Showing papers on "Describing function published in 2007"
TL;DR: Analysis of chattering in such systems with unmodeled based on the Lyapunov theory and the describing function method and various approaches to reduce chattering are described including methods based on relay control gain adaptation.
535 citations
TL;DR: In this paper, the authors derived both static and dynamic models that include quantization effects and used them to explain the origins of limit-cycle oscillations in dc-dc converters with a single voltage feedback loop.
Abstract: In digitally controlled dc-dc converters with a single voltage feedback loop, the two quantizers, namely the analog-to-digital (A/D) converter and the digital pulse-width modulator (DPWM), can cause undesirable limit-cycle oscillations. In this paper, static and dynamic models that include the quantization effects are derived and used to explain the origins of limit-cycle oscillations. In the static model, existence of dc solution, which is a necessary no-limit-cycle condition, is examined using a graphical method. Based on the generalized describing function method, the amplitude and offset-dependent gain model of a quantizer is applied to derive the dynamic system model. From the static and dynamic models, no-limit-cycle conditions associated with A/D, DPWM and compensator design criteria are derived. The conclusions are illustrated by simulation and experimental examples
367 citations
TL;DR: A systematic approach to the chattering analysis in systems with second-order sliding modes is developed, finding sufficient conditions for the existence of orbitally stable periodic motions in terms of the properties of corresponding Poincare maps.
Abstract: A systematic approach to the chattering analysis in systems with second-order sliding modes is developed. The neglected actuator dynamics is considered to be the main cause of chattering in real systems. The magnitude of oscillations in nonlinear systems with unmodeled fast nonlinear actuators driven by second-order sliding-mode control generalized suboptimal (2-SMC G-SO) algorithms is evaluated. Sufficient conditions for the existence of orbitally stable periodic motions are found in terms of the properties of corresponding Poincare maps. For linear systems driven by 2-SMC G-SO algorithms, analysis tools based on the frequency-domain methods are developed. The first of these techniques is based on the describing function method and provides for a simple approximate approach to evaluate the frequency and the amplitude of possible periodic motions. The second technique represents a modified Tsypkin's method and provides for a relatively simple, theoretically exact, approach to evaluate the periodic motion parameters. Examples of analysis and simulation results are given throughout this paper.
363 citations
TL;DR: Recorded transient waveforms show that the closed-loop converter is capable of not only responding to the reference input change as required by the design specifications, but also stabilizing the output effectively under disturbances from both the output and the input.
Abstract: A nonlinear model for a phase-controlled series-parallel resonant converter is developed using the extended describing function method and d-q decomposition. The model is linearized and reduced using the balanced model reduction technique. Based on the reduced model and taking into account the zero-order hold delay and the computation delay in the sampled-data system, a digital controller for the converter is designed. The controller is implemented with a digital signal processor (DSP). The closed-loop converter with the DSP controller is built and tested experimentally. Recorded transient waveforms show that the closed-loop converter is capable of not only responding to the reference input change as required by the design specifications, but also stabilizing the output effectively under disturbances from both the output and the input
69 citations
TL;DR: An accurate modeling of the series-parallel resonant converter operating in self-sustained oscillation mode including the overlap effects of the output rectifying stage due to the leakage inductance of the transformer is presented.
Abstract: An accurate modeling of the series-parallel resonant converter operating in self-sustained oscillation mode including the overlap effects of the output rectifying stage due to the leakage inductance of the transformer is presented. This paper presents a systematic procedure to study the aforementioned effects on the converter dynamic and steady-state performance. Such information is critical in designing isolated high-frequency resonance-based voltage-regulator modules for powering future subvoltage very large scale integration circuits such as microprocessors. The extended describing function technique is used to extract the steady-state characteristics in order to get an optimum converter design. Averaging state-space techniques are employed to derive a small-signal model that can describe the converter dynamics accurately. Analytical and simulation results are given. Finally, a 1-kW experimental prototype is built to verify the validity of the proposed work
48 citations
TL;DR: In this article, a methodology based on the multifrequency model with describing functions was developed to investigate the phenomenon of beat-frequency oscillation in phase currents, and the influences from the interleaving operation were identified.
Abstract: Under high-frequency repetitive load transients, multiphase voltage regulators suffer from beat-frequency oscillations of the phase currents. To investigate this issue, this paper develops a methodology based on the multifrequency model with describing functions. The phenomenon of beat-frequency oscillation in phase currents is explained. The influences from the interleaving operation are identified. Using peak-current control as an example, the magnitude of beat-frequency oscillations is predicted by a proposed model that considers sideband information. To reduce this oscillation without impacting the voltage regulation, several solutions are proposed. Experimental and simulation results are provided as verification of the analyses and solutions.
45 citations
TL;DR: A set of general expressions is derived from a single symmetrical relay feedback test for identification of a class of process transfer functions and the parameters of open loop stable nonminimum phase transfer function models may be obtained from simple measurements made on the limit cycle.
Abstract: A set of general expressions is derived from a single symmetrical relay feedback test for identification of a class of process transfer functions. Using these expressions the parameters of open loop stable nonminimum phase transfer function models may be obtained from simple measurements made on the limit cycle. For comparison, the conventional describing function based identification formulae are presented. Fourier series based curve fitting with the options of nonlinear least squares method and trust-region algorithm is used to measure limit cycle parameters in the presence of measurement noise. Examples are given to illustrate the value of the proposed method
30 citations
TL;DR: In this article, an indirect multi-term harmonic balance method is proposed to determine periodic solutions of nonlinear force instead of displacement in a single-degree-of-freedom mechanical system.
28 citations
TL;DR: The small-signal analysis shows that the MOS Colpitts oscillator is described by a third order characteristic equation, and it is shown that the amplitude self-modulation (squegging) in the considered oscillators is absent for any conducting angle of the transistor.
Abstract: The small-signal analysis shows that the MOS Colpitts oscillator is described by a third order characteristic equation. The procedure for finding the second order approximation is defined, and the solution corresponding to this approximation is found. Then the equations for transistor transconductance describing function are analyzed, and the design procedure corresponding to the "convenient" operation point is given. The same equations are also used for the analysis of amplitude stability in this oscillator. It is shown that the amplitude self-modulation (squegging) in the considered oscillator is absent for any conducting angle of the transistor.
28 citations
09 Jul 2007
TL;DR: This work presents results on the stability of time-delay systems under reset control by developing a generalization of previous stability results of reset systems without delays, and also the case of delay-independent stability.
Abstract: This work presents results on the stability of time-delay systems under reset control. The case of delay-dependent stability is addressed by developing a generalization of previous stability results of reset systems without delays, and also the case of delay-independent stability. Stability conditions are given by using a set of LMIs and passivity properties. These conditions guarantee that the reset action does not destabilize the base LTI system. Finally, limitations and future directions are discussed by describing function analysis.
21 citations
TL;DR: In this article, the authors introduce the concept of phase deviation to compensate for the span between the critical point and the oscillation point, so that the ultimate gain and ultimate frequency can be accurately obtained using only a single relay feedback test.
Abstract: Describing function analysis has been widely adopted in many existing identification methods that are based on relay feedback tests. In these methods, the critical point of the Nyquist curve is usually approximated to the oscillation point. Nevertheless, such an approximation may result in poor accuracy for identification, in particular, for those processes with large dead time. In this paper, we introduce the concept of phase deviation to compensate for the span between the critical point and the oscillation point, so that the ultimate gain and ultimate frequency can be accurately obtained using only a single relay feedback test. The phase deviation is then analytically derived, with which all the model parameters can be identified without any prior information of the dead time or static gain. Numerical examples are given to illustrate the proposed method.
TL;DR: In this paper, the authors show that limit cycle Floquet multipliers and the related bifurcation phenomena can be estimated through a spectral approach, entirely based on the describing function technique.
Abstract: Spectral techniques, like harmonic balance, are classical numerical tools for designing nonlinear oscillators and microwave circuits. Recently these techniques have been exploited for investigating complex dynamics in nonlinear systems. In this manuscript we firstly show that limit cycle Floquet's multipliers and the related bifurcation phenomena can be estimated through a spectral approach, entirely based on the describing function technique. Then we consider some significant case studies, and we show that our method yields more accurate results than the other describing function-based approaches proposed in the literature
Journal Article•
TL;DR: In this article, the authors presented an INS grade, surface micro-machined differential resonant accelerometer (DRXL) manufactured by an epitaxially grown thick polysilicon process, which generates a differential digital output upon an applied acceleration, in which frequency transition is measured due to gap dependent electrical stiffness change.
Abstract: This paper presents an INS (Inertial Navigation System) grade, surface micro-machined differential resonant accelerometer (DRXL) manufactured by an epitaxially grown thick polysilicon process. The proposed DRXL system generates a differential digital output upon an applied acceleration, in which frequency transition is measured due to gap dependent electrical stiffness change. To facilitate the resonance dynamics of the electromechanical system, the micromachined DRXL device is packaged by using the wafer level vacuum sealing process. To test the DRXL performance, a nonlinear self-oscillation loop is designed based on the extended describing function technique. The oscillation loop is implemented using discrete electronic elements including precision charge amplifier and hard feedback nonlinearity. The performance test of the DRXL system shows that the sensitivity of the accelerometer is 24 ㎐/g and its long term bias stability is about 2 ㎎ (1σ) with dynamic range of σ 70g.
TL;DR: A multimodel scheme is designed for a triggering tunnel-diode circuit that improves the transient behavior during the transition time period after switching from a stable system equilibrium point to another one which is known as a triggering process.
Abstract: A multimodel scheme is designed for a triggering tunnel-diode circuit. The scheme improves the transient behavior during the transition time period after switching from a stable system equilibrium point to another one which is known as a triggering process. Each model is obtained by a linearization of the circuit near an equilibrium point. Moreover, each of these models can be described as a combination of two other transfer functions describing the linearized plant behavior near two different equilibrium points. The scheme chooses online the model with the best tracking performance in order to generate the control law. Different reference transfer functions are proposed with the aim of generating the desired transient in the triggering process. Some simulations show the usefulness of this scheme.
26 Dec 2007
TL;DR: In this paper, a nonlinear feedback control strategy, utilizing a gain scheduling obtained from the aforementioned model-based identification, is implemented for compensating the error of a conventional cascade controller.
Abstract: Harmonic drives (HD) are high-ratio, compact torque transmission systems. However, due to frictional effects and internal flexibility, a HD shows unique (torsional) stiffness behavior characterized by a stiffening spring in parallel with a hysteresis element, which is modeled using the Maxwell-slip model. As a result of this complex nonlinear behavior, a classical linear control strategy obviously does not perform very well, in a system with HD component, for achieving accurate positioning. This paper aims to characterize the dynamic behavior in mechanical system with HD and, more specifically, to use this knowledge to design effective control schemes. Modeling of the system is achieved via the Describing Function approach, applied to the hysteresis element, to yield the stiffness and damping in function of the amplitude of motion. Afterwards, a nonlinear feedback control strategy, utilizing a gain scheduling obtained from the aforementioned model-based identification, is implemented for compensating the error. The results show high performance in regard to rise time, positioning error and robustness as compared to a conventional cascade controller.
26 Dec 2007
TL;DR: In this article, a fuzzy-PD controller is used to replace the PD controller for the electro-hydraulic servo systems subject to nonlinear friction, in accordance with the known data with the use of PD controller, a robust methodology using fuzzy logic is proposed to have better tracking performance.
Abstract: In this paper, the focus is on the system stability information, namely the stable region and stability boundary explored by the use of the parameter plane method, the stability equation method, and the describing function method. A PD controller, under the consideration of stability, is first designed for the electro-hydraulic servo systems subject to nonlinear friction. Then, in accordance with the known data with the use of PD controller, a robust methodology using the fuzzy logic is proposed to have better tracking performance. Finally, the fuzzy-PD controller is used to replace the PD controller.
02 Jul 2007
TL;DR: In this article, a new approach in generating a periodic motion for underactuated mechanical systems using variable structure controllers without computing a reference trajectory is proposed, in order to find explicit expressions of the controller gain parameters in terms of the desired frequency and amplitude, is made through frequency-domain tool, particularly with describing function method.
Abstract: A new approach in generating a periodic motion for underactuated mechanical systems using variable structure controllers without computing a reference trajectory is proposed in this paper. The design procedure, in order to find explicit expressions of the controller gain parameters in terms of the desired frequency and amplitude, is made through frequency-domain tool, particularly with describing function method. Performance issue of a self-excited system, applied to a double pendulum, is illustrated by experiments.
TL;DR: In this paper, the second-order sliding mode observer dynamics are analyzed in the frequency domain and the analysis is based on the describing function method and the concept of the equivalent gains of nonlinear functions of the super-twisting algorithm.
Abstract: The second-order sliding mode observer dynamics are analyzed in the frequency domain. The so-called super-twisting algorithm is utilized for generating the second-order sliding mode in the observer dynamical system. The frequency response of the observer dynamics is obtained and used as a characteristic of the observer. The analysis proposed is based on the describing function method and the concept of the equivalent gains of nonlinear functions of the super-twisting algorithm.
TL;DR: An admissible specification-oriented Kharitonov region is found directly on the controller parameter plane and constitutes all of the feasible controller gain sets to achieve robust prevention of limit cycle for the considered uncertain nonlinear control systems.
Abstract: A new method is proposed to compute all feasible robust stabilizing controllers for preventing the generation of limit cycle of nonlinear control systems with parametric uncertainties both in the linear plant and nonlinearity. The describing function analysis method is employed to approximate the behaviors of the nonlinearity. The Kharitonov theorem is utilized to characterize parametric uncertainties in the linear plant and nonlinearity. Necessary conditions for limit cycles are established. Boundaries for the generation of limit cycle and boundaries for asymptotic stability are portrayed exploiting the stability equation method. The region for prescribed limit cycle behavior and the region for asymptotic stability are located. An admissible specification-oriented Kharitonov region is found directly on the controller parameter plane. The region is non-conservative and constitutes all of the feasible controller gain sets to achieve robust prevention of limit cycle for the considered uncertain nonlinear control systems. The way to tune the controller gains is suggested. Finally, for comparison purpose, two illustrative examples proposed in the literature are given to show how the proposed algorithm can be effectively applied to tune a robust controller to achieve a prescribed limit cycle behavior and accomplish robust limit cycle amplitude suppression and prevention.
DOI•
01 Jan 2007
TL;DR: In this thesis an alternative concept for frequency based nonlinear system analysis is presented, the required measurement techniques are described and some application examples are shown.
Abstract: In modern positioning systems, accuracy and speed requirements have increased significantly. These accuracies can only be realized if account is given to nonlinear system behavior in both the mechanical and the control design. This requires additional tools for frequency based identification of nonlinear system behavior since existing tools either are either too limited to successfully describe nonlinear behavior or the results are very difficult to interpret and as such do not relate to the background of the intended user. In this thesis an alternative concept for frequency based nonlinear system analysis is presented, the required measurement techniques are described and some application examples are shown. The method is applicable for the class of causal, stable, time-invariant non-linear systems which have a harmonic response to a sinusoidal excitation. This new concept is the generalization of the Sinusoidal Input Describing Function to Higher Order Sinusoidal Input Describing Functions (HOSIDF) as it yields the magnitude and phase relations between the individual higher harmonics in the response signal and the sinusoidal excitation signal, both as function of magnitude and frequency of the excitation signal. An essential element in the HOSIDF theory is the concept of the Virtual Harmonics Expander (VHE). This nonlinear function describes the transformation of a single sinusoid into an infinite amount of harmonics, each with equal amplitude as the input signal and with a phase equal to the phase of the input signal times the harmonic number. Nonlinear systems belonging to the class can be modeled as a parallel connection of an (infinite) amount of HOSIDF describing quasi-linear subsystems in series with the VHE. Two measurement methods for nonparametric identification of HOSIDF are presented. The Fast Fourier Transform based method on fast fourier transforms shows ideal characteristics due to its perfect selectivity. The IQ (In phase-Quadrature phase) demodulation method has limited performance due to non perfect selectivity. The bias in the HOSIDF estimates caused by harmonic components in the input signal is analyzed and a compensation algorithm is presented to reduce this bias. Accept- ing harmonic distortion in the excitation signal allows the application of non-constant amplitude-time profiles for testing. It is demonstrated that a ramped amplitude-time signal reduces the required settling time of the digital filters used in the IQ methode. The capabilities of the HOSIDF technique are demonstrated in a real measurement in which the stick to gross sliding transition of a mechanical system with dry friction is captured as function of frequency. The odd HOSIDF clearly reveal this transition which is not possible with the Frequency Response Function technique. From the HOSIDF the pre-sliding displacement and the friction-induced stiffness are determined and the friction force which must be present in the stick-phase is calculated. Validation with force measurements shows excellent agreement. Special attention is paid to the determination of the HOSIDF of a nonlinear plant operating in feedback. In a controlled systemthe harmonics generated by the non-linear system will be fed back to the input, changing the sinusoidal excitation into an harmonic excitation. Two different solutions are presented to deal with this problem. The first method applies a numerical compensatie techniques to compensate the bias caused by the harmonic components in the excitation signal. The secondmethod uses amodified repetitive control scheme to suppress the harmonic components in the excitation signal. The effectiveness of both methods is tested in simulation experiments of a mass operating in feedback subjected to Coulomb friction, Stribeck-effect and hysteresis in the pre-sliding regime. The friction forces are modeled with the modified Leuven friction model. The results are compared with the HOSIDF measured under open loop condition and both methods yield correct results. It is shown that by rearranging the repetitive control loop, the output signal of a class of stable, time-invariant nonlinear systems becomes sinusoidal as response to an harmonic excitation. For this class of signals Higher Order Sinusoidal Output Describing Functions (HOSODF) can be defined as the dual of the HOSIDF. The HOSODF describe magnitude and phase relations between the individual higher harmonics in the input signal and the sinusoidal output signal, both as function of magnitude and frequency of the output signal. The required dual of the Virtual Harmonics Expander is defined as the Virtual Harmonics Compressor. This nonlinear function describes the transformation of an infinite amount of harmonics into a single sinusoid. Finally, an application example shows the extreme sensitivity of the HOSIDF technique for changes in friction characteristics, indicating interesting opportunities for application in the field of machine condition monitoring. De eisen die gesteld worden aan de snelheid en positioneringsnauwkeurigheid van moderne positioneringssystemen zijn significant toegenomen. Deze nauwkeurigheden kunnen alleen maar gerealiseerd worden als met niet-lineair systeemgedrag rekening wordt gehouden in zowel het mechanische als het regeltechnische ontwerp. In tegenstelling tot de tijddomein gebaseerde systeemidentificatie is de moderne regeltechniek op frequentiedomein technieken gebaseerd. Maar de transformatie van niet-lineaire tijddomeinmodellen naar het frequentiedomein is nietmogelijkmet alleen lineaire technieken. Dit vereist extra gereedschappen ten behoeve van de frequentiedomein gebaseerde identificatie van niet-linear systeemgedrag omdat de bestaande gereedschappen ofwel te beperkt zijn om met succes niet-linear gedrag te beschrijven ofwel resultaten leveren in een formaat dat moeilijk te interpreteren is en niet aansluit bij de achtergrond van de gebruiker. In dit proefschrift wordt een alternatief concept gepresenteerd voor een op frequentiedomeintechnieken gebaseerde niet-lineaire systeemanalyse. Eveneens worden de vereiste meetmethodes beschreven en enkele toepassingsvoorbeelden getoond. De methode is van toepassing op de klasse I gedefinieerd als de klasse van causale, stabiele, tijdsinvariante, niet-lineaire systemen welke een harmonische responsie hebben ten gevolge van een sinusvormige excitatie. Dit nieuwe concept is de generalisatie van de Sinusoidal Input Describing Function tot de Higher Order Sinusoidal Input Describing Functions (HOSIDF). De HOSIDF beschrijven de magnitude- en faserelaties die bestaan tussen de afzonderlijke hogere harmonische componenten in het responsiesignaal en de sinusvormige excitatie, allen als functie van amplitude en frequentie van dat excitatiesignaal. In de HOSIDF theorie wordt een essentiele plaats ingenomen door het begrip Virtual Harmonics Expander (VHE). Deze niet-lineaire functie beschrijft de transformatie van een zuiver sinusvormig signaal in een oneindige reeks harmonischen, elk met identieke amplitude gelijk aan de amplitude van het ingangssignaal en een fase gelijk aan de fase van het ingangssignaal maal het rangnummer van de harmonische component. Systemen die behoren tot de klasse I kunnen gemodelleerd worden als een parallel schakeling van een (oneindig) aantal HOSIDF in serie met de VHE. Twee meetmethodes voor de niet-parametrische identificatie van HOSIDF worden gepresenteerd. De op Fast Fourier
01 May 2007
TL;DR: The BIMBO identification method can be used for the online identification of the digital filter in a sigma-delta modulator's loop and this parameter estimation method only requires the knowledge of the bit stream at the modulator output.
Abstract: We show how the BIMBO identification method can be used for the online identification of the digital filter in a sigma-delta modulator's loop. This parameter estimation method only requires the knowledge of the bit stream at the modulator output. This approach does not rely on an approximation of the comparator (such as additive white noise or describing function approximation). Moreover, it requires no amplitude measurement, as opposed to most other methods, and, thus, it involves no additional electronics: it is therefore very economical to implement it practically. The method is described from a theoretical point of view and an experimental validation is given.
01 Dec 2007
TL;DR: The fuzzy controller can be linearized by utilizing the describing function method with experiments and the stability analysis of the vehicle lateral control system with the variations of velocity and friction is carried out by the use of parameter plane method.
Abstract: In this paper, the robust stability analysis of a fuzzy vehicle lateral system with perturbed parameters is presented. Firstly, the fuzzy controller can be linearized by utilizing the describing function method with experiments. After the describing function is obtained, the stability analysis of the vehicle lateral control system with the variations of velocity and friction is then carried out by the use of parameter plane method. Afterward some limit cycle loci caused by the fuzzy controller can be easily pointed out in the parameter plane. Computer simulation shows the efficiency of this approach.
24 Jun 2007
TL;DR: In this paper, two models for MOV protected series capacitors are evaluated on simplified circuits with a single machine system and an infinite bus, and the evaluation of the model is done using an equal area criterion and numerical solutions of nonlinear differential equations.
Abstract: This paper presents assessments of two models for MOV protected series capacitors. The first model is empirical and the second is analytically developed by solving the nonlinear system and applying the describing function (DF). The models are assets in short-circuit and transient stability studies. For short-circuit analysis, symmetrical and unsymmetrical faults are considered on simplified circuits using a compensation technique. For the transient stability analysis, the model is evaluated on simplified circuits with a single machine system and an infinite bus. The evaluation of the model is done using an equal area criterion and numerical solutions of nonlinear differential equations. Comparing the model with the results obtained from electromagnetic transient simulations and the empirical model proves the accuracy and flexibility of the DF model.
27 Jun 2007
TL;DR: The procedure is illustrated on determination of describing function, and corresponding error estimation, of nonlinear structure consisting of fuzzy element and typical nonlinear element.
Abstract: Engineering procedure for approximate analytical determination of describing function of nonlinear systems with odd static characteristics is presented in the paper. Generalized mathematical expressions for determining such describing function with error estimation are given. The procedure is illustrated on determination of describing function, and corresponding error estimation, of nonlinear structure consisting of fuzzy element and typical nonlinear element.
26 Dec 2007
TL;DR: Computer aided harmonic linearization, as experimental practical method for determining static characteristics and describing functions is described, which introduces an inherent error resulting from the difference between original static characteristic and corresponding linearly approximated static characteristic.
Abstract: Harmonic linearization, also known as describing function analysis is well known method for analysis of SISO systems. Computer aided harmonic linearization, as experimental practical method for determining static characteristics and describing functions is described. The use of linearly approximated static characteristic in determination of describing function introduces an inherent error resulting from the difference between original static characteristic and corresponding linearly approximated static characteristic. Error estimation and comparison of approximated characteristic error with the actual error is illustrated on several examples.
TL;DR: The paper deals with the performance analysis of harmonically forced nonlinear systems of Lur'e type and its applications in medicine, physics, and engineering.
Journal Article•
TL;DR: In this article, the authors address the problem of testing continuous-time nuclear-pulse shapers using Oscillation-Based Test (OBT) to avoid the partition in low-order sections.
Abstract: This paper addresses the problem of testing continuous-time nuclear-pulse shapers using Oscillation-Based Test (OBT). The proposal is to convert the whole systems into non-linear oscillators, avoiding the partition in low-order sections. The de- sign of the oscillators is very simple because the non- linear elements are mathematically modeled using the describing function approach, and the study of the oscillators is made using techniques of linear sys- tems. The test strategy presents high fault coverage and requires only one test session. The last charac- teristic allows reducing the time required for execut- ing the test and the complexity of the test controller. The OBT schemes are validated using deviation and catastrophic fault models.
TL;DR: A describing function method for limit cycle analysis of a system with a rate limiter that can predict the limit cycle occurrence more exactly is proposed, which is applicable to a prediction of a rate-limiter based PIO (Pilot-Induced Oscillation) of an aircraft.
Abstract: This paper proposes a describing function method for limit cycle analysis of a system with a rate limiter. First, the rate limiter is defined and its describing function is derived in a strict manner.Then, the stability analysis using the describing function of the rate limiter is executed. In many cases of limit cycles, the output of the rate limiter becomes triangular, and in order to deal with such cases by the describing function, a new method to modify an open-loop transfer function is introduced. This method can predict the limit cycle occurrence more exactly, which is applicable to a prediction of a rate-limiter based PIO (Pilot-Induced Oscillation) of an aircraft.
TL;DR: In this article, a graphical technique is proposed to synthesize robust controllers to suppress the amplitude of the limit cycle persisting in uncertain nonlinear control systems using the stability equation method.
Abstract: A graphical technique is proposed to synthesize robust controllers to suppress the amplitude of the limit cycle persisting in uncertain nonlinear control systems using the stability equation method. First, the describing function analysis method is employed to approximate the behavior of the nonlinearity and the celebrated Kharitonov theorem is utilized to characterize the plant variations. Accordingly, a family of Kharitonov polynomials is obtained for limit cycle analysis. Further, decomposing each of the vertex Kharitonov polynomials into real part and imaginary parts results in two related stability equations. By solving the two stability equations, families of constant limit cycle amplitude loci are plotted. These loci isolate the parameter plane into several limit cycle regions. Therefore, an admissible specification-oriented parameter region is found directly on the controller coefficient plane. Furthermore, the overlapped region of the admissible parameter region for each Kharitonov polynomial is called the Kharitonov region. The Kharitonov region constitutes all of the feasible controller parameter sets to achieve robust limit cycle amplitude suppression for the entire uncertain nonlinear control system. Hence, the controller can be designed more flexibly. Additionally, a way to tune the robust controller gains is suggested. Finally, the distinguished advantages of the proposed algorithm as compared with other proposed methods are demonstrated by two examples proposed in the literature for comparison.
29 Oct 2007
TL;DR: By exploiting the method of multiple scales a set of ordinary differential equations governing the amplitude and phase dynamics is derived and permit to study accurately weakly connected oscillatory networks.
Abstract: The aim of this work is to study periodic oscillations and bifurcations in cellular nonlinear networks composed by oscillatory cells and connected through arbitrary couplings. In order to characterize each oscillator by using amplitude and phase variables, a method based on a generalized version of the describing function technique is proposed. Furthermore, by exploiting the method of multiple scales a set of ordinary differential equations governing the amplitude and phase dynamics is derived. The results also permit to study accurately weakly connected oscillatory networks. Finally, the method is compared to a spectral technique, based on the harmonic balance approach, by considering a chain of Chua's circuits.