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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
Linearization of Nonlinear Mathematical Models
Graphical Interpretation
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
=
=
Linearization of Nonlinear Mathematical Models
Graphical Interpretation
( )xxmff −=− xmf =
Linearization of Nonlinear Mathematical Models
( )
( ) ( ) ( ) +−+−+=
=
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
Linearization of Nonlinear Mathematical Models
( )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−
Linearization of Nonlinear Mathematical Models
( )xxmyyy −==−
Eq. 2 rewrites as
xx −yy −Term directly change to .Thus, eq.3 is linear equation for nonlinear equation
Eq.3xmf =
Linearization of Nonlinear Mathematical Models
( )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
Linearization of Nonlinear Mathematical Models
( ) ( )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
Linearization of Nonlinear Mathematical Models
Example: Pendulum oscillator modelTorque & angular displaement
sinMglT =
Relationship between T and such as nonlinear system
Linearization of Nonlinear Mathematical Models
:Pendulum oscillator model
( )00
0
sin
−
−
=
MgLTT
Select operating point at 0= 0. Thus, linear approximation is
when T0=0, Thus
( )( ) MgLMgLT =−= 0cos
Linearization of Nonlinear Mathematical Models
:Pendulum oscillator model
MgLT =
Linearization for pendulum oscillator model in the range
( ) ( )4 4 −
