System Identification Using Laguerre Basis Functions

This document describes how to use discrete time Laguerre basis functions, and their associated z-transforms, to construct a system identification process using experimentally determined Laguerre expansion coefficients of the input and output sequences. The process is derived using a new Product Propert} for the discrete time Laguerre basis functions. The system identification process is "linear in the parameters"; it does not require assumptions/knowledge of the poles locations of the system under test. An example is presented using data generated by a system that has appeared in the recent literature. The procedure naturally produces equations that can be used to determine if the chosen model order is correct or if its order needs to be increased. Constraints can easily be incorporated.

System Identification Using Laguerre Basis Functions

This document describes how to use discrete time Laguerre basis functions, and their associated z-transforms, to construct a system identification process using experimentally determined Laguerre expansion coefficients of the input and output sequences. The process is derived using a new Product Propert} for the discrete time Laguerre basis functions. The system identification process is "linear in the parameters"; it does not require assumptions/knowledge of the poles locations of the system under test. An example is presented using data generated by a system that has appeared in the recent literature. The procedure naturally produces equations that can be used to determine if the chosen model order is correct or if its order needs to be increased. Constraints can easily be incorporated.

A gyrokinetic model for the plasma periphery of tokamak devices

A gyrokinetic model is presented that can properly describe large and small amplitude electromagnetic fluctuations occurring on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and the arbitrarily large deviations from thermal equilibrium that are present in the plasma periphery of tokamak devices. The formulation of the gyrokinetic model is based on a second-order accurate description of the single charged particle dynamics, derived from Lie perturbation theory, where the fast particle gyromotion is decoupled from the slow drifts assuming that the ratio of the ion sound Larmor radius to the perpendicular equilibrium pressure scale length is small. The collective behaviour of the plasma is obtained by a gyrokinetic Boltzmann equation that describes the evolution of the gyroaveraged distribution function. The collisional effects are included by a nonlinear gyrokinetic Dougherty collision operator. The gyrokinetic model is then developed into a set of coupled fluid equations referred to as the gyrokinetic moment hierarchy. To obtain this hierarchy, the gyroaveraged distribution function is expanded onto a Hermite–Laguerre velocity-space polynomial basis. Then, the gyrokinetic equation is projected onto the same basis yielding the spatial and temporal evolution of the Hermite–Laguerre expansion coefficients. A closed set of fluid equations for the lowest-order coefficients is presented. The Hermite–Laguerre projection is performed accurately at arbitrary perpendicular wavenumber values. Finally, the self-consistent evolution of the electromagnetic fields is described by a set of gyrokinetic Maxwell equations derived from a variational principle where the velocity integrals are explicitly evaluated.

On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain

Herein, three important theorems were stated and proved. The first relates the modified generalized Laguerre expansion coefficients of the derivatives of a function in terms of its original expansion coefficients; and an explicit expression for the derivatives of modified generalized Laguerre polynomials of any degree and for any order as a linear combination of modified generalized Laguerre polynomials themselves is also deduced. The second theorem gives new modified generalized Laguerre coefficients of the moments of one single modified generalized Laguerre polynomials of any degree. Finally, the third theorem expresses explicitly the modified generalized Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its modified generalized Laguerre coefficients. Some spectral applications of these theorems for solving ordinary differential equations with varying coefficients and some specific applied differential problems, by reducing them to recurrence relations in their expansion coefficients of the solution are considered.