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This paper presents a general methodology for finding stability equivalence regions, of a wide class of linear time-invariant systems with irrational transfer functions, inside a parametric space. The proposed methodology can be applied to distributed-parameter, time-delay and fractional systems. Unlike rational transfer functions which have only a finite number of poles, irrational transfer functions may generally possess an infinite number of poles, branch points and even essential singularities. Due to this, stability of such systems is more difficult to analyze. Two variants of the new methodology are presented. The first one analyzes stability equivalence along a curve in the parametric space, starting from a given parametric point. The second one finds the maximal stability equivalence region in the parametric space around a given parametric point. Both methodologies are based on iterative application of Rouché's theorem. They are illustrated on several examples, including heat diffusion equation and generalized time-fractional telegrapher's equation, which exhibit special functions such as sinh and cosh of s , the Laplace variable of order 0.5. [ABSTRACT FROM AUTHOR]

The present work extends and generalizes the notion of frequency-distributed (FD) representation to a broad class of linear, stationary, continuous-time systems. On one hand, the proposed FD representation can be seen as a generalization of the diffusive representation, which is primarily utilized in the context of fractional order systems. Alternatively, it can also be seen as an extension to the Jordan canonical form, which is used as one of the main theoretical tools when analyzing finite-dimensional systems. Sufficient conditions under which FD representation can be achieved are derived. The proposed approach ensures real-valued state functions and output weights even when applied to oscillatory systems, and in a wast majority of cases manages to avoid utilization of generalized functions. Potential applications include simulation, representation theory and stability analysis, control synthesis, etc. All considerations have been illustrated by numerical examples. [ABSTRACT FROM AUTHOR]