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In The Name of Allah The Most Beneficent The Most Merciful

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ECE 4545:Control Systems

Lecture:Reduction of Multiple Subsystems

Engr. Ijlal HaiderUoL, Lahore

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Outline

Control TalkReducing Block DiagramsSignal Flow GraphReducing Signal Flow GraphsMason’s Rule

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.1The space shuttleconsists of multiplesubsystems. Can you identify those that are control systems, or parts of control systems? © NASA-Houston.

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.2Components of a block diagram fora linear, time-invariantsystem

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.3a. Cascadedsubsystems;b. equivalent transferfunction

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.4Loading in cascadedsystems

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.5a. Parallelsubsystems;b. equivalenttransferfunction

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.6Feedback Loop

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.7Block diagramalgebra for summingjunctions—equivalent forms for moving a blocka. to the left past asumming junction;b. to the right past asumming junction

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.8Block diagram algebra for pickoff points—equivalent forms for moving a blocka. to the left past a pickoff point;b. to the right past a pickoff point

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.17Signal-flow graph components:a. system;b. signal;c. interconnection of systems and signals

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.18Building signal-flowgraphs:a. cascaded systemnodes (from Figure 5.3(a));b. cascaded systemsignal-flow graph;c. parallel systemnodes (from Figure 5.5(a));d. parallel systemsignal-flow graph;e. feedback systemnodes (from Figure5.6(b));f. feedback systemsignal-flow graph

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©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Figure 5.19Signal-flow graphdevelopment:a. signal nodes;b. signal-flow graph;c. simplified signal-flow graph

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Mason’s RuleLoop gain. The product of branch gains found by traversing 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. For examples of loop gains. There are four loop gains:

1. G2(s)Hi(s) 2. G4{s)H2{s)3. G4(s)G5(s)H3(s)4. G4(s)G6(,(s)H3(s)

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Mason’s RuleForward-path gain. The product of gains found for demonstrating Mason's rule by traversing a path from the input node to the output node of the signal-flow graph in the direction of signal flow. There are two forward-path gains:1. G1(s)G2(s)G3(s)G4(s)G5(s)G7(s)2. G1(s)G2(s)G3(s)G4(s)G6(s)G1(s)

Nontouching loops. Loops that do not have any nodes in common. Loop G2(s)Hi(s) does not touch loops G4(s)H2(s), G4(s)G5(s)H3(s), and G4(s)G6(s)H3(s).

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Mason’s RuleNon-touching-loop gain. The product of loop gains from non-touching loops taken two, three, four, or more at a time. The product of loop gain G2{s)Hi{s) and loop gain G4(s)H2(s) is a non-touching-loop gain taken two at a time.In summary, all three of the non-touching-loop gains taken two at a time are

1. [G2(s)H1(s)][G4(s)H2(s)]2. [G2(s)H1(s)][G4(s)G5(s)H3(s)]3. [G2(s)H1(s)][G4(s)G6(s)H3(s)]

The product of loop gains [G4(s)G5(s)H3(s)][G4(s)G6(s)H3(s)] is not a non-touching loop gain since these two loops have nodes in common. In our example there are no Non-touching-loop gain

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Mason’s Rule

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Thank You!