Since Goldfarb's original work on describing functions, a considerable number of papers have been published in which the describing functions of particular nonlinearities have been derived. It appears however that little effort has been made to classify the nonlinearities. Since the describing function method is one of the more powerful methods available at present to analyze nonlinear feedback systems, it appears desirable to collect the expressions for the describing functions of a few different types of nonlinearities in one paper. It is the purpose of this paper to derive the describing functions of two general types of nonlinearities and show how the describing functions of many other practical types of nonlinearities for which the describing function analysis is valid naturally follow.
[Show abstract][Hide abstract] ABSTRACT: A graphical technique is presented for determining the closed loop response of nonlinear control systems driven by sinusoidal inputs. The nonlinear portion of the system is represented by its conventional describing function, which may be frequency dependent as well as amplitude dependent. The linear portion of the system is represented by its complex frequency response function G(jomega) . A transparent overlay is used to mechanize a functional transformation similar to that performed by a Nichols chart, allowing rapid determination of the system output for sinusoidal inputs. The accuracy of the method is limited by the accuracy of the describing function approximation. In addition to offering a rapid solution to what has been regarded as a time consuming problem, the method gives the designer sufficient insight into the behavior of the system to allow the intelligent choice of compensating networks to improve system performance. A numerical example is used as a vehicle for discussion of compensation, and experimental results are presented to verify the analysis.
IRE Transactions on Automatic Control 08/1962; DOI:10.1109/TAC.1962.1105464
[Show abstract][Hide abstract] ABSTRACT: Recently, considerable attention has been devoted to the analysis of high-order systems containing severe nonlinearities separated by linear functions, particularly those using bang-bang or on-off controllers. These have proven to be very satisfactory in the attitude control systems of manned or unmanned spacecraft and satellites. This paper develops describing functions for a particularly complicated multiple nonlinearity: a tri-stable (bang-bang with dead zone) characteristic, followed by a linear integrator with a constrained range of integration. It should be noted that constraining range of integration is not equivalent to simple limiting of an integrator's output. This system of nonlinearities has not previously been treated in the literature, although it is founds for example, in satellite attitude controllers where on-off torques give rise to constrained momentum wheel angular velocities or in guided missile hydraulic actuators with bang-bang power spool and control surface position limits. This frequency-variant nonlinearity has three distinct modes of operation and, therefore, is quite different from the usual single nonlinearity considered for describing function application. The describing function and boundary equation for each mode are derived in the paper, and numerical examples are given. The analytical results were found to agree well with results from an analog computer simulation. These describing functions may be used to size power actuators ands therefores should be very useful for preliminary systems design.
IEEE Transactions on Automatic Control 11/1967; DOI:10.1109/TAC.1967.1098681 · 2.78 Impact Factor
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