Exponential Stability Analysis of Linear (Irrational) Systems in the Parametric Space

Speaker:
Rachid Malti (Université de Bordeaux, IMS - UMR 5218 CNRS, France)

Abstract:
This talk presents some new results, concerning robust stability analysis of LTI systems having irrational transfer functions which cover a wide variety of linear systems including distributed parameter systems that are solutions of partial differential equations, time-delay rational systems, and fractional order systems. 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. First, it is proven that, under mild hypotheses, new poles may appear to the right of a vertical axis of abscissa γ (when γ = 0: imaginary axis) through a continuous variation of parameters only if existing poles to the left of γ 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 robust estimation algorithm of the SCS is formulated as an interval constraint satisfaction problem and solved using guaranteed methods from interval arithmetics. 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. This work is currently under review for a possible publication in an IFAC journal (Malti et al. 2023).

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