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 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. Kálmán.[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 reviews of the topic can be found in [5] and in Chapter 3 of.[6]
Multivariable Kalman–Yakubovich–Popov lemmaedit
Given with for all and controllable, the following are equivalent:
for all
there exists a matrix such that and
The corresponding equivalence for strict inequalities holds even if is not controllable. [7]
Referencesedit
^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. PMC299777. 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.{{cite book}}: CS1 maint: multiple names: authors list (link)
^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. S2CID 120970123.
^Brogliato, B. and Lozano, R. and Maschke, B. and Egeland, O. (2020). Dissipative Systems Analysis and Control (3rd ed.). Switzerland AG: Springer Nature.{{cite book}}: CS1 maint: multiple names: authors list (link)
^Anders Rantzer (1996). "On the Kalman–Yakubovich–Popov lemma". Systems & Control Letters. 28 (1): 7–10. doi:10.1016/0167-6911(95)00063-1.
April 27, 2024
kalman, yakubovich, popov, lemma, result, system, analysis, control, theory, which, states, given, number, displaystyle, gamma, vectors, hurwitz, matrix, pair, displaystyle, completely, controllable, then, symmetric, matrix, vector, satisfying, displaystyle, d. The Kalman Yakubovich Popov lemma is a result in system analysis and control theory which states Given a number g gt 0 displaystyle gamma gt 0 two n vectors B C and an n x n Hurwitz matrix A if the pair A B displaystyle A B is completely controllable then a symmetric matrix P and a vector Q satisfying A T P P A Q Q T displaystyle A T P PA QQ T P B C g Q displaystyle PB C sqrt gamma Q exist if and only if g 2 R e C T j w I A 1 B 0 displaystyle gamma 2Re C T j omega I A 1 B geq 0 Moreover the set x x T P x 0 displaystyle x x T Px 0 is the unobservable subspace for the pair C A 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 reviews of the topic can be found in 5 and in Chapter 3 of 6 Multivariable Kalman Yakubovich Popov lemma editGiven A R n n B R n m M M T R n m n m displaystyle A in mathbb R n times n B in mathbb R n times m M M T in mathbb R n m times n m nbsp with det j w I A 0 displaystyle det j omega I A neq 0 nbsp for all w R displaystyle omega in mathbb R nbsp and A B displaystyle A B nbsp controllable the following are equivalent for all w R displaystyle omega in mathbb R cup infty nbsp j w I A 1 B I M j w I A 1 B I 0 displaystyle 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 leq 0 nbsp there exists a matrix P R n n displaystyle P in mathbb R n times n nbsp such that P P T displaystyle P P T nbsp and M A T P P A P B B T P 0 0 displaystyle M left begin matrix A T P PA amp PB B T P amp 0 end matrix right leq 0 nbsp The corresponding equivalence for strict inequalities holds even if A B displaystyle A B nbsp is not controllable 7 References edit 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 a href Template Cite book html title Template Cite book cite book a CS1 maint multiple names authors list link 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 S2CID 120970123 Brogliato B and Lozano R and Maschke B and Egeland O 2020 Dissipative Systems Analysis and Control 3rd ed Switzerland AG Springer Nature a href Template Cite book html title Template Cite book cite book a CS1 maint multiple names authors list link Anders Rantzer 1996 On the Kalman Yakubovich Popov lemma Systems amp Control Letters 28 1 7 10 doi 10 1016 0167 6911 95 00063 1 Retrieved from https en wikipedia org w index php title Kalman Yakubovich Popov lemma amp oldid 1174561304, wikipedia, wiki, book, books, library,