Frequency Response of Linear Systems

8/10/2019 Frequency Response of Linear Systems

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Chemical Process Control

ZK Nagy 1

Frequency Responseof Linear Systems

CHE 456Process Dynamics and

Control

Prof. Zoltan K. Nagy

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Background:Last time we saw

First order systems (time domain and s domain)

Second order systems (time domain and s domain)

Linear/locally linear systems

More complex systems


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Lecture Overview

Introduction to the frequency domain

Understand the importance of periodic signals forthe dynamic analysis of linear systems(frequency response )

Generate the frequency response of typical linearsystems

Analyze the dynamic behavior of typical linear

systems based on their frequency response Identify dynamic behavior of a process based on

its frequency response

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Objectives for Today

Whatis and whyis frequency analysis important ?

Frequency response of typical linear systems:

First order

Pure gain

Integrating

Second order

Nth order

Systems with zeros

Time delay

Examples

Summary


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Frequency response What ?

Frequency response = steady state

response of a system to a sinusoidal

input

It is a valuable tool in the analysis and

design of control systems

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Frequency response Why ?!

Many natural phenomena are sinusoidal in nature (mechanical,electrical systems)

Lots of periodic signals

Examples: 24 hour variations in cooling water temperature, 50-Hzelectrical noise (in Europe)

Any periodic signal can be represented by a series ofsinusoidal components

Systems respond differently to slowly or fast changing signals

Slowly changing inputFast changing input

input

output

time time


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For many electro-mechanical systems, frequencyresponses are informative and natural representation ofsystem dynamics (characteristics)

Audio system

Electric signal Sound wave signal

AR

frequency20 Hz 20 KHz

ExpensivespeakerCheap

speaker

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Equalizer

Raw signal Processed signal

AR

frequency20 Hz 20 KHz

Frequency response ofthe equalizer dependson its settings


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Tacoma Narrows Bridge Failure (1940)

Importance of periodic inputs on the dynamic behavior

of the process

input output

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Mechanical structures

Wind speed Tower displacement


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Mechanical structures - continued

DVFB = Direct Velocity Feedback)

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Frequency Response Function

Concerned with sinusoidal inputs over a SWEEP of frequencies

G(s)

linear system

outputgeneral input

For linear systems we can evaluate directly using Transfer Functions!

Replace s with j to obtain Frequency Response Function :

G(j)

linear system

outputinput

sinusoidalsinusoidal


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Have we met before ?

Remember your chemistry classes: UV, IR spectra:

ChemicalMolecule

EM radiationwith different G(j)

IR/UV spectrum = f()

IR/UV spectra are

Frequencyresponsefunctions

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Frequency response

Fundamental theorem of linear systems:

If a sinusoidal input with frequency is applied to a stable

linear system G (s) then the response (output) approachesa sinusoidal motion with frequency .

That is:Long-term response of any linear systemto a sinusoidal input is a time-shifted sinusoid with a different amplitude than the input, but the same

frequency:

G(s)

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Y sK

s

A

s

c

s

c

s j

c

s j

p! "

#

#

$ $

# $ %

Towards Frequency Response

Response of process to sinusoidal inputs

Partial fraction expansion

e.g., first-order process

What happens to the output response when wechange the amplitude and the frequency of thesinusoidal input?

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first term - leads to time exponential (transient response) dies away after (4-5) x time constant

last two terms - sustained sinusoid

' ' ' '% & "#$% &( (#% %& " & "! # ' #

( ' )' '

% &( (

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* * * * + * +

this term dies out for large tInverting: (indiv. exercise)

this term persists1tan ( )

note: is not a function of tbut of and .


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Ultimate Periodic Response:

For large t, y(t) is also sinusoidal,

output sine is attenuated by

(fast vs.

slow )#

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Tendency of real physical systems to have smallAmplitude ratio for sufficiently large

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Frequency response

Long-term response of any linear systemto a sinusoidal input isa time-shifted sinusoid with a different amplitude than the input,

but the same frequency:

G(s)

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For general linear systems directly from Transfer Function!

Memorize

Remember Topic 1(complex numbers)


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Shortcut Method for Finding

the Frequency Response

Step 1. &'( s=j*+ G!s" (, ,-(.*+ /

Step 2. 0.(*,+.1*2' G!j" 3 4!j"56!j"7 8' 9.+( (,

':;' D'+,A*+.(,' .A;1*(ED' .=' .+H1' ,@ G(s).


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M*+D (>' @ +EA'' D'+,A*+.(,

.( *=P #j

Class Example (Solution)

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9>'' &>,


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Hendrik W. Bode1905 - 1982

Frequency response of a dynamic

system is a summarization of its

(ultimate) responses to pure sine wave

inputs over a range of frequencies .

A special graph, called the Bodediagramor Bode plot, provides aconvenient display of the frequency

response characteristics of a transfer

function model.

Frequency response analysisBode diagram

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Frequency response analysisBode diagram

Frequency response of a dynamic system is a summarization of its (ultimate)

responses to pure sine wave inputs over a range of frequencies .

Bode diagram plot:

log(MR) vs. log()

Phase angle vs. log()

Process steady-state gain (K)and time constant () areoften used for scaling (easiercomparison between differentsystems) :

Magnitude Ratio:MR=AR/K

log ()Angular frequency x time constant () [rad]


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Frequency response of First-order system

l.f. asymptote

0 MR1BIG MR-log

log(MR) - log

l.f. and h.f. asymptotesintersect in:

Corner (natural)

frequency: n= 1/

for which: = -45

Slope = -1

$ $L0

#

K

# (.+

Phase angle of firstorder system isasymptotic to 0 at lowfrequencies and to -90at high frequencies(first order lag) [rad]

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Frequency response of Pure gain system

G(s) = K

AR = K

= 0

Capillary system: Proportional Controllerh = RF

[rad]


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Frequency response of an Integratingsystem

G(s) = K/s

AR = K/

= -tan-1() = -90

Storage (surge) tankwith an outlet pump

Slope = -1

[rad]

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General Second order Systems

Examples of physical systems:

Overdamped:

Two noninteracting first-ordersystems (tanks) in series

Two interacting first-ordersystems (tanks) in series


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Examples of physical systems:

Underdamped:

Plucked guitar string Pendulum fromgrandfathers clock

Look forward: IMPORTANCE in Controller design

Spring shock absorber

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&"0

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0 MR1

MR-2log

Natural frequency: n= 1/

0 0

-180

=1 -90 for all

Resonance:

hump for 1

'( ( '1

l.f. asymptote

Slope = -2

.-/ '

(

' (20

Application: Design of organ pipesvs. vibration of automobiles

[rad]


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Tacoma Narrows Bridge Failure (1940)

Importance of periodic inputs on the dynamic behavior of the

process

Driving frequency = natural frequency of the systems

LARGE oscillations

input output

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Mechanical structures

Wind speed Tower displacement


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H.f. asymptote a straight line of slope N

L.f. asymptote the horizontal line: MR = 1

Phase angle plot starts at 0 at =0 and approaches

asymptotically -(90N) at very high frequencies

The complete bode diagram of the Nthorder systemhas the properties of the lower order components

will have similar character as the ones studied

Generalization of frequency response forNthorder systems

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Frequency response of several systems inseries

Frequency response characteristics of complex linearsystems can be obtained by summing up the log(AR)and phase angles of the individual contributing systems.

G1(s) G2(s) Gn(s)

Overall TF:

G(s)= G1(s) G2(s) Gn(s)

G(j)= G1(j) G2(j) Gn(j) ( '( '% & * % &* * % &* * % &*

3+ + +

3, + , + ' , + ' , + '

( '% &

( '% & * % &* * % &** % &* * % &* 3+ +

3, + , + ' , + , + , + '

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( ' 3


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Class exercise

What can you tell about the process thathas the following frequency response?

[rad]

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Systems with Zeros

G(s) = s+ 1

Slope = 1

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(

(

,-$ 4

"0

LHP and RHP zeros same AR

LHP zero improves with 90for each zero at high

RHP zero worsens with -90for each zero at high

( *

( *

First order lead

Can such a physical system exist ?

[rad]


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Example process with inverseresponse behavior: drum boiler

In the long run, the level is expected to increase, because we have

increased the feed material without changing the heat supply

But immediately afterthe cold water has been increased, a drop in the

drum liquid temperature is observed, which causes the bubbles to collapse

and the observed level to reduce

Disturbance:step increase in the cold

feedwater flowrate

Output:level in the boiler

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Systems with time delay

Typical example: plug flow in a pipe

Input(u)

Output(y

)

= dead time (time delay)

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Systems with time delay

System with transport lag is difficult to analyze for stabilityreasons (e.g. by Routh test)

It is also difficult to simulate

Approximation can be used applying Taylor series expansion (leadto lead-lag system, i.e. with nominator dynamics)

( +'

( +'* *'

*

First order Pad:

Low frequency first order Padeis satisfactory

In practice also used with firstorder FOPDT to identify higherorder dynamics

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Filter Design Why?

Noise is the non-repeatable componentof a measurement

Causes: turbulence, imperfect mixing,electrical interference, etc.

Slow variations lowfrequency

fast variations highfrequency


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Filter Design

A filter is a device that impedes the passage ofsignals whose frequencies fall within a bandcalled the stop band

It permits frequencies those in the pass bandthrough relatively unchanged

In signal processing, it removes unwanted partsof the signal, such as random noise, or to

extract useful parts of the signal EXAMPLES

Low pass filter

High pass filter

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Filter Design

1

MR

0

Ideal LPF

1

MR

0

Ideal HPF

Overall MR is the product of the MR of the filter and the rest of theprocess

Design filter to have MR = 0 in particular frequency bands to filterthe signal with this frequency

In practice this behavior is not possible. More complicated design(high order systems, lead-lag, etc.)

Simplest filter used is first order with small timeconstant Which filter type is approximated with the first order system?


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Filter Design

Ideal LPF

Ideal HPF

How should the phase angle look like for the ideal filters?

In practice ideal filter dynamics is not possible. Morecomplicated design (high order systems, lead-lag, etc.)

Simplest filter used is first order with small timeconstant

Which filter type is approximated with the first order system?

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Filter Design

Low pass filter

High pass filter

FilterRaw signal Filtered signal


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Filter Design - example

Yellowfilter

White light Yellow light

red orange yellow green blue violet

AR

frequency

Gain (AR) is ~0 for all frequenciesother than those for the yellow light

1

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What can we do with this tool ?

Model identification (frequency responseis fingerprint for process)

Design Process

Asses stability

Controller Design


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Class Exercise: Design Example 1

Problem: Find volume of tank necessaryto dampen oscillations in inlet feedconcentration of amplitude 200 g/ m3 downto 20 g/m3

feed flow rate is 1 m3/min

inlet period = 5 min (frequency?)

f

$J

$

- angular frequency (radians/min)

T period (min)

f frequency (min-1)

Hint:

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Class Exercise: Design Example 1

Help:

Tank - first-order process

Need an amplitude ratio of 0.1

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s

KsG

p


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Solution

find time constant and then minimumvolume:

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Class Exercise 2

4"(/

( (B"( .' 'B"( .

'"(

Sine disturbance with

Amplitude = 1 mol/l

Frequency = 0.20 rad/min

Must have fluctuations

< 0.1 mol/l

We know: V1 = 10 l

V2 = 20 l

F= 2 l/min

Solution:


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Summary

Frequency response characteristics (AR and )

characterize the dynamic behavior of the process

Analytical expressions for AR and can be derived

from TR

Bode diagram convenient way to represent

frequency response of systems

Studied the frequency response characteristics of

various class of systems

Next:Frequency response analysis useful techniquefor stability analysis and controller design

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Sample Exam Questions From ThisLecture

What can you tell about theprocess that has thefollowing frequencyresponse?

Give an example ofunderdamped first order

process.


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Suggestion for Self-study

Draw the Bode diagram of different linear systems.Exchange them with friends and discuss the dynamicbehavior of the systems based on their frequencyresponses.

Sketchthe Bode diagram for the systems with thetransfer function given below. Be sure to identifyclearly on the sketch all the important distinguishingcharacteristics.

'% &

(

*&', *

* *

Documents

Frequency Response of Linear Systems