Showing papers on "Describing function published in 2009"

Process Identification and PID Control

Abstract: Preface. Part One Basics of Process Dynamics. 1 Mathematical Representations of Linear Processes. 1.1 Introduction to Process Control and Identification. 1.2 Properties of Linear Processes. 1.3 Laplace Transform. 1.4 Transfer Function and State-Space Systems. Problems. 2 Simulations. 2.1 Simulating Processes Composed of Differential Equations. 2.2 Simulating Processes Including Time Delay. 2.3 Simulating Closed-Loop Control Systems. 2.4 Useful Numerical Analysis Methods. Problems. 3 Dynamic Behavior of Linear Processes. 3.1 Low-Order Plus Time-Delay Processes. 3.2 Process Reaction Curve Method. 3.3 Poles and Zeroes. 3.4 Block Diagram. 3.5 Frequency Responses. Problems. Part Two Process Control. 4 Proportional-Integral-Derivative Control. 4.1 Structure of Proportional-Integral-Derivative Controllers and Implementation in Computers/Microprocessors. 4.2 Roles of Three Parts of Proportional-Integral-Derivative Controllers. 4.3 Integral Windup. 4.4 Commercial Proportional-Integral-Derivative Controllers. Problems. 5 Proportional-Integral-Derivative Controller Tuning. 5.1 Trial-and-Error Tuning. 5.2 Simple Process Identification Methods. 5.3 Ziegler-Nichols Tuning Rule. 5.4 Internal Model Control Tuning Rule. 5.5 Integral of the Time-Weighted Absolute Value of the Error Tunning Rule for a First-Order Plus Time-Delay Model (ITAE-1). 5.6 Integral of the Time-Weighted Absolute Value of the Error Tunning Rule for a Second-Order Plus Time-Delay Model (ITAE-2). 5.7 Optimal Gain Margin Tuning Rule for an Unstable Second-Order Plus Time-Delay Model (OGM-unstable). 5.8 Model Reduction Method for Proportional-Integral-Derivative Controller Tuning. 5.9 Consideration of Modeling Errors. 5.10 Concluding Remarks. Problems. 6 Dynamic Behavior of Closed-Loop Control Systems. 6.1 Closed-Loop Transfer Function and Characteristic Equation. 6.2 Bode Stability Criterion. 6.3 Nyquist Stability Criterion. 6.4 Gain Margin and Phase Margin. Problems. 7 Enhanced Control Strategies. 7.1 Cascade Control. 7.2 Time-Delay Compensators. 7.3 Gain Scheduling. 7.4 Proportional-Integral-Derivative Control using Internal Feedback Loop. Problems. Part Three Process Identification. 8 Process Identification Methods for Frequency Response Models. 8.1 Fourier Series. 8.2 Frequency Response Analysis and Autotuning. 8.3 Describing Function Analysis. 8.4 Fourier Analysis. 8.5 Modified Fourier Transform. 8.6 Frequency Response Analysis with Integrals. Problems. 9 Process Identification Methods for Continuous-Time Differential Equation Models. 9.1 Identification Methods Using Integral Transforms. 9.2 Prediction Error Identification Method. Problems. 10 Process Identification Methods for Discrete-Time Difference Equation Models. 10.1 Prediction Model: Autoregressive Exogenous Input Model and Output Error Model. 10.2 Prediction Error Identification Method for the Autoregressive Exogenous Input Model. 10.3 Prediction Error Identification Method for the Output Error Model. 10.4 Concluding Remarks. Problems. 11 Model Conversion from Discrete-Time to Continuous-Time Linear Models. 11.1 Transfer Function of Discrete-Time Processes. 11.2 Frequency Responses of Discrete-Time Processes and Model Conversion. Problems. Part Four Process Activation. 12 Relay Feedback Methods. 12.1 Conventional Relay Feedback Methods. 12.2 Relay Feedback Method to Reject Static Disturbances. 12.3 Relay Feedback Method under Nonlinearity and Static Disturbances. 12.4 Relay Feedback Method for a Large Range of Operation. Problems. 13 Modifications of Relay Feedback Methods. 13.1 Process Activation Method Using Pulse Signals. 13.2 Process Activation Method Using Sine Signals. Problems. Appendix Use of Virtual Control System. A.1 Setup of the Virtual Control System. A.2 Examples. Index.

Generating Self-Excited Oscillations via Two-Relay Controller

Abstract: A tool for the design of a self-excited oscillation of a desired amplitude and frequency in linear plants by means of the variable structure control is proposed. An approximate approach based on the describing function method is given, which requires that the mechanical plant should be a low-pass filter - the hypothesis that usually holds when the oscillations are relatively fast. The proposed approach is demonstrated via the controller design and experiments on the Furuta pendulum.

Describing function based methods for predicting chaos in a class of fractional order differential equations

Abstract: This paper deals with two different methods for predicting chaotic dynamics in fractional order differential equations. These methods, which have been previously proposed for detecting chaos in classical integer order systems, are based on using the describing function method. One of these methods is constructed based on Genesio–Tesi conjecture for existence of chaos, and another method is introduced based on Hirai conjecture about occurrence of chaos in a nonlinear system. These methods are restated to use in predicting chaos in a fractional order differential equation of the order between 2 and 3. Numerical simulation results are presented to show the ability of these methods to detect chaos in two fractional order differential equations with quadratic and cubic nonlinearities.

Describing function of two masses with backlash

Abstract: This paper analyzes the dynamical properties of systems with backlash and impact phenomena based on the describing function method. It is shown that this type of nonlinearity can be analyzed in the perspective of the fractional calculus theory. The fractional dynamics is compared with that of standard models.

Modeling of V 2 Current-Mode Control

Abstract: Recently, the V2 type of constant on-time control has been widely used to improve light-load efficiency. In V2 implementation, the nonlinear PWM modulator is much more complicated than usual, since not only is the inductor current information fed back to the modulator, but the capacitor voltage ripple information is also fed back to the modulator. Generally speaking, there is no sub-harmonic oscillation in constant on-time control. However, the delay due to the capacitor ripple results in sub-harmonic oscillation in V2 constant on-time control. So far, there has been no accurate model to predict instability issue due to the capacitor ripple. This paper presents a new modeling approach for V2 constant on-time control. The power stage, the switches and the PWM modulator are treated as a single entity and modeled based the describing function method. The model for the V2 constant on-time control achieved by the new approach can accurately predict sub-harmonic oscillation. Two solutions are discussed to solve the instability issue. The extension of the model to other types of V2 current-mode control is also shown in the paper. Simulation and experimental results are used to verify the proposed model.

Fractional describing function of systems with Coulomb friction

Abstract: This paper studies the describing function (DF) of systems constituted by a mass subjected to nonlinear friction. The friction force is decomposed into two components, namely, the viscous and the Coulomb friction. The system dynamics is analyzed in the DF perspective revealing a fractional-order behavior. The reliability of the DF method is evaluated through the signal harmonic contents.

New Modeling Approach for Current-Mode Control

Abstract: Constant on-time current-mode control has been widely used to improve light-load efficiency, because it can reduce the switching frequency to save switching-related loss. Therefore, an accurate model for constant on-time control is indispensable to system design. However, available models for constant on-time control are unable to provide accurate physical insight and predict the system response, especially the possible sub-harmonic oscillation in the V2 type implementation. This paper introduces a new modeling approach for constant on-time control. The inductor, the switches and the PWM modulator are treated as a single entity and modeled based on the describing function method. The fundamental difference between constant on-time control and peak current-mode control is analyzed using the proposed model. The new modeling approach can be extended to other current-mode controls as well. A simple equivalent circuit representation is proposed for easy understanding of current-mode control. Simulation and experimental results are used to verify the proposed model.

Generating self-excited oscillations for underactuated mechanical systems via two-relay controller

Abstract: A tool for the design of a periodic motion in an underactuated mechanical system via generating a self-excited oscillation of a desired amplitude and frequency by means of the variable structure control is proposed First, an approximate approach based on the describing function method is given, which requires that the mechanical plant should be a linear low-pass filter–the hypothesis that usually holds when the oscillations are relatively fast The method based on the locus of a perturbed relay systems provides an exact model of the oscillations when the plant is linear Finally, the Poincare map's design provides the value of the controller parameters ensuring the locally orbitally stable periodic motions for an arbitrary mechanical plant The proposed approach is shown by the controller design and experiments on the Furuta pendulum

Numerical Evaluation of Limit Cycles of Aeroelastic Systems

Abstract: This paper focuses on the analysis of limit-cycle oscillations of aeroelastic systems with multiple lumped nonlinearities. It aims at a comprehensive investigation capable of identifying limit cycles and their stability. The goal is achieved by using an incremental complexity approach. At the beginning, a solution based on dual-input describing functions is sought, to find both symmetric and asymmetric cycles approximated to their first harmonic. The related stability is investigated afterward by extending the single-input describing function “quasi-static” method. Such an approach is simple and quite similar to well-established existing methods used to evaluate linear flutter conditions directly. If higher harmonics are required, an extended harmonic balance based on a numerical minimization in the frequency domain is adopted and the stability of the computed solutions is then determined by using Floquet theory. The presented approach is applied to several nonlinear aeroelastic examples and validated by comparing stable limit cycles with solutions obtained through direct time marching integrations.

A Method for Estimating the Noise Level of Unstable Combustion Based on the Flame Describing Function

Abstract: Practical combustion is often coupled by the system acoustics and under some conditions the process may become unstable giving rise to oscillations of finite amplitude and to discrete tone noise radiation. The objective of this investigation is to model the nonlinear features of unstable combustion to predict limit cycle amplitudes and thus get an estimation of the radiated noise level under such regimes of operation. The present study is based on an analysis derived in a companion paper (Noiray et al. (2008)) which contains a complete nonlinear description of the thermo-acoustic coupling encountered in an unconfined reaction zone layout. Theoretical predictions are successfully compared with a set of measurements. It is shown that nonlinear phenomena such as frequency shifts, triggering or hysteresis may be suitably predicted by an analytical model based on the flame describing function (FDF). The objective of the present paper is to make use of this method to investigate nonlinear acoustic-combustion dy...

The Role of Nonlinear Acoustic Boundary Conditions in Combustion/Acoustic Coupled Instabilities

Abstract: Triggering, frequency shifting, mode switching and hysteresis are commonly encountered during self-sustained oscillations in combustors. These mechanisms cannot be anticipated from classical linear stability analysis and the nonlinear flame response to incident flow perturbations is often invoked to interpret these features. However, the flame may not be solely responsible for nonlinearities. Recent studies indicate that interactions with boundaries can be influenced by the perturbation level and that this needs to be considered. The nonlinear response of acoustic boundary conditions to flow perturbations is here exemplified in two configurations which typify practical applications. The first corresponds to a perforated plate backed by a cavity conveying a bias flow and the second corresponds to a set of flames stabilized at a burner outlet. These systems are submitted to acoustic perturbations of increasing amplitudes as can be encountered during unstable operation. It shown that these terminations can be characterized by an impedance featuring an amplitude dependent response. The classical linear impedance Z(ω ) is then replaced by its nonlinear counterpart an Impedance Describing Function (IDF), which depends on the perturbation level input Z(ω , |p′ | or |u′ |). Using this concept, it is shown that the passive perforated plate optimized to damp instabilities of small amplitudes may eventually loose its properties when submitted to large sound pressure levels and that the flame response shifts when the amplitude of incoming flow perturbations is amplified. The influence of these nonlinear elements on the stability of a generic burner is then examined using a methodology which extends a previous analysis based on the Flame Describing Function (FDF) to systems with complex flow interactions at the boundaries.Copyright © 2009 by ASME

Non linear and Robust Stability Analysis for ATV Rendezvous Control

Abstract: ATV Rendezvous controller shall insure a high performance level as well as a high safety level. This study presents the design and the validation of this controller using baseline analysis tools and modern robust analysis tools, namely μ-analysis and discrete time describing function. In this paper, emphasis is put on validation methodology. The first mean of validation relies on linear and single axis stability analysis. Then we have also used a novel robust approach based on worst case approach coupled with multi axes analyses that is detailed here. Finally, non linearity effects are analyzed to prevent from unexpected or unstable limit cycle. One major innovation is that non linear methodology takes into account discrete time constraint. Furthermore, one way to prevent from unstable initial conditions was found based on use of saturated feedback in the controller and applied to all ATV attitude and position control functions.

A new statistical approach for nonlinear analysis of limit cycle amplifiers

Abstract: Limit cycle loops are becoming popular due to their potential for higher efficiency and linearity, as data convertors and/or amplifiers. Therefore the analysis of this architecture in different aspects is essential. In this paper a statistical orthogonalization based approach for analysis of limit cycle amplifiers, excited by a Gaussian random signal is developed and the advantages over conventional methods are discussed. The validity of the proposed approach is verified by comparison between the circuit envelope simulation and the results achieved from the proposed algorithm.

Elimination of limit cycles in HVAC systems using the describing function method

Abstract: In heating, ventilating and air conditioning systems, the presence of undesirable limit cycles is a common problem. This paper outlines the application of the well known describing function method to the analysis and synthesis of a temperature control system used for air-conditioning an office room. PI-controllers are most commonly used in this field of application. In the present case, an existing standard controller leads to limit cycles around the desired room temperature. The so-called inward approach in combination with the describing function method proves to be an innovative tool for eliminating oscillations while preserving desired performance specifications.

Limit cycles induced in type-1 linear systems with PID-type of relay feedback

Abstract: This article investigates the limit cycles within type-1 linear systems under PID-type of relay feedback. The problem is generalised from the identification of friction models of servo mechanical systems via limit-cycle experiments under dual-channel relay feedback. Locations of limit cycles are given so that the exact durations between two consecutive switchings of relays can be determined via numerical computation. After this, local stability of limit cycles can be checked via the Jacobian of Poincare map. Examples are analysed using proposed theorems.

A study of oscillation for signal stabilization of nonlinear system

Abstract: The phenomena of desynchronization, synchronization, and forced oscillation has been investigation using describing function theory for a two input and two output nonlinear system containing saturation-type nonlinearities and subjected to high-frequency deterministic signal for the purpose of limit cycle quenching. The analytical results have been compared with the results of digital simulation/Matlab-Simulink for a typical example varying the nonlinear element.

A new approach for nonlinear metric estimation of limit cycle amplifiers

Abstract: Limit Cycle amplifiers are promising solutions to combine high linearity and efficiency, thanks to the efficient switch mode amplifier inside. Therefore an insightful and efficient analysis method is required for these systems. In this paper, a novel Hermite series based statistical approach for nonlinear analysis of the limit cycle amplifier system with real Gaussian excitation is presented. The proposed algorithm gives both in-band and out of band nonlinear distortions through them same approach. The approach is applied to an 802.11 g OFDM signal to calculate EVM, ACPR and output power. The method is validated by good agreement between the extracted results and ADS circuit envelope results.

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

Abstract: A 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

Describing function analysis of Dahl model friction

Abstract: Mechanical systems subject to friction hysteresis exhibit amplitude dependent frequency responses. This paper derives the describing function (DF) of the Dahl friction model subject to a sinusoidal input. This DF is useful for analysis and modeling of systems subject to friction hysteresis from frequency response data.

Inducing oscillations in an inertia wheel pendulum via two-relays controller: Theory and experiments

Abstract: A tool for generating a self-excited oscillations for an inertia wheel pendulum by means of a variable structure controller is proposed. The original system is transformed into the normal form for exact linearization. The design procedure, based on Describing Function (DF) method, allows for finding the explicit expressions of the two-relays controller gain parameters in terms of the desired frequency and amplitude. Necessary condition for orbital asymptotic stability of the output of the exactly linearized system is derived. Performance issues of the system with self-excited oscillations are validated with experiments.

Tracking and decoupling of multivariable non-linear systems

Abstract: This paper presents a new systematic procedure for decoupling and command tracking of multivariable, non-linear, and unstable systems. The design methodology is based on stabilisation of the multivariable system followed by generating its describing function models; two algebraic procedures for decoupling and tracking are used. Finally, the design must be verified by a non-linear simulation to make sure that approximations made during design are valid. The major outcome of this research is that controllers of a general class of non-linear systems may be designed taking into account the amplitude dependency features of that system. This amplitude dependency is the most important characteristic of a non-linear system. This is the first paper in the area of simultaneous decoupling and signal tracking of multivariable non-linear systems that is based on the application of the describing function technique and two other algebraic techniques. In fact, this is the first work in the area of decoupling of general ...

Characterization and Attenuation of Sandwiched Deadband Problem Using Describing Function Analysis and Application to Electrohydraulic Systems Controlled by Closed-Center Valves

Abstract: Unlike input deadband, the sandwiched deadband between actuator and plant dynamics is very difficult to be explicitly compensated for due to the proceeding actuator dynamics whose effect may not be negligible. The paper presents a practical way to overcome the design conservativeness of existing methods in dealing with sandwiched deadband. Specifically, a describing function based nonlinear analysis method is proposed to characterize the effect of the sandwiched deadband on the stability and performance of the overall closed-loop system. The analysis results can be used to determine the highest closed-loop bandwidth that can be achieved without inducing residual limit cycles and instability. Optimal controller parameters can then be found to maximize the achievable closed-loop control performance. The technique is applied to an electrohydraulic system controlled by closed-center valves and a nonlinear feedback controller. Simulation results showed severe oscillations as the feedback control gains are increased to the predicted threshold values. Comparative experimental results also showed the effectiveness of the proposed method in reducing the conservativeness of traditional design and the improved closed-loop control performance in implementation.

Design of a Fuzzy Gain Scheduling Controller Having Input Saturation: A Comparative Study

Abstract: A fuzzy gain scheduling proportional controllers that exhibits the improved performance than the conventional linear fixed gain controller having input saturation is proposed in this paper. We proposed an adaptive anti-windup control system in which the gain of a proportional controller is dynamically tuned by a fuzzy PD system. The methodology employed in the analysis is based on the describing function (DF), which is determined the range of the universal discourse for the output of the fuzzy PD system. Other approaches are the linear quadratic regulator (LQR) method and combined back-calculation and generalized conditioning anti-windup. In this paper, a comparison study of these three approaches for design anti-windup controller is made. The simulation results show the differences and confirm the availability of the proposed design approaches.

A describing function method for evaluating the statistics of the harmonics scattered from a non-linear device in a mode stirred chamber

Abstract: The statistics of the received radiation scattered from a non-linear device within a reverberation chamber are examined by measurement and Monte-Carlo simulation. A Describing Function formulation is used for the non-linear analysis that enables the complete dynamic range of the non-linear process to be described. The work is a precursor to developments in diagnostic immunity measurements for digital electronic hardware.

Analysis of adaptive interference cancellation system with limiter

Abstract: The interference cancellation ratio (ICR) is influenced by the zero offsets of the devices in the weight branches. The gains in weight branches are moved to error feedback loop to guarantee ICR without influencing the convergence speed. The limiter is used to protect the multipliers thus the system becomes a nonlinear system. Through the describing function method and simulation, the stability, dynamic characteristics and steady-state characteristics are analyzed. With limiter, the stability of the system is uninfluenced. The steady-state characteristics are related with the parameters of limiter, the gain, the time constant of the low-pass and the amplitude of the signal. The convergence speed will decrease under the conditions of under damping and critical damping. If system is over-damped, the initial gain determines the growth of convergence speed. In this paper, a criterion to judge whether steady-state characteristics varies and the proper rule to increase the convergence speed with over damping are obtained.

The performance of different dither signals in nonlinear systems

Abstract: Dither is a high frequency signal which is introduced into a nonlinear system with the object of augmenting stability, quenching undesirable limit cycle, and altering the apparent nonlinear element characteristics. In this paper, we investigate the performance of various high frequency dither signals with different wave forms in nonlinear feedback systems. Meanwhile, an analog experimental device is set up to simulate the nonlinear systems, which consists of dither and a nonlinear element such as a relay or a limiter. A describing function and an equivalent nonlinearity method are used to evaluate the dither performance. The performance of various dither signals having a certain wave form is examined through these studies, and the results could evaluate effectiveness of these dither-injection techniques into nonlinear feedback systems.