Strong stability of discrete-time systems

The paper introduces a new notion of stability for internal (state-space) autonomous system descriptions in discrete-time, referred to as strong stability which extends a parallel notion introduced in the continuous-time case. This is a stronger notion of stability compared to alternative definitions (asymptotic, Lyapunov), which prohibits systems described by natural coordinates to have overshooting responses for arbitrary initial conditions in state-space. Three finer notions of strong stability are introduced and necessary and sufficient conditions are established for each one of them. The class of discrete-time systems for which strong and asymptotic stability coincide is characterized and links between the skewness of the eigen-frame and the violation of strong stability property are obtained. Connections between the notions of strong stability in the continuous and discrete-domains are briefly discussed. Finally strong stabilization problems under state and output feedback are studied. The results of the paper are illustrated with a numerical example

Matrix powers in finite precision arithmetic

SIGLEGBUnited Kingdo

Matrix Powers In Finite Precision Arithmetic

If A is a square matrix with spectral radius less than 1 then A k ! 0 as k !1, but the powers computed in finite precision arithmetic may or may not converge. We derive a sufficient condition for f l(A k ) ! 0 as k !1 and a bound on kf l(A k )k, both expressed in terms of the Jordan canonical form of A. Examples show that the results can be sharp. We show that the sufficient condition can be rephrased in terms of a pseudospectrum of A when A is diagonalizable, under certain assumptions. Our analysis leads to the rule of thumb that convergence or divergence of the computed powers of A can be expected according as the spectral radius computed by any backward stable algorithm is less than or greater than 1