In The Name of Allah The Most Beneficent The Most Merciful 1

ECE 4545: Control Systems Lecture: Reduction of Multiple Subsystems Engr. Ijlal Haider UoL, Lahore 2

Outline  Control Talk  Reducing Block Diagrams  Signal Flow Graph  Reducing Signal Flow Graphs  Mason’s Rule 3

©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 5.1 The space shuttle consists of multiple subsystems. 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 5.2 Components of a block diagram for a linear, time- invariant system

©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 5.3 a. Cascaded subsystems; b. equivalent transfer function

©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 5.4 Loading in cascaded systems

©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 5.5 a. Parallel subsystems; b. equivalent transfer function

©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 5.6 Feedback Loop

©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 5.7 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

©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 5.8 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

©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 5.17 Signal-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 5.18 Building signal-flow graphs: a. cascaded system nodes (from Figure 5.3(a)); b. cascaded system signal-flow graph; c. parallel system nodes (from Figure 5.5(a)); d. parallel system signal-flow graph; e. feedback system nodes (from Figure 5.6(b)); f. feedback system signal-flow graph

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©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering, 3/e Figure 5.19 Signal-flow graph development: a. signal nodes; b. signal-flow graph; c. simplified signal-flow graph

Mason’s Rule 16 Loop 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)

Mason’s Rule 17 Forward-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).

Mason’s Rule 18 Non-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

Mason’s Rule 19

Thank You! 20