ME_446_-_Lec_09-Signal_Flow_Graphs

ME_446_-_Lec_09-Signal_Flow_Graphs - Signal Flow Graphs...

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Unformatted text preview: Signal Flow Graphs 446-9 446- Prof. Neil A. Duffie University of Wisconsin-Madison Wisconsin- 9 ® Neil A. Duffie, 1996 All rights reserved 1 Incorrect Block Diagram Manipulations s Control Kp + Kc + Process Kp s(τ p s + 1) R(s) + - C(s) R(s) + - s Control Kp + Kc + Process Kp C(s) s(τ p s + 1) 2 9 Correct Block Diagram Manipulations s K cK p Control + Kc Process Kp s(τ p s + 1) R(s) + s K cK p + C(s) Control + + Kc Process Kp s(τ p s + 1) 3 + R(s) 9 C(s) Block Diagram Reduction K cK p R(s) 1+ s K cK p s( τ p s + 1) K cK p 1+ s( τ p s + 1) C(s) R(s) R(s) ⎛ ⎞⎛ ⎞ K cK p ⎜1+ s ⎟ ⎜ ⎟ ⎜ K K ⎟ ⎜ s( τ s + 1) + K K ⎟ ⎝ c p⎠⎝ p c p⎠ C(s) 9 4 Block Diagram with Disturbance Input Command R(s) + Disturbance D(s) + C(s) + G1(s) G2(s) H(s) • A disturbance input is an unwanted or unavoidable input signal that affects a system’s output. Examples: - load torque in motor control 9 - open door in room climate control 5 Superposition: C(s) = ∆Cr(s) + ∆Cd(s) G1(s) + D(s) G2(s) H(s) Note sign change! R(s) + 9 ∆Cd(s) + G2(s) H(s) + C(s) G1(s) ∆Cr(s) 6 Disturbance Portion Redrawn D(s) + G1(s) G2(s) H(s) + R(s) + 9 ∆Cd(s) C(s) G1(s) H(s) G2(s) + ∆Cr(s) 7 Reduced Block Diagram D(s) G 2 ( s) 1 + G1(s)G 2 (s)H(s) ∆Cd(s) + + C(s) R(s) G1 ( s ) G 2 ( s ) 1 + G1(s)G 2 (s)H(s) ∆Cr(s) 8 9 Individual Transfer Functions With R(s) = 0: G 2 ( s) C( s ) = D(s) 1 + G1(s)G 2 (s)H(s) With D(s) = 0: C( s ) G1 ( s)G2 (s) = R(s) 1 + G1(s)G 2 (s)H(s) Transfer equation: G (s )G 2 (s )R(s ) + G 2 ( s)D( s) C(s) = 1 1 + G1(s)G 2 (s)H(s) 9 9 Utility of Signal Flow Graphs • Alternative to block diagram approach - may be better for complex systems - good for highly interwoven systems - system variables represented as nodes - branches (lines) between nodes show relationships between system variables • The “flow graph gain formula” (Mason) allows the system transfer function to be directly computed without manipulation or reduction of the diagram. 10 9 Basic Signal Flow Graph R(s) + H(s) E(s) G(s) C(s) R(s) Input node 9 1 E(s) G(s) -H(s) C(s) Output node 11 Signal Flow Graph Example R(s) + E(s) F(s) + G1(s) D(s) + Q(s) + H(s) D(s) G2(s) C(s) G2(s) C(s) R(s) 1 E(s) 1 F(s) G1(s) 1 Q(s) 9 -1 -H(s) 12 Terms for Mason’s Gain Formula • Path: A branch or sequence of branches that can be traversed from one node to another. • Loop: A closed path, along which no node is met twice, that originates and terminates in the same node. • Nontouching: Two loops are nontouching if they do not share a common node. 9 • Gain: Refers, in this case, to the product of transfer functions. 13 Mason’s Gain Formula: P∆ O( s) =∑ k k I(s) ∆ k • Pk = the gain of the kth forward path between I(s) and O(s). • ∆ = 1 - (sum of all individual loop gains) + (sum of gain products of all combinations of 2 nontouching loops) - (sum of gain products of all combinations of 3 nontouching loops) +… 9 • ∆k = value of ∆ for that part of graph nontouching the kth forward path. 14 Example of Gain Formula Use R(s) 1 E(s) 1 F(s) G1(s) 1 D(s) G2(s) C(s) Q(s) -1 -H(s) • Assume R(s) = 0, desire to find the transfer function C(s)/D(s). • There is only one forward path between D(s) and C(s), therefore k = 1. 9 • There are two loops. They are touching. 15 Example of Gain Formula Use R(s) 1 E(s) 1 F(s) G1(s) 1 D(s) G2(s) C(s) Q(s) -1 P1 = G2(s) ∆ = 1 - [-G1(s)G2(s)H(s) - G1(s)G2(s)] ∆1 = 1 (Both loops touch the kth path) path) 9 16 -H(s) Example of Gain Formula Use R(s) 1 E(s) 1 F(s) G1(s) 1 D(s) G2(s) C(s) Q(s) -1 P∆ C( s ) =∑ k k D(s) k ∆ G 2 (s) C(s) = D(s) 1 + G1(s)G 2 (s)H(s) + G1(s)G2 (s) 9 -H(s) 17 ...
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