Christophe Farges

Brief paper: Pseudo-state feedback stabilization of commensurate fractional order systems

This paper addresses the problem of pseudo state feedback stabilization of commensurate fractional order systems. In the proposed approach, Linear Matrix Inequalities (LMI) formalism is used to check if the pseudo state matrix eigenvalues belong to the non convex fractional system stability region of the complex plane. A new LMI stability condition is first proposed. Based on this condition, a necessary and sufficient LMI method for the design of stabilizing controllers is given. Its efficiency is evaluated on an inverted fractional pendulum stabilization problem.

On observability and pseudo state estimation of fractional order systems

In this paper, fractional order system observability is discussed. A representation of these systems that involves a classical linear integer system and a system described by a parabolic equation is used to define the system real state and to conclude that the system state is approximately observable. However, it is also shown that the pseudo state space representation, usually encountered in the literature for fractional systems, can be used to design Luenberger like observers that permits an estimation of important variables in the system. Such an observer is finally used in order to estimate the temperature in a thermal system from a temperature measure at a different abscissa.

A stability test for non-commensurate fractional order systems

This paper presents a necessary and sufficient condition to evaluate non-commensurate fractional order systems Bounded Input, Bounded Output stability. This condition is based on an algorithm that relies on a recursively defined closed-loop realization of the system and involves Cauchys theorem. Its efficiency is attested by several numerical examples.

On computation of H ∞ norm for commensurate fractional order systems

This paper tackles the problem of H-infinity (H∞) norm computation for a commensurate Fractional Order System (FOS). First, H∞ norm definition is given for FOS and Hamiltonian matrix of a FOS is computed. Two methods based on this Hamiltonian matrix are then proposed to compute the FOS H∞ norm: one based on a dichotomy algorithm and another one on LMI conditions. The LMI conditions are based on the Generalized LMI characterization of axes in the complex plane on which the Hamiltonian matrix eigenvalues must not appear to ensure a FOS norm less than predefined value. The accuracy of the proposed methods is proved on the computation of the modulus margin of a CRONE passive car suspension.

Robust stability analysis and stabilization of fractional order polytopic systems

This paper addresses the problem of robust pseudo state feedback stabilization of commensurate fractional order polytopic systems (FOS). In the proposed approach, Linear Matrix Inequalities (LMI) formalism is used to check if the pseudo state matrix eigenvalues belong to the FOS whatever the value of the uncertain parameters. The paper focuses particularly on the case 0 < nu < 1 as the stability region is non convex and associated LMI condition is not as straightforward to obtain as in the case 1 < nu < 2. The quadratic stabilisation problem involving a single matrix in order to prove stability of the closed loop system is first addressed. Additional variables are then introduced in order to decouple system matrices from the ones proving stability of the closed loop system. This decoupling allows using parameter dependant stability matrices and leads to less conservative results as attested by a numerical example.

Approximation of a fractional order model by an integer order model: a new approach taking into account approximation error as an uncertainty:

In order to solve some analysis or control problems for fractional order models, integer order approximations are often used. However, in many works, approximation error is not taken into account, leading to results that cannot be guaranteed for the initial fractional order model. The objective of the paper is thus to provide a new methodology that takes into account approximation error and leads to rewriting the fractional order model as an uncertain integer order model.

Transistor thermal fractional modeling for junction temperature estimation

Abstract : In this paper, the thermal behavior of a power transistor mounted on a dissipator is considered in order to estimate the transistor temperature junction only using a measure of the dissipator temperature. The thermal transfers between the electric power applied to the transistor, the junction temperature and the dissipator temperature are characterized by two fractional transfer functions. These models are then used in a Luenberger like observer to estimate the transistor junction temperature.