Tom T. Hartley

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

HARDWARE IMPLEMENTATION OF FRACTIONAL-ORDER SYSTEMS AS INFINITE IMPULSE RESPONSE FILTERS

Abstract In this paper we approximate fractional-order systems as IIR filters using field programmable gate arrays (FPGAs). The resulting implementations are low cost and high speed, with guaranteed stability. Our method implements the IIR approximation as a parallel combination of first-order sections, and quantizes each filter coefficient in fixed-point according to its own independent contribution to the frequency response. This makes it possible to explore the tradeoffs between system quality and hardware cost easily, with a short design time.

Conditions for stable and causal conjugate-order systems

The stability properties of the fundamental linear conjugate-order system are studied. The values of the complex order for which this system is stable and causal are determined. It is shown that all stable, causal systems have orders that lie within unit-radius circles centered at ±1 or on the imaginary axis in the order plane. Plots are given to illustrate these results for several examples. It is shown that as the system bandwidth moves to large or small values relative to unity, the stability region in the order plane becomes more fragmented.

Estimation of the Hankel Singular Values of Fractional-Order Systems

This paper applies the Rayleigh-Ritz method to approximating the Hankel singular values of fractional-order systems. The algorithm is presented, and estimates of the first ten Hankel singular values of G(s) = 1 /(sq +1 ) for several values of q ∊ (0 , 1 ] are given. The estimates are computed by restricting the operator domain to a finite-dimensional space. The Hankel-norm estimates are found to be within 15% of the actual values for all q ∊ (0 , 1 ].Copyright