查看“︁阿依熱爾曼猜想”︁的源代码

'''阿依熱爾曼猜想'''（Aizerman's conjecture）或'''阿依熱爾曼問題猜想'''（Aizerman problem）是[[非線性控制]]的猜想，認為一線性系統有非線性的回授，不過是在一個扇形的線性區間內，若線性系統在此扇形線性區間都穩定，則整個系統都會穩定。

阿依熱爾曼猜想在一維系統成立，在二維系統是全域穩定的充份必要條件，而針對維度大於3的情形，這個猜想已找到反證<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 DF method |accessdate=2019-04-04 |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><ref name=JCSSI2011/>，不過後來因此推導出（有效的）[[非線性控制#絕對穩定性問題|非線性控制全域穩定性準則]]。

==阿依熱爾曼猜想的數學描述==
考慮一個系統，其中包括一個純量非線性的函數
:<math>
  \frac{dx}{dt}=Px+qf(e),\quad e=r^*x \quad x\in\mathbb R^n,
</math>
:其中P是常數n×n矩陣、q和r是常數n維向量、∗ 是轉置算子、f(e)是純量函數，且 f(0)=0。假設非線性函數f是有扇型區間的上下限，也就是存在實數<math>
 k_1
</math>及<math>
 k_2
</math>，滿足<math>
 k_1 <k_2
</math>，且函數<math>
 f
</math>滿足
:<math>
 k_1  < \frac{f(e)}{e}< k_2, \quad \forall \; e \neq 0.
</math>

阿依熱爾曼猜想就是指此系統在全域穩定（有唯一穩定點，而且是全域[[吸引子]]）若所有在f(e)=ke, k ∈(k1,k2)下的線性系統都是漸近穩定。

存在阿依熱爾曼猜想的反例，非線性函數在線性穩定的範圍內，且系統除了唯一的穩定平衡點外，還有穩定的週期解—[[隱蔽振盪]]。<ref name=JCSSI2011>{{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 = 511–543
 | url = http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf
 | doi = 10.1134/S106423071104006X
 | access-date = 2019-04-04
 | 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-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 = 475–481
 | url = http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf
 | doi = 10.1134/S1064562411040120
 | access-date = 2019-04-04
 | 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 name=2011-IFAC-hidden-survey>{{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 = 2494–2505
 | 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 = 2019-04-04
 | 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
 }}</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|
doi = 10.1142/S0218127413300024}}
</ref>

[[卡爾曼猜想]]是強化版本的阿依熱爾曼猜想，在非線性回授的部份要求回授的微分需在線性穩定區間內，結果也存在反例。

==參考資料==
{{reflist}}

==延伸閱讀==
*{{cite journal
 | author = Atherton, D.P.
 | author2 = Siouris, G.M.
 | year = 1977
 | title = Nonlinear Control Engineering
 | journal = Systems, Man and Cybernetics, IEEE Transactions on
 | volume = 7
 | issue = 7
 | pages = 567–568
 | url = http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4309773
 | accessdate = 2008-06-30
 | doi = 10.1109/TSMC.1977.4309773
 | archive-date = 2019-07-13
 | archive-url = https://web.archive.org/web/20190713061129/https://ieeexplore.ieee.org/document/4309773/?arnumber=4309773
 | dead-url = no
 }}

==外部連結==
*{{cite web
|url=http://www.math.spbu.ru/user/nk/PDF/Harmonic_balance_Absolute_stability.pdf
|title=Counterexamples to Aizerman's and Kalman's conjectures and describing function method
|accessdate=2019-04-04
|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
}}

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