The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number , two n-vectors B, C and an n x n Hurwitz matrix A, if the pair is completely controllable, then a symmetric matrix P and a vector Q satisfying exist if and only if Moreover, the set is the unobservable subspace for the pair . The lemma can be seen as a generalization of the Lyapunov equation in stability theory. It establishes a relation between a linear matrix inequality involving the state space constructs A, B, C and a condition in the frequency domain.
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| - Kalman–Yakubovich–Popov lemma (en)
- Лемма Кальмана — Якубовича — Попова (ru)
- Kalman–Yakubovich–Popov引理 (zh)
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| - Лемма Ка́льмана — По́пова — Якубо́вича — один из основополагающих результатов в области теории управления, связанный с устойчивостью нелинейных систем управления и линейно-квадратичной оптимизацией. Лемма имеет репутацию одного из наиболее трудных для доказательства результатов в теории управления. Существуют доказательства с помощью методов алгебры, комплексного анализа, оптимального управления и выпуклого программирования. (ru)
- Kalman–Yakubovich–Popov引理（Kalman–Yakubovich–Popov lemma）是系統分析及控制理论的結果，其中提到：給定一數，二個n維向量B, C，及n x n的赫維茲穩定矩陣 A（所有特徵值的實部都為負值），若具有完全可控制性，則滿足下式的對稱矩陣P和向量Q 存在的充份必要條件是 而且，集合是的不可觀測子空間。 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年由寫出且證明，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。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，而也獨立得到相同結論。在中有針對此一主題的廣泛探討。 (zh)
- The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number , two n-vectors B, C and an n x n Hurwitz matrix A, if the pair is completely controllable, then a symmetric matrix P and a vector Q satisfying exist if and only if Moreover, the set is the unobservable subspace for the pair . The lemma can be seen as a generalization of the Lyapunov equation in stability theory. It establishes a relation between a linear matrix inequality involving the state space constructs A, B, C and a condition in the frequency domain. (en)
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| - The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number , two n-vectors B, C and an n x n Hurwitz matrix A, if the pair is completely controllable, then a symmetric matrix P and a vector Q satisfying exist if and only if Moreover, the set is the unobservable subspace for the pair . The lemma can be seen as a generalization of the Lyapunov equation in stability theory. It establishes a relation between a linear matrix inequality involving the state space constructs A, B, C and a condition in the frequency domain. The Kalman–Popov–Yakubovich lemma which was first formulated and proved in 1962 by Vladimir Andreevich Yakubovich where it was stated that for the strict frequency inequality. The case of nonstrict frequency inequality was published in 1963 by Rudolf E. Kálmán. In that paper the relation to solvability of the Lur’e equations was also established. Both papers considered scalar-input systems. The constraint on the control dimensionality was removed in 1964 by Gantmakher and Yakubovich and independently by Vasile Mihai Popov. Extensive review of the topic can be found in. (en)
- Лемма Ка́льмана — По́пова — Якубо́вича — один из основополагающих результатов в области теории управления, связанный с устойчивостью нелинейных систем управления и линейно-квадратичной оптимизацией. Лемма имеет репутацию одного из наиболее трудных для доказательства результатов в теории управления. Существуют доказательства с помощью методов алгебры, комплексного анализа, оптимального управления и выпуклого программирования. (ru)
- Kalman–Yakubovich–Popov引理（Kalman–Yakubovich–Popov lemma）是系統分析及控制理论的結果，其中提到：給定一數，二個n維向量B, C，及n x n的赫維茲穩定矩陣 A（所有特徵值的實部都為負值），若具有完全可控制性，則滿足下式的對稱矩陣P和向量Q 存在的充份必要條件是 而且，集合是的不可觀測子空間。 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年由寫出且證明，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。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，而也獨立得到相同結論。在中有針對此一主題的廣泛探討。 (zh)
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