Olivier Cois

Solving an inverse heat conduction problem using a non-integer identified model

Abstract An inverse heat conduction problem in a system is solved using a non-integer identified model as the direct model for the estimation procedure. This method is efficient when some governing parameters of the heat transfer equations, such as thermal conductivity or thermal resistance, are not known precisely. Reliability of the inversion depends on the precision of the identified model. From considerations on the analytical solutions in simple cases and on the definition of non-integer (or fractional) derivative, the non-integer model appears to be the most adapted. However, some experiments do need to be carried out on the physical thermal system before it can be identified. An application that consists in estimating the heat flux in a turning tool insert during machining is presented. First, identification is performed using a specific apparatus that permits a simultaneous measurement of temperature and heat flux in the insert. Then, during machining, heat flux can be estimated from temperature using this identified model.

Non Integer Model from Modal Decomposition for Time Domain System Identification

Abstract This article deals with modeling and identification of non integer systems in the time domain. An identification method is developed providing estimation of both model coefficients and differentiation orders. The non integer state-space representation is defined, and a modal decomposition of non integer systems is given. From the modal decomposition, also called developed modal form, a new identification model is defined. This model is parametrized by eigenmode parameters, three for each mode: the modal coefficient, the eigenvalue and the differentiation order which is common to all modes. These parameters are then estimated by an output error identification method using the Marquardt algorithm. Two examples illustrate this identification method: a simulated noised non integer system identification, and a real thermal system identification. In each case, the system is accurately identified by a three eigen mode non integer model. An integer ARX model with the same number of parameters, used for the identification of the real thermal system, gives unsatisfactory results.

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.

H2 Norm of Fractional Differential Systems

The H2 norm, or the impulse response energy, of fractional differential explicit systems is computed, and complements Astrom’s algebraic formula for classical rational systems.

Modal Placement Control Method for Fractional Systems: Application to a Testing Bench

This paper is dedicated to modal placement control for fractional systems. The influence of the complex plane eigenvalue locations on their time response is discussed for two elementary fractional systems. A dynamic output feedback modal placement control method is proposed. It is applied to the temperature control of a thermal system whose dynamic model is obtained through system identification.© 2003 ASME

SYSTEM IDENTIFICATION USING FRACTIONAL HAMMERSTEIN MODELS

Abstract Identification of continuous-time non-linear systems characterised by fractional order dynamics is studied. The Riemann-Liouville definition of fractional differentiation is used. A new identification method is proposed through the extension of Hammerstein-type models by allowing their linear part to belong to the class of fractional models. Fractional models are compact and so are used here to model complex dynamics with few parameters.

Restricted-Complexity Controller with Crone Control-System Design and Closed-Loop Tuning

The benchmark problem, “Design and optimisation of restricted complexity controllers”, for an active suspension system is studied. An initial fractional controller based on the first and third generation Crone methodologies is designed first. The high-level parameters of the rational controller are then fine-tuned using a tuned in closed-loop approach. The optimisation technique uses the power spectral density of the closedloop simulation of the residual force to be minimised. This final controller is finally assessed with real-time implementation.

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, ∞[.