Loeb's Criterion Research Articles

Design of the second‐order robust exact differentiator: A describing function approach

AbstractThe paper proposes a design methodology for the second‐order robust exact differentiator in the presence of small time‐delays characterizing the time of signal processing. With this aim, the describing function is used to provide several sets of the gains ensuring high accuracy in the reconstruction of the first‐order derivative of smooth signals, that is, minimizing the main harmonic of chattering. The proposal fulfills the Loeb's criterion: if self‐excited oscillations appear in the trajectories of the delayed differentiator, they are orbital and asymptotically stable. Besides the global finite‐time stability of the error is proved for the non‐delayed differentiator under such gains settings by using a homogeneous Lyapunov function. Numerical examples are provided to illustrate the results.

Static breakdown threshold modeling of quasi-uniform gas gaps with a focus on the PDIV of contacting enameled wire pairs

This paper provides a theoretical and conceptual framework for the determination of static breakdown inception thresholds in quasi-uniform gas gaps bounded by dielectric layers of thickness s and relative permittivity . The special case of uncoated metallic electrodes is included in the limit . Moreover, a review of breakdown mechanisms and the underlying physical processes in quasi-uniform gas gaps is provided, and the applicability of the associated breakdown criteria is discussed. The results include a parametrization of the partial discharge inception voltage of wedge-shaped air gaps at atmospheric pressure as a function of the reduced coating thickness . The predicted results for this prototypical insulation geometry agree well with a broad array of literature data and own measurements. Finally, a physically motivated parametrization of the ionization threshold K(d) is suggested for atmospheric pressure air in terms of secondary electron feedback by energetic photons and ion-enhanced field emission. This contrasts with the ad hoc introduction of modified ‘streamer constants’ often found in literature to explain breakdown voltages that deviate from the predictions of the classical Raether–Meek–Loeb criterion.

Design of a proportional integral derivative‐like continuous sliding mode controller

AbstractA describing function‐based design for the continuous sliding mode controller in the proportional integral derivative (PID) form (PID‐CSMC) is provided. Two sets of the PID‐CSMC gains are suggested. The Harmonic Balance is used to predict the amplitude and frequency of the main harmonic of chattering caused by the presence of fast‐parasitic dynamics in the closed‐loop. Predicted values of amplitude and frequency allow to compute the average power needed to maintain the trajectories of the system into real sliding modes. The methodology for selection of suboptimal PID‐CSMC gains consists of amplitude of chattering and average power minimization, taking into account the presence of a critically damped actuator parameterizing the effects of parasitic dynamics. A novel homogeneous Lyapunov function proves that suggested sets of the PID‐CSMC gains ensure, in theory, finite‐time stability of the perturbed double integrator without parasitic dynamics, and exact compensation of Lipschitz perturbations. On the other hand, the Loeb's criterion allows to ensure that the PID‐CSMC with suggested gains generates orbitally asymptotically stable fast‐oscillations in the presence of fast‐actuators.

Analysis of orbital stability in lure system, based on dynamic harmonic balance

A criterion of orbital stability of a limit cycle in a Lure system is formulated using the dynamic harmonic balance (DHB) principle and the describing function (DF) method. It is demonstrated that a more precise formulation of orbital stability than that provided by the Loeb's criterion can be produced. This enhancement is achieved through elimination of the assumption that is used in Loeb's criterion derivation. It is demonstrated in the paper that this assumption does not hold. An example of analysis is given.

Loeb's criterion and ‘latch-up’ effect in second order oscillators with single OA

The aim of this paper is to show that Loeb's criterion in 2nd-order oscillators with a single operational amplifier enables a linear interpretation. The ‘latch-up’ effect in these systems can be also predicted by a linear theory

On Limit Cycles of Feedback Systems which Contain a Hysteresis Nonlinearity

In this paper, we concern ourselves with the stability of limit cycles in feedback systems which have hysteresis nonlinearities. Although the quasi-static analysis of limit cycles (Loeb criterion) predicts, in most cases correctly, the stability properties of limit cycles, it is well known that analyses which are based on the method of describing functions may lead to erroneous conclusions. In this paper, we show to what extent the describing function method can be given a rigorous mathematical basis. We show that for a specific example, the main result of this paper predicts correctly the stability of a limit cycle while the Loeb criterion yields an incorrect result. Also, we show that our analysis explains to a certain extent the presence of distortions in solutions of the class of feedback systems considered herein. In arriving at the main result of this paper, use is made of several known facts for functional differential equations and of a result on integral manifolds.

On the stability of limit cycles in nonlinear feedback systems: Analysis using describing functions

In this paper we establish computable conditions for the stability and instability of limit cycles in nonlinear feedback systems. In the proof of the present results, we make use of several novel transformations, of averaging, and of a result on integral manifolds and we assume that we can establish the existence of limit cycles by means of the describing function method. Our results, which in part justify the popular quasistatic stability analysis of limit cycles (Loeb's criterion), are significantly different from existing results dealing with the stability analysis of limit cycles. We demonstrate the applicability of our results by means of specific examples.

Some observations on single-element control sine-wave oscillators

A new generalized approach to the study of the variable frequency sine-wave oscillators in active RC filters having a single-element control facility is presented. The constraints on the filter functions which are amenable for generation of such oscillations with feedback are derived and a number of exemplary circuits are given and also their frequency and gain limits in tables. The oscillation stability is also studied following Loeb's criterion of limit cycle stability for systems having saturation characteristics in the active parts. The circuits are also shown to yield VCO's with voltage controlled FET as the frequency controlling parameter, The possible configurations of the amplifier circuits with operational amplifiers as the building block are considered including a discussion on bandwidth limitation.

Limit cycles in high-order nonlinear systems

This paper investigates the stability of limit cycles in nonlinear feedback systems with several describing function (d.f.) predicted limit cycle solutions. The investigations showed that for a single-frequency limit cycle the necessary conditions for stability of Loeb and Gelb could be satisfied yet the limit cycle was unstable. Also the simple and often used graphical criterion was found to be incorrect. On the other hand, the incremental-describing function (i.d.f.) method used for both asynchronous and synchronous perturbations gave sufficient conditions, within the expected d.f. accuracy, for all the situations. It is further shown that the synchronous i.d.f. and the Loeb criterion are normally identical. The combined mode limit cycle oscillations were also satisfactorily predicted by a twoinput d.f. and i.d.f.

Stability of oscillations in low-order nonlinear systems

Loeb's criterion and the incremental-input describing function (IDF) stability technique are compared for general nonlinear second- and third-order systems. Situations in which the two stability techniques disagree are determined and studied in detail. The IDF stability technique is related directly to the variational approach for a general system, and Loeb's criterion is related to that for second-order systems. The IDF stability technique is shown to involve the approximation of time averaging in addition to the inherent describing function (DF) approximation. Discrepancies between the two are found to coincide with relatively poor DF approximations. One technique cannot be considered better than the other in applications-they should be used together for proper interpretation.

A general stability criterion of the oscillations in nonlinear systems

A general stability criterion of the oscillations in systems containing a memoryless nonlinearity is derived. A new more general form of Loeb's criterion is also obtained by considering the variation of the generalized transfer function of the nonlinear element with frequency.

Limit cycles in time-delay relay systems†

A system is described for which Loeb's criterion for the determination of stability of limit cycles fails. Details of an investigation into the reasons for this failure are given. The Hamel locus is extended to include time-delay in a general way and the results applied to the above system along with associated sampled-data techniques for the accurate determination of stability. The cause of failure is discussed in detail.

The Stability of Oscillations in Nonlinear Networks

In this correspondence, a necessary condition for the stability of oscillations in nonlinear networks is proved. It is shown that this condition is necessary and sufficient for second-order systems; it is also equivalent to the well-known Loeb criterion for second-order systems.