Kalman–Yakubovich–Popov lemma

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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[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[edit]

Given with for all and controllable, the following are equivalent:

  1. for all
  2. there exists a matrix such that and

The corresponding equivalence for strict inequalities holds even if is not controllable. [6]


References[edit]

  1. ^ Yakubovich, Vladimir Andreevich (1962). "The Solution of Certain Matrix Inequalities in Automatic Control Theory". Dokl. Akad. Nauk SSSR. 143 (6): 1304–1307.
  2. ^ 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. doi:10.1073/pnas.49.2.201. PMC 299777.
  3. ^ 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.CS1 maint: Multiple names: authors list (link)
  4. ^ Popov, Vasile M. (1964). "Hyperstability and Optimality of Automatic Systems with Several Control Functions". Rev. Roumaine Sci. Tech. 9 (4): 629–890.
  5. ^ 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.
  6. ^ "Anders Rantzer" (1996). "On the Kalman–Yakubovich–Popov lemma". Systems & Control Letters. 28 (1): 7–10. doi:10.1016/0167-6911(95)00063-1.