Chap02b.advanced process control.pdf

8/14/2019 Chap02b.advanced process control.pdf

1/15

Advance Topics in Process Control

CHE 522 1

CHAPTER 2

SISO Linear

System Analysis

4. Analysis of first order systems

A First order system is that whose output is modeled by a first orderdifferential equation.

(1) first order lag (self-regulating)

)(10 tbuyadt

dya =+

)(11

0 tu

a

by

dt

dy

a

a=+

)(tukydt

dyp=+

where1

0

a

a= ;

1a

bkp =

Taking Laplace transform with all variables are in deviation form we get:

)(1

)( sus

ksy

p

+=

The term, (time constant) and kp(static gain) characterize the first ordersystem.

Chemical Engineering Department

King Saud University, 2003


2/15


CHE 522 2

Time constant: characterize the speed of response of a first-order system.

It is a measure of the time necessary for a process to adjust to a change in

the input.

Static gain: characterize the sensitivity of the output to the input signal.

Dynamic response:

The dynamic response for a step change in uof magnitude of a: u(s)=a/s

)1()( / = tp eakty

The transient response for step change is shown in Figure (2).

0 T

u(t)

y(t)

ime

a

kpa

Figure 1: Response of a first-order lag system to step change in the input

Dynamic characteristics:

For a step change in the input, the process reaches a new steady state. The ultimate value of the output is kpa, which can be found from the

last Equation by setting t

The time constant, can be found be setting t = in the last equationwhich gives y = 0.632 kpa. Then from Figure2, the time needed for y

to reach 0.632 kpais . The smaller the value of , the steeper is the initial response of the

output.

The larger static gain of a process, the larger steady state value of itsoutput for the same input change.




3/15


CHE 522 3

0 Time

y(t)

kp1

a kp1

kp2< kp1

kp2

a kp2

0 T

y(t)

ime

kpa

1

2

1< 2

Figure 2: Effect of static gain, time constant on the response of first-order lag system

(2) Pure capacitive system (integrator)

)(0 tbudt

dya =

Example: Liquid storage tank with fixed outlet flow

FtFthA i = )()(&

F

As

sF

As

sh i1

)(1

)( = ; kp1/A

)()( sus

ksy

p=

Dynamic response:

For a step change in uof magnitude ofa: u(s) = a/s

2)( s

ak

sy

p

=

which have the following time response:

atkty p=)(

The transient response looks like this:




4/15


CHE 522 4

0 T

u(t)

ime

y(t)

kp

a

Figure 3: Response of an integrator first-order system to a step change

Apparently, the ultimate value for the output is not achievable, yas t.

A pure capacitive process causes serious control problem because itcan not balance itself.

For small change in the input, the output grows continuously. Thisattribute knows as non-self-regulating (integrator) process.

5. Analysis of second order system

A second order system is that whose output is modeled by a second orderdifferential equation.

)(212

2

0 tbuyadt

dya

dt

yda =++

)(22

1

2

2

2

0 tua

by

dt

dy

a

a

dt

yd

a

a=++

)(22

22 tuky

dt

dy

dt

ydp=++

)(12

)(22

suss

ksy

p

++=

= natural period of oscillation (characteristic time constant)




5/15


CHE 522 5

= damping ratiokp = static gain

Sources of second order dynamics in the chemical industries come from a

series of first-order systems, or a processing system with a controller.

Dynamic response:For input step change of magnitude of a, u(s) = a/swe have:

))((

/

)12()(

21

2

22 pspss

ak

sss

aksy

pp

++=

where+

=

121p

=

121p

Case A: (over-damped) when > 1, we have two distinct and real poles

+

= )1sinh(

1)1cosh(1)( 2

2

2/ tteakty tp

Case B: (critically-damped) when = 1, we have two equal poles

+= /11)( tp e

takty

Case C: (under-damped) when < 1, we have two complex conjugatepoles

+

= )sin(

1

11)(

/

2wteakty

tp

The dynamic response for all cases is shown in Figure (4)




6/15


CHE 522 6

0 T

u(t)

ime

y(t)

1

Figure 4: The response of second order system to step change

The over-damped response is sluggish and resembles a little theresponse of a first-order system. IT becomes more sluggish with

larger values of . The critically damped response is faster than the over-damped one. The under-damped response is faster than the others, but oscillates.

The oscillatory behavior becomes pronounced with smaller values .

Characteristics of under-damped system

Overshoot, A/B

=

21exp/BA

Decay ratio, C/A

=

21

2exp/AC

Period of oscillation

==

21frequencyradian




7/15


CHE 522 7

21

2

== noscillatioofperiodT

Rise time, tr: is the time the output takes to first reach the new steadystate value Process time, tp: is the time required for the output to reach its first

peak.

Settling time, ts: is the time required for the output to reach within5% of the new steady state value.

y

A

B

C

T

Timetr tp ts Figure 5: characteristics of second-order under-damped response




8/15


CHE 522 8

6. Delay time

The delay time known as transportation time is basically the time required

for a material to move a specific distant. However, time delay is an inherentproperty of any chemical process.

sesG =)(

t

y(t)

delay time

u(t)

Sources:

The use of chromatography to measure composition of liquids orgases is another source of dead time.

Flow in a long pipe. Time lag produced from staged processes.

Therefore, a first-order system with dead-time is:

1)(

1 +

=

s

KesG

s

Therefore, a second-order system with dead-time is:

12)(

2 ++=

ss

KesG

s




9/15


CHE 522 9

7. Approximation of higher order systems

Systems with order higher than one can be represented by:

==

+==

n

in

ii

i

s

KsGsG

1

1

1)(

)()(

It can be approximated by low order transfer function with dead time as

follows:

1)(

1 +=

s

KesG

s

where 1is the dominant time constant and the dead time is:

==

n

ii

2

8. Transfer function and Block diagram

Transfer function relates the process input to the process output in a linear

mathematical function. Therefore,

Transfer Function: is the Laplace transform of the output,y(s) divided bythe Laplace transform of the process input, u(s), with all initial conditions

are zero (deviation form)

)()()(

susysG =

Example: For the storage tank we have:

)(1)(

)(sG

s

K

sF

shp+

= ; with =AR, K = 1/R

Example: For the heated tank we have:




10/15


CHE 522 10

)(1)(

)(sG

s

K

sQ

sTp+

= ; with = V/F, K = 1/FCp

Example: For the CSTR we have:

)(1)(

)(sG

s

K

sC

sCp

Ai

A +

= ; with = V/(F+Vk), K =F/(F+Vk)

8.1 Properties of Transfer Function

Order: The order of the system is the highest derivative of the output

variable in the defining differential equation. For Transfer Function, it isthe highest power ofsin the denominator.

Pole: is the root of the denominator of the transfer function, i.e. the root ofthe characteristic polynomial. It directly determines:

The stability of the system (positive poles) The potential of periodic transient (imaginary poles)

Zero: is the root of the numerator of the transfer function. It determines aninverse response (positive zero).

Casuality: A physical system is causalwhen the order of the denominatoris greater than the numerator, and when the transfer function goes to 0 as s

, the system is hence strictly proper. If the transfer function contains

es or the order of numerator is higher than the denominator, then the

system is non-casual or not realizable because the current values of thesystem depends on the future values of the variables.

Steady state gain: is the steady state value of the transfer function, is

evaluated by settings= 0 in the stable transfer function.

8.2 Effect of poles and zeros

The poles and zeros of a transfer function affect the dynamic of a process.

Consider a particular transfer function:




11/15


CHE 522 11

)12)(1()(

221 +++

=ssss

KsG

The poles, i.e. the roots of the characteristic equationare:

01=s

12

1

=s

2

2

2

3

1

+

= js

2

2

2

4

1

= js

The poles can be represented in the complex plane as follows:

Real axis

imaginaryaxis

XX

X

X

0

1

1

2

2

21

2

21

+

Figure 6: roots of the characteristic equation of G(s)

Complex poles indicate the response will contain sine and cosinemodes, i.e. will exhibit oscillation.

Negative poles will result in a stable decaying response.Chemical Engineering Department



12/15


CHE 522 12

Positive poles indicate that the response will have unstable mode.

The transfer function can be written as follows:

)())((

)())(()(

21

21

n

m

n

m

pspsps

zszszs

a

bsG

=

L

L

Positive zero leads to inverse response. Zero-pole cancellation occurs when a zero has exactly the same

numerical value as a pole.

A zero can exert a profound effect on the coefficient of responsemode.

8.3 Block diagrams

Block diagram is a graphical representation of transfer functions and their

interactions.

Block diagram assist the engineer in determining the quantitative aspects of

dynamic performance and in understanding the qualitative features of the

system.

y(s)= G(s)u(s)

G(s)u(s) y(s)

y(s)=A+By(s)

B

+

+

A

y(s)=G1(s)G2(s)u(s)




13/15


CHE 522 13

G1(s)

u(s) y(s)G

2(s)

y(s)=G1(s)A G2(s)BG

1(s)

y(s)A

B

+

_

G2(s)

y(s) = G1(s)(AB)

y(s)A

B

+

_

G1(s)

8.4 Input-output relation

Direct action: an increase in the input lead to an increase in the output.

I/Iu y

Reverse action: an increase in the input lead to a decrease in the output.

I/Du y




14/15


CHE 522 14

Fi

F

h

Fi

F

h

A series of I/I blocks always result in I/I action:

I/Iu yI/I

An odd series of I/D blocks results in I/D action:

I/Iu yI/D

An Even series of I/D blocks results in I/I action:

I/Du yI/D




15/15


CHE 522 15

It is important that the overall system has an I/D action (negative

feedback), therefore, the controller action must be adjusted such that the

overall action of the closed-loop be I/D or reveres action.

Alternatively:

vekk pc +=



Chap02b.advanced process control.pdf

Documents

Transcript of Chap02b.advanced process control.pdf