Linearization of Nonlinear Mathematical Models

Linearization of Nonlinear Mathematical

Models

Linearization of Nonlinear Mathematical Models

Graphical Interpretation

( ) point operating,:let −fxP

( ) point typical,:Alet −fx

l1 – line connecting point P,A

( ) ( ) xtxtx −= ( ) ( ) ftftf −=และ



Let point A very is closed to point P.Thus, value Δx and Δf have very small value.We can be estimatedslope of line l1 and l2 as equal.

xxdx

dfm

=

=



( )xxmff −=− xmf =


( )

( ) ( ) ( ) +−+−+=

=

2

2

2

!2

1 xx

dx

fdxx

dx

dfxf

xfy

Consider input x(t) and output y(t). Thus, relationship between input and output are

( )xfy =

Condition: mean values are and presents in Taylor series formyx,

Eq.1


( )xxmyy −+=

( )

xxdx

dfm

xfy

=

=

=

where derivativeIf is very small value. We aren’t considered high order term, thus

2 2, , are evaluated at df dx d f dx x x=

when

Eq.2

x x−


( )xxmyyy −==−

Eq. 2 rewrites as

xx −yy −Term directly change to .Thus, eq.3 is linear equation for nonlinear equation

Eq.3xmf =


( )21, xxfy =

Consider output y(t) as function of input x1 and x2

Eq.4

( ) ( ) ( )

( ) ( )( )

( )

−

+

−−

+−

+

−

+−

+=

2

222

2

2

2211

21

22

112

1

2

22

2

11

1

21

2!2

1

,

xxx

f

xxxxxx

fxx

x

f

xxx

fxx

x

fxxfy


( ) ( )222111 xxmxxmyyy −+−==−

Consider nonlinear system with output y(t) and function input of x1 and x2

( )

2211

2211

,2

2

,1

121 , ,,

xxxx

xxxx

x

fm

x

fmxxfy

==

==

=

==

when


Example: Pendulum oscillator modelTorque & angular displaement

sinMglT =

Relationship between T and such as nonlinear system


:Pendulum oscillator model

( )00

0

sin

−

−

=

MgLTT

Select operating point at 0= 0. Thus, linear approximation is

when T0=0, Thus

( )( ) MgLMgLT =−= 0cos


:Pendulum oscillator model

MgLT =

Linearization for pendulum oscillator model in the range

( ) ( )4 4 −

Linearization of Nonlinear Mathematical Models

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