Toward the Development of Iteration Procedures for the Interval-Based Simulation of Fractional-Order Systems

Abstract

In many fields of engineering as well as computational physics, it is necessary to describe dynamic phenomena which are characterized by an infinitely long horizon of past state values. This infinite horizon of past data then influences the evolution of future state trajectories. Such phenomena can be characterized effectively by means of fractional-order differential equations. In contrast to classical <em>linear</em> ordinary differential equations, <em>linear</em> fractional-order models have frequency domain characteristics with amplitude responses that deviate from the classical integer multiples of ±20 dB per frequency decade and, respectively, deviate from integer multiples of ±π/2 in the limit values of their corresponding phase response. Although numerous simulation approaches have been developed in recent years for the numerical evaluation of fractional-order models with point-valued initial conditions and parameters, the robustness analysis of such system representations is still a widely open area of research. This statement is especially true if interval uncertainty is considered with respect to initial states and parameters. Therefore, this paper summarizes the current state-of-the-art concerning the simulation-based analysis of fractional-order dynamics with a restriction to those approaches that can be extended to set-valued (interval) evaluations for models with bounded uncertainty. Especially, it is shown how verified simulation techniques for integer-order models with uncertain parameters can be extended toward fractional counterparts. Selected linear as well as nonlinear illustrating examples conclude this paper to visualize algorithmic properties of the suggested interval-based simulation methodology and point out directions of ongoing research.

Verified Integration of Differential Equations with Discrete Delay

“…As future work, we plan to extend the exponential enclosure technique to fractional-order differential equations by replacing the exponential terms with socalled Mittag-Leffler functions [23,24]. Fractional-order models have a large practical relevance in the context of control and state estimation of electrochemical energy converters and storage elements such as fuel cells and batteries.…”

Transformation of Uncertain Linear Systems with Real Eigenvalues into Cooperative Form: The Case of Constant and Time-Varying Bounded Parameters

“…In addition, further research should be directed towards extensions for arbitrarily fast changes of parameters occurring in a time-or event-triggered framework as well as to interfacing the proposed methodology with techniques for gain scheduling control of uncertain dynamic systems [33]. Moreover, possible generalizations towards the simulation of uncertain fractional-order differential equations will be investigated [46][47][48].…”

“…Considering the parameters listed in the previous subsection, it can be shown that there does not exist a common, point-valued transformation matrix Θ according to Section 2.3 that simultaneously leads to a Metzler matrix representation for the transformed system dynamics stated in Equation (15) and at the same time preserves the asymptotic stability of the original system (46).…”

“…When appending the uncertain parameters to the state vector x(t), introduced in (45), and accounting for the time invariance of these parameters by the so-called integrator disturbance modelṗ = 0 in the state equations, the solver verifyode does not at all succeed in computing guaranteed state bounds. This results from a division by zero when substituting the Taylor model representations of both uncertain parameters into the system model (46). This problem is independent whether identical Taylor model orders are chosen for all four state variables or if the Taylor model order for the interval parameters is reduced to its minimum value 1 as described in [43].…”

Verification and Reachability Analysis of Fractional-Order Differential Equations Using Interval Analysis

“…For further general discussions about modeling dynamic systems by using fractional-order representations, their identification, as well as their potential use in control engineering tasks, the reader is referred to [15]. Moreover, the paper [18] provides an overview of the state-of-the-art concerning already existing approaches for a verified simulation of fractional-order models. These include the definition of differential inclusions, the exploitation of cooperativity (i.e., monotonicity of solutions with respect to initial conditions) and positivity of state variables in simulation as well as state estimation, nonlinear time transformations to cast fractional-order models into equivalent integer-order representations, and, finally, also the extension of Picard iteration procedures [12] to non-integer-order models.…”