On the existence of a common solution to the Lyapunov equations

• Autorzy: Białas S. , Góra M.

Identyfikatory

DOI

10.1515/bpasts-2015-0018

• Języki publikacji: EN

Abstrakty

EN

In this paper, a system of Lyapunov equations A*i P + PAi = −Qi (i = 1, . . . ,m), (A) is considered in which Ai are given n × n complex matrices, Qi are unknown n × n Hermitian positive definite matrices and P, if any, is a common solution to the Lyapunov equations (A). Both sufficient and necessary and sufficient conditions are derived for the existence of such a matrix P. Examples are presented to illustrate the results.

Słowa kluczowe

EN

common quadratic Lyapunov functions; switched system; robust stability

PL

funkcje kwadratowe Lapunowa; system przełączników; wysoka stabilność

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2015
• Tom: Vol. 63, nr 1
• Strony: 163--–168
• Opis fizyczny: Bibliogr. 10, wykr., rys., tab., fot.

Twórcy

autor

Białas S.

• The School of Banking and Management, 4 Armii Krajowej St., 30-150 Kraków, Poland

autor

Góra M.

• AGH University of Science and Technology, Faculty of Applied Mathematics, Al. A. Mickiewicza 30, 30-059 Krakow, Poland

Bibliografia

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• [3] Y. Mori, T. Mori, and Y. Kuroe, “A solution to the common Lyapunov function problem for continuous-time systems”, Proc. 36th IEEE Conf. on Decision and Control 4, 3530-3531 (1997).
• [4] N. Cohen and I. Lewkowicz, “Convex invertible cones and the Lyapunov equation”, Linear Algebra Appl. 250 (1), 105-131 (1997).
• [5] R. Shorten and K.S. Narendra, “Necessary and sufficient conditions for common quadratic Lyapunov functions for a pair of stable LTI systems whose system matrices are in companion form”, IEEE Trans. Automat. Control 48 (4), 618-621 (2003).
• [6] R.N. Shorten and K.S. Narendra, “Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear timeinvariant systems”, Int. J. Adaptive Control and Signal Processing 16 (10), 709-728 (2002).
• [7] N. Cohen and I. Lewkowicz, “A pair of matrices sharing common Lyapunov solutions - a closer look”, Linear Algebra Appl. 360 (1), 83-104 (2003).
• [8] T.J. Laffey and H. ˇSmigoc, “Common solution to the Lyapunov equation for 2×2 complex matrices”, Linear Algebra Appl. 420 (2-3), 609-62 (2007).
• [9] T.J. Laffey and H. ˇSmigoc, “Tensor conditions for the existence of a common solution to the Lyapunov equation”, Linear Algebra Appl. 420 (2-3), 672-685 (2007).
• [10] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 2007