Sharif Digital Repository

This paper deals with the problem of designing the PI λDμ-type controllers for minimum-phase fractional systems of rational order. In such systems, the powers of the Laplace variable, s, are limited to rational numbers. Unlike many existing methods that use numerical optimisation algorithms, the proposed method is based on an analytic approach and avoids complicated numerical calculations. The method presented in this paper is based on the asymptotic behaviour of fractional algebraic equations and applies a delicate property of the root loci of the systems under consideration. In many cases, the resulted controller is conveniently in the form of P, Iλ, PDμ or PIλDμ. Four design examples are... 

This paper introduces a system with switching multi-model structure which can generate chaos. Sub-models in this structure are fractional-order linear systems with any desired commensurate order less than 1. It shows that this system is capable of demonstrating chaotic behavior if its parameters and switching rule are suitably chosen. The structure of the proposed system is defined in a general form; consequently various chaotic attractors can be created by this system with different choices of order, parameters and switching rule. Numerical simulations illustrate behavior of the introduced system in some different situations  

This study deals with the properties of magnitude-frequency responses in fractional order systems. Using Phragmén-Lindelöf theorem in complex analysis, it is shown that the supremum of the magnitude-frequency response of a fractional system with a commensurate order less than one cannot be greater than that of its integer order bounded-input, bounded-output stable counterpart. Further results are also obtained on magnitude-frequency response of stable/unstable fractional order systems. Moreover, it is found that the supremum (infimum) of the magnitude-scaling frequency of the family of fractional order systems having a fixed structure and different orders in the range (0, 2) is a piecewise... 

In approximation of fractional order systems, a significant objective is to preserve the important properties of the original system. The monotonicity of time frequency responses is one of these properties whose preservation is of great importance in approximation process. Considering this importance, the issues of monotonicity preservation of the step response and monotonicity preservation of the magnitude-frequency response are independently investigated in this paper. In these investigations, some conditions on approximating filters of fractional operators are found to guarantee the preservation of step magnitude-frequency response monotonicity in approximation process. These conditions... 

In this paper, based on the stability theorems in fractional differential equations, a necessary condition is given to check the existence of 1-scroll, 2-scroll or multi-scroll chaotic attractors in a fractional order system. This condition is proposed for incommensurate order systems in general, but in the special case it converts to the condition given in the previous works for the commensurate fractional order systems. Though the presented condition is only a necessary (and not sufficient) condition for the existence of chaos it can be used as a powerful tool to distinguish for what parameters and orders of a given fractional order system, chaotic attractors can not be observed and for... 

In this paper, we propose a controller based on active sliding mode theory to synchronize chaotic fractional-order systems in master-slave structure. Master and slave systems may be identical or different. Based on stability theorems in the fractional calculus, analysis of stability is performed for the proposed method. Finally, three numerical simulations (synchronizing fractional-order Lü-Lü systems, synchronizing fractional order Chen-Chen systems and synchronizing fractional-order Lü-Chen systems) are presented to show the effectiveness of the proposed controller. The simulations are implemented using two different numerical methods to solve the fractional differential equations. © 2007... 

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 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 this paper, by introducing a new general state-space form for uncertain linear time-invariant fractional-order systems subjected to interval and polytopic uncertainties, two problems including robust stability analysis and robust stabilization of the presented systems are investigated. Subsequently, two sufficient conditions in terms of several linear matrix inequalities for the problems mentioned are concluded as two separate theorems. It is assumed that the fractional order α is a known constant belonging to 0 < α < 1. Simulation results of two different numerical examples demonstrate that the provided sufficient conditions are applicable and effective for tackling robust stability and... 

In this paper, issues related to the identifiability of a fractional order system having its input and output frequency contents are discussed. The effects of the commensurate order α in the identifiability of the model structure and model parameters are analytically studied. It is shown that both identifiabilities (model structure and model parameters) are reduced remarkably for smaller values of α. This phenomenon is observed even though the input signals are rich enough and system belongs to the model set. Our understanding is that the problem arises since differences among different members of the model set fall beyond the practically recognizable precision range. The issue is more... 

This paper proposes a novel representation of uncertain LTI fractional order systems based on the state-space model which contains both interval and polytopic uncertainties. First, a set of linear matrix inequalities, which are sufficient conditions, are presented for analyzing the robust stability of the mentioned systems. Then, some sufficient conditions are obtained for designing a feedback gain matrix to tackle the robust stabilization of the considered systems. Note that the concluded conditions of this paper are valid for fractional systems with a given constant derivative order α in 1 ≤ α < 2 and also, can be employed conservatively for α in 0 < α < 1. Finally, through two numerical... 

This article presents a robust adaptive fractional order proportional integral derivative controller for a class of uncertain fractional order nonlinear systems using fractional order sliding mode control. The goal is to achieve closed-loop control system robustness against the system uncertainty and external disturbance. The fractional order proportional integral derivative controller gains are adjustable and will be updated using the gradient method from a proper sliding surface. A supervisory controller is used to guarantee the stability of the closed-loop fractional order proportional integral derivative control system. Finally, fractional order Duffing–Holmes system is used to verify... 

This paper is devoted to the analysis of fractional order Van der Pol system studied in the literature. Based on the existing theorems on the stability of incommensurate fractional order systems, we determine parametric range for which a fractional order Van der Pol system with a specific order can perform as an undamped oscillator. Numerical simulations are presented to support the given analytical results. These results also illuminate a main difference between oscillations in a fractional order Van der Pol oscillator and its integer order counterpart. We show that contrary to integer order case, trajectories in a fractional Van der Pol oscillator do not converge to a unique cycle. © 2009... 

In this article, a fault-tolerant adaptive neural network fractional controller has been proposed for a class of uncertain multi-input single-output (MISO) incommensurate fractional-order non-strict nonlinear systems subject to five different types of unknown input nonlinearities, infinite number of actuators failures and arbitrary independent time-varying output constraints. The barrier Lyapunov function (BLF)-based backstepping technique and fractional Lyapunov direct method (FLDM) have been used to design the controller and establish system stability. To tackle the incommensurate derivatives problem, in each step of the backstepping technique, an appropriate Lyapunov function has been...