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Linearization of Non

This document discusses linearization of nonlinear models. It provides examples of nonlinear phenomena that make nonlinear systems unpredictable using linear models. The document then explains that linearization allows using linear equations to approximate the behavior of a nonlinear system near an equilibrium point, but the accuracy decreases further from that point. It outlines the limitations of linearization. Finally, it presents the steps to derive a linearized model from a nonlinear model using Taylor series expansion, including obtaining the steady state, subtracting it, and expressing the model in deviation variable form to get the state-space representation.

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JAN JERICHO MENTOY
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74 views5 pages

Linearization of Non

This document discusses linearization of nonlinear models. It provides examples of nonlinear phenomena that make nonlinear systems unpredictable using linear models. The document then explains that linearization allows using linear equations to approximate the behavior of a nonlinear system near an equilibrium point, but the accuracy decreases further from that point. It outlines the limitations of linearization. Finally, it presents the steps to derive a linearized model from a nonlinear model using Taylor series expansion, including obtaining the steady state, subtracting it, and expressing the model in deviation variable form to get the state-space representation.

Uploaded by

JAN JERICHO MENTOY
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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 Linearization of Non-Linear Models

Most of real control systems are non-linear. Non-linear system follows nonlinear phenomena that takes
place in the presence of nonlinearity, making them indescribable or unpredictable by linear models

Example of Nonlinear Phenomena

 Finite Escape Time


 Multiple Isolated equilibria
 Limit Cycles
 Subharmonic, harmonic or almost periodic oscillation
 Chaos
 Multiple modes of Behavior

 Linearization
Linearization is an important tool in analyzing non-linear model system. It allows the utilization of linear
equations to estimate a point (equilibrium point) in a nonlinear function, the further from that point the
greater the likelihood of error.

Limitations of linearization process

 It can only predict the local behavior of the nonlinear system in the vicinity
 Nonlinear system dynamics is much richer compared to linear system dynamics

 Linear System
Linear system is defined in terms of system excitation (input) and response (output). In equation,
excitation could be expressed in terms of x(t), response as y(t)

Linear system satisfies the properties of superposition and homogeneity

 Superposition
- For every excitation, there must be a response
 Homogeneity
- The response y(t) of a linear system to a constant multiple B of an input x(t) must be
equal to the response to the input multiplied by the same constant

Linear equations occur at nominal steady state conditions.


STEPS IN LINEARIZATION PROCESS
Derivation of Non Linear Equation to Deviation Variable Form

Supposed, an equation is stated

dy
=f [ x ( t ) , y ( t ) ]
dt
Step 1: Obtain steady-state model by setting the derivatives to zero. Specify the nominal state
conditions

0=f [ x̄+ ȳ ]
Wherein x̄, ȳ is the nominal steady-state operating point

Step 2 Subtract the steady state equations from the differential equation

dy
− 0=f [ x ( t ) , y ( t ) ] +b −[f [ x̄+ ȳ ] +b ]
dt
dy
=f [ x ( t ) , y ( t ) ] − f [ x̄+ ȳ ]
dt
Linearize the equation

dy df
dt
=
dx[ df
]
¿ x̄ , ȳ ( x ( t ) − x̄ ) + ¿ x̄ , ȳ ( y ( t ) − ȳ ) + f [ x̄ + ȳ ] − f [ x̄+ ȳ ]
dy

dy df
dt
=
dx[ df
¿ x̄ , ȳ ( x ( t ) − x̄ ) + ¿ x̄ , ȳ ( y ( t ) − ȳ )
dy ]
Substitute the deviation variables
x’ = x(t) - x̄ y’ = y(t) - ȳ

[ ]
'
d ( y + ȳ) df df
= ¿ x̄ , ȳ ( x ' )+ ¿x̄ , ȳ ( y ' )
dt dx dy

Step 3 : Express model in deviation form


Since ȳ = 0

[ ]
'
dy df df
= ¿ x̄ , ȳ ( x ' )+ ¿x̄ , ȳ ( y ' )
dt dx dy
 Taylor Series Expansion in Linearization Process
A non-linear function, which is a function of independent variables, can be expanded in terms of infinite
series around the given point/s of linearization through Taylor series

However, since higher order differentials have an infinitesimal value, their values are disregarded
leaving the equation with:

Supposed, an equation is stated with a one state variable, one input variable, and an output function

dx
=f [ x ( t ) ,u ( t ) ]
dt

y= g(x,u)
Using Taylor Series Expansion

Let xs and us be the steady state operation point

dx df
dt
=
dx [ df
¿ xs,us ( x − xs ) + ¿ xs , us ( u −us )+ f [ xs +us ]
du ]
y=
[ dg
dx
¿ xs , us ( x − xs )+
dg
¿ ( u −us )+ g (xs , us)
du xs ,us ]
Derive : f[xs +us] = 0 ; g(xs, us) = ys

x=x − xs ;u=u −us ; y= y − ys


g(xs, us) = ys
d ( x − xs) df
dt
=
[
dx
df
¿ xs ,us ( x ) + ¿xs ,us ( u )
du ]
d ( x ) df
dt
=
dx [ df
¿ xs , us ( x )+ ¿ xs ,us ( u )
du ]
y=
[ dg
¿
dx xs , us
( x )+
dg
¿ ( u )+ ys
du xs ,us ]
y − ys=
[ dg
¿
dx xs , us
( x )+
dg
¿
du xs , us
( u) ]
y=
[ dg
¿
dx xs ,us
( x )+
dg
¿ (u )
du xs ,us ]
Convert the model into state-space form
df df dg dg
Let a= ¿ xs ,us , b= ¿xs ,us , c= ¿ xs , us , d= ¿ (u)
dx du dx du xs, us

d(x)
=[ a ( x ) +b ( u ) ]
dt
y= [ c ( x ) +d (u ) ]

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