Rachid Malti

Dynamic SISO and MISO system approximations based on optimal Laguerre models

A general procedure, based on the knowledge of the input and the output signals, is proposed to approximate the prescribed linear time-invariant (LTI) systems by means of optimal Laguerre models. The main contribution of this paper is to apply the Newton Raphsons iterative technique to compute the so-called optimal Laguerre pole in a continuous time case (or optimal time scale factor in a discrete-time case) and especially to show that the gradient and the Hessian can be expressed analytically. Moreover, the excitations used are not limited to the ones that ensure the orthogonality of the outputs of Laguerre filters (i.e., Dirac delta or white noise) as is usually done in existing methods, however persistently exciting input signal(s) are used. The proposed procedure, will be directly formulated for SISO systems, SISO systems being a special case with the number of inputs equal to one. The proposed algorithm has direct applications in system identification, model reduction, and noisy modeling.

CRONE control system design toolbox for the control engineering community: tutorial and case study

Fractional-order differentiation offers new degrees of freedom that simplify the design of high-performance dynamic controllers. The CRONE control system design (CSD) methodology proposes the design of robust controllers by using fractional-order operators. A software toolbox has been developed based on this methodology and is freely available for the international scientific and industrial communities. This paper presents both the CRONE CSD methodology and its implementation using the toolbox. The design of two robust controllers for irrigation canals shows how to use the toolbox.

Optimality conditions for the truncated network of the generalized discrete orthonormal basis having real poles

This paper deals with the synthesis of optimality conditions for the truncated network of the generalized orthonormal basis in the case where all the poles belong to the set of real numbers. These conditions are brought to a very simple form, but their solutions are not trivial. They generalize the optimality conditions for the truncated Laguerre network and are very attractive in system identification, model representation, and model reduction frameworks.

H2-norm of fractional transfer functions of implicit type of the first kind

This paper studies the H2-norm, or impulse response energy, of fractional transfer functions of implicit type. The analytical expression of the H2-norm is first derived for an elementary fractional transfer function of the first kind with a single real pole. Series connection of such a transfer function with a pure fractional integrator and with another implicit transfer function of the first kind are then studied. Results developed in the paper are finally used to derive a criterion to evaluate the quality of an integer order approximation for an implicit type fractional order model of the first kind.

Approximation of a Fractance by a Network of Four Identical RC Cells Arranged in Gamma and a Purely Capacitive Cell

In this paper a storage I-element and a fractance are associated, thus defining a fractional system. In order to achieve the fractance, an approximation by a network of 4 identical RC cells arranged in gamma and a purely capacitive cell is proposed, thus defining a rational system. When compared to each other, the dynamic behaviors of the fractional and the rational systems show an excellent superposition of frequency and time-domain responses. Moreover, the robustness of stability margins obtained with both systems is illustrated versus variations of the I-element.

Resonance and Stability Conditions for Fractional Transfer Functions of the Second Kind

Fractional transfer functions of the second kind are studied in this chapter. First, stability conditions are established in terms of pseudo-damping factor and commensurable order. Then, resonance conditions are determined. An open-loop analysis of the equivalent closed-loop transfer function of the second kind confirms the analysis.

ℋ 2 -norm of a class of fractional transfer functions suited for modeling diffusive phenomena

This paper focuses on the ℋ2-norm of a class of implicit fractional order transfer functions well suited to describe input-output behaviour of diffusive systems. First, analytical expression of the ℋ2-norm of this kind of transfer function is established. This result is then used to evaluate the quality of an integer order approximation of such an implicit fractional transfer function.

Experiment design in system identification using fractional models

This paper tackles the problem of experiment design in system identification using elementary fractional models. It is based on a frequency domain approach and answers the question: what is the best sinusoidal input signal maximizing the D-optimality criterion of the Fisher information matrix in different contexts? First, a single parameter is assumed to be unknown (any of the parameters of the elementary transfer functions). Then multiple parameters and their combinations are supposed to be unknown.

Application of the principal fractional meta-trigonometric functions for the solution of linear commensurate-order time-invariant fractional differential equations

A new and simplified method for the solution of linear constant coefficient fractional differential equations of any commensurate order is presented. The solutions are based on the R-function and on specialized Laplace transform pairs derived from the principal fractional meta-trigonometric functions. The new method simplifies the solution of such fractional differential equations and presents the solutions in the form of real functions as opposed to fractional complex exponential functions, and thus is directly applicable to real-world physics.

Bilateral decomposition of a time function into laguerre series. Application to LTI system identification

This paper deals with the estimation of linear time invariant (LTI) systems using Laguerrien models. The Laguerre functions used in this paper are presented in their general form, i.e. using two exponential factors. The main contribution of this paper is the development of an algorithm which performs the decomposition of a signal into Laguerre series for negative as well as positive time domains. These two decompositions are then aggregated to form a unique bilateral expansion. The method is then used to decompose input and output signals defined for positive as well as negative time (the case of non zero initial conditions of the state process is considered) into their bilateral Laguerre expansions in order to compute the parameters of the transfer function of the system that needs to be identified.