Describing function

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

Automatic tuning of systems with one or two unstable poles

Abstract: Stabilizability and identification of systems with one or two unstable poles is addressed. Two limit points of conditional stability are characterized using closed-loop dynamics with a relay and a novel nonlinear element. An algorithm is developed to analyze information from the resultant stable limit cycles in the process input and output. An easily tuned cascade PID control structure is then proposed to stabilize the system and achieve desirable performance. The proposed autotuning technique is tested with good results on a wide variety of unstable systems.

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.

Feedback control of limit cycle amplitudes

Abstract: A bifurcation control problem of modifying the amplitudes of limit cycles via feedback is studied. A graphical approach reminiscent to the familiar describing function method is developed for validating the harmonic balance approximations of both the amplitude and the frequency of the system oscillatory outputs, starting from the Hopf bifurcation mechanism. The second, fourth, and sixth-order harmonic balance approximations provide a sequential graphical testing for the convergence of the oscillatory outputs, thereby yielding an accurate approximation of the desired limit cycles of small amplitudes. The knowledge of degenerate Hopf bifurcations and the associate Poincare normal forms are useful for formulating the control objective: to capture small-amplitude oscillatory system outputs and to avoid unstable equilibria or other complicated limit sets. A power system example is included for illustration.

On the Application of the Describing Function Technique to the Bifurcation Analysis of Nonlinear Systems

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

Using characteristic loci in the Hopf bifurcation

Abstract: The Hopf bifurcation theorem is used by physicists, biologists and others interested in oscillations in systems described by differential equations. This paper presents a version of the theorem applicable directly to feedback systems, using the method of characteristic loci in an interpretation which looks rather like a multiple-loop version of the graphical describing function method. Frequency, amplitude and stability of oscillations about an equilibrium state are predicted by an iterative process which starts with a 2nd-order harmonic balance and successively increases the number of harmonics taken into account. The results from each stage of the process are used to define a curve in the complex plane from which the results of the next stage can be read off; consequently, the user has a visual indication of the accuracy and rate of convergence. The main theoretical limitation is that the nonlinear elements in the system must be continuously differentiable at least once more than the order of the highest harmonic used, whereas the main practical limitation is that the formula for the curve becomes extremely complicated if more than about six harmonics are considered.