Sinusoidal-input Describing Function Models Research Articles

New Methodology for Control of Nonlinear Unstable Multivariable Systems

AbstractIn this paper, a new methodology for control of nonlinear unstable multivariable systems is developed. The method involves stabilization of the nonlinear plant followed by generation of sinusoidal input describing function (SIDF) models of the stabilized closed-loop system. With the known SIDF model of the closed-loop system and the known stabilizing controller, the model of the unstable nonlinear multivariable system is extracted; then a controller for the model is designed. Finally, the design is verified by simulation. Stability is demonstrated by successful generation of SIDF models of the designed closed-loop system for various amplitudes of excitation. The method is applied to a two-input/two-output (TITO) model of a nonlinear unstable robotic arm, and the results are compared with an alternative approach.

Design of Dual-Range Linear Multivariable Controllers for Nonlinear Systems with Application to UAV Control

In this research, the previous design procedure for single-variable dual-range linear controllers for highly nonlinear systems is extended to the multivariable case. The procedure is based on generating the sinusoidal input describing function models of the plant followed by selection of two of these models for optimization of the dual-range linear controller. A single multivariable controller is designed by optimization to achieve a stable closed-loop system that satisfies a set of user-defined performance measures. Finally, the design is verified by digital simulation of the nonlinear plant and linear controller. Stability is demonstrated by successful generation of the sinusoidal input describing function models of the final closed-loop system. The procedure developed and the associated software are applied to bank-angle control of an unmanned aerial vehicle (UAV) with a discontinuous nonlinear term. It is shown that performance of a dual-range linear controller is superior to that of both a single-range and a multirange linear controller, and it competes with two other nonlinear controllers.

Non-linear proportional and rate feedback design for highly non-linear systems

A new procedure for the design of non-linear proportional and rate feedback controllers based on a previously proposed concept is developed. The procedure is to obtain the sinusoidal-input describing function models of the system, followed by determination of the amplitude-dependent rate feedback gains. Then, this amplitude-dependent gain function is inverted to obtain the non-linear function describing a non-linear rate feedback gain. Then, the amplitude-dependent proportional gains that desensitize the overall open-loop system are determined via optimization. Finally, the obtained amplitude-dependent proportional gains are inverted to obtain the actual non-linear function describing the required non-linear behaviour of the proportional gain. Bounded-input–bounded-output stability is demonstrated by successful generation of the closed-loop describing function models, and the procedure is applied to a servo problem; the results are compared with four other non-linear controller design procedures that were reported in the open literature.

Design of nonlinear H∞ controllers using describing functions with application to servomechanism

AbstractA systematic procedure for design of nonlinear H∞ controllers for amplitude‐dependent nonlinear plants is presented. The procedure is based on describing function concepts involving sinusoidal input describing function models of the nonlinear plant followed by application of an inverse describing function technique to arrive at nonlinear gains of the H∞ controller. The procedure is applied to a servomechanism problem to demonstrate the typical results that may be achieved. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

CAE Tools for Nonlinear Systems Analysis and Design Based on Sinusoidal-Input Describing Functions

Abstract We report on recent progress in the development of a computer-aided engineering (CAE) environment for nonlinear control system analysis and design based on sinusoidal-input describing function (SIDF) methods. Several major additions have been made to our nonlinear controls CAE software: ACSL macros were developed to allow the generation of SIDF models of nonlinear plants in a manner analogous to that of the SIMNON-based software developed earlier, and MATLAB routines were developed for the analysis of these models and for the design of general nonlinear controllers based on them. This software provides an integrated tool set for treating very general nonlinear systems with no restrictions on system order, number of nonlinearities, configuration, or nonlinearity type. Based on the new software presented here, the use of SIDF-based nonlinear control system analysis and design methods is substantially easier to carry out and more powerful than before

Identification of frequency domain models for nonlinear systems

This paper presents a computer-aided engineering approach for identification of linear models from a set of frequency response data. The approach is based on a new system identification technique. The primary intention of the developed system identification technique and the associated software is to identify linear models for nonlinear systems whose input/output behaviour is characterized by their corresponding sinusoidal-input describing function models. However, the technique may also be applied to identification of linear models from experimental frequency response data. At present, the identification approach and the associated software is limited to single-output, linear, deterministic, and time-invariant systems. A computer-aided engineering environment based on the developed system identification technique has also been developed. The software is developed on a Harris-800 super-minicomputer and a Tektronix 4115B high resolution, raster, and color graphics terminal.

Controller Design For Nonlinear Systems Based on Simultaneous Stabilization Theory and Describing Function Models

A new systematic and algebraic linear control system design procedure for use with highly nonlinear plants is developed. This procedure is based on simultaneous stabilization theory and sinusoidal-input describing function models of the nonlinear plant, and is presently applicable to single-input single-output, time-invariant, deterministic, stable, and continuous-time systems which are representable in standard state-variable differential equation form. Three software utilities to implement the controller design procedure are also outlined. This method and the associated software is applied to a position control problem of the sort encountered in robotics, and the results are compared with those previously obtained using both linear and nonlinear PID control.