3
STEADYSTATE OSCILLATIONS
IN
NONLINEAR SYSTEMS
3.0
INTRODUCTION
The preceding chapter introduced the notion of a sinusoidalinput describing
function (DF).
Some of the implications of this type of linearization are
discussed there.
Here we apply the D F to the study of steadystate oscilla
tions.
For D F utilization to be meaningful, certain conditions must be ful
filled by the nonlinearity, and also by the system in which the nonlinearity is
present:
1.
The nonlinear element is time4nvariant.l
2.
No subharmonics are generated by the nonlinearity in response to a
sinusoidal input.
3.
The system filters nonlinearity output harmonics to the extent that only a
trivial quantity is fed back.
Condition
3
is the socalled$Iter hypothesis.
We shall have a great deal to
say about it in this chapter, for this heuristically motivated requirement is
certainly a basic factor underlying
DF
success or failure.
Certain periodically varying nonlinearities are also admissible.
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DETERMINATION
OF
LIMIT CYCLES
Ill
Three types of steadystate oscillations are of interest:
1.
Forced oscillations
2.
Conservative free oscillations
3.
Limit cycles
For our present purpose a
forced
oscillation is taken to be a systematic
response whose frequency is precisely the forcing signal frequency and
whose amplitude depends on the forcing signal amplitude.
Forced oscilla
tions are encountered in the study of frequency response, in which connection
we find a useful application for the DF.
The next two oscillation types are
behavioral modes of unforced systems.
A
conservative free oscillation
is an
initial conditiondependent
periodic mode associated with nondissipative
(conservative) systems.
A
continuous
range of conservativefreeoscillation
amplitudes and frequencies may be possible in a given system.
Limit
cycle
denotes an
initial conditionindependent
periodic mode occurring in dissipa
tive (nonconservative) systems.
Only a
discrete
set of limit cycle amplitudes
and frequencies may exist in
a
given system.
We shall find the D F a most
powerful tool for the study of both types of unforced behavior of nonlinear
systems.
3.1
DETERMINATION OF LIMIT
CYCLES
The limit cycle phenomenon is deserving of special attention since it is apt to
occur in any physical nonlinear system.
A
limit cycle can be desirable, for
example, by providing the vibration (dither) which minimizes frictional effects
in mechanical systems.
In fact, it can be absolutely necessary for proper
system performance, as we shall see when studying a certain adaptive control
scheme in Chap.
6.
On the other hand, a limit cycle can cause mechanical
failure of a control system (destructive limit cycle) or operator discomfort
and other undesirable effects, as in an aircraft autopilot.
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 Spring '04
 EricFeron
 The Land, LTI system theory, Linear system, Nonlinear system, limit cycle, R SYSTEMS

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