View
223
Download
0
Category
Preview:
Citation preview
7/25/2019 System Oheads
1/17
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
School of Chemical Engineering and Advanced Materials
Newcastle University3
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
7/25/2019 System Oheads
2/17
2
School of Chemical Engineering and Advanced Materials
Newcastle University4
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
== )()(
)()(
)(
School of Chemical Engineering and Advanced Materials
Newcastle University5
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
School of Chemical Engineering and Advanced Materials
Newcastle University6
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
7/25/2019 System Oheads
3/17
3
School of Chemical Engineering and Advanced Materials
Newcastle University7
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
School of Chemical Engineering and Advanced Materials
Newcastle University8
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
School of Chemical Engineering and Advanced Materials
Newcastle University9
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
7/25/2019 System Oheads
4/17
4
School of Chemical Engineering and Advanced Materials
Newcastle University10
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
School of Chemical Engineering and Advanced Materials
Newcastle University11
Unit step inputUnit step input
School of Chemical Engineering and Advanced Materials
Newcastle University12
Responses of systems with different gainsResponses of systems with diff erent gains
7/25/2019 System Oheads
5/17
5
School of Chemical Engineering and Advanced Materials
Newcastle University13
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
School of Chemical Engineering and Advanced Materials
Newcastle University14
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:
School of Chemical Engineering and Advanced Materials
Newcastle University15
11stst
order systemsorder systems -- different time constantsdifferent time constants
ssU
sY
+=
1
1
)(
)(
1
2
1 < 2 < 3
3
7/25/2019 System Oheads
6/17
6
School of Chemical Engineering and Advanced Materials
Newcastle University16
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)(
)(
School of Chemical Engineering and Advanced Materials
Newcastle University17
Different timeDifferent time--delaysdelays
with delay
without delay
time-delay
School of Chemical Engineering and Advanced Materials
Newcastle University18
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 ==)(
)()(
)(
)(
7/25/2019 System Oheads
7/17
7
School of Chemical Engineering and Advanced Materials
Newcastle University19
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 == )()()()( )(
School of Chemical Engineering and Advanced Materials
Newcastle University20
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
School of Chemical Engineering and Advanced Materials
Newcastle University21
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
7/25/2019 System Oheads
8/17
8
School of Chemical Engineering and Advanced Materials
Newcastle University22
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
School of Chemical Engineering and Advanced Materials
Newcastle University23
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 =
)(
)()(
School of Chemical Engineering and Advanced Materials
Newcastle University24
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)
7/25/2019 System Oheads
9/17
9
School of Chemical Engineering and Advanced Materials
Newcastle University25
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
School of Chemical Engineering and Advanced Materials
Newcastle University26
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
School of Chemical Engineering and Advanced Materials
Newcastle University27
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
=
+
7/25/2019 System Oheads
10/17
10
School of Chemical Engineering and Advanced Materials
Newcastle University28
Zeros and ini tial responsesZeros and ini tial responses
G s as
s s( )
( )( )=
++ +
1
1 2 1 3
School of Chemical Engineering and Advanced Materials
Newcastle University29
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
School of Chemical Engineering and Advanced Materials
Newcastle University30
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
7/25/2019 System Oheads
11/17
11
School of Chemical Engineering and Advanced Materials
Newcastle University31
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/
School of Chemical Engineering and Advanced Materials
Newcastle University32
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/
School of Chemical Engineering and Advanced Materials
Newcastle University33
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)
7/25/2019 System Oheads
12/17
12
School of Chemical Engineering and Advanced Materials
Newcastle University34
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
School of Chemical Engineering and Advanced Materials
Newcastle University35
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
=
+ +
School of Chemical Engineering and Advanced Materials
Newcastle University36
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
=
+ +
7/25/2019 System Oheads
13/17
13
School of Chemical Engineering and Advanced Materials
Newcastle University37
22ndnd order overdamped systemsorder overdamped systems
School of Chemical Engineering and Advanced Materials
Newcastle University38
11stst order & 2order & 2ndnd order overdamped systemorder overdamped system
School of Chemical Engineering and Advanced Materials
Newcastle University39
Initial responses of 1Initial responses of 1stst
& 2& 2ndnd
order systemsorder systems
7/25/2019 System Oheads
14/17
14
School of Chemical Engineering and Advanced Materials
Newcastle University40
22ndnd order oscillatory systemsorder oscillatory systems
School of Chemical Engineering and Advanced Materials
Newcastle University41
=1
=0.5
=2
Damping factorDamping factor2
2 2( )
2
n
n n
G ss s
=
+ +
School of Chemical Engineering and Advanced Materials
Newcastle University42
Characterising oscillatory responsesCharacterising oscillatory responses Overshoot
Decay ratio
Rise time
Settling time/Response time
Period of oscillation
7/25/2019 System Oheads
15/17
15
School of Chemical Engineering and Advanced Materials
Newcastle University43
OvershootOvershoot
Overshoot = A/C =
21exp
School of Chemical Engineering and Advanced Materials
Newcastle University44
Decay ratioDecay ratio
Decay ratio = B/A =
21
2exp
School of Chemical Engineering and Advanced Materials
Newcastle University45
Rise timeRise time
Rise time: time taken to first reach
final equilibrium value
7/25/2019 System Oheads
16/17
16
School of Chemical Engineering and Advanced Materials
Newcastle University46
Settling time/Response timeSettling time/Response time
Settling time: time taken reach andremain within 5% of final equilibrium
value
School of Chemical Engineering and Advanced Materials
Newcastle University47
Period of oscillationPeriod of oscillation
PPeriod of oscillation (P): time
elapsed between successive peaks
[time/cycle]
School of Chemical Engineering and Advanced Materials
Newcastle University48
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
7/25/2019 System Oheads
17/17
School of Chemical Engineering and Advanced Materials
Newcastle University49
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
Recommended