Firas Khemane

Brief paper: Stability and resonance conditions of elementary fractional transfer functions

Elementary fractional transfer functions are studied in this paper. Some basic properties of elementary transfer functions of the first kind are recalled. Then, two main results are presented regarding elementary fractional transfer functions of the second kind, written in a canonical form and characterized by a commensurate order, a pseudo-damping factor, and a natural frequency. First, stability conditions are established in terms of the pseudo-damping factor and the commensurate order, as a corollary to Matignons stability theorem. They extend the previous result into conditions that are simpler to check. Then, resonance conditions are established numerically in terms of the commensurate order and the pseudo-damping factor and give interesting information on the frequency behavior of fractional systems. It is shown that elementary transfer functions of the second kind might have up to two resonant frequencies. Moreover, three abaci are given allowing to determine the pseudo-damping factor and the commensurate order for, respectively, a desired normalized gain at each resonance, a desired phase at each resonance, and a desired normalized first or second resonant frequency.

Robust estimation of fractional models in the frequency domain using set membership methods

In this paper, the usual definition of Grunwald-Letnikov fractional derivative is first extended to interval derivatives in order to deal with uncertainties in the differentiation orders. The Laplace transform of interval derivatives is computed and its monotonicity is studied in the frequency domain. Next, the main objectives of this paper are presented as the implementation of three methods for set membership parameters estimation of fractional differentiation models based on complex frequency data. The first one uses a rectangular inclusion function with rectangle sides corresponding to real and imaginary parts of the complex frequency response; the second one uses a polar inclusion function and the gain/phase representation; the third one uses a circular inclusion function with disk representation. Each inclusion function introduces pessimism differently. It is shown that all three approaches are complementary and that the results can be merged to obtain a smaller feasible solution set. The proposed methods can be applied to estimate parameters of certain/uncertain linear time variant/invariant systems.

Approximation of a Fractance by a Network of Four Identical RC Cells Arranged in Gamma and a Purely Capacitive Cell

In this paper a storage I-element and a fractance are associated, thus defining a fractional system. In order to achieve the fractance, an approximation by a network of 4 identical RC cells arranged in gamma and a purely capacitive cell is proposed, thus defining a rational system. When compared to each other, the dynamic behaviors of the fractional and the rational systems show an excellent superposition of frequency and time-domain responses. Moreover, the robustness of stability margins obtained with both systems is illustrated versus variations of the I-element.

Resonance and Stability Conditions for Fractional Transfer Functions of the Second Kind

Fractional transfer functions of the second kind are studied in this chapter. First, stability conditions are established in terms of pseudo-damping factor and commensurable order. Then, resonance conditions are determined. An open-loop analysis of the equivalent closed-loop transfer function of the second kind confirms the analysis.

Set Membership Estimation of Fractional Models in the Frequency Domain

The main objective of this paper is to estimate the whole set of feasible parameters of a fractional differentiation model, based on gain and phase frequency data. All parameters, including differentiation orders, are expressed as intervals and then estimated using a bounded error approach. A contraction method named forward-backward propagation is first applied to reduce the initial searching space. Then, a set inversion algorithm named SIVIA is applied on the reduced searching space to obtain the whole set of feasible parameters. One of the interesting points of this study is to show the separate contribution of gain and phase data on the final estimation.

Comparison between two set membership methods for frequency domain system identification using fractional models

The main objective of this paper is to estimate the whole set of feasible parameters of a fractional differentiation model, through two methods based on complex frequency data. The first one uses a rectangular inclusion function with rectangle sides corresponding to real and imaginary parts of the complex frequency response; whereas the second one uses a polar inclusion function and the magnitude/phase representation. Each inclusion function introduces some pessimism differently. It is shown that the two approaches are complementary and that the results can be merged to obtain the smallest feasible solution set.