Simplifying Complex Systems through Block Diagram Reduction

1. CH 5 Reduction of Multiple Subsystems 5.1 Introduction • 實際系統複雜 由許多子系統組成 (1子系統 1方塊) • 分別求子系統的數學模型 連結各方塊呈現整體系統 如何預測? Tp , Ts , Tr , %OS with step input

2. 系統表示法: Frequency domain → Block diagrams Time domain → Signal-flow graphs

3. 本章目標 1. 化簡複雜 block diagrams → 註1 2. 由Ge(s) →預測 Tp , Ts , Tr , %OSwith step input 3. Find step response for Ge(s) 4. Design sys. gain → desired transient response 註:討論Linear, Time-Invariant Systems 即 線性非時變系統 註2

4. 註1:化簡成單一方塊的理由

5. 註2: 系統形態 Linear, Time-Invariant Systems 線性非時變系統 Linear, Time-varying Systems 線性時變系統 當a(t),b(t),k(t) 為變動的常數 Non-linear, Time-varying Systems 非線性時變系統 當a(t),b(t),k(t) 為變動的非線性函數 e.g. a(t) = sinωt 非線性非時變系統

6. Figure 5.2Components of a block diagram for a linear, time-invariant system 5.2 Block Diagrams • 符號介紹 加法器 撿取點

7. Figure 5.3b equivalent transfer function Figure 5.3a Cascaded subsystems • Cascade Systems(分類1)

8. 兩子系統串連 要注意 Loading 問題 Figure 5.4Loading in cascaded systems

9. Figure 5.5 a. Parallel subsystems; b. equivalent transfer function • Parallel Systems(分類2)

10. Feedback Systems(分類3) Figure 5.6a. Feedback control system; b. simplified model; c. equivalent transfer function

11. 化簡 → Block Diagram Algebra 證明: 重要! 重要 公式!

12. 化簡 → Block Diagram Algebra 證明: 重要! 重要 公式!

13. 化簡 → 重要 公式! 化簡 → 化簡 →

14. G(s) H(s): open-loop T.F. loop gain

15. Block Diagram Algebra (for summing junctions ) Figure 5.7Block diagram algebra for summing junctions — equivalent forms for moving a blocka. to the left past a summing junction; b. to the right past a summing junction

16. Block Diagram Algebra (for pickoff points ) 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

17. Example 5.1方塊圖化簡 Figure 5.9 Figure 5.10 a. collapse summing junctions; b. form equivalent cascaded system in the forward path and equivalent parallel system in the feedback path; c. form equivalent feedback system and multiply by cascaded G1(s)

22. Example 5.2 P. 1/4 Figure 5.11

23. Example 5.2 P. 1/4 Figure 5.12 a

24. Example 5.2 P. 2/4 Figure 5.12 a Figure 5.12 b

26. Example 5.2 P. 3/4 Figure 5.12 b Figure 5.12 c

27. Example 5.2 P. 4/4 Figure 5.12 c Figure 5.12 d Figure 5.12 e

28. 5.3 Analysis and Design of Feedback Systems Figure 5.14 Second-order feedback control systemK 為變數 ↙ Example 5.3Figure 5.15Feedback system 求Tp , Ts , %OS Find ζ, ωn first

29. Example 5.4 Figure 5.16 求k=?Such that %OS=10% gain design for transient response H.W. 1 Skill-Assessment Exercise 5.2H.W. 2 Problem 5 Fig. P5.5H.W. 3 Problem 17 Fig. P5.17Have a good time.