Algebraic Conditions for Stability Analysis of Linear Time‐Invariant Distributed Order Dynamic Systems: A Lagrange Inversion Theorem Approach

Taghavian, Hamed; Tavazoei, Mohammad Saleh

doi:10.1002/asjc.1780

Cited by 11 publications

(3 citation statements)

References 31 publications

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“…The stability contours are often impossible to express via elementary functions, which makes the stability tests of DO systems more complicated than their constant- and integer-order counterparts. In this regard, the Lagrange inversion theorem was utilized in [ 345 ] to obtain explicit expressions for the stability contours. Several interesting properties of these stability curves such as the slope of the tangent at very high and very low frequencies, convexity, inability to cut itself, location in the first and fourth quadrants, and shifting and enhancement of the area of the stability via multiplication of suitable functions to the strength distribution, have been presented in [ 346 , 347 , 348 ].The above mentioned properties of the stability boundaries were used in [ 347 ] to present a remarkable framework for the robust stability analysis of DO LTI systems with uncertain strength distributions and dynamic matrices.…”

Section: Applications To Control Theorymentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

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Section: Applications To Control Theorymentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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“…In contrast to classical integer‐order calculus, fractional‐order calculus considers more general memory properties, which allow to model a more comprehensive class of dynamical systems 1,2 . In this sense, distributed‐order calculus makes it possible to further extend these properties to analyse diverse physical and engineering systems 3–8 . Distributed‐order calculus generalises integer‐ and fractional‐order calculus, 9 and it has been extensively studied in recent decades, 10 for instance, in diffusion processes 11–15 .…”

Section: Introductionmentioning

confidence: 99%

Robust stabilisation of distributed‐order systems

Muñoz‐Vázquez

Fernández‐Anaya

Sánchez‐Torres

et al. 2022

Math Methods in App Sciences

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Funding informationThere are no funders to report for this submission.This paper presents the design of a novel methodology for robust stabilisation of distributed-order systems, which are subject to both matched and mismatched disturbances. Matched disturbances are coped through a nonlinear controller, while mismatched disturbances are rejected by a pseudo-state feedback, whose gain is adjusted by solving alinear matrix inequality. Since nonsmooth techniques induce not-necessarily integer-order differentiable solutions, the dynamical systems under consideration are defined through an operator that extends the distributed-order differentiation of integer-order differentiable functions to the case of not necessarily integer-order differentiable ones. Numerical simulations highlight the reliability of the proposed scheme.

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“…Fractional calculus is a natural generalization of classical integer order calculus, whose inception can be traced back to 300 years ago. It is well known that fractional calculus has been widely applied to system modelling [1,2], stability analysis [3,4], controller synthesis [5,6], and optimization algorithm [7,8], etc. Due to the great efforts devoted by researchers, a large number of valuable results have been reported on fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions

Wei

Wang

Tse

et al. 2019

Asian Journal of Control

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The paper focuses on the numerical approximation of nabla fractional order systems with the conditions of nonzero initial instant and nonzero initial state. First, the inverse nabla Laplace transform is developed and the equivalent infinite dimensional frequency distributed models of discrete fractional order system are introduced. Then, resorting the nabla Laplace transform, the rationality of the finite dimensional frequency distributed model approaching the infinite one is illuminated. Based on this, an original algorithm to estimate the parameters of the approximate model is proposed with the help of vector fitting method. Additionally, the applicable object is extended from a sum operator to a general system. Three numerical examples are performed to illustrate the applicability and flexibility of the introduced methodology.

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Algebraic Conditions for Stability Analysis of Linear Time‐Invariant Distributed Order Dynamic Systems: A Lagrange Inversion Theorem Approach

Cited by 11 publications

References 31 publications

Applications of Distributed-Order Fractional Operators: A Review

Applications of Distributed-Order Fractional Operators: A Review

Robust stabilisation of distributed‐order systems

Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions

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