Control Systems (CS)

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1 Control Systems (CS) Lecture-0- Signal Flow raphs Dr. Imtiaz ussain Associate Professor Mehran University of Engineering & Technology Jamshoro, Pakistan URL :

2 Outline Introduction to Signal Flow raphs Definitions Terminologies Examples Mason s ain Formula Examples Signal Flow raph from Block Diagrams Design Examples

3 Introduction Alternative method to block diagram representation, developed by Samuel Jefferson Mason. Advantage: the availability of a flow graph gain formula, also called Mason s gain formula. A signal-flow graph consists of a network in which nodes are connected by directed branches. It depicts the flow of signals from one point of a system to another and gives the relationships among the signals.

4 Fundamentals of Signal Flow raphs Consider a simple equation below and draw its signal flow graph: y The signal flow graph of the equation is shown below; ax x a y Every variable in a signal flow graph is designed by a Node. Every transmission function in a signal flow graph is designed by a Branch. Branches are always unidirectional. The arrow in the branch denotes the direction of the signal flow. 4

5 Signal-Flow raph Models Y ( s) ( s) R ( s) ( s) R ( s) Y ( s) ( s) R ( s) ( s) R ( s) 5

6 Signal-Flow raph Models r and r are inputs and x and x are outputs a x a x r x a x a x r x 6

7 Signal-Flow raph Models x o is input and x 4 is output x x x x 4 ax dx fx hx 0 0 bx ex gx cx x 0 a b f x c d x g e x h x 4 7

8 Construct the signal flow graph for the following set of simultaneous equations. There are four variables in the equations (i.e., x,x,x,and x 4 ) therefore four nodes are required to construct the signal flow graph. Arrange these four nodes from left to right and connect them with the associated branches. Another way to arrange this graph is shown in the figure. 8

9 Terminologies An input node or source contain only the outgoing branches. i.e., X An output node or sink contain only the incoming branches. i.e., X 4 A path is a continuous, unidirectional succession of branches along which no node is passed more than ones. i.e., X to X to X to X 4 X to X to X 4 X to X to X 4 A forward path is a path from the input node to the output node. i.e., X to X to X to X 4, and X to X to X 4, are forward paths. A feedback path or feedback loop is a path which originates and terminates on the same node. i.e.; X to X and back to X is a feedback path. 9

10 Terminologies A self-loop is a feedback loop consisting of a single branch. i.e.; A is a self loop. The gain of a branch is the transmission function of that branch. The path gain is the product of branch gains encountered in traversing a path. i.e. the gain of forwards path X to X to X to X 4 is A A A 4 The loop gain is the product of the branch gains of the loop. i.e., the loop gain of the feedback loop from X to X and back to X is A A. Two loops, paths, or loop and a path are said to be non-touching if they have no nodes in common. 0

11 Consider the signal flow graph below and identify the following a) Input node. b) Output node. c) Forward paths. d) Feedback paths (loops). e) Determine the loop gains of the feedback loops. f) Determine the path gains of the forward paths. g) Non-touching loops

12 Consider the signal flow graph below and identify the following There are two forward path gains;

13 Consider the signal flow graph below and identify the following There are four loops

14 Consider the signal flow graph below and identify the following Nontouching loop gains; 4

15 Consider the signal flow graph below and identify the following a) Input node. b) Output node. c) Forward paths. d) Feedback paths. e) Self loop. f) Determine the loop gains of the feedback loops. g) Determine the path gains of the forward paths. 5

16 Input and output Nodes a) Input node b) Output node 6

17 (c) Forward Paths 7

18 (d) Feedback Paths or Loops 8

19 (d) Feedback Paths or Loops 9

20 (d) Feedback Paths or Loops 0

21 (d) Feedback Paths or Loops

22 (e) Self Loop(s)

23 (f) Loop ains of the Feedback Loops

24 (g) Path ains of the Forward Paths 4

25 Mason s Rule (Mason, 95) The block diagram reduction technique requires successive application of fundamental relationships in order to arrive at the system transfer function. On the other hand, Mason s rule for reducing a signal-flow graph to a single transfer function requires the application of one formula. The formula was derived by S. J. Mason when he related the signal-flow graph to the simultaneous equations that can be written from the graph. 5

26 Mason s Rule: The transfer function, C(s)/R(s), of a system represented by a signal-flow graph is; n P i i i C( s) Where R( s) n = number of forward paths. P i = the i th forward-path gain. = Determinant of the system i = Determinant of the i th forward path is called the signal flow graph determinant or characteristic function. Since =0 is the system characteristic equation. 6

27 Mason s Rule: n P C( s) i i R( s) = - (sum of all individual loop gains) + (sum of the products of the gains of all possible two loops that do not touch each other) (sum of the products of the gains of all possible three loops that do not touch each other) + and so forth with sums of higher number of non-touching loop gains i = value of Δ for the part of the block diagram that does not touch the i- th forward path (Δ i = if there are no non-touching loops to the i-th path.) i 7

28 Systematic approach. Calculate forward path gain P i for each forward path i.. Calculate all loop transfer functions. Consider non-touching loops at a time 4. Consider non-touching loops at a time 5. etc 6. Calculate Δ from steps,,4 and 5 7. Calculate Δ i as portion of Δ not touching forward path i 8

29 Example#: Apply Mason s Rule to calculate the transfer function of the system represented by following Signal Flow raph Therefore, There are three feedback loops C R P P L 4, L 4, L 4 9

30 Example#: Apply Mason s Rule to calculate the transfer function of the system represented by following Signal Flow raph There are no non-touching loops, therefore = - (sum of all individual loop gains) L L L

31 Example#: Apply Mason s Rule to calculate the transfer function of the system represented by following Signal Flow raph Eliminate forward path- = - (sum of all individual loop gains)+... = Eliminate forward path- = - (sum of all individual loop gains)+... =

32 Example#: Continue

33 Example#: Apply Mason s Rule to calculate the transfer function of the system represented by following Signal Flow raph P P. Calculate forward path gains for each forward path. P (path ) and P (path ). Calculate all loop gains. L, L, L, L. Consider two non-touching loops. L L L L 4 L L 4 L L

34 4. Consider three non-touching loops. None. 5. Calculate Δ from steps,, L L L L L L L L L L L L Example#: continue

35 5 Example#: continue Eliminate forward path- 4 L L L L Eliminate forward path

36 P P s R s Y ) ( ) ( 6 Example#: continue s R s Y ) ( ) (

37 Example# Find the transfer function, C(s)/R(s), for the signal-flow graph in figure below. 7

38 Example# There is only one forward Path. P ( s) ( s) ( s) 4( s) 5( s) 8

39 Example# There are four feedback loops. 9

40 Example# Non-touching loops taken two at a time. 40

41 Example# Non-touching loops taken three at a time. 4

42 Example# Eliminate forward path- 4

43 Example#4: Apply Mason s Rule to calculate the transfer function of the system represented by following Signal Flow raph P P P P s R s C i i i ) ( ) ( There are three forward paths, therefore n=. 4

44 Example#4: Forward Paths P A A A A76 P A A A A A76 P A 7 44

45 Example#4: Loop ains of the Feedback Loops L A A A4A4 L L A 54A45 L A 4 65A56 L A 5 76A67 L6 A 77 L A A 7 4 4A L A A A67 L L 9 A7A57A45A4A 0 A7A67A56A45A4A 45

46 Example#4: two non-touching loops L L L L 4 L L 5 L L 6 L L 4 L L 5 L L 6 L L 8 L L 5 L 4 L 6 7 L L 6 L 4 L 7 L 5 L L 7 L 8 L L 8 46

47 Example#4: Three non-touching loops L L L L 4 L L 5 L L 6 L L 4 L L 5 L L 6 L L 8 L L 5 L 4 L 6 7 L L 6 L 4 L 7 L 5 L L 7 L 8 L L 8 47

48 From Block Diagram to Signal-Flow raph Models Example#5 R(s) E(s) X - X - X - 4 C(s) R(s) - E(s) X X X 4 C(s)

49 ; ) ( P ) ( ) ( s R s C R(s) - 4 C(s) - - X X X E(s) From Block Diagram to Signal-Flow raph Models Example#5 49

50 Example#6 R(s) - E(s) X X Y + Y + C(s) X - - R(s) E(s) C(s) X Y - Y

51 Example#6 - - R(s) E(s) C(s) X Y 7 loops: X Y non-touching loops : 5

52 Example#6 X - - R(s) E(s) C(s) X Y Y Then: Δ 4 4 forward paths: p ( ) p ( ) ( ) Δ p Δ p4 Δ4 Δ 5

53 Example#6 We have C( s) R( s) pk k 4 5

54 Example-7: Determine the transfer function C/R for the block diagram below by signal flow graph techniques. The signal flow graph of the above block diagram is shown below. There are two forward paths. The path gains are The three feedback loop gains are No loops are non-touching, hence Because the loops touch the nodes of P, hence ence the control ratio T = C/R is Since no loops touch the nodes of P, therefore 54

55 Example-6: Find the control ratio C/R for the system given below. The signal flow graph is shown in the figure. The two forward path gains are The five feedback loop gains are There are no non-touching loops, hence All feedback loops touches the two forward paths, hence ence the control ratio T = 55

56 Design Example# V ( s) I( s) I( s) R Cs V s) I ( s) R ( CsV ( s) CsV( s) I( s) Cs Cs R V (s) (s) I V (s) 56

57 Design Example# F Ms X k( X X ) 0 M s X k( X X) k X 57

58 Design Example# 58

59 Design Example# 59

60 To download this lecture visit END OF LECTURES-0-60

SFG and Mason s Rule : A revision

SFG and Mason s Rule : A revision Andersen Ang 2016-Nov-29 SFG and Mason s Rule Vu Pham Review SFG: Signal-Flow Graph SFG is a directed graph SFG is used to model signal flow in a system SFG can be used