Pierre Melchior

Fractional differentiation for edge detection

In image processing, edge detection often makes use of integer-order differentiation operators, especially order 1 used by the gradient and order 2 by the Laplacian. This paper demonstrates how introducing an edge detector based on noninteger (fractional) differentiation can improve the criterion of thin detection, or detection selectivity in the case of parabolic luminance transitions, and the criterion of immunity to noise, which can be interpreted in term of robustness to noise in general.

Consideration of obstacle danger level in path planning using A* and fast-marching optimisation: comparative study

Obstacle danger level is taken into consideration in path planning using fractional potential maps. This paper describes the two optimisation methods tested: the A* algorithm and the Fast-Marching technique. The efficiency of the two approaches is illustrated and compared through a vehicle path planning application in a fixed obstacle environment. A* is a heuristically ordered research algorithm and is complete and admissible. Fast-Marching provides a convex map without local minima and permits real-time evaluation of optimal trajectories. A vehicle path planning application is considered in a fixed obstacle environment. A specific danger level is given to each obstacle. The obtained continuous curve trajectories are compared.

Fractional Multi-models of the Frog Gastrocnemius Muscle

In this article, frog gastrocnemius muscles are studied, and a multi-model identification presented. A transfer function is defined with few parameters to simulate striated muscle (Gastroctnemius) behaviour, for inclusion in a future real-time salamander computer model. A two-model structure permits description of both contraction and relaxation properties. Here, two physiological influences, fatigue and fibre types, are taken into account. A multi-model structure for each fibre type (fast (IIB), intermediate (IIA) and slow (I) fibres) is also used for inclusion in a future agonist-antagonist structure computer model.

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.

The CRONE aproach: Theoretical developments and major applications

Abstract This article deals with fractal robustness. Of physical origin, this robustness represents the insensitivity of damping in nature, resulting from the combination of fractality and non integer differentiation. The study case is a natural robust phenomenon: the relaxation of water on a porous dyke whose damping ratio is independent of the mass of moving water. The dynamic model which governs this phenomenon is a non integer order linear differential equation for which the natural frequency and the damping ratio of the oscillatory mode of the solution are determined. A remarkable result is that damping ratio is exclusively linked to differential equation non integer order which is dictated by the fractal dimension of the dyke. Damping robustness is illustrated by two isodamping half-straight lines in the operational plane, and by a frequency template in the Nyquist plane. Then, the principle of the CRONE suspension, the synthesis method and the performances are developed. This suspension system is called the CRONE suspension because of the link with the second-generation CRONE control. The transposition to automatic control of the two isodamping half-straight lines, defines the robust strategy used by the second generation CRONE control. Defining a generalization of the second generation CRONE control, the great principles of the third generation CRONE control are also given. Finally, the problem of time domain identification by non integer model is studied. Among the whole of the methods developed in the CRONE team, a single one is dealt with in this paper. This one is based on the sampling of a non integer differential equation through the Grunwald numeric approximation of the non integer derivative of a time function. An application is carried out within the context of the identification of the dynamic behaviour of a thermal system. As far as the CRONE approach applications are concerned, a first part deals with the first ones showing the performance improvement. A second part is limited to a non exhaustive list of the other applications.

Robust path tracking using flatness for fractional linear MIMO systems: A thermal application

This paper deals with robust path tracking using flatness principles extended to fractional linear MIMO systems. As soon as the path has been obtained by means of the fractional flatness, a robust path tracking based on CRONE control is presented. Flatness in path planning is used to determine the controls to apply without integrating any differential equations when the trajectory is fixed (in space and in time). Several developments have been made for fractional linear SISO systems using a transfer function approach. For fractional systems, especially in MIMO, developments are still to be made. Throughout this paper, flatness principles are applied using polynomial matrices for fractional linear MIMO systems. To illustrate the robustness performances, a third-generation multi-scalar CRONE controller is compared to a PID one.

Fractional multi-models of the gastrocnemius frog muscle

Abstract Frog gastrocnemius muscles are studied, and a multi-model identification presented. A transfer function is here defined with few parameters to simulate striated muscle (Gastroctnemius) behaviour, for inclusion in a future real-time salamander computer model. A two-model structure permits description of both contraction and relaxation properties. Here, two physiological influences, fatigue and fibre types, are taken into account. A multi-model structure for each fibre type (Fast (IIB), Intermediate (IIA) and Slow (I) fibres) is also used for inclusion in a future agonist-antagonist structure computer model.

Towards an object oriented CRONE toolbox for fractional differential systems

Abstract The CRONE Toolbox, developed since the nineties by the CRONE team, is a Matlab Toolbox dedicated to fractional calculus. It is currently evolving towards an object oriented version, which allows many enhancements. Three main user classes, dedicated to fractional systems representations namely fractional transfer functions (frac_tf), fractional zeros poles and gain (frac_zpk), and fractional state-space (frac_ss), are developed. All three user classes are children of a parent class (frac_lti) which contains some common attributes of fractional systems. Among enhancements of the object programming of the CRONE toolbox, is the overloading of basic operators (+, –, x, /, .x, …) and standard Matlab scripts (lsim, bode, nichols, …) for the new classes. As a consequence, an end user familiar with standard Matlab operators and scripts can use straightforwardly the CRONE toolbox. The main objective of this paper is to present class diagrams and principle features of the object oriented CRONE toolbox, which can be downloaded at http://cronetoolbox.ims-bordeaux.frcronetoolbox.ims-bordeaux.fr .

Fractional Motion Control: Application to an XY Cutting Table

In path tracking design, the dynamic of actuators must be taken intoaccount in order to reduce overshoots appearing for small displacements.A new approach to path tracking using fractional differentiation isproposed with its application on a XY cutting table. It permits thegeneration of optimal movement reference-input leading to a minimum pathcompletion time, taking into account both maximum velocity, accelerationand torque and the bandwidth of the closed-loop system. Fractionaldifferentiation is used here through a Davidson–Cole filter. Amethodology aiming at improving the accuracy especially on checkpointsis presented. The reference-input obtained is compared with splinefunction. Both are applied to an XY cutting table model and actuatoroutputs compared.

Path planning by fractional differentiation

In path planning design, potential fields can introduce force constraints to ensure curvature continuity of trajectories and thus facilitate path-tracking design. The parametric thrift of fractional potentials permits smooth variations of the potential in function of the distance to obstacles without requiring design of geometric charge distribution. In the approach we use, the fractional order of differentiation is the risk coefficient associated to obstacles. A convex danger map towards a target and a convex geodesic distance map are defined. Real-time computation can also lead to the shortest minimum danger trajectory, or to the least dangerous of minimum length trajectories.