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The aim of this paper is to provide a survey on the recently obtained results which are useful in time response analysis of fractional-order control systems. In this survey, at first some results on error signal analysis in fractional-order control systems are presented. Then, some previously obtained results which are helpful for system output analysis in fractional-order control systems are summarized. In addition, some results on the analysis of the control signal and the system response to the load disturbances in fractional-order control systems are reviewed  

The aim of this note is to highlight one of the basic differences between fractional order and integer order systems. It is analytically shown that a time invariant fractional order system contrary to its integer order counterpart cannot generate exactly periodic signals. As a result, a limit cycle cannot be expected in the solution of these systems. Our investigation is based on Caputo's definition of the fractional order derivative and includes both the commensurate or incommensurate fractional order systems. © 2009 Elsevier Ltd. All rights reserved  

This paper deals with realization of fractional-order impedance functions by passive electrical networks composed of a fractional capacitor and some RLC components. The necessary and sufficient conditions for the existence of such a realization are found in a general case. Also for impedance functions described by a class of fractional-order transfer functions, the realizability conditions are stated as some algebraic conditions on the parameters of the transfer function. Moreover, a procedure is proposed for implementation of such impedance functions by passive electrical networks composed of a fractional capacitor and some RLC components. Numerical examples are presented to show the... 

In this paper the derivation of Kalman filter for discrete time-stochastic fractional system is investigated. Based on a novel cumulative vector form model for fractional systems, a general Kalman filter is introduced. The validity of the proposed method has been compared with a previously presented method via simulation results. It is shown that this method can be better applied for discrete time stochastic fractional systems with slower dynamics  

In this paper, an effective way is proposed for reduction of oscillations in the response of dynamical systems. In this regard, it is analytically shown that the undesirable oscillations in the response of dynamical systems can be reduced using a simple input shaping fractional order filter. The effectiveness of the proposed fractional calculus based technique is numerically verified in reduction of oscillations in a mass-spring-damper system, a low-pass Chebyshev filter, and a PI-controlled two-mass drive system  

This paper offers a systematic framework for the design of suboptimal integer-order controllers based on fractional-order structures. The proposed approach is built upon the integer-order approximations that are traditionally used to implement fractional-order controllers after they are designed. Accordingly, the fractional-order structures are exploited to derive a suitable parameterization for a fixed-structure integer-order controller. The parameters, which describe the structure of the fixed-order integer controller, are the same as those used to describe the original fractional structure. Optimal tuning is then performed in the parameter space of the fractional structure and the... 

This paper deals with a new fractional calculus based method to stabilize fixed points of single-input 3D systems. In the proposed method, the control signal is determined by fractional order integration of a linear combination of the system linearized model states. The tuning rule for this method is based on the stability theorems in the incommensurate fractional order systems. The introduced technique can be used in suppression of chaotic oscillations. To evaluate the performance of the proposed technique in practical applications, it has been experimentally applied to control chaos in two chaotic circuits  

Our main interest in the present article is to consider a fractional program with both linear and quadratic equation in numerator and denominator with second order cone (SOC) constraints. With a suitable change of variable, we transform the problem into a second order cone programming (SOCP) problem. For the quadratic fractional case, using a relaxation, then the problem is reduced to a semi-definite optimization (SDO)) program. The problem is solved with SDO relaxation and the obtained results are compared with the interior point method (IPM), sequential quadratic programming approach (SQP), active set, genetic algorithm. It is observe that the SDO relaxation method is much more accurate... 

Our main interest in the present article is to consider a fractional program with both linear and quadratic equation in numerator and denominator with second order cone (SOC) constraints. With a suitable change of variable, we transform the problem into a second order cone programming (SOCP) problem. For the quadratic fractional case, using a relaxation, then the problem is reduced to a semi-definite optimization (SDO)) program. The problem is solved with SDO relaxation and the obtained results are compared with the interior point method (IPM), sequential quadratic programming approach (SQP), active set, genetic algorithm. It is observe that the SDO relaxation method is much more accurate... 

This paper presents a new method to control chaos in fractional order systems based on the fractional control theory. The proposed controller is a fractional P1 (PIα) controller and can locally stabilize unstable equilibrium points of a class of chaotic fractional order systems. Using the ideas available in the chaos control methods such as On-Grebogi-Yorke (OGY), this local stabilization can be extended to the global stabilization. The controller has simple structure and its parameters can be determined by pole placement technique. To illustrate its capability, the proposed controller is applied to control chaos in the fractional order unified system. Numerical simulations confirm the... 

Some efficient interior-point methods (IPMs) are based on using a self-concordant barrier function related to the feasibility set of the underlying problem. Here, we use IPMs for solving fractional programming problems involving second order cone constraints. We propose a logarithmic barrier function to show the self concordant property and present an algorithm to compute ε—solution of a fractional programming problem. Finally, we provide a numerical example to illustrate the approach. © 2021. All Rights Reserved  

In this paper the derivation of Kalman filter for discrete time-stochastic fractional system is investigated. Based on a novel cumulative vector form model for fractional systems, a general Kalman filter is introduced. The validity of the proposed method has been compared with a previously presented method via simulation results. It is shown that this method can be better applied for discrete time stochastic fractional systems with slower dynamics  

Fractional order models have been widely used in modeling and identification of thermal systems. General model in this category is considered as the model of thermal systems in this paper, and a fractional order controller is proposed for controlling such systems. The proposed controller is a generalization for the traditional PI controllers. The parameters of this controller can be obtained by using a recently introduced tuning method which can simultaneously ensure the following three requirements: desired phase margin, desired gain crossover frequency, and flatness of the phase Bode plot at this frequency. In this paper, it is found whether simultaneously achieving the mentioned... 

Control input energy efficiency is an important issue which should be considered in designing any control system. Due to the importance of this subject, in the present paper fractional order control systems are studied in the viewpoint of control input energy efficiency. In this study, the divergent terms of the control input energy function of fractional order control systems are obtained. It is shown that these terms have a significant role in the amount of the energy injected to the plant by the controller. Finally, two examples are provided to demonstrate the usefulness of the presented results in the paper  

In this note, we study on the existence and uniqueness of a positive solution to the following doubly singular fractional problem: {(-Δ)su=K(x)uq+f(x)uγ+μinΩ,u>0inΩ,u=0in(RN\u03a9).Here Ω ⊂ RN (N> 2 s) is an open bounded domain with smooth boundary, s∈ (0 , 1) , q> 0 , γ> 0 , and K(x) is a positive Hölder continuous function in which behaves as dist (x, ∂Ω) -β near the boundary with 0 ≤ β< 2 s. Also, 0 ≤ f, μ∈ L1(Ω) , or non-negative bounded Radon measures in Ω. Moreover, we assume that 0<βs+q<1, or βs+q>1 with 2 β+ q(2 s- 1) < (2 s+ 1). For s∈(0,12), we take advantage of the convexity of Ω. For any γ> 0 , we will prove the existence of a positive weak (distributional) solution to the above... 

In this paper, conditions for checking the realizability of fractional-order impedance functions by passive networks composed of a fractional element (either a fractional capacitor or a fractional inductor) and some RLC components are derived. To this end, at first the newly obtained conditions for realizability of fractional-order impedance functions by a passive network composed of a fractional capacitor and some RLC components are extended to include the case that the polynomials involving in the impedance function can have roots on the imaginary axis. Then, the necessary and sufficient conditions are found on a fractional-order impedance function to be realized by a passive network...