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The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number \( \gamma >0 \) , two n-vectors B, C and an n x n Hurwitz matrix A, if the pair ( (A,B) is completely controllable, then a symmetric matrix P and a vector Q satisfying

\( {\displaystyle A^{T}P+PA=-QQ^{T}} \)

\( {\displaystyle PB-C={\sqrt {\gamma }}Q} \)

exist if and only if

\( \gamma+2 Re[C^T (j\omega I-A)^{-1}B]\ge 0 \)

Moreover, the set \( \{x: x^T P x = 0\} \) is the unobservable subspace for the pair \( {\displaystyle (C,A)}. \)

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[1] where it was stated that for the strict frequency inequality. The case of nonstrict frequency inequality was published in 1963 by Rudolf E. Kalman[2]. 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[3] and independently by Vasile Mihai Popov[4]. Extensive review of the topic can be found in [5].
Multivariable Kalman–Yakubovich–Popov lemma

Given \ A \in \R^{n \times n}, B \in \R^{n \times m}, M = M^T \in \R^{(n+m) \times (n+m)} \) with \( \det(j\omega I - A) \ne 0 \) for all \( \omega \in \R \) and (A, B) controllable, the following are equivalent:

for all \( \omega \in \R \cup \{\infty\} \)

\( \left[\begin{matrix} (j\omega I - A)^{-1}B \\ I \end{matrix}\right]^* M \left[\begin{matrix} (j\omega I - A)^{-1}B \\ I \end{matrix}\right] \le 0 \ \)

there exists a matrix \( P \in \R^{n \times n} \ such that \( P = P^T \) and

\( M + \left[\begin{matrix} A^T P + PA & PB \\ B^T P & 0 \end{matrix}\right] \le 0. \)

The corresponding equivalence for strict inequalities holds even if ( A, B) is not controllable. [6]


References

Yakubovich, Vladimir Andreevich (1962). "The Solution of Certain Matrix Inequalities in Automatic Control Theory". Dokl. Akad. Nauk SSSR. 143 (6): 1304–1307.
Kalman, Rudolf E. (1963). "Lyapunov functions for the problem of Lur'e in automatic control" (PDF). Proceedings of the National Academy of Sciences. 49 (2): 201–205. Bibcode:1963PNAS...49..201K. doi:10.1073/pnas.49.2.201. PMC 299777. PMID 16591048.
Gantmakher, F.R. and Yakubovich, V.A. (1964). Absolute Stability of the Nonlinear Controllable Systems, Proc. II All-Union Conf. Theoretical Applied Mechanics. Moscow: Nauka.
Popov, Vasile M. (1964). "Hyperstability and Optimality of Automatic Systems with Several Control Functions". Rev. Roumaine Sci. Tech. 9 (4): 629–890.
Gusev S. V. and Likhtarnikov A. L. (2006). "Kalman-Popov-Yakubovich lemma and the S-procedure: A historical essay". Automation and Remote Control. 67 (11): 1768–1810. doi:10.1134/s000511790611004x.

Anders Rantzer (1996). "On the Kalman–Yakubovich–Popov lemma". Systems & Control Letters. 28 (1): 7–10. doi:10.1016/0167-6911(95)00063-1.

B. Brogliato, R. Lozano, M. Maschke, O. Egeland, Dissipative Systems Analysis and Control, Springer Nature Switzerland AG, 3rd Edition, 2020 (chapter 3, pp.81-262), ISBN 978-3--030-19419-2

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