NONLINEAR SYSTEMS

By

Nivedhan V S (PWE14015)

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.

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..

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.

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.

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.

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.

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.

Thanking you…