Robotics Research Laboratory 1 Chapter 2 Discrete System Analysis – Discrete Signals

Robotics Research Labo-ratory

1

Chapter 2

Discrete System Analysis – Discrete Signals


2

Sampling of Continuous-time Signals

How to treat the sampling process mathematically ?

Output of sampler

( )f t*( )pf t

Continuous-time analog signal

T

For convenience, uniform-rate sampler(1/T) with finite sampling duration (p) is assumed.

T

p1

time

( )f t*( )pf t

where p(t) is a carrier signal (unit pulse train)

sampler


3

This procedure is called a pulse amplitude modulation (PAM)

carrier signal p(t)

( )f t*( ) ( ) ( )pf t f t p t

PAM

The unit pulse train is written as

k

p t u t kT u t kT p p T

( ) ( ) ( ) where

By Fourier series

0

2( ) , ( )

1and ( )

s

s

jnω tn sn

T jnω tn

πp t C e ω sampling frequency

T

C p t e dtT

0

1 1 s

s

jnω pp jnω t

ns

eC e dt

T jnω p

sjnω p

sn

s

nω ppC e

T nω p

2sin( / 2)

/ 2

magnitude phase

or


4

0

sin( / 2),

/ 2s

ns

nω pp pC C

T nω p T

* ( ) ( ) ( ) ( ) sjnω tp nn

f t f t p t C f t e

* *

*

( ) { ( )}

( )

( )

( )

s

s

p p

jωtp

jnω t jωtnn

jnω t jωtnn

F jω f t

f t e dt

C f t e e dt

C f t e e dt

F

s 2s 3s 4s0-s-2s-3s-4s

2

T

2

p

ω


5

Let { ( )} ( ) where { ( )} ( )sjnω tse f t F jω jnω f t F jω F F

* *( ) { ( )} ( )p p n snF jω f t C F jω jnω

F

c s/20-c-s /2

1

|F(j)|

c : Cutoff frequency

* ( ) ( ) sjnω t jωtp nn

F jω C f t e e dt

Spectrum of Input Signal


6

0-s2π

p

* ( )pF jω

1

2 s cω ω

s0-s

* ( )pF jω1

2 s cω ω

ω

frequency folding

s ω

Spectrum of Output Signal

2π

p


7

Theorem : Shannon’s Sampling Theorem

To recover a signal from its sampling, you must sample at least twice the highest frequency in the signal.

2 2since and c s

c s

π πω ω

T T

or 22

sc c s

ωω ω ω

c s c si e T T T T 1. ., 2 or 2/ 2 : Nyquist frequency (or folding frequency)N sω ω

Remarks:

i) A practical difficulty is that real signals do not have Fourier

transforms that vanish outside a given frequency band.

To avoid the frequency folding (aliasing) problem,

it is necessary to filter the analog signal before sampling.

Note: Claude Shannon (1917 - 2001)


8

ii) Many controlled systems have low-pass filter characteristics.

iii) Sampling rates > 10 ~ 30 times of the BW of the system.

iv) For the train of unit impulses ( )δ t kt1

( ) where sjnω tn nn n

δ t kt C e CT

1* ( ) ( )snF jω F jω jnω

T

s0-s

1

T

ω-2s


9

Remarks :

i) Impulse response of ideal low-pass filter (non-causality)

Ideal low-pass filter is not realizable in a physical system.

How to realize it in a physical system ? ZOH or FOH

( )X s( )δ t *( )X s Ideal filter

G(j)

( )Y s

sampling

c0-c

1

|X(j)|

c0-c

1/T

|X*(j)|

c0-c

|Y(j)|

reconstruction

1/T

T


10

0 1

1 71

8 8Z Z s Zex f H f H f H )

0 1 1 ( )sf f f alias of f

ii) (Aliasing) It is not possible to reconstruct exactly a continuous-time signal in a practical control system once it is sampled.


11

If the continuous-time signal involves a frequency compo-nent equal to n times the sampling frequency

(where n is an integer), then that component may not appear in the sampled signal.

sω( )f t

iii) (Hidden oscillation)


12

2 2) ( ) sin sin3 if 3 / sec

3s ss

π πex f t t t ω rad T

ω


13

Signal Reconstruction

How to reconstruct (approximate) the origi-nal signal from the sampled signal?

- ZOH (zero-order hold)

- FOH (first-order hold)


14

ZOH (Zero-order Hold)2

21

( )( ) ( ) ( )( ) ( )

!where ( )

f kTf t f kT f kT t kT t kT

kT t k T

1

1

f t f kT kT t k T

f t f kT f kT t kT kT t k T

in ZOH, ( ) ( ) , ( )

in FOH, ( ) ( ) ( )( ) , ( )

Ttimek k+1k-1

zero-order hold reconstruction


15

/2

( / )

( ) ( ) ( )

1 1( )

1 sin( / 2)( )

sin( / )2

/

1

s

ho

Tsho

jωT jωT

ho

jπ ω ωs

s

Ts

s

g t u t u t T

G s es s

e ωT eG jω

jω

e

ω

πω ωπe

ω πω

s

ω

phase lag


16

0

π

2π

3π

ω

22

2

( ) sin

sin2

ωTj

ωTho

ωT

G jω e

ωT

ω2ωs ωs2ωs 3ωs0

Ideal low-pass filter2( )

s

πT

ω

0.637T

( )hoG jω


17

Remarks:

i) The ZOH behaves essentially as a low-pass filter.

ii) The accuracy of the ZOH as an extrapolator depends greatly

on the sampling frequency, .

iii) In general, the filtering property of the ZOH is used almost

exclusively in practice.

ωs


18

( ) ( ) '( )( )

( ) (( 1) ) = ( ) ( ), ( 1)

f t f kT f kT t kT

f kT f k Tf kT t kT kT t k T

T

k-1 k k+1

T

FOH (First-order Hold)


19

When k =0, (0) ( )( ) (0)

impulse (0) 1, ( ) 0

1 ( ) 1 ( ), 0

hf

f f Tf t f t

Tf f T

f t t g t t TT

( ) (0)1, ( ) ( ) ( )

(0) 1, ( ) 0

1 ( ) 1 ( ), 2

hf

f T fk f t f T t T

Tf f T

f t t g t T t TT

2, ( ) 0

( ) 0, (2 ) 0

k f kT

f T f T

( ) 0 ( ), 2

hff t g t t T


20

2

1

0

-1 T 2T 3T time

2

2

( ) 1 ( ) 2 1 ( )

2 1 ( 2 )

1( ) 1

hf

Tshf

t t Tg t u t u t T

T T

t Tu t T

T

TsG s e

Ts


21

2 sω 3 sω ω

2

s

π

ω

2 sω ω

π

2πfirst

zero

Large lag(delay) in high frequency makes a system unstable

( )hfg jωfirst

zero

3 sω

sω

sω0

0


22

Remark:

At low frequencies, the phase lag produced by the ZOH exceeds that of FOH, but as the frequencies become higher, the opposite is true


23

Z-transform

( ) ( ) ( )

( ) 1

f t δ t a dt f a

δ τ dτ

* *

*

( ) ( ) 1

( ) ( )

( ) ( )

( ) ( )

sτ

sτ

sτ

skT

δ t δ τ e dτ

r t r t e dτ

r τ δ τ kT e dτ

R s r kT e

L

L

( )r t * ( ) ( ) ( )r t r t δ t kT

T


24

1

Since is an irrational function,

define or ln

skT

sTTz e s z

e

T

T

s z

s z

k

R z z - transform of r t

r t or r kT

r t

R s

R z r kT z

1

1

*ln

*ln

( ) ( )

( ) ( )

[Laplace transform of ( )]

[ ( )]

( ) ( )

Z Z


25

iii) If 0, the sequence {rk} is said to be z-transformable.

ii) The series in z-1 has a radius of convergence such that the

series converges absolutely when | z-1 |<

1

1.

lim

1 lim

k

kk

k kk

i e zρ

rratio test ρ

r

root test ρr

i) Because R(z) is a power series in z-1 , the theory of power series may be applied to determine the convergence of

the z-transform.


26

Z-transform of Elementary Functionsi) Unit pulse function

0

1 , 0( )

0 , 0

( ) ( ) 1k

k

ke k

k

E z e k z z

Remark: unit impulse = 1[1]Z

0

1 2

11

1 , 0( )

0 , 0

( ) ( )

1

1 , 1

1

, 11

k k

k k

ke k

k

E z e k z z

z z

zz

zz

z

ii) Unit step function


27

0 0

1 2 3

1 2 3 4

1 2 3

1 2 3 1

1

1 2 2

, 0( )

0 , 0

( )

( 2 3 )

( ) ( 2 3 4 )

( ) (1 2 3 4 )

( )( 1) (1 ), 1

( )(1 ) ( 1)

k k

k k

kT ke k

k

E z kTz T kz

T z z z

E z T z z z z

zE z T z z z

E z z T z z z z

z zE z T T

z z

iii) Ramp function


28

iv) Polynomial function

1

0 0

11

, 0( )

0 , 0

( ) ( )

1 , 1

1

,

k

k k k

k k

r ke k

k

E z r z rz

rzrz

zz r

z r

v) Exponential function

1 2 2

11

, 0( )

0 , 0

( ) 1

1 , 1

1

,

rkT

rT rT

rrT

rrT

e ke k

k

E z e z e z

e ze zz

z ez e


29

0

0

1 1

1

1 2

2

sin , 0( )

0 , 0

( ) sin

2

1 1 1

2 1 1

sin

1 2 cossin

2 cos 1

k

k

j kT j kTk

k

j T j T

kT ke k

k

E z kTz

e ez

j

j e z e z

z T

z T zz T

z z T

vi) Sinusoidal function


30

Remark: Refer Table 2-1 in pp.29-30 (Ogata)

Also, refer Appendix B.2 Table in pp. 702-703(Franklin)

0

0

1 1

1

1 2 2

2 2

cos , 0( )

0 , 0

( ) cos

2

1 1 1

2 1 1

(1 cos )

1 2 cos( cos )

2 cos

k

k k

k

j kT j kTk k

k

j T j T

r kT ke k

k

E z r kTz

e er z

e rz e rz

z r T

z T r zz z r T

z z T r


31

Correspondence with Continuous Sig-nals

s-plane z-planez=esT

( ) ( 2 )sT σ jω T σT jωT σT j ωT πkz e e e e e e


32

①

②③

④ ⑤

j

0

j

0

-2 1

ReZ

ImZ

①② ③

④⑤ 0 1-1

ReZ

ImZ

0 1

1Te2Te

2sω

-j

2sω

j

2sω

j

2sω

-j


33

j

0 ReZ

ImZ

0

fixed

j

0 ReZ

ImZ

1

2sj

2sj

2sj

2j1j

1j2j

TjTee 2TjTee 1

TjTee 1

2s

4

3 s

4s

s 0


34

Important Properties and Theorems of the z-transform

1. Linearity

1 2

1 2

1 2

1 2

1 2 1 2

( ) ( )

( ) ( )

( ) ( )

( ) { ( )}

Assume ( ) ( )

( ) ( ) ( ) ( )

k

k k

i i

αf k βf k

αf k βf k z

α f k z β f k z

α f k β f k

F z f k

αf k βf k αF z βF z

Z

Z{ } Z

Z

Z

2. Time Shifting

( )

( ) ( ) ,

( )

( )

k

j n

n

f k n f k n z k n j

f j z

z F z

Z


35

k

n k n

n

n k n

k

n nn k n k k

k k k

nn k

k

f k n f k n z

z f k n z

z F z

f k k

f k n z f k n z

z f k n z f k z f k z

z F z f k z

( )

( )

0

1 1( )

0 0 0

1

0

( ) ( )

( )

( )

If ( ) 0 for 0

( ) ( )

( ) ( ) ( )

( ) ( )

Z

Z


36

1 2 1 2( ) ( ) ( ) ( )f k f k l F z F z Z

4. Scaling

( ) ( )kr f k F rz Z

5. Initial Value Theorem

0 1 2( ) ( ) (0) (1) (2)

(0) lim ( )

k

k

z

F z f k z f z f z f z

f F z

2 2

)

( 1) ( ) (0)

( 2) ( ) (0) (1)

ex

x k zX z zx

x k z X z z x zx

Z

Z

3. Convolution


37

6. Final Value Theorem

k z z

F z z z F z

f k z F z z F z

1

1 1

If ( ) converges for 1 and all poles of ( 1) ( ) are inside

the unit circle, then

lim ( ) lim(1 ) ( ) lim( 1) ( )

1 2

1 2

11

11

1

1

1

)

( ) (0) (1) (2) ( )

( 1) (0) (1) ( 1)

( )

lim ( ) ( ) ( )

lim ( ) lim lim ( )

k n k

n

k n k

n

k n

n

k kn n

n nz

k n

nk k z

pf

f n z f f z f z f k z

f n z f z f z f k z

z f n z

f n z z f n z f k

f k f n z z

1

1 1

1 1

( )

lim ( ) ( ) lim(1 ) ( )

k n

n

z z

f n z

F z z F z z F z


38

1 1

11

11 1

)

( ) 1 0

1 1 ( ) z

1 1 ( ) 1

1 ( ) lim(1 ) ( ) lim 1 1

1

at

aTaT

aTz z

ex

f t e a

F z ez e z

f

zf z F z

e z

Remark: Refer Table 2-2 in p. 38.(Ogata)

Also, refer Appendix B.1 Table in p.701(Franklin)


39

T

Continuous-time domain

S-plane Z-plane

Discrete-time domain

LZ

- 1Z

( )r t *( )r t

*( )R s

sTz e

1( ln )s z

T

*( )R z


40

1 1

1

1 1

1) ( )

( 1)

( ) ( ) 1 , 0

1 1 ( ) 1

(1 ) (1 )

(1 ) (1 )

(1 )(1 ) ( 1)( )

t

tT

T T

T T

ex X ss s

x t X s e t

X z ez e z

e z e z

z e z z z e

- 1L

Z


41

T

T

Ts

T

s

s

Ts s

ex

e zX s X z

s s z z e

X sX z X s

z zs s

s s s sz e z e

z

z e

0 1

)

1 (1 ) ( ) ( )

( 1) ( 1)( )

( ) ( ) residue of at pole of ( ) ( )

1 1 lim lim ( 1)

( 1) ( 1)( ) ( )

( )

T

T

T

z z

z z e

e z

z z e

( 1) ( )

(1 )

( 1)( )

Remark: Refer the derivation of ( ) in pp. 83-85 (Ogata's)


42

Inverse z-transform

i) Power Series Method (Direct Division)

ii) Computational Method :

- MATLAB Approach - Difference Equation Approach

iii) Partial Fraction Expansion Method

iv) Inversion Integral Method


43

Example 1) Power Series Method

1 2

1

1 2

1 2

( ) (0) (1) (2)

1( )

2 1 1.5 0.5

1 2.5 3.252

(0) , (1) (2.5), (2) (3.25)2 2 2

U z u u z u z

T zU z

z zT

z z

T T Tu u u


44

Example 2) Computational Method


45


46

Example 3) Partial Fraction Expansion Method

1

1 2

1

1 1

1 1

1( )

2 1 1.5 0.5

1

2 (1 )(1 0.5 )

1 1 0.53

2 , 2

1 3( ) 4

2 2 2

k

k

T zU z

z z

T z

z z

A B

z zT

A T B

Tu k A B

Remark:

1

1( ) ( )

1kx kT a X z

az


47

Example 4) Inverse Integral Method :

1

( ) { ( )} ( )

1( ) ( )

2

( ) { ( )} ( )

1( ) ( ) ( )

2

st

c j st

c j

k

k

c

F s f t f t e dt

f t F s e dsj

F z f k f k z

f k F z F z z dzj

- 1

L

Z

Z

where c is a circle with its center at the origin of the z plane such that all poles of F(z)zk-1 are inside it.

1

1 2

-1( ) ( ) ( )

ik

m

i

m

f k residue of F z at pole z z of z F z

K K K


48

Case 1) simple pole

1( ) ( ) |kz aresidue z a F z z

Case 2) m multiple poles

11

1

2

2

11

2

12

21

1 2

1 ( ) ( ) |

( 1)!

)

( )( 1) ( )

( )( 1) ( )

( ) ( 1) ( )

mm k

z am

aT

kk

aT

k

iaTi

dresidue z a F z z

m dz

ex

zF z

z z e

zF z z

z z e

zf k residue of at pole z z

z z e

K K


49

1

1

2

( 1)

2

2

12

21

1

lim ( )( 1) ( )

(1 )

1

1 lim ( 1)

(2 1)! ( 1) ( )

lim

aT

aT

kaT

aTz e

a k T

aT

k

aTz

z

K residue of simple pole at z e

zz e

z z e

e

e

K residue of multiple poles at z

d zz

dz z z e

d

1

1

21

2

( 1) ( ) lim

( )

1 1

1 (1 )

k

aT

k aT k

aTz

aT aT

z

dz z e

k z z e z

z e

k

e e


50

1 2

( 1)

2

( )

1 1 , 0,1,2,

1 (1 )

a k T

aT aT

f k K K

k ek

e e

Documents

Robotics Research Laboratory 1 Chapter 2 Discrete System Analysis – Discrete Signals