Theory of nonlinear control

Abstract

In the introduction the author discusses the nonlinearity in Nature as exemplified by a simple pendulum, inverse square laws of gravitation and electromagnetism, electronics, and planetary motion.Nonlinear feedback control systems have been analyzed in symposia sponsored by Polytechnic Institute of Brooklyn, American Society of Mechanical Engineers, and Institute of Radio Engineers. Even nonlinear system synthesis has been attempted. Part II discusses the three stages in the study of physical systems: (1) Development of linear system analysis and idealization of actual systems as linear; (2) Recognition of nonlinear systems and analysis of such systems as they actually are; and (3) Introduction of the new concept of nonlinear control and improvement of the performance of linear and nonlinear systems. Under (3) it is recognized that besides linear compensation of systems with nonlinearities the logical alternative might be the nonlinear feedback control of physical systems for either stability or optimum response. Part I I I starts with a review of linearizing techniques developed by Kochenburger, Johnson, Klotter, and others, known as the describing function method. Other quasi-linearization, equivalent linearization, and piecewise linearization techniques by Chen, Booton, Bass, and Stern are discussed. Part IV deals with the phase-space, the switching surface, and related concepts. The method of simultaneous phase-plane equations in a multi-degree-of-freedom system is especially adaptable to nonlinear feedback control system analysis. The phase-space approach has been used by Kalman, Hopkin, and others for the design of nonlinear servomechanisms. Part V covers new analytic techniques and new transforms.Papers by Pipes, Madwed, L. J. Lewis, Gillies, Stout, E. Weber, and others are reviewed. Taylor-Cauchy transforms for nonlinear systems are presented. Part VI discusses theory of nonlinear systems, including recent developments in stability theory and criteria, optimum nonlinear control systems, sampled-data systems, and nonlinear systems with random igputs.These recent developments give substantial support to the author's statement: “The newest aspect of feedback control is the development of the theory of nonlinear control.”