Type of response

Common input changes

1. Step Input

A sudden change in a process variable can be approximated by a step change of magnitude, M:

• Special Case: If M = 1, we have a “unit step change”. We give it the symbol, S(t).

• Example of a step change: A reactor feedstock is suddenly switched from one supply to another, causing sudden changes in feed concentration, flow, etc.

The step change occurs at an arbitrary time denoted as t = 0.

We can approximate a drifting disturbance by a ramp input:

2. Ramp Input

• Industrial processes often experience “drifting disturbances”, that is, relatively slow changes up or down for some period of time.

• The rate of change is approximately constant.

Examples:

1. Reactor feed is shut off for one hour.2. The fuel gas supply to a furnace is briefly interrupted.

0

h

URP tw Time, t

3. Rectangular PulseIt represents a brief, sudden change in a process variable:

4. Sinusoidal InputProcesses are also subject to periodic, or cyclic, disturbances. They can be approximated by a sinusoidal disturbance:

sin0 for 0

(5-14)sin for 0

tU t

A t t

Examples:

1. 24 hour variations in cooling water temperature.

where: A = amplitude, ω = angular frequency

sin 2 2( ) AU ss

Response of first order system• First order differential equation

• General first order transfer function

)()()(01 tbXtYa

dttdYa

ctbxtyadt

tdya )()()(01

)()()( tKXtYdt

tdY

)(1

)( sXsKsY

0

01

//abKaa

inputoutput/response

Smith & Corripio pg 41

sx

sKsY

1)(

1.Step response

)(1

)( sXsKsY

5.2.1 page 108 Seborg,Inverse laplace

Response in time domain,y(t)

Coughanowr page 79

So, do you understand

how process constant,

corresponds to 0.632?1 2 3𝜏 4 5

• All first order systems forced by a step function will have a response of this same shape.

Step response for first order system

• To calculate the gain and time constant from the graph

xyK

Gain,

Time constant, – value of t which the response is 63.2% complete

2. Ramp response

)(1

)( sXsKsY

21)(

sa

sKsY

y(t)=??

Inverse Laplace, SeborgPage 110

Ramp response for first order system

The normalized output lags the input by exactly one time constant

Rampinput Ramp

output

Seborg page 111

22)(

s

sU

222

2210

22p

sss

1ss1sK

)s(Y

1K

1K

1K

22p

2

22p

1

22

2p

0

3. Sine input

By partial fraction decomposition,

)tsin(1

Ke

1K

)t(y22

pt22

p

Where )(tan 1

)(1

)( sUsKsY

First order response to the sine wave

Response with time delay

X(t)

Y(t)

t=0 t=t0

Θ=Time delay/dead time

1. Step response

)(1

)(0

sXs

KesYst

First-order-plus-dead-time (FOPDT)

Response of second order system• Second order differential equation

• General second order transfer function

ctbxtyadt

tdyadt

tyda )()()()(012

2

2

)()()()(012

2

2 tbXtYadt

tdYadt

tYda

)()()(2)(2

22 tKXtY

dttdY

dttYd

)(12

)(22

sXss

KsY

0

20

1

0

1

0

2

22

abK

aaa

aaaa

1+)s(+sK=G(s)

212

21

21

1s2sK=G(s) 22

21

21

2=

2nd order ODE model(overdamped)

Composed of two first order subsystems (G1 and G2)

roots: 12

damped critically 1dunderdampe 10

overdamped 1

11)(

21

21

ssKK

sY

1. Step response

)(12

)(22

sXss

KsY

s

xss

KsY

12

)( 22

Second Order Step Changea. Overshoot – fraction of the final steady-state change

by which the first peak exceeds this change

b. time of first maximum-time required for the output to reach its first maximum value

c. decay ratio-ratio which the amplitude of the sine wave is reduced during one complete cycle

21pt

2

22

2exp1

c aa b

os=2

exp1

ab

d. period of oscillation, P – time between two successive peaks of the response.

e. Rise time, tr – time taken for the process output to first reach the new steady state value.

f. Settling time – time it takes for the output to come within a band of the final steady-state value and remain in this band

2

2

1p

Ideal response: The desired process response is achieved at an instantaneous time.

SP1

PV1

PV2

SP2

Time

Idealresponse

process responses under automatic control.Terminology

© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)

Stable: The process response stabilized at (near) the set point .

SP1

PV1

PV2

SP2

Time

Idealresponse

Terminology

© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)

Unstable: The process response could not be stabilized at the set point.

SP1

PV1

PV2

SP2

Time

Idealresponse

Terminology

© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)

SP1

SP2

PV

Time01. LCL = Lower control (quality) limit.

2. UCL = Upper control (quality)

limit.

Out of spec

Out of spec LCL

UCL

Quality limits: A range, set values above and below the set point, whereby the process is allowed to oscillate. Product quality is acceptable within these limits.

Terminology

© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)

QAD

Underdamped

Overdamped

Oscillatory

Offset

Various shapes of process responses under automatic control.

© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)

Settling criteria: A response curve that meet any of the following criteria (criterion) is considered settle.

1. Response time2. Settling time3. Rise time4. Quarter Amplitude Damping (QAD)5. Quality limits (BEST for product quality control)6. No overshoot or no undershoot (BEST for

temperature and pH control)7. Minimum IAE, ITSE, etc.

Unit 1: Process settling criteria

Terminologies

© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)