Describing function

About: Describing function is a research topic. Over the lifetime, 1742 publications have been published within this topic receiving 26702 citations.

A Robust Relay Feedback Structure for Processes Under Disturbances: Analysis and Applications

Abstract: In this paper, a robust relay feedback structure is used to provide a stable oscillation under large static disturbances or drift. This relay structure is composed of a block to remove static disturbance or drift followed by a relay. The robust relay structure is analyzed using the describing function method. Furthermore, based on Poincare map, the conditions for the existence and local stability of the limit cycle are obtained. In order to estimate multiple frequency points of the process, a step change and periodic square waves with frequencies lower than the ultimate frequency are applied at the process input. Thus, these frequencies information are used to identification of first-order plus time delay models over a wider frequency region. For these models, a phase adjustment is proposed from the frequencies information of the additional signal. From simulated and experimental studies, it is verified that the robust relay structure provides a symmetrical oscillation at the process output, independent of the disturbance in the process input and iterations between the loops.

Prediction of limit cycles in nonlinear systems with reset controllers using describing function

Abstract: In nonlinear systems, limit cycles may exsist in some's parameters and may cause some unexpected damage. This paper proposes a design method of a reset controller for the Reset control of limit cycles. We consider a plant consisting of a linear system with a nonlinear feedback element and show a condition of the parameter for which limit cycles exists, using describing functions of the nonlinear element and the reset controller. The adjustable parameter of the reset controller is determined by the intersection of the inverse Nyquist locus of the linear system and the existence region of the limit cycles on the complex plain. We apply the proposed design method of the reset controller to the van der Pol equation and show its effectiveness.

A double input describing function approach for stability analysis in centerless grinding under interrupted cut

Abstract: This article presents a novel stability analysis of plunge centerless grinding. The analysis considers the nonlinearity associated to wheel-workpiece detachment under large waviness. The loss of contact is approximated by a harmonic linearization, in the frequency domain, by a double input describing function (DIDF). The stability analysis shows the effects of clipping and structural compliance: both clearly produce a waviness with a quasi-integer number of lobes. Our approach removes the need of additional hypotheses, sometime found in the literature. Under increasing lobes amplitude, clipping reduces waviness growth rate until a limit cycle is reached. Numerical and experimental verifications are provided.

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.

Modelling of the PRC-LCC resonant topology with a capacitor as output filter using EDF

Abstract: In this paper, the extended describing function (EDF) and the generalised averaging modelling techniques have been applied to a PRC-LCC resonant AC/DC power conversion topology with a capacitor as output filter. In this way, a mathematical large signal model has been obtained to describe the topology dynamic behaviour. This large signal model consists of a set of nonlinear differential equations which are solved numerically with the help of a computer. The resulting algorithm is faster than a straight PSpice simulation and free of convergence problems. From this dynamical model, a very simple steady state model has also been deduced. In the paper it is shown and particularised for the operation along the optimum switching tine. Finally, the accuracy of the presented models is verified with a wide set of experimental results.