Estimation of Heat Source Term and Thermal Diffusion in Tokamak Plasmas Using a Kalman Filtering Method in the Early Lumping Approach

Abstract

14 pagesInternational audience— In this paper, early lumping estimation of space-time varying diffusion coefficient and source term for a non-homogeneous linear parabolic partial differential equation (PDE) describing Tokamak plasma heat transport is considered. The analysis of this PDE is achieved in a finite dimensional framework using the cubic b-splines finite element method with the Galerkin formulation. This leads to a finite dimensional linear time-varying state-space model with unknown parameters and inputs. The Extended Kalman Filter with Unknown Inputs Without Direct Feed-through (EKF-UI-WDF) is applied to estimate simultaneously the unknown parameters and inputs and an adaptive fading memory coefficient is introduced in the EKF-UI-WDF, to deal with time varying parameters. Conditions under which the direct problem is well posed and the reduced order model converges to the initial one are established. Insilico and real data simulations are provided to evaluate the performances of the proposed technique

Distributed Control of Coupled Inhomogeneous Diffusion in Tokamak Plasmas

“…Corollary 1. If the conditions of Theorem 1 are satisfied, the system (11) withũ = 0, boundary conditions (12) and initial condition (13) is globally exponentially stable. The convergence rate of the system satisfies…”

“…for all x ∈ [0, 1], where: 17) and the elements in the A 1 (x) matrix are given in (18). Then the time derivativeV of V defined in (14) along the solutions of (11) and (12)…”

“…Nevertheless, some of the transport coefficients are not well known and empirical models are developed for these coefficients. For example, estimation methods for heat transport coefficients are proposed in [11], [12], [13].…”

Discrete-Time Kalman Filter Design for Linear Infinite-Dimensional Systems

“…Then, traditional state estimation and/or control methods from finite-dimensional system theory can be exploited for the resulting early lumped systems. The widely-utilized early lumping methods are reported as: modal analysis (Galerkin's method) [47,48], finite difference [49], proper orthogonal decomposition (POD) [50,51], finite element method (FEM) [52], FEM Galerkin cubic b-splines method [53], etc. In contrast, late lumping means that one conducts observer, filter, and/or controller design steps on the original infinite-dimensional space and applies spatial approximation later for simulation purposes, where the backstepping method [54,55], the adjoint-based approach [56], and the Crank-Nicolson method [57] are widely used.…”

Multi-experiment state-space identification of coupled magnetic and kinetic parameters in tokamak plasmas

“…Only some empirical models are developed for these coefficients. An estimation of the thermal diffusion and of the source term of the heat transport model is presented in [10] and simplified model for the temperature profile obtained by using neural networks is presented in [11].…”