automatic control system iv. time domain performance stability
TRANSCRIPT
Automatic Control System
IV.
Time domain performance
Stability
Transfer functions of closed loop
Gp2(s)( )e s
( )w s
( )y sGr(s)
Gw(s)
Gp1(s)Ga(s)Gc(s)
Gt(s)
( )r s
1 2
1 2
( )1
c a p pry
c a p p t
G G G GyW s
r G G G G G t2p1pacre GGGGG1
1
r
e)s(W
2
1 2
( )1
w pwy
c a p p t
G GyW s
w G G G G G2
1 2
( 1)( )
1
w p t
wec a p p t
G G GeW s
w G G G G G
( )My s
Steady-state characteristic of the feedback system
R
Y
r(t)
y(t)
t
t
The unit step response of the feedback systems shows the quality parameters in time domain.
WP2
WP1
Dynamic response of the feedback system
Time domain performance specification(rv-fv) = Steady-state error
Tolerance band
Settling time
Required value of response (rv)
Final value (fv) of response It’s 100%
Peak value (pv)
Rise time
10%
90%
Second peak value (spv)
fv
fvpvootOver
100sh%
0
)( dttyfvIa
Stability of a closed loop system
Definition of stability of linear system: Energised the closed-loop system input a short impulse which move the system from it’s steady-state position and wait. When the transient response has died and the system remove the original position or remain inside a predicted defined small area of the original position the system is stable and linear.
Definition of stability of non-linear system: Energised the closed-loop system input a short impulse which move the system from it’s steady-state position and wait. If the system remove and remain inside a predicted defined small area of the original position the system is stable.
If the transient response hasn’t died the system is unstable.
)(sGc ( )PFG s)s(G)s(G1
)s(G)s(G)s(W
)s(r
)s(y
PFC
PFCyr
Stability examination from closed loop transfer function
)(sGC )(sGPF
nm
m
m
mn
n
n dt
trdb
dt
tdrbtrbtya
dt
tdya
dt
tyda
)()()()(
)()(1001
011
1
011
1
)()(
)(
)()(
1
)()(
)()(
)(
asasasa
bsbsbsb
sNsD
sN
sDsN
sDsN
sWsr
syn
nn
n
mm
mm
yr
r(s) y(s)
Unit feedback model
)(
)()()()(0 sD
sNsGsGsG PFC
)()()()(
)()()()(
011
1
011
1
srbssrbsrsbsrsb
syassyasysasysam
mm
m
nnm
nn
Continuation of stability examination from closed loop
)(sGc )(sGp
n
ii
m
jjm
ps
zsb
1
1
)(
)(
n
i
ticTransient
ieKtx1
)(
The transient solution is given bz the roots of the differential equation’s characteristic equation.
0011
1 aaa n
nn
The roots are equal the denominator of closed loop transfer function and so it gives transient solution too.
0011
1 asasas n
nn
Definition of stability in frequency domain
)(sGc )(sGp
The system is stable if the all real part of the rootsof denominator of the transfer function is negative.
0)()(1 sGsG pc
n
1ii
m
1jjm
)ps(
)zs(b
)s(r
)s(y
The pi are named the poles and zj are named the zeros
Re
Implanes _
The root loci depends on the arrangement of poles and zeros and changing a parameter. The default: K runs from zero value to infinite. The loci shows the actual the roots’ positions at actual parameters’ value.
Stability examination from closed loop’s transfer function
)(sGc )(sGp
The root locus diagram
Poles arrangement of actual value of K
Poles and zeros arrangement
)()(
)(
)()(
1
)()(
)(sKNsD
sNK
sDsN
K
sDsN
KsWrc
Re
Im
planes _
Advantages of stability examination from closed loop’s transfer function
Poles and zeros arrangement
Acceptable region for poles
Minimum damping ratio ofthe second order parts of transfer function
Maximum settling time
Using the results derived for first andsecond-order systems enables a boundeddesign region to be drawn on the s planewithin which the poles produce anacceptable performance.
Stability examination from transfer function of opened-loop
)(sGc )(sGp
)s(G)s(G)s(G 0pc
Opened-loop transfer function
01)(0)(1 jjGsG olol From definition of stability:180j)(j
0 e1e)(A0j1)j(G
The frequency at which the open-loop system gain is unity is termedthe gain-crossover frequency. The frequency at which the open-loop system phase-shift is -180° is termed the phase-crossover frequency.
The system is stable if plots of the opened-loop transfer functionat the gain-crossover frequency the phase-shift less than -180° andat the phase-crossover frequency the amplitude gain less than unity.
Gain margin and phase margin
Opened-loop transfer function)()(01)( j
ol eAjjG
The gain margin is the reciprocal of open-loop gain at the phase-crossoverfrequency. Expressed in decibel the gain margins is: -20log(open-loop’s gain at the phase-crossover frequency) dB
The phase margin is the difference of the phase-shift of the system and -180° at the gain-crossover frequency.
The system is stable if plots of the opened-loop transfer function at thegain-crossover frequency has a positive the phase margin and at the phase-crossover frequency has a positive the gain margin.
Stability examination from transfer function of opened-loop
)(j0 e)(A0j1)j(G
Plots on j (s) planeis named Nyquist diagram.
This is a point on the s plane.
If the area of the s plane whichis bordered by the positive realaxis and the Nyquist plotsdoesn’t cover this point than the system is stable.
Independent plots the amplitude gainand phase-shift on lg axis is namedBode diagram.
It is the x axis line on the amplitude plotsand the -180 phase-shift line on thephase-shift plots.
If the gain and phase margin criterion istrue the system is stable.
Advantages of working with Bode plots
n
1ii
m
1jj
i0
)sT1(
)s1(
s
K)s(GBode form of
opened-loop transfer function
Bode plots of systems in series simply add.Bode’s important phase-gain relationship is given.A much wider range of system behaviour can be displayed
on a single plot.Dynamic compensator design can be based on Bode plots.
Using MATLAB
Start of stability examination K=1
First using root’s loci. Enabled damping ratio: 0.72, 0.45, 0,23
Secondly using Bode plots. Enabled phase shift: -120, -135, -150
Define the time domain performance specifications.
K)101)(51)(1(
1
sss
Summary questions • Definition of stability of linear and non-linear systems.• Which methods have used the closed-loop transfer function to
examine the stability a control system?• Which methods have used the opened-loop transfer function to
examine the stability a control system?• Explain the following terms: phase margin, gain margin.• Explain what is meant by the term “dead time” in control and how
it may affect the stability of a control loop.