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 Ge(s) → 預測 Tp , Ts , Tr , %OS with 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 1

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) = sinωt

非線性非時變系統

註 2: 系統形態

5.2 Block Diagrams

• 符號介紹

Figure 5.2Components 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 5.6a. 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 5.7Block 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

• Block Diagram Algebra (for summing junctions )

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

• 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 G1(s)

Figure 5.11

Example 5.2 P. 1/4

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 5.3Figure 5.15Feedback system

↙

求 Tp , Ts , %OS Find ζ, ωn first

Example 5.4 Figure 5.16 求 k=? Such that %OS=10% gain design for transient response

H.W. 1 Skill-Assessment Exercise 5.2H.W. 2 Problem 5 Fig. P5.5H.W. 3 Problem 17 Fig. P5.17Have a good time.