# Nonlinear Control

Contributed by Ljubo Vlacic.

**1883-1892**- Lindstedt ([129]) and Poincare ([130]) tackled the problem of finding a limit cycle solution for some of the second-order nonlinear differential equations.

- Lindstedt ([129]) and Poincare ([130]) tackled the problem of finding a limit cycle solution for some of the second-order nonlinear differential equations.
**1892**- Poincare ([130]) 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 ([134]).

- 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 ([134]).
**1915**- 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 ([135]).
- 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 gears ([136]).
- 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 ([136]).

- 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 ([136]).
**1918**- Continuing Duffing's seminal work ([133]), various forms of harmonic balance technique were used to study both free and forced oscillations in the second-order nonlinear differential equations ([176]).

- Continuing Duffing's seminal work ([133]), various forms of harmonic balance technique were used to study both free and forced oscillations in the second-order nonlinear differential equations ([176]).
**1941**- Minorsky ([137]) made a brief reference to nonlinear control problems and the possibility of using Lyapunovâ€™s method.

- Minorsky ([137]) made a brief reference to nonlinear control problems and the possibility of using Lyapunovâ€™s method.
**1949-1958**- 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 ([170]) and Tsypkin ([171]) to develop techniques for accurate measurement of the limit cycle in such systems. Further details can be found in [168], [169], [172].

- 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 ([170]) and Tsypkin ([171]) to develop techniques for accurate measurement of the limit cycle in such systems. Further details can be found in [168], [169], [172].
**1949-1958**- For detailed discussions on this topic and latest developments in nonlinear control see [177], [178], [179], [180], [181], [182], [183].

- For detailed discussions on this topic and latest developments in nonlinear control see [177], [178], [179], [180], [181], [182], [183].
**1950s**- Phase plane technique was the main focus in studying the nonlinear differential equations with many papers and books appearing during this year ([138], [139], [140], [141], [142]). 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. [143], [144], [145], [146], [147].
- Goldfarb, Dutilh, Oppelt, Kochenburger and Daniell appeared to have independently used the describing function in studying nonlinear differential equations ([148]). 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 ([176]).

- Goldfarb, Dutilh, Oppelt, Kochenburger and Daniell appeared to have independently used the describing function in studying nonlinear differential equations ([148]). 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 ([176]).
**1954**- The problem of examining nonlinear systems with random inputs was first pioneered by Booton ([160]) 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 ([161], [162], [164], [165]).

- The problem of examining nonlinear systems with random inputs was first pioneered by Booton ([160]) 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 ([161], [162], [164], [165]).
**1954-1955**- The extension of the describing function theory to determine the stability of any predicted limit cycle ([150]) and to utilize the describing function to determine the forced harmonic response of a nonlinear system ([151], [152]).

- The extension of the describing function theory to determine the stability of any predicted limit cycle ([150]) and to utilize the describing function to determine the forced harmonic response of a nonlinear system ([151], [152]).
**1956**- In attempts to study the occurrence of nonlinear phenomena in control loops by extending the describing function, particularly servomechanism, West et al. ([156]) realized that the response of nonlinear elements to two harmonic inputs had to be examined.

- In attempts to study the occurrence of nonlinear phenomena in control loops by extending the describing function, particularly servomechanism, West et al. ([156]) realized that the response of nonlinear elements to two harmonic inputs had to be examined.
**1956-58**- 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 ([176]). Other alternatives either placed nonlinearity in series or parallel with the inherent system or used nonlinear integrators for specific problems ([154], [155]).

- 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 ([176]). Other alternatives either placed nonlinearity in series or parallel with the inherent system or used nonlinear integrators for specific problems ([154], [155]).
**1957-58**- The incremental describing function was used to assess the stability of a limit cycle ([157], [158]).

- The incremental describing function was used to assess the stability of a limit cycle ([157], [158]).
**1962-1943**- Van der Pol ([131], [132]) and Krylov and Bogoliubov ([132]) introduced averaging methods for obtaining solutions to the second-order nonlinear differential equations.

- Van der Pol ([131], [132]) and Krylov and Bogoliubov ([132]) introduced averaging methods for obtaining solutions to the second-order nonlinear differential equations.