Unformatted text preview: x6. FrequencyResponse Design Methods 1. Frequency Response:
We recall inverse Laplace transform:
2
3
K5
K
4
L,1 6 s , j! + s + j! 7 = 2jK j cos!ot + 6 K :
o
o
Consider sinusoidal input signal ut = Au cos!ot
applied to a plant Gs. Then the output in sdomain is s
Y s = GsU s = Gs s2Aus 2 = Gs s + j!Aus , j!
+ !o
o
o
K + K + terms generated by Gs:
=
s , j!o s + j!o
The K value can be computed from
sG
1
K = s!j!os , j!oY s = s!j!o Au+ j!s = 2 AuGj!o
lim
lim
s
o
= jK j = AujGj!oj=2; 6 K = 6 Gj!o: 1 By inverse Laplace transform, yt = 2jK j cos!ot + 6 K + terms generated by Gs
= AujGj!oj cos!ot + 6 Gj!o + terms of Gs:
If Gs is stable, then the steadystate response is ysst = AujGj!oj cos!ot + 6 Gj!o: Observation: The steadystate response is also sinusoidal. But
its amplitude is ampli ed by a factor of jGj!oj, and its phase
is shifted by 6 Gj!o. Example: The gure below shows ut = cos!ot dashed line, and ysss = 2 cos!ot + =4 solid line with !o = 1. Thus
Gj!o = 26 45o.
2 1.5 1 0.5 0 −0.5 −1 −1.5 −2
0 2 4 6 8 2 10 12 14 16 De nition: Gj! is termed frequency response with jGj!j magnitude response, and 6 Gj!o phase response.
Because every signal is a composition of sinusoidal functions,
jGj!j and 6 Gj!o characterizes spectral information of the
system, and they uniquely determine the behavior of the system.
Consider feedback system:
r(t) e(t) D(s) G(s) y(t)
η(t) Then there hold DGs
1
E s = 1 + DGs Rs; Y s = , 1 + DGs N s: Some primitive goals are: tracking reference signal in operating
frequency range: 0; !r , and noise rejection in frequency range
!n; 1. It thus requires ideally
1
j1 + DGj!j = 0; if 0 ! !r ; jDGj!j = 0; if ! ! :
n
j1 + DGj!j
3 Performance in Frequency Domain
Since perfect tracking and noise rejection is not possible, and
1
DGs = 1; 8! 2 R;
+
1 + DGs 1 + DGs
performance requirement are often given by
8 1
jS j!j = j1 + DGj!j : ;
1;
8
DGj!j
jT j!j = j1 j+ DGj!j : 1;
; for ! 2 0; !r
for ! 2 !n 1;
for ! 2 0; !r
for ! 2 !n 1: The performance requirement on S s is shown in the following
gure by assunming !n !r :
 S(j ω)
1+ δ
1 δ
δ ε
ωr ωn 4 Sensitivity Consideration:
Recall sensitivity:
change in T
1
S s = change in G = 1 + DGs ; that is the same as transfer function from reference input to tracking error output. Hence the tracking error performance is consistent with sensitivity requirement.
Translation to Loopshaping:
We have that jS j!j is guaranteed by
jDGj!j 1 + 1 ;
and jT j!j is guaranteed by jDGj!j 1 + : We thus translated performance requirement on sensitivity to loop
transfer function:
8
1 + ,1;
! 2 0; !r ;
jLj!j = jDGj!j :
1 + ,1; ! 2 !n; 1: 5 Loopshaping Design
A typical loopshape is shown next:
DG(jw) The lowerleft shaded area represents the performance requirement on sensitivity, and tracking, and the upperright shaded area
represents the performance requirement on noise rejection. The
loop transfer function jLj!j = jDGj!j need avoid the forbidden region shaded areas in order to satisfy the performance
requirement. This is called loopshaping design. Compensator
Ds will be designed to satisfy the design speci cations. 6 Logrithm plots:
In engineering pratice, linear scale plots are not used. Bode
invented logrithm plots where both frequency and magnitude
jGj!j in logrithm scale.
In horizontal frequency axis, the length is measured with ratio:
log!2 , log!1 = log!2=!1:
In vertical axis, the length is also measured with ratio that
has unit dB:
10 dB of jGj!j = 10 20 log10jGj!j:
See the picture below:
dB
80 (10 4 ) 60 (10 3 ) 40 (10 2 ) 20 (10) ω 2ω 4ω 7 8ω 16 ω Real factor s + z , z 0: 8 20 log10z ;
20 log10jj! + z j = :
20 log10!; !
;! z;
z: Straight line approximation gives with GL = 20 log10z : 20 dB GL z Real factor s + p,1, p 10z 0: 8 20 log10jj! + pj,1 = : ,20 log10p; !
,20 log10!; ; ! GL
20 dB p 10p 8 p;
p: largest approximation error takes place at corner frequency
!o = z or !o = p that has gain p 20 log10j!o + j!oj = 20 log10 2!o
= 3 + 20 log10!o = 3 + GL:
Thus the approximation error is at most 3dB. The recti ed
curve is depicted with dashed line as in previous page. Summary: the magnitude bode plot has a gain GL for small frequency with zero slope, and it increases slope by 20dB dec after
corner frequency if corresponding zero; and it decreases slope by
20dB dec after corner frequency if corresponding pole. Example: Find magnitude Bode plot for 10 + 5 + 100
Gs = s + 0s1s +s1s + 50
: Solution: We sketch magnitude Bode plot as follows:
Step 1: We mark corner frequency on horizontal axis with
corresponding to pole, and to zero.
Step 2: We compute low frequency gain: GL = 20 log10 jG0j = 20 log105000=5 = 60dB:
9 Step 3: We begin at frequency ! = 0:01 with zero slope,
and decrease its slope by ,20dB dec after it comes across 0.1
corresponding to pole, and decrease its slope by ,20dB dec
again after it comes across 1 corresponding to pole. Increase
its slope by 20dB dec after it comes across 5 corresponding
to zero until high frequency ! = 1000.
Rectify the plot with 3dB correction.
dB
60 40 20 log( ω)
2 1 0 1 20 40 10 2 3 Magnitude Bode Plots for Type ` Systems
Type ` system has a transfer function:
1s
Gs = s` K + + z1 s + zm ;
s p1 s + pn
where zi 6= 0 and pk 6= 0 for 1 i m and 1 k n. We have
the following modi cations:
Low frequency gain is replaced by 0
jKz1z2 zmj 1 :
@
A
GL = 20 log10 B !` jp p p j C
n
L 12 Initial slope is now ,` 20dB=dec. Example: Find magnitude Bode plot for s
Gs = 10s + 5s + 100
s + 1s + 50 Solution: We sketch magnitude Bode plot as follows:
Step 1: We mark corner frequency on horizontal axis with
corresponding to pole, and to zero.
Step 2: We compute low frequency and its gain: !L = 0:1 minfjz1j; jzmj; jp1j; ; jpnjg = 0:1;
0
10 5 100 1
GL = 20 log10 @ 0:1 1 50 A = 60dB:
11 Step 3: We compute its initial slope: ,20dB=dec.
Step 4: We begin at frequency !L; GL = 0:1; 60 with
slope of ,20dB=dec, and draw straight line by increasing !.
Step 5: We decrease its slope by ,20dB dec after it comes
across 1 corresponding to pole, and increase its slope by
20dB dec after it comes across 5 corresponding to zero
until high frequency ! = 1000.
Rectify the plot with 3dB correction.
dB
60 40 20 log( ω)
1 0 1 20 40 12 2 3 Magnitude Bode Plots for Complex Poles
Consider prototype system: 2
!n
Gs = s2 + 2
! s + !2 :
n
n Straight line approximation: 2
!n ;
Gs s + ! 2
n if 1. Thus we have zero slope straight line for !
and has a ,40dB=dec slope after ! crossing !n.
dB 0 20
ωn /10 ωn 40 13 10ωn !n, Correction at !n: To rectify the plot, 6= 1 needs take into
consideration. Since
Gs = s 2 + 1 s + 1 :
2 !n
!n
The maximum error at !n is:
1
20 log10 jGj!nj = 20 log10 :
2 dB 0 20
ωn /10 ωn 40 14 10ωn Example: Sketch maginitude Bode plot for
s
Gs = ss2002s+ 1 :
2+
+ 100
Corner frequencies are 1; 10; 10, and
2
s2 + 2s + 100 = s2 + 2
!ns + !n with !n = 10, and = 2=2!n = 2=20 = 0:1:
The low and high freqeuncies are thus !L = 0:1; !H = 100:
Compute low frequency gain: GL = 20 log10200=10 = 26dB:
Since it is type 1 system, the initial slope is ,20dB=dec. We
begin at
!L; GL = 0:1; 26
and draw stright line with initial slope ,20dB=dec.
After crossing corner frequency 1 corresponding to zero, we
increase slope by 20dB=dec, and after crossing frequency 10
corresponding double poles, we decrease slope by 40dB=dec. 15 After completion of straight line approximation, we make correction at corner frequencies. At 1, the correction is 3dB, as
it corresponds to real zero. At 10, the correction is
20 log102 ,1 = 20 log105 = 14dB:
The complete plot is shown next with dashed line for corrected
Bode plot. 20 0 20
0.1 1 10 40 16
100 Phase Bode Plot:
Straightline approximation for real pole: Gs = s K p ; K 0; p = !o 0:
+ Then its argument at s = j! is given by
8 ,5:7o; ! = 0:1!o;
6 Gj! = , tan,1!=!o = ,45o;
! = !o ;
: ,84:3o; ! = 10!o:
Since 6 Gj! = 0 for !
!o, and 6 Gj! = ,90o for ! !o, a stright line approximation is given as follows: 0.1ω o 45 90 ωo o o 17 10ω o Correction: 8 6 ,5:7o; ! = 0:1!o;
Gj! = :
,84:3o; ! = 10!o; which are at the two end points: one decade above, and one
decade below ! = !o. We also note that the following three
points are coincide with the staight line approximation:
8 6 ,45o; ! = !o;
Gj! = ,9o; ! = 0:16!o;
: ,81o; ! = 6:4!o : Thus we have the following correction, together with the
straight line approximation.
0
−10
−20
−30
−40
−50
−60
−70
−80
−90 −1
10 0 10 18 1 10 Phase plot for real zero: Gs = s + z , z = !o
6 0: Gj! = tan,1!=!o that is positive. Thus we easily get a phase plot as follows
with the correction:
90
80
70
60
50
40
30
20
10
0 −1
10 0 10 1 10 For zero corner frequency: Gs = K=s, or Gs = s, then
8 90o;
6 Gj! =
: ,90o; if corresponding to zero;
if corresponding to pole: 19 Discussion of Phase Plot:
We consider only the transfer function having all poles and
zeros on negative real axis, including the origin.
Each corner frequency corresponding to zero contributes a
slope of 45o=dec, and the slope is e ective for only two decades
centered at the corner freqeuncy.
Each corner frequency corresponding to pole contributes a
slope of ,45o=dec, and the slope is e ective for only two
decades centered at the corner freqeuncy.
Initial phase at the low freqeuncy is
8 0o ;
,180o;
PhL =
l 90o;
: ,l 90o; if G0 0;
if G0 0;
if Gs has l zero at the origin;
if Gs has m pole at the origin: The initial slope is zero if !L is smaller than 10 of the
smallest of the corner freqeuncy. 20 Example: Sketch phase Bode plot for Gs == 10s + 5s + 100
ss + 1s + 50 Solution: We sketch phase Bode plot as follows:
Step 1: We mark corner frequency on horizontal axis with
corresponding to pole, and to zero.
Step 2: Since we have one pole at the origin, the low frequency
phase is given by:
PhL = ,90o:
Step 3: We begin at frequency ! 0:1 with a phase of ,90o,
and draw straight line with zero slope.
If ! crosses ! = 0:1 that is within a decade of the corner
frequency 1 corresponding to a pole, decrease its slope by
,45o=dec.
As frequency increases to ! = 0:5, it is within a decade of
corner frequency of 5 corresponding to zero, increase its slope
by 45o=dec. 21 As frequency arrives at ! = 5, the corner frequency at 50 has
the e ect, that corresponds to a pole. In this case, decrease
the slope by ,45o=dec.
If ! crosses ! = 10 that is beyond a decade of the corner
frequency 1, the slope generated by this pole has no e ect that
eliminates the ,45o=dec generated by the corner frequency
at 1. Since ! = 10 is now within a decade of corner freqeuncy
at 100 corresponding to a zero, we need increase its slope by
45o=dec. The total e ect is that we increase the slope by
90o=dec.
As frequency increases to ! = 50 that is beyond a decade of
the corner frequency at 5 corresponding to a zero, its slope
has no e ect. Thus we decrease its slope by 45o=dec.
As frequency arrives at ! = 500, the corner frequency at 50
has no e ect, that corresponds to a pole. We thus increase
its slope by 45o=dec.
Finally we reach ! = 1000. The corner frequency at 100 now
has no e ect. Hence we increase its slope by 45o=dec. 22 Note that after ! = 1000, no corner frequency has no contribution to the slope of the straight line approximation. We
should come back to zero slope.
Rectify the plot with corrections. Another way to make correction is to compute actual phase at ! = 0:1; 0:5; 1; 5; 10; 50; 100;
and connect them smoothly with straight line approximation
as the guidance.
0.1 1 10 100 1000 90 180 We note the nal phase, or phase at the high frequency is: PhH = ,n , m 90o;
where n is the number of zeros, and n the number of poles.
For our example, n = 3 and m = 2. Thus the nal phase is
,90o.
23 Phase Bode Plot for Complex Roots:
The complex roots are treated as double roots at natural frequency !n. The correction is more involved for the phase plot
as the damping ratio determined the steepness at the corner frequency. An easy way to make correction is to compute the phase
angles at several frequencies near the natural frequency, and connect these points with a smooth curve. Example: Sketch phase plot for s
Gs = ss2002s+ 1 :
2+
+ 100 The corner frequencies are: 1; 10. Thus we set !L = 0:1; !H = 100;
and mark corner frequencies, as well as !L and !H .
Since we have one pole at the origin, PhL = ,90o.
We begin at !L; PhL. The left of this point is a straight
line with zero slope. The straight line to the right has a slope
of 45o=dec, generated by the zero corresponding to corner
freqeuncy at 1. 24 As ! increases to ! = 1, the double corner frequency at
10 corresponding to poles generates ,90o=dec which yields
,45o=dec for the slope.
As ! passess 10, the slope generated by the corner frequency
at 1 is not e ective. We thus have ,90o=dec for the slope
which disappears after ! 100.
Since n = 3, and m = 1, the nal phase is ,180o.
Correction are easily made at ! = 0:1 that should be 5:7o
higher than ,90o. It is also easy to make correction at 1 that
should be 11:4o lower, and at ! = 100 that should be 11:4o
higher due to double corner frequency.
0.1 1 10 45 90 135 180 25 100 Example: Sketch both magnitude and phase plots for
1000ss2 + s + 10
Gs = s + 12s + 10s + 100 :
The numerator has a pair of complex roots with p
pp !n = 10; = 1=2 10: Thus the corner frequencies are 1; 1; 10; 10; 10; 100. The
low and high frequencies are chosen as !L = 0:1, and !H =
100.
For magbitude plot, the low frequency gain is GL = 20 log101000 0:1 10=10 100 = 0dB:
Since there one zero at the origin, we begin with slope of
20dB=dec. We change the slope at 1 by decreasing slope by
40dB=dec as it corresponds to double pole. Hence the slope
is ,20dB=dec after ! = 1. p The slope comes back to 20dB=dec at ! = 10 due to double
zero. After ! = 10, the slope decreases to zero due to pole.
For ! 100, it rolls o at ,20dB=dec due to the pole at 10. 26 For phase plot, we have that PhL = 90o; ! !L;
due to one zero at the origin. p For ! 2 0:1; 0:1 10 , the slope is ,45o=dec due to double
corner freqeuncy at 1. p For ! 2 0:1 10; 1 , the slope is zero due to the cancellation
p
e ect of the double corner frequency at 1 and 10.
For ! 2 1 10 , the corner frequency at 10 is also in e ect
that gives ,45o=dec slope. p For ! 2 10; 10 10 , the double corner frequency at 1 is not
e ective, but all others are that give zero slope. p For ! 2 10 10; 100 , the slope is generated by only pole at
10.
The nal phase is ,90o as n = 4 and m = 3. 27 The magnitude and phase plots are given as follows:
dB
20 1000
0.1 1 10 100 20
degree
90 90 28 Read Example 6.5 of page 355. 29 ...
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 Fall '10
 GU
 Frequency, Signal Processing, Decibel, corner frequency

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