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On common quadratic Lyapunov functions for stable discrete-time LTI systems. (English) Zbl 1067.93057

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.

MSC:

93D30 Lyapunov and storage functions
93C55 Discrete-time control/observation systems
39A11 Stability of difference equations (MSC2000)
93B17 Transformations
93B60 Eigenvalue problems
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