Computation of Fractional Order Derivative and Integral via Power Series Expansion and Signal Modelling

The three techniques of s-to-z transform, power series expansion (PSE) and signal modelling are combined to develop a new procedure for efficiently computing the fractional order derivatives and integrals of discrete-time signals. A mapping function between the s-plane and the z-plane is first chosen, and then a PSE of this mapping function raised to fractional order is performed to get the desired infinite impulse response of the ideal digital fractional operator. Finally, the desired impulse response is modelled as the impulse response of a linear invariant system whose rational transfer function is determined using deterministic signal modelling techniques. Three non-iterative techniques, namely Padé, Prony and Shanks’ methods have been considered in this paper. Using Al-Alaoui’s rule as s-to-z transform, computation examples show that both Prony and Shanks’ method can achieve more accurate fractional differentiation and integration than Padé method which is equivalent to continued fraction expansion technique.     

Fractional Order Disturbance Observer for Robust Vibration Suppression

For the first time, the fractional order disturbance observer (FO-DOB) is proposed for vibration suppression applications such as hard disk drive servo control. It has been discovered in a recently published US patent application (US20010036026) that there is a tradeoff between phase margin loss and strength of the low frequency vibration suppression. Given the required cutoff frequency of the low pass filter, also known as the Q-filter, it turns out that the relative degree of the Q-filter is the major tuning knob for this tradeoff. The solution in US20010036026 was based on an integer order Q-filter with a variable relative degree. This actually motivated the use of a fractional order Q-filter. The fractional order disturbance observer is based on the fractional order Q-filter. The implementation issue is also discussed. The nice point of this paper is that the traditional DOB is extended to the fractional order DOB with the advantage that the FO-DOB design is now no longer conservative nor aggressive, i.e., given the cutoff frequency and the desired phase margin, we can uniquely determine the fractional order of the low pass filter.     

Identification of Fractional Systems Using an Output-Error Technique

An original method for modeling, simulation and identification of fractional systems in the time domain is presented in this article. The basic idea is to model the fractional system by a state-space representation, where conventional integration is replaced by a fractional one with the help of a non-integer integrator. This operator is itself approximated by a N-dimensional system composed of an integrator and of a phase-lead filter. An output-error technique is used in order to estimate the parameters of the model, including the fractional order N. Simulations exhibit the properties of the identification algorithm. Finally, this methodology is applied to the modeling of the dynamics of a real heat transfer system.     

The Time-Scaled Trapezoidal Integration Rule for Discrete Fractional Order Controllers

In this paper, the time-scaled trapezoidal integration rule for discretizing fractional order controllers is discussed. This interesting proposal is used to interpret discrete fractional order control (FOC) systems as control with scaled sampling time. Based on this time-scaled version of trapezoidal integration rule, discrete FOC can be regarded as some kind of control strategy, in which strong control action is applied to the latest sampled inputs by using scaled sampling time. Namely, there are two time scalers for FOC systems: a normal time scale for ordinary feedback and a scaled one for fractional order controllers. A new realization method is also proposed for discrete fractional order controllers, which is based on the time-scaled trapezoidal integration rule. Finally, a one mass position 1/s^k control system, realized by the proposed method, is introduced to verify discrete FOC systems and their robustness against saturation non-linearity.     

Fractional relaxation and the time-temperature superposition principle

Relaxation processes in complex systems like polymers or other viscoelastic materials can be described by equations containing fractional differential or integral operators. In order to give a physical motivation for fractional order equations, the fractional relaxation is discussed in the framework of statistical mechanics. We show that fractional relaxation represents a special type of a non-Markovian process. Assuming a separation condition and the validity of the thermo-rheological principle, stating that a change of the temperature only influences the time scale but not the rheological functional form, it is shown that a fractional operator equation for the underlying relaxation process results.     

Modified characteristic finite difference fractional step method for moving boundary value problem of percolation coupled system

For the coupled system with moving boundary values of multilayer dynamicsof fluids in porous media,a characteristic finite difference fractional step scheme appli-cable to the parallel arithmetic is put forward.Some techniques,such as the change ofregions,the calculus of variations,the piecewise threefold quadratic interpolation,themultiplicative commutation rule of difference operators,the decomposition of high orderdifference operators,and the prior estimates,are adopted.The optimal order estimatesin the l2norm are derived to determine the error in the approximate solution.This nu-merical method has been successfully used to simulate the flow of migration-accumulationof the multilayer percolation coupled system.Some numerical results are well illustratedin this paper.     

Stabilization of fractional order systems using a finite number of state feedback laws

In this paper, the stabilization of linear time-invariant systems with fractional derivatives using a limited number of available state feedback gains, none of which is individually capable of system stabilization, is studied. In order to solve this problem in fractional order systems, the linear matrix inequality (LMI) approach has been used for fractional order systems. A shadow integer order system for each fractional order system is defined, which has a behavior similar to the fractional order system only from the stabilization point of view. This facilitates the use of Lyapunov function and convex analysis in systems with fractional order 1<q<2. To this end, an extremum-seeking method is used for obtaining Lyapunov function and defining a suitable sliding sector in order to enable switching between available control gains for system stabilization. Consequently, using the LMI approach in fractional order systems, necessary and sufficient conditions are provided for stabilization of systems with fractional order 1<q<2 using a limited number of available state feedback gains which lead to variable structure control.     

On Fractional Adaptive Control

Introducing fractional operators in the adaptive control loop, and especially in Model Reference Adaptive Control (MRAC), has proven to be a good mean for improving the plant dynamics with respect to response time and disturbance rejection. The idea of introducing fractional operators in adaptation algorithms is very recent and needs to be more established, that is why many research teams are working on the subject. Previously, some authors have introduced a fractional model reference in the adaptation scheme, and then fractional integration has been used to deal directly with the control rule. Our original contribution in this paper is the use of a fractional derivative feedback of the plant output, showing that this scheme is equivalent to the fractional integration, one with a certain benefit action on the system dynamical behaviour and a good robustness effect. Numerical simulations are presented to show the effectiveness of the proposed fractional adaptive schemes.     

Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control

With the increasingly deep studies in physics and technology,the dynamics of fractional order nonlinear systems and the synchronization of fractional order chaotic systems have become the focus in scientific research.In this paper,the dynamic behavior including the chaotic properties of fractional order Duffing systems is extensively investigated.With the stability criterion of linear fractional systems,the synchronization of a fractional non-autonomous system is obtained.Specifically,an effective singly active control is proposed and used to synchronize a fractional order Duffing system.The numerical results demonstrate the effectiveness of the proposed methods.     

Modeling and Analog Realization of the Fundamental Linear Fractional Order Differential Equation

This paper provides a rational function approximation of the irrational transfer function of the fundamental linear fra- ctional order differential equation, namely, whose transfer function is given by for 0<m<2. Simple methods, useful in system and control theory, which consists of approximating, for a given frequency band, the transfer function of this fractional order system by a rational function are presented. The impulse and step responses of this system are derived and simple analog circuit which can serve as fundamental analog fractional order system is also obtained. Illustrative examples are presented to show the exactitude and the usefulness of the approximation methods.