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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  

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 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  

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  

It has been pointed out that the numerical simulation results presented in the above paper are not consistent with reality. The reason for this inconsistency has analytically been clarified in this note. © 2009 IEEE  

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  

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... 

This paper deals with proportional stabilization and closed-loop step response identification of the fractional order counterparts of the unstable first order plus dead time (FOPDT) processes. At first, the necessary and sufficient condition for stabilizability of such processes by proportional controllers is found. Then, by assuming that a process of this kind has been stabilized by a proportional controller and the step response data of the closed-loop system is available, an algorithm is proposed for estimating the order and the parameters of an unstable fractional order model by using the mentioned data  

This paper deals with a method recently proposed for tuning fractional order [proportional-derivative] (FO-[PD]) controllers. Using this tuning method, the tuned FO-[PD] controller can ensure the desired phase margin, the desired gain crossover frequency, and the flatness of the phase Bode plot at such a frequency. In the present paper, the achievable region of this tuning method in the gain crossover frequency-phase margin plane is obtained analytically. Also, the continuity of this region and uniqueness of the tuned parameters are investigated. Moreover, the achievable region of the aforementioned tuning method in the presence of time delay in the feedback loop is found  

This paper deals with asymptotic swarm stabilization of fractional order linear time invariant swarm systems in the presence of two constraints: the input saturation constraint and the restriction on distance of the agents from final destination which should be less than a desired value. A feedback control law is proposed for asymptotic swarm stabilization of fractional order swarm systems which guarantees satisfying the above-mentioned constraints. Numerical simulation results are given to confirm the efficiency of the proposed control method  

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... 

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... 

In this paper via a novel method of discretized continuous-time Kalman filter, the problem of synchronization and cryptography in fractional-order systems has been investigated in presence of noisy environment for process and output signals. The fractional-order Kalman filter equation, applicable for linear systems, and its extension called the extended Kalman filter, which can be used for nonlinear systems, are derived. The result is utilized for chaos synchronization with the aim of cryptography while the transmitter system is fractional-order, and both the transmitter and transmission channel are noisy. The fractional-order stochastic chaotic Chen system is then presented to apply the... 

This paper presents an adaptive controller to achieve consensus tracking for the fractional-order linear time invariant swarm systems in which the matrices describing the agent dynamics and the interactive dynamics between agents are unknown. This controller consists of two parts: an adaptive stabilizer and an adaptive tracker. The adaptive stabilizer guarantees the asymptotic swarm stability of the considered swarm system. Also, the adaptive tracker enforces the system agents to track a desired trajectory while achieving consensus. Numerical simulation results are presented to show the effectiveness of the proposed controller. Copyright  

Five different approaches are presented to assign characteristic ratios for commensurate fractional order systems having order in (0,0.5]. Through the indirect methods, a closed-loop or plant transfer function is converted to a commensurate order one with an order greater than 0.5 so that the previously designed CRA method by the authors is applicable. The first method among the proposed direct ones is based on increasing the order of the desired closed-loop transfer function that allows the employment of positive characteristic ratios. In the second method the closed-loop response is sped up by augmenting an appropriate zero. The final method uses negative characteristic ratios to reach the... 

When an integer order linear time invariant system possesses unrepeated pure imaginary poles it can generate oscillatory response which is represented by invariant closed contours in the phase plane. In linear time invariant fractional order systems with the same property, due to their special characteristics, this behavior will be more complicated and the contours would not be invariant. In this paper we will investigate the behavior of fractional order systems under such conditions  

In this note some points to paper [L. Pan, W. Zhou, J. Fang, D. Li, Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control, Commun Nonlinear Sci Numer Simulat 2010;15:3754-3762] are presented. Hereby, we illustrate that the way that authors in [1] treat with fractional version of Lyapunov stability theorem suffers lack of a correct justification