Linear Systems Approximation: Understanding Homogeneity and Superposition

1. Linear System It is important to approximate a nonlinear system as a linear system. A linear system exhibits properties of Superposition: Homogeneity:

2. Linear System It is important to approximate a nonlinear system as a linear system. A linear system exhibit properties of Superposition: Homogeneity:

3. Linear vs Nonlinear

4. Nonlinearities

5. Nonlinear system We must find the equilibrium point Using Taylor series Neglecting higher order term

6. about Linearize Example Thus

7. Signal Flow Graph and Mason Rule

12. Figure 2.31 Two-path interacting system. (a) Signal-flow graph. (b) Block diagram.

13. P1=G1G2G3G4 (path 1) P2=G5G6G7G8 (path 2) There are four self loops: Loops L1 and L2 do not touch L3 and L4, Removing loops that touch path 1, Removing loops that touch path 2,

14. Using the block diagram,

16. Figure 2.33 Multiple-loop system.

17. Find the paths, Find the feedback loops: Loop L5 does not touch L4 or L7, and loop L3 does not touch L4