A passivity based stability measure for discrete 3-D IIR system realizations

Abstract

Increasing miniaturization of electronic systems and nano-technology enables more and more the realization of real time 3-D spatio-temporal signal processing and control systems. However, while stability is well understood for 1-D systems, open problems remain in the higher dimensional case, e.g. for 3D system realizations, especially with respect to computational complexity and thus hardware effort. We propose a basic and thus fast stability test for systems that can be given in a standard 3-D Roesser-like state space model. Most practical systems can be represented in that manner or at least be transformed into an equivalent system so that the test is applicable to most real world problems. The test itself is inspired by passivity assumptions which for example guarantee stability of electrical reactance networks. It represents a sufficient condition for stability and comprises eigenvalue computations of a sum of Gramian matrices of the same size as the system matrix, which leads in this step to the same complexity as a 1-D stability test. The latter can thus also be applied to adaptive systems, where the system matrix is periodically changing.