system oheads

7/25/2019 System Oheads

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1

1School of Chemical Engineering and Advanced Materials

Newcastle University

SYSTEM CHARACTERISTICSSYSTEM CHARACTERISTICS

School of Chemical Engineering and Advanced Materials

Newcastle University2

ScopeScope

Different forms of transfer functions

Parameters of transfer functions

Responses of typical systems

Terminology



Transfer functionsTransfer functions Relates the dynamic behaviour of an output

Y(s) to an input U(s)

General form:

( )s

n

s

n

esAs

sB

easasasas

bsbsbsbsG

sU

sY

=

++++

++++==

)(

)(

)()(

)(

01

1

1

01

1

1


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Transfer FunctionsTransfer Functions

assumed that the polynomials A(s) and B(s)

do not have common factors

is the time-delay

n denotes the system type

if n=1, system is a Type 1 system

if n=3, system is a Type 3 system

s

n esAs

sB

sGsU

sY

== )()(

)()(

)(



Transfer functionsTransfer functions

Specific forms:

s

n e

pspspss

zszszssG

=

)())((

)())(()(

21

21

Pole-zero form

sn esasasas

sbsbsbKsG

+++

+++= )1()1)(1(

)1()1)(1()(

21

21

Time-constantform



TimeTime--constant formconstant form For the time-constant form:

the parameters of the transfer function are:

system gain, K

time-constants of lag terms,

time constants of lead terms,

time-delay,

system type, n

s

n e

sasasas

sbsbsbKsG

+++

+++=

)1()1)(1(

)1()1)(1()(

21

21

ia

ib


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PolePole--zero formzero form

For the pole-zero form

the parameters of the transfer function are:

zeros,

poles,

time-delay,

system type, n

s

n e

pspspss

zszszssG

=

)())((

)())(()(

21

21

iz

ip



Parameters of transfer functionsParameters of transfer functions

The parameters of different transfer function

representations are related

Transfer function parameters characterise the

behaviour of an output to an input

speed of response

whether the response is stable

whether the response is oscillatory equilibrium points



GainGain

The gain determines the degree of modification

(scaling) that an input will be subject to by the

system

if K>1, then amplification

if K


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GainGain

It is defined as:

Can be obtained analytically by setting all the

s terms to zero (equivalent to steady-state)

Can be determined experimentally using step-

tests

s

n esasasas

sbsbsbK

sG

+++

+++= )1()1)(1(

)1()1)(1(

)(21

21

inputinchangefinal

outputinchangefinal=K



Unit step inputUnit step input



Responses of systems with different gainsResponses of systems with diff erent gains


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TimeTime--constantconstant

Has units of time

Determines the speed of response

Also known as residence time

For a pure first-order system, the time

constant is the time taken for the system to

reach 63.2% of its final change in value



Time constant of 1Time constant of 1stst order systemorder system

Why 63.2% of its final change in value?

dy t

dty t Ku t

( )( ) ( )+ =

s

K

sU

sYsG

+==

1)(

)()(

y t K e t( ) /= 1

=== tKeKty when632.01)( 1

Time domain solution:



11stst

order systemsorder systems -- different time constantsdifferent time constants

ssU

sY

+=

1

1

)(

)(

1

2

1 < 2 < 3

3


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TimeTime--delaysdelays

Also called dead-time

The time taken, after the system has been

perturbed, before the system starts to react

It is a measure of the systems time inertia

A 1st-order system, with gain K; time-constant ;and time delay of magnitude is given by

)()()(

=+ tKutydt

tdy

s

Ke

sU

sY s

+=

1)(

)(



Different timeDifferent time--delaysdelays

with delay

without delay

time-delay



System typeSystem type

Indicated by the integer 'n'.

If n=0, then G(s) is classified as a 'type-0' system

If n=1, then G(s) is a 'type-1' system, etc.

Type 1 systems and above are said to have

integrating properties, and have significant

implications in process control systems

s

n esAs

sBsG

sU

sY ==)(

)()(

)(

)(


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System orderSystem order

The order of the system is given by the order of

the denominator polynomial plus the integer 'n'.

It is equivalent to the order of the differential

equation that gives rise to the transfer function

after Laplace transformation

sn e

sAs

sBsG

sU

sY == )()()()( )(



Poles and ZerosPoles and Zeros

Poles: roots of the denominator

Zeros: roots of the numerator

Can be either real or complex

Complex poles or zeros always occur ascomplex-conjugate pairs

s

n e

pspspss

zszszssG

=

)())((

)())(()(

21

21

iz

ip



ZerosZeros

roots of the numerator polynomial, i.e. those

values of 's' which sets G(s) to zero

their values determine the shape of the initial

response of the system

s

n e

pspspss

zszszssG

=

)())((

)())(()(

21

21


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PolesPoles

roots of the denominator polynomial, i.e. those

values of 's' which sets G(s) to infinity

determine whether system response is

oscillatory

determine the stability of the system

sn e

pspspss

zszszssG

= )())(()())(()(

21

21



Characteristic Polynomial & EquationCharacteristic Polynomial & Equation

The characteristic polynomial is the polynomial

in the denominator of transfer functions

Setting the characteristic polynomial to zerogives the characteristic equation

Poles are those values of s that satisfy the

characteristic equation

s

n esAs

sBsG =

)(

)()(



The sThe s--planeplane How zeros and poles affect a systems

response depends on their positions in the s-

plane

Imaginary

Real

s-plane

Right Half Plane

(RHP)

Left Half Plane

(LHP)


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Zeros and ini tial responsesZeros and ini tial responses

The values of transfer function zeros affect thesystems initial responses

In general, the presence of zeros increases thespeed of response

Under certain conditions, zeros cause inverse

responses to occur

s

n e

pspspss

zszszssG

=)())((

)())(()(21

21



Zeros and inverse responsesZeros and inverse responses

An inverse response is one where the initial

response is in a direction opposite to that which

it eventually settles out



Zeros and inverse responsesZeros and inverse responses A system exhibiting inverse response has at

least one zero with a positive real part

It is caused by components of the system with

responses that oppose each other

)(1 sG)(sY)(sU

)(2 sG

+

)21(

1

)31(

2)(

sssG

+

++

=

1

2( )

(1 3 )G s

s=

+

2

1( )

(1 2 )G s

s

=

+


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Zeros and ini tial responsesZeros and ini tial responses

G s as

s s( )

( )( )=

++ +

1

1 2 1 3



Poles and system stabili tyPoles and system stabilit y

A system is stable if, for a bounded input, the

output response is also bounded

Bounded-input bounded-output (BIBO) stability

The stability of systems represented by Laplace

transfer functions are governed by the positions

of their poles in the s-plane

A system is unstable if it has a pole with a +vereal part or

A system is stable if all its poles have ve real

parts



Stability: a 1Stability: a 1stst

order exampleorder example

Has no zeros, has 1 pole

Has time domain solution:

G s K

s( )=

+1

==+ /101 ss

= /1)( teKty


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Stability: a 1Stability: a 1stst order exampleorder example

Pole is:

If is positive, pole is negative (in LHP)

exp(-t/ ) is a decaying function as time, t, tends towards infinity, y(t) settles to a

equilibrium value of K

response is stable

[ ]= /1)( teKty (1/ )( )

1 (1/ )K KG s

s s= =+ +

1/



Stability: a 1Stability: a 1stst order exampleorder example

Pole is:

If is negative, pole is positive (in RHP)

exp(-t/ ) is an increasing function as time, t, tends towards infinity, y(t) also tends

towards infinity

response is unstable

= /1)( teKty(1/ )

( )1 (1/ )

K KG s

s s

= =

+ +

1/



StabilityStability A system represented by a Laplace transfer

function is stable only if all its poles lie in the

left-half of the s-plane

Imaginary

Real

s-plane

Right Half Plane

(RHP)

Left Half Plane

(LHP)


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22ndnd order systemsorder systems

Simplest example of higher order systems

Systems may be naturally 2nd order

More usually due to combinations of 2 1st order

components

)21(

1.

)31(

1

)().()( 21

ss

sGsGsG

++=

=

)(1 sG

)(sY)(sU

)(2 sG



General 2General 2ndnd order transfer functionorder transfer function

Because the characteristic polynomial is a 2nd

order polynomial

the transfer function has 2 poles

the poles can be real or complex,

depends on and n (pronounced zeta) is the damping factor n is the natural frequency

2

2 2( )

2

n

n n

G ss s

=

+ +



Poles of 2Poles of 2ndnd

order systemorder system

There are 3 possible cases

Poles are real and distinct (not equal)

overdamped, non-oscillatory response

Poles are real and equal

critically damped, fastest non-oscillatory response

Poles are complex

underdamped, oscillatory response

2

2 2( )

2

n

n n

G ss s

=

+ +


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22ndnd order overdamped systemsorder overdamped systems



11stst order & 2order & 2ndnd order overdamped systemorder overdamped system



Initial responses of 1Initial responses of 1stst

& 2& 2ndnd

order systemsorder systems


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22ndnd order oscillatory systemsorder oscillatory systems



=1

=0.5

=2

Damping factorDamping factor2

2 2( )

2

n

n n

G ss s

=

+ +



Characterising oscillatory responsesCharacterising oscillatory responses Overshoot

Decay ratio

Rise time

Settling time/Response time

Period of oscillation


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OvershootOvershoot

Overshoot = A/C =

21exp



Decay ratioDecay ratio

Decay ratio = B/A =

21

2exp



Rise timeRise time

Rise time: time taken to first reach

final equilibrium value


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Settling time/Response timeSettling time/Response time

Settling time: time taken reach andremain within 5% of final equilibrium

value



Period of oscillationPeriod of oscillation

PPeriod of oscillation (P): time

elapsed between successive peaks

[time/cycle]



Frequency of oscillationFrequency of oscillation Has units

(f ) number of oscillations per unit time, e.g.cycles per second (Hertz)

( ) radians per unit time

The natural frequency, n, is the frequency atwhich the system response oscillates when

there is no damping

Pf /1= 21/22 === nPf


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Oscillatory responsesOscillatory responses

Will only occur if transfer function has complex

poles

Complex poles occur as complex conjugate

pairs

First order systems will never give oscillatory

responses

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