Showing papers on "Describing function published in 1990"

The influence of thruster dynamics on underwater vehicle behavior and their incorporation into control system design

Abstract: A nonlinear parametric model of a torque-controlled thruster is developed and experimentally confirmed. The model shows that the thruster behaves like a sluggish nonlinear filter, where the speed of response depends on the commanded thrust level. A quasi-linear analysis which utilizes describing functions shows that the dynamics of the thruster produce a strong bandwidth constraint and a limit cycle, which are both commonly seen in practice. Three forms of compensation are tested, utilizing a hybrid simulation combining an instrumented thruster with a real-time mathematical vehicle model. The first compensator, a linear lead network, is easy to implement and greatly improves performance over the uncompensated system, but does not perform uniformly over the entire operating range. The second compensator, which attempts to cancel the nonlinear effect of the thruster, is effective over the entire operating range but depends on an accurate thruster model. The final compensator, an adaptive sliding controller, is effective over the entire operating range and can compensate for uncertainties or the degradation of the thruster. >

Pulse Modulated Control Synthesis for a Flexible Spacecraft

Abstract: The describing function method is employed for the nonlinear control analysis and design of a flexible spacecraft equipped with pulse modulated reaction jets. The method provides a means of characterizing the pulse modulator in terms of its gain and phase for structural mode limit cycle analysis. Although the describing function method is inherently inexact and is not widely used in practice, a new way of utilizing it for practical control design problems is presented. It is shown that the approximations inherent in the method can be accounted as a modeling uncertainty for the nonlinear control robustness analysis. The pulse modulated control system of the INTELSAT 5 spacecraft is used as an example to illustrate the concept and methodology developed in the gaper. The nonlinear stability margins predicted by the describing function analysis are verified from nonlinear simulations.

Frequency analysis of time-interval-modulated switched networks

Abstract: A Fourier analysis of the time-interval-modulated switched networks is undertaken using time-varying system theory. The result is a linearized describing function approach which determines the small-signal control-to-output (fundamental Fourier component) frequency response. The proposed algorithm is general and exact, and the results are developed in closed form. The intended application area is that of switching DC-to-DC converters, where use of the algorithm is demonstrated in deriving an exact analytical expression for the control-to-output transfer function of pulse-width-modulated converters. Experimental results are presented, verifying the modeling technique. >

Synthesis of Nonlinear Controllers with Rate Feedback via Sinusoidal-Input Describing Function Methods

Abstract: In this paper, we report on recent advances in the design of fully nonlinear controllers for amplitude-sensitive nonlinear plants using sinusoidal-input describing function methods. This work includes the development of a new nonlinear controller synthesis approach that includes derivative action in an inner-loop feedback path (nonlinear rate feedback), and its application to a motor + load model with torque saturation and stiction. This approach is capable of treaing nonlinear systems of a very general nature, with no restrictions as to system order, number of nonlinearities, configuration, or nonlinearity type; additionally, the techniques can be generalized for the design of nonlinear controllers of different structures. The end result is a closed-loop nonlinear control system that is relatively insensitive to reference-input amplitude.

Effects of biodynamic coupling on the human operator model

Abstract: Describing function models of the human operator in manual tracking tasks developed so far, have essentially been restricted to a static cockpit environment using fixed based simulators. This paper addresses the problem whether, and how, the response of the human operator and his associated describing function are influenced by cockpit motion induced by his own control commands. It is suggested, that the motion induced biodynamic stick feedthrough affecting the inner kinesthetic control loop, can be interpreted as a modification of the dynamics of the controlled vehicle. By assuming that the crossover model remains valid under conditions of vehicle motion, an extended analytical model of the human operator describing function is derived. Computer simulation of this analytical model provides reference characteristics of the describing function and remnant noise. A validation of this model, accomplished by extensive dynamical tests on a moving base simulator is described. The excellent match of the analytical and experimental characteristics obtained both in magnitude and shape substantiates the notion of the extended human operator model and suggests that it may contribute to improved overall pilot-aircraft system design.

Frequency stability in resonator-stabilized oscillators

Abstract: A simplified stability analysis of resonator-stabilized oscillators is carried out by using the describing function approach. On this basis a criterion for the evaluation and optimization of the frequency stabilization introduction in an oscillator by a resonating element with a large quality factor is proposed. In particular, a frequency-stabilization index which can be conveniently used in the design of highly stable oscillators is defined. The validity of this performance index has been verified in the design of microwave oscillators using dielectric resonators as frequency-stabilizing elements. >

Systematic approach to linear approximation of non-linear stochastic systems Part 1. Asymptotic expansions

Abstract: A new technique is presented for the analysis of non-linear stochastic systems in which they are represented by a series of non-linearly coupled linear systems. Considerable insight is gained in the approximation of non-linear stochastic systems by linear systems and in the use of describing functions. A measure of the error and second-order correction terms for the linear approximation are obtained. The analysis is extended to the non-zero mean case.

Synthesis of Linear PID Controllers for Nonlinear Multivariable Systems

Abstract: We report on a new, systematic approach to the synthesis of linear multivariable controllers for use with multiple-input/multiple-output (MIMO) nonlinear systems. The approach is based on describing function models of the system followed by numerical optimization in the frequency domain. The end result is a closed-loop feedback system which is insensitive to the amplitude level of the excitation signal, and which approximates a set of user-defined performance criteria with minimum mean square error. We demonstrate the procedure by solving an example problem.

Computer-aided design and analysis for a class of nonlinear systems

Abstract: A description is given of a computer-aided control system design package, NLCON, that supports analysis and design functions for single-input/single-output nonlinear control systems with a single nonlinearity. The single nonlinearity can be one of seven nonlinearities commonly encountered in mechanical systems, and its location within the system is not restricted. The controller design benefits from designer experience, but it is facilitated by the software package, since program operation is highly interactive and the interface is user friendly. The analysis functions supported include sinusoidal-input describing functions and limit-cycle analysis, frequency-response determination, and jump-resonance analysis. Design functions include procedures for compensator modification to modify limit-cycle and jump-resonance behavior. A design advisory is provided for some simple design situations. Examples to illustrate various facets of program operation are presented. >

Non-uniform error-sampled control system

Abstract: A nonuniform ADC (analog-to-digital converter) is shown to be a viable option for a system designed for zero or small steady state errors. The choice of quantization levels is based on the ease of hardware implementation, speed of response, and the idea that, in control systems designed for zero steady state error, it should be sufficient to know the error with certain precision around zero, but that away from zero the same precision need not be necessary. A describing function analysis of the effect of the nonlinearity due to nonuniform quantization is presented. Some simulation results are also included. >

Harmonic Describing Function: Application to Microwave Oscillator's Design

Abstract: A classic description of negative-resistance devices through the first harmonic approach (Describing Function, DF) yields to results that can be insufficiently accurate or even inexact since the harmonic content in the control signals cannot be avoided a priori by the designer, so the basic condition for the application of DF is not fulfilled. In this paper, the use of a new approach, the "Harmonic Describing Function" (HDF), is proposed to study the nonlinear device behaviour without restrictions in the harmonic content of signals involved in the analysis. HDF, evaluated by means of harmonic balance technique, describes the nonlinear device together with the linear network in which it is integrated, so the harmonic content of signals appears as a result of the analysis.

Analysis and reduction of an infinite dimensional chaotic system

Abstract: A singularly perturbed nonlinear time delay system is considered. It is shown that as the system becomes more singular, it evolves through a series of bifurcations into chaotic behavior. Describing functions are used to predict when the initial bifurcations occur. Based on the attractor dimension, reduced-order finite-dimensional models are obtained that qualitatively reproduce the system dynamics. >

Systematic approach to linear approximation of non-linear stochastic systems Part 2. Filtering hypothesis

Abstract: A complete understanding of the accuracy for describing function methods for nonlinear stochastic systems is obtained in terms of a 'filtering hypothesis’. The insight gained is exploited to justify the use of ∥ΦgΦω∥2 as a measure of error in approximating non-linear systems. The non-linear quadratic gaussian (NLQG) design methodology is shown to be consistent with the use of describing functions being reasonably accurate. Various approximation schemes for estimating the accuracy of linearized representations of non-linear systems are developed. It is concluded that a systematic approach to the use of describing functions has been developed and that they constitute a useful approach to a wide range of non-linear systems.

An analytical approach to stability analysis of systems with two nonlinear elements

Abstract: Stability analysis of systems with one nonlinear element is commonly performed by making use of the describing function principle and Nyquist or Nicholas diagrams. A procedure is described whereby the analysis method is extended to systems with two nonlinear elements. The proposed procedure is verified by stability analysis of a position control system with a saturation element and a dead zone as dominant nonlinearities. The limit cycle parameters predicted with the analysis procedure correspond very well with simulation results of the system. Using the results obtained from the analysis, a linear compensator is designed in order to suppress the limit cycle. >

Coupled modal sliding mode control of vibration in flexible structures

Abstract: The sliding mode control algorithm is developed for the coupled modal space control of a flexible structure when the bounds on the system parameters' errors are known. An explicit method to construct the desired sliding hyperplanes for the coupled modal sliding mode control is formulated. A boundary layer is used around each sliding hyperplane to eliminate the chattering phenomenon. Three types of steady-state solutions for the closed-loop system inside the boundary layers are found: the zero solution (origin of the state space), the constant nonzero solution, and the limit cycle. The amplitudes, phase angles, and the frequency of the limit cycle have been estimated by the describing function approach. The modal displacements corresponding to the constant nonzero solution have been obtained analytically. The stability of the zero solution has been examined by the linearized system analysis. Using a flexible tetrahedral truss structure, numerical examples are presented to verify the theoretical analyses.

Predicting the Autonomous Behaviour of Simple Nonlinear Feedback Systems

Abstract: The paper examines in depth two approaches, namely the describing function and Tsypkin methods, for predicting the autonomous behavior of simple nonlinear feedback systems. Both procedures are supported by software which, in the case of the describing function method, allows iteration to the exact limit cycle solution and, for both methods, enables display of resulting limit cycle waveforms. One advantage of the Tsypkin methods, which is applicable primarily to relay systems, is that the exact stability of the limited cycle solution can be found. It is shown how this may be helpful in indicating the possibility of chaotic motion. Several examples are given to show the advantages and limitations of the software implementations of the methods.

Nicely nonlinear modelling: a df approach with application to “linearized” process control

Abstract: Roughly, a model is said to be nicely nonlinear if it is readily linearizable by output feedback. This paper is primarily concerned with “control-oriented nicely nonlinear modelling”; namely, with the problem of finding, within a specific class of nicely nonlinear models, the one that best fits with significant aspects of the (external) behavior of a given plant, relative to a particular control problem defined on it. The approach proposed here is based on the least square fitting, over a given frequency range, of the plant “nominal” describing function. As an application, a simulation study concerning a binary distillation column is presented. Possible extensions and some current investigations are briefly discussed in the concluding remarks.