Sampled-data systems

Ingegneria dell ’Automazione - Sistemi in Tempo Reale

Selected topics on discrete-time and sampled-data systems

Luigi Palopoli

palopoli@sssup.it - Tel. 050/883444

Outline

Effects of delay

Sampled-data systems

Discrete equivalents

An example

Consider a first order system ˙x = λx + u Γ₁ = R_T

mT e^ληdη = ^eλT−_λ^eλmT , Γ2 = ^emλT_λ ⁻¹ Augmented discrete-time system:

2 4

x(k + 1) z(k + 1)

3 5 =

2 4

e^{λT eλT−}_λ^eλmT

0 0

3 5

2 4

x(k) z(k)

3 5 +

2 4

e^mλT−1 λ

1

3

5u(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

e^λT + k^emλT_λ ^{−1 eλT−}_λ^eλmT

k 0

3 5z

Stability can be studied- via Jury criterion - on

z² − (e^λT + ke^mλT − 1

λ )z − ke^λT − e^λmT λ

An example

Consider a first order system ˙x = λx + u Γ₁ = R_T

mT e^ληdη = ^eλT−_λ^eλmT , Γ2 = ^emλT_λ ⁻¹ Augmented discrete-time system:

2 4

x(k + 1) z(k + 1)

3 5 =

2 4

e^{λT eλT−}_λ^eλmT

0 0

3 5

2 4

x(k) z(k)

3 5 +

2 4

e^mλT−1 λ

1

3

5u(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

e^λT + k^emλT_λ ^{−1 eλT−}_λ^eλmT

k 0

3 5z

Stability can be studied- via Jury criterion - on

z² − (e^λT + ke^mλT − 1

λ )z − ke^λT − e^λmT λ

An example

Consider a first order system ˙x = λx + u Γ₁ = R_T

mT e^ληdη = ^eλT−_λ^eλmT , Γ2 = ^emλT_λ ⁻¹ Augmented discrete-time system:

2 4

x(k + 1) z(k + 1)

3 5 =

2 4

e^{λT eλT−}_λ^eλmT

0 0

3 5

2 4

x(k) z(k)

3 5 +

2 4

e^mλT−1 λ

1

3

5u(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

e^λT + k^emλT_λ ^{−1 eλT−}_λ^eλmT

k 0

3 5z

Stability can be studied- via Jury criterion - on

z² − (e^λT + ke^mλT − 1

λ )z − ke^λT − e^λmT λ

An example

Consider a first order system ˙x = λx + u Γ₁ = R_T

mT e^ληdη = ^eλT−_λ^eλmT , Γ2 = ^emλT_λ ⁻¹ Augmented discrete-time system:

2 4

x(k + 1) z(k + 1)

3 5 =

2 4

e^{λT eλT−}_λ^eλmT

0 0

3 5

2 4

x(k) z(k)

3 5 +

2 4

e^mλT−1 λ

1

3

5u(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

e^λT + k^emλT_λ ^{−1 eλT−}_λ^eλmT

k 0

3 5z Stability can be studied- via Jury criterion - on

z² − (e^λT + ke^mλT − 1

λ )z − ke^λT − e^λmT λ

An example

Consider a first order system ˙x = λx + u Γ₁ = R_T

mT e^ληdη = ^eλT−_λ^eλmT , Γ2 = ^emλT_λ ⁻¹ Augmented discrete-time system:

2 4

x(k + 1) z(k + 1)

3 5 =

2 4

e^{λT eλT−}_λ^eλmT

0 0

3 5

2 4

x(k) z(k)

3 5 +

2 4

e^mλT−1 λ

1

3

5u(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

e^λT + k^emλT_λ ^{−1 eλT−}_λ^eλmT

k 0

3 5z

Stability can be studied- via Jury criterion - on

z² − (e^λT + ke^mλT − 1

)z − ke^λT − e^λmT

Solution

a 2 < 1 ⇒ k > − _e

_−e ^λ

a 2 > −1 − a 1 ⇒ k < −λ

a 2 > −1 + a 1 ⇒ (

k > −λ _2e

^e

_−e ⁺¹

₋₁ if m > ^log

λT 2

λT

k > _−2e

^e

_+e ⁺¹

₊₁ λ Otherwise

Solution

a 2 < 1 ⇒ k > − _e

_−e ^λ

a 2 > −1 − a 1 ⇒ k < −λ a 2 > −1 + a 1 ⇒

(

k > −λ _2e

^e

_−e ⁺¹

₋₁ if m > ^log

λT 2

λT

k > _−2e

^e

_+e ⁺¹

₊₁ λ Otherwise

Solution

a 2 < 1 ⇒ k > − _e

_−e ^λ

a 2 > −1 − a 1 ⇒ k < −λ a 2 > −1 + a 1 ⇒

(

k > −λ _2e

^e

_−e ⁺¹

₋₁ if m > ^log

λT 2

λT

k > _−2e

^e

_+e ⁺¹

₊₁ λ Otherwise

Sampled-data systems

So far we have seen discrete-time systems Computer controlled systems are actually a mix of discrete-time and continuous time

systems

We need to understand the interaction

between different components

Ideal sampler

Ideal sampling can intuitively be seen as generated by multiplying a signal by a

sequence of dirac’s δ

Properties of ^δ

R + ∞

−∞ f (t)δ(t − a)dt = f (a) R ^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

−∞

r

(τ )e

dτ = R

−∞

P

−∞

r(τ )δ(τ − kT )dτ =

= P

−∞

R

−∞

r(τ )δ(τ − kT )dτ = P

−∞

r(kT )e

= R

(s) The L-transform of the sampled-data signal r can be

achieved from the Z-transform of the sequence r(kT ) by choosing z = e

Backward or forward shifting yields different samples:

the sampling operation is not time-invariant (but it is

Spectrum of the Sampled Signal

Fourier Series transform of the sampling signal

P_+∞

−∞δ(t − kT ) = P_+∞

−∞ C_ne^j2nπt/T Cn = R_{T /2}

−T /2

P_+∞

−∞ δ(t − kT )e^−j2nπt/Tdτ = R _{T /2}

−T /2 δ(t)e^−j2nπt/Tdτ = _T¹ → P_+∞

−∞δ(t − kT ) = _T¹ P_+∞

−∞e^j2nπt/T Spectrum of r^∗

L[r^∗(t)] = R _+∞

−∞ r(t)_T¹ P_+∞

−∞e^j2nπt/Te^−sTdt =

= _T¹ P_+∞

−∞ r(t)R_+∞

−∞ e⁻(s−j2nπ/T )tdt =

= _T¹ P_+∞

−∞ R(s − j2πn/T )

Example

−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

ω

|R(j ω)|

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

0 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 not attainable from samples!)

−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 be reconstructed from samples collected with a period

such that _T ¹ ≥ 2B

Band-limited signals have infinite duration;

many signals of interest have infinite bandwidth

Typically a low-pass filter is used to

de-emphasize higher frequencies

Data Extrapolation

If the following hypotheses hold

the signal has ^limited bandwidth B

the signal is sampled at frequency f

=

≥ 2B then the signal can be reconstructed using an ideal lowpass filter L(s) with

|L(jω)| =







T if ω ∈ [−

,

] 0 elsewhere.

Signal l(t) is given by:

1 Z _pi/T sin(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−τ_T 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 some sample before the reconstruction

Not viable in control applications since large delays jeopardise stability

Extrapolation via ZOH

Zoh transfer function

Zoh(jω) = ^1−e_jω^jωT = e^{−jωT /2} n

e^{jωT /2}−^{−jωT /2} 2j

o 2j jω =

= T e^{jωT /2}sinc(^ωT₂ ) Amplitude:

|Zoh(jω)| = T |sinc(ωT 2 )|

Phase:

∠Zoh(jω) = −ωT 2

Sinusoidal signal

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

−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) = _π ¹ sin(t) The spectrum is given by

R(jω) = j(δ(ω − 1) − δ(ω + 1))

Suppose we sample it with period T = 1 The spectrum of

r ^∗ (t) = 1/T P _+∞

k=−∞ R(jω − 2nπ)

Example - I

−8 −6 −4 −2 0 2 4 6 8

0 0.5 1 1.5 2

−8 −6 −4 −2 0 2 4 6 8

0 0.5 1 1.5 2

|R(j ω) |

|R^*(j ω)|

Example - II

Sampling and ZoH extrapolation orignate spurious harmonics (called impostors)

Taking adavantage of the low pass behaviour of the plant we can restrict to the first

harmonic

v ₁ (t) = 1

π sinc(T/2) sin(t − T/2) The sample-and-hold operation can be

thought of (at a first approximation) as the

introduction of delay of T/2