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Chapter 2. Mathematical Foundation1

Chapter 2. Mathematical Foundation

1. Complex-Variable Concept

2. Frequency-Domain Plots

3.Differential Equations

4. Laplace transform

5. Inverse Laplace transform by partial-fraction expansion

6. Summary

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Chapter 2. Mathematical Foundation2

2-1 Complex Variable

z x j y= + ejz R R = =

cos

sin

x R

y R

=

=

2 2

1tan

R x y

yx

= +

=

cos sinje j = + * jz x j y Re = =

rectangular form polar form

2* 2 2 2z zz x y R= = + =

2 3 4, , , , nj j j j z

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Chapter 2. Mathematical Foundation3

2-1 Complex Variable

( ) : ~ ratioal functionG s C C

2

10( 2)( )

( 1)( 3)

sG s

s s s

+=

+ +zeropole

order of pole simple pole

2 2

10( 2)( )

( 2 2)( 3)

sG s

s s s

+=

+ + +s C

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Chapter 2. Mathematical Foundation4

2-2 Frequency-Domain Plots

Polar Plot, Bode Plot, Magnitude-phase Plot

magnitude, phase( ) ( ) ( )G j G j G j =

Fig 2-8

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Chapter 2. Mathematical Foundation5

2-3 Differential Equations

linear ODE1

1 1 01( ) ( ) ( ) ( ) ( )

n n

nn nd y t d y t dy t a a a y t f t

dt dt dt

+ + + + =L

2

2

( )sin ( ) 0

d tm mg t

dt

+ =l nonlinear ODE

2

1 0 02

( ) ( )( ) ( )

d y t dy t a a y t b f t

dt dt + + =

state equation & output equation

1 2

( )( ) ( ); ( ) ( )

dy tx t y t x t y t

dt= = =&

1 2

2 0 1 0 0 1 1 2 0

( ) ( ) ( );( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

x t y t x tdy t

x t y t a y t a b f t a x t a x t b f tdt

= =

= = + = +

& &

& &&

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Chapter 2. Mathematical Foundation6

2-3 Differential Equations

x Ax Bu= +&

state equation

; ;n q px R y R u R

1 1

0 1 02 2

( ) ( )0 1 0 ( )( ) ( )

x t x t f ta a bx t x t

= +

&

&

[ ] [ ]1

2

( )( ) 1 0 0 ( )

( )

x ty t f t

x t

= +

output equation

y Cx Du= +

~ ; ~ ; ~ ; ~A n n B n p C q n D q p

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Chapter 2. Mathematical Foundation7

2-4 Laplace transform

0( ) ( )stF s f t e dt

= ( ) ( )f t F s

( ) ( )kf t kF s

1 2 1 2( ) ( ) ( ) ( )f t f t F s F s+ +

( ) ( ) (0)d

f t sF s fdt

1 ( 1)( ) ( ) (0) (0)n

n n n

n

df t s F s s f f

dt

L

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Chapter 2. Mathematical Foundation8

2-4 Laplace transform

0

1

( ) ( )

t

f d F ss 1 1

1 10 0 0

1( ) ( )

nt t t

n nf d dt dt F s

s

L L

( ) ( ) ( )Tssf t T u t T e F s

( ) ( )te f t F s

1 2 1 2( ) ( ) ( ) ( )f t f t F s F s

1 2 1 2( ) ( ) ( ) ( )f t f t F s F s

convolution

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Chapter 2. Mathematical Foundation9

2-4 Laplace transform

0lim ( ) lim ( )t s

f t sF s

=

0lim ( ) lim ( )t s

f t sF s

=

Initial-value theorem

Final-value theorem

If ( ) does not have poles on or to the right of the imaginary axis,sF s

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Chapter 2. Mathematical Foundation10

2-5 Inverse Laplace transform

partial-fraction expansion

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Chapter 2. Mathematical Foundation11

2-6 Application of the Laplace Transform

Ex2-6-11

( ) ( ) ( )y t y t f t

+ =&(0) 0; ( ) ( )sy f t u t= =

1 1 1( ) ( ) ( )1

( )

sY s Y s Y ss

s s

+ = =

+

( ) ( )1

t

Y s y t ess

= =

+

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Chapter 2. Mathematical Foundation12

2-6 Application of the Laplace Transform

Ex2-6-1

2 22 ( )n n n sy y y u t + + =&& & (0) (0) 0y y= =&

2

2 2 2 2

21

( ) ( 2 ) 2

n n

n n n n

s

Y s s s s s s s

+= =

+ + + +

0 1