Automation Systems. Lecture 4 - Block Diagram Models. Jakub Mozaryn. Institute of Automatic Control and Robotics, Department of Mechatronics, WUT

Transcription

1 Lecture 4 - Block Diagram Models Institute of Automatic Control and Robotics, Department of Mechatronics, WUT Warszawa, 2018

2 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 the systems 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.

3 Introduction Figure 1: Example of block diagram model

4 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 and the signs of signals are distinguished. z = u y (2)

5 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 areonly 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 diagram to a simpler one.

6 Types of connections in the block diagram models Connection type Transfer function Block diagram Serial connection (chain) G(s) = G 1 (s)g 2 (s) Parallel connection G(s) = ±G 1 (s)±g 2 (s) Negative feedback loop G(s) = ±G 1 (s) 1 + G 1 (s)g 2 (s) Positive feedback loop G(s) = ±G 1 (s) 1 G 1 (s)g 2 (s)

7 Block diagram transformations - pickoff points Moving pickoff point ahead of the block Changing the order of pickoff points Moving pickoff point behind the block

8 Block diagram transformations - summary junctions Moving a summary junction behind a block Moving a summary junction ahead of a block Separation of a multiinput summary junction Changing the order of summary junctions

9 Block diagram transformations - pickoff point and summary junctions y(s) = u 1 (s) u 2 (s) (3)

10 Block diagram transformations - example 1, solution 1 Simplify the following block diagram where: 1 and 2 - summary junctions.

11 Block diagram transformations - example 1, solution 1 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).

12 Block diagram transformations - example 1, solution 1 where finally G ( s) = G (s) = G 1 (s) G (s) = G 1 (s) 1 G 1 (s)g 2 (s) [ ] G 1 (s) G 1 (s) 1 G 1 (s)g 2 (s) = 1 + G 1 (s) 1 G 1 (s)g 2 (s) (4) (5) (6)

13 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). G ( s) = [1 + G 1 (s)] 1 1 G 1 (s)g 2 (s) = 1 + G 1 (s) 1 G 1 (s)g 2 (s) (7)

14 Multi-input components - example 1 Where: x 1, x 2, y - displacements. Equation of motion: y(s) = b a + b x 1(s) + a a + b x 1(s) (8)

15 Multi-input components - example 2 Figure 2: Hydraulic servodrive - with spool valve

16 Multi-input components - example 2 Figure 3: Hydraulic servodrive - with spool valve Where: x 1, x 2, y - displacements. Equation of motion: y(s) = 1 Ts (x 1(s) + x 2 (s)) (9)

17 Multi-input components - example 3 Equation of motion: Figure 4: Absorber: A - surface, Q - flow, x 1, x 2, y, - displacements, C - spring constant, α - valve constant y(s) = Ts Ts + 1 x 1(s) + 1 Ts + 1 x 2(s)

18 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.

19 Construction of block diagram model - Example 1

20 Construction of block diagram models - Example 1 Transfer function G(s) = 1 Ts = b a b a + b 1 T a + b s + 1 a Static characteristic y = a b x a a + b 1 Ts =

21 Construction of block diagram models - Example 2

22 Construction of block diagram models - Example 2

23 Construction of block diagram models - Example 2 Substitution Transfer function G(s) = A = a a + b e e + b b 1 Ts a + b 1 + A 1 Ts = b a + b 1 Ts + A (10) (11) Static characteristic y = b A(a + b) x (12)

24 Lecture 4 - Block Diagram Models Institute of Automatic Control and Robotics, Department of Mechatronics, WUT Warszawa, 2018