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

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

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 /

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

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!:

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.

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.

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

.

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

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

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

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

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.

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)

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.

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!

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

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

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

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

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