Automation Systems
Lecture 4 - Block Diagram Models
Jakub Mozaryn
Institute of Automatic Control and Robotics, Department of Mechatronics, WUT
Warszawa, 2016
Introduction
Block diagram model
Block diagram model (structural): Graphical representation of interrelationships between the parts of analyzed system, ie. there are given directions of signal flow and the relationships between input and output signals of all components of the analyzed system.
A block diagram, of either a single element or a complex system, is a form of a mathematical description of its function. It clearly expresses the dependence of the output signals from the input signal, if there are known informations about properties (the transfer functions) of its components.
Block diagrams consists of unidirectional, operational blocks that represent the transfer function of variables of interest.
Introduction
Figure 1 : Example of block diagram model
Elements of block diagrams
Block: A rectangle with arrows representing input and output signals. Inside rectangle the transfer function is written.
y (s) = G (s)u(s) (1) Pickoff point (information point):
Represents device that allow to retrieve the information and send it to several branches of the system.
Summary junction: represents the device that allow an algebraic summation of signals - the signs of signals are distinguished.
z = u − y (2)
Types of connections in the block diagram models
Using appropriate transformations, the block diagram representation can be often reduced to a simplified block diagram with fewer blocks than a original one, in which there are only four 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.
Types of connections in the block diagram models
Connection type Transfer function Block diagram
Serial connection
(chain) G (s) = G1(s)G2(s)
Parallel connection G (s) = ±G1(s)±G2(s)
Negative feedback
loop G (s) = ±G1(s)
1 + G1(s)G2(s)
Positive feedback
loop G (s) = ±G1(s)
1 − G1(s)G2(s)
Block diagram transformations - pickoff points
Moving pickoff point ahead of the block
Changing the order of pickoff points
Moving pickoff point behind the block
Block diagram transformations - summary junctions
Moving a summary junction behind a block
Moving a summary junction ahead of a block
Separation of a multi-input summary junction
Changing the order of summary junctions
Block diagram transformations - pickoff point and summary junctions
y (s) = u1(s) − u2(s) (3)
Block diagram transformations - example 1, solution 1
where: 1 and 2 - summary junctions.
Block diagram transformations - example 1, solution 1
where: 1 and 2 - summary junctions.
The block diagram can be simplified using the following rules: a) moving summary junction (2) behind the block, b) changing the order of
summary junctions (1) and (2).
Block diagram transformations - example 1, solution 1
where
G0(s) = 1 + 1
G1(s) (4)
G00(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)
Block diagram transformations - example 1, solution 2
The block diagram can be simplified using the following rules: a) moving summary junction (1) ahead of the block, b) changing the order of summary junctions (1) and (2).
Block diagram transformations - exercise 1
Block diagram transformations - exercise 2
Multi-input components - example 1
Where: x1, x2, y - displacements.
Equation of motion:
y (s) = b
a + bx1(s) + a
a + bx1(s) (8)
Multi-input components - example 2
Figure 2 : hydraulic servodrive - with spool valve Where: x1, x2, y - displacements.
Equation of motion:
Multi-input components - example 2
Figure 3 : hydraulic servodrive - with spool valve
Multi-input components - example 3
where: x1(t), x2(t), y (t) - displacements.
Equation of motion:
y (s) = Ts
x1(s) + 1
x2(s)
construction of block diagram models
The block diagram enables to determine the role and place of each element present in the system and how this element influences the processing of information.
In order to construct the block diagram model, the following steps should be taken:
1 Identify interactions, caused by changes in the value of the input signal.
2 Distinguish the elemets that process these interactions (blocks in the block diagram).
3 Determine the transfer fuctions of distinguished elements.
REMARK: The number of elements present in the block diagram may be larger than the number of structural elements in the block diagram - since some components may be influenced by more than one input.
Construction of block diagram model - Example 1
Construction of block diagram models - Example 1
Transfer function G (s) = 1
Ts b a + b
1 1 + a
a + b 1 Ts
=
= b a
1 Ta + b
a s + 1 Static characteristic
y = a bx
Construction of block diagram models - Example 2
Construction of block diagram models - Example 2
Construction of block diagram models - Example 2
Substitution
A = a
a + b − e
e + b (10)
Transfer function
G (s) = b a + b
1 Ts
1 + ATs1 = b a + b
1
Ts + A (11)
Static characteristic
Construction of block diagram models - Excercise 1
Where:
x (t), y (t), α1(t), α2(t), α(t), β(t), γ(t) - displacements, a, b, c - lenghts,
T1, T2- time constants.
Determine: block diagram, transfer function, static characteristic.
Automation Systems
Lecture 4 - Block Diagram Models
Jakub Mozaryn
Institute of Automatic Control and Robotics, Department of Mechatronics, WUT
Warszawa, 2016
