Mason, Oliver; Shorten, Robert On common quadratic Lyapunov functions for stable discrete-time LTI systems. (English) Zbl 1067.93057 IMA J. Appl. Math. 69, No. 3, 271-283 (2004). The main result of this paper is the following. Let \(A_1,A_2\) be two \(n\times n\) matrices, \(n\geq 2\), having all their eigenvalues inside the unit disk. Assume that a unique solution \(P\geq 0\) exists to the non-strict Lyapunov equations \[ A_i^TPA_i - P = - Q_i \leq 0\;,\;i=1,2 \] with \(Q_i\) both of rank \(n-1\). Under these assumptions at least one of the matrix products \(C(A_1)C(A_2)\) or \(C(A_1)C(A_2)^{-1}\) has a real negative eigenvalue, where \(C(A)=(A-I)(A+I)^{-1}\). Equivalently this signifies that at least one of the matrix pencils \(C(A_1) + \gamma C(A_2)\) or \(C(A_1) + \gamma C(A_2)^{-1}\), \(\gamma\geq 0\), is singular. Reviewer: Vladimir Răsvan (Craiova) Cited in 8 Documents MSC: 93D30 Lyapunov and storage functions 93C55 Discrete-time control/observation systems 39A11 Stability of difference equations (MSC2000) 93B17 Transformations 93B60 Eigenvalue problems Keywords:common quadratic Lyapunov function; switched discrete system; stability; Cayley transform; singular eigenvalue; singular matrix pencils PDFBibTeX XMLCite \textit{O. Mason} and \textit{R. Shorten}, IMA J. Appl. Math. 69, No. 3, 271--283 (2004; Zbl 1067.93057) Full Text: DOI Link