Sharif Digital Repository

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... 

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 stabilizing unstable periodic orbits (UPO) in a chaotic fractional order system is studied. Firstly, a technique for finding unstable periodic orbits in chaotic fractional order systems is stated. Then by applying this technique to the fractional van der Pol and fractional Duffing systems as two demonstrative examples, their unstable periodic orbits are found. After that, a method is presented for stabilization of the discovered UPOs based on the theories of stability of linear integer order and fractional order systems. Finally, based on the proposed idea a linear feedback controller is applied to the systems and simulations are done for demonstration of controller performance... 

In tins paper, stabilizing the unstable periodic orbits (UPO) in a chaotic fractional order system called Van der Pol is studied. Firstly, a technique for finding unstable periodic orbit in chaotic fractional order systems is stated. Then by applying tins technique to the van der Pol system, unstable periodic orbit of system is found. After that, a method is presented for stabilization of the discovered UPO based on theories stability of the linear integer order and fractional order systems. Finally, a linear feedback controller was applied to the system and simulation is done for demonstration of controller performance  

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  

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  

A fractional order PID controller is designed to stabilize fractional delay systems with commensurate orders and multiple commensurate delays, where the time delays in the system may belong to several distinct intervals. Moreover, the controller parameters should belong to given intervals. In order to stabilize the system, the D-subdivision method is employed to choose the stabilizing set of the controller parameters from their available values. Furthermore, the nearest values of the obtained stabilizing set to their mean values are selected as the controller parameters so that a non-fragile controller is concluded. Two numerical examples evaluate the proposed control design method  

This paper deals with irrational transfer functions having monotonic nondecreasing step responses. Firstly, some results on the monotonicity of step responses in irrational transfer functions describing fractional- or distributed-order systems are presented. Then, some conditions guaranteeing the existence of monotonic nondecreasing step responses in more general forms of irrational transfer functions are found. Various examples are brought to show the usefulness of the obtained results in time response analysis of fractional/distributed-order systems. The achievements of the paper can be used in the design of control systems having monotonic step responses  

This study deals with the investigation of the magnitude-frequency responses of all-pole fractional order systems in the viewpoint of extrema existence in these responses. In this investigation, a sufficient algebraic condition, two necessary algebraic conditions, and a necessary and sufficient algebraic condition are obtained to guarantee the non-existence of extrema in the magnitude-frequency response of all-pole fractional order systems. Some examples are presented to show the effectiveness of the results of the paper  

Finding the oscillatory region in the order space is one of the most challenging problems in nonlinear fractional-order systems. This paper proposes a method to find the possible oscillatory region in the order space for a nonlinear fractional-order system. The effectiveness of the proposed method in finding the oscillatory region and special order sets placed in its boundary is confirmed by presenting some examples  

In this paper, we address global non-fragile control and synchronization of a new fractional order chaotic system. First we inspect the chaotic behavior of the fractional order system under study and also find the lowest order (2.49) for the introduced dynamics to remain chaotic. Then, a necessary and sufficient condition which can be easily extended to other fractional-order systems is proposed in terms of Linear Matrix Inequality (LMI) to check whether the candidate state feedback controller with parameter uncertainty can guarantee zero convergence of error or not. In addition, the proposed method provides a global zero attraction of error that guarantees stability around all existing... 

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  

The undamped oscillations of a scalar fractional neutral system are studied in this paper. For this purpose, it is proved that this kind of systems can have the oscillatory behavior. The necessary and sufficient condition is proposed to determine that there exists a delay value for which the system oscillates. The frequency, amplitude and required delay for the undamped oscillations depend on the fractional order. However, the behavior of dependencies is not specified. Indeed, when the system has oscillatory behavior, it is bounded-input bounded-output stable for delay less than the required delay. Moreover, when the order is approaching to an integer value, the necessary and sufficient... 

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 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  

It has been known that finding a minimal pseudo state space realization for a fractional order transfer function is helpful in circuitry implementation of such a transfer function with the minimum number of fractional capacitors. Considering this importance, the present paper deals with finding minimal realizations for some classes of fractional order transfer functions. To this end, at first some upper bounds are obtained for the minimal inner dimension of a fractional order transfer function. By considering these upper bounds and also the lower bounds, previously presented in literature, the minimal inner dimension is exactly found for some classes of fractional order transfer functions.... 

This paper presents a new method for assessing the bounded input bounded output stability of a class of fractional delay systems with commensurate orders and multiple commensurate delays of retarded type. In the proposed method, first, by mapping the principal sheet of the Riemann surface and a pseudo-delay transformation, an auxiliary polynomial is generated. Then, this auxiliary polynomial is employed to find roots of the characteristic equation on the imaginary axis. The properties of the root path close to these roots are used to identify intervals of delay values, in which the system is stable. The obtained results are illustrated via some numerical examples  

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... 

This paper studies undamped oscillations generated by marginally stable fractional order linear time invariant (LTI) systems. In this study, an analytical approach is proposed to be used for harmonic analysis of such oscillations in marginally stable commensurate order LTI systems. Also, it is shown that the Q-semi norm of the limit sets for a trajectory of these systems can be analytically determined based on the Q-semi norm of the initial condition, where Q is a specific matrix. Moreover, this result is extended to the wider class of rational order systems. Finally, some numerical examples are presented to demonstrate the use of the paper results  

This paper considers a closed-loop system consisting of a fractional/integer order system and a fractional PID controller. Assuming that the uncertain coefficients of the fractional PID controller lie in some known intervals independently (i.e. that controller is a member of an interval family), the paper presents some easy to use theorems to investigate the robust bounded-input bounded-output stability of the resultant closed-loop system. Moreover, a finite frequency bound required in drawing the related graphs has also been provided. Finally, some numerical examples are presented to illustrate the results