N9-digital-sampling - Lecture notes 5 PDF

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Notes on digital sampling....

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Part 9 Digital Control - concept and sampling

• Continuous time, discrete time and digital control • Basic structure of a computer-control system – Sampling and A/D conversion – Digital controller – D/A conversion and ZOH

• Sampling and z-Transforms • Control design methods 1

1

Continuous time, discrete time and digital control

r(t)

Continuous controller

y(t) • • • •

r(kT)

Discrete controller

y(kT)

Continuous time (in time and value) Discrete time (dis-continuous in time) Digital control (dis-continuous in both time and values) Difference between latter two is negligible 2

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1

A basic continuous control system

Continuous Controller u(t)

r(t)

ym(t)

Plant

y(t)

Sensor

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3

Basic structure of a computer (digital)-control system r(kT)

Digital Controller

D/A

u(t)

y(t) Plant

Clock

ym(kT)

A/D

• A/D: Analogue-to-Digital Converter • D/A: Digital-to-Analogue Converter • Clock: Internal Timer

Sensor ym(t)

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4

2

Sampling and A/D: A/D

Analogue (voltage) Continuous

Digital signal

T

Analogue(voltage) Discrete

D/A and ZOH: D/A Digital signal

ZOH Analogue (voltage) Continuous

Analogue(voltage) Discrete

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A/D: Sampling process

Analogue (voltage) Continuous

A/D Digital signal

T

Analogue(voltage) Discrete

x(t) x(t)

x(kT)

x(kT)

0

T

2T

3T

time

0

T

2T

3T

time6

6

3

A/D - Sampling rate / sampling frequency

x(t) Variable intervals x(kT) 0

T

2T

time

3T

x(t) Regular intervals, but at a faster rate

x(kT) 0 T 2T 3T

7

time

7

D/A – continuous signal to plant

D/A Digital signal

ZOH Analogue (voltage) Continuous

Analogue(voltage) Discrete

x(kT)

x(t)

0

T

2T

3T

time

8

8

4

Sampled Sequences f(t)

f(kT) 0

time

T 2T 3T

{ f(kT)}= { f (0), f (T ), f (2T ), f (3T ), { f (k )}= { f (0), f (1), f ( 2), f (3),

!}

!}

𝑓𝑘 = 𝑓0, 𝑓1, 𝑓2, 𝑓3, … , 𝑓𝑛, … fk

fn

…eq-1 …eq-2 …eq-3

f (n )

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Sampling from math functions, • in general, replacing t by kT, • T is sampling interval, • k is the sequence (integer numbers = 0, 1, 2, 3….)

f (t ) = 1,

t >0

…eq-1

𝑓𝑘 = 1, 1, 1, … …

f ( t) = t ,

t >0 𝑓𝑘 = 𝑘𝑇

…eq-2

…eq-3 …eq-4

𝑓𝑘 = 0, 𝑇, 2𝑇, 3𝑇 , … …

…eq-5

10

5

Sampling from math functions, • in general, replacing t by kT, • T is sampling interval, • k is the sequence (integer numbers = 0, 1, 2, 3….)

f (t ) = e- 5t

…eq-1

𝑓𝑘 = 𝑒 &'+(

…eq-2

𝑓𝑘 = 1, 𝑒 &'( , 𝑒 &)*( , 𝑒 &)'( , … … 𝑓(𝑡) = sin(3𝑡)

…eq-3

…eq-4

𝑓𝑘 = sin(3𝑘𝑇)

…eq-5

𝑓𝑘 = 0, sin(3𝑇) , sin 6𝑇 , … … .

…eq-6

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Design approaches for digital control systems

Plant

Plant

Controller design In continuous-time

Discrete-time form of the plant rep.

A number of conversion methods Convert to Discrete time

Controller design In discrete time 12

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Practice P9-A 1) Find the sampled sequence of the following continuous signals, when the sampling interval T=0.1 and T=0.5 2) Compare the sequences with different sampling intervals and offer your thought on the differences

𝑓1(𝑡) = sin(2𝜋𝑡) 𝑓2(𝑡) = sin(10𝜋𝑡) 𝑓3(𝑡) = 𝑒 &, 𝑓4(𝑡) = 𝑒 &',

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