Showing papers on "Describing function published in 1994"

Quantitative design of a class of nonlinear systems with parameter uncertainty

Abstract: This paper considers the case in which a linear time-invariant (LTI) but uncertain plant suffers from nonlinearities y=n(x) which can be expressed as y=Kn+η(x), |η(x)|≤M, with K, a possibly uncertain scalar. This covers a large and very important class of nonlinearities encountered in practice such as friction, backlash, dead zone and quantization. Quantitative design techniques are presented for this class for the satisfaction of specifications. Special attention is paid to the avoidance of limit cycles using describing function theory, although the design method is also amenable of application using other stability criteria such as the circle criteria. Numerical examples are developed illustrating the design procedure.

Disk drive pivot nonlinearity modeling. I. Frequency domain

Abstract: This paper describes studies done at HP Labs on the actuator pivot bearing nonlinearity of a small disk drive. The nonlinear frictional behavior of the pivot bearing varies with actuator position, from drive to drive, and with time and temperature. This nonlinear behavior has made the traditional linear disk drive models inadequate. The paper shows how the swept-sine/describing function method was used to develop a nonlinear model of the pivot. This model departs from the classical friction models, but does a good job of matching laboratory frequency domain measurements.

PID control in the presence of static friction: exact and describing function analysis

Abstract: PID position control in the presence of static friction is shown to lead to a frictional limit cycle for all stabilizing combinations of P, I and D parameters. An idealized Coulomb+static friction model, dimensional analysis and the study of all possible solutions to the differential equations are used to achieve analytic results for this nonlinear problem. Three theorems establishing the predictions of the describing function are also presented.

Extension of the covariance control principle to nonlinear stochastic systems

Abstract: An approach for the constrained variance control of nonlinear stochastic systems, which uses the theory of covariance control of linear stochastic systems, is proposed. The nonlinear stochastic systems are linearised by the use of describing functions. Two cases of nonlinear systems are considered: zero-mean nonlinear stochastic systems and nonzero-mean nonlinear stochastic systems. The application of this approach to a position servomechanism is illustrated by a numerical example.

Spacecraft attitude control system performance using pulse-width pulse-frequency modulated thrusters

Abstract: : Many current satellites employ on-off thrusters to accomplish attitude control tasks which may include initial acquisition, rotational maneuvers, and on-orbit stabilization. This work shows that the use of pulse-width pulse-frequency (PWPF)-modulated thrusters provides several important advantages over conventional bang- bang thruster control methods, including less thruster activity and closer-to-linear actuation. The PWPF modulator is implemented in simulations using the Matrix/System build software package. Simulations assuming a rigid spacecraft are first performed to compare the performance of the PWPF-modulated thrust controller with that of conventional bang-bang and time-optimal bang-bang controllers. The discussion is then extended to the case of a spacecraft with structural flexibility, as is encountered quite often in three-axis stabilized vehicles with large fold-out solar arrays. Simulations for comparison of the controllers are performed using the flexible spacecraft dynamics model. The control loop design in the presence of flexibility and possible interaction with the PWPF modulator nonlinearity are addressed. Using a describing function model of the modulator, stability margin with respect to the structural mode limit cycle is predicted. Simulations are then conducted to verify the predicted stability margin.

Analysis of an electrohydraulic servo system subject to stribeck-type friction

Abstract: The main objective of this paper is to study the phenomena of the electrohydraulic servo system under the influence of the Stribeck-type friction. Owing to the nonlinear nature of this friction, a sustained oscillation, or limit cycle, might appear in the system behaviour. The system is first divided into two parts, the linear and the nonlinear. For the nonlinear Stribeck-type friction, a flexible model is proposed to represent the whole friction family, and the describing function of this nonlinear friction is generated. The system characteristic equation is established by correlating the frequency response of the linear part and the describing function of the nonlinear part. The existence and stability of the limit cycle are predicted based on the characteristic equation. Finally, by using the sinusoidal, step and ramp input, respectively, the numerical simulation is adopted to observe the system response. All predicted limit cycles can be verified by the simulation results, and the effects of stable an...

Comparison of small-signal modelling techniques for the series-parallel resonant converter

Abstract: Three techniques are employed to derive small-signal models for the series-parallel resonant power converter over a range of operating conditions which includes the discontinuous capacitor voltage mode. A comparison of the results with experimental data is undertaken to quantify the limitations and accuracy of each technique. The three techniques are: sampled-data; extended describing function; and an approximate method.

Describing function-based analysis of a nonlinear hydraulic transmission line

Abstract: This paper presents an analytical method of predicting and eliminating limit cycle conditions in hydraulic drive systems, where backlash in the actuator seal is the dominant nonlinearity. It is shown that limit cycle conditions are a strong function of certain plant parameters. The elimination of limit cycle conditions may thus be achieved by the alteration of certain inherent plant parameters. Fluid flow in a hydraulic pipe is modeled using transmission line theory. The sinusoidal-input describing function is used to indicate which values of the plant parameters would result in limit cycle conditions. The analytical results were verified on an experimental test bench. >

Phase shifter: a nonlinear circuit for direct measurement of phase margin in feedback control systems

Abstract: A novel nonlinear circuit called a "Phase Shifter" is introduced which accurately measures the phase margin of a linear system operating in a feedback control loop. The measurement is made by inserting the phase shifter in the control loop, causing it to undergo self-sustained oscillations (limit cycle) of relatively low amplitude. It is shown that sustained oscillations occur at the gain crossover frequency, and the phase margin is equal to the phase lag of the inserted circuit. A complete analysis of the amplitude and frequency of oscillation, which employs the classical sinusoidal-input describing function, is also given and the stability of the limit cycle and its neighborhood, using the theory of integral manifolds, is included. >

H/sub /spl infin// control analysis and design for nonlinear systems

Abstract: The /spl mu/ analysis method is applied to a class of multivariable separable autonomous nonlinear feedback systems using sinusoidal input describing function (SIDF). Some properties of the /sup /spl prop//-norm and structured singular values (SSVs) are shown to hold for the new quasilinear system. The model uncertainty induced by the SIDF approximation is incorporated in the control design as an input multiplicative perturbation. The condition for the absence of limit cycle is derived. The model is then used to design a bank of H/sub /spl infin// controllers with a simple switching scheme to select the appropriate controller for any operating range. The simulation results are presented to illustrate the performance of the controller.

Air/fuel ratio controller design using the describing function approach

Abstract: In this paper, a nonlinear fuel-injection controller is designed using the describing function method The controller is designed to achieve a commanded air-to-fuel ratio within 145 to 149, which guarantees optimum operational characteristics The controller is robust to changing throttle maneuvers, modeling errors and time delay Using the describing function method, simple analog and digital controllers are designed to meet the control objectives The simplicity of the controllers makes them particularly suitable for practical implementation using hardware or firmware

Stability of fuzzy control systems

Abstract: Stability of a class of fuzzy control systems is analyzed by replacing the fuzzy controller with an appropriate describing function. Nonlinear analytical expressions for fuzzy controllers are derived. These expressions are numerically approximated by describing functions. The Nyquist stability criterion is used to determine the stability of the fuzzy control system. Numerical examples illustrate the methodology. >

The describing function perturbation method for the dominant behaviour analysis of adiabatic negative-conductance oscillators

Abstract: Negative-conductance oscillators are termed adiabatic when their transient operation signal can be represented as a narrowband-modulated Fourier series with a finite number only of dominant components, including the DC one. the method presented here allows one to perform the complete analysis of such oscillators in a rigorous shortened manner. Indeed, it merges the approach of the describing function technique into the classical frame of perturbation theory, so as to replace the oscillator model in the instantaneous variables with a simplified one in the dominant variables, i.e. the complex envelopes of the dominant components. More precisely, the starting model comprises the input impedance and the open-circuit voltage of the Thevenin equivalent one-port of the linear network in the complex frequency domain and the voltage/current characteristic of the non-linear resistor in the time domain. the final model instead comprises as many such triads as there are dominant components. Each triad is made up of an input impedance and an open-circuit voltage in the baseband complex frequency domain and a voltage/current describing characteristic in the slow time domain. Standard circuit analysis tools are then used to obtain the equations of the dominant dynamics, steady state and dynamical stability of the oscillator. To facilitate reading the paper, as soon as a new formula is proposed or derived, the result of its application to the well-known van der Pol oscillator is also provided. In the end, to highlight the simplicity and efficiency of the method, a much more complex case (an eighth-order oscillator with three dominant components) is analysed in detail.

New topologies of switching mode oscillators and their applications

Abstract: Switching-mode-oscillators constitute a unique class in the field of oscillators and also in the field of switching-mode-invertors. The paper treats relatively new oscillators in the class. They can be entitled as parallel resonant switching mode oscillators. Furthermore, flexible methods of triggering their MOSFET switches through ring transformers are discussed. Analysis of the system's useful limit cycle dynamics is helped by resorting to the describing function method.

Frequency-domain simulation of power electronics circuits by using describing functions in a modified nodal approach

Abstract: A new approach for AC analysis of switching-mode power converters is presented. It is based on the use of two concepts: the bifrequency transfer function, which relates the input and output spectrums of the power electronics circuit, and the describing function. In order to calculate the input-to-output or control-to-output frequency response, small sinusoidal perturbations in either the line voltage or the control variable are considered. The fundamental component of the output variable at the perturbation's frequency is determined by using the zeroth order term of the Fourier transform of the impulse response, allowing then for the deduction of the line and control describing functions. No additional approximations (as neglecting higher order terms, etc) are required. The method is applied for the simulation of a switched-capacitor-based DC-to-DC converter.