自控原理课件3(英文版)

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3-2 Time response of first-order system
Figure 5-3 Unit-ramp response of the system shown in Figure
5–1(a).
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3-2 Time response of first-order system
r(t) 1 t 2 1(t)
1 t2 2
t 0
2
0
t 0
r (t )
L{1 2
t2
1(t)}
1 s3
0
t
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3-1. Introduction
Function imagine
t
t
0
Original function of Time-domain
image
relation
Image function
Complex-domain relation
Example
Unit pulse
f t t
Impact
1
Recoil force
Electric pulse
1t
t
0
t
t
0
t2 2
t
0
Unit step
f
t
1 0
t0 t0
Unit ramp
f
t
t 0
t0 t0
Unit acceleration
t2
Response due to unit step signal
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3-2 Time response of first-order system
c(t)
1
1
eT
t
c(0)
1
T
T
Figure 5-2 Exponential response curve.
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3-2 Time response of first-order system
Analysis
General Concepts (chap1)
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Time-DomainMethod (Chap5)
Mathematical Model (chap3)
Root-Locus Method (chap6,7)
Performance Specifications 稳、准、快
Frequency-Domain Method (chap8,9)
c(t) ctr (t) css (t)
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3-1. Introduction
Absolute stability, relative stability, and steady-state error
The most important characteristic of the dynamic behavior of a control system is absolute stability, that is, whether the system is stable or unstable.
response curve
微分
微分
t) 1
1(t)
t
End of 3-2
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3-3 Time response of second-order system
P224
Standard Form of Second-Order Systems
R(s)
n2
C(s)
s(s 2ns)
C(s) R(s)
C(s) (s) R(s) 1 1 1 1 Ts 1 s s s 1 T
c(t) L1[
1
1
]
1
1
eT
t
Ts 1 s
when t T t 3T
c(t) 1 e1 0.632 c(t) 0.95
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t 4T c(t) 0.982
12
3-2 Time response of first-order system
Transient response: goes from the initial state to the final state.
Steady state response: the system output behaves as t approaches infinity.
Thus the system response c(t) may be written as:
f
t
2
t0
0 t 0
1 s
df dt
1 s2
1 s3
Switch signal
s Constant velocity position tracking
8
3-1. Introduction
Transient Response and Steady-State Response.
The time response of control system consists of two parts: the transient response and the steady-state response.
Unit-impulse response of first-order system
For the unit-impulse input, R(s)=1, so
(s) C(s) 1 R(s) Ts 1
C(s) 1 Ts 1
c(t) 1 et /T , for t 0 T
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3-2 Time response of first-order system
s2 2ns n2 0
-1 0 1
According to value of , the closed poles (characteristic
roots) of second-order system are:
(1) when 1, the system is overdamped. It
T: Time constant
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3-2 Time response of first-order system
Unit step response of first-order system
(s) C(s) 1 R(s) Ts 1
partial fractions expansion
Unit ramp response for first-order system
(s) C(s) 1 R(s) Ts 1
C(s)
11 Ts 1 s2
1 s2
T s
T2 Ts 1
Taking the inverse Laplace transform of equation, we obtain
s2
n2 2n s
n2
ξ: Damping ratio, 阻尼比
ωn: Undamped natural frequency，无阻尼自然振 荡频率，简称自然频率
The unit step response for this system
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C(s)
s2
n2 2n s
n2
1 s
?
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3-3 Time response of second-order system
Unit ramp response for first-order system
c(t) t T Tet /T , for t 0
The error signal e(t) is then
e(t) r(t) c(t) T (1 et /T )
As t approaches infinity et/T approaches zero, and thus the error signal e(t) approaches T
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3-1. Introduction P219
Characteristics of Time-Domain Method
Time domain analysis is the basic analytic method and the foundation for complex domain method and frequency domain method. (1) Analyzing system in time domain is direct and accurate. (2) It presents the whole information of the system time response. (3) but, tedious to solve differential equations.
C(s)
n2
1
n2
(S S1)(S S2 ) S [S n ( 2 1)][S n ( 2 1)]S
A1
A2
A3
s s n ( 2 1) s n ( 2 1)
Chapter 3 Transient and SteadyState Response Analyses
重点掌握： 1、一、二阶系统时域响应及性能指标的计算； 2、稳定性定义、条件、判据； 3、系统稳态误差的计算。
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Tasks and the Structure of the Course
has 2 different real negative poles, the closed-
loop transfer function can be rewritten as:
s2
s1
s1
n
n
2 1
s2 n n 2 1
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3-3 Time response of second-order system
Figure 5-4 Unit-impulse response of the system shown
in Figure 5–1(a).
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3-2 Time response of first-order system
r(t)§R3(s).2C.3(s)=一F(阶s) R系(s)统的典c型(t) 响应
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3-1. Introduction
Typical test signals
1. Unit impulse function
t0
(t) 0 t0
(t)dt 1
(t)
0
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பைடு நூலகம்
L{ (t)} 1
t
5
3-1. Introduction
2. Unit step function
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End of 3-1
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3-2 Time response of first-order system
P221
Figure 5-1 (a) Block diagram of a first-order system; (b) simplified block diagram.
Thus the transfer function of a first-order system is (s) C(s) 1 R(s) Ts 1
r(t) 1(t) 1
0
t 0 t 0
L{1(t)} 1 s
3. Unit ramp function
t t0
r(t) t 1(t)
0 t0
r (t )
1 0
r (t )
L{t
1(t)}
1 s2
0
t
t
3-1. Introduction
4. Unit parabolic function
State-Space Method (chap11,12)
Design
The structure of the course
2
Chapter 3 Transient and Steady-State Response Analyses
Main contents
3-1. Introduction 3-2. First-order Systems 3-3. Second-order Systems 3-4. Higher-order Systems 3-5. Stability Analysis of Linear Systems 3-6. The Steady-State Error of Linear Systems 3-7. Summary 3-8. Transient response analysis with matlab
Given an absolute stable system, we will discuss further the relative stability of the system.
If the output of a system at steady state does not exactly agree with the input, the system is said to have steady-state error. This error is indicative of accuracy of the system.