Continuous Time System Identification Using Shifted Chebyshev Polynomials

A new instrumental variables based identification procedure is introduced to estimate linear and nonlinear continuous time models using a shifted Chebyshev basis in the presence of noise. Orthogonal polynomials have been used by many authors in continuous time system analysis, identification and control. By introducing an operational matrix continuous time differential equations can be transferred to algebraic forms, from which the parameters can be estimated using least squares. In the system identification problem, however, the application of most of these approaches in the literature have ignored noise or assumed that the noise level is unrealistically low. In this letter shifted Chebyshev polynomials will be applied to the identification of continuous time linear and nonlinear systems. If the noise contained in the measured output were ignored the estimates would be biased. An Instrumental Variable method is introduced to overcome the bias problem

Identification of time-varying nonlinear systems using Chebyshev polynomials

AbstractThis paper is devoted to an identification of time-varying nonlinear systems using Chebyshev polynomials of the first kind. For systems being linear relatively unknown functional parameters, a method of approximate determination of these parameters has been worked out. As the identification problems are ill-posed to solve the obtained redefinite system of linear algebraic equations, a regularization method is used

Chebyshev Series Expansion of Inverse Polynomials

An inverse polynomial has a Chebyshev series expansion 1/\sum(j=0..k)b_j*T_j(x)=\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear combinations of simple functions of the polynomial roots. If the first k of the coefficients a_n are known, the others become linear combinations of these with expansion coefficients derived recursively from the b_j's. On a closely related theme, finding a polynomial with minimum relative error towards a given f(x) is approximately equivalent to finding the b_j in f(x)/sum_(j=0..k)b_j*T_j(x)=1+sum_(n=k+1..oo) a_n*T_n(x), and may be handled with a Newton method providing the Chebyshev expansion of f(x) is known.Comment: LaTeX2e, 24 pages, 1 PostScript figure. More references. Corrected typos in (1.1), (3.4), (4.2), (A.5), (E.8) and (E.11

On modelling transitional turbulent flows using under-resolved direct numerical simulations: The case of plane Couette flow

Direct numerical simulations have proven of inestimable help to our understanding of the transition to turbulence in wall-bounded flows. While the dynamics of the transition from laminar flow to turbulence via localised spots can be investigated with reasonable computing resources in domains of limited extent, the study of the decay of turbulence in conditions approaching those in the laboratory requires consideration of domains so wide as to exclude the recourse to fully resolved simulations. Using Gibson's C++ code ChannelFlow, we scrutinize the effects of a controlled lowering of the numerical resolution on the decay of turbulence in plane Couette flow at a quantitative level. We show that the number of Chebyshev polynomials describing the cross-stream dependence can be drastically decreased while preserving all the qualitative features of the solution. In particular, the oblique turbulent band regime experimentally observed in the upper part of the transitional range is extremely robust. In terms of Reynolds numbers, the resolution lowering is seen to yield a regular downward shift of the upper and lower thresholds Rt and Rg where the bands appear and break down. The study is illustrated with the results of two preliminary experiments.Comment: 20 pages, 9 figures. Accepted on August 24, 2010, to appear in TCF

Preconditioning complex symmetric linear systems

A new polynomial preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to Conjugate Orthogonal Conjugate Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. Moreover, to reduce the computational cost, an inexact variant based on incomplete Cholesky decomposition or orthogonal polynomials is proposed. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the method completes the description of the preconditioner.Comment: 26 pages, 4 figures, 4 table