stability in control system

1. NONLINEAR SYSTEMS
By
Nivedhan V S (PWE14015)

2. INTRODUCTION
 Many practical systems are sufficiently nonlinear so that
the important features of their performance may be
completely overlooked if they are analyzed and
designed through linear techniques.
 The mathematical models of the nonlinear systems are
represented by nonlinear differential equations.
 Hence, there are no general methods for the analysis
and synthesis of nonlinear control systems.
 For such systems we must necessarily employ special
analytical, graphical and numerical techniques which
take account of system nonlinearities.

3. BEHAVIOUR
 The most important feature of nonlinear systems is that
nonlinear systems do not obey the principle of
superposition.
 Due to this reason, in contrast to the linear case, the
response of nonlinear systems to a particular test signal is
no guide to their behavior to other inputs.
 The nonlinear system response may be highly sensitive to
input amplitude.
 Hence, in a nonlinear system, the stability is very much
dependent on the input and also the initial state.
 The o/p of the nonlinear system will have harmonics and
sub-harmonics when excited by sinusoidal signals.
 It will exhibit various phenomena like jump resonance, limit
cycle, frequency entrainment, asynchronous quenching etc..

4.  In the frequency response of the nonlinear systems,
the amplitude of the response may jump from one
point to another for decreasing or increasing values
of frequency is called jump resonance.
 The subharmonic oscillations are nonlinear
steady state oscillations whose frequencies are an
integral submultiple of the forcing frequency.
 The limit cycles are oscillations of the response of
nonlinear system with fixed amplitude and
frequency.
 The frequency of limit cycle is entrained by the
forcing frequency within certain band of frequencies
is called frequency entrainment.

5.  In a nonlinear system that exhibits a limit cycle of
frequency ωl, it is possible to quench the limit cycle
oscillation by forcing the system at frequency ωq is
called asynchronous quenching.

6. CLASSIFICATION:
 Incidental Nonlinearities - Those which are
inherently present in the system like saturation,
dead zone, friction etc., and
 Intentional Nonlinearities – Those which are
deliberately inserted into the system to modify the
system characteristics i.e, to improvement the
system performance or/and to simplify the
construction of the system.

7. NONLINEARITIES
 The output is proportional to input for a limited
range of input signals, when input exceeds this
range, the output tends to become nearly constant.
This phenomenon is called saturation.
 All devices when driven by large signals, exhibit the
phenomenon of saturation due to limitations of their
physical capabilities.
 Example are electronics amplifier, output of sensors
which measuring the position, velocity, temperature
etc.,
 Many physical devices do not respond to small
signals, i.e, if the input amplitude is less than some
small value, there will be no output. The region in
which the output is zero is called dead zone.

8.  A relay is a nonlinear power amplifier which can
provide large power amplification inexpensively and
is therefore deliberately introduced in control
systems.
 A relay controlled system can be switched abruptly
between several discrete states which are usually
off, full forward and full reverse.
 Relay controlled systems find wide applications in
the control field.
 Some nonlinearities such as the torque-speed
characteristics of a servomotor, transistor
characteristics etc., are functions of more than one
variable. Such nonlinearities are called
multivariable nonlinearities.

9. Thanking you…