查看“︁描述函數”︁的源代码

'''描述函數'''（describing function）是[[控制系統]]中用近似方式處理[[非線性系統]]的方法，由{{link-en|Nikolay Mitrofanovich Krylov|Nikolay Mitrofanovich Krylov}}及[[尼古拉·尼古拉耶维奇·博戈柳博夫|尼古拉·博戈柳博夫]]在1930年代提出<ref name="Krylov">{{cite book
  | last = Krylov
  | first = N. M.
  | author2 = N. Bogoliubov
  | title = Introduction to Nonlinear Mechanics
  | publisher = Princeton Univ. Press
  | year = 1943
  | location = Princeton, US
  | pages = 
  | url = http://libra.msra.cn/Publication/3271320/introduction-to-nonlinear-mechanics
  | doi = 
  | id = 
  | isbn = 0691079854
  | access-date = 2019-05-02
  | archive-date = 2013-06-20
  | archive-url = https://archive.today/20130620041142/http://libra.msra.cn/Publication/3271320/introduction-to-nonlinear-mechanics
  | dead-url = yes
  }}</ref><ref name="Blaquiere">{{cite book   
  | last = Blaquiere
  | first = Austin 
  | title = Nonlinear System Analysis
  | publisher = Elsevier Science
  | location = 
  | pages = 177
  | url = https://books.google.com/books?id=LC2_lK9HZQgC&pg=PA177&dq=krylov+bogoliubov
  | doi = 
  | id = 
  | isbn = 0323151663}}</ref>，後來由Ralph Kochenburger延伸<ref name="Kochenburger">{{cite journal
  | last = Kochenburger
  | first = Ralph J.
  | title = A Frequency Response Method for Analyzing and Synthesizing Contactor Servomechanisms
  | journal = Trans. AIEE
  | volume = 69
  | issue = 1
  | pages = 270–284
  | publisher = American Institute of Electrical Engineers
  | location = 
  | date = January 1950
  | url = http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=5060149&contentType=Journals+%26+Magazines
  | issn = 
  | doi = 10.1109/t-aiee.1950.5060149
  | id = 
  | accessdate = June 18, 2013
  | archive-date = 2016-03-04
  | archive-url = https://web.archive.org/web/20160304195729/http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=5060149&contentType=Journals+%26+Magazines
  | dead-url = no
  }}</ref>。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[[线性时不变系统理论|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有關頻率的資訊，也許也包括振幅相關的資訊。此情形下，描述函數有點類似在描述回授系統的[[滑動模式控制|滑動模式]]。

[[File:Function-block-harmonic-balance.png|thumb|center|600px|在諧波平衡下的非線性系統]]
利用低通濾波器的假設，系統響應可以表示為一組[[正弦曲線]]中的一個弦波。此情形下，系統可以表示為弦波描述函數（SIDF）<math>H(A,\,j\omega)</math>，是對振幅為A，頻率為<math>\omega</math>的弦波輸入的系統響應。SIDF是描述線性函數[[传递函数]] <math>H(j\omega)</math>的變體。在準線性系統中，輸入信號為弦波時，其輸出也是相同頻率的弦波，但其振幅及相位的關係可以用<math>H(A,\,j\omega)</math>表示。以此觀點來看，許多系統在弦波輸入下的響應雖不一定是純弦波，但大部份輸出能量集中在是和輸入信號相同的頻率<math>\omega</math>，因此可以近似為準線性系統。其原因是這類系統在其本質上有[[低通滤波器|低通]]或是[[带通滤波器|带通]]的特性，因此高次的諧波受到了抑制，也有可能是特意加入了{{link-en|濾波 (信號處理)|filter (signal processing)|濾波器}}。弦波描述函數（SIDF）的重要用途之一是消除弦波[[电子振荡器]]的非理想訊號。

考慮非線性系統<math>u = f(xd).</math>，在弦波輸入<math>xd=Asin \omega t</math>下，其描述函數可以表示為<math>H(A,\,j\omega)=g+jb,</math>，其中的實部g及虛部b可以表示如下：

: <math>g(A,\,j\omega)= \frac{1}{\pi A} \int_0^{2\pi} f(A \sin \omega t) \sin \omega t d \omega t</math>

: <math>b(A,\,j\omega)= \frac{1}{\pi A} \int_0^{2\pi} f(A \sin \omega t) \cos \omega t d \omega t</math>
也有其他型式的描述函數，例如水平輸入以及高斯雜訊輸入的描述函數。描述函數無法完整的描述系統，不過多半已可以處理像是控制或是穩定性的問題。描述函數最適用於分析非線性程度相對輕微的系統。此外，[[高階弦波輸入描述函數]]（HOSIDF）描述非線性系統在弦波輸入下，其各階諧波成份的振幅及相位。高階弦波輸入描述函數是描述函數是延伸版本，用在響應的非線性程度非常明顯的場合。

==注意事項==
在許多種類的系統中，描述函數都可以產生一定準確度的結果，不過也有些情形會失效。例如在一些很強調其高階諧波特性的系統，描述函數就不一定能發揮作用。Tzypkin曾經用[[起停式控制]]系統為例說明過<ref>{{cite book|last=Tsypkin|first=Yakov Z.|title=Relay Control Systems|publisher=Cambridge: Univ Press|year=1984}}</ref>。另一個比較簡單的例子是由非反相[[施密特触发器]]加上反相[[積分器]]組成的閉迴路振盪器，以積分器的輸出為施密特触发器的輸入。施密特触发器的輸出是方波，而積分器的輸出是[[三角波]]，的三角波的波峰恰好就是方波的切換點。振盪器中的這兩部份輸出都落後輸入90度。若是用描述函數來處此一電路，施密特触发器的輸入會變成頻率為其基頻的弦波，通過触发器也會有延遲，但會比90度要小（弦波觸發触发器的時機會比三角波要快），因此描述函數下，系統振盪的方式會和原系統的不同<ref name="LurieEnright2000">{{cite book|author1=Boris Lurie|author2=Paul Enright|title=Classical Feedback Control: With MATLAB|url=https://archive.org/details/classicalfeedbac0000luri|year=2000|publisher=CRC Press|isbn=978-0-8247-0370-7|pages=[https://archive.org/details/classicalfeedbac0000luri/page/298 298]–299}}</ref>。

若是符合[[阿依熱爾曼猜想]]或[[卡爾曼猜想]]的條件，可能利用描述函數找不到週期解<ref>{{cite journal
 | author1 = Leonov G.A.
 | author2 = Kuznetsov N.V.
 | year = 2011
 | title = Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems
 | journal = Doklady Mathematics
 | volume = 84
 | number = 1
 | url = http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf
 | pages = 475&ndash;481
 | doi = 10.1134/S1064562411040120
 | access-date = 2019-05-02
 | archive-date = 2016-03-04
 | archive-url = https://web.archive.org/web/20160304053548/http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf
 | dead-url = no
 }},</ref><ref>{{cite web |url=http://www.math.spbu.ru/user/nk/PDF/Harmonic_balance_Absolute_stability.pdf |title=Aizerman's and Kalman's conjectures and describing function method |access-date=2019-05-02 |archive-date=2016-03-04 |archive-url=https://web.archive.org/web/20160304112151/http://www.math.spbu.ru/user/nk/PDF/Harmonic_balance_Absolute_stability.pdf |dead-url=no }}</ref>，相反的，也有週期解的反例（[[隱藏振盪]]）。因此描述函數的應用也需要確認是否適合<ref>{{cite journal
 | author1 = Bragin V.O.
 | author2 = Vagaitsev V.I.
 | author3 = Kuznetsov N.V.
 | author4 = Leonov G.A.
 | year = 2011
 | title = Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits
 | journal = Journal of Computer and Systems Sciences International
 | volume = 50
 | number = 4
 | pages = 511&ndash;543
 | url = http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf
 | doi = 10.1134/S106423071104006X
 | access-date = 2019-05-02
 | archive-date = 2016-03-04
 | archive-url = https://web.archive.org/web/20160304045017/http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf
 | dead-url = no
 }}</ref><ref name=2011-IJBC-Hidden-attractors>{{cite journal | 
author1 = Leonov G.A. | 
author2 = Kuznetsov N.V. | 
year = 2013 | 
title = Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits | 
journal = International Journal of Bifurcation and Chaos | 
volume = 23 | 
issue = 1 | 
pages = art. no. 1330002 | 
url = http://www.worldscientific.com/doi/pdf/10.1142/S0218127413300024 | 
doi = 10.1142/S0218127413300024 | 
access-date = 2019-05-02 | 
archive-date = 2019-03-24 | 
archive-url = https://web.archive.org/web/20190324122428/https://www.worldscientific.com/doi/pdf/10.1142/S0218127413300024 | 
dead-url = no }}</ref>。

==參考資料==
{{Reflist|2}}

==延伸閱讀==
{{refbegin}}
* N. Krylov and N. Bogolyubov: ''Introduction to Nonlinear Mechanics'', Princeton University Press, 1947
* A. Gelb and W. E. Vander Velde: [http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-30-estimation-and-control-of-aerospace-systems-spring-2004/readings/#Downloadable  ''Multiple-Input Describing Functions and Nonlinear System Design''] {{Wayback|url=http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-30-estimation-and-control-of-aerospace-systems-spring-2004/readings/#Downloadable |date=20200522213557 }}, McGraw Hill, 1968.
* James K. Roberge, ''Operational Amplifiers: Theory and Practice,'' [http://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/textbook/MITRES_6-010S13_chap06.pdf chapter 6: Non-Linear Systems] {{Wayback|url=http://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/textbook/MITRES_6-010S13_chap06.pdf |date=20190502145854 }}, 1975; free copy courtesy of <!-- [[MIT OpenCourseWare]] -->[[MIT OpenCourseWare]] 6.010 (2013); see also (1985) video recording of Roberge's lecture on [http://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/course-videos/lecture-15-describing-functions/ describing functions] {{Wayback|url=http://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/course-videos/lecture-15-describing-functions/ |date=20190502145901 }}
* 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)
{{refend}}
==相關條目==
*{{le|諧波平衡法|Harmonic Balance}}
== 外部連結==
*[http://www.ee.unb.ca/jtaylor/Publications/EEncyc_final.pdf Electrical Engineering Encyclopedia: Describing Functions] {{Wayback|url=http://www.ee.unb.ca/jtaylor/Publications/EEncyc_final.pdf |date=20111003003202 }}

{{DEFAULTSORT:Describing Function}}
[[Category:非線性控制]]