Showing papers on "Describing function published in 2010"

New Modeling Approach and Equivalent Circuit Representation 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 or predict system response very well. This paper introduces a new modeling approach for constant on-time control. The inductor, the switches, and the pulsewidth-modulated modulator are treated as a single entity and modeled based on the describing function method. The fundamental difference between constant on-time control and constant-frequency peak-current-mode control is analyzed through the proposed model. This proposed modeling method can be easily extended to other current-mode controls, including V2 controls. A simple equivalent circuit representation is proposed for the sake of easy understanding and simulation of current-mode controls. Simulation and experimental results are used to verify the proposed model.

Modeling of ${\rm V}^{2}$ Current-Mode Control

Abstract: Recently, V2 constant on-time control has been widely used to improve light-load efficiency. In the 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 also the capacitor voltage ripple information. Generally speaking, there is no subharmonic oscillation in constant on-time control. However, the delay due to the capacitor ripple results in subharmonic oscillation in V2 constant on-time control. So far, there has been no accurate model to predict the instability issue due to the capacitor ripple. This paper presents a new modeling approach for the V2 constant on-time control. The power stage, the switches, and the PWM modulator are treated as a single entity and modeled based on the describing function method. The model for the V2 constant on-time control achieved by the new approach can accurately predict subharmonic oscillations. Two solutions are discussed to solve the instability issue. The extension of the model to other types of V2 current-mode control and multiphase application is also shown in this paper. Simulation and experimental results verify the proposed model.

Multiband Hysteresis Modulation and Switching Characterization for Sliding-Mode-Controlled Cascaded Multilevel Inverter

Abstract: In this paper, a generalized multiband hysteresis modulation and its characterization have been proposed for the sliding-mode control of cascaded H-bridge multilevel-inverter (CHBMLI)-controlled systems A frequency-domain method is proposed for the determination of net hysteresis bandwidth for a given desired maximum switching frequency of the inverter The switching transition concept of Tsypkin's method and the describing function of nonlinear relay have been used for the derivation of results A hierarchical switching algorithm has been suggested for the modular cells of the cascaded multilevel inverter The hierarchy of each cell is swapped sequentially to provide the self-balancing capability in case the dc-link voltage is supported by the capacitors The simulation and experimental verification of the derived results are provided through a single-phase distribution static compensator (DSTATCOM) model The application in the three-phase system has been shown through simulation studies on a 33-kV distribution-system compensation using DSTATCOM Verification on both single- and three-phase systems is obtained using a five-level cascaded-multilevel-inverter topology

Modeling and Compensation of Ripples and Friction in Permanent-Magnet Linear Motor Using a Hysteretic Relay

Abstract: The tracking performance of a motion system based on a permanent-magnet linear motor (PMLM) is greatly affected by nonlinearities present, such as force ripples and frictions. Although various identification and compensation schemes haven been reported for the PMLM, to the authors' best knowledge, no direct and unified modeling method for force ripples and friction is available till now. This paper proposes a new method to identify various linear and nonlinear parameters in PMLM, using a hysteretic relay feedback. Dual-input describing functions are imported to leverage on the biased limit cycles generated by even nonlinearities due to force ripple. The explicit formulae, derived from the harmonic balance condition, enable direct computation of model parameters with a minimum number of relay experiments. Simulation results and real-time experiments will be presented to verify the practical appeal of proposed method in precision motion control.

Algorithm for constructing counterexamples to the Kalman problem

Abstract: The method of harmonic linearization and describing function method, numerical methods, and the applied bifurcation theory together discover new opportunities for analysis of hidden periodic oscillations (with basin of attraction which does not contain neighborhoods of equilibria) of control systems. In the present paper new algorithms for construction of counterexamples to Aizerman's conjecture and Kalman's conjecture is suggested.

Describing Function Analysis of Limit Cycles in a Multiple Flame Combustor

Abstract: A recently developed nonlinear flame describing function (FDF) is used to analyze combustion instabilities in a system where the feeding manifold has a variable size and where the flame is confined by quartz tubes of variable length. Self-sustained combustion oscillations are observed when the geometry is changed. The regimes of oscillation are characterized at the limit cycle and also during the onset of oscillations. The theoretical predictions of the oscillation frequencies and levels are obtained using the FDF. This generalizes the concept of flame transfer function by including dependence on the frequency and level of oscillation. Predictions are compared with experimental results for two different lengths of the confinement tube. These results are, in turn, used to predict most of the experimentally observed phenomena and in particular, the correct oscillation levels and frequencies at limit cycles.

Analysis of a fractional order Van der Pol-like oscillator via describing function method

Abstract: In this paper, the behavior of a fractional order Van der Pol-like oscillator is investigated using a describing function method. A parametric function for the boundary between oscillatory and nonoscillatory regions of this system is extracted. The analytical results are evaluated by numerical simulations which demonstrate sufficient reliability of the proposed analyzing method.

Design oriented model for constant on-time V 2 control

Abstract: V 2 type constant on-time control is a practical implementation for current mode control. Equivalent series resistor (ESR) of the output capacitors is used as the sensing resistor for inductor current. Along with inductor current information, capacitor voltage is also fed back to the modulator. Previously CPES proposed a model for V2 control based on describing function. Although the model is very accurate, the derivation process is complicated. This paper tries to propose a simplified but less accurate model for practical design purpose. The new modeling method separates current feedback information and capacitor voltage feedback information. The analytical results are verified using Simplis simulation.

Nonlinearities in Industrial motion stages - detection and classification

Abstract: Detection and classification of nonlinearities in motion systems becomes of increasing importance with high demands on (closed loop) performance. In this paper two methods are compared that aim to measure both the linearized dynamics and the influence of nonlinearities. First, a broadband signal is used to measure a linear approximation of the systems dynamics. This method uses multisine signals with identical amplitude spectrum, but randomly distributed phases. Averaging over multiple periodic responses to the same signal and over multiple realizations of the random phase multisine allows the computation of the level of nonlinearities and external disturbances separately. This yields both a linear approximation of the systems dynamics and the amount of nonlinear ‘disturbance’ as a function of frequency. Second, single sine based measurements are used to measure the Higher Order Sinusoidal Input Describing Functions (HOSIDF) of the system under test. HOSIDFs describe the response of the system by describing not only the ‘direct’ response (gain and phase shift) of the system at the input frequency, but by describing the response at higher harmonics of the input frequency as well. This yields a quantitative measure of the power generated by nonlinearities at harmonics of the input frequency as a function of this frequency and the signal amplitude. In the paper these methods are utilized to acquire a non-parametric model for an industrial high precision stage. The effects of and sources for nonlinear influences are discussed for this particular case as well.

Characterization and quenching of friction-induced limit cycles of electro-hydraulic servovalve control systems with transport delay.

Abstract: This paper develops a systematic and straightforward methodology to characterize and quench the friction-induced limit cycle conditions in electro-hydraulic servovalve control systems with transport delay in the transmission line. The nonlinear friction characteristic is linearized by using its corresponding describing function. The delay time in the transmission line, which could accelerate the generation of limit cycles is particularly considered. The stability equation method together with parameter plane method provides a useful tool for the establishment of necessary conditions to sustain a limit cycle directly in the constructed controller coefficient plane. Also, the stable region, the unstable region, and the limit cycle region are identified in the parameter plane. The parameter plane characterizes a clear relationship between limit cycle amplitude, frequency, transport delay, and the controller coefficients to be designed. The stability of the predicted limit cycle is checked by plotting stability curves. The stability of the system is examined when the viscous gain changes with respect to the temperature of the working fluid. A feasible stable region is characterized in the parameter plane to allow a flexible choice of controller gains. The robust prevention of limit cycle is achieved by selecting controller gains from the asymptotic stability region. The predicted results are verified by simulations. It is seen that the friction-induced limit cycles can be effectively predicted, removed, and quenched via the design of the compensator even in the case of viscous gain and delay time variations unconditionally.

Parametric Study for Dynamics of Spacecraft with Local Nonlinearities

Abstract: Various kinds of connectors existing in most spacecraft are usually nonlinear and could strongly affect the dynamic characteristics of the spacecraft. For dynamic analysis, the spacecraft are generally idealized as finite-element models that often have huge numbers of degrees of freedom, while their connectors are spatially localized. In addition to forced harmonic response, the influences of the connectors' nonlinear parameters and/or the excitation levels on the response are also significant among major design considerations. However, it is computationally expensive to repetitively perform the analysis and computation for studying the effect of modifying nonlinear parameters and/or changing excitation levels with a direct method, especially for large-scale structures. Accordingly, an approach based on the pseudoarclength continuation scheme, the describing function, and linear receptance data is developed in the present paper. With linear receptance data, the computational efficiency of the pseudoarclength continuation scheme can be significantly enhanced and only associated with nonlinear degrees of freedom that usually constitute a small part of large-scale structures with local nonlinearities. A finite-element model of a satellite with nonlinear connectors is used to study the relationship among the response of the satellite's payloads, the excitation level, and the connectors' physical parameters.

Prediction of Thermoacoustic Limit Cycles during Premixed Combustion using the Modified Galerkin Approach

Abstract: The present work extends the Generalized Instability Model, which solves the pressure wave equation via the modified Galerkin method, to include nonlinear heat release models in the form of Flame Describing Functions as well as acoustic losses at the boundaries. These extensions are required for limit cycle prediction in premixed combustors. The governing equations are presented in detail followed by the implementation of two Flame Describing Functions obtained experimentally from a bluff body burner and a swirl stabilized burner. For both test cases, multiple operating points are analyzed, with the model predicting correctly the stability in all cases. Good correspondence between modeled and measured limit cycle amplitudes for the unstable modes is also obtained. However, a parametric study shows that the predicted limit cycle amplitude has a strong dependence on the heat release model and on the acoustic losses present in the system. This indicates that both the energy gain due the combustion process and the acoustic losses through the boundaries need to be determined accurately for limit cycle prediction.

The memristor as controller

Abstract: Recently a fourth circuit element, predicted four decades ago, has been discovered by HP researchers. This device, known as the memristor, is basically a resistance with memory that can be used as a tunable gain in control systems. In this paper the memristor experimental characteristic is modeled with a describing function and used to predict oscillations in closed loop systems with linear plants.

Nonlinear Controller Synthesis for Single-Input Multiple-Output Systems with Application to Cruise Missile

Abstract: The purpose of this paper is to present a new systematic procedure for synthesis of nonlinear controllers for single-input multiple-output systems. The procedure is based on several describing function models of the plant, and the nonlinear gains are obtained via an inverse describing function technique. The procedure and associated software are applied to control the bank angle of a cruise missile. The results are compared with a linear controller and another nonlinear controller that was previously reported in the open literature; it is shown that the developed procedure has resulted in a less conservative design (i.e., performance is not sacrificed to assure absolute stability).

A method to select correct stimuli levels for S-functions behavioral model extraction

Abstract: S-functions and other frequency-domain behavioral models based on describing functions rely on the linearization of the response of a nonlinear RF or microwave component around a set of large-signal operating points. To assure the validity of the linearization principle, and thus the validity of the model, one has to select appropriate stimuli levels during the experiments which are performed to extract the model parameters. In this work we propose a formal procedure to identify the correct small-signal power levels using a system identification approach, i.e. taking the uncertainty of the measured data into account. The method is based on the analysis of the cost function of an errors-in-variables (EIV) estimator obtained for simple models.

Using Describing Functions for Limit-Cycle-Oscillation Analysis Applied to Aeroservoelastic Models with Free-Play

Abstract: Nonlinearities in aeroservoelastic systems have been shown to be a source of Limit Cycle Oscillations (LCO). Aerodynamic control surfaces and actuators can have excessive free- play in connecting joints and hinges. This nonlinearity has the potential to cause LCO associated with aeroservoelastic vibrations, impacting handling qualities, ride quality and can cause structural fatigue. Due to the negative impacts of LCO, absolute free-play limits have been established by the Joint Services Guidance Military specification. These military specifications have also been adopted by the FAA for commercial aircraft. This stringent requirement can be overly conservative and is very difficult and costly to meet. Analytical tools using describing functions have been developed for establishing free-play limits on aeroservoelastic aircraft systems without adverse consequences, justifying relaxation or validating the conservative requirements. It is shown that LCO frequency and amplitude can be accurately estimated using describing functions and an analytical model of an aeroservoelastic system with free-play. Novel analysis techniques are also demonstrated, whereby the free-play occurrence and limit cycle behavior can be visualized using time- varying frequency content. Validation of the methods is done with an aeroservoelastic stabilator control surface model via nonlinear simulation with actual free-play limits.

Input impedance modeling of multipulse rectifiers by double-fourier series method

Abstract: Rectifier input impedance models are required for stability analysis of ac power systems with significant amount of rectification loads. Such stability analysis is particularly important for mobile and autonomous systems, such as aircraft and ship power systems, where high-power, multipulse rectifiers often dominates the loads. Averaging techniques can be applied to develop input impedance models for PWM rectifiers. To model the input impedance of line-frequency rectifiers, a direct linearization technique was developed recently and has been applied to model direct multipulse rectifiers without using inter-phase transformers. This paper extends the previous work and uses double-Fourier series method to develop describing (mapping) functions for the rectifier bridge. The double-Fourier method overcomes the limitations of conventional Fourier analysis which requires the perturbation frequency to be commensurable with the line fundamental frequency. The method is also applied to model the input impedance of multipulse rectifiers with interphase transformers. Analytical input impedance models are presented and validated by numerical simulation and experimental results.

Nonlinear Interactions of Multiple Linearly Unstable Thermoacoustic Modes

Abstract: We investigate the dynamics of thermoacoustic systems with multiple linearly unstable modes. If a linear analysis reveals more than one mode with positive growth rate, nonlinear methods have to be used to determine the existence and stability of steady-state oscillations. One possible way to engage this problem is a first-order harmonic balance approach based on describing function representations for the flame response. In contrast to the case of a single unstable mode, the nonlinearity output to multiple sinusoidal components with different frequencies and amplitudes has to be known. Based on this approach, we present conditions for the existence and stability of single- or multi-mode steady-state oscillations. We apply this method to a thermoacoustic model system having two linearly unstable modes. By varying one of the system parameters, we find stable and unstable single-mode steady-states as well as unstable simultaneous oscillations. Associated with the stability of the single-mode limit cycles, we...

Multiharmonic Response Analysis of Systems With Local Nonlinearities Based on Describing Functions and Linear Receptance

Abstract: Direct time integration methods are usually applied to determine the dynamic response of systems with local nonlinearities. Nevertheless, these methods are computationally expensive to predict the steady state response. To significantly reduce the computational effort, a new approach is proposed for the multiharmonic response analysis of dynamical systems with local nonlinearities. The approach is based on the describing function (DF) method and linear receptance data. With the DF method, the kinetic equations are converted into a set of complex algebraic equations. By using the linear receptance data, the dimension of the complex algebraic equations, which should be solved iteratively, are only related to nonlinear degrees of freedom (DOFs). A cantilever beam with a local nonlinear element is presented to show the procedure and performance of the proposed approach. The approach can greatly reduce the size and computational cost of the problem. Thus, it can be applicable to large-scale systems with local nonlinearities.

On the design and synthesis of limit cycles using switching linear systems

Abstract: This article deals with the design and synthesis of limit cycles for a class of switched linear systems. This work is motivated by an application in the context of electrical networks, but the methodologies can be applied in other engineering fields as well. Several methods for the design and synthesis of limit cycles are presented. Based on a monodromy matrix associated with a periodically switched system, a design method and two feedback strategies are developed. The first strategy is based on linear feedback design using pole placement. The second strategy is based on the observation that certain state-dependent switching strategies can be implemented by means of a simple nonlinear output feedback controller. Advantages of this latter strategy are not only the ease with which the switching strategy can be implemented, but also the fact that classical techniques may also be used to ascertain the stability of the resulting limit cycle. Our next contribution is the development of a novel frequency domain-...

Idle speed control using a describing-function approach

Abstract: A new systematic method for idle speed control of uncertain automobile engines with a time delay is developed. The method is based on obtaining a describing-function model of the internal combustion engine. This is followed by designing a classical controller via an algebraic approach. The design method and the associated software are applied to a specific automobile engine, and it is shown that the results compete with the observer-based design of Bengea et al. Stability is demonstrated by successful generation of describing-function models of the final non-linear closed-loop system.

Analysis of FLC with Changing Fuzzy Variables in Frequency Domain

Abstract: This paper discusses a simple method for analyzing FLC in frequency domain based on describing function. Since nonlinear characteristics of FLC make it difficult FLC analysis, it usually requires a big deal of trial-and-error procedures based on computer simulation. The proposed method is simple and easy to understand, because it is based on the Nyquist stability criterion used to analyze absolute and relative stability, phase and gain margin of a linear system. To linearize in frequency domain, a describing function for FLC is derived by using a piecewise linearization of the FLC response plot. This describing function is represented as a function of magnitude of input sinusoid and nonlinear parameters x1 and x2 which change consequence fuzzy variables and nonlinearity of FLC. The describing function is redefined without the magnitude of sinusoid input because maximum values of the describing function can explain the stability of the system. This redefined describing function is used to get minimum stability characteristic, an absolute stability, phase margin and gain margin, of FLC. Using this function, we can explicitly figure out various characteristic of FLC according to x1 and x2 in frequency domain. In this work, we suggest a minimum phase margin (MPM) and a minimum gain margin (MGM) for FLC which can be used to determine whether the system is stable or not and how stable it is. For simplicity, we use one-input FLC with three rules. For various nonlinear response of FLC, changing fuzzy variables of a consequence membership function is used. Simulation results show that these parameters are effective in analyzing FLC.

Design of compensators for second order sliding modes

Abstract: In this paper, compensators for the Twisting and Sub-Optimal algorithms are designed The compensation is aimed at the increase of the open-loop gain of the system with respect to the averaged variables (at low frequencies) achieved through the mechanism of the increase of the equivalent gains kn of the controller nonlinearities The describing function method is used for the development of the system model An example of design and simulations are presented

Can Describing Function Technique predict Bifurcations in Thermoacoustic Systems

Abstract: Modeling thermoacoustic instabilities using inputs from experimental data usually employs a two-part approach: the response of the flame to perturbations in the flow is first obtained in terms of a flame transfer function, which then used as a source term in the acoustic equation. Within the framework of the two-part approach, an approximate modal analysis technique, known as describing function technique is used to model the nonlinearities in the flame dynamics. In this paper, we examine the consequences of employing a two-part approach in predicting the nonlinear behavior of a thermoacoustic system. Further, we examine the applicability of describing function technique for predicting the bifurcations of thermoacoustic systems. A toy model of thermoacoustic system was designed in order to analyze the effects of the two-part approach. It is found that the two-part approach modifies the dynamics of the system and cannot capture the intricate coupling between the combustion processes and acoustic field. A ducted diffusion flame model and a ducted premixed flame model are then studied as representative problems of thermoacoustic instability in combustors. The bifurcation plots obtained from the predictions using describing function technique were compared with the bifurcation plots obtained from the simulations in time domain for these models. The results show that the system behavior predicted by the describing function technique is quantitatively as well as qualitatively different from those obtained from time evolutions.

On the characterization of limit cycle modes in oversampled data converters

Abstract: This paper studies the appearance of periodic and quazi-periodic modes, denoted as limit cycles, in the operation of closed-loop oversampled data converters. It demonstrates that the properties of these limit cycles are largely determined by the parameterization of the system loop filter, the amplitudes and the frequencies of the applied input signals, and the applied sampling frequency. For the analysis, quantization in amplitude and quantization in time are seen as separate functions. The quantizer is represented with sampled describing functions and a quazi-linear time-variant model for the loop operation is introduced and illustrated with simulation examples.

Frequency domain based feed forward tuning for friction compensation

Abstract: In high precision motion control, performance is often limited by the presence of nonlinearities. In this study, the presence of nonlinear influences in a high precision transmission electron microscope stage is investigated using broadband multisine signals. These measurements yield the nature and level of nonlinearities as well as the best linear approximation of the dynamics. By quantitatively measuring the level of nonlinear influences, this method indicates the relevance of improved modeling. Next, the nonlinear influences are modeled explicitly by measuring the higher order sinusoidal input describing functions (HOSIDF) of the system which describe the 'direct' response of the system at the input frequency as well as at harmonics of the input frequency. Application of this technique yields a structured way to design Coulomb friction feed forward in the presence of nonlinearities. This procedure linearizes the input-output dynamics by applying feed forward and measuring the HOSIDFs which indicate the remaining nonlinear effects. Application of this technique yields a structured way to design feed forward in the presence of nonlinearities.