Inverse Laplace transformation can be denoted
f (t ) = inverse Laplace transformation of F ( s ) = L [F ( s )]
−1
or
1 c + j∞ st f (t ) = ∫c− j∞ F (s)e ds 2πj
d s = dt
1 = s
∫
t
0
dt
Important Laplace Transform Pairs
1 L[1(t )] = s
L[ t ] =
n
n! s n +1
L[e
− at
1 ]= s+a
L [sin ω t ] =
ω
s L[cos ω t ] = 2 2 s +ω
s2 + ω 2
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Inverse Laplace Transformation
y
y 0 + ∆y
y0
df dx
f (x )
x0
∆y = k∆x
A
x
x 0 x0 + ∆x
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df So we get: y = f ( x ) = f ( x 0 ) + dt
Set Δy=f(x)-f(x0), so we have
df ∆y = ∆x dx x 0 df = k Set dx x 0
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Principle of superposition
Superposition Property
system
y1
y2
r2
r +r2 1
system
y1 + y2
system
Homogeneity Property
r
system
y
ar
system
ay
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Example
(1) (2) (3)
y = kx
(1)
The inverse Laplace transform of Eq.(1)is
y (t ) = 2 e
−t
−e
−2 t
Example 2
d y dy +4 + 3 y = 2 r (t ) 2 dt dt
When the initial conditions are
y (0) = 1 ,
y = kx + b
y=x
2
Does not satisfy the homogeneity property Does not satisfy the superposition property
y = y 0 + ∆ y Equation (2) can be rewritten
When x = x0 + ∆x and as
∆x
x0
y
y0 + ∆y
y0
df dx x0
f (x)
∆y = k∆x
We get Δy=kΔx
A
x
Or
y=kx
x0 x0 + ∆x
The Laplace Transform
Definition
If a function of time,f(t),satisfy
∞ 0
∫
f ( t ) e − σ 1 t dt < ∞
a1 = −1, a2 = 0, a3 = −1
The transfer function of linear systems
definition Transfer function: The ratio of the Laplace transform of the output variable to the Laplace transform of the input variable,with all initial conditions assumed to be zero.
y 0 + ∆ y = kx
0
+ k∆ x + b
We have or
∆ y = k∆ x
y = kx
Linearization of Weak Nonlinear Characteristic
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Linearization using Taylor series expansion about the operating point( Equilibrium Position) The output-input nonlinear characteristic of y=f(x) is illustrated in the following figure:
3
Types of mathematical models
1、Differential Equation 2、Transfer Function 3、Frequency Response 4、State Equation 5、Difference Equation
4
Differential Equations of Physical Systems
Differential Equations for Ideal Elements
(1) Electrical Resistance
i R U
U i = R
6
(2) Electrical Capacitance
i C U
i = C
dU dt
(3)Electrical Inductance
i L U
U
di = L dt = M dv dt
(4) Mass block
F v M
F
(5) Spring
F x1 k x2
F = k ( x1 − x2 )
(6) Damper
F v1 b v2
F = b ( v1 − v 2 )
examples
Example1 :RLC circuit
R r(t) i(t) L C c(t)
How to get the differential equatThe differential equations describing the dynamic performance of a physical system are obtained by utilizing the physical laws of the process. Step1:确定系统中各元件的输入、输出变量。 确定系统中各元件的输入、输出变量。 确定系统中各元件的输入 Step2:按信号传递顺序列写微分方程。 按信号传递顺序列写微分方程。 按信号传递顺序列写微分方程 Step3:化简（线性化、消去中间变量），写出输入、输出变 化简（线性化、消去中间变量），写出输入、 化简 ），写出输入 量间的数学表达式。 量间的数学表达式。 5
when
ɺ r (t ) = 0 , y(0) = y0 , y (0) = 0
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We can get
( Ms + b ) y 0 p(s) Y (s) = = 2 Ms + bs + k q(s)
when
k/M =2 ,
b/ M = 3 ,
y0 = 1
Then Y(s) becomes
s+3 2 1 Y (s) = = − ( s + 1)( s + 2 ) s + 1 s + 2
We have the Laplace transformation for function f(t),is
F (s) =
∫
∞
0
f ( t ) e − st dt = L { f ( t )}
17
The Laplace variable s can be considered to be differential operator so that,we have
Example 1
M d y (t ) dy + b + ky ( t ) = r ( t ) 2 dt dt
2
The Laplace transform of the equation is
ɺ M[s2Y (s) − sy(0) − y(0)] + b[sY(s) − y(0)] + kY(s) = R(s)
2
Definition of Mathematical model of system
Mathematical model:Descriptions of the behavior of a system using mathematics. 描述系统的输入、输出变量以及系统内 部各个变量之间的数学表达式。
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Important Theorems of Laplace Transform
(1) linearity
L[k1 f1 (t ) ± k2 f 2 (t )] = k1F1 ( s) ± k2 F2 ( s)
(2) differentiation
d f (t ) k k −1 ( k −1) L[ ] = s F (s) − s f (0) − ⋯⋯ f (0) k dt
Chapter 2 Mathematical Models of Systems
重点掌握： 微分方程 传递函数 系统结构图及结构图化简 梅逊公式
1
Main contents
Differential Equations of Physical Systems. The Laplace Transform and Inverse Transform. The Transfer function of Linear Systems. Block Diagram and Block Diagram Reduction. Mason’s gain formula