1. سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم دکتر حسين بلندي- دکتر سید مجید اسما عیل زاده

2. Recap. State Space Equation : –Canonical Forms, –Transfer Function, Block Diagram 2

3. Signal-Flow Graph Models

4. 4 Outline Terms and concepts Mason’s signal-flow gain formula Numerical examples

5. 5 A signal-flow graph A diagram consisting of nodes that are connected by several directed branches. A graphical representation of a set of linear relations.

6. 6 The basic elements of a signal- flow graph branch - a unidirectional path segment, which relates the dependency of an input and an output variable. nodes - the input and output points or junctions. path - a branch or continuous sequence of branches that can be traversed from one node to another node.

7. 7 All branches leaving a node will pass the nodal signal to the output node of each branch ( uniderectionally ). The summation of all signals entering a node is equal to the node variable. The relation between each variable is written next to the directional arrow.

8. 8 A loop - a closed path that originates and terminates on the same note, and along the path no node is met twice. Two loops are said to be nontouching if they do not have a common node. Two touching loops share one or more common nodes.

9. 9 Block and branch of DC motor

10. 10 Two-input, two-output system

11. 11 Y 1 (s) = G 11 (s) R 1 (s) + G 12 (s) R 2 (s) Y 2 (s) = G 21 (s) R 1 (s) + G 22 (s) R 2 (s) G ik - transfer function relating the i-th output to the k-th input

12. 12 Interconnected system

13. 13

14. 14 The flow graph is a pictorial method of writing a system of algebraic equations so as to indicate the interdependencies of the variables.

15. 15 A set of simultaneous equations 1. Write the system equations in the form X 1 = A 11 X 1 + A 12 X 2 + …+ A 1n X n X 2 = A 21 X 1 + A 22 X 2 + …+ A 2n X n ….…………………………………… X m = A m1 X 1 + A m2 X 2 + …+ A mn X n Note: An equation for X 1 is not required if X 1 is an input node.

16. 16 2. Arrange the m or n (whichever is larger) nodes from left to right. 3. Connect the nodes by the appropriate branches A 11, A 12, etc. 4. If the desired output node has outgoing branches, add a dummy note and unity branch. 5. Rearrange the nodes and/or loops in the graph to achieve pictorial clarity.

17. 17 Set of simultaneous algebraic equations a 11 x 1 + a 12 x 2 + r 1 = x 1 a 21 x 1 + a 22 x 2 + r 2 = x 2 r 1, r 2 - input variables x 1, x 2 - output variables x 1 (1 - a 11 ) + x 2 (- a 12 ) = r 1 x 1 ( - a 21 ) + x 2 (1 - a 22 ) = r 2

18. 18

19. 19 Mason’s signal-flow gain formula T ij (s) = ∑ k P ijk ∆ ijk /∆ P ijk = k-th path from variable x i to variable x j ∆ = determinant of the graph ∆ ijk = cofactor of the path P ijk ∑ k = all possible k path from x i to x j

20. 20 Step by step construction y 2 = a 12 y 1 + a 32 y 3 y 3 = a 23 y 2 + a 43 y 4 y 4 = a 24 y 2 + a 34 y 3 + a 44 y 4 y 5 = a 25 y 2 + a 45 y 4

21. 21 The nodes representing the variables y 1,y 2,y 3,y 4 and y 5 are located in order from left to right. The first equation states that y 2 depends upon two signals a 11 y 1 and a 32 y 3. The second equation states that y 3 depends upon the sum of a 23 y 2 and a 43 y 4,therefore a branch of gain a 23 is drawn from node y 2 to y 3 and a branch of gain a 43 is drawn from y 4 to y 3, with directions of the branches indicated by arrows. Similarly, with consideration of the third and fourth equation.

22. 22 Mason’s signal-flow gain formula ∆ = 1 - (sum of all different loop gains) + ( sum of the gain products of all combinations of two nontouching loops) - ( sum of the gain products of all combinations of three nontouching loops) + …,

23. 23 ∆ ijk = cofactor of the path P ijk is the the determinant with the loops touching the k-th path removed.

24. 24 T(s) = Y(s)/R(s) T(s) = Y(s)/R(s) = ∑ k P k ∆ k /∆

25. 25 The path gain or transmittance P k is defined as continuous succession of branches that are traversed in the direction of the arrows and with no node encountered more than once. A loop is defined as a closed path in which no node is encountered more than ones per traversal.

26. 26 Ex. 2.8 Interacting system

27. 27 The paths connecting the input R(s) and output Y(s) path 1 P 1 = G 1 G 2 G 3 G 4 path 2 P 2 = G 5 G 6 G 7 G 8

28. 28 Four self-loops L 1 = G 2 H 2 L 2 = H 3 G 3 L 3 = G 6 H 6 L 4 = G 7 H 8 Loops L 1 and L 2 do not touch L 3 and L 4

29. 29 The determinant : ∆ = 1 - (L 1 + L 2 + L 3 + L 4 ) + ( L 1 L 3 +L 1 L 4 + L 2 L 3 + L 2 L 4 )

30. 30 The cofactor along path 1 is evaluated by removing the loops that touch path 1 from ∆. L 1 = L 2 = 0 ∆ 1 = 1 - (L 3 +L 4 )

31. 31 The cofactor along path 2 is evaluated by removing the loops that touch path 2 from ∆. L 3 = L 4 = 0 ∆ 2 = 1 - (L 1 +L 2 )

32. 32 The transfer function of the system T(s) = Y(s)/R(s) = (P 1 ∆ 1 + P 2 ∆ 2 )/∆

33. 33 Ex. 2.7 Block diagram reduction

34. 34

35. 35

36. 36 Ex. 2.7 Mason’s signal-flow gain P 1 = G 1 G 2 G 3 G 4 L 1 = - G 2 G 3 H 2 L 2 = G 3 G 4 H 1 L 3 = - G 1 G 2 G 3 G 4 H 3

37. 37 All the loops have common nodes and therefore are all touching. The path P 1 touches all the loops, so ∆ 1 = 1 T(s) = Y(s)/R(s) = P 1 ∆ 1 /(1 - L 1 - L 2 - L 3 )

38. 38 P 1 = G 1 G 2 G 3 G 4 L 1 = - G 2 G 3 H 2 L 2 = G3 G4 H1 L3 = - G1 G2 G3 G4 H3 T(s) = Y(s)/R(s) = P 1 ∆ 1 /(1 - L 1 - L 2 - L 3 ) G 1 G 2 G 3 G 4 /(1+ G 2 G 3 H 2 - G3 G4 H1- G1 G2 G3 G4 H3)

39. 39 T(s) = G 1 G 2 G 3 G 4 /(1 - G3 G4 H1 + G 2 G 3 H 2 + G1 G2 G3 G4 H3)

40. 40 Ex. 2.11 A complex system

41. 41 The forward paths: P 1 = G 1 G 2 G 3 G 4 G 5 G 6 P 2 = G 1 G 2 G 7 G 6 P 3 = G 1 G 2 G 3 G 4 G 8

42. 42 Ex. 2.11 A complex system

43. 43 The feedback loops: L 1 = - G 2 G 3 G 4 G 5 H 2 L 2 = - G 5 G 6 H 1 L 3 = - G 8 H 1 L 4 = - G 7 H 2 G 2 L 5 = - G 4 H 4 L 6 = - G 1 G 2 G 3 G 4 G 5 G 6 H 3 L 7 = - G 1 G 2 G 7 G 6 H 3 L 8 = - G 1 G 2 G 3 G 4 G 8 H 3

44. 44 The determinant and cofactors: ∆ = 1 - (L 1 + L 2 + L 3 + L 4 + L 5 + L 6 + L 7 + L 8 ) + ( L 5 L 7 + L 5 L 4 + L 3 L 4 ) ∆ 1 = ∆ 3 = 1 ∆ 2 = 1 - L 5 = 1 + G 4 H 4 Loop L 5 does not touch loop L 7 or loop L 4, and loop L 3 does not touch loop L 4 ; but all other loops touch.

45. 45 The transfer function: T(s) = Y(s)/R(s) = (P 1 ∆ 1 + P 2 ∆ 2 + P 3 ∆ 3 )/∆

46. 46 Important properties of SF-G A SF-G applies only to linear systems. The equations for which a SF-G is drawn must be algebraic equations in the form of effects as function of causes. Nodes are used to represent variables. Normally, the nodes are arranged from left to right, following a succession of causes and effects through the system.

47. 47 Signal travel along branches only in the direction described by the arrows of the branches. The branch directing from node y k to y j represents the dependence of variable y j upon y k, but not the reverse. A signal y k traveling along a branch between nodes y k and y j is multiplied by the gain of the branch, a kj, so that a signal a kj y k is delivered at note y j.

48. 48 In the case when the system is represented by a set of integrodifferencial equations, we must first transform these into Laplace transform equations and then rearrange the latter into the form of Y j (s) = ∑G kj (s) Y k (s) for k, j =1,2,…,N

49. 49 Summary An alternative use of T(s) models in S-FG was investigated. Mason’s SF-G formula was found to be useful for obtaining the relationship between system variables in complex feedback system. Mason’s SF-G formula provides the relationship between system variables without any reduction or manipulation of the flow graph.

50. Time Domain Performance Specification 50

51. Test Input Signal Since the actual input signal of the system is usually unknown, a standard test input signal is normally chosen. Commonly used test signals include step input, ramp input, and the parabolic input. 51

52. General form of the standard test signals r(t) = t n R(s) = n!/s n+1 52

53. Test signals r(t) = A t n n = 0 n = 1 n = 2 r(t) = Ar(t) = Atr(t) = At 2 R(s) = 2A/s 3 R(s) = A/s R(s) = A/s 2 53

54. Table 5.1 Test Signal Inputs Test Signal r(t) R(s) Step position r(t) = A, t > 0 = 0, t < 0 R(s) = A/s Ramp velocity r(t) = At, t > 0 = 0, t < 0 R(s) = A/s 2 Parabolic acceleration r(t) = At 2, t > 0 = 0, t < 0 R(s) = 2A/s 3 54

55. Test inputs vary with target type parabola ramp step 55

56. Steady-state error Is a difference between input and the output for a prescribed test input as 56

57. Application to stable systems Unstable systems represent loss of control in the steady state and are not acceptable for use at all. 57

58. Steady-state error: a) step input, b) ramp input 58

59. Time response of systems c(t) = c t (t) + c ss (t) The time response of a control system is divided into two parts: c t (t) - transient response c ss (t) - steady state response 59

60. Transient response All real control systems exhibit transient phenomena to some extend before steady state is reached. lim ct(t) = 0 for t  60

61. Steady-state response The response that exists for a long time following any input signal initiation. 61

62. Poles and zeros of a first order system Css(t)Ct(t) 62

63. Poles and zeros 1.A pole of the input function generates the form of the forced response ( that is the pole at the origin generated a step function at the output). 2.A pole of the transfer function generate the form of the exponential response 3. The zeros and poles generate the amplitudes for both the transit and steady state responses 63

64. A pole on the real axis generate an exponential response of the form Exp[-  t] where -  is the pole location on real axis. The farther to the left a pole is on the negative real axis, the faster the exponential transit response will decay to zero. Effect of a real-axis pole upon transient response 64

65. Evaluating response using poles Css(t ) Ct(t) 65