1. Chapter 7 Steady-State Errors
穩態誤差

2. 7.1 Introduction 控制系統設計 3規格: Transient response 暫態反應
(Tp , Ts , Tr , %OS )
Stability 穩定度
Steady-state errors 穩態誤差, e(∞)
System discussed: stable system only.

3. 討論3類系統的控制誤差 位置控制； 等速度控制； 等加速度控制。
Figure 7.1 Test inputs for steady-state error analysis and design vary with target type

4. 位置控制指令 等速度控制指令 等加速度控制指令
3 Inputs (即3種指令) : Step input(位置) ；Ramp input(等速度) ；Parabolic input(等加速度)
位置控制指令
等速度控制指令
等加速度控制指令
Table 7.1 Test waveforms for evaluating steady-state errors of position control systems

6. Figure 7. 2 Steady-state error e(∞) a
Figure 7.2 Steady-state error e(∞) a. step input; output1: e(∞)=0 output2: e(∞)=constant b. ramp input output1: e(∞)=0 output2: e(∞)= constant output3: e(∞)= ∞ unstable

7. Error 定義: e(t) = Input (t) - Output (t) E(s) = R(s) – C(s)
Steady-state error 定義:
e(∞) = Input (∞) - Output (∞) at time domain
e(∞) = lims→0 s E(s) (by final value theorem)

8. Figure 7. 4 e(∞) 由system configuration and input 決定: a
Figure e(∞) 由system configuration and input 決定: a. finite steady-state error for a step input; Csteady-state = K esteady-state K↑ esteady-state ↓ esteady-state = 0 → impossible b. zero steady-state error for step input esteady-state = 0 (∵ 系統具積分器)

9. 7.2 Steady-State Error for Unity Feedback Systems (case 1) 1/2
E(s) = R(s) – C(s) where C (s) = G(s)E(s) → E(s) = R(s)/(1 + G(s)) 註：e(∞) 由system configuration and input 決定
G(s)
R(s)

10. 7.2 Steady-State Error for Unity Feedback Systems (case 1) 2/2
E(s) = R(s) – C(s) where C (s) = G(s)E(s) → E(s) = R(s)/(1 + G(s)) → e(∞) = lims→0 S E(s) or e(∞) = lims→0 S R(s)/(1 + G(s)) 註：Steady-State Error

11. 7.2 Steady-State Error for Unity Feedback Systems (case 2)
E(s) = R(s) – C(s) where C (s) = T(s)R(s) → E(s) = R(s) [1 － T(s)] → e(∞) = lims→0 S E(s) or e(∞) = lims→0 S R(s)[1 － T(s)]
e(∞) 由system configuration and input 決定

12. Figure 7.8 Feedback control system for defining system type
定義 of System Type: n=0 Type 0 system n=1 Type 1 system n=2 Type 2 system

13. ※ 求 esteady-state under 3 input signals 1/4
For step input R(s)=1/S
e(∞) = lims→0 S R(s)/(1 + G(s)) ←公式
= lims→0 S (1/S)/(1 + G(s))
= lims→0 1/(1 + G(s))
= 1/(1 + lims→0 G(s))
if wish e(∞) = 0 → then lims→0 G(s) = ∞

14. ※ 求 esteady-state under 3 input signals 2/4
For step input (續)
e(∞) = 1/(1 + lims→0 G(s))
if wish e(∞) = 0 → then lims→0 G(s) = ∞
lims→0 G(s) = ∞ if n≧1 for
n≧1 stands for 1 integrator in the forward path
i.e. system type ≧1 to derive e(∞) = 0

15. ※ 求 esteady-state under 3 input signals 3/4
For ramp input R(s)=1/S2
e(∞) = lims→0 S R(s)/(1 + G(s))
= lims→0 S (1/S2)/(1 + G(s))
= lims→0 1/S(1 + G(s))
= 1/ lims→0 SG(s)
if wish e(∞) = 0 → then lims→0 SG(s) = ∞
lims→0 sG(s) = ∞ if n≧2 for
n≧2 stands for 2 integrators in the forward path
i.e. system type ≧2 to derive e(∞) = 0

16. ※ 求 esteady-state under 3 input signals 4/4
For parabolic input R(s)=1/S3
e(∞) = lims→0 S R(s)/(1 + G(s))
= lims→0 S (1/S3)/(1 + G(s))
= lims→0 1/ S2(1 + G(s))
= 1/ lims→0 s2G(s)
if wish e(∞) = 0 → then lims→0 s2G(s) = ∞
lims→0 s2G(s) = ∞ if n≧3 for
n≧3 stands for 3 integrators in the forward path
i.e. system type ≧3 to derive e(∞) = 0

17. 公式彙總 指令不同 求 esteady-state 公式不同 For step input R(s)=1/S
e(∞) = 1/(1 + lims→0 G(s))
For ramp input R(s)=1/S2
e(∞) = 1/ lims→0 SG(s)
For parabolic input R(s)=1/S3
e(∞) = 1/ lims→0 s2G(s)

18. Example 7. 2 不同指令下 求 esteady-state Figure 7
Example 不同指令下 求 esteady-state Figure 7.5 Feedback control system for system with no integrator
type 0 系統
R(s) = 5u(t) = 5/S e(∞) = 5/(1 + lims→0 G(s)) = 5/21 R(s) = 5tu(t) = 5/S e(∞) = 5/ lims→0SG(s) = 1/0 = ∞ R(s) = 5t2u(t) = 10/S3 e(∞) = 5/ lims→0 S2G(s) = 1/0 = ∞ type 0系統 只能執行位置控制 產生有限誤差; 無法執行速度及加速度控制

19. Example 7.3 Figure 7.6 Feedback control system for system with no one integrator
type1 系統
R(s) = 5u(t) = 5/S e(∞) = 5/(1 + lims→0 G(s)) = 0 R(s)= 5tu(t) = 5/S e(∞) = 5/ lims→0SG(s) = 1/20 = finite R(s)= 5t2u(t) = 10/S3 e(∞) = 10/ lims→0 S2G(s) = 1/0 = ∞ type 1系統 執行位置控制 無誤差產生; 執行速度控制 產生有限誤差; 無法執行加速度控制 H.W.: Skill-Assessment Exercise 7.1

20. 7.3 Static Error Constants and System Type
定義: Static Error Constants Kp Kv Ka
For step input R(s) = 1/s
e(∞) = 1/(1 + lims→0 G(s)) = 1/1+Kp
Kp = lims→0 G(s) position error constant
For ramp input R(s) =1/s2
e(∞) = 1/ lims→0SG(s) = 1/Kv
Kv = lims→0 SG(s) velocity error constant
For parabolic input R(s) = 1/s3
e(∞) = 1/ lims→0 s2G(s) = 1/Ka
Ka = lims→0 S2G(s) acceleration error constant

21. Example 7. 4 利用Static Error Constants 求解 Figure 7
Example 利用Static Error Constants 求解 Figure 7.7 Feedback control systems 求3系統之steady-state error? /3
Type 0 system For step input: R(s) = 1/s Kp = lims→0G(s)= e(∞) = 1/ /(1+Kp) = For ramp input: R(s) =1/s2 Kv = lims→0 sG(s) = e(∞) = 1/Kv = ∞ For parabolic input: R(s) = 1/s3 Ka = lims→0s2G(s)= e(∞) = 1/Ka = ∞

22. Example 7.4 Figure 7.7 Feedback control systems 求3系統之steady-state error? 2/3
(b) Type 1 system For step input: R(s) = 1/s Kp = lims→0G(s)= ?/0 = ∞ e(∞) = 1/(1+Kp) = 0 For ramp input: R(s) =1/s2 Kv = lims→0 sG(s) = 30000/960 = e(∞) = 1/Kv = For parabolic input: R(s) = 1/s3 Ka = lims→0s2G(s)=0*?= e(∞) = 1/Ka = ∞

23. Example 7.4 Figure 7.7 Feedback control systems 求3系統之steady-state error? 3/3
(c) Type 2 system For step input: R(s) = 1/s Kp = lims→0 G(s)= ?/0 = ∞ e(∞) = 1/ (1+Kp) = 0 For ramp input: R(s) =1/s2 Kv = lims→0 sG(s) = ?/0 = ∞ e(∞) = 1/Kv = 0 For parabolic input: R(s) = 1/s3 Ka = lims→0 s2G(s) = e(∞) = 1/Ka =

24. Table 7.2 Relationships between input, system type, static error constants, and steady-state errors
Static Error Constants: Kp Kv Ka 決定系統之 e(∞) ; 其可為steady-state error 之規格
H.W.: Skill-Assessment Exercise 7.2

25. 7.4 Steady-State Error Specifications
Example 7.5
Given Kv=1000 → draw ?? conclusions
1. Stable system
2. Ramp input
3. Type 1 system

26. Example 7.6 Find K=? → e(∞) = 10%
Type 1 system (已知)
有限的e(∞) → Ramp input (已知)
e(∞) = 1/ Kv = 0.1 → Kv = 10
Kv = lims→0 SG(s) = k*5 / 6*7*8 = 10
→ k = 672
自修 Skill-Assessment Exercise 7.3

27. 7.5 Steady-State Error for Disturbances 1/3
Figure 7.11 Feedback control system showing disturbance
2 inputs R(s) & D(s)
C(s) =﹝E(s)G1(s) ＋ D(s) ﹞G2(s)
= E(s)G1(s)G2(s) ＋ D(s)G2(s)
E(s) = R(s) – C(s)
→ R(s) –E(s) = E(s)G1(s)G2(s) ＋ D(s)G2(s)
E(s)G1(s)G2(s) ＋ E(s) = R(s) – D(s)G2(s)
E(s)(1+G1(s)G2(s)) = R(s) – D(s)G2(s)

28. 7.5 Steady-State Error for Disturbances 2/3
E(s)(1+G1(s)G2(s)) = R(s) – D(s)G2(s)

29. 7.5 Steady-State Error for Disturbances 3/3
DC gain of G1(s)
eD(∞)↓ (分母變大) if DC gain of G1(s)↑
or DC gain of G2(s)↓

30. D(s) = step disturbance
Example Fig 如下 自修
D(s) = step disturbance
Find eD(∞) = ?
H.W. : Skill-Assessment Exercise 7.4

31. Figure 7.12 Figure 7.11 system rearranged to show disturbance as input and error as output, with R(s) = 0
-C(s) = E(s)

32. 7.6 Steady-State Error for Nonunity Feedback Systems

33. 7.7 Sensitivity H.W. : Skill-Assessment Exercise 7.6
Defination
Examples:
H.W. : Skill-Assessment Exercise 7.6

34. Se:a = ﹝a/(a/k)﹞﹝δ(a/k)/δa﹞= 1
Example 7.11
Figure 7.19
Find Se:a = ? Se:k = ?
R(s) = Ramp input = 1/s2 →
e(∞) = 1/kv = 1/(k/a) = a/k
Se:a = ﹝a/(a/k)﹞﹝δ(a/k)/δa﹞= 1
Se:k = ﹝k/(a/k)﹞﹝δ(a/k)/δk﹞= -1