Automation Systems - Lecture 4 - Block Diagram...

Automation SystemsLecture 4 - Block Diagram Models

Jakub Mozaryn

Institute of Automatic Control and Robotics, Department of Mechatronics, WUT

Warszawa, 2020

Jakub Mozaryn Automation Systems

Introduction

Block diagram model

Block diagram model (structural): Graphical representation of inter-relationships between the parts of analyzed system, ie. there are givendirections of signal flow and the relationships between input and outputsignals of all components of the analyzed system.

A block diagram, of either a single element or a complex system, is aform of a mathematical description of the systems function. It clearlyexpresses the dependence of the output signals from the input signal, ifthere are known informations about properties (the transfer functions) ofits components.

Block diagrams consists of unidirectional, operational blocks that rep-resent the transfer function.


Introduction

Figure 1: Example of block diagram model


Elements of block diagrams

Block: A rectangle with arrows representinginput and output signals. Inside rectanglethe transfer function is written.

y(s) = G (s)u(s) (1)

Pickoff point (information point): Repre-sents device that allow to retrieve the infor-mation and send it to several branches of thesystem.

Summary junction: represents the devicethat allow an algebraic summation of signalsand the signs of signals are distinguished.

z = u − y (2)


Types of connections in the block diagram models

Using appropriate transformations, the block diagram representationcan be often reduced to a simplified block diagram with fewer blocksthan a original one, in which there are only 4 types of connections,called elementary connections.

Elemetary connections are:

1 serial connection (chain, cascade),

2 parallel connection,

3 negative feedback loop,

4 positive feedback loop.

There are also several rules that allow to trasform a complex block diagramto a simpler one.


Types of connections in the block diagram modelsConnection type Transfer function Block diagram

Serial connection(chain) G (s) = G1(s)G2(s)

Parallel connection G (s) = ±G1(s)±G2(s)

Negative feedbackloop

G (s) =±G1(s)

1 + G1(s)G2(s)

Positive feedbackloop

G (s) =±G1(s)

1 − G1(s)G2(s)


Block diagram transformations - pickoff points

Moving pickoff pointahead of the block

Changing the order ofpickoff points

Moving pickoff pointbehind the block


Block diagram transformations - summary junctions

Moving a summaryjunction behind ablock

Moving a summaryjunction ahead of ablock

Separation of a multi-input summary junc-tion

Changing the order ofsummary junctionsJakub Mozaryn Automation Systems

Block diagram transformations - pickoff point andsummary junctions

y(s) = u1(s) − u2(s) (3)


Block diagram transformations - example 1, solution 1

Simplify the following block diagram

where: 1 and 2 - summary junctions.



The block diagram can be simplified using the following rules: a) movingsummary junction (2) behind the block, b) changing the order of summaryjunctions (1) and (2).



where

G′(s) = 1 +1

G1(s)(4)

G′′(s) =G1(s)

1 − G1(s)G2(s)(5)

finally

G(s) =

[1 +

1

G1(s)

]G1(s)

1 − G1(s)G2(s)=

1 + G1(s)

1 − G1(s)G2(s)(6)



The block diagram can be simplified using the following rules: a) mov-ing summary junction (1) ahead of the block, b) changing the order ofsummary junctions (1) and (2).

G(s) = [1 + G1(s)]1

1 − G1(s)G2(s)=

1 + G1(s)

1 − G1(s)G2(s)(7)


Multi-input components - Lever

Where: x1, x2, y - displacements.

Equation of motion:

y(s) =b

a + bx1(s) +

a

a + bx1(s) (8)


Multi-input components - Hydraulic servodrive

Figure 2: Hydraulic servodrive - with spool valve


Multi-input components - Hydraulic servodrive

Figure 3: Hydraulic servodrive - with spool valve

Where: x1, x2, y - displacements.

Equation of motion:

y(s) =1

Ts(x1(s) + x2(s)) (9)


Multi-input components - Absorber

Figure 4: Absorber/Damper


Multi-input components - Absorber

Figure 5: Absorber: A - surface, Q -flow, x1, x2, y , - displacements, C -spring constant, α - valve constant

Equation of motion:

y(s) =Ts

Ts + 1x1(s) +

1

Ts + 1x2(s)


Construction of block diagram models

The block diagram enables to determine the role and place of each elementpresent in the system and how this element influences the processing ofinformation.

In order to construct the block diagram model, the following steps shouldbe taken:

1 Identify interactions, caused by changes in the value of the inputsignal.

2 Distinguish the elemets that process these interactions (blocksin the block diagram).

3 Determine the transfer fuctions of distinguished elements.

REMARK: The number of elements present in the block diagram may belarger than the number of structural elements in the block diagram - sincesome components may be influenced by more than one input.


Construction of block diagram model - Mechanicalfeedback 1


Construction of block diagram models - Mechanicalfeedback 1

Transfer function

G (s) =1

Ts

b

a + b

1

1 +a

a + b

1

Ts

=

=b

a

1

Ta + b

as + 1

Static characteristic

y =a

bx



SubstitutionA =

a

a + b− e

e + b(10)

Transfer function

G (s) =b

a + b

1Ts

1 + A 1Ts

=b

a + b

1

Ts + A(11)

Static characteristic

y =b

A(a + b)x (12)


Automation SystemsLecture 4 - Block Diagram Models

Jakub Mozaryn

Institute of Automatic Control and Robotics, Department of Mechatronics, WUT

Warszawa, 2020


Automation Systems - Lecture 4 - Block Diagram...

Documents

Transcript of Automation Systems - Lecture 4 - Block Diagram...