View source for Kalman–Yakubovich–Popov lemma

The '''Kalman–Yakubovich–Popov lemma''' is a result in [[system analysis]] and [[control theory]] which states: Given a number <math>\gamma > 0</math>, two n-vectors B, C and an n x n [[Hurwitz matrix]] A, if the pair <math>(A,B)</math> is completely [[controllability|controllable]], then a symmetric matrix P and a vector Q satisfying
:<math>A^T P + P A = -Q Q^T</math>

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

exist if and only if
:<math>
\gamma+2 Re[C^T (j\omega I-A)^{-1}B]\ge 0
</math>
Moreover, the set <math>\{x: x^T P x = 0\}</math> is the unobservable subspace for the pair <math>(C,A)</math>.

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]]<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>  where it was stated that for the strict frequency inequality. The case of nonstrict frequency inequality was published [[Rudolf Kalman|in 1963 by Rudolf E. Kalman]]<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>. 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<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> and independently by [[Vasile M. Popov|Vasile Mihai Popov]]<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>. Extensive review of the topic can be found in <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>.

==Multivariable Kalman–Yakubovich–Popov lemma==
Given <math>A \in \R^{n \times n}, B \in \R^{n \times m}, M = M^T \in \R^{(n+m) \times (n+m)}</math> with <math>\det(j\omega I - A) \ne 0</math> for all <math>\omega \in \R</math> and <math>(A, B)</math> controllable, the following are equivalent: 
<ol type="i">
<li>for all <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>there exists a matrix <math>P \in \R^{n \times n}</math> such that <math>P = P^T</math> and 
:<math>M + \left[\begin{matrix} A^T P + PA & PB \\ B^T P & 0 \end{matrix}\right] \le 0. </math>
</li>
</ol>
The corresponding equivalence for strict inequalities holds even if <math>(A, B)</math> is not controllable. <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> 



==References==
{{Reflist}}

{{DEFAULTSORT:Kalman-Yakubovich-Popov Lemma}}
[[Category:Lemmas]]
[[Category:Stability theory]]

'''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