自控原理总复习

2
s
1, 2
= — ζ ωn ±ωn
ζ
2
－1
The step response for the four cases :
0 , =1 , 1 , 0 1
s
— 1, 2 n n

2
－1
When = 0 the undamped case
S1, 2 = ±jωn
Module 4 Second-order Systems
(二阶系统)
Standard form of the second-order system:
C s 2 2 Rs s 2 n s n
2 n
There are two characteristic parameters here: ζ — the damping ratio (阻尼比） ωn — the undamped natural frequency (无阻尼自然频率)
Closed-loop transfer function
R
ω s + 2ζ ω
2
2 n n
C
s + ωn
2
Fig.4.14 Generalized closed-loop transfer function
Open-loop transfer function
R(s)
-
ss 2 n
C
H1 / G3
H1
H3
⇓
G2G3 1 G2G3 H 2 G2 H1
R
s1 -
s2 +
G1
+ s3
C
H1
H3
Module 3
First-order Systems
(一阶系统)
❃The Generalized First-Order Systems:
• Transfer function
C ( s) 1 R( s ) 1 s
• Transducer: 变送器 • Differencing junction: 差分连接点，减法器
Module 2
Transfer Functions and Block Diagram Algebra （传递函数和方框图运算）
Block Diagram Algebra
A. The simplification of a complex feedback control systems
A 1t
t
At
At
2
A sin t
First-Order Feedback Systems
a. Open loop control
the response of a unit step input
R(s)
K S 1
C(s)
ct K 1 e t
－1
When >1 the overdamped case
= — ζ ± ω 1, 2 n ωn
ζ
2
－1 = －ωn (ζ
ζ
2
－1)
C(t)
1
t
s
1, 2
= — ζ ωn ±ωn
ζ
2
－1
When 0 < < 1 the underdamped case
s
1, 2
= — ζ ωn ±ωn
Module 5 Second-Order System Time -Domain Response （二阶系统时域响应）
Relationship Between System Poles and Transient Response
C s 2 2 Rs s 2n s n

b. Closed loop control
R
K
1 1 s
C
Fig.3.10 First-order feedback control system
K K K S 1 1 K G S K S 1 1 S 1 S 1 1 K

2 n
C(s)
◆ Step Response
When
the second-order system is subjected to a step input 2 S + 2ζ ω 1 ω n = － C ( s) = 2 2 S (s + 2ζ ωn s + ωn) S (S + ζ ωn) + ω (1－ζ )
e
n t 2
ζ
2
－1
c t 1
1
sin d t
C(t)
1
t
表3-2 二阶系统的阻尼系数与单位阶跃响应关系
阻尼系数 特征方程根 根在复平面上位置
j
单位阶跃响应
y (t)

=0
s
(无阻尼)
1,2
j n
j
S1
n
0
1 s2
j
n
0
t
0 <

j
<1
2 n
Characteristic equation:
s 2n s 0
2 2 n
s1, 2 n j n 1
2
Closed-loop pole (conjugate pair)
a. Serial Elements
U(s)
G1 s
V(s)
G2 s
W(s)
G3 s
X(s)
Fig.2.3 Block diagram of serial elements U(s)
G1G2G3 s
X(s)
Fig.2.4 Reduction of serial elements
b. Paralel Elements
RULE # PROCESS ORIGINAL BLOCK DIAGRAM
x
EQUIVALENT BLOCK DIAGRAM
1
Combining serial blocks
Combining serial blocks
G1
G2
y
x
G1G2
y
x
G1
+ +
y x
G1+G2
k´< k , ´<
Let τ =1 and K=1,
Let τ =1 and K=10,
1 0.909 c(t)
c(t ) 0.5(1 e )
2 t
ct 0.909 1 e
r(t)

11t

K=10
0.5
open-loop response
K=1
t Fig.3.11 Step response of feedback system
Important observations: (P45.)
1. The feedback system has a steady-state error between the input and output. 2. The feedback system response is faster than the open loop system. If K is increased to 10, output is close to 1 Steady-state error is less than 10%,and 10 times the response speed.
自控原理总复习
控制理论主要研究内容:
一 控制系统的分析
1 稳定性 2 动态特性 3 稳态特性
二 控制系统的设计
Generalized feedback control system
Forward path Input + Error Differencing Controller Control signal Output Plant
d. Movement of a summing junction or a
pickoff (分支）point
4
Moving a summing Junction ahead of a block Moving a summing Junction ahead of a block Moving a pickoff point ahead of a block
x x x
+ +
G
+ +
z
G
z
y
1/G
x
+
y
x
+ +
5
G
z y
G
z
+
y
G
x
G
y
y
x
G 1/G
y
y
6
x
G
y
x
G
x
y
Moving a pickoff point past a block
1/G
e. Multiple inputs
R + D C
G1
-
G2 H1
Fig.2.8 Linear system with two inputs
s1,2 — n
j n
S1
2
j n
1
2
y (t) 1
欠阻尼
1
0 s2
j n
1
2
0
t

j
=1 y (t)
s1,2 — n
(临界阻尼)