Transcript Automatic Control

Automatic Control
By
Dr. / Mohamed Ahmed Ebrahim Mohamed
E-mail: [email protected]
Web site: http://bu.edu.eg/staff/mohamedmohamed033
Dr. / Mohamed Ahmed Ebrahim Mohamed
1
• Introduction.
2
• Frequency Response Definition.
3
• Bode Plot Definition.
4
• Frequency Response Plot.
5
• Viewpoints of analyzing control system behavior.
6
• Logarithmic coordinate.
7
• Bode Plot Construction.
What is frequency response of a system?
 The frequency response of a system is
defined as the steady-state response of the
system to a sinusoidal input signal.
 The sinusoid is a unique input signal, and the
resulting output signal for a linear system, as
well as signals throughout the system, is
sinusoidal in the steady-state.
What is Bode Plot?
Bode Plot is a (semi log) plot of the
transfer function magnitude and phase
angle as a function of frequency.
Polar Plots
• The transfer function can be separated into magnitude and
phase angle information
H(j) = |H(j)| Φ(j)
e.g., H(j)=Z(j)
Viewpoints of analyzing control
system behavior
•
•
•
•
•
•
Routh-Hurwitz (s    j )
Root locus (s    j )
Bode diagram (plots) (s  j )
Nyquist plots (s  j )
Nicols plots (s  j )
Time domain
y(t )  B sin(t   )
r (t )  A sin t
L.T.I system
Magnitude:
B
A
Phase:

y (t )
r (t )
G(s)
＋
－
Steady state response
H(s)
Y (s)
G(s)

R( s) 1  G ( s) H ( s)
Magnitude:
G( j )
1  G( j ) H ( j )
s    j  s  j
Phase:
G ( j )
[1  G ( j ) H ( j )]
Decade :
dec  log10
2
1
Octave :
2
oct  log2
1
dB

1
2
3 4
10
20
100
• The gain magnitude is many times expressed in terms of decibels (dB)
dB = 20 log10 A
where A is the amplitude or gain
– a decade is defined as any 10-to-1 frequency range
– an octave is any 2-to-1 frequency range
20 dB/decade = 6 dB/octave
Y ( s)
k ( s  z1 )(s  z2 ) 

R( s) ( s  p1 )(s  p2 )(s 2  as  b) 
GH (dB)
Case I : k
Magnitude:
0 .1
1
10

k dB  20log k (dB)
GH
1800
Phase:
 0
k   o
180
o
,k  0
,k  0
900

1
sp
Case II :
Magnitude:
1
( j ) p
0 .1
dB
1
( j ) p
p2
p 1
 20 p log (dB)
1
10

GH
Phase:

GH (dB)
 (90o )  p
900
 900
 1800
p 1
p2

Case III :
sp
p2
GH (dB)
Magnitude:
( j ) p
dB
p 1
 20 p log  (dB )
0 .1
GH
Phase:
0
180
0
( j )  (90 )  p
p
90
o
 900
 1800
1
10

p2
p 1

a
1
or ( s  1) 1
( s  a)
a
Case IV :
a 1
Magnitude:
(1  j

a

) 1
dB
GH (dB)
 20 log 1  ( ) 2
a

 10 log[1  ( ) 2 ]
a
  a 

0 .1
1800
  a  1  j1  dB  10log 2  3.01
900
GH
Phase:
(1  j
a
)  0  tan
0
1

 900
a
 1800

  a   0  GH  tan 1 0  0o
a

  a     GH   tan 1   90 o
a
10
 0  dB  10 log 1  0
a
 

  a  1  j   dB  20log
a a
a
dB  [20 log  20 log a]

1

  a  450

( s  a)
or
a
Case V :
1
( s  1)
a
a 1
Magnitude:
(1  j

a

)
dB
GH (dB)
 20 log 1  ( ) 2
a

 10 log[1  ( ) 2 ]
a
  a 

0 .1
  a  1  j1  dB  10log 2  3.01
GH
1800
900
Phase:
(1  j
  a 
  a 
a

a

a
10
 0  dB  10 log 1  0
a
 

  a  1  j   dB  20 log
a a
a
dB  20 log  20 log a

1

)  tan
1

 900
a
 1800
 0  GH  tan 1 0  0o
   GH  tan 1   90 o
  a  450

 n2
T ( s)  2
s  2 n s   n2
Case VI :
T ( j ) 
T ( j ) 
 n2
( n   2 )  2 j n
2
1
 2

(1  ( ) )  j 2
n
n
 T ( j )   t an1
2n
2
( n   2 )

n
1
 T ( j )   t an
 2
1 ( )
n
2



,
 1
0 ,
 1

0
n


0
n





0
1
T ( j )    20 log(2 ) ,
 1 T ( j )    90 ,
n

 180o  n






40
log(
)
,

1
,
 1



n
n

n
  n
Example :
50(s  2)
T (s ) 
s (s  10)
1 s  2 10
T ( s)  10( )(
)(
)
s
2
s  10
Minimum phase system
k ( s  z1 ) 
T ( s)  n
, zi  0, pi  0
s ( s  p1 ) 
Type 0 : (i.e. n=0)
0dB/dec
GH (dB)
T ( s) 
k p p1
(s  p1 )
20log K p  A
A
0.1 p1
p1
10p1

Type I : (i.e. n=1)
kv p1
T (s) 
s ( s  p1 )
GH (dB)
-20dB/dec
A
20log Kv  A
Kv
20 log
 0dB
j 0
0  kv
10p1
0.1 p1 1
p1
0
-40dB/dec

Type 2 : (i.e. n=2)
-40dB/dec
ka p1
T ( s)  2
s ( s  p1 )
GH (dB)
A
20log Ka  A
Ka
20log
 0dB
2
( j0 )
0  ka
2
0
0.1 p1 1
p1
-60dB/dec
10p1

Relative Stability
A transfer function is called minimum phase when all the poles and zeroes are
LHP and non-minimum-phase when there are RHP poles or zeroes.
Minimum phase system
Stable
The gain margin (GM) is the distance on the bode magnitude plot from
the amplitude at the phase crossover frequency up to the 0 dB point.
GM=-(dB of GH measured at the phase crossover frequency)
The phase margin (PM) is the distance from -180 up to the phase at the
gain crossover frequency. PM=180+phase of GH measured at the gain
crossover frequency
Open loop transfer function :
Closed-loop transfer function :
G( s) H ( s)
1  G( s ) H ( s )
Open loop Stability  poles of G( s) H ( s) in LHP
Im
Closed-loop Stability 
poles of
in left side of (-1,0)
RHP
G( s) H ( s)
Re
(1,0)
(0,0)
 0dB
(1,0)  
0

180

GH (dB)

g
G.M.>0
GH
1800
Stable system
0
90

 900
 180
0
P.M.>0
p
Gain crossover frequency:
g
phase crossover frequency:
p
GH (dB)
G.M.<0
g

Unstable system
GH
1800
Stable system
900

 900
 180
0
p
P.M.<0
Unstable system
• Straight-line approximations of the Bode plot may be drawn
quickly from knowing the poles and zeroes
– response approaches a minimum near the zeroes
– response approaches a maximum near the poles
• The overall effect of constant, zero and pole terms
Term
Magnitude
Break
Constant (K)
N/A
Zero
Pole
Asymptotic
Magnitude Slope
Asymptotic
Phase Shift
0
0
upward
+20 dB/decade
+ 90
downward
–20 dB/decade
– 90
• Express the transfer function in standard form


K  j  (1  j1 ) 1  2 2 ( j 2 )  ( j2 ) 2 
H( j ) 
(1  ja ) 1  2 b ( jb )  ( jb ) 2 
N

• There are four different factors:
–
–
–
–
Constant gain term, K
Poles or zeroes at the origin, (j)±N
Poles or zeroes of the form (1+ j)
Quadratic poles or zeroes of the form 1+2(j)+(j)2

• We can combine the constant gain term (K) and the N pole(s)
or zero(s) at the origin such that the magnitude crosses 0 dB
at
K
1/ N
Pole :


K
0 dB
( j ) N
Zero : K ( j ) N
 0 dB  (1 / K )1 / N
• Define the break frequency to be at ω=1/ with magnitude at
±3 dB and phase at ±45°
Magnitude Behavior
Factor
Constant
Low
Freq
Break
Asymptotic
Phase Behavior
Low
Freq
Break
Asymptotic
20 log10(K) for all frequencies
0 for all frequencies
Poles or
zeros at origin
±20N dB/decade for all
frequencies with a crossover of
0 dB at ω=1
±90(N) for all frequencies
First order
(simple) poles
or zeros
0 dB
±3N dB
at ω=1/
±20N
dB/decade
0
±45(N) with
slope ±45(N)
per decade
±90(N)
Quadratic
poles or zeros
0 dB
see ζ at
ω=1/
±40N
dB/decade
0
±90(N)
±180(N)
where N is the number of roots of value τ
Single Pole & Zero Bode Plots
Gain
ωp
Gain
0 dB
+20 dB
–20 dB
0 dB
ωz
ω
ω
Phase
0°
+90°
–45°
+45°
–90°
0°
ω
Pole at
ωp=1/
One
Decade
Phase
One
Decade
Assume K=1
20 log10(K) = 0 dB
ω
Zero at
ωz=1/
• Further refinement of the magnitude characteristic for first
order poles and zeros is possible since
Magnitude at half break frequency:
|H(½b)| = ±1 dB
Magnitude at break frequency:|H(b)| = ±3 dB
Magnitude at twice break frequency: |H(2b)| = ±7 dB
• Second order poles (and zeros) require that the damping ratio
( value) be taken into account; see Fig. 9-30 in textbook
• We can also take the Bode plot and extract the transfer
function from it (although in reality there will be error
associated with our extracting information from the graph)
• First, determine the constant gain factor, K
• Next, move from lowest to highest frequency noting the
appearance and order of the poles and zeros
Frequency Response Plots
Bode Plots – Real Poles (Graphical Construction)
Frequency Response Plots
Bode Plots – Real Poles
Frequency Response Plots
Bode Plots – Real Poles
Gain and Phase Margin
Let's say that we have the following system:
where K is a variable (constant) gain and G(s) is the plant under consideration.
The gain margin is defined as the change in open loop gain required to make
the system unstable. Systems with greater gain margins can withstand greater
changes in system parameters before becoming unstable in closed loop. Keep
in mind that unity gain in magnitude is equal to a gain of zero in dB
The phase margin is defined as the change in open loop phase shift required to
make a closed loop system unstable.
The phase margin is the difference in phase between the phase curve and -180
deg at the point corresponding to the frequency that gives us a gain of 0dB (the
gain cross over frequency, Wgc).
Likewise, the gain margin is the difference between the magnitude curve and
0dB at the point corresponding to the frequency that gives us a phase of -180
deg (the phase cross over frequency, Wpc).
Gain and Phase Margin
-180
Examples - Bode
Examples - Bode
Examples – Bode
Mohamed Ahmed
Ebrahim