描述函数
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描述函数(describing function)也称为谐波平衡法,是控制系统中的方法,由Nikolay Mitrofanovich Krylov及尼古拉·博戈柳博夫在1930年代提出[1][2],后来由Ralph Kochenburger延伸[3],是用近似方式来分析非线性控制问题的作法。描述函数是以准线性为基础,是用会依输入波形振幅而变化的线性时不变传递函数来近似非线性系统的作法。依照定义,真正线性时不变系统的传递函数不会随输入函数的振幅而变化(因为是线性系统)。因此,其和振幅的相依性就会产生一群的线性系统,这些系统结合起来的目的是为了近似非线性系统的特性。描述函数是少数广为应用来设计非线性系统的方法,描述函数是在分析闭回路控制器(例如工业过程控制、伺服机构、电子振荡器)的极限环时,常见的数学工具。
原理
考虑一个慢速,稳定的线性系统,其回授路径中有不连续(但有分段连续)的非线性特性(例如有饱和的放大器、或是有死区效应的元件)。在非线性元件上看到的连续区域会视线性系统的振幅而定。若线性系统输出的振幅变小,其非线性元件的特性可能又会变换到另一个区域。这种在二个连续区间之间的切换会造成周期性的振荡。描述函数方式法目的是要预设这些振荡的特性(也就是其基频),作法是假设慢速系统特性类似低通滤波器或带通滤波器,会将能量集中在单一频率。即使输出波形有多个不同的模态,描述函数仍可以提供有关频率的资讯,也许也包括振幅相关的资讯。此情形下,描述函数有点类似在描述回授系统的滑动模式。
利用低通滤波器的假设,系统响应可以表示为一组正弦曲线中的一个弦波。此情形下,系统可以表示为弦波描述函数(SIDF),是对振幅为A,频率为的弦波输入的系统响应。SIDF是描述线性函数传递函数 的变体。在准线性系统中,输入信号为弦波时,其输出也是相同频率的弦波,但其振幅及相位的关系可以用表示。以此观点来看,许多系统在弦波输入下的响应虽不一定是纯弦波,但大部分输出能量集中在是和输入信号相同的频率,因此可以近似为准线性系统。其原因是这类系统在其本质上有低通或是带通的特性,因此高次的谐波受到了抑制,也有可能是特意加入了滤波器。弦波描述函数(SIDF)的重要用途之一是消除弦波电子振荡器的不理想讯号。
考虑非线性系统,在弦波输入下,其描述函数可以表示为,其中的实部g及虚部b可以表示如下:
也有其他型式的描述函数,例如水平输入以及高斯噪声输入的描述函数。描述函数无法完整的描述系统,不过多半已可以处理像是控制或是稳定性的问题。描述函数最适用于分析非线性程度相对轻微的系统。此外,高阶弦波输入描述函数(HOSIDF)描述非线性系统在弦波输入下,其各阶谐波成分的振幅及相位。高阶弦波输入描述函数是描述函数是延伸版本,用在响应的非线性程度非常明显的场合。
注意事项
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.[4] A fairly similar example is a closed-loop oscillator consisting of a non-inverting 施密特触发器 followed by an inverting 积分器 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.[5]
Also, in the case where the conditions for 阿依热尔曼猜想 or 卡尔曼猜想 are fulfilled, there are no periodic solutions by describing function method,[6][7] but counterexamples with periodic solutions (隐藏振荡) are well known. Therefore, the application of the describing function method requires additional justification.[8][9]
参考资料
- ^ Krylov, N. M.; N. Bogoliubov. Introduction to Nonlinear Mechanics. Princeton, US: Princeton Univ. Press. 1943. ISBN 0691079854.
- ^ Blaquiere, Austin. Nonlinear System Analysis. Elsevier Science. : 177. ISBN 0323151663.
- ^ Kochenburger, Ralph J. A Frequency Response Method for Analyzing and Synthesizing Contactor Servomechanisms. Trans. AIEE (American Institute of Electrical Engineers). January 1950, 69 (1): 270–284 [June 18, 2013]. doi:10.1109/t-aiee.1950.5060149.
- ^ Tsypkin, Yakov Z. Relay Control Systems. Cambridge: Univ Press. 1984.
- ^ Boris Lurie; Paul Enright. Classical Feedback Control: With MATLAB. CRC Press. 2000: 298–299. ISBN 978-0-8247-0370-7.
- ^ Leonov G.A.; Kuznetsov N.V. Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems (PDF). Doklady Mathematics. 2011, 84 (1): 475–481. doi:10.1134/S1064562411040120.,
- ^ Aizerman's and Kalman's conjectures and describing function method (PDF).
- ^ Bragin V.O.; Vagaitsev V.I.; Kuznetsov N.V.; Leonov G.A. Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits (PDF). Journal of Computer and Systems Sciences International. 2011, 50 (4): 511–543. doi:10.1134/S106423071104006X.
- ^ Leonov G.A.; Kuznetsov N.V. Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. International Journal of Bifurcation and Chaos. 2013, 23 (1): art. no. 1330002. doi:10.1142/S0218127413300024.
延伸阅读
- 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)