Showing papers on "Describing function published in 1989"

Actuator limitations on achievable manipulator impedance

Abstract: The effects of actuator torque/speed constraints on achievable manipulator mechanical impedance are discussed. Beginning with a simplified dynamic model, a describing function analysis is used to examine the constraints for linear behavior as well as the effective closed-loop impedance resulting from the actuator nonlinear characteristics. It is shown that the closed-loop manipulator impedance is a blend of two extreme impedances, and the degree of nonlinearity in the actuator controls the proportion of these two components in the overall dynamic response. The extension to multiple-degree-of-freedom manipulators is also discussed, revealing behavioral properties similar to the single-degree-of-freedom case. The relevance of this simple analysis to manipulator system preliminary design and feasibility studies is stressed. >

A feedback model of millimeter wave harmonic oscillators and its application

Abstract: In this paper a feedback model of second harmonic oscillators is developed. By using describing functions of nonlinearity of active devices, the performances of second harmonic oscillators are studied. Frequency dependence of I–V characteristics of active element are taken into account. The ratio of maximum output power of second harmonic to fundamental is given. The maximum harmonic locking bandwidth of injected harmonic oscillator is derived. The theoretical prediction is compared with experimental results.

Feedback Design from the Zero Dynamics Point of View

Abstract: The development of a comprehensive feedback design methodology for nonlinear control systems, similar both in scope and in intuitive appeal to classical automatic control, is a long sought after goal in modern systems and control theory. The classical approaches to the control of finite dimensional linear systems made heavy use, in one or another form, of superposition properties of linear systems. In frequency domain terms, the response to fairly general input signals can be determined in terms of the superposition of the response to input sinusoids of arbitrary frequencies which can be conveniently encoded in terms of a frequency response, or transfer, function. Partial results, using “describing function” methods, similar in spirit to frequency domain methods, have been obtained for certain nonlinear systems; e.g., it is sometimes possible to estimate the frequency of a nonlinear oscillation using describing functions. Nonetheless, straightforward attempts to extend inherently linear properties, such as superposition, apply only in fairly special situations, falling far short of goals such as feedback stabilization, asymptotic tracking or disturbance attenuation for broad classes of nonlinear control systems.

Designing autonomous relay systems with chaotic motion

Abstract: An investigation has been conducted, using simulation, of the existence of chaotic motion in several relay feedback systems. Particular emphasis has been placed on determining when chaotic motion might exist from a knowledge of the unstable limit cycles predicted by the Tsypkin method, the largest amplitude unstable limit cycle being approximately sinusoidal. It is shown that this sinusoidal limit cycle can be calculated quite accurately by the describing function method, and it is found essentially to bound the region of chaotic motion. The chaotic motion gives the appearance of jumps between two or more of the unstable limit cycles found within the region. These unstable limit cycles exhibit two or more oscillations per half period, or spirals if viewed on a phase plane, and their peak amplitudes can be predicted approximately from consideration of the relay switching levels and the DC gain of the linear transfer function. >

Parallel linkage steering for an automated guided vehicle

Abstract: The computer-aided design of suitable control strategies for parallel linkage steering of an experimental automated guided vehicle is described. The microprocessor-based steering control system is modeled, and stability characteristics using proportional control are investigated by means of describing functions. Although the approximation is coarse, the analysis indicates an upper bound for gain and suggests derivative action for improved control. For better accuracy, a flexible and fast simulation program that makes possible an efficient search for optimal control laws for operation of the vehicle under the constant speed mode and slow-down mode is developed. Time-independent control laws are investigated, and solutions are found that give tolerable tracking over 90 degrees arcs of 0.5 m radius. The simulation results show that time-dependent control laws are essential for improved, nonoscillatory tracking. >

Stability and Performance of a Control System with an Intelligent Limiter

Abstract: All mechanical systems have saturation nonlinearity in actuators or in final control elements. When controllers have integral action, reset windup can cause instability as well as make the system performance unsatisfactory. In this study, an intelligent limiter which needs no tuning of parameters is tested with the PDF controller used for control of a second-order plant. This paper presents analysis of the stability of the system using the describing function method and the Nyquist Stability theorem. The improvement of system performance by the limiter is illustrated by computer simulations.

Frequency-domain modeling of nonlinear multivariable systems

Abstract: A frequency-domain technique for input/output (I/O) characterization of stable, multivariable, and highly nonlinear systems (e.g., industrial robots, aerospace vehicles, chemical processes) is presented. We require only that the nonlinear system is representable in state-variable differential equation form, and that it is possible to integrate the system of equations numerically when input signals are sinusoidal. Otherwise, the technique is not restricted with respect to system order, number of nonlinearities, configuration, or nonlinearity type. The I/O characterization technique involves determining the gain and phase of the nonlinear system response to sinusoidal inputs of various excitation amplitudes at a set of user-defined discrete frequencies. These sinusoidal-input describing function (S1DF) models are obtained by exciting all input channels at one time with sinusoids of different but nearly equal frequencies, integrating the dynamic equations of motion over time, and simultaneously performing a Fourier analysis (evaluating Fourier integrals) on the output signals after they are at steady-state. Repeating this procedure for various amplitude-levels of the excitation signal will result in a number of matrix sinusoidal-input describing function I/O models.

Frequency domain relaxation of power system dynamics

Abstract: A highly parallel frequency-domain waveform relaxation method is presented to solve transient dynamics of nonlinear systems. The method attempts to exploit parallelism offered by the frequency-domain matrix equations used to represent and solve the systems. A comparison with various other methods shows that this is a unique method in that it solves transient dynamics of nonlinear systems entirely in the frequency domain while being suitable for large-scale parallel computation. The method treats linear and nonlinear parts of the system individually. Linear blocks are easily solved in parallel in the frequency domain since superposition holds and each frequency can be handled independently of the other. Nonlinear blocks are approximated by polynomials and are solved in parallel using the describing function matrix technique. The frequency-domain-relaxation (FDR) has been used to solve for the transient response of two power system machine models. Results obtained by this method are compared to the results obtained by a conventional time-domain iteration method. It is observed that there is a reduction in the number of parallel steps required by the FDR method, which is dependent on the model and the ratio of data points required by the two methods to give solutions of comparable accuracy. >

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 paper. The nonlinear stability margins predicted by the describing function analysis are verified from nonlinear simulations.

Solutions incorporating magnetic hysteresis to the subharmonic problem in power systems

Abstract: One possibility of taking care of the subharmonics in high speed protection is to filter the fundamental frequency components from the fault transients and use them for fault detection and discrimination. This is correct only if the phases and amplitudes of fundamental components during and post-subharmonic periods remain the same in transient waveforms. An attempt is therefore made to establish this aspect by a detailed study of a typical electrical network presenting subharmonics. The Dual Input Describing Function is adopted to show the subharmonic boundaries and the influence of system parameters on subharmonic oscillations. Magnetic hysteresis is incorporated in the analysis. A fifth order nonlinearity and one-third subharmonic are assumed. The theory could be easily extended to any order of nonlinearity and to any subharmonic frequency and also to a combination of subharmonics. The analysis gives a set of nonlinear algebraic equations. These equations are solved by Newton's technique using ...

Describing function method applied to solution of nonlinear heat conduction equation

Abstract: Describing functions have traditionally been used to obtain the solutions of systems of ordinary differential equations. The describing function concept has been extended to include the non-linear, distributed parameter solid heat conduction equation. A four-step solution algorithm is presented that may be applicable to many classes of nonlinear partial differential equations. As a specific application of the solution technique, the one-dimensional nonlinear transient heat conduction equation in an annular fuel pin is considered. A computer program was written to calculate one-dimensional transient heat conduction in annular cylindrical geometry. It is found that the quasi-linearization used in the describing function method is as accurate as and faster than true linearization methods.

A laboratory exercise on nonlinear compensation using a microprocessor

Abstract: A laboratory exercise on a compensation technique using flexible digital tools is presented. The method for the generation of a special kind of switching-type double-valued characteristic using a microprocessor is described. This characteristic is used to stabilize and improve the system performance of a linear control system. A mathematical analysis leads to conclusions about the determination of the nonlinear element characteristics to improve the system response. An approach based on the describing function (DF) method leads to a compensation technique in the complex plane. The circuit, the theoretical analysis, and experimental results for the compensation of a double integrator plant are presented. >