A unified framework for exponential stability analysis of irrational transfer functions in the parametric space

Abstract

This paper presents a unified framework for exponential stability analysis of linear stationary systems with irrational transfer functions in the space of an arbitrary number of unknown parameters. Systems described by irrational transfer functions may be of infinite dimension, typically having an infinite number of poles and/or zeros, rendering their stability analysis more challenging as compared to their finite-dimensional counterparts. The analysis covers a wide class of distributed parameter systems, time delayed systems, or even fractional systems. First, it is proven that, under mild hypotheses, new poles may appear to the right of a vertical axis of abscissa $\gamma$ (imaginary axis, when $\gamma =0$) through a continuous variation of parameters only if existing poles to the left of $\gamma$ cross the vertical axis. Hence, by determining parametric values for which the crossing occurs, known as stability crossing sets (SCS), the entire parametric space is separated into regions within which the number of right-half poles (including multiplicities) is invariant. Based on the aforementioned result, a constraint satisfaction problem is formulated and a robust estimation algorithm, from interval arithmetics that uses contraction and bisection, is used to solve it. Applications are provided for determining the SCS of (i) a controlled parabolic 1D partial differential equation, namely the heat equation, in finite and semi-infinite media, (ii) time-delay rational systems with distributed and retarded type delays, (iii) fractional systems, providing stability results even for incommensurate differentiation orders.