introduction to automatic control the laplace transform li huifeng tel:82339276...
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The Laplace Transform
The Laplace Transform
Li Huifeng
Tel:82339276
Email:[email protected]
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The Laplace Transform
Module objectives
• When you have completed this module you should be able to: – Apply the Laplace transform to differential
equations. – Solve linear differential equations. – Apply the main theorems of the Laplace
transform. – Know how useful this techniques is to handle
dynamical systems
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The Laplace Transform
Subsections
• Definition
• Correspondences of the Laplace transform
• Main theorems of the Laplace transform
• The inverse Laplace transform
• Solving linear differential equations using the Laplace transform
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The Laplace Transform
How to understand LT
• The Laplace transform is conceptually similar to multiplication via logarithms
• log(axb) = log(a) + log(b)
• To multiply a by b– Compute logarithms of a and b– Add these logarithms – Inverse logarithm of sum gives product of a and
b.
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The Laplace Transform
Linear differential equation
Time domain solution
Laplace Solution
Laplace transformed Equation
Convolution
Time Domain
Laplace Domain
Laplace Transform
Inverse Laplace Transform
Algebraic manipulation
t
o
t
o
dftfdtfftftf )()()()()()( 212121
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The Laplace Transform
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The Laplace Transform
Definition函数 f(t), t 为实变量,如果线性积分
存在,则称其为函数 f(t) 的拉普拉斯变换。变换后的函数是复变量 s 的函数,记作 F(s)或 L[f(t)] ,即:
称 F(s)为 f(t) 的变换函数或象函数,而 f(t)为 F(s)的原函数。
为复变量)jwsdtetf st
()(0
0
)()()]([ dtetfsFtfL st
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The Laplace Transform
Correspondences of the Laplace transform
• Table 2.1: Corresponding elements of the Laplace transform or Textbook Page 21
)]([)(
)]([)(
)()(
1 sFLtf
tfLsF
sFtf
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The Laplace Transform
几种典型函数的拉氏变换
t
1. 1(t)单位阶跃函数
0t0
0t1)t(1)t(f
其拉氏变换为:
s
1dte1)s(F)]t(f[L
0
st
f(t)
0
其数学表达式为:
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The Laplace Transform
t
2. 单位斜坡函数
0t
0t
0
t)t(1t)t(f
其拉氏变换为:
20
st
s
1dtet)s(F)]t(f[L
f(t)
0
其数学表达式为:
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The Laplace Transform
3. 等加速函数
0t
0t
0
t2
1)t(f
2
其拉氏变换为:
30
st2
s
1dtet
2
1)s(F)]t(f[L
其数学表达式为:
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The Laplace Transform
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The Laplace Transform
4. 指数函数
0t
0t
0
e)t(f
t
其拉氏变换为:
s
1dtee)s(F)]t(f[L
0
stt
其数学表达式为:
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The Laplace Transform
5. 正弦函数
0t
0t
0
wtsin)t(f
其拉氏变换为:
220
st
ws
wdtewtsin)s(F)]t(f[L
其数学表达式为:
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The Laplace Transform
6. 余弦函数
0t0
0twtcos)t(f
其拉氏变换为:
220
st
ws
sdtewtcos)s(F)]t(f[L
其数学表达式为:
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The Laplace Transform
7. 单位脉冲函数
0t
0t0)t()t(f
其拉氏变换为:
其数学表达式为:
1)s(F)]t(f[L
1dt)t(定义:
t
)(t
0
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The Laplace Transform
)(
1)(sincos)(
jssFtjtetf tj
)(
1)(sincos)(
jssFtjtetf tj
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The Laplace Transform
Euler equation
2cos
tjtj eet
j
eet
tjtj
2sin
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The Laplace Transform
)(cos
22
s
st
)(sin
22
s
t
Equating real & imaginary parts yields:
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The Laplace Transform
))((cos
22
s
ste t
))((sin
22
ste t
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The Laplace Transform
典型函数的拉氏变换形式
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The Laplace Transform
Main theorems of the Laplace transform
① Superposition theorem( 叠加定理 ):
)()()}()({ 22112211 sFasFatfatfaL
各函数和的拉氏变换=各函数拉氏变换的和
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The Laplace Transform
② Similarity theorem: ( 比例尺改变 )
0)(1
)}({ aa
sF
aatfL
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The Laplace Transform
③ Real Shifting theorem( 延时定理 ):
0)()}({ asFeatfL as
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The Laplace Transform
提示: f(t) 相当于 t· 1(t) 在时间上延迟了一个值。
)(tf
t0
)(1 tt )(1)( tt
的拉氏变换。求 )t(1)t()t(f
)](1)[()( ttLsF
ses
2
1
解:
例 1
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The Laplace Transform
④ Complex Shifting theorem( 衰减定理) :
0)()}({ aasFtfeL at
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The Laplace Transform
的拉氏变换。求 wtsine)t(f at
]sin[)( wteLsF at
22)( was
w
解:
例 2
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The Laplace Transform
⑤ Derivative theorem:
)0(f)0(fs)0(fs)s(Fs]dt
)t(fd[L
)0(f)0(sf)s(Fs]dt
)t(fd[L
)0(f)s(sF]dt
)t(df[L
)1n('2n1nn
n
n
'2
2
2
拉氏变换将原函数求导数的运算转换为“象函数乘 s 后减初值”的代数运算。
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The Laplace Transform
⑥ Complex differentiation theorem: ( 不要求掌握 )
k
kkk
ds
sFdtftL
)()1()}({
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The Laplace Transform
⑦ Integral theorem:
)0(fs
1)0(f
s
1)s(F
s
1]dt)t(f[L
)0(fs
1)0(f
s
1)s(F
s
1]dt)t(f[L
)0(fs
1)s(F
s
1]dt)t(f[L
)n()1(
nn
n
)2()1(
22
2
)1(
拉氏变换将原函数求积分的运算转换为“象函数除以 s 后加初值”的代数运算。
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The Laplace Transform
⑧ Initial value theorems:
)0()(lim)(lim0
ftfssFts
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The Laplace Transform
Given:
22 5)1(
)2()(
s
ssF
Find f(0)
1)26(2
2lim
2512
2lim
5)1(
)2(lim)(lim)0(
2222
222
2
2
22
sssss
ssss
ss
ss
s
ssssFf
ss s
s
解:
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The Laplace Transform
⑨ final value theorems:
)()(lim)(lim0
ftfssFts
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The Laplace Transform
Given:
ttesFnotes
ssF t 3cos)(
3)2(
3)2()( 21
22
22
Find )(f .
03)2(
3)2(lim)(lim)(
22
22
s
ssssFf
0s0s
解:
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The Laplace Transform
⑩ Convolution in the time domain: ( 不要求掌握 )
t
o
t
o
dftfdtfftftf )()()()()()( 212121
)()()}()({ 2121 sFsFtftfL
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The Laplace Transform
⑪ Convolution in the frequency domain: ( 不要求掌握 )
jc
jc
dppsFpFj
tftfL )()(2
1)}()({ 2121
拉氏变换性质的证明
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The Laplace Transform
The inverse Laplace transform
j
j
ts1 dte)s(Fj2
1)t(f)]s(F[L
定义: 由象函数 F(s) 求其原函数 f(t) 的运算
称为拉氏反变换。
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The Laplace Transform
Partial Fraction Expansion思路: 将 F(s) 分解成一些简单的有理分式函数之和,
然后由拉氏变换表一一查出对应的反变换函数,即得所求的原函数 f(t) 。
)}({)}({)}({)}({
)()()()(
12
11
11
21
sFLsFLsFLsFL
sFsFsFsF
n
n
)()()()( 21 tftftftf n
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The Laplace Transform
。为正数,且均为实数,及式中,
的一般形式为:
nmn,m
b,b,ba,a,a
asasas
bsbsbsb
)s(A
)s(B)s(F
)s(F
m10n21
n1n
1n
1
n
m1m
1m
1
m
0
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The Laplace Transform
将 F(s) 的分母多项式 A(s) 进行因式分解,即写为:
A(s) = (s - s1)(s - s2)…(s - sn)
式中, s1, s2, …sn为 A(s) = 0 的根。
分两种情况讨论:1.A(s) = 0 无重根2.A(s) = 0 有重根
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The Laplace Transform
1. A(s)=0 无重根情况
n
n
i
i
ss
C
ss
C
ss
C
ss
CsF
2
2
1
1)(
可将 F(s) 换写为 n 个部分分式之和,每个分式的分母都是 A(s) 的一个因式。
n
i i
i
ss
CsF
1
)(
关键问题 : 确定每个部分分式中的待定常数 Ci 。
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The Laplace Transform
确定待定常数 Ci
)()(lim sFssC issii
iss
i sA
sBC
)(
)('
或
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The Laplace Transform
n
i i
i
ss
CLtfsFL
1
11 ][)()]([
n
i
ts
iieC
1
代入 Ci 即可求得 f(t):
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The Laplace Transform
34
2)(
)(
2
ss
ssF
sF 的拉氏反变换求
确定待定系数
所以
例 3
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The Laplace Transform
要点:
明确四个步骤,语言表述清晰
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The Laplace Transform
分子分母同阶的例子:
所以
例 4
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The Laplace Transform
分母有复数根的例子:
例 5
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The Laplace Transform
求 Ci
所以 欧拉方程
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The Laplace Transform
欧拉方程
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The Laplace Transform
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The Laplace Transform
2. A(s)=0 有重根情况
:)(
,,, 211
和可展成如下部分分式之则为单根。阶重根,为设
sF
sssms nmm
)()(
)()()()(
1
1
1
1
1
1
1
1
n
n
m
m
m
m
m
m
ss
C
ss
C
ss
C
ss
C
ss
CsF
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The Laplace Transform
确定待定常数 Ci
)()(lim 11
sFssC m
ssm
)]()[(lim 111
sFssds
dC m
ssm
)]()[(lim
!
11
1
sFssds
d
jC m
j
j
ssjm
)]()[(lim)!1(
111
)1(
11
sFssds
d
mC m
m
m
ss
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The Laplace Transform
)3()1(
2)(
)(
2
sss
ssF
sF 的原函数。求例 6
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The Laplace Transform
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The Laplace Transform
Solving linear differential equations using the Laplace
transform
Linear differential equation
Time domain solution
Laplace Solution
Laplace transformed Equation
Convolution
Time Domain
Laplace Domain
Laplace Transform
Inverse Laplace Transform
Algebraic manipulation
1
2
3
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The Laplace Transform
1
2
3
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The Laplace Transform
三个步骤 :
① 代入初始条件对微分方程进行拉氏变换;② 解变换方程 ( 代数方程 ) ,求出响应函数的拉
氏变换式;③ 用部分分式法求拉氏反变换,得到微分方程
的解。
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The Laplace Transform
方程。试用拉氏变换法求解该,初始条件为设 ,2)0(,2)0()(16)(
)()(6)(
5)(
2
2
yyttu
tutydt
tdy
dt
tyd
分析 :
)()(6)(
5)(
2
2
tutydt
tdy
dt
tyd
)()]([ sYtyL
例 7
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The Laplace Transform
)()(6)(
5)(
2
2
tutydt
tdy
dt
tyd
)0()(])(
[ yssYdt
tdyL
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The Laplace Transform
)()(6)(
5)(
2
2
tutydt
tdy
dt
tyd
)0()0()(])(
[ 2
2
2
ysysYsdt
tydL
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The Laplace Transform
)()(6)(
5)(
2
2
tutydt
tdy
dt
tyd
stLtuL
6)](16[)]([
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The Laplace Transform
解:
Step1: For a ODE w/initial conditions, apply Laplace transform to each term.
ssYyssYysysYs
6)(6)0(5)(5)0()0()(2
sssYssYsYs
6122)(6)(5)(2
Step2: Solve for Y(s)
)65(
6122)(
2
2
sss
sssY
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The Laplace Transform
Step3: Apply partial fraction expansion to obtain inverse Laplace transform.
2
541)(
sssssY
)0(541)( 23 teety tt
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The Laplace Transform
0)0(')0(2862
2
yyydt
dy
dt
yd ODE w/initial conditions
Apply Laplace transform to each term
Solve for Y(s)
Apply partial fraction expansion
Apply inverse Laplace transform to each term
ssYsYssYs /2)(8)(6)(2
)4()2(
2)(
ssssY
)4(4
1
)2(2
1
4
1)(
sss
sY
424
1)(
42 tt eety
例 8
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The Laplace Transform
OVERVIEW
• This module is a mathematical section to establish a base for the theory of control systems.
• This is a tool and it is indispensable as most of linear system dynamics are described in a mapped space that can only be understood when the main theorems of the Laplace transform are known.
• Special focus is put on the solution of differential equations using the Laplace transform.
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The Laplace Transform
Homework
• 见WORD 文件:拉氏变换作业