Ingegneria dell ’Automazione - Sistemi in Tempo...

Ingegneria dell ’Automazione -Sistemi in Tempo Reale

Selected topics on discrete-time andsampled-data systems

Luigi Palopoli

[email protected] - Tel. 050/883444

Ingegneria dell ’Automazione - Sistemi in Tempo Reale – p.1/19

Outline

Effects of delay

Sampled-data systems

Discrete equivalents


An example

Consider a first order system x = λx + u

Γ1 =R T

mT eληdη = eλT−eλmT

λ, Γ2 = emλT

−1λ

Augmented discrete-time system:

2

4

x(k + 1)

z(k + 1)

3

5 =

2

4

eλT eλT−eλmT

λ

0 0

3

5

2

4

x(k)

z(k)

3

5 +

2

4

emλT−1

λ

1

3

5 u(k)

Control by static feedback u(k) = kx(x) yields the closed loop dynamic:

2

4

x(k + 1)

z(k + 1)

3

5 =

2

4

eλT + k emλT−1

λeλT

−eλmT

λ

k 0

3

5 z

Stability can be studied- via Jury criterion - on

z2 − (eλT + kemλT − 1

λ)z − k

eλT − eλmT

λ


Solution

a2 < 1 ⇒ k > − λ

eλT−eλmT

a2 > −1 − a1 ⇒ k < −λ

a2 > −1 + a1 ⇒{

k > −λ eλT +1

2eλmT−eλT−1if m >

log 1+eλT

2

λT

k > eλT +1

−2eλmT +eλT +1λ Otherwise


Sampled-data systems

So far we have seen discrete-time systems

Computer controlled systems are actually amix of discrete-time and continuous timesystems

We need to understand the interactionbetween different components


Ideal sampler

Ideal sampling can intuitively be seen asgenerated by multiplying a signal by asequence of dirac’s δ


Properties of δ

∫ +∞

−∞ f(t)δ(t − a)dt = f(a)∫

t

−∞ δ(τ)dτ = 1 → L[δ(t)] = 1


L-trasform of r∗

Using the above properties it is possible to write:

L[r∗(t)] =∫ +∞

−∞r∗(τ)e−sτdτ =

∫ +∞

−∞

∑+∞−∞ r(τ)δ(τ − kT )dτ =

=∑+∞

−∞

∫ +∞

−∞r(τ)δ(τ − kT )dτ =

∑+∞−∞ r(kT )e−sKT = R∗(s)

The L-transform of the sampled-data signal r can beachieved from the Z-transform of the sequence r(kT )

by choosing z = e−sT

Backward or forward shifting yields different samples:the sampling operation is not time-invariant (but it islinear)


Spectrum of the Sampled Signal

Fourier Series transform of the sampling signal

P+∞

−∞δ(t − kT ) =

P+∞

−∞Cnej2nπt/T

Cn =R T/2−T/2

P+∞

−∞δ(t − kT )e−j2nπt/T dτ =

R T/2−T/2

δ(t)e−j2nπt/T dτ = 1T

→P+∞

−∞δ(t − kT ) = 1

T

P+∞

−∞ej2nπt/T

Spectrum of r∗

L[r∗(t)] =R +∞

−∞r(t) 1

T

P+∞

−∞ej2nπt/T e−sT dt =

= 1T

P+∞

−∞r(t)

R +∞

−∞e−(s−j2nπ/T )tdt =

= 1T

P+∞

−∞R(s − j2πn/T )


Example

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ω

|R(j

ω)|

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ω

|R* (jω

)|

2 π /T


Aliasing

The spectrum might be altered (i.e., signal notattainable from samples!)

0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

sin(2 π t)

sin(2 π t/3)

samples collected with T = 3/2


Shannon theorem

If the signal has a finite badwidth then the signal can bereconstructed from samples collected with a periodsuch that 1

T≥ 2B

Band-limited signals have infinite duration;many signals of interest have infinitebandwidth

Typically a low-pass filter is used tode-emphasize higher frequencies


Data Extrapolation

If the following hypotheses hold

the signal has limited bandwidth B

the signal is sampled at frequency fs = 1T≥ 2B

then the signal can be reconstructed using an ideallowpass filter L(s) with

|L(jω)| =

T if ω ∈ [− πT, π

T]

0 elsewhere.

Signal l(t) is given by:

l(t) =1

2π

Z pi/T

−pi/TTejωT dω =

sin(πt/T )

πt/T= sinc(πt/T )


Data Extrapolation I

The reconstructed signal is computed as follows:

r(t) = r∗(t) ∗ l(t) =R +∞

−∞r(τ)

P

δ(τ − kT )sinc π(t−τT

dτ =

=P+∞

−∞r(kT )sinc π(t−kT )

T

The function sinc is not causal and has infinite duration

In communication applications

The duration problem can be solved truncating the signal

The causality problem can be solved introducing a delay and collecting somesample before the reconstruction

Not viable in control applications since large delays jeopardise stability


Extrapolation via ZOH

Zoh transfer function

Zoh(jω) = 1−ejωT

jω= e−jωT/2

n

ejωT/2−

−jωT/2

2j

o

2jjω

=

= TejωT/2sinc( ωT2

)

Amplitude:

|Zoh(jω)| = T |sinc(ωT

2)|

Phase:

∠Zoh(jω) = −ωT

2


Sinusoidal signal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

sin(2 π t)

ZoH extrapolation


Example

Consider the signal r(t) = 1π

sin(t)

The spectrum is given byR(jω) = j(δ(ω − 1) − δ(ω + 1))

Suppose we sample it with period T = 1

The spectrum ofr∗(t) = 1/T

∑+∞k=−∞ R(jω − 2nπ)


Example - II

Sampling and ZoH extrapolation orignatespurious harmonics (called impostors)

Taking adavantage of the low pass behaviourof the plant we can restrict to the firstharmonic

v1(t) =1

πsinc(T/2) sin(t − T/2)

The sample-and-hold operation can bethought of (at a first approximation) as theintroduction of delay of T/2


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