查看“Kalman–Yakubovich–Popov引理”的源代码

'''Kalman–Yakubovich–Popov引理'''（Kalman–Yakubovich–Popov lemma）是{{link-en|系統分析|system analysis}}及[[控制理论]]的結果，其中提到：給定一數<math>\gamma > 0</math>，二個n維向量B, C，及n x n的[[赫維茲矩陣|赫維茲穩定矩陣]] A（所有特徵值的實部都為負值），若<math>(A,B)</math>具有完全[[可控制性]]，則滿足下式的對稱矩陣P和向量Q
:<math>A^T P + P A = -Q Q^T</math>

:<math> P B-C = \sqrt{\gamma}Q</math>

存在的充份必要條件是
:<math>
\gamma+2 Re[C^T (j\omega I-A)^{-1}B]\ge 0
</math>
而且，集合<math>\{x: x^T P x = 0\}</math>是<math>(C,A)</math>的不可觀測子空間。

此引理可以視為是穩定性理論[[李亞普諾夫方程]]的推廣。建構了由[[狀態空間 (計算機科學)|狀態空間]]A, B, C建構的[[线性矩阵不等式]]以及其[[頻域]]條件的關係。

Kalman–Popov–Yakubovich引理最早是在1962年由{{link-en|Vladimir Andreevich Yakubovich|Vladimir Andreevich Yakubovich}}寫出且證明<ref>{{Cite journal|last=Yakubovich|first=Vladimir Andreevich|date=1962|title=The Solution of Certain Matrix Inequalities in Automatic Control Theory|url=|journal=Dokl. Akad. Nauk SSSR |volume=143 |issue=6|pages=1304–1307|via=}}</ref>，當時列的是嚴格的頻率不等式。允許等於的不等式是由[[鲁道夫·卡尔曼]]在1963年提出<ref name="Kalman1963">{{cite journal|author=Kalman, Rudolf E.|year=1963|title=Lyapunov functions for the problem of Lur'e in automatic control|url=http://www.pnas.org/content/49/2/201.full.pdf|journal=Proceedings of the National Academy of Sciences|volume=49|issue=2|pages=201–205|doi=10.1073/pnas.49.2.201|pmc=299777}}</ref>。在該文中也建立了Lur'e方程可解性的關係。兩篇都是針對純量輸入系統。其控制維度的限制是在1964年被Gantmakher和Yakubovich放寬的<ref>{{Cite book|title=Absolute Stability of the Nonlinear Controllable Systems, Proc. II All-Union Conf. Theoretical Applied Mechanics|last=Gantmakher, F.R. and Yakubovich, V.A.,|first=|publisher=Nauka|year=1964|isbn=|location=Moscow|pages=}}</ref>，而{{link-en|Vasile M. Popov|Vasile Mihai Popov}}也獨立得到相同結論<ref>{{Cite journal|last=Popov|first=Vasile M.|date=1964|title=Hyperstability and Optimality of Automatic Systems with Several Control Functions|url=|journal=Rev. Roumaine Sci. Tech.|volume=9 |issue=4|pages=629–890|via=}}</ref>。在<ref>{{Cite journal|last=Gusev S. V. and Likhtarnikov A. L.|first=|date=2006|title=Kalman-Popov-Yakubovich lemma and the S-procedure: A historical essay|url=|journal=Automation and Remote Control|volume=67 |issue=11|pages=1768–1810|via=}}</ref>中有針對此一主題的廣泛探討。

==多變數Kalman–Yakubovich–Popov引理==
給定<math>A \in \R^{n \times n}, B \in \R^{n \times m}, M = M^T \in \R^{(n+m) \times (n+m)}</math>，其中 <math>\det(j\omega I - A) \ne 0</math>針對所有<math>\omega \in \R</math>，且<math>(A, B)</math>有可控制性，則以下的敘述是等價的： 
<ol type="i">
<li>針對所有<math>\omega \in \R \cup \{\infty\} </math> 
:<math> \left[\begin{matrix} (j\omega I - A)^{-1}B \\ I \end{matrix}\right]^*   M   \left[\begin{matrix} (j\omega I - A)^{-1}B \\ I \end{matrix}\right] \le 0 </math></li>
<li>存在一矩陣<math>P \in \R^{n \times n}</math>使得<math>P = P^T</math>且 
:<math>M + \left[\begin{matrix} A^T P + PA & PB \\ B^T P & 0 \end{matrix}\right] \le 0. </math>
</li>
</ol>
即使<math>(A, B)</math>不具有可控制性，對應上式的嚴格不等式仍成立<ref name=Rantzer1996>{{cite journal
|title = On the Kalman–Yakubovich–Popov lemma 
|journal = Systems & Control Letters
|volume = 28
|number = 1
|pages = 7–10
|year = 1996
|doi = 10.1016/0167-6911(95)00063-1
|author = "Anders Rantzer"
}}</ref>。 

==參考資料==
{{Reflist}}

{{DEFAULTSORT:Kalman-Yakubovich-Popov Lemma}}
[[Category:引理]]
[[Category:稳定性理论]]