Stability Analysis of Limit Cycles in Non Linear Feedback Systems Using Describing Functions: Improved Results

Ordinary differential equations

ABRAMSON Information theory and coding BATTIN Astronautical guidance BLACHMAN Noise and its effect on communication BREMER Superconductive devices BROXMEYER Inertial navigation systems GELB AND VANDER VELDE Multiple-input describing functions and nonlinear system design GILL Introduction to the theory of finite-state machines HANCOCK AND WINTZ Signal detection theory HUELSMAN Circuits, matrices, and linear vector spaces KELSO Radio ray propagation in the ionosphere MERRIAM Optimization theory and the design of feedback control systems MUUM Biological control systems analysis NEWCOMB Linear multiport synthesis PAPOULIS The fourier integral and its applications R. N. BRACEWELL) STEINBERG AND LEQUEUX (TRANSLATOR Radio astronomy WEEKS Antenna engineering PREFACE The theory of automatic control has been advanced in important ways during recent years, particularly with respect to stability and optimal control. These are significant contributions which appeal to many workers, including the writers, because they answer important questions and are both theoretically elegant and practically useful. These theories do not, however, lay to rest all questions of importance to the control engineer. The designer of the attitude control system for a space vehicle booster which, for simplicity, utilizes a rate-switched engine gimbal drive, must know the characteristics of the limit cycle oscillation that the system will sustain and must have some idea of how the system will respond to attitude commands while continuing to limit-cycle. The designer of a chemical process control system must be able to predict the transient oscillations the process may experience during start-up due to the limited magnitudes of important variables in the system. The designer of a radar antenna pointing system with limited torque capability must be able to predict the rms pointing error due to random wind disturbances on the antenna, and must understand how these random disturbances will influence the behavior of the system in its response to command inputs. But more important than just being able to evaluate how a given system will behave in a postulated situation is the fact that these control engineers must design their systems to meet specifications on important characteristics. Thus a complicated exact analytical tool, if one existed, would be of less value to the designer than an approximate tool which is simple enough in application to give insight into the trends in system behavior as a function of system parameter values or possible compensations, hence providing the basis for system design. As an analytical tool to answer questions such as these in a way …

https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-30-estimation-and-control-of-aerospace-systems-spring-2004/readings/gelb_ch1_ocra.pdf

Multiple-Input Describing Functions and Nonlinear System Design

/pdf/oscillations-in-nonlinear-systems-4361fg4y6i.pdf

Oscillations in Nonlinear Systems

Nonlinear Automatic Control

The well-known graphical describing function procedure can be simply modified to provide a completely reliable method for predicting whether or not certain kinds of nonlinear feedback system can oscillate. The modified method is easy to use and quantifies, in a natural way, the usual intuitive ideas about describing function reliability. In addition to the usual graphs, the user has to draw a band which measures the amount of uncertainty introduced by the approximations inherent in the method; in return for this extra work, the method gives error bounds for oscillation predictions, as well as ranges of frequency and amplitude over which oscillation is impossible. The main restriction is that the nonlinear element must be single valued and have bounded slope.

Stability Analysis of Limit Cycles in Non Linear Feedback Systems Using Describing Functions: Improved Results

References

Ordinary differential equations

Multiple-Input Describing Functions and Nonlinear System Design

Oscillations in Nonlinear Systems

Nonlinear Automatic Control

Describing functions revisited

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