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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]

In this paper, a particular discrete-time nonlinear and time-invariant system represented as a vector difference equation is analyzed for its stability properties. The motivation for analyzing this particular system is that it models gated recurrent unit neural networks commonly used and well known in machine learning applications. From the technical perspective, the analyses exploit the systems similarities to a convex combination of discrete-time systems, where one of the systems is trivial, and thus, the overall properties are mostly dependent on the other one. Stability results are formulated for the nonlinear system and its linearization with respect to the systems, in general, multiple equilibria. To motivate and illustrate the potential of these results in applications, some particular results are derived for the gated recurrent unit neural network models and a connection between local stability analysis and learning is provided. [ABSTRACT FROM AUTHOR]

Classical telegrapher's equation is generalized in order to account for the hereditary nature of polarization and magnetization phenomena of the medium by postulating fractional order constitutive relations for capacitive and inductive elements in the elementary circuit, as well as by the two topological modifications of Heaviside's elementary circuit, referred as series and parallel, differing in the manner in which the effect of charge accumulation effect along the line is taken into consideration. Frequency analysis of generalized telegrapher's equations is performed, with a particular emphasis on the asymptotic behavior for low and high frequencies. It is found that, like Heaviside's elementary circuit, parallel topology leads to low-pass frequency characteristics, while the series topology leads to band-pass characteristics. It is also demonstrated that the logarithmic phase characteristics are linear functions of frequency, being suitable for determining some of the fractional differentiation orders in generalized telegrapher's equations. [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]

The article presents remarks on the paper "A Commentary on Fractionalization of Multi-Compartmental Models," by A. Dokoumetzidis and colleagues, focusing on the mass balance for multi-compartmental models, a form of nonlinear compartmental model. It mentions the preservation of the mass balance in the equations presented. It points out a printing error in a one equation. It recommends the presentation of another possible generalization of the system of equations treated in the article.

This study presents a new two compartmental model and its application to the evaluation of diclofenac pharmacokinetics in a small number of healthy adults, during a bioequivalence trial. In the model the integer order derivatives are replaced by derivatives of real order often called fractional order derivatives. Physically that means that a history (memory) of a biological process, realized as a transfer from one compartment to another one with the mass balance conservation, is taken into account. This kind of investigations in pharmacokinetics is founded by Dokoumetzidis and Macheras through the one compartmental models while our contribution is the analysis of multi-dimensional compartmental models with the applications of the two compartmental model in evaluation of diclofenac pharmacokinetics. Two experiments were preformed with 12 healthy volunteers with two slow release 100 mg diclofenac tablet formulations. The agreement of the values predicted by the proposed model with the values obtained through experiments is shown to be good. Thus, pharmacokinetics of slow release diclofenac can be described well by a specific two compartmental model with fractional derivatives of the same order. Parameters in the model are determined by the least-squares method and the Particle Swarm Optimization (PSO) numerical procedure is used. The results show that the fractional order two compartmental model for diclofenac is superior in comparison to the classical two compartmental model. Actually this is true in general case since the classical one is a special case of the fractional one. [ABSTRACT FROM AUTHOR]

Corrections to the article "A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac," by Jovan K. Popovi&#0107 and colleagues such as the Experimental section and References section that was published in the 2010 issue are presented.