Showing papers on "Describing function published in 2012"

Small Signal Modeling of LLC Resonant Converters Based on Extended Describing Function

Abstract: Due to the advantages of low switching loss and high efficiency, LLC resonant converters have been widely used. Based on the extended describing function concept, the small-signal model of an LLC resonant converter is derived in this paper. The equivalent circuit of small-signal model is illustrated so that the corresponding frequency response can be easily obtained by using Is Spice simulation. The well agreement with experimental measurements has verified the accuracy of the derived small-signal model.

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...

Nonlinear Phenomena in Thermoacoustic Systems With Premixed Flames

Abstract: Nonlinear analysis of thermoacoustic instability is essential for the prediction of the frequencies, amplitudes, and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper, an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, however, it can be used to calculate the amplitudes of limit cycles and their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modeled using a nonlinear kinematic model based on the G-equation, while the acoustics of planar waves in the tube are governed by linearized momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state for single-mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitude-dependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, along with twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact, resulting in a larger limit cycle amplitude. Multimode simulations show that, in some situations, the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems. [DOI: 10.1115/1.4023305]

Analysis of a Self-Oscillating Bidirectional DC–DC Converter in Battery Energy Storage Applications

Abstract: This paper combines three different methods for the analysis of a self-oscillating bidirectional dc-dc converter under hysteresis control. First, the describing function method is used to predict the steady-state limit cycle, along with its oscillation amplitude and frequency. Second, the Tsypkin method is applied to provide more precise information on the dynamical behavior of the system. To complete the study, a sliding-mode approach is used to provide more insight into the system response in terms of its parameters. A comparative study among the results obtained from the different approaches in the frequency and time domain is given. Finally, an experimental prototype validates the theoretical and the numerical predictions.

A sliding-mode control scheme for llc resonant DC/DC converter with fast transient response

Abstract: In this paper, a sliding-mode control scheme for LLC resonant dc/dc converter is presented. The proposed controller operates at two fixed switching frequencies with sliding-mode control implementation. The large-signal dynamic model and steady-state characteristics are explored using the extended describing function method. The parameters design and the sliding motions are discussed on the phase plane. The proposed control scheme significantly improves the dynamic response against transient load changes while preserving the closed-loop stability and the attractive features of LLC resonant converter. The simulation and experimental results demonstrate the validity of the sliding-mode control strategy and the fast transient response compared with the conventional PI control.

Instabilities in digitally controlled voltage-mode synchronous buck converter

Abstract: In this paper, instabilities and bifurcation behavior in a DC–DC digitally controlled voltage-mode synchronous buck converter (SBC) are studied. The bifurcation diagram of the system under a proportional-integral (PI) compensation is presented. Following the system mathematical description in the continuous time domain, the z-domain pulse transfer function is derived in terms of system parameters. The stability range of the design parameter is obtained in closed form by using the Jury test combined with the describing function (DF) method. Experimental measurements from a laboratory prototype are used to validate the theoretical predictions and the numerical simulations.

Parameter tuning and chattering adjustment of Super-Twisting sliding mode control system for linear plants

Abstract: A simple procedure for tuning the parameters of the Super-Twisting (STW) second-order sliding mode control (2-SMC) algorithm, used for the feedback control of uncertain linear plants, is presented. When the plant relative degree is higher than one, it is known [10] that a self-sustained periodic oscillation takes place in the feedback system. The purpose of the present work is that of illustrating a systematic procedure for control tuning based on Describing Function (DF) approach guaranteeing pre-specified frequency and magnitude of the resulting oscillation. The knowledge of the plant's Harmonic Response (magnitude and phase) at the desired chattering frequency is the only required prior information. By means of a simulation example, we show the effectiveness of the proposed procedure.

Analysis and design of average current mode control using describing function-based equivalent circuit model

Abstract: A small signal model for average current mode control based on equivalent circuit is proposed. The model uses the three-terminal equivalent circuit model based on linearized describing function method to include the feedback effect of the side-band frequency components of inductor current. It extends the results obtained in peak-current model control to average current mode control. The proposed small signal model is accurate up to half switching frequency, predicting the sub-harmonic instability. The proposed model is verified using SIMPLIS simulation and hardware experiments, showing good agreement with the measurement results. Based on the proposed model, a new feedback design guideline is presented. The proposed design guideline is compared with several conventional, widely used design criteria to highlight its virtue. By designing the external ramp following the proposed design guideline, quality factor of the double poles at half of switching frequency in control-to-output transfer function can be precisely controlled. This helps the feedback design to achieve widest control bandwidth and proper damping.

Quantification of Valve Stiction and Dead Band in Control Loops Based on the Harmonic Balance Method

Abstract: The presence of stiction in control valves often causes oscillations in control loops, with negative effects on quality and cost of goods. To address this issue, it is necessary to quantify this stiction to decide about maintenance or to implement compensators that can improve control loop performance until the next plant stop. The describing function (DF) method is a well-known scheme to predict the period and amplitude of limit cycles in control loops, requiring the knowledge of linear and nonlinear parameters of the system model. In the present study, a method has been proposed to estimate these nonlinear parameters using the parameters of linear model and amplitude and period of limit cycle produced by nonlinearity. A procedure is proposed to overcome the case of unknown process model. In addition, generalization of this method to the case of parametric uncertainties in the linear part of the control loop has also been presented. The result is a simple and efficient algorithm that can be easily extended to other nonlinearities. Furthermore, the conditions for existence and uniqueness of solution for dead band and stiction estimations have been obtained. The usefulness of the proposed method has been demonstrated through simulations and its applications to a pilot plant and to real industrial data.

A Frequency-Based Model for Limit Cycle and Spur Predictions in Bang-Bang All Digital PLL

Abstract: In this work, a frequency-based model is presented to examine limit cycle and spurious behavior in a bang-bang all-digital phase locked loop (BB-ADPLL). The proposed model considers different type of nonlinearities such as quantization effects of the digital controlled oscillator (DCO), quantization effects of the bang-bang phase detector (BB-PD) in noiseless BB-ADPLLs by a proposed novel discrete-time model. In essence, the traditional phase-locked model is transformed into a frequency-locked topology equivalent to a sigma delta modulator (SDM) with a dc-input which represents frequency deviation in phase locked state. The frequency deviation must be introduced and placed correctly within the proposed model to enable the accurate prediction of limit cycles. Thanks to the SDM-like topology, traditional techniques used in the SDM nonlinear analysis such as the discrete describing function (DDF) and number theory can be applied to predict limit cycles in first and second-order BB-ADPLLs. The inherent DCO and reference phase noise can also be easily integrated into the proposed model to accurately predict their effect on the stability of the limit cycle. The results obtained from the proposed model show good agreement with time-domain simulations.

Modeling of resonant inverters with high harmonic content using the extended describing function method

Abstract: This paper analyzes the extended describing function technique (EDF) for modeling a resonant power converter applied to induction heating applications. Since in this specific application operating conditions may be far from resonance, this paper is focused on extending the model to several harmonics and verifying its validity range. The main goal is to obtain accurate control-to-power transfer functions which predict the resonant inverter small-signal behaviour. A simulation test bench is proposed and evaluated for measuring the open-loop duty cycle and frequency-to-power transfer function plots from time domain simulations of the switching model. Finally, the accuracy of the model is analyzed.

A describing function approach to the design of robust limit-cycle controllers

Abstract: The design of robust limit-cycle controllers is introduced for autonomous systems with separable SISO nonlinearities. The objective is to design a controller to secure specified robust oscillation amplitude and frequency. The method consists of quasi-linearization of the nonlinear element via a Describing Function (DF) approach and then shaping the loop to reach desired limit-cycle characteristics. As the DF method is used, loop shaping takes place in the Nyquist plot. An example is given to illustrate the robustness of the controlled system to uncertainties in the linear subsystem model.

Multiple-input describing function technique applied to design a single channel ON-OFF controller for a spinning flight vehicle:

Abstract: In this study, the guidance and control problem of a single-channel spinning missile is investigated. The missile utilizes a single ON-OFF actuator to drive a pair of control surfaces (e.g. elevato...

Substructuring with Nonlinear Subcomponents: A Nonlinear Normal Mode Perspective

Abstract: Substructure coupling techniques allow one to predict the response of an assembly from dynamic models for each subcomponent. Linear substructures are routinely used in analysis (e.g. the Craig-Bampton method) to reduce the computational cost of vibration, noise and load predictions for structures. They also provide a designer with insight into the influence that each subcomponent has on the assembled system’s natural frequencies, mode shapes and damping. These concepts are currently limited to assemblies of linear substructures. This work explores substructuring with nonlinear subcomponents (substructures), using the nonlinear normal modes of each substructure to seek to understand how those contribute to the nonlinear modes of the assembly. The goal is to extend the insights that have been developed over the past several years for linear substructures, to nonlinear ones. A specific type of describing function model, which captures the variation of the structure’s natural frequencies and mode shapes with energy is introduced, which is here dubbed a Representative Linear Modal Model (RLMM). The results show that this type of model can be used with linear substructuring techniques to effectively predict the dynamics of a nonlinear assembly, suggesting that linear substructuring concepts may be applied advantageously to nonlinear assemblies.

Relay Based Identification of Systems

Abstract: In this paper, describing functions (DF) of relay with hysteresis is used for identification of systems. Describing function analysis is widely adopted in relay feedback-based identification methods because of the ease of computation involved and the general usefulness of the method. In process control systems, the noises come from measuring devices, control valves or the process itself. Dur ing the relay feedback experiments, amplitude of the limit cycle output is often corrupted with noises which may even fail the test. To overcome the possible failure, a relay with hysteresis is considered in the proposed identification method. Both off-line and on-line identification methods are presented to show the advantages of the online identification. The identification method is developed for a first order plus time delay (FOPDT) system although the method can be extended for higher order systems in a straightforward manner. A Simulation example is included to illustrate the use of the proposed method for both off-line and on-line identification.

Nonlinear dynamics and control analysis of combustion instabilities based on the “Flame Describing Function” (FDF)

Abstract: This thesis is concerned with an investigation of combustion instabilities in premixed combustors. This problem has been the subject of a continuous effort in relation with the many issues encountered in practical systems like those used in propulsion and energy production. Combustion instabilities usually arise from the coupling between combustion and acoustic eigenmodes of the system. In most cases such resonances lead to vibrations, structural fatigue and intensified heat fluxes to the chamber walls. The first part of this thesis pursues the development of prediction methods for combustion instabilities and the associated nonlinear phenomena such as limit cycles establishment, triggering, mode switching and hysteresis. The aim is to delineate physical mechanisms and develop analytical methods dedicated to prediction. The theoretical framework relies on the “harmonic balance” formalism well known in the domain of control and which has been adopted more recently in combustion instability studies carried out at EM2C, CNRS laboratory. Through this concept, it is possible to take into account the evolution of the flame response as a function of amplitude. This flame response, depending on frequency and amplitude, extends the flame transfer function principle and is designated as the “Flame Describing Function” (FDF). The development of the FDF framework is pursued in the present study. The experimental setup which exemplifies combustion instabilities and serves to validate the method has generic features as it comprises in an idealized version, all the parts found in practical systems : a feeding manifold delivering a mixture of methane and air, a multipoint injector made of a perforated plate anchoring a collection of small laminar conical flames and a flame tube made of quartz which confines the combustion zone. The downstream boundary of the system is open. This device allows a simplified analysis and provides a wide variety of configurations through the continuous modification of the feeding manifold length which is bounded by a piston on the upstream and through changes of the flame tube lengths. Systematic comparison between theoretical results and well controlled experiments is undertaken. Depending on the geometry, the setup exhibits a large variety of unstable modes which are classified in terms of their limit cycle behavior using tools from dynamical system theory. It is shown that limit cycles with constant amplitude are well predicted by the unified FDF methodology. For some configurations, the experiment reveals limit cycles characterized by time variable amplitude and frequency. One finds situations where the oscillation is coupled by multiple modes leading either to regular amplitude variations or more irregular evolutions with a “galloping” pattern as a function of time. For this special type of limit cycle, the FDF indicates the range of the onset, but is not able to fully describe these complex limit cycles. These oscillations require a time domain state space analysis which is not addressed in this manuscript. The experimental database may be of value for further work in this direction. The second part of this thesis deals with control methods for instabilities. One specifically considers damping systems relying on perforated plates biased by a flow (BFP : “Bias Flow Perforate”). These systems are particularly interesting because they can be used to cancel low frequency oscillations which are otherwise difficult to reduce through passive control methods. This BFP design relies on recent work carried out at EM2C, CNRS laboratory which extends the frequency range where the system is effective. The experimental study and the associated FDF calculations are used to delineate the possibilities of such systems and uncover conditions required for an effective damping of oscillations. This study provides indications on the practical application of BFPs.

Steady-state analysis of series resonant converter using extended describing function method

Abstract: This paper presents the steady-state analysis of series resonant converter (SRC) using the extended describing function (EDF) method. The EDF method is used to express non-linear terms of switching waveforms as linear descriptors to develop a linear model of the converter. The converter under consideration consists of a half-bridge inverter and a full-wave rectifier. The steady-state analysis in the frequency-domain is presented. The characteristics such as input-to-output-voltage transfer function and input impedance are obtained by first harmonic approximation of the converter state variables. Experimental results are given for a laboratory prototype of the SRC operating at 24 VDC input, 10 VDC output, output power of 15 W, and at a switching frequency of 110 kHz to verify the theoretical analysis.

Parametric Identification of Nonlinearity from Incomplete FRF Data Using Describing Function Inversion

Abstract: Most engineering structures include nonlinearity to some degree. Depending on the dynamic conditions and level of external forcing, sometimes a linear structure assumption may be justified. However, design requirements of sophisticated structures such as satellites require even the smallest nonlinear behavior to be considered for better performance. Therefore, it is very important to successfully detect, localize and parametrically identify nonlinearity in such cases. In engineering applications, the location of nonlinearity and its type may not be always known in advance. Furthermore, in most of the cases, test data will be incomplete. These handicaps make most of the methods given in the literature difficult to apply to engineering structures. The aim of this study is to improve a previously developed method considering these practical limitations. The approach proposed can be used for detection, localization, characterization and parametric identification of nonlinear elements by using incomplete FRF data. In order to reduce the effort and avoid the limitations in using footprint graphs for identification of nonlinearity, describing function inversion is used. Thus, it is made possible to identify the restoring force of more than one type of nonlinearity which may co-exist at the same location. The verification of the method is demonstrated with case studies.

Computer-aided Nonlinear Control System Design: Using Describing Function Models

Abstract: A systematic computer-aided approach provides a versatile setting for the control engineer to overcome the complications of controller design for highly nonlinear systems. Computer-aided Nonlinear Control System Design provides such an approach based on the use of describing functions. The text deals with a large class of nonlinear systems without restrictions on the system order, the number of inputs and/or outputs or the number, type or arrangement of nonlinear terms. The strongly software-oriented methods detailed facilitate fulfillment of tight performance requirements and help the designer to think in purely nonlinear terms, avoiding the expedient of linearization which can impose substantial and unrealistic model limitations and drive up the cost of the final product. Design procedures are presented in a step-by-step algorithmic format each step being a functional unit with outputs that drive the other steps. This procedure may be easily implemented on a digital computer with example problems from mechatronic and aerospace design being used to demonstrate the techniques discussed. The authors commercial MATLAB-based environment, available separately from insert URL here, can be used to create simulations showing the results of using the computer-aided control system design ideas characterized in the text. Academic researchers and graduate students studying nonlinear control systems and control engineers dealing with nonlinear plant, particularly mechatronic or aerospace systems will find Computer-aided Nonlinear Control System Design to be of great practical assistance adding to their toolbox of techniques for dealing with system nonlinearities. A basic knowledge of calculus, nonlinear analysis and software engineering will enable the reader to get the best from this book.

Insights from control theory into deep brain stimulation for relief from Parkinson's disease

Abstract: Using ideas from control theory. i.e., the root locus method, Lyapunov's theorem of the first approximation, the describing function, Nyquist stability theory and the concept of the equivalent nonlinearity associated with dither injection in a nonlinear feedback loop, the phenomenon of quenching of pathological neural oscillations by deep brain stimulation is explored. The model used contains a second order unstable, linear, dynamical system, in a negative feedback loop with a nonlinearity comprising a linear gain in parallel with a “signed square”. This mimics, what is referred to by Alim Louis Benabid, the great pioneer of deep brain stimulation as “excitation of inhibitory pathways that lead to functional inhibition”. Describing function analysis is used to give a very close estimate of the inherent, almost sinusoidal oscillation, which is quenched by deep brain stimulation. The relationship between the critical amplitude of deep brain stimulation (expressed either in volts or milliamps) and the fractional pulse width needed for quenching the oscillation is derived. This is fitted as closely as possible to experimental results by Benabid et al., by minimizing a sum of squared error index.

Nonlinear Flame Describing Function Analysis of Galloping Limit Cycles Featuring Chaotic States in Premixed Combustors

Abstract: Nonlinear prediction of combustion instabilities in premixed systems is undertaken on a generic configuration featuring an adjustable feeding manifold length, a multipoint injector composed of a perforated plate and a flame confinement tube. By changing the feeding manifold or flame tube lengths, the system exhibits different types of combustion regimes for the same flow operating conditions. Velocity, pressure and heat release rate measurements are used to examine oscillations during unstable operation. For many operating conditions, a limit cycle is reached at an essentially fixed oscillation frequency and quasi-constant amplitude. In another set of cases, the system features other types of oscillations characterized by multiple frequencies, amplitude modulation and irregular bursts which can be designated by “galloping” limit cycles or GLC. These situations are explored in this article. Imaging during GLCs indicates that the flame is globally oscillating but that the cycle is irregular. Prediction of these special oscillation states is tackled within the Flame Describing Function (FDF) framework. It is shown that it is possible to predict with a reasonable degree of agreement the ranges where a quasi-constant amplitude limit cycle will be established and ranges where the oscillation will be less regular and take the form of a galloping limit cycle. It is found that the FDF analysis also provides indications on the bounding levels of the oscillation envelope in the latter case.Copyright © 2012 by ASME

TITO system identification using relay with hysteresis

Abstract: In this paper an on-line identification of two input two output (TITO) system using describing functions (DF) of relays with hysteresis is presented. In relay based system identification, describing function is widely adopted because of the ease of computation involved and general usefulness of the method. Measurement noise is an important issue in an identification problem. During the relay feedback experiments, amplitude of the limit cycle output is often corrupted with noises which may even fail the test. To overcome the possible failure, a relay with hysteresis is considered in the proposed identification method. The identification method developed for a second order plus time delay (SOPDT) system is extended for the TITO system. Simulation examples are included to illustrate usefulness of the proposed method for on-line identification of TITO systems.

A novel model-free prediction of power quality problems via DSTATCOM

Abstract: The distribution static compensator (DSTAT-COM) has benefitted from computer aided-design simulations But its performance has been hampered by inadequacies in linear models of the power quality problem due to variations in system parameters, uncertainties and often missing gradient information To address this problem, this paper applies model-free predictive control (MFPC) and describing functions in nonlinear modelling to DSTAT-COM control, using a simplex optimisation approach to online tuning The MFPC offers the advantage of bypassing the computationally over-burdened state-space model and can sufficiently handle constraints, while arresting voltage quality violations online at high speeds against changes in customers loads The effectiveness and efficiency of the novel method is compared against the proportional-integralderivative as well as the fuzzy proportional-derivative controllers in a simulink environment

Describing function analysis of uncertain fuzzy vehicle control systems

Abstract: In this paper, some useful frequency domain methods including describing function, parameter space, and Kharitonov approach are applied to analyze the stability of an uncertain fuzzy vehicle control system for limit cycle prediction. A systematic procedure is proposed to solve this problem. The fuzzy controller can be linearized by the use of classical describing function firstly. By doing so, it is feasible to treat the stability problem of a fuzzy control system as linear one. In order to consider the robustness of a fuzzy vehicle control system, parameter space method and Kharitonov approach are then employed for plotting the stability boundaries. Furthermore, the effect of transport delay is also addressed. More information of limit cycles can be obtained by this approach. This work shows that the limit cycles caused by a static fuzzy controller can be easily suppressed if the system parameters are chosen carefully.

Describing function analysis of systems with NN compensated odd symmetric nonlinear actuators

Abstract: Paper deals with NN compensated odd symmetric nonlinear actuators. Most actuators are nonlinear sporting odd symmetric nonlinearities such as deadzone and saturation. One of ways of dealing with such actuators is to try to compensate for nonlinearities by static neural networks. Compensated actuators can improve behavior of the system, but then arises the problem of stability analysis because compensated nonlinearity is now complex nonlinearity not described in common literature. One way of dealing with such problem is to perform stability analysis via describing function method. Paper describes the method for compensating nonlinearities, recording describing function and performing stability analysis.

Non-linear proportional–integral–derivative synthesis for unstable non-linear systems using describing function inversion with experimental verification:

Abstract: This paper demonstrates a new non-linear proportional–integral–derivative (PID) controller synthesis approach using a describing function inversion technique for unstable systems where a mathematical model may not be available. The approach is applied to an inverted pendulum experimental set-up whose dynamic behaviour is very sensitive to the amplitude level of excitation. The procedure involves stabilization of the unstable system followed by generation of the describing function models of the stabilized closed-loop system. Then, the corresponding unstable open-loop frequency domain models at various operating regimes are extracted. A controller at nominal conditions is designed, followed by obtaining the corresponding desired open-loop frequency domain model. A set of controllers that force the open-loop behaviour of the system mimic, which is desired at various operating regimes, is designed by optimization. Finally, the controller gains are inverted using a describing function inversion technique foll...