sinusoids, phasors and resonance - bu · dr. ayman yousef introduction • we have been working...

Automatic Control Dr. Ayman Yousef

Introduction

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Introduction• We have been working with individual subsystems represented

by a block with its input and output.• More complicated systems, however, are represented by the

interconnection of many subsystems.• Since the response of a single transfer function can be

calculated, we want to represent multiple subsystems as asingle transfer function.

• In this chapter, multiple subsystems are represented in twoways: as block diagrams and as signal-flow graphs.

• Signal-flow graphs represent transfer functions as lines, andsignals as small circular nodes.

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Block Diagrams

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Block Diagrams

When multiple subsystems are interconnected, a few more schematicelements must be added to the block diagram. These new elements aresumming junctions and take off points.

Take off pointSumming point4


Block diagram componentsBlock Diagrams

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Closed-Loop Transfer FunctionBlock Diagrams

C(s) = G(s) E(s)= G(s) [R(s) – B(s)]= G(s) [R(s) – H(s)C(s)]= G(s)R(s) – G(s)H(s)C(s)

G(s)R(s) = C(s) + G(s)H(s)C(s)G(s)R(s) = C(s)[1 + G(s)H(s)]

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Feedback FormThe typical feedback system, is shown in figure below.

Block Diagrams

a simplified model is shown in the following Figure.

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Block Diagram Reduction

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Cascade Form

Block Diagrams

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Parallel Form

Block Diagrams

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Moving Blocks to Create Familiar Forms

Moving summing point to RIGHT

Block Diagrams

Moving summing point to LEFT

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Moving Blocks to Create Familiar Forms

Block Diagrams

Take off summing point to RIGHT

Take off summing point to LEFT

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Block Diagram Reduction Rules

Block Diagrams

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Basic rules with block diagram transformation Block Diagrams

Basic rules with block diagram transformation

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Reduce the system shown to a single T.F.

Example 1

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Solution

Example 1 (cont’d)

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18



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R_+

_+1G 2G 3G

1H

2H

++

C

Example 2Reduce the system shown to a single T.F.

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R_+

_+

1G 2G 3G

1H

1

2

GH

++

C


Solution

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R_+

_+

21GG 3G

1H

1

2

GH

++

C


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R_+

_+ 21GG 3G

1H

1

2

GH

++

C


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R_+

_+

121

21

1 HGGGG

− 3G

1

2

GH

C


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R_+

_+

121

321

1 HGGGGG

−

1

2

GH

C


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R_+

232121

321

1 HGGHGGGGG+−

C


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R

321232121

321

1 GGGHGGHGGGGG

++−

C


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Example 3

Reduce the block diagram shown to a single T.F.

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Solution


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Reduce the system shown to a single T.F.

Example 4

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Solution


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Summary

This Lecture:– Block Control Diagram Components– Block Control Diagram Reduction Rules– Examples

Next Lecture:– Signal Flow Graphs

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Slide Number 1Slide Number 2IntroductionSlide Number 4Block DiagramsSlide Number 6Closed-Loop Transfer FunctionFeedback FormSlide Number 10Cascade FormParallel FormMoving Blocks to Create Familiar FormsMoving Blocks to Create Familiar FormsSlide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Example 3Slide Number 30Slide Number 31Slide Number 32Slide Number 33SummarySlide Number 35What is Signal Flow Graph (SFG)?Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Rules for drawing of SFG from Block diagramsignal flow graphSlide Number 45Slide Number 46Slide Number 47Slide Number 48Slide Number 49Slide Number 50Slide Number 51Slide Number 52Slide Number 53Slide Number 54Signal flow graphs of linear systemsSlide Number 56Slide Number 57Slide Number 58Slide Number 59Slide Number 60Slide Number 61Slide Number 62Mason's Rule� (Gain formula) Mason’s Gain Formula Slide Number 65Slide Number 66Slide Number 67Slide Number 68Slide Number 69Slide Number 70Slide Number 71Summary

sinusoids, phasors and resonance - bu · dr. ayman yousef introduction • we have been working...

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