Jocelyn Sabatier

LMI stability conditions for fractional order systems

After an overview of the results dedicated to stability analysis of systems described by differential equations involving fractional derivatives, also denoted fractional order systems, this paper deals with Linear Matrix Inequality (LMI) stability conditions for fractional order systems. Under commensurate order hypothesis, it is shown that a direct extension of the second Lyapunovs method is a tedious task. If the fractional order @n is such that 0<@n<1, the stability domain is not a convex region of the complex plane. However, through a direct stability domain characterization, three LMI stability analysis conditions are proposed. The first one is based on the stability domain deformation and the second one on a characterization of the instability domain (which is convex). The third one is based on generalized LMI framework. These conditions are applied to the gain margin computation of a CRONE suspension.

Embedded Fractional Nonlinear Supercapacitor Model and Its Parametric Estimation Method

This paper deals with a supercapacitor (SC) onboard model and its online identification procedure for embedded applications. To take into account its nonlinear behavior, the strategy used in this paper consists, as the first step, in the application of the porous electrode theory to the SC and the approximation of the resulting model. A set of fractional linear systems which are represented by a differential equation involving fractional derivatives is then obtained. Each element of this set represents the behavior of the SC only around one operating voltage. A global nonlinear model is then deduced through an integration method. An online identification procedure has been developed for this nonlinear model. This time identification is based on the mean least square method. Its time behavior has been compared with that of an SC cell for a specific current profile with different levels.

Fractional systems state space description: some wrong ideas and proposed solutions

This paper highlights several misinterpretations that arise in the field of fractional systems analysis using a representation known in the literature as “state space description”. Given these misinterpretations, some results already published and based on this description are questionable. Thus alternative descriptions are proposed.

On Lead-Acid-Battery Resistance and Cranking-Capability Estimation

With hybrid and electric vehicle developments, battery-monitoring systems have to meet the new requirements of the automobile industry. This paper deals with one of them, the batterys ability to start a vehicle, also called battery crankability, through battery-resistance estimation. A fractional-order model obtained by system identification is used to estimate the internal resistance of lead-acid batteries. Fractional-order modeling permits an accurate simulation of the battery electrical behavior with a low number of parameters. Moreover, the high-frequency gain of the fractional model is directly linked to the battery resistance. A resistance-estimation method based on a frequency-invalidation method is, thus, proposed. It is demonstrated that the batterys available power that defines battery crankability is correlated to the battery resistance. Thus, a battery-crankability estimator using the battery resistance is suggested. Validation tests are carried out with various batteries. This estimator cannot be embedded in a microcontroller due to the linear-matrix-inequality-based optimization algorithm in the invalidation-model method used. A simplified algorithm is finally proposed, and its efficiency is proved.

ON STABILITY OF COMMENSURATE FRACTIONAL ORDER SYSTEMS

This paper proposes a new proof of the Matignons stability theorem. This theorem is the starting point of numerous results in the field of fractional order systems. However, in the original work, its proof is limited to a fractional order ν such that 0 < ν < 1. Moreover, it relies on Caputos definition for fractional differentiation and the study of system trajectories for non-null initial conditions which is now questionable in regard of recent works. The new proof proposed here is based on a closed loop realization and the application of the Nyquist theorem. It does not rely on a peculiar definition of fractional differentiation and is valid for orders ν such that 1 < ν < 2.

Lithium-ion batteries modeling

This paper deals with lithium-ion batteries modeling. From an electrochemical model available in the literature, several assumptions and simplifying hypotheses are proposed in order to get a simpler but accurate model. The obtained model is based on a fractional transfer function resulting in the resolution of a partial differential equation that describes the lithium ion diffusion inside the electrodes. The model involves only three parameters and a polynomial that fits the open circuit voltage of the battery. A method to estimate the model parameters and the polynomial is proposed. A single discharge test (constituted of several discharge steps) is required to operate the parameter estimation method. The accuracy for battery voltage prediction of the resulting model is evaluated with various tests. A simplified model is obtained from a lithium-ion battery electrochemical model.The proposed model is a single-electrode model.Fractional differentiation is used for a low number of parameters in the model.A relative error less than 0.5% on the voltage is obtained under various conditions.The model simplicity and accuracy are interesting for use in automobile BMS.

Bode optimal loop shaping with CRONE compensators

Ideal Bode characteristics give a classical answer to optimal loop design for linear time invariant feedback control systems in the frequency domain. This work recovers eight-parameter Bode optimal loop gains, providing a useful and simple theoretical reference for the best possible loop shaping from a practical point of view. The main result of the paper is to use CRONE compensators to make a good approximation and in addition a way for the synthesis of the Bode optimal loop. For that purpose, a special loop structure based on second and third generation CRONE compensators is used. As a result, simple design relationships will be obtained for tuning the proposed CRONE compensator.

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.

A Mathematical Model for the Simulation of New and Aged Automotive Lead-Acid Batteries

Today, it is possible to design a mathematical model of lead-acid battery on a laptop from scratch with MATLAB. It still takes time to develop, but nowadays these models alone cannot be considered innovations anymore because anybody is able to do the same with the appropriate tools. More than ever, the actual innovation lies in the answer given to the most important question: a model to do what? Here, the models main goal is to simulate the flooded battery behavior in all vehicle life cycles. Thus, it seems legitimate to validate them under different operating conditions with experimental data to prove that the simulation results are reliable. It is challenging but feasible, notably by making a correct selection of the parameter value ranges and studying how they influence the output. Another goal is to understand the battery behavior changes due to the physicochemical processes involved at various rates and temperatures to explain Peukerts law and the impact of temperature on vehicle cranking. The final objective of this model is to find parameter changes corresponding to the main aging phenomena to simulate new and also used batteries.

Fractional Models for Thermal Modeling and Temperature Estimation of a Transistor Junction

The thermal behavior of a power transistor mounted on a dissipator is considered in order to estimate the transistor temperature junction using a measure of the dissipator temperature only. 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 Control Output Observer (COO) to estimate the transistor junction temperature.