查看“︁卡爾曼猜想”︁的源代码

[[File:Block scheme of control system.jpg|thumb|
圖1：控制系統的方塊圖，其中的''G''(''s'')是線性傳遞函數，而''f''(''e'')是單值連續可微的函數]]

'''卡爾曼猜想'''（Kalman's conjecture）或'''卡爾曼問題'''（Kalman problem）是已找到反例的[[猜想]]，是針對[[非線性控制]]系統，其中有一個純量非線性元素，此系統在線性穩定區間內的絕對穩定性。卡爾曼猜想是[[阿依熱爾曼猜想]]的加強版本，也是{{link-en|Markus–Yamabe猜想|Markus–Yamabe conjecture}}的特例。卡爾曼猜想雖已證實為否，不過帶出了（有效的）[[非線性控制|絕對穩定性的充份準則]]。

== 卡爾曼猜想的數學描述（卡爾曼問題）==
[[鲁道夫·卡尔曼]]在1957年的論文<ref name=1957-Kalman>{{cite journal
 | author = Kalman R.E. 
 | year = 1957
 | title = Physical and Mathematical mechanisms of instability in nonlinear automatic control systems 
 | journal = Transactions of ASME
 | volume = 79
 | number = 3
 | pages = 553&ndash;566
}}
</ref>中提到：
<blockquote>若圖1中的''f''(''e'')用''e''乘上常數''K''取代，''K''對應''f''<nowiki>'</nowiki>(''e'')中所有的可能值，發現閉迴路系統在所有''K''值下都收斂。在直覺上會認為此系統是單調穩定的，也就是說，所有暫態的解都會收斂到唯一、穩定的臨界點。
</blockquote>

卡尔曼的描述可以寫成以下的猜想<ref name=2011-DAN-Kalman>{{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
 |pages        = 475&ndash;481
 |url          = http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf
 |doi          = 10.1134/S1064562411040120
 |access-date  = 2019-05-11
 |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>：
<blockquote>
考慮一個有單一純量非線性函數的函數
:<math>  
  \frac{dx}{dt}=Px+qf(e),\quad e=r^*x \quad x\in R^n,
</math>
其中''P''是常數的''n''×''n''矩陣，''q''、''r''是常數的''n''維向量，∗是轉置符號，''f''(''e'')是純量函數，''f''(0)&nbsp;=&nbsp;0。假設，''f''(''e'')是可微分函數，而且滿足以下條件
:<math>
 k_1 < f'(e)< k_2. \,
</math>
。則卡爾曼猜想是指此系統在大區域穩定（也就是其唯一駐點為全域[[吸引子]]）若配合''f''(''e'')&nbsp;=&nbsp;''ke'', ''k''&nbsp;∈&nbsp;(''k''<sub>1</sub>,&nbsp;''k''<sub>2</sub>)的所有線性系統都是漸近穩定。
</blockquote>

[[阿依熱爾曼猜想]]要求非線性導數的條件，而卡爾曼猜想要求非線性本身要在線性區間內。

卡爾曼猜想在''n''&nbsp;≤&nbsp;3時成立，若是''n''&nbsp;>&nbsp;3，存在有效生成反例的作法<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 = 5
 | 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 = 2019-05-11
 | 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 = 2019-05-11 | 
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>：非線性的導數在線性穩定區間內，存在唯一穩定的平衡點，以及一個穩定的周期解（[[隱蔽振盪]]）。

在離散系統下，卡爾曼猜想只在n=1時成立，在''n''&nbsp;≥&nbsp;2時可以建構反例<ref>{{cite journal
 |author1=Carrasco J. |author2=Heath W. P. |author3=de la Sen M. | year = 2015
 | title = Second-order counterexample to the Kalman conjecture in discrete-time
 | journal = 2015 European Control Conference
}}
</ref><ref>{{cite journal
 |author1=Heath W. P. |author2=Carrasco J |author3=de la Sen M. | year = 2015
 | title = Second-order counterexamples to the discrete-time Kalman conjecture
 | journal = Automatica | doi = 10.1016/j.automatica.2015.07.005
}}
</ref>。

== 參考資料 ==
{{Reflist}}

== 延伸閱讀 ==
*{{cite journal
 | author1 = Leonov G.A.
 | author2 = Kuznetsov N.V.
 | year = 2011
 | title = Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems
 | journal = IFAC Proceedings Volumes (IFAC-PapersOnline)
 | volume = 18
 | number = 1
 | pages = 2494&ndash;2505
 | url = http://www.math.spbu.ru/user/nk/PDF/2011-IFAC-Hidden-oscillations-control-systems-Aizerman-problem-Kalman.pdf
 | doi = 10.3182/20110828-6-IT-1002.03315
 | access-date = 2019-05-11
 | archive-date = 2020-07-09
 | archive-url = https://web.archive.org/web/20200709161140/https://www.math.spbu.ru/user/nk/PDF/2011-IFAC-Hidden-oscillations-control-systems-Aizerman-problem-Kalman.pdf
 | dead-url = no
 }}

== 外部連結 ==
* [http://www.math.spbu.ru/user/nk/PDF/Harmonic_balance_Absolute_stability.pdf Analytical-numerical localization of hidden oscillation in counterexamples to Aizerman's and Kalman's conjectures]{{Wayback|url=http://www.math.spbu.ru/user/nk/PDF/Harmonic_balance_Absolute_stability.pdf |date=20160304112151 }}
* [http://maplecloud.maplesoft.com/application.jsp?appId=5742954790518784 Discrete-time counterexample in Maplecloud]{{Wayback|url=http://maplecloud.maplesoft.com/application.jsp?appId=5742954790518784 |date=20160528205728 }}

[[Category:已證否的猜想]]
[[Category:非線性控制]]