COMMON QUADRATIC LYAPUNOV FUNCTIONS FOR TWO STABLE MATRICES

Yıl 2022, Cilt: 10 Sayı: 1, 18 - 26, 25.02.2022

Şerife Yılmaz

https://doi.org/10.20290/estubtdb.1057006

Cited By: 1

Öz

In this paper we consider the problem of existence and evaluation of common solutions to Lyapunov equations for switched systems consisting of two real or complex matrices. Conditions for existence and solution algorithm based on the gradient of a matrix function are given. Number of examples are provided.

Anahtar Kelimeler

Lyapunov function, Common solution, Hurwitz stability

KARARLI İKİ MATRİS İÇİN ORTAK KUADRATİK LYAPUNOV FONKSİYONU

Öz

Bu çalışmada gerçel ya da kompleks iki matris ile verilen anahtarlamalı doğrusal sistemler için Lyapunov denklemlerinin ortak çözümlerini bulma problemi ele alınmıştır. Çözümün varlığına ve bulunması için gradyent yöntemini esas alan bir algoritmaya ilişkin koşullar verilmiştir. Sonuçlar yeterli sayıda örneklerle desteklenmiştir.

Anahtar Kelimeler

Lyapunov function, Common solution, Hurwitz stability

Kaynakça

• [1] Liberzon D. Switching in Systems and Control. Boston MA: Birkauser, 2003.
• [2] Lin H, Antsaklis PJ. Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results. IEEE Transactions on Automatic Control, 2009; 54(2): 308-322.
• [3] Shorten RN, Narendra KS. Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time-invariant systems. International Journal of Adaptive Control and Signal Processing, 2002; 16: 709-728.
• [4] Laffey TJ, Šmigoc H. Common solution to the Lyapunov equation for 2×2 complex matrices. Linear Algebra and its Applications, 2007; 420: 609-624.
• [5] Horn RA, Johnson CR. Matrix Analysis. Cambridge: Cambridge University Press, 1985.
• [6] Dzhafarov V, Büyükköroğlu T, Yılmaz Ş. On one application of convex optimization to stability of linear systems. Trudy Instituta Matematiki i Mekhaniki Ur0 Ran, 2015; 21(2): 320-328.
• [7] Liberzon D, Tempo R. Common Lyapunov functions and gradient algorithms. IEEE Transactions on Automatic Control, 2004; 49(6): 990-994.
• [8] Büyükköroğlu T, Esen Ö, Dzhafarov V. Common Lyapunov functions for some special classes of stable systems. IEEE Transactions on Automatic Control, 2011; 56(8): 1963-1967.