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CHAPTER 2

UNIVERSITI MALAYSIA PERLIS fakulti teknologi kejuruteraan Jabatan kejuruteraan elektriK. CHAPTER 2. PLT305 -The Basic of Control Theory. Objectives of Chapter. Block Diagram. Control Signal Signal Flow Graph. The Laplace Transform.

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CHAPTER 2

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  1. UNIVERSITI MALAYSIA PERLIS fakultiteknologikejuruteraan JabatankejuruteraanelektriK CHAPTER 2 PLT305 -The Basic of Control Theory PLT 305 : CONTROL SYSTEMS TECHNOLOGY

  2. Objectives of Chapter • Block Diagram. • Control Signal • Signal Flow Graph. • .

  3. The Laplace Transform The Laplace Transform of a function, f(t), is defined as; Eq A The Inverse Laplace Transform is defined by Eq B *notes

  4. Cont… An important point to remember: • The above is a statement that f(t) and F(s) are transform pairs. What this means is that for each f(t) there is a unique F(s) and for each F(s)there is a unique f(t).

  5. Cont… Transform Pairs: f(t) F(s)

  6. Block Diagram • A block diagram of a system is a practical representation of the functions performed by each component and of the flow of signals. • Cascaded sub-systems: Transfer Function G(s) Input Output

  7. Cont … • Transfer function is the ratio of the output over the input variables. • The output signal can then be derived as; C = GR (a) Multi-variables. Figure 2.0: Block Diagram. Figure 2.1: Block Diagram of Summing Point.

  8. Cont’d… (b) Block Diagram Summing point. Figure 2.3: Block Diagram of Summing Point.

  9. Cont’d… (c) Linear Time Invariant System. Figure 2.4: Components of a Block Diagram for a Linear, Time-Invariant System.

  10. (d) Cascade System. Figure 2.5: Cascade System and the Equivalent Transfer Function. Figure 2.6: Parallel System and the Equivalent Transfer Function.

  11. (e) Summing Junction. (f) Pickoff Points. Figure 1.12: Block diagram algebra for pickoff points— equivalent forms for moving a block (a) to the left past a pickoff point; (b) to the right past a pickoff point.

  12. Basic rules with block diagram transformation

  13. Cont …

  14. Cont …

  15. Example 1 Combining block in cascade Eliminating a feedback loop

  16. Example 2 1. Eliminating a feedback loop 2. Combining block in cascade 3. Eliminating a feedback loop

  17. Example 3

  18. Exercise 1 • Hint: • Apply moving a pickoff point behind the block • Apply eliminating a feedback loop Answer:

  19. Exercise 2 Simplify this diagram

  20. Control Signal E(s) error signal R(s) reference signal Y(s) output signal C(s) output signal B(s) output signal from feedback • Open loop transfer function: • Feedforward transfer function:

  21. Cont … • Feed forward transfer function, • Feedbacktransfer function, • Error, E(s)transfer function,

  22. Cont … (1) • Close-Loop transfer function, (2) (3) Substitute (3) in (1) (4) (5) (6)

  23. Signal Flow Graph. • Multiple subsystem can be represented in two ways; (a) Block Diagram. (b) Signal Flow Graph. • The block diagram. • The signal flow graph,

  24. Cont’d… • Signal flow graph consists only branches which represent system and nodes which represents signals. • Variables are represented as nodes. • Transmittance with directed branch. • Source node: node that has only outgoing branches. • Sink node: node that has only incoming branches.

  25. Cont…

  26. Cont… • Parallel connection, • Signal-flow graph components: (a) system; (b) signal; (c) interconnection of systems and signals

  27. Cont’d… Cascade Parallel Feedback Signal-flow graph development: (a) signal nodes; (b) signal-flow graph; (c) simplified signal-flow graph

  28. Example 4: Signal Flow Graph. Given the block diagram, find the signal flow graph.

  29. Solution:

  30. Mason’s rule where  Total transmittence for every single loop.  Total transmittence for every 2 non-touching loops.  Total transmittence for every 3 non-touching loops.  Total transmittence for every m non-touching loops.  Total transmittence for k paths from source to sink nodes.

  31. Mason’s rule where:  Total transmittence for every single non-touching loop of ks’ paths.  Total transmittence for every 2 non-touching loop of ks’ paths.  Total transmittence for every 3 non-touching loop of ks’ paths.  Total transmittence for every n non-touching loop of ks’ paths.

  32. Example 5: Mason’s Rule Given the block diagram, find the transfers function (Y(s)/R(s)).

  33. 1. The path connection: P1 = G1G2G3G4 P2 = G5G6G7G8 • The loop gains: • L1 = G2H2 • L2 = G3H3 • L3 = G6H6 • L4 = G7H7 • The non-touching taken two at a time: • L1L3 • L1L4 • L2 L3 • L2L4

  34. Compute : • =1-[L1+ L2+ L3+ L4]+[L1L3 + L1L4 + L2L3+L2L4] • Compute k: • Path 1: 1=1- (L3+L4) • Path 2: 2=1- (L1+L2)

  35. Block diagram

  36. G1(s) G2(s) G3(s) G4(s) G5(s) R(s) C(s) V4(s) V3(s) V2(s) V1(s) H2(s) H1(s) G6(s) G8(s) G7(s) V6(s) V5(s) H4(s) Exercise 2 Given the block diagram, find the transfers function (C(s)/R(s)).

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