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Control Theory (035188)lecture no. 13
Leonid Mirkin
Faculty of Mechanical EngineeringTechnion IIT
Outline
Motivating example
Sampling in frequency domain
The Sampling Theorem (Whittaker-Kotelnikov-Shannon)
Sampled-data controllers with ZOH in frequency domain
Antialiasing filtering
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Deadbeat controller for double integrator
Let P.s/ D 1s2
. Its discretized (nonminimum-phase) version is
NP ./ D h2
2
C 12 2C 1:
Consider the design of deadbeat controller by pole placement and Sylvestermatrix (like in Lecture 5). We choose bi-proper 1-order controller, assigningNcl./ D 3 (always possible), via solving for controller parameters2664
1 0 0 0
2 1 h2=2 01 2 h2=2 h2=20 1 0 h2=2
3775
MS2
26641010
3775 D26641
0
0
0
3775 :
This results in 1 D 1, 0 D 34 , 1 D 52h2 , 0 D 32h2 , so that the controller is
NC./ D 2h25 34C 3:
Deadbeat controller for double integrator: 2DOF design
The discrete complementary sensitivity is then
NT ./ D .5 3/.C 1/43
(poles are roots of Ncl./ and zeros are zeros of NP ./ and NC./) with staticgain NT .1/ D 1. Stable zero at D 0:6 can be canceled by prefilter
NF ./ D 25 3
(we want to keep NF .1/ D 1), which corresponds to the (analysis) scheme
h2
2
C1.1/2
2
h2534C3
2
53NrNuNy
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Noise-free simulation
1
s2HZOH
SIdl
2
h2534C3
2
53NrNuuy
Then,
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
Time
Plan
t out
put,
y(t)
0 1 2 3 4 5 6 7 8 9 10
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
Time
Cont
rol s
igna
l, u(t
)
Simulations with measurement noise
1
s2HZOH
SIdl
2
h2534C3
2
53NrNuuy
n
Now,
0 10 20 30 40 50 60 70 80 90 1000.2
0
0.2
0.4
0.6
0.8
1
1.2
Time, t
Plan
toutpu
t,y.t/,an
dno
ise,
n.t/
0 10 20 30 40 50 60 70 80 90 1000.2
0
0.2
0.4
0.6
0.8
1
1.2
Time, t
Plan
toutpu
t,y.t/,an
dno
ise,
n.t/
n.t/ D sin. 2ht / n.t/ D sin. 2:2
ht /
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Simulations with measurement noise (contd)
0 10 20 30 40 50 60 70 80 90 1000.2
0
0.2
0.4
0.6
0.8
1
1.2
Time, t
Plan
toutpu
t,y.t/,an
dno
ise,
n.t/
0 10 20 30 40 50 60 70 80 90 1000.2
0
0.2
0.4
0.6
0.8
1
1.2
Time, t
Plan
toutpu
t,y.t/,an
dno
ise,
n.t/
Oops,I steady-state output response has different frequencies than input.
This is impossible in continuous-time LTI systems. Explanations?
Outline
Motivating example
Sampling in frequency domain
The Sampling Theorem (Whittaker-Kotelnikov-Shannon)
Sampled-data controllers with ZOH in frequency domain
Antialiasing filtering
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A weird function
Consider analog signal f .t/ with spectrum (Fourier transform) F.!/:
F.!
/
!0 !s!s 2!s2!s
and define, for some h > 0, function
Fh.!/ 1h
Xk2Z
F.! C !sk/; where !s 2h
Fh.!
/
!0 !s!s 2!s2!s
A weird function: Fourier series
Fh.!
/
!0 !s!s 2!s2!s
As Fh.!/ is !s-periodic,Xk2Z
F.! C !s C !sk/ DXk2Z
F.! C !s.k C 1// DXk2Z
F.! C !sk/;
we can bring in its Fourier series expansion
Fh.!/ DXk2Z
ckej 2!s k! D
Xk2Z
ckejkh! ;
where Fourier coefficients are calculated as
ck D 1!s
Z !s=2!s=2
Fh.!/e jkh!d!:
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A weird function: calculating Fourier coefficients
With some extra efforts:
ck D 1!s
Z !s=2!s=2
1
h
Xi2Z
F.! C !si/e jkh.!C!si/d!
D 12
Xi2Z
Z !s=2!s=2
F.! C !si/e jkh.!C!si/d!
D 12
Xi2Z
Z !siC!s=2!si!s=2
F.!/e jkh!d!
D 12
Z 11
F.!/e jkh!d! (remember, f .t/ D 12
Z 11
F.!/e j!td!)
D f .kh/:
A weird function: Fourier series (contd)
Thus, we end up with
Fh.!/ DXk2Z
f .kh/e jkh! DXk2Z
f .kh/e jkh! :
Comparing this with DTFT of sequence f Nfkg,NF ./ D
Xk2Z
Nfke jk ;
we conclude that
Fh.!/ D 1h
Xk2Z
F.! C !sk/; where !s D 2h is sampling frequency;
is the DTFT of the sampled signal Nfk f .kh/ modulo scaling, D !h:NF ./ D Fh.!h/:
Occasionally, we also write NF .!/, understanding by ! scaled , i.e., h.
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Spectrum of sampled signal
By spectrum of a discrete signal Nf we understand itsI DTFT, NF ./, in 2 ;
(as DTFT periodic, there is no new information outside this range anyway).
In other words, spectrum of sampled signals is NF .!/ in ! 2 !s=2; !s=2:
N F.!/
!0 !s!s 2!s2!s !n!n
Frequency
!N !s2D h
called the Nyquist frequency.
Spectrum of sampled signal: aliasing
N F.!/
!0
NF .0/
!0 !1!1 !2!2
Spectrum of Nf at each discrete frequency 0, i.e., NF .0/, is aI blend of analog frequency responses at !k 0h C !sk, 8k 2 Z.
Blending means information lost as we can no longer tell F.!i / from F.!j /in their effect on NF .0/ (unless we know their dependencies).
We say thatI every discrete frequency 0 2 ; is an alias of all !k, k 2 Z.
and call this phenomenon aliasing.
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Examples of aliasing
Consider signals f1.t/ D sin4t
and f2.t/ D sin94t
sampled at h D 1:
0 1 2 3 4 5 6 7 8 9 10
1
0
1
!
0 1 2 3 4 5 6 7 8 9 10
1
0
1
Sampling frequency is !s D 2 , so thatI both !0 D 4 and !1 D 94 D !s C !0 have aliases at 0 D !0
and, consequently, produce the same sampled signal.
Deadbeat control (contd)
Return to our example at the beginning:
0 10 20 30 40 50 60 70 80 90 1000.2
0
0.2
0.4
0.6
0.8
1
1.2
Time, t
Plan
toutpu
t,y.t/,an
dno
ise,
n.t/
0 10 20 30 40 50 60 70 80 90 1000.2
0
0.2
0.4
0.6
0.8
1
1.2
Time, t
Plan
toutpu
t,y.t/,an
dno
ise,
n.t/
n.t/ D sin. 2ht / n.t/ D sin. 2:2
ht /
Now we see thatI frequency ! D 2
hhas alias at ! D 0 and
I frequency ! D 2:2h
has alias at ! D 0:2h
.
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Examples of aliasing (contd)
Wagon-wheel effect:0rpm
(shot with 12 FPS frame rate)
Moire pattern:downsampling!
Frequency folding
N F.!/
!0 !s!s 2!s2!s !n!n
If F.!/ 2 R for all ! (and F.!/ D F.!/), we may consider only ! 0 andthen NF .!/ can be produced through folding procedure:
0 !N
F.!/
2!N 3!N 4!N 5!N
!
0 !N 3!N 5!N
!
NF .!/
0 !N
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Instability of the ideal sampler
Let f .t/ be a signal with
F.!/ D 1p!2 C 1:
By Parseval,
Ef D 12
Z 11
1
!2 C 1d! D1
2:
Frequency response of Nfk D f .kh/ is
NF ./ D 1h
Xk2Z
1p.=hC 2=h k/2 C 1 D
Xk2Z
1p. C 2k/2 C h2 D1
for every 2 ; . This means that Nf is unbounded, i.e., thatI the ideal sampler SIdl is (energetically) unstable1.1In fact, SIdl bounded only for signals with sufficiently fast decay of their spectra (faster
than 1=j!j at high frequencies for real-valued spectra), i.e., with vanishing high-frequencyharmonics.
Outline
Motivating example
Sampling in frequency domain
The Sampling Theorem (Whittaker-Kotelnikov-Shannon)
Sampled-data controllers with ZOH in frequency domain
Antialiasing filtering
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Reconstruction from sampling
Let analog signal f .t/ be sampled with period h. We want to know
1. can f .t/ be precisely reconstructed from Nfk D f .kh/?2. if yes, how f .t/ can be reconstructed?
Bandlimited signals
One class of signals, for which the reconstruction question can be certainlyanswered, is the class of bandlimited signals, i.e.,I signals having zero spectrum 8j!j > !b for some !b > 0
(the smallest such !b called the signal bandwidth).
Example 1:
F.!
/
!0 !b!b
Example 2:
F.!
/
!0 !b!b
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What happens when sampled with !N < !b
F.!
/
!0 !b!b
#
N F.!/
!0 !n!n
I frequency responses at aliased frequencies might blend in NF .!/I precise reconstruction impossible (at least, in general)
What happens when sampled with !N !b
F.!
/
!0 !b!b
#
N F.!/
!0 !n!n
I F.!/ is preserved in NF .!/ with no distortionI no information lost under sampling H) F.!/ reconstructable
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What happens when sampled with !N !b (contd)
F.!
/
!0 !b!b
#
N F.!/
!0 !n!n
I F.!/ is still preserved in NF .!/ with no distortionI no information lost under sampling H) F.!/ reconstructable
How to reconstruct bandlimited signal
Let y.t/ be bandlimited signal with !b !N. Then F.!/ D h NF .!/ and
f .t/ D 12
Z 11
F.!/e j!td!
D h2
Z !N!N
NF .!/e j!td! D h2
Z !N!N
Xk2Z
f .kh/e jkh!e j!td!
DXk2Z
f .kh/1
2!N
Z !N!N
e j.tkh/!d! DXk2Z
f .kh/sin.!N.t kh//!N.t kh/
DXk2Z
f .kh/ sinc.!N.t kh//:
The function
sinc.!N t / Dt
1
h h
is the impulse response of the ideal low-pass filter with bandwidth !N.
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The Sampling Theorem
Theorem (Whittaker-Kotelnikov-Shannon)Let f .t/ be analog bandlimited signal with bandwidth !b. If !b !N, f .t/can be perfectly reconstructed from its sampled measurements Nfk D f .kh/and then the reconstructor (hold or D/A device) is
f .t/ DXk2Z
Nfk sinc.!N.t kh//:
Thats how it works:
N f kk3 2 1 1 2 3
#f.t/
t3h 2h h h 2h 3h
The Sampling Theorem: some observations
Reconstruction of sampled signals using the Sampling Theorem is not quitepractical for many applications. Indeed:I ideal reconstructor is not causal
(i.e., we have to collect all data before processing; impossible in feedback loops)
I signals we deal with are never bandlimited
For these reasons different methods used various applications:
Control applications typically use simple ZOH-like reconstructors becausereconstructorsI must be causal,I should not introduce too much phase lag,I should be simple (for on-line implementation).
Signal/image processing applications use more complicated reconstructors,like polynomial splines etcetera because reconstructorsI may have some degree of non-causality (delays tolerable)
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For curious: perfect reconstruction with !N < !b
F.!
/
!0 !b!b
#
N F.!/
!0 !n!n
I F.!i / D 0 for all i but one (either i D 0 or i D 1) H) no blendingI hence, F.!/ reconstructable (consider this a homework assignment)
Outline
Motivating example
Sampling in frequency domain
The Sampling Theorem (Whittaker-Kotelnikov-Shannon)
Sampled-data controllers with ZOH in frequency domain
Antialiasing filtering
-
Sample-and-hold circuit
Consider cascade of ideal sampler (SIdl) and zero-order hold (HZOH):
SIdlHZOH yNyu
In time domain, it acts as
u.khC / D y.kh/; 8k 2 ZC; 2 0; h/:
Impulse-train interpretation
Consider now yet another system:
1eshs
yy
P
u
Herey.t/ D
Xi2Z
.t ih/y.t/ DXi2Z
.t ih/y.ih/
called impulse-train modulated signal.
Because the response of 1eshs
to .t kh/ is 1.t kh/ 1.t kh h/,
u.khC / DXi2Z
1. C .k i/h/ 1. C .k i 1/h/y.ih/
D 1./ 1. h/y.kh/D y.kh/; 8k 2 Z; 2 0; h/;
which is exactly the i/o map of the sample-and-hold circuit.
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Impulse-train model
SIdlHZOH yNyu 1eshs
yy
P
u
Advantage:I we can use continuous-time machinery (transform models)
in analysis of sampled-data (hybrid continuous/discrete) systems.
Fourier transform of impulse-train modulated signal is
Y.!/ DXk2Z
e j!khy.kh/ D Yh.!/ 1h
Xk2Z
Y.! C 2!Nk/:
Hence, for both schemes
U.!/ D 1 e j!h
j!Yh.!/ D 1 e
j!h
j!h
Xk2Z
Y.! C 2!Nk/:
Sample-and-hold circuit in frequency domain
SIdlHZOH yNyu
Thus, in frequency domain it acts as
U.!/ D 1 e j!h
j!Yh.!/;
whereI Yh.!/ is 2!N-periodic and
I the magnitude of the frequency response 1e j!hj! of ZOH is
!0 !n!n 2!n2!n 4!n4!n
h1 e j!h
j!
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Sample-and-hold circuit in frequency domain: example 1
SIdlHZOH yNyu
Y.!/:
Y.!
/
!0 !n!n
Yh.!/:
Yh.!
/
!0 !n!n 2!n2!n 4!n4!n
U.!/:
jU.!
/j
!0 !n!n 2!n2!n 4!n4!n
Sample-and-hold circuit in frequency domain: example 2
SIdlHZOH yNyu
Y.!/:
Y.!
/
!0 !n!n
Yh.!/:
Yh.!
/
!0 !n!n 2!n2!n
U.!/:
jU.!
/j
!0 !n!n 2!n2!n
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Sample-and-hold circuit in frequency domain: example 3
SIdlHZOH yNyu
Y.!/:
Y.!
/
!0 !n!n
Yh.!/:
Yh.!
/
!0 !n!n 2!n2!n 4!n4!n
U.!/:jU
.!/j
!0 !n!n 2!n2!n 4!n4!n
Including discrete-time system
General sampled-data controller is
SIdlNC./HZOH yNyNuu
It can be shown that in this case
U.!/ D 1 e j!h
j!NC.e j!h/Yh.!/:
Thus, NC./ processes each period of Yh.!/ in the same fashion, so thatI NC.e j!h/Yh.!/ is still a 2!N-periodic function of ! 2 R
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Outline
Motivating example
Sampling in frequency domain
The Sampling Theorem (Whittaker-Kotelnikov-Shannon)
Sampled-data controllers with ZOH in frequency domain
Antialiasing filtering
Sample-and-hold circuit: sources of distortion
Y.!/:
Y.!
/
!0 !n!n
Yh.!/:
Yh.!
/
!0 !n!n 2!n2!n 4!n4!n
U.!/:
jU.!
/j
!0 !n!n 2!n2!n 4!n4!n
Transformation y.t/ 7! u.t/ introduces distortions caused byI blending, because of aliasing during samplingI reshape in the reconstruction stage
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Antialiasing filtering: idea
The problem is thatI aliasing causes high-frequency signals (e.g., measurement noise) to
appear in low-frequency range.
To prevent this,I antialiasing low-pass filter must be places between measured signal
and A/D converter
with the purpose to filter out signal spectrum above !N.
SIdlHZOH Fa.s/ yyfNyu
Antialiasing filtering in action
SIdlHZOH Fa.s/ yyfNyu
Y.!/: Y.!/
!0 !n!n
Yf .!/:
Y f.!
/
!0 !n!n
Yh.!/:
Yh.!
/
!0 !n!n 2!n2!n 4!n4!n
U.!/:
jU.!
/j
!0 !n!n 2!n2!n 4!n4!n
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Antialiasing filtering in action (contd)
Non-control examples of antialiasing filtering:
w/o antialiasing filter with antialiasing filter
Antialiasing filtering (contd)
Antialiasing filters, however,I introduce additional phase lag,
so we have to balance filtering out frequencies above !N and phase lag atcrossover frequency. This means that the need to use antialiasing filterI imposes limitation on attainable closed-loop bandwidth.
Motivating exampleSampling in frequency domainThe Sampling Theorem (Whittaker-Kotel'nikov-Shannon)Sampled-data controllers with ZOH in frequency domainAntialiasing filtering
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