Mathieu Moze

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.

LMI Tools for Stability Analysis of Fractional Systems

The main point when dealing with Linear Matrix Inequalities (LMI) is convexity. However, with state space representation of fractional systems, the stability domain for a fractional order 0 < ν < 1 is not convex. The classical stability condition thus cannot be extended to fractional systems. In this paper, three LMI based methods are used to characterize stability. The first is based on the second Lyapunov method and provides a sufficient but non-necessary condition. The second and new method provides a sufficient and necessary condition, and is based on a geometric analysis of a fractional system stability domain. The third method is more conventional but involves non strict LMI. A comparison of the first two methods is provided.Copyright

An overview of the CRONE approach in system analysis, modeling and identification, observation and control

Abstract The aim of the paper is to present the fundamental definitions connected to fractional differentiation and to present an overview of the CRONE approach in the fields of system analysis, modeling and identification, observation and control. Industrial applications of fractional differentiation are also described in this paper. Some recent developments are also presented.

Fractional order polytopic systems: robust stability and stabilisation

This article addresses the problem of robust pseudo state feedback stabilisation 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 stability domain whatever the value of the uncertain parameters. The article focuses particularly on the case of a fractional order ν such that 0 < ν < 1, as the stability region is non-convex and associated LMI condition is not as straightforward to obtain as in the case 1 < ν < 2. In relation to the quadratic stabilisation problem previously addressed by the authors and that involves a single matrix to prove stability of the closed loop system, additional variables are then introduced to decouple system matrices in the closed loop system stability condition. This decoupling allows using parameter-dependent stability matrices and leads to less conservative results as attested by a numerical example.

On Fractional Systems H &#8734; , -Norm Computation

Two methods are proposed in this paper for fractional system H∞-norm computation. These methods are extensions to fractional systems of well-known methods for integer systems. The first is based on singular value properties of a linear system and is applied on an academic example. In the second, two extensions of the real bounded lemma derived directly from Lyapunovs theory are deduced. The first method is applied on an academic example.

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.

On bounded real lemma for fractional systems

Abstract Two state space “like” representation based methods for fractional systems L 2 -gain computation are proposed in this paper. The first is based on an approach already presented in the literature and leads to a new theorem. The theorem is based on the location of the eigenvalues of a matrix issued from the state space “like” representation and is then converted using Riccati theory into an LMI constraint to give the second theorem. Its formulation is similar to the well known bounded real lemma whereas it does not guarantee stability. The theorems are finally applied to car suspension analysis for the computation of modulus margins. Prospects of this study are in the fields covered by the usual bounded real lemma such as H infin; control, thus aiming at straightforward extension to fractional systems.

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.

Air-Fuel Ratio Control of an Internal Combustion Engine Using CRONE Control Extended to LPV Systems

An extension of the CRONE control method to LPV systems is presented in the paper. A LPV controller whose parameters are scheduled on those of the system is determined from small gain theorem applied with a LFT formulation of the system. This approach permits open loop insensitivity to varying parameters, enabling its optimal parameterization for robustness purposes. The method is finally applied to air/fuel ratio control of internal combustion engine. The approach is validated from simulation results.

Simple and Robust Experiment Design for System Identification Using Fractional Models

This paper tackles the problems of simple and robust experiment design for system identification using elementary fractional models. It is based on a frequency domain approach and allows to determine the best sine input signal maximizing D-optimality criterion of the parameters inverse covariance matrix in different contexts? First, a single parameter (any of the parameters of the elementary fractional model) is assumed to be unknown. Next, any combination of two and then three parameters are supposed to be unknown. Finally, the problem of robust experiment design is treated when a bounded interval of the estimated parameters is known, in the same contexts.