no simple method for evaluating the accuracy of Uncertainty throughout the analysis about the accuracy. * That is, the nonlinear differential equation governing the system is typically Is related to the inaccuracy of the method and, in particular, to the The describing-function method, which can be remedied to a considerable extent with machine aids, "The third and most basic difficulty Systems F behaves as a low-pass filter with the result that the higherĪside from at least two computational difficulties'' associated with Smaller amplitude than the fundamental and, third, in most feedback Present j second, the harmonics of the output of '1/1 are ordinarily of The first harmonic approximation is often "justified" on three grounds:įirst, no significant subharmonic components of x(t) are ordinarily Is valid, the usual frequency response methods can be employed. Input signal level, and to the extent that the describing-function approximation (sometimes called the "first harmonic approximation") Thus the nonlinear element is treated as an element with a gain that varies with Input (this ratio is called the describing function for '1/1). Hence it is assumed that the input to the nonlinearĭevice is a sinusoid and that '1/1 is characterized by the ratio of the fundamental component of its output to the amplitude of the sinusoidal The describing-function approach simplifies the problem by assuming that the output is periodic and that the only significant frequency component of the output is that component at the The well-known special techniques applicable to second-order systemsĬannot be used. Typically the system is of high order* so that The Sinusoidal Response of the System in Fig. 1 the input signal y is related to the output signal x by the The normalization !( a + (j) = 1 permits some simplification in the Space ffi into itself is said to be a contraction if there exists a number Let ffi = be an arbitrary metric space." A mapping A of the Results relating to the functional equation that governs the behavior The remaining sections are concerned with mathematical Some assumptions and notation, and discuss the describing-function Section III we describe the physical system to be studied, introduce Some mathematical preliminaries are considered in Section II. Under which subharmonic response components and self-sustained The use of the describing-function method. The intuitive engineering arguments that are often employed to justify THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1964ĭeriv e an upper bound on the mean-square error incurred by applying T However, Borne interesting relevant ideas have been presented by Johnson.t * The describing-function technique was discovered independently by engineers to an arbitrary periodic input with the same period, and we We present conditions under which there exists a unique periodic response, with a given In this paper we study a broad class of nonlinear control systemsĬontaining a single memoryless nonlinear element. Has been no rigorous discussion of its validity. Regard to predicting the existence of self-sustained oacillations.! there Is ordinarily restricted to systems containing only one nonlinear element, it is assumed that the response is periodic, with only the component at the input frequency significant.Īlthough the describing-function technique is of considerable practical value and indeed is one of the most powerful analytical toolsĪvailable to the control system synthesist, it appears that, except with This approach, * which is applicable to systems of any order but which Response of nonlinear control systems to sinusoidal input signals. The describing-function technique is often used to determine the Subharmonic response components and self-sustained oscillations cannot Conditions are also presented under which
The expression for the error reflects the intuitive engineering arguments that are often employed to justify the use of the describing-function method. To an arbitrary periodic input with the same period, and we derive an upperīound on the mean-square e7'1'Or incurred by applying the describingfunction technique. Under which there exists a unique periodic response, with a given period, In this paper we study a broad class of nonlinecr controt systems containing a single memorflese tumlinear element.
0 kommentar(er)
