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# Control chap3

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### Control chap3

1. 1. CONTROL SYSTEMS THEORY Reduction of Multiple Subsystems CHAPTER 3 STB 35103
2. 2. Objectives  To reduce a block diagram of multiple subsystems to a signal block representing the transfer function from input to output
3. 3. Introduction  Before this we only worked with individual subsystems represented by a block with its input and output.  Complex systems are represented by the interconnection of many subsystems.  In order to analyze our system, we want to represent multiple subsystems as a single transfer function.
4. 4. Block diagram  A subsystems is represented as a block with an input and output and a transfer function.  Many systems are composed of multiple subsystems. So, we need to add a few more schematic elements to the block diagram.   Summing junction Pickoff points
5. 5. Block diagram
6. 6. Block diagram  Summing junction   Output signal, C(s), is the algebraic sum of the input signals, R1(s), R2(s) and R3(s). Pickoff point  Distributes the input signals, R(s), undiminished, to several output points.
7. 7. Block diagram  There are three topologies that can be used to reduce a complicated system to a single block.    Cascade form Parallel form Feedback form
8. 8. Block diagram  Cascade form    a. cascaded subsystem b. equivalent transfer function Equivalent transfer function is the output divided by the input.
9. 9. Block diagram  Parallel form  Parallel subsystems have a common input and output formed by the algebraic sum of the outputs from all of the subsystems.
10. 10. Block diagram  Feedback form  It is the same as the closed loop system that we learn in Chapter 1. a. closed loop system  b. closed loop, G(s)H(s) is open loop transfer function 
11. 11. Block diagram  Moving blocks to create familiar forms  Cascade, parallel and feedback topologies are not always apparent in a block diagram.  You will learn block moves that can be made in order to establish familiar forms when they almost exist. I.e. move blocks left and right past summing junctions and pickoff points.
12. 12. Block diagram Block diagram algebra for summing junctions— equivalent forms for moving a block a. to the left past a summing junction; b. to the right past a summing junction
13. 13. Block diagram 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
14. 14. Block diagram Block diagram reduction via familiar forms Example: Reduce the block diagram to a single transfer function.
15. 15. Block diagram Solution: Steps in solving Example 5.1: a. collapse summing junctions; b. form equivalent cascaded system in the forward path and equivalent parallel system in the feedback path; c. form equivalent feedback system and multiply by cascaded G1(s)
16. 16. Block diagram Block diagram reduction by moving blocks Example: Reduce the system shown to a single transfer function.
17. 17. Block diagram Solution: First, move G2(s) to the left past the pickoff point to create parallel subsystems, and reduce the feedback system consisting of G3(s) and H3(s).
18. 18. Block diagram Second, reduce the parallel pair consisting of 1/g2(s) and unity and push G1(s) to the right past the summing junction, creating parallel subsystems in the feedback.
19. 19. Block diagram Third, collapse the summing junctions, add the two feedback elements together, and combined the last two cascaded blocks.
20. 20. Block diagram Fourth, use the feedback formula to obtain figure below Finally multiply the two cascaded blocks and obtain the final result.
21. 21. Block diagram Exercise: Find the equivalent transfer function, T(s)=C(s)/R(s)
22. 22. Solution   Combine the parallel blocks in the forward path. Then, push 1/s to the left past the pickoff point. Combine the parallel feedback paths and get 2s. Apply the feedback formula and simplify
23. 23. Block diagram reduction rules  Summary
24. 24. Block diagram reduction rules
25. 25. Signal-Flow graphs Alternative method to block diagrams.  Consists of   (a) Branches   Represents systems (b) Nodes  Represents signals
26. 26. Signal-Flow graphs  Interconnection of systems and signals Example  V(s)=R1(s)G1(s)-R2(s)G2(s)+R3(s)G3(s) 
27. 27. Signal-Flow graphs  Cascaded system Block diagram Signal flow
28. 28. Signal-Flow graphs  Parallel system Block diagram Signal flow
29. 29. Signal-Flow graphs  Feedback system Block diagram Signal flow
30. 30. SFG Question  Given the following block diagram, draw a signal-flow graph
31. 31. Solution
32. 32. Mason’s rule  What?   A technique for reducing signal-flow graphs to single transfer function that relate the output of system to its input. We must understand some components before using Mason’s rule     Loop gain Forward-path gain Nontouching loops Nontouching-loop gain
33. 33. Mason’s rule  Loop gain  Product of branch gains found by going through a path that starts at a node and ends at the same node, following the direction of the signal flow, without passing through any other node more than once.     G2(s)H1(s) G4(s)H2(s) G4(s)G5(s)H3(s) G4(s)G6(s)H3(s)
34. 34. Mason’s rule  Forward-path gain  Product of gains found by going through a path from the input node of the signal-flow graph in the direction of signal flow.   G1(s)G2(s)G3(s)G4(s)G5(s)G7(s) G1(s)G2(s)G3(s)G4(s)G6(s)G7(s)
35. 35. Mason’s rule  Nontouching loops  Loops that do not have any nodes in common.  Loop G2(s)H1(s) does not touch loops G4(s)H2(s), G4(s)G5(s)H3(s) and G4(s)G6(s)H3(s)
36. 36. Mason’s rule  Nontouching-loop gain  Product of gains form nontouching loops taken two, three, four, or more at a time. [G2(s)H1(s)][G4(s)H2(s)]  [G2(s)H1(s)][G4(s)G5(s)H3(s)]  [G2(s)H1(s)][G4(s)G6(s)H3(s)] 
37. 37. Mason’s rule  The transfer function, C(s)/R(s), of a system represented by a signal-flow graph is C (s) G (s) = = R( s) ∑T ∆ k k k ∆ k = number of forward path Tk = the kth forward - path gain
38. 38. Mason’s rule ∆ k = formed by eliminating from ∆ those loop gains that touch the kth forward path. ∆ = 1 - Σ loop gains + Σ nontouching loop gains taken two at a time − Σ nontouching loop gains taken three at a time + Σ nontouching loop gains taken four at a time 
39. 39. Example  Draw the SFG representation
40. 40. Solution  SFG
41. 41. Solution
42. 42. Mason’s rule Question  Using Mason’s rule, find the transfer function of the following SFG
43. 43. Solution
44. 44. Exercise 1  Apply Mason’s rule to obtain a single transfer function
45. 45. Exercise 2 1. Reduce to a single transfer function (BDR) 2. Draw the SFG representation 3. Apply Mason’s rule to obtain the transfer function