Contributed by Ljubo Vlacic.
- Lindstedt () and Poincare () tackled the problem of finding a limit cycle
solution for some of the second-order nonlinear differential equations.
- Poincare () introduced the phase
plane method in studying the second-order nonlinear differential
equations. This method became the dominant approach and a valuable tool
available to control engineers from the late 1930s.
- Following Poincare, many contributions were made to the field
of phase plane topology such as information on singular points and the
structure of trajectories near them, and conditions for the existence
of limit cycles ().
- Stability theory for linear differential equations was
established around the work of Poincare but little was done on the
general nonlinear differential equations case as Lyapunov's original
work was neglected ().
- Major research efforts into the effects of nonlinearity in
control systems was carried out at MIT as Bush and his colleagues
studied nonlinear differential equations considering differential
analyzers using mechanical integrators. These efforts included
implementation of various designs by Hazen including some using relays
being aware of the limitations on the performance due to backlash in
- Motivated by wartime, the need for accurate fire control
systems led to significant work on servomechanism in the western world.
The phase plane methods and describing function approach were being
used to study the nonlinear effects ().
- Continuing Duffing's seminal work (),
various forms of harmonic balance technique were used to study both
free and forced oscillations in the second-order nonlinear differential
- Minorsky () made a brief reference
to nonlinear control problems and the possibility of using
- In studying relay systems, it was realized that the output
from a relay, once it had switched, became independent of the input
which led Hamel () and Tsypkin ()
to develop techniques for accurate measurement of the limit cycle in
such systems. Further details can be found in ,
- For detailed discussions on this topic and latest
developments in nonlinear control see , , , ,
, , .
- Phase plane technique was the main focus in studying the
nonlinear differential equations with many papers and books appearing
during this year (, ,
, , ). However, different nonlinear effects in
specific second-order systems were investigated and understood later;
nonlinear effects like the effects of torque saturation, nonlinearities
in the error channel, backlash, friction, and relay control in
second-order systems, optimum control using relays, chattering in relay
systems. Detailed coverage of these developments were published, see
e.g. , , , , .
- Goldfarb, Dutilh, Oppelt, Kochenburger and Daniell appeared
to have independently used the describing function in studying
nonlinear differential equations (). The
method is identical to a harmonic balance approach, where the first
harmonic only is balanced, but was developed in a way more suitable for
use in feedback control, in which nonlinear systems were modeled in
terms of interconnected blocks of static nonlinear and transfer
function elements ().
- The problem of examining nonlinear systems with random inputs
was first pioneered by Booton () who
approximated the nonlinearity by a linear gain such that the error
between the nonlinearity output and that from the linear gain with the
same random input (Gaussian) was minimum. Other related materials and
contributions can be found in (, , , ).
- The extension of the describing function theory to determine
the stability of any predicted limit cycle ()
and to utilize the describing function to determine the forced harmonic
response of a nonlinear system (, ).
- In attempts to study the occurrence of nonlinear phenomena in
control loops by extending the describing function, particularly
servomechanism, West et al. () realized that
the response of nonlinear elements to two harmonic inputs had to be
- To avoid limit cycles predicted by the describing function
method, a common procedure was to change the open-loop dynamics so that
no intersection existed between the loci of the system and the
specialized describing function on the Nyquist diagram ().
Other alternatives either placed nonlinearity in series or parallel
with the inherent system or used nonlinear integrators for specific
problems (, ).
- The incremental describing function was used to assess the
stability of a limit cycle (, ).
- Van der Pol (, )
and Krylov and Bogoliubov () introduced
averaging methods for obtaining solutions to the second-order nonlinear