/
Digital and Non-Linear Control Digital and Non-Linear Control

Digital and Non-Linear Control - PowerPoint Presentation

kittie-lecroy
kittie-lecroy . @kittie-lecroy
Follow
356 views
Uploaded On 2018-09-21

Digital and Non-Linear Control - PPT Presentation

Digital System Frequency Response and Modeling 1 Lecture Outline Sampling Theorem Frequency Response ADC Model DAC Model Combined Models 2 Sampling Theorem Sampling is necessary for the processing of analog data using digital elements ID: 674692

sampling dac function analog dac sampling analog function transfer response adc subsystem system digital model step theorem frequency time impulse signal transform

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "Digital and Non-Linear Control" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

Slide1

Digital and Non-Linear Control

Digital System Frequency Response and Modeling

1Slide2

Lecture Outline

Sampling Theorem

Frequency Response

ADC Model

DAC ModelCombined Models

2Slide3

Sampling Theorem

Sampling is necessary for the processing of analog data using digital elements.

Successful digital data processing requires that the samples reflect the nature of the analog signal and that analog signals be recoverable from a sequence of samples.

3Slide4

Sampling Theorem

Following figure shows two distinct waveforms with identical samples.

Obviously, faster sampling of the two waveforms would produce distinguishable sequences.

4Slide5

Sampling Theorem

The band limited signal with

can be reconstructed from the discrete-time waveform

if and only if the sampling angular frequency

satisfies (so sampling frequency needs to double the required reconstruction frequency)

 

5

 

 

 Slide6

Selection of Sampling Frequency

A general signal often has a finite “effective bandwidth” beyond which its spectral components are negligible.

This allows us to treat physical signals as band limited and choose a suitable sampling rate for them based on the sampling theorem.

In practice, the sampling rate chosen is often larger than the lower bound specified in the sampling theorem.

A rule of thumb is to choose

as

 

6

 Slide7

Frequency Response

Consider a transfer function

of an LTI system. For the discrete input

, the steady state output is

.

In other words, setting

in

gives magnitude and phase change. Refer to page 40-41 in the textbook.

 

7Slide8

Example 1

Compute

the steady state response for the following system

due to the sampled sinusoidal

Set

, so

the steady state response is

 

8Slide9

Digital Control System Modeling

A common configuration of digital control system is shown in following figure.

9Slide10

ADC Model

Assume that

ADC outputs are exactly equal in magnitude to their inputs (i.e., quantization errors are negligible)

The ADC yields a digital output instantaneously

Sampling is perfectly uniform (i.e., occur at a fixed rate)The ADC can be modeled as an ideal sampler with sampling period

T.

10

T

t

u

*

(

t

)

0

t

u

(

t

)

0Slide11

0

t

u

*

(

t

)

t

u

(

t

)

0

T

t

δ

T

(

t

)

0

×

modulating

pulse(carrier)

modulated

wave

Modulation

signal

u

(

t

)

u*

(

t

)

Sampling ProcessSlide12

DAC Model

Assume that

DAC outputs are exactly equal in magnitude to their inputs.

The DAC yields an analog output instantaneously.

DAC outputs are constant over each sampling period (ZOH).

The input-output relationship of the DAC is given by

12

 

u

(k)

u

(

t

)

u

h

(

t

)Slide13

DAC Model

Unit impulse response of ZOH

13Slide14

DAC Model

As shown in figure the impulse response is a unit pulse of width T.

A pulse can be represented as a positive step at time zero followed by a negative step at time

T

.

Using the Laplace transform of a unit step and the time delay theorem for Laplace transforms,

14

 

 Slide15

DAC Model

Thus, the transfer function of the ZOH is

15

 

 

 Slide16

DAC, Analog Subsystem, and ADC Combination

Transfer Function

The cascade of a DAC, analog subsystem, and ADC is shown in following figure. The time difference between the negative step and positive step is T.

16Slide17

DAC and

Analog Subsystem

Using the DAC model, and assuming that the transfer function of the analog subsystem is

G

(s), the transfer function of the DAC and analog subsystem cascade is

17

 

 Slide18

DAC and

Analog Subsystem

The corresponding impulse response is

The impulse response is the analog system step response minus a second step response delayed by one sampling period.

18

 

 

 Slide19

DAC and Analog Subsystem

19

 Slide20

DAC and

Analog Subsystem

Inverse Laplace yields

Where

 

20

 

 Slide21

DAC, Analog Subsystem, and ADC Combination

Transfer Function

The analog response is sampled to give the sampled impulse response

(ADC

part)

By

z

-transform, we can obtain the

z

-transfer function of the DAC (zero-order hold), analog subsystem, and ADC (ideal sampler) cascade.

21

 

 Slide22

DAC, Analog Subsystem, and ADC Combination

Transfer Function

Z-Transform is given as

The * in above equation is to emphasize that sampling of a time function is necessary before

z

-transformation.

Having made this point, the equation can be rewritten more conveniently as

22

 

 

 

 

 Slide23

Example 2

Find G

ZAS

(

z) for the transfer function of the system given as

Rewrite transfer function in standard form

23

Solution

 

 Slide24

Example 2

Where

and

Now we know

Therefore,

The corresponding partial fraction expansion is

 

24

 

 

 

 Slide25

Example 2

Using the

z

-transform table, the desired

z-domain transfer function is

25

 

 

 

 Slide26

Example 3

Find G

ZAS

(

z) for the transfer function of the system given as

Rewrite transfer function in standard form

26

Solution

 

 Slide27

Example 3

Where

and

Now we know

Therefore,

The corresponding partial fraction expansion is

 

27

 

 

 

 Slide28

Example 3

The desired

z

-domain transfer function can be obtained as

28