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This paper investigates the step responses of fractional order systems in the viewpoint of extrema existence in such responses. It is proven that a fractional order system with a commensurate order between zero and one has an extrema-free step response if its integer counterpart has such a step response. In addition, it is shown that the step response of a stable fractional order system with a commensurate order between one and two cannot be monotonic. Based on these achievements, some further results on the step response of different classes of fractional order systems are presented  

The effects of using frequency-domain approximation in numerical simulation of fractional-order systems are analytically studied. The main aim in the study is to determine the number, location and stability property of the equilibriums in a fractional-order system and its frequency-based approximating counterpart. The comparison shows that the original fractional-order system and its frequency-based approximation may differ from each other in some or all issues considered in the study. Unfortunately, these differences can lead to wrong consequences in some special cases such as detecting chaos in the fractional-order systems. It is shown that using the frequency-domain approximation methods... 

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  

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.... 

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 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  

An approach is proposed to control transient response of fractional order systems with maximum permissible control signal. This goal is achieved using a newly suggested characteristic ratio assignment method. Based on the proposed method, the generalized time constant τ and the characteristic ratios including their pattern, an adjustable parameter β, and the product of two successive characteristic ratios ρ are determined such that predefined level of overshoot and time specification of closed loop step response are obtained while the control signal is confined to a pre assigned maximum magnitude. The raised issue is solved by defining an optimization problem in which the design parameters... 

In this work, a novel method is proposed to study the BIBO stability of a fractional delay system. The characteristic equation of a fractional delay system with some transcendental terms has infinitely many roots. Applying D-subdivision method and the Rekasius substitution divide one-dimensional parametric space of the time delay to infinite intervals with finite unstable roots. The number of unstable roots in each interval is calculated with the definition of root tendency on the boundary of each interval. Two illustrative examples are presented to confirm the proposed method results  

In this paper, a class of the direct methods for discretization of fractional order transfer functions is studied in the sense of stability preservation. The stability boundary curve is exactly determined for these discretization methods. Having this boundary helps us to recognize whether the original system and its discretized model are the same in the sense of stability. Finally, some illustrative examples are presented to evaluate achievements of the paper  

This paper deals with estimation of fractional order and pole locator in fractional order systems. The estimation is based on Bode diagram of the system that is obtained using input and output measurements. Here the magnitude diagram is approximated with number of straight lines depending on the level of complexity and in consequence a very good estimation of fractional order and acceptable approximations of pole locators are determined. Relying on the proposed method, complexity of fractional order system identification which is mostly due to the estimation of fractional order is substantially resolved. Some example simulation results are provided to explain the work and show its... 

In this paper, based on a stability theorem proved for linear fractional-order systems, a scheme for robust synchronization of two perturbed fractional-order Chen systems is proposed. In the proposed scheme, both master and slave systems are considered to be involved with external disturbances having unknown values. It is analytically shown that any set of bounded external disturbances can be damped by the proposed method, where synchronization error will be forced and then kept inside a ball around the origin. Since during the design procedure the radius of this ball could be easily chosen by the designer, the synchronization can be done with any desired accuracy. The proposed method can be... 

In this paper, the concept of minimal state space realization for a fractional order system is defined from the inner dimension point of view. Some basic differences of the minimal realization concept in the fractional and integer order systems are discussed. Five lower bounds are obtained for the inner dimension of a minimal state space realization of a fractional order transfer function. Also, the concept of optimal realization, which can be a helpful concept in practice, is introduced for transfer functions having rational orders. An algorithm is suggested to obtain the optimal realizations of rational order transfer functions. The introduced concept might be used to get minimal... 

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  

In this paper, the computation of robustness margin for linear time invariant fractional order systems is studied. For the definition of robustness margin, we employ the one introduced for polynomials (i.e. integer order) and extend it to fractional order functions. Using the well known concept of the value set and knowing its shape for the intended functions, this paper presents an easy way to obtain the robustness margin for fractional order systems. To illustrate the results, a numerical example is provided  

This paper deals with the stability problem in LTI fractional order systems having fractional orders between 1 and 1.5. Some sufficient algebraic conditions to guarantee the BIBO stability in such systems are obtained. The obtained conditions directly depend on the coefficients of the system equations, and consequently using them is easier than the use of conditions constructed based on solving the characteristic equation of the system. Some illustrations are presented to show the applicability of the obtained conditions. For example, it is shown that these conditions may be useful in stabilization of unstable fractional order systems or in taming fractional order chaotic systems  

This paper deals with the identification of fractional-order systems through orthogonal rational functions. Fractional systems are characterized by their non-exponential aperiodic multimodes; therefore, fractional orthogonal rational functions provide better approximating models with fewer parameters. In spite of the fact that the Laguerre-based model is simple, it is, to some extent, deficient at high frequencies. Motivated by this objective, the use of a Legendre basis which has progressive pole locations and can be expected to perform better at high frequencies is studied  

Integral performance indices as quantitative measures of the performance of a system are commonly used to evaluate the performance of designed control systems. In this paper, it is pointed out that due to existence of non-exponential modes in the step response of a fractional-order control system having zero steady state error, integral performance indices of such a system may be infinite. According to this point, some simple conditions are derived to guarantee the finiteness of different integral performance indices in a class of fractional-order control systems. Finally, some numerical examples are presented to show the applicability of the analytical achievements of the paper  

This paper deals with issues related to the use of rational approximations in the simulation of fractional-order systems and practical implementations of fractional-order dynamics and controllers. Based on the mathematical formulation of the problem, a descriptor model is found to describe the rational approximating model. This model is analyzed and compared with the original fractional-order system under the aspects which are important in their simulation and implementation. From the results achieved, one can determine in what applications the use of rational approximations would be unproblematic and in what applications it would lead to fallacious results. In order to clarify this point,... 

This paper investigates the robust stability analysis of fractional-order interval systems with multiple time delays, including retarded and neutral systems. A bound on the poles of fractional-order interval systems of retarded and neutral type is obtained. Then, the concept of the value set and zero exclusion principle is extended to these systems, and a necessary and sufficient condition is produced for checking the robust stability of them. The value set of the characteristic equation of the systems is obtained analytically and, based on it, an auxiliary function is introduced to check the zero exclusion principle. Finally, two numerical examples are given to illustrate the effectiveness... 

In this paper, the frequency domain-based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation. First, the stability boundary curve is exactly determined for those methods. Then, this boundary is analyzed and compared with an accurate (ideal) boundary in different frequency ranges. Also, the critical regions in which the stability does not preserve are determined. Finally, the analytical achievements are confirmed via some numerical illustrations. © 2008 Society for Industrial and Applied Mathematics