Jay L. Adams

Energy storage and loss in fractional-order circuit elements

The efficiency of a general fractional-order circuit element as an energy storage device is analysed. Simple expressions are derived for the proportions of energy that may be transferred into and then recovered from a fractional-order element by either constant-current or constant-voltage charging and discharging. For a half-order element, it is shown that the efficiency of the charging phase of the cycle is equal to the efficiency of the discharging phase. The results demonstrate the duality of the fractional capacitive and inductive elements, in that the efficiency of one under constant-current cycling is the same as the efficiency of the other under constant-voltage cycling, and vice-versa.

Finite-Time Controllability of Fractional-Order Systems

In this paper the conditions that lead to a system output remaining at zero with zero input are considered. It is shown that the initialization of fractional-order integrators plays a key role in determining whether the integrator output will remain at a zero with zero input. Three examples are given that demonstrate the importance of initialization for integrators of order less than unity, inclusive. Two examples give a concrete illustration of the role that initialization plays in keeping the output of a fractional-order integrator at zero once it has been driven to zero. The implications of these results are considered, with special consideration given to the formulation of the fractional-order optimal control problem.Copyright

Identification of Complex Order-distributions

This article discusses the identification of fractional systems using the concepts of complex order distribution. Based on the ability to define systems using complex order-distributions, it is shown that system identification in the frequency domain using a least-squares approach can be performed. A mesh is created to cover an area in the complex-order plane. The weighting of each block in the order-plane is selected to minimize the square-error between the frequency response of the system and the identified system. The identified systems have real time responses. Four examples, including both pure real-order systems and pure complex-order systems, are presented to demonstrate the utility of the identification method.

Conjugated-Order Differintegrals

This paper introduces the concept of conjugated-order differintegrals. These are fractional derivatives whose orders are complex conjugates. These conjugate-order differintegrals allow the use of complex-order differintegrals while still resulting in real time-responses and real transfer-functions. Both frequency responses and time responses are developed. The conjugated differintegral is shown to be a useful representation for control design. An example is presented to demonstrate its utility.© 2005 ASME

Complex-Order Distributions

This paper develops the concept of the complex order-distribution. This is a continuum of fractional differintegrals whose order is complex. Two types of complex order-distributions are considered, uniformly distributed and Gaussian distributed. It is shown that these basis distributions can be summed to approximate other complex order-distributions. Conjugated differintegrals are an essential analytical tool applied in this development due to their associated real time-responses. An example is presented to demonstrate the complex order-distribution concept. This work enables the generalization of fractional system identification to allow the search for complex order-derivatives that may better describe real time behaviors.Copyright

Modeling Ultracapacitors as Fractional-Order Systems

Ultracapacitors display long-term transients lasting for many months. This paper shows that these long-term transients can be accurately represented using a fractional-order system model for the ultracapacitor impedance. Time-domain data is used to determine the impedance transfer-function coefficients. A circuit realization for the ultracapacitor is given which explicitly shows the fractional-order component. These long-term transient models will allow the development of improved ultracapacitor management systems.

Time-Varying Initialization and Corrected Laplace Transform of the Caputo Derivative

Abstract This paper derives the time-varying initialization function for the Caputo derivative of any order u >0 where m –1 u m , m = 1,2,…. The derivative is redefined to include this initialization function. Then, the Laplace transform for the redefined Caputo derivative is determined which corrects (supplants) that given for the derivative in the literature and properly accounts for time-varying initialization effects. The paper generalizes previous results for order 0 u

On the Energy Stored in Fractional-Order Electrical Elements

In this paper, fractional-order electrical elements are considered as energy storage devices. They are studied by comparing the energy available from the element to do future external work, relative to the energy input into the element in the past. A standard circuit realization is used to represent the fractional-order element connected to an inductor with a given initial current. This circuit realization is used to determine the energy returned by both capacitive and inductive fractional-order elements of order between zero and one. Plots of the energy stored versus time are provided. The major conclusion is that fractional-order elements tend to rapidly dissipate much of their input energy leaving less energy for doing work in the future.Copyright

FRACTIONAL-ORDER SYSTEM IDENTIFICATION USING COMPLEX ORDER-DISTRIBUTIONS

Abstract This paper discusses the identification of fractional systems using the concepts of complex order-distribution. Based on the ability to define systems using complex order-distributions, it is shown that system identification in the frequency domain using a least-squares approach can be performed. Four examples are presented to show the utility of the identification method.

Teaching freshmen VHDL-based digital design

Industry demand for highly-skilled digital VLSI and embedded systems engineers has made the teaching of design a challenging task for undergraduate educators. Increasingly challenging electronic design automation (EDA) environments call for a variety of skills in engineering graduates which necessitate not only industrial skills but also fundementals in basic science and digital design. These urgent needs are addressed by creating a digital logic design course taught at the freshman level that introduces students to VHDL design of digital VLSI systems while including core concepts. Critical thinking, logic synthesis and circuit innovation take priority over conventional analysis techniques. This case study includes example design projects that have been successfully implemented at The University of Akron.