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DUAL-INPUT DESCRIBING b FUNCTION (DIDF) 6.0 INTRODUCTION The two-sinusoid-input describing function (TSIDF) of Chap. 5 is certainly again brought to mind by the title of this chapter, as are all other describing functions which simultaneously accommodate two nonlinearity input waveforms. Because of the specific utility of the particular describing function of this class to be discussed presently, however, we reserve for it the otherwise general appellation dual-input describing function, and related abbreviation DIDF. TSIDF application to nonlinear control systems is conceptually limited by requiring that only sinusoids be described in the nonlinearity test input. For this reason an alternative linearization accommodating two inputs was sought by researchers interested, among other things, in the approximate forced response behavior of certain nonlinear systems. The DIDF, as we shall now see, is a physically motivated linearization of a nonlinearity which readily permits study, among other things, of the forced responses of limit cycling nonlinear systems. In what follows it is required that command
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298 DUALINPUT DESCRIBING FUNCTION (DIDF) inputs do not cause the limit cycle to terminate, an assumption that can be verified when under suspicion. The nonlinearity input is then comprised of the limit cycle and a component due to the command input. By assuming that the component due to the command input varies little during a limit cycle period, one can formulate a nonlinearity linearization similar in concept to the TSIDF, but far simpler to calculate. Hence the DIDF model input waveform is a bias plus a sinusoid, the latter component effectively serving to linearize the nonlinearity gain to the former. The idea of linearizing nonlinear characteristics by means of an additive sinusoid is not a new one. MacColl (Ref. 12) described the use of such a signal in a motor-drive system incorporating a relay. Loeb (Ref. 10) has suggested that any nonlinear system can be treated in this manner. Lozier (Ref. 11) is credited with a method of treating oscillating control systems using this linearization, an interpretation later independently arrived at by Li and Vander Velde (Ref. 9) in connection with limit cycling adaptive feedback control system applications. Other papers by Gelb (Refs. 3-5) followed this interpretation and further developed the dynamic charac- terization of limit cycling systems. Oldenburger experimentally discovered the effects of an additive high-frequency low-amplitude input to a control system, and subsequently provided analytical justification via DIDF consider- ations for these effects in several papers on "signal stabilization" (Refs. 16-18). Popov (Ref. 20) and later Popov and Pal'tov (Ref. 21) have published books in which DIDF harmonic linearization is treated.
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This note was uploaded on 02/03/2012 for the course AERO 16.30 taught by Professor Ericferon during the Spring '04 term at MIT.

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