Multidimensional BIBO stability and Jury’s conjecture

Abstract

Twenty years ago E. I. Jury conjectured by analogy to the case of digital filters that a two-dimensional analog filter is BIBO stable if its transfer function has the form H = 1/P where P is a very strict Hurwitz polynomial (VSHP). In more detail he conjectured that the impulse response of the filter is an absolutely integrable function. However, he did not specify the exact equations of these filters and did not prove the existence of the impulse response. In the present paper we generalise Jury’s conjecture to arbitrary proper transfer functions H = Q/P where P is a bivariate VSHP and prove this generalisation. In particular, we show the existence of a suitable impulse response or fundamental solution for any multivariate proper rational function. However, this impulse response is a measure and not a function. We have not succeeded to prove an analogue of Jury’s conjecture in higher dimensions than two yet, but we propose a new conjecture in context with the robustly stable multivariate polynomials investigated by Kharitonov et. al. For the discrete case we prove that the structurally stable rational functions after Bose, Lin et al. coincide with the stable rational functions discussed in context with the stabilisation of discrete input/output systems. These rational functions are BIBO stable, but the converse is not true as established by several authors.