) Controller

The controller output at the nth instant is given by

n p n n n n

m K e b m b= + = +

The analog controller output is

m = Kp

e + b

where, Kp = proportional gain,bn = fixed bias.

Realization of the P controller:

Kp

+

bn

mn

en

m'n


6/49

Realization of Digital Controllers(through discrete approximation of analog controllers)

Proportional-Integral (PI) Controller


1.

t

p o

i

m K e e dt bT

= + +

= m+ b, where Ti is the integral time and1 .

t

p o

i

m K e e dt T

= +

Let the integration term be .t

o I e dt =

The integration term at the nth instant is .n

n o I e dt

= where is the sampling interval

. .n n

o ne dt e dt

= + for n = 1,2,3,..

Now,( )1

1. .n n

no oe dt e dt I

= =

Then,


7/49

t

e

( )1

t n

t n

e

e

=

=

( )1n n

This area may be approximated by the shaded rectangle (called the method of

rectangular integration) as

Pl Controller

The second term of the above relation represents the area

under the curve e for (n 1) t n.


8/49

Using the notation en for t ne = 1n n n I I e= +,

t

e

( )1

t n

t n

e

e

=

=

( )1n n

Thus,

( )1.

n

n t nedt e

=

Pl Controller


9/49

Similarly the output at the (n 1)th instant may be expressed as

1n n n I I e= +

Now, output m at the nth instant may be expressed as

where

Pl Controller

The difference between these two outputs is

( )1 1 11

n n p n n n n

i

m m K e e I I T

= +


10/49

1 1 1, say

o n n na e a e m = + +

where, 1o pi

a K T

= + and

1 pa K=

Pl Controller

or,1 11n n p p n n

i

m e K K e m

T

= + +

1 1

n

n n p n n

i

e

m m K e e T

= +

( )1 1 11

n n p n n n n

i

m m K e e I I

T

= +

Substituting the value of (I

n I

n-1) for rectangular integration, [(I

n I

n-1) = e

n]


11/49

Realization of the PI Controller

a1

ao

+

+

m'n

delay

m'n-1

bn

(bias)

mn

en-1

delay

en

errorsequence

Controlleroutput

sequence

Controller output at the nth instant:

mn= mn+ bn

mn= a0en+ a1en-1 + mn-1

Problem: Develop a digital PI Controller using Trapezoidal rule for integration.


12/49

Velocity or incremental form of PI controller

The controller output is proportional to the derivative of a standard PIcontroller and it may be expressed as (without bias):

1 1 1 1n n n n n o n nm m m m m a e a e = = = +

Velocity form of controller is useful when the actuator is some kind ofadder (integral action), like a stepping motor.


13/49

Software Realization of the PI Controller

a1

a0 + +mp

b

m

e1

e

# include < studio.h>void main (void)

{float e = 0, e1, m, mp = 0;float a0, a1, b;float adc (void) ; // digitized error

void dac (float m ) ; // analog outputa0 = - - - - - - - - ; // Kp [1 + /Ti ]a1 = - - - - -- - - ; // -Kpb = - - - - - - - - - ; // bias


14/49


a1

a0 + +mp

b

m

e1

e

for (;;) // continuous loop{ // loop time is the sampling interval e1 = e;e = adc ();mp = mp + a0*e + a1*e1;m = mp + b;

// provision for saturationif (m < 0) m = 0;if (m > 100) m = 100;dac (m);}

}


15/49


a1

a0 + +mp

b

m

e1

e

float adc (void) // Analog-to-digital conversion{float v;

scanf (%f, &v); // for (keyboard) simulationreturn v; // (to be replaced for actual} // realization)

void dac (float m) // Digital-to-analog conversion{

printf (%f\n, m); // for (VDU) simulation} // (to be replaced for actual realization)


16/49


17/49

PD Controller

Thus the controller output at the nth instant is

1

1

1 1

or 1

n n

n p n d n

p dd

n p n n n

o n n n

e em K e T b

K TT

m K e e b

a e a e b

= + +

= + +

= + +

where 1 do p

Ta K

= +

and

1

p dK T

a

=


18/49

Realization of the PD Controller

a1

ao

+ +

bn

(bias)

mn

en-1

delay

en

(error) Controlleroutput

Problem: Develop a c program for software realization of the PD Controller.


19/49

PD Controller

Provision for anti-derivative kick

To avoid derivative action from a sudden change in set-point, the derivative actionis generally derived from the measured output.

Now, e = r c

then, ,de dr dc

dt dt dt

=

,dc

dt= assuming set-point r is constant.

Thus, the controller output may be expressed as,

p d

dcm K e T b

dt

= +


20/49


21/49

Realization of PD Controller with anti-derivative kick

Problem: Develop a c program for software realization of the PD Controllerwith anti-derivative kick

ao

po

p1

+ +

en

error

cn

(processoutput)

cn-1

delay

bn

(bias)

mn

(Controller output)


22/49

Realization of Digital Controllers

(through discrete approximation of analog controllers)

Proportional-Integral-Derivative (PID) Controller


1

' (say)

t

p do

i

dem K e edt T b

T dt

m b

= + + +

= +

where1

't

p do

i

dem K e edt T

T dt

= + +

The controller output (without bias) at the nth instant, using backward differencealgorithm, is

1n n nn p n d

i

I e em K e T

T

= + +


23/49


24/49

( )11 1 2 12n n d

n n p n n n n n

i

I I T m m K e e e e e

T

= + + +

Using rectangular integration algorithm, 1n n n I I e =

then, ( )1 1 1 22d

n n p n n n n n n

i

Tm m K e e e e e e

T

= + + +

or,1 2 1

21 1

p dd dn n p n p n n

i

K TT Tm e K e K e m

T

= + + + + +

or, 1 1 2 2 1n o n n n nm a e a e a e m = + + +

PID Controller


25/49

1 1 2 2 1n o n n n nm a e a e a e m = + + +

1

1

2 1

do p

i

dp

Ta KT

Ta K

= + +

= +

where,

and2

p dK Ta

=

PID Controller


26/49

Realization of the PID Controller

1 1 2 2 1n o n n n nm a e a e a e m = + + +

a

2

a1

ao

+ +

bn

(bias)

mn

m'n-1

m'n

en-2

en-1

en

mn = mn + bnand


27/49


28/49

Techniques used for anti-integral windup

By saturating or limiting the integral value

By resetting the integral value to zero

By omitting the integral term

By adaptive adjustment of controller parameters


29/49

Some anti-integral windup schemes

Stop integration when PI/PID controller internal output (prior to thesaturation block) exceeds the saturation limits

+ Kp +100

1000Process

+_

+

n

Saturation

r e

+_

X

Multiplier

0

11/0

Stop integration ifsaturated

C

Scheme for PI Controller

hardswitching Scheme

i

p

sT

K


30/49

+ Kp +100

1000Process

+

_+

n

Saturation

r e

+_

X

Multiplier

0

11/0

Stop integration ifsaturated

C

Scheme for PI Controller

hardswitching Scheme

i

p

sT

K

Performance of anti-integral windup scheme

PI control without anti-integral windup PI control with anti-integral windup


31/49


Reduce integration gradually as PI/PID controller internal outputexceeds the saturation limits

+ Kp +100

1000Process

+_

+

Saturation

r e

+_

p

i

K

T

C

softswitching Scheme

+

G

G : a constant

s

1


32/49


Clegg integrator the integrator is set to zero (reset) when the errorcrosses zero

_

+

C

reset control

resetSW

error integrator outputR


33/49


34/49

Automatic/Manual modes of Operations

ControllerFinal

Control element

Manual input or commnd

Auto

Man

me

error

vProcess

If there is any difference between the controller output and the manual command, abumpoccurs in the process output when the switch position is altered.

To provide bump-less transferfrom auto-to-manual change over, specialarrangements may be made for set-point initialization.

The manual command is driven to equal the controller output when the loop is inAUTO mode.


35/49


ControllerFinal

Control element

Manual input or commnd

Auto

Man

me

error

vProcess

When the loop is in MANual mode, if there is a steady error existing due to anydifference between the set-point of the controller and the process output (under

manual control), integral term, in case of PI and PID controllers, may wind-up to alarge value, and consequently anti-integral wind-up is necessary for suchsituations.

To provide bump-less transferfor all the operating modes, incremental or velocityfrom of controller is used with an additional integrator.


36/49


37/49


38/49


Digital

incremental

controller

Integrator

Incremental manual command

Auto

Man

m'ne

n +

bn

(bias)

m'n m

n

Scheme for bump-less transfer

The presence of integrator at the output ensures a smooth output variation even

when the actual manual command is different from the actual controller outputunder closed-loop control.


39/49

Realization of the incremental type PID Controller

a

2

a1

ao

+ +

bn

mn

m'n-1

m'n

en-2

en-1

en +

m'n

Incremental manualCommand

MAN

AUTO

Incremental orVelocity

Controller

Integrator


40/49

Automatic tuning of PID Controllers the Relay autotuner

This is based on a special technique for determining the critical gain Kc and critical

time period Tc of the process loop.

Kc is the gain margin of the process loop.

Controller parameters Kp, Ti and Td are calculated according to Ziegler Nichols(Z-N) rulefor a stable time response.

Suitable for processes with non-zero dead-time.

Z-N settings

gain margin: 2

Controller Kp Ti Td

P Controller 0.5 Kc

PI Controller 0.45 K c Tc/1.2

PID Controller 0.6 Kc Tc/2 Tc/8


41/49


The critical gain Kc and critical time period Tc are determined from an experimentwith relay (switching element) feedback.

+

-D

0

D

Processe v

Relay

r = 0

set pointc+

_

The relay control provides ON / OFF control of the process.

The input r is set to zero.

The output c oscillates around a mean value of zero (limit-cycle oscillations).

The process is driven by a square wave of amplitude D.


42/49


+

-D0

D

Processe v

Relay

r = 0

set pointc+

_

Assuming the process to be a low-pass system, the process output c containsmainly the fundamental component.

Thus the error signal e becomes sinusoidal,

e = A sint


43/49


The relay output v may be found out as follows:

+

-D0

D

Processe v

Relay

r = 0set point

c+

_

v

D

0e

- D

time

e0 A

- A

v

D

0time

- D

2

2

e = A sint


44/49


+

-D0

D

Processe v

Relay

r = 0set point

c+

_

D

- D

2

v

time

The Fourier series of the relay output (v) may be expressed as:

4 1 1sin sin 3 sin 5 .......3 5

Dv t t t

= + + +

The process practically attenuates all higher harmonics other than the fundamental.

Then the process gain at frequency becomes

( )

4 4

output amplitude A AG

Dinput amplitude D

= = =


45/49


+

-D0

D

Processe v

Relay

r = 0set point

c+

_

( )

44

output amplitude A AG

Dinput amplitude D

= = =

Process gain at frequency :

Now, to maintain steady oscillations at = c, the loop gain is 1 (consideringnegative feedback).

Thus the gain of the relay controller (i.e. the critical gain) at = c is

( )( )c c

1 4as K . G 1

c

c

DK

AG

= = =

Also G (c) = , as relay phase shift is zero.

.


46/49


+

-D0

D

Processe v

Relay

r = 0set point

c+

_

Thus by knowing the relay amplitude D and by measuring the amplitude A of the

process output c, critical gain Kc may be determined .

4

c

D

K A

=

Tc may be estimated by measuring the frequency of the output oscillation .2

c

c

T

=


47/49

Block diagram of the Relay autotuner

D

0

D

Relay

+PID

ControllerProcess

Manualcommand

r +

_

eM

A v

T

c

Modeswitch

M Manual positionA Auto positionT Tune position

(The Satt Control Autotuner by Satt Control, Sweden)

At first, the process is brought to equilibrium state ( zero error), bysetting a constant control signal in manual mode.

The tuning is then activated by pushing the mode switch to tune position.


48/49

Modified relay characteristic for a non-zero set-point

Manualsetting

v

0e

v

0t

Controlleroutput forzero error

D = 0.1vmax

The relay amplitude D is initially set to 10% of the controller output-range.

This amplitude is adjusted after one and a half period to give oscillation of 2% of the mean output.

This ensures minimum disturbance at the process output due to tuning.

This adjustment is done by measuring the change in output during the first one and a half period.

The modified relay amplitude is stored for the next tuning operation.


49/49

D0

D

Relay

+PID

ControllerProcess

Manualcommand

r +

_

eM

A v

T

c

Modeswitch

M Manual positionA Auto positionT Tune position

The system is automatically switched to Auto mode after estimating the

critical gain Kc and critical time period Tc during first 5 period of oscillation.

The parameters of PID controller (viz. Kp, Ti and Td ) are determined from Kcand Tc according to Z-N rule.

Satt Control Autotuner

digital controllers_anjan raksit

Documents