# National Taiwan University

Τεχνίτη Νοημοσύνη και Ρομποτική

24 Νοε 2013 (πριν από 4 χρόνια και 5 μήνες)

101 εμφανίσεις

National Taiwan University

Communication Engineering

Term Paper (Tut ori al )

Non
-
Li near Ti me I nvar i ant

Sys t em Anal ys i s

R 㤸㤴㈰㐸

~
2

~

Non
-
Linear Time In
variant System Analysis

Hsin
-
Kai Huang

(

)

E
-
mail

r98942048@ntu.edu.tw

Nation Taiwan University, Taipei, Taiwan, ROC

Abstract

The equations governing the behav
ior of dynamic systems are usually nonlinear.
Even in cases where a linear approximation is justified, its range of validity is likely to
be limited.

The engineer faced with the design or operation of dynamic systems,
especially the control engineer, must
understand the various modes of operation that
a system may exhibit. Usually, a system is designed to yield operation in a certain
mode and, at the same time,
suppression

of some other modes. A typical example is
the design of a servo exhibiting asymptotic

stability of the response to every constant
input but which cannot go into self
-
oscillations. Unlike

linear systems, nonlinear
systems can exhibit different behavior at different signal levels. The fact that a system
is nonlinear, however, may not necessa
are frequently introduced to yield optimal performance in a system. It is the
objective of this tutorial to discuss some of the fundamental properties of nonlinear
systems and to illustrate some of the inheren
t problems, as well as considerations
needed when dealing with the analysis or design of non
-
linear time invariant
systems.

Keywords

Nonlinear time invariant system,
phase
-
space, stability analysis,

approximate
method, describing function, Krylov
-

Bogoli
ubov asymptotical method.

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3

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Contents

Abstract
………………………………………………………………………………
..2

1.

Introduction
……………………………………………………………………
..4

1.1

Linear and Nonlinear Systems
……………………………………
4

1.2

Time
-
Varying and Time
-
Invariant
Systems
…………………
.5

1.3

Systems With and Without Memory
…………………………
.5

1.4

Linear Time Invariant Systems
……………………………………
5

1.5

Nonlinear Time Invariant Systems
……………………………
..6

2.

Introduction to Analysis of Nonlinear System
……………………
6

2.1

The Phase
-
space approach
…………………………………………
6

2.2

The stability analysis
…………………………………………………
.7

2.3

Approximate methods
……
…………………………………………
.8

3.

Approximate Analysis methods
………………………………………
10

3.1

Mathematical Description of Nonlinear Systems
……
..10

3.2

Describing Function
…………………………………………………
14

3.3

Krylov
-
Bogoliubov Asymptotical Method
…………………
14

4.

The Applications
……………………………………………………………
..16

5.

C
onclus
ion
………………………………………………………………………
19

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4

~

6.

References
…………………
………………
……………………………………
19

1.

Introduc
t
i
on

Every system can be characterized by its ability to accept an input such as
voltage, pressure, etc. and to produce an output in response to this input. An
example is a fi
lter whose input is a signal corrupted by noise and interference and
whose output is the desired signal. So, a system can be viewed as a process that
results in transforming input signals into output signals.

First of all, we review the concept of systems

by discussing the classification of
systems according to the way the system interacts with the input signal. This
interaction, which defines the model for the system, can be linear or nonlinear,
time
-
invariant or time varying, memoryless or

with memory, c
ausal or noncausal,
stable or unstable, and deterministic or nondeterministic. We briefly review the
properties of each of these classes.

1.1

Linear and Nonlinear Systems

When the system is linear, the superposition principle can be applied. This
important fa
ct is the reason that the techniques of linear
-
system analysis have been
so well developed. The superposition principle can be stated as follows. If the
input/output relation of a system

is

x(t)
-
> y(t),
αx
1
(t)+βx
2
(t)
-
>αy
1
(t)+βy
2
(t)

Then

the system is li
near. So, a system is said to be nonlinear if
this equation

is not
valid.

Example

Consider the voltage divider shown
in Figure 1 with R
1
=R
2
.
For

input
x(t) and output y(t), this is a linear system. The input/output relation can be written
as

2 1
( ) ( ) ( )
1 2 2
R
y t x t x t
R R
 

On the other hand, if R1 is a voltage
-
dependent resistor such that R1=x(t)R2, then
the system is nonlinear. The input/output relation in this case can be written as

2 ( )
( ) ( )
( ) 2 2 ( ) 1
R x t
y t x t
x t R R x t
 
 

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5

~

Figure1

1.2

Time
-
Varying and Time
-
Invariant Syst
ems

A system is said to be time
-
invariant if a time shift in the input signal causes the
same time shift in the output signal. If y(t) is the output corresponding to input x(t), a
time
-
invariant

system will have y(t
-
t
0
) as the output when x(t
-
t
0
) is the ou
tput. So, the
rule used to compute the system output does not depend on time at which the input
is applied.

1.3

Systems With and Without Memory

For most systems, the inputs and outputs are functions of the independent
variable. A system is said to be memoryle
ss, if the present value of the output
depends only on the present value of the input. For example, a resistor is a
memoryless system, since with input x(t) taken as the current and output y(t) take as
the voltage, the input/output relationship is y(t)=Rx(
t), where R is the resistance.
Thus, the value of y(t) at any instant depends only on the value of x(t) at that time.
On the other hand, a capacitor is an example of a system with memory.

1.4

Linear Time Invariant Systems

We have discussed a number of basic s
ystem properties. Linear time
-
invariant
systems play a fundamental role in signal and system analysis because of the many
physical phenomena that can be modeled. A linear time
-
invariant(LTI) system is
completely characterized by its
impulse

response h(t) a
nd o
utput y(t) of
a LTI system
is the convolution of the input x(t) with the
impulse

response of the system

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6

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( ) ( ) ( ) ( ) ( )
y t x t h t x h t d
  


   

1.5

Nonlinear Time Invariant Systems

With nonlinear systems, we cannot count on the above nice properties.
The
nonlinear ti
me invariant

system is a system, whose operator is time
-
invariant but
depends

on the input.
For

example
,

a square amplifier

is nonlinear time invariant

system, provided

y
(
t
) =
O
[
x
(
t
)]
x
(
t
) =
ax
2
(
t
)
.

Other examples are rectifiers, oscillators,
phase
-
looked l
oops (PLL), etc. Note that all real electronic systems become
practically nonlinear owing to saturation.

Because of the difficulties involved in
nonlinear analysis, approximation methods are commonly used.

2.

Introduction to Analysis of Nonlinear System

Nonl
inear systems with either inherent nonlinear characteristics or
nonlinearities deliberately introduced into the system to improve their dynamic
characteristics have found wide application in the most diverse fields or engineering.
linear system analysis is to obtain a comprehensive picture,
quantitative if possible, but as least qualitative, of what happens in the system if the
variables are allowed, or forced, to move far away from the operating points. This is
called the global, o
r in
-
the
-
large, behavior. Local, or in
-
the
-
small, behavior of the
system can be analyzed on a linearized model of the system.

So, the local behavior can be investigated by rather general and efficient linear
methods that are based upon the powerful super
position and homogeneity
principles. If linear methods are extended to the investigation of the global behavior
of a nonlinear system, the results can be erroneous both
quantitatively

and
qualitatively since the nonlinear characteristics may be essential b
ut the linear
methods may fail to reveal it. Therefore, there is a strong emphasis on the
development of methods and techniques for the analysis and design of nonlinear
system.

2.1

The Phase
-
space approach

The phase
-
space, or more specifically the phase
-
plane
, approach has been used
for solving problems in mathematics and
physics

at least since Poincare. The
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7

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approach gives both the local and the global behavior of the nonlinear system and
provides an exact topological account of all possible system motions und
er various
operating conditions. It is convenient, however, only in the case of second
-
order
equations, and for high
-
order cases the phase
-
space

approach is cumbersome to use.

Nevertheless, it is a powerful concept underlying the entire theory of ordinary

differential equations (linear or nonlinear, time varying or time invariant). It can be
extended to the study of high
-
order differential equations in those cases where a
reasonable approximation can be made to find an equivalent second
-
order equation.
How
ever, this may lead to either erroneous conclusions about the essential system
behavior, such as stability and instability, or various practical difficulties such as time
scaling.

2.2

The stability analysis

The stability analysis of
nonlinear

systems, which i
s heavily based on the work of
Liapunov, is a powerful approach to the qualitative study of the system global
behavior. By this approach, the global behavior of the system is investigated utilizing
the given form of the nonlinear differential equations but

without explicit knowledge
of their solutions. Stability is an inherent feature of wide classes of systems, thus
system theory is largely devoted to the stability concept and related methods of
analysis
.

Stability analysis, however, does not constitute a

complete satisfactory theory
for the design of
nonlinear

systems. The stability conditions, which are often hard to
determine, are sufficient but usually not necessary. This comes from the fact that the
given equations are reformulated for the application

of the stability analysis. In that
reformulation certain information about the specific system characteristics is lost and,
unfortunately
, the amount of information th
at is lost cannot be estimated.

For example, if a
nonlinear

system is found to be stabl
e for a certain range of
parameter values, it is not possible to predict how far from that range the parameter
value can be chosen without affecting the system stability. Furthermore the system
can be unstable and still be satisfactory for practical applic
ations. For example, a
system can exhibit stable periodic oscillations and therefore be unstable. However, in
the application of the system, these oscillations may not be observed because their
amplitude is sufficiently small and the perturbations permanen
tly acting on the
system are large enough to drive the system far from the periodic oscillations.

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8

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2.3

Approximate methods

Approximate methods for solving pr
oblems in mathematical physics
have been
received with much interest by engineers and have promptly ob
tained wide diffusion
in diverse fields of system engineering. The basic merit of approximate methods
consists in their being direct and
efficient
, and they permit a simple evaluation of the
solution for a wide class of problems arising in the analysis of
nonlinear

oscillations.

The application of computer techniques and system simulations has given
strong emphasis to those approximate methods which employ rather straightforward
and realizable solution procedures and calculations. These methods enable a si
mple
estimation of how different system structures and parameters influence the salient
system dynamic characteristics. The application of a computer simulation can then
provide the actual solution of the design problem. If the system behavior is not
satis
factory, or if the computer solution does not agree with predicted characteristics,
the approximate methods can again be applied to guide the next step in the system
simulation and also achieve a better solution of the analysis problem, If we
interchange t
hese two steps
---
that is, apply the approximate methods and then the
computer simulation
---
the design converges eventually to a final satisfactory solution,
This philosophy in the analysis of
nonlinear

systems can give improved results not
only in a specif
ic system but also in the related class of systems, and thus has an
important generality in system theory and application.

It is of particular significance to classify the nonlinear problem before a specific
technique is applied to its
solution
. Thus it i
s necessary to evaluate the potential of
both the exact and the approximate methods before they are tested on the actual
problem. This involves engineering experience and ingenuity in choosing the
appropriate design technique and procedure. If an exact met
hod is to be applied, we
should be aware of the fact that it may require that a sequence of simplifications be
introduced

in the original problem.

In the simplifications, certain vital characteristics of the original problem can be
lost
---
for example, th
e reduction of the order of a differential equation through
neglect of one of the system parameters. Then the approximate solution of the
original problem may represent more appropriately the actual situation and be of
more use in the design. In addition,
the approximate methods normally yield more
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9

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-
offs or the structural and
parameter changes that might enhance the overall system characteristics.

On the other hand, the exact methods can reveal vari
ous subtle phenomena in
nonlinear

system behavior that cannot be discovered by the approximate methods. It
can be concluded that in a majority of practical problems both the exact and the
approximate methods should be applied to obtain a satisfactory solut
ion of the
nonlinear

system design problem, and the versatility of the designer in various
solution procedures is a prerequisite for a
successful

system analysis and design.

In the application of the approximate methods, a significant problem is the
estim
ation of their accuracy. A certain degree of accuracy is necessary to guarantee
the applicability of the method involved, and to ensure the validity of both the
qualitative conclusions and the
quantitative

results obtained by the approximate
analysis. The
accuracy problem,
however
, involves various mathematical difficulties,
and the designer is forced to use simple and practical approximate methods despite
some pessimism about the validity of the methods and promising results have been
obtained
in solving t
he accuracy problem
.

Among the approximate methods used for the analysis of
nonlinear

oscillations,
the Krylov
-
Bogoliubov asymptotical method stands out because of its usefulness in
system engineering problems. The original method not only enables the
det
-
state periodic oscillations, but also gives in evidence the
transient process
corresponding

to small amplitude perturbations of the oscillations.
The latter is of particular interest in system
design
, where the transient process is
oft
en the ultimate goal. However, the method is applicable to systems described by
second
-
order nonlinear differential equations.

The approximate method to be used
in the analysis of nonlinear systems along
with the parameter plane concept is the harmonic li
nearization method, often called
the describing function method or the method of harmonic balance.

The harmonic
linearization is heavily based on the Krylov
-
Bogoliubov approach, and be applied to
nonlinear

systems described by high
-
order differential equat
ions.

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10

~

3.

A
pproximate Ana
lysis methods

In this section, we will present
several

methods for approximately analyzing a
given nonlinear system. Because a closed
-
form analytic solution of a nonlinear
differential equation is usually impossible to obtain, meth
ods for carrying out an
approximate analysis are very useful in practice. The methods
presented here fall
into three categories

Describing function methods consist of replacing a
nonlinear

element within
the system by a linear element and carrying out further analysis. The utility
of these methods is in predicting the existence and stabil
ity of limit cycles, in
predicting jump resonance, etc.

Numerical
solution

methods are specifically aimed at carrying out a
numerical solution of a given nonlinear differential equation using a
computer.

Singular perturbation methods are especially well su
ited for the analysis of
systems where the inclusion or exclusion of a particular component changes
the order of the system. For example, in an
amplifier
, the inclusion of a stray
capacitance in the system model increases the order of the dynamic model
by
one.

The above three types of methods are only some of the many varieties of
techniques that are available for the approximate analysis of nonlinear systems.
Moreover, even with regard to the three subject areas mentioned above, the
presentation here only

scratches the
surface
, and references are given, at
appropriate places, to works that treat the subjects more thoroughly.

3.1

Mathematical Description of Nonlinear Systems

In general a
nonlinear

system consists of linear and
nonlinear

elements. The
linear e
lements are described by linear differential equations. The nonlinear
elements, which are normally very limited in number, are described by a nonlinear
function or differential equation

relating the input and output of the element. The
n
onlinear input
-
outp
ut relationship can have a rather arbitrary form. The parameter
plane analysis to be presented is restricted to a certain class of these relationships.

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11

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In treating a real system as linear, we assume that the system is linear in a
certain range of operati
on. The signals appearing in various points of the system are
such that the superposition principle is justified. However, if signals in the system go
beyond the range of linear operation and, for example, become either very large or
very small, the charac
teristics of the system elements can be essentially different
form the linearized

characteristics and the system must be treated as nonlinear. Such
cases are illustrated graphically by the characteristics shown
in Figure2
, where x
denotes the input to the
element and the output is given by the value of the function
F(x). If the output of the element is denoted by y, the input
-
output relationship can
be written analytically as

( )
y F x

Figure2

Certain nonlinear
characteristics

can be gi
ven in analytical form. For example,
the characteristic of Figure 2 can be analytically described by

,| |
( )
,| |
kx x S
F x
csignx x S

 

 

 

The characteristic is linear with slope k=c/S for inputs less than S, and it exhibits
saturation for input magnitudes greater t
han S.

In
various

practical applications the
nonlinear

characteristic is obtained
experimentally
, and an adequate analytical expression cannot be justified. On the
other hand, some characteristics are conveniently expressed analytically, whereas a
graphic
al interpretation is not possible.

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12

~

Now, only single
-
valued
nonlinear

characteristics have been discussed; in the
characteristics of Figure2, to each value of the input x there is one and only
one

value of the output y=F(x). The characteristics in Figure3,

which have a hysteresis
loop, are multi
-
valued nonlinear characteristics.

Figure3

The hysteresis property can be such that the loop dimensions depend on the
magnitude of the input signal. It is also to be noted that
hysteresis

type
nonlinear

characte
ristics cannot be completely described by the function y=F(x) since the
output y inherently depends on the direction of change

in the magnitude of the
input x. If the rate of change in x is greater than zero, the right
-
hand side loop
represents the nonline
ar characteristic, and vice versa. Thus the adequate
description of the hysteresis type of nonlinearities should be expressed as

(,),
d
y F x signsx s
dt
 

R
ather than as y=F(x).

Figure4

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13

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Besides the analytical description of nonlinear
elements

and syste
ms, it is
essential to consider the structure of the system, which is usu
ally given in familiar
block diagram or signal flow graph form. The structure of the system displays certain
inherent features of nonlinear systems that are not apparent in the analyt
ical
description.

The basic nonlinear system with one nonlinear element n is shown in Figure4. It
should be noted that the function F(x, sx) associated with n does not necessarily
represent the nonlinear element as described by a nonlinear differential eq
uation as

(,)
(,)
( )
Tsy y F x sx
F x y sy y Kx
F x sy y Kx
 
 
 

To make the analysis easier, the nonlinear function F(x, sx) may be isolated in
the system, while all the linear relations are

joined in the block G(s). For example, if
the nonlinear element n is described, it can be s
plit into two equations

( 1)
(,)
Ts y z
z F x sx
 

Then the
equations

are associated with the other linear elements of the system,
and the function F(x, sx) is isolated in the block n. Naturally, the function F(x, sx) does
not represent the nonlinear ele
ment n and therefore will be called the nonlinearity.

The linear elements may coupled in an arbitrary way to make the equivalent
transfer function G(s), whose order is not theoretically limited as far as the
parameter plane analysis is concerned. However,

certain restrictions on the nature of
the function G(s) are imposed in order to justify the application of the approximate
analysis. According to the block diagram of Figure4, the transfer
function

G(s) is

( )
( )
( )
C s
G s
B s

The function f=f(t),
which may be either a desired input signal

or an undesired
perturbation, is applied somewhere in the linear part of the system. The block
diagram of Figure4 may represent a nonlinear system having two nonlinear elements
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14

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connected is cascade, providing it i
s possible to isolate the two related nonlinearities
and join them in one
equivalent

block.

3.2

Describing Function

Among the methods used for stability analysis and investigation of sustained
nonlinear oscillations, sometimes called a limit cycle, the descri
bing function
generally stands out because of its usefulness in engineering problems of control
system analysis. The describing function technique can be successfully applied to
systems other than control whenever the sustained oscillations, which are base
d on
some nonlinear phenomena, represent possible operating conditions.

The theoretical basis of the describing function analysis lies in the van der Pol
method of slowly varying coefficients as well as in the methods of harmonic balance
and equivalent li
nearization for solving certain problems of nonlinear mechanics. The
analysis has been further developed in the work of Goldfarb with the emphasis on
nonlinear phenomena in feedback systems.

For presenting the concept of describing function method, a nonl
inear time
invariant system with a block diagram of Figure4 is considered. The block n
represents the isolated nonlinearity described by a given
function
, F(x, sx). The linear
part of the system is presented by a known transfer
function

G(s) = C(s)/B(s). T
he
external forcing function f=f(t) is identically zero for all values of time t. Thus the free
oscillations in the system are determined by a nonlinear homogeneous differential
equation

( ) ( ) (,) 0,
d
B s x C s F x sx s
dt
  

3.3

Krylov
-
Bogoliubov Asymptotical Method

Sin
ce the parameter plane analysis of nonlinear oscillations is based upon the
concept and results of the Krylov
-

Bogoliubov asymptotical method, the fundamental
aspects of the method. Then the derivations involved in further extensions and
applications of th
e method can be more easily followed. Furthermore, the method is
highly applicable to practical problems of nonlinear oscillations and represents a
basis for other approximate methods in nonlinear analysis, particularly the de
scribing
function technique.

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15

~

The basis of approximate analysis of nonlinear oscillations is the small
parameter method introduced in connection with the three
-
body problem of
celestial mechanics. The fundamental concept and certain solution procedures have
been p
ostulated in a general

form by Poincare. In this method a second
-
order
nonlinear

differential equations describing the oscillations has been formulated so
that it incorporates a small parameter. The parameter is small in the sense that it
represents a number of sufficiently sma
ll absolute value.

For a zero value of the parameter the nonlinear operation reduces to a linear
equation, the solution of which is a harmonic oscillation. The solution of the linear
equation is called the generating solution. The essential idea of the m
ethod is to
assume

the solution of the nonlinear differential equation in the fo
rm of an infinite
power series.

Then, by substituting the solution into the original differential
equation
, a
recursive system of linear nonhomogeneous differential equations
with constant
coefficients is obtained. Based upon the generating solution, the recursive system
can be solved by elementary calculations up to a desired degree of accuracy. The
small parameter method has proved useful for solving numerous problems in phys
ics
and the technical sciences.

By considering certain nonlinear phenomena in electron tube oscillators,

van
der Pol proposed the method of slowly varying coefficients for evaluation of the
related
periodic

oscillations. This method is a variant of the th
e small parameter
method, which is heavily based upon the consideration of the first harmonic in the
Fourier series expansion of the nonlinear
function
, this being the keystone in the
describing
function

analysis.

Furthermore, not only is the method conve
nient for the identification of
periodic solutions of second
-
order nonlinear differential equations, but it also places
in evidence the manner in which the possible periodic solutions are established, after
small amplitude perturbations, around the solutio
n. The method, however, has been
based on a rather intuitive approach and only the first approximation has been
considered. From the approach it is not clear how the higher approximations can be

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16

~

4.

The Applications

Recall that a system
Y
(
μ, t
) =
T
[
X
(
μ, t
)]
is called memoryless iff the output
Y
(
μ, t
)
is a function of the input
X
(
μ, t
)
only for the same time instant. For example,
Y
(
μ, t
) =
X
(
μ, t

τ
)
and
Y
(
μ, t
) =
X
2
(
μ, t
)
are memoryless systems. Note that
Y
(
μ, t
) =
X
2
(
μ, t
)
is
nonline
ar. We are here interested in memoryless nonlinear systems whose input and
output are both real
-
valued and can be characterized by
Y
(
μ, t
) =
g
(
X
(
μ, t
))
where
g
(
x
)
is a function of
x
.

Figure5

A nonlinear memoryless system

For memoryless nonlinear system
s,

* if
X
(
μ, t
)
is
Strict
-
Sense Stationary processes
, so is
Y
(
μ, t
)
;

* if
X
(
μ, t
)
is stationary of order
N
, so is
Y
(
μ, t
)
;

* if
X
(
μ, t
)
is
Wide
-
Sense Stationary processes
,
Y
(
μ, t
)
may not be

stationary in any sense.

Therefore, the second
-
moment descr
iption of
X
(
μ, t
)
is not sufficient

for
second
-
moment description for memoryless nonlinear systems.

Examples of Nonlinearity:

* Full
-
Wave Square Law:
g
(
x
) =
ax
2
.

* Half
-
Wave Linear Law:
g
(
x
) =
ax

u
(
x
)
with
u
(
x
)
a unit step function.

* Hysteresis Law

* H
a
rd Limiter

* Soft Limiter

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17

~

Most of the nonlinear analytical methods concentrate on the second order
statistical description of input and output processes, namely autocorrelations and
power spectrums. One famous approach is the direct method which deals
with
probability density functions and is good for use if
X
(
μ, t
)
is Gaussian.

Consider the memoryless nonlinear system
Y
(
μ, t
) =
g
(
X
(
μ, t
))
where

the
first
-
order and second
-
order densities of input process
X
(
μ, t
)
, namely

f
X
(
x
;
t
)
and
f
X
(
x
1
, x
2
;
t
1
, t
2
)
, are given.

Now, the following statistics of output
Y
(
μ,
t
)
can be obtained

(;) (;) | ( ) |
Y x i i
allroots
f y t f x t J x

{ (,)} ( ) (;)
n n
x
E Y u t g x f x t dx



1 2 1 2
1 2 1 2 1 2 1 2
(,) { ( (,)) ( (,))}
( ) ( ) (,;,)
Y
x
R t t E g X u t g X u t
g x g x f x x t t dx dx
 


 

Consider some examples below.

Full
-
Wave Square Law Device:
Y
(
μ, t
) =
aX
2
(
μ, t
)
with
a >
0
.

1
(;) [ (/;) (/;), 0
2
Y x x
f y t f y a t f y a t y
ay
   

And F
Y
(y;

t) = 0
for y<0.

Square
-
La
w Detector:

Figure
6

~
18

~

The zonal LPF allows a spectral band to pass undistorted and everything

else is
filtered perfectly.

Assume further that
X
(
μ, t
)
is a narrowband WSS Gaussian

noise process
n
(
μ, t
)
with zero mean and autocorrelation
R
X
(
τ
)
. Thus,

X
(
μ
, t
)
can be expressed in the
polar form

(,) (,) (,)cos(2 (,))
c
X u t n u t v u t f t u t
 
  

For fixed
t
,
v
(
μ, t
)
and
θ
(
μ, t
)
are independent random variables where

v
(
μ, t
)
is
Rayleigh distributed with

2
(;) exp{ }, 0
(0) 2 (0)
v
n n
v v
f v t v
R R
  

and
f
v
(
v
;
t
) = 0
for
v <
0
, and
θ
(
μ, t
)
is uniform in

[0
,
2
π
)
. Here,

σ
2
n

=
R
n
(0) =
E
{
n
2
(
μ, t
)
}
and
E
{
v
2
(
μ, t
)
}
= 2
R
n
(0)
.
Note that the zonal LPF filters out all information about
frequency and phase.

When the band
width

of the zonal LPF is much smaller than
f
c
,
we have

2 2
2 2
(,) (,)cos (2 (,))
[ (,) (,)cos(4 2 (,))]
2
c
c
Y u t av u t f t u t
a
v u t v u t f t u t
 
 
 
  

2
(,) (,)
2
a
Z u t v u t

Consider further that
S
n
(
f
) =
A
for
|
f

f
c
|
2

< B/
2
and
|
f
+
f
c
|
2
< B/
2

and
S
n
(
f
) = 0
otherwise, with
A,B >
0
. Zonal bandwidth is assumed

larger than
2
B
.

Figure
7

In the case,
R
X
(0) =
R
n
(0) = 2
AB
and the following can be obtained.

~
19

~

2
(2 )!
{ (,)} { (,)} ( (0))
2!
n n n n n
n
n
n
E Y u t a E X u t a R
n
 

2 2 2 2 2
{ (,)} { (,)} { (,)} 8
Var Y u t E Y u t E Y u t a A B
  

2
{ (,)} { (,)} 2
2
a
E Z u t E v u t aAB
 

5.

C
onclusion

The development of
nonlinear

methods faces real difficulties for various reasons.
There are no universal mathematical methods for the solution of
nonlinear

differential equat
ions which are the mathematical models of nonlinear system. The
methods deal with specific classes of nonlinear equations and have only limited
applicability to system analysis. The classification of a given system and the choice of
an appropriate method o
f analysis are not at all an easy task. Furthermore, even in
simple nonlinear problem, there are numerous new phenomena qualitatively
different from those expected in linear system behavior, and it is impossible to
encompass all these phenomena in a single

and unique method of analysis.

Although there is no universal approach to the analysis of
nonlinear

systems, by
excluding specific techniques we can still conclude that the nonlinear methods
generally fall under one of three following approached

t
he phase
-
space topological
techniques, stability analysis method, and the approximate methods of
nonlinear

a
nalysis. This classification of the
nonlinear

methods is rather subjective but can be
useful in systematization of their review.

6.

References

[1]

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Bell System Techn. J.
, 13, 1

18,

1934.

[2]

Bogoliubov N.N., and Mi
tropolskiy Yu.A.,
Asymptotic Methods in the Theory

of
Non
-
Linear Oscillations
, New York, Gordon and Breach, 1961.

[3]

Director, S.W., and Rohrer, R.A.,
Introduction to Systems Theory
, McGraw
-
Hill,
New York, 1972.

[4]

Doyle J.C., Francis B.A., Tannenbaum A.
R.,
Feedback Control Theory
, Macmillan

~
20

~

Publishing Company, New York, 1992.

[5]

Dulac, H.,
Signals, Systems, and Transforms
, 3
rd
ed., Prentice Hall, New
-
York,

2002.

[6]

Gelb, A., and Velde, W.E.,
Multiple
-
Input Describing Functions and Nonlinear
System Des
ign
, McGraw
-
Hill, New York, 1968.

[7]

Guckenheimer, J., Holmes, P.,
Nonlinear Oscillations, Dynamical Systems, and
Bifurcations of Vector Fields
, 7
th
printing, Springer
-
Verlag, New
-
York, 2002.

[8]

Hayfeh A.H., and Mook D.T.,
Nonlinear Oscillations
, New Yor
k, John Wiley

&
Sons, 1999.

[9]

Haykin, S., and Van Veen, B.,
Signals and Systems,
2
nd
ed., New
-
York, Wiley

&
Sons, 2002.

[10]

Hilborn, R.,
Chaos and Nonlinear Dynamics: An Introduction for Scientistsand
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, 2
nd
ed., Oxford University Press, New
-
Yor
k, 2004.

[11]

Jordan D.W., and Smith P.,
Nonlinear Ordinary Differential Equations: An
Introduction to Dynamical Systems
, 3
rd
ed., New York, Oxford Univ. Press,

1999.

[12]

Khalil H.K.,
Nonlinear systems
, Prentice
-

River,
2002.

[13]

Rugh W. J.,
Nonlinear System Theory: The Volterra/Wiener Approach
, Baltimore,

John Hopkins Univ. Press, 1981.

[14]

Samarskiy, A.A., and Gulin, A.V.,
Numerical Methods
, Nauka, Moscow, 1989.

[15]

Sandberg, I.W., On the response of nonolinea
r control systems to periodic

input
signals,
Bell Syst. Tech. J.
, 43, 1964.

[16]

Sastry, S.,
Nonlinear Systems: Analysis, Stability and Control
, Springer
-
Berlag,

New York, 1999.

[17]

Shmaliy, Yu. S.,
Continuous
-
Time Signals,
Springer, Dordrecht, 2006.

[18]

Verhulst, F.,
Nonlinear Differential Equations and Dynamic Systems
,
Springer
-
Verlag, Berlin, 1990.

[
19
]

Wiener, N.
Response of a Non
-
Linear Device to Noise

Laboratory, M.I.T., Cambridge, MA, Apr. 1942.

[2
0
]

Zames, G., Realizabil
ity conditions for nonlinear feedback systems,
IEEE Trans.
Circuit Theory
, Ct
-
11, 186

194, 1964.