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

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

本章目標 1. 化簡複雜 block diagrams → 2. 由 G e (s) → 預測 T p, T s, T r, %OS with step input 3. Find step response for G e (s) 4. Design sys. gain → desired transient response 註 : 討論 Linear, Time-Invariant Systems 即 線性非時變系統 註1 註1 註2 註2

註 1 : 化簡成單一方塊的理由

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) = sin ω t 非線性非時變系統 註 2 : 系統形態

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

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

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

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

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

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

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

化簡 → 重要 公式 ! 化簡 → 化簡 →

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

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 Block Diagram Algebra (for summing junctions )

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 Block Diagram Algebra (for pickoff points )

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 G 1 (s)

Figure 5.11 Example 5.2 P. 1/4

Figure 5.12 a

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

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

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

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

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