François Levron

Brief paper: Synthesis of fractional Laguerre basis for system approximation

Fractional differentiation systems are characterized by the presence of non-exponential aperiodic multimodes. Although rational orthogonal bases can be used to model any L2[0,∞[ system, they fail to quickly capture the aperiodic multimode behavior with a limited number of terms. Hence, fractional orthogonal bases are expected to better approximate fractional models with fewer parameters. Intuitive reasoning could lead to simply extending the differentiation order of existing bases from integer to any positive real number. However, classical Laguerre, and by extension Kautz and generalized orthogonal basis functions, are divergent as soon as their differentiation order is non-integer. In this paper, the first fractional orthogonal basis is synthesized, extrapolating the definition of Laguerre functions to any fractional order derivative. Completeness of the new basis is demonstrated. Hence, a new class of fixed denominator models is provided for fractional system approximation and identification.

Frequency-domain synthesis of a filter using Viete root functions

This paper proposes a noniterative method to fit a rational transfer function to a specified frequency response. A method using a smoothing of width-modulated pulses of the phase asymptotic-diagram is modified to provide an exact algebraic method. This is then used to synthesize a robust controller and a nonrational transfer function with a fractional differentiation order.

CRONE control of continuous linear time periodic systems: Application to a testing bench

Frequency methods only are used here for the study and control of continuous linear time periodic systems. Using time varying frequency responses defined by L. A. Zadeh in the 1950s, the second generation CRONE control is extended to the control of linear time periodic systems. This control strategy ensures, for the closed-loop system, a near stationary behavior, performances set by the designer, and robustness of performances to gain variations of the plant. An application of the proposed control strategy to a testing bench shows its efficiency.

Orthonormal basis functions for modeling continuous-time fractional systems 1

Abstract The classical Laguerre functions are known to be divergent as soon as their differentiation order is non-integer. They are hence inappropriate for representing fractional differentiation systems. A complete orthogonal basis, having fractional differentiation orders and a single pole, is synthesized. It extends the well-known definition of Laguerre functions to fractional systems. Hence a new class of fixed denominator models is provided for system identification. Fourier coefficients are computed using a least squares method. The least squares error is plotted versus the differentiation order and the pole, in an example, and shows that an optimal differentiation order may be located away from an integer number. Hence, the use of the synthesized basis is fully justitied.

Third Generation Crone Control of Continuous Linear Time Periodic Systems

Abstract Third generation CRONE control based on complex non integer differentiation results from an optimal approach of control. Its aim is to ensure the robustness of the stability degree of the control loop at the time of a reparametration of the plant In this paper, third generation CRONE control is extended to the control of linear time periodic systems. This extension is possible through a generalisation of the Nyquist theorem, and through the use of transfer functions for linear time periodic system modelling : time varying transfer function and hannonic transfer function.

Frequency Identification by Non Integer Model

Abstract This article deals with frequency identification by non integer model. The problem consists in interpolating a frequency response of system using implicit generalized derivative transmittance. The solution of this problem is given by two strategies, one linear and the other nonlinear. Linear approach determines the parameters of model by the minimization of a criterion based on the prediction error of intermediate integer model. Nonlinear approoch determines directly parameters from the minimization of a criterion using the gradient mcthod. Two applications illustrate these two different approachs.

ENERGY OF FRACTIONAL ORDER TRANSFER FUNCTIONS

Abstract The objective of the paper is to compute the impulse response energy of a fractional order transfer function having a single mode. The differentiation order n , defined in the sense of Riemann-Liouville, is allowed to be a strictly positive real number. A necessary and sufficient condition is established on n , in order for the impulse response to belong to the Lebesgue space L 2 [0, ∞[of square integrable functions on [0, ∞[.

Propagative Recursive Distributed Parameter Systems and Non Integer Differentiation

Abstract This paper studies the link between non integer differentiation and the class of partial differential equations which describes propagative recursive distributed parameter systems. Through an input-output approach, it shows that these systems can be characterised by a particular admittance : a continued fraction. After the study of this continued fraction, it gives a theorem which highlights the link of the class of partial differential equations considered with non integer differentiation. It shows that the non integer order, and only this, imposes frequency behaviour in a well defined interval. It finally proposes an illustrative example : the characterisation of the behaviour of a hydraulic pipe.