CH. 6 Stability 穩定度

•控制系統設計 3規格 : Transient response 暫態反應 (Tp , Ts , Tr , %OS ) Stability Stability 穩定度穩定度 Steady-state errors 穩態誤差

6.1 Introduction

• Stable system Stable system 定義定義 Natural (transient) response → 0 as t →∞

• Unstable systemUnstable system:

Natural (transient) response → ∞ as t →∞

• Marginally stable systemMarginally stable system: Natural (transient) response → neither decays nor

grows

Figure 6.1 Closed-loop poles and response:a. stable system; closed-loop poles at LHP (stands for Left Half Plane)

由閉回路系統之極點位置判別是否為穩定系統

Underdamped responses two complex poles:

Figure 6.1 Closed-loop poles and response:b. unstable system at least 1 closed-loop pole at RHP (Right Half Plane)

由閉回路系統之極點位置判別是否為穩定系統

2 closed-loop poles on imaginary axis → unstable system

C(t) = A tn cos(ωt+φ)

註 : marginally stable 1 closed-loop pole on imaginary axis

Figure 6.2Common cause of problems in finding closed-loop poles: a. original system; b. equivalent system

由因式分解求 closed-loop poles → 困難 ( 高階系統 ) → Routh-Hurwitz Criterion

T(s) Closed-loop T.F.

• Closed-loop T.F. T(s) = N(s)/D(s)

• Stable system 之條件 (1) D(s) = ( s + ai ) = ( s + a1 ) … ( s + an )=0

ai ＞ 0 穩定系統之條件 → D(s) 係數符號需一致

註 1: 若 D(s) 缺項或係數之符號不同→不穩定系統

註 2: D(s) 缺項 , 如 s2+1 = 0 缺 s 項

(2) 若 D(s) 無缺項且係數之符號相同 → 仍無法確認是否為穩定系統 → Routh-Hurwitz Criterion

• ( s + 1 ) ( s + 2) = s2 + 3s + 2

• ( s + 1 ) ( s + 2) ( s + 3)

= (s2 + 3s + 2) ( s + 3)

= s3 + 6s2 + 11s + 6

aaii ＞＞ 0 0 穩定系統 → 穩定系統 → 係數符號一致係數符號一致 → → 不缺項不缺項

6.2 Routh-Hurwitz Criterion

Table 6.1Initial layout for Routh table

Figure 6.3Equivalent closed-looptransfer function

T(s)

Table 6.2 Completed Routh table

Intepret the Routh table: 由 Routh table 判讀 closed-loop poles 在 RHP 之數目 closed-loop polesclosed-loop poles 在在 RHPRHP 之數目 之數目 = first column = first column 符號變更之次數符號變更之次數

如 Routh table’s first column 無符號變更 → stable system

• Ex. 6.1 Fig. 6.4

Table 6.3Completed Routh table

first column 符號變更之次數 = 2

→ 2 closed-loop poles at RHP→ unstable systemunstable system

6.3 Routh-Hurwitz Criterion: Special Cases Case 1: Zero Only in the First Column 1/2

Table 6.4 Completed Routh table

Example 6.2

Example 6.2

Table 6.5 Determining signs in first column of a Routh table with zero as first element in a row

2 sign changes → 2 closed-loop poles at RHP → unstable system

6.3 Routh-Hurwitz Criterion: Special Cases Case 1: Zero Only in the First Column 2/2

Example 6.4

Closed-loop T.F. T(s) = 10/ D(s)

D(s) = s5+7s4+6s3+42s2+8s+56

6.3 Routh-Hurwitz Criterion: Special Cases Case 2: Entire Row is Zero 1/6

Example 6.4 D(s) = s5+7s4+6s3+42s2+8s+56

Table 6.7 Routh table

Even polynomial: p(s) = s4+6s2+8

→ dP/ds = 4s3+12s+0 → 代入 s3 row → complete the Routh table

6.3 Routh-Hurwitz Criterion: Special Cases Case 2: Entire Row is Zero 2/6

→→

↓↓

Example 6.4 Closed-loop T.F. T(s) = 10/ D(s)

D(s) = s5+7s4+6s3+42s2+8s+56

Table 6.7 Completed Routh table

6.3 Routh-Hurwitz Criterion: Special Cases Case 2: Entire Row is Zero 3/6

→→

Example 6.4Example 6.4 poles 所在區域的判讀分兩部分

Table 6.7 Completed Routh table

6.3 Routh-Hurwitz Criterion: Special Cases Case 2: Entire Row is Zero 4/6

→→↓↓

↑↑

even polynomials

6.3 Routh-Hurwitz Criterion: Special Cases Case 2: Entire Row is Zero 5/6

Figure 6.5 Root positions to generate even polynomials: A , B, C, or any combination

註註 :: p(s) 之根對稱於原點

6.3 Routh-Hurwitz Criterion: Special Cases Case 2: Entire Row is Zero 6/6

Example 6.4Table 6.7 first column of Routh table

註 1: 檢驗 檢驗 p(s) p(s) 以下以下 , , 第第 11 列有列有無變號無變號 →無變號→無 closed-loop poles 在 RHP→ 無 closed-loop poles 在 LHP → 4 closed-loop poles 在 jω 軸上 註 2: 檢驗 檢驗 p(s) p(s) 以上以上 , , 第第 11 列有列有無變號無變號 1 closed-loop pole 在 LHP → → unstable system

→→↓↓

↑↑

系統穩定度判別

Example 6.5 Closed-loop T.F. T(s) = 10/ D(s)

D(s) = s8+s7+12s6+22s5+39s4+59s3+48s2+38s+20

6.3 Routh-Hurwitz Criterion: Special Cases Case 2: Entire Row is Zero 1/4

Example 6.5

D(s) = s8+s7+12s6+22s5+39s4+59s3+48s2+38s+20

Table 6.8 Routh table

Even polynomial: p(s) = s4+3s2+2

→ dP/ds = 4s3+6s+0 → 代入 s3 row → complete the Routh table

6.3 Routh-Hurwitz Criterion: Special Cases Case 2: Entire Row is Zero 2/4

↓

→

Table 6.8 Completed Routh table

6.3 Routh-Hurwitz Criterion: Special Cases Case 2: Entire Row is Zero 3/4

Example 6.5 Closed-loop T.F. T(s) = 10/ D(s) D(s) = s8+s7+12s6+22s5+39s4+59s3+48s2+38s+20

←←

6.3 Routh-Hurwitz Criterion: Special Cases Case 2: Entire Row is Zero 4/4

Example 6.5 Table 6.8 first column of Routh table

系統穩定度判別註 : 檢驗 p(s) p(s) 以下以下 , 第 1 列有無變號 → 無變號 → 無 closed-loop poles 在 RHP → 無 closed-loop poles 在 LHP → 4 4 closed-loopclosed-loop poles poles 在在 jjωω 軸軸上上註 : 檢驗 p(s) p(s) 以上以上 , 第 1 列有無變號 → 2 變號 →2 closed-loop pole 在 RHP → 2 closed-loop pole在 LHP → → unstable systemunstable system

←←

• 自修題 : Ex. 6.6 - Ex. 6.8

• H.W.: Skill-Assessment Exercise 6.1

Skill-Assessment Exercise 6.2

Skill-Assessment Exercise 6.3

Example 6.9 Stability Design via Routh-Hurwitz

Figure 6.10 Feedback control system

T(s) = k/ D(s) D(s) = s3+18s2+77s+k

For stable system:No sign change in the first column

→ k ＜ 1386 stable sys→ k ＞ 1386 unstable sys→ k = 1386 unstable sys (via even polynomial, 2 roots on jω axis, 1 root at LHP, unstable system )

Example 6.10 Factoring via Routh-Hurwitz

Factor the polynomial s4+3s3+30s2+30s+200Table 6.17

Even polynomial p(s) = s2+10

s4+3s3+30s2+30s+200 = (s2+10)(s2+3s+20)

→

Table 6.19 case study Routh table for antenna control

求 K 範圍 such that system is stable.