2. Three basic forms G1G1 G2G2 G2G2 G1G1 G1G1 G2G2 G1G1 G2G2 G1G1 G2G2 G1G1 G1G1 G2G2 1+ cascade parallelfeedback

3. 2.6 block diagram models (dynamic) 2.6.2.2 block diagram transformations behind a block x1x1 y G ± x2x2 ± x1x1 x2x2 y G G Ahead a block ± x1x1 x2x2 y G x1x1 y G ± x2x2 1/G 1. Moving a summing point to be:

4. 2.6 block diagram models (dynamic) 2. Moving a pickoff point to be: behind a block G x1x1 x2x2 y G x1x1 x2x2 y 1/G ahead a block G x1x1 x2x2 y G G x1x1 x2x2 y

5. 2.6 block diagram models (dynamic) 3. Interchanging the neighboring— Summing points x3x3 x1x1 x2x2 y ＋ － x1x1 x3x3 y ＋ － x2x2 Pickoff points y x1x1 x2x2 y x1x1 x2x2

6. 2.6 block diagram models (dynamic) 4. Combining the blocks according to three basic forms. Notes: 1. Neighboring summing point and pickoff point can not be interchanged ！ 2. The summing point or pickoff point should be moved to the same kind ！ 3. Reduce the blocks according to three basic forms ！ Examples:

7. Moving pickoff point G1G1 G2G2 G3G3 G4G4 H3H3 H2H2 H1H1 a b G4G4 1 G1G1 G2G2 G3G3 G4G4 H3H3 H2H2 H1H1 Example 2.17

8. G2G2 H1H1 G1G1 G3G3 Moving summing point Move to the same kind G1G1 G2G2 G3G3 H1H1 G1G1 Example 2.18

9. G1G1 G4G4 H3H3 G2G2 G3G3 H1H1 Disassembling the actions H1H1 H3H3 G1G1 G4G4 G2G2 G3G3 H3H3 H1H1 Example 2.19

10. Chapter 2 mathematical models of systems 2.7 Signal-Flow Graph Models Block diagram reduction ——is not convenient to a complicated system. Signal-Flow graph —is a very available approach to determine the relationship between the input and output variables of a sys- tem, only needing a Mason’s formula without the complex reduc- tion procedures. 2.7.1 Signal-Flow Graph only utilize two graphical symbols for describing the relation- ship between system variables 。 Nodes, representing the signals or variables. Branches, representing the relationship and gain Between two variables. G

11. b 2.7 Signal-Flow Graph Models Example 2.20: x4x4 x3x3 x2x2 x1x1 x0x0 h f g e d c a 2.7.2 some terms of Signal-Flow Graph Path — a branch or a continuous sequence of branches traversing from one node to another node. Path gain — the product of all branch gains along the path.

12. 2.7 Signal-Flow Graph Models Loop —— a closed path that originates and terminates on the same node, and along the path no node is met twice. 2.7.3 Mason’s gain formula Loop gain —— the product of all branch gains along the loop. Touching loops —— more than one loops sharing one or more common nodes. Non-touching loops — more than one loops they do not have a common node.

13. 2.7 Signal-Flow Graph Models

14. Example 2.21 x4x4 x3x3 x2x2 x1x1 x0x0 h f g e d c b a

15. 2.7 Signal-Flow Graph Models 2.7.4 Portray Signal-Flow Graph based on Block Diagram Graphical symbol comparison between the signal-flow graph and block diagram: and Block diagramSignal-flow graph G(s)

16. G1G1 G4G4 G3G3 2.7 Signal-Flow Graph Models Example 2.22 － － － C(s) R(s) G1G1 G2G2 H2H2 H1H1 G4G4 G3G3 H3H3 E(s) X1X1 X2X2 X3X3 R(s)C(s) －H2－H2 －H1－H1 －H3－H3 X1X1 X2X2 X3X3 E(s) 1G2G2

17. 2.7 Signal-Flow Graph Models R(s) －H2－H2 1 G4G4 G3G3 G2G2 G1G1 1 C(s) －H1－H1 －H3－H3 X1X1 X2X2 X3X3 E(s)

18. 2.7 Signal-Flow Graph Models G1G1 G2G2 + － + － － － + C(s) R(s) E(s) Y2Y2 Y1Y1 X1X1 X2X2 － 1 -1 1 -1 -1 -1 -1 1 1 G1G1 G2G2 1 R(s) E(s) C(s) X1X1 X2X2 Y2Y2 Y1Y1 Example 2.23

19. 2.7 Signal-Flow Graph Models 1 -1 1 -1 -1 -1 -1 1 1 G1G1 G2G2 1 R(s) E(s) C(s) X1X1 X2X2 Y2Y2 Y1Y1 7 loops: 3 ‘2 non-touching loops’ :

20. 2.7 Signal-Flow Graph Models 1 -1 1 -1 -1 -1 -1 1 1 G1G1 G2G2 1 R(s) E(s) C(s) X1X1 X2X2 Y2Y2 Y1Y1 Then: 4 forward paths:

21. 2.7 Signal-Flow Graph Models We have