Fractional Elementary Bicomplex Functions in the Riemann–Liouville Sense

Coloma, Nicolás; Teodoro, Antonio Di; Ochoa-Tocachi, Diego; Ponce, Francisco

doi:10.1007/s00006-021-01165-0

Cited by 11 publications

(9 citation statements)

References 28 publications

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“…Let α ∈ (0, 1). Then we consider the real Riemann-Liouville fractional polynomials defined as (see [48], [50]):…”

Section: B Behavior Of the Fractional Parameter αmentioning

confidence: 99%

A New Fractional Reduced-Order Model-Inspired System Identification Method for Dynamical Systems

Gude,

Di Teodoro,

Camacho

et al. 2023

IEEE Access

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This paper presents a new method for identifying dynamical systems to get fractionalreduced-order models based on the process reaction curve. This proposal uses information collected from the process. It can be applied to processes with an S-shaped step response that can be considered with fractional behavior and a fractional order range of α ∈ [0.5, 1.0]. The proposed approach combines obtaining model fractional order using asymptotic properties of the Mittag-Leffler function with time-based parameter estimation by considering two arbitrary points on the process reaction curve. The improvement in terms of accuracy of the identified FFOPDT model is obtained due to a more accurate estimation of α parameter. This method is characterized by its effectiveness and simplicity of implementation, which are key aspects when applying at an industrial level. Several examples are used to illustrate the effectiveness and simplicity of the proposed method compared to other well-established methods and other approaches based on the process reaction curve. Finally, it is also implemented on microprocessor-based hardware to demonstrate the applicability of the proposed method to identify the fractional model of a thermal process.INDEX TERMS fractional-order systems, process identification, fractional first-order plus dead-time model.

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“…Let α ∈ (0, 1). Then we consider the real Riemann-Liouville fractional polynomials defined as (see [48], [50]):…”

Section: B Behavior Of the Fractional Parameter αmentioning

confidence: 99%

A New Fractional Reduced-Order Model-Inspired System Identification Method for Dynamical Systems

Gude,

Di Teodoro,

Camacho

et al. 2023

IEEE Access

Self Cite

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“…Consider the real Riemann–Liouville fractional polynomials defined for α ∈ (0,1) and a j + > 0 as (see: ,,, ) normalΦ j α , m : = ( x j − a j ) false( m + 1 false) α − 1 normalΓ ( α ) normalΓ ( false( m + 1 false) α ) , m ∈ double-struckN 0 , j = 0 , 1 Using the definition 2, (eqs and). Then we have…”

Section: Introductionmentioning

confidence: 99%

A Hybrid Control Framework for Chemical Processes with Long Time Delay: Theory and Experiments

Di Teodoro,

Herrera,

Rincon

et al. 2024

ACS Omega

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This paper proposes a hybrid control framework based on internal model concepts, sliding mode control methodology, and fractional-order calculus theory. As a result, a modified Smith predictor (SP) is proposed for nonlinear systems with significant delays. The particular predictive approach enhances the sliding mode control (SMC) controller's transient responses for dead-time processes, and the SMC gives the predictive structure robustness for model mismatches by combining the previous methods with fractional order concepts; the result is a dynamical sliding mode controller. A numerical example is considered to evaluate the performance of the proposed approach, where a step change, external disturbance, and parametric uncertainty test are performed. A real application in the TCLab Arduino kit is presented; the proposed method presented good performance with a little amount of chattering, and in the disturbance rejection case, the overshoot increased with an aggressive response; in both cases, better tuning parameters can improve the process response and the controller action.

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“…Gu et al 27 suggested bicomplex partial metric space, its properties, and applications by considering some examples. A development of fractional bicomplex calculus in the Riemann-Liouville sense, based on the modification of the Cauchy-Riemann operator using the one-dimensional Riemann-Liouville derivative in each direction of the bicomplex basis and elementary functions such as analytic polynomials, exponential, trigonometric, and some properties of these functions, was introduced by Coloma et al 28 With this, if we talk about fractional calculus, then we can say that fractional calculus is not a new topic. Its history is almost as old as that of classical calculus.…”

Section: Introductionmentioning

confidence: 99%

Riemann–Liouville fractional operators of bicomplex order and its properties

Goswami

Kumar

2022

Math Methods in App Sciences

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In this paper, we construct the Riemann-Liouville fractional integral and differential operator of bicomplex order and illustrate some examples to calculate the fractional integration and differentiation of bicomplex order of some elementary bicomplex-valued functions. Also, we discuss some properties of these operators by proving analogues of the semigroup property, Leibniz rule, chain rule, etc.

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Fractional Elementary Bicomplex Functions in the Riemann–Liouville Sense

Cited by 11 publications

References 28 publications

A New Fractional Reduced-Order Model-Inspired System Identification Method for Dynamical Systems

A New Fractional Reduced-Order Model-Inspired System Identification Method for Dynamical Systems

A Hybrid Control Framework for Chemical Processes with Long Time Delay: Theory and Experiments

Riemann–Liouville fractional operators of bicomplex order and its properties

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