描述函數

描述函數（describing function）是控制系統中的方法，由Nikolay Mitrofanovich Krylov及尼古拉·博戈柳博夫在1930年代提出，後來由Ralph Kochenburger延伸，是用近似方式來分析非線性控制問題的作法。描述函數是以準線性為基礎，是用會依輸入波形振幅而變化的线性时不变传递函数來近似非線性系統的作法。依照定義，真正线性时不变系統的传递函数不會隨輸入函數的振幅而變化（因為是線性系統）。因此，其和振幅的相依性就會產生一群的線性系統，這些系統結合起來的目的是為了近似非線性系統的特性。描述函數是少數廣為應用來設計非線性系統的方法，描述函數是在分析閉迴路控制器（例如工業過程控制、伺服機構、电子振荡器）的极限环時，常見的數學工具。

原理

考慮一個慢速，穩定的線性系統，其回授路徑中有不連續（但有分段連續）的非線性特性（例如有飽和的放大器、或是有死區效應的元件）。在非線性元件上看到的連續區域會視線性系統的振幅而定。若線性系統輸出的振幅變小，其非線性元件的特性可能又會變換到另一個區域。這種在二個連續區間之間的切換會造成週期性的振荡。描述函數方式法目的是要預設這些振盪的特性（也就是其基頻），作法是假設慢速系統特性類似低通滤波器或带通滤波器，會將能量集中在單一頻率。即使輸出波形有多個不同的模態，描述函數仍可以提供一些有關頻率的資訊，也許也包括一些振幅相關資訊。此情形下，描述函數有點類似在描述回授系統的滑動模式。

在諧波平衡下的非線性系統

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Using this low-pass assumption, the system response can be described by one of a family of 正弦曲線s; in this case the system would be characterized by a sine input describing function (SIDF) $H(A,\,j\omega )$ giving the system response to an input consisting of a sine wave of amplitude A and frequency $\omega$. This SIDF is a modification of the 传递函数 $H(j\omega )$ used to characterize linear systems. In a quasi-linear system, when the input is a sine wave, the output will be a sine wave of the same frequency but with a scaled amplitude and shifted phase as given by $H(A,\,j\omega )$. Many systems are approximately quasi-linear in the sense that although the response to a sine wave is not a pure sine wave, most of the energy in the output is indeed at the same frequency $\omega$ as the input. This is because such systems may possess intrinsic 低通滤波器 or 带通滤波器 characteristics such that harmonics are naturally attenuated, or because external 濾波s are added for this purpose. An important application of the SIDF technique is to estimate the oscillation amplitude in sinusoidal 电子振荡器s.

Other types of describing functions that have been used are DFs for level inputs and for Gaussian noise inputs. Although not a complete description of the system, the DFs often suffice to answer specific questions about control and stability. DF methods are best for analyzing systems with relatively weak nonlinearities. In addition the 高階弦波輸入描述函數s (HOSIDF), describe the response of a class of nonlinear systems at harmonics of the input frequency of a sinusoidal input. The HOSIDFs are an extension of the SIDF for systems where the nonlinearities are significant in the response.

注意事項

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Although the describing function method can produce reasonably accurate results for a wide class of systems, it can fail badly for others. For example, the method can fail if the system emphasizes higher harmonics of the nonlinearity. Such examples have been presented by Tzypkin for 起停式控制 systems. A fairly similar example is a closed-loop oscillator consisting of a non-inverting 施密特触发器 followed by an inverting 積分器 that feeds back its output to the Schmitt trigger's input. The output of the Schmitt trigger is going to be a 方波form, while that of the integrator (following it) is going to have a 三角波form with peaks coinciding with the transitions in the square wave. Each of these two oscillator stages lags the signal exactly by 90 degrees (relative to its input). If one were to perform DF analysis on this circuit, the triangle wave at the Schmitt trigger's input would be replaced by its fundamental (sine wave), which passing through the trigger would cause a phase shift of less than 90 degrees (because the sine wave would trigger it sooner than the triangle wave does) so the system would appear not to oscillate in the same (simple) way.

Also, in the case where the conditions for 阿依熱爾曼猜想 or 卡爾曼猜想 are fulfilled, there are no periodic solutions by describing function method, but counterexamples with periodic solutions (隱藏振盪) are well known. Therefore, the application of the describing function method requires additional justification.

延伸閱讀

• N. Krylov and N. Bogolyubov: Introduction to Nonlinear Mechanics, Princeton University Press, 1947
• A. Gelb and W. E. Vander Velde: Multiple-Input Describing Functions and Nonlinear System Design, McGraw Hill, 1968.
• James K. Roberge, Operational Amplifiers: Theory and Practice, chapter 6: Non-Linear Systems, 1975; free copy courtesy of MIT OpenCourseWare 6.010 (2013); see also (1985) video recording of Roberge's lecture on describing functions
• P.W.J.M. Nuij, O.H. Bosgra, M. Steinbuch, Higher Order Sinusoidal Input Describing Functions for the Analysis of Nonlinear Systems with Harmonic Responses, Mechanical Systems and Signal Processing, 20(8), 1883–1904, (2006)

外部連結

• Electrical Engineering Encyclopedia: Describing Functions