A unified framework for robust stability analysis of linear irrational systems in the parametric space
Résumé
This paper presents a unified framework for robust stability analysis of linear stationary systems with irrational transfer functions in the space of an arbitrary number of unknown parameters. Systems of this kind are encountered when analyzing distributed parameter systems, time delayed systems of both retarded and neutral type, or even fractional systems. Systems described by irrational transfer functions are 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. First, it is proven that, under mild hypotheses, new poles with positive real parts may appear through a continuous variation of parameters only if existing stable poles cross the imaginary 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 their multiplicity) is invariant. Based on the aforementioned result, a robust estimation algorithm, formulated as an interval constraint satisfaction problem, is solved using guaranteed methods, for determining the SCS. The developed algorithm is applied for assessing stability 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.
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