Spatiotemporal System Identification with Spectral Methods

ABSTRACT

A system identification technique based on spectral methods is proposed for spatiotemporal systems governed by partial differential equations. To the best of our knowledge, there is no evidence of spectral methods having been used for inverse problems before this work. Thus, the main contribution of this work is to introduce spectral methods into the realm of inverse problems.
As for the identification part, the structure of the system is modeled with a nonlinear PDE of polynomial type allowing terms of arbitrary derivative order and nonlinearity degree. The boundaries are assumed to be periodic which is the case for a lot of problems like when the problem is not related to the boundaries, or when the coordinate space is periodic, as is the case for angle variables in polar or spherical coordinates.
To discretize the assumed continuous structure with periodic boundaries and arrive at an appropriate regression form, global interpolants are introduced and, with their aid, Fourier spectral differentiation operators are derived for spatial derivative approximations.
Applying finite difference weights, as is traditionally done for the aforementioned discretization, can be numerically ill-conditioned (especially when dealing with high order derivatives) leading to severe cancellations and loss of significant digits in derivative approximations. On the other hand, spectral methods yield “infinitely” accurate derivative approximations and what’s more, that is achieved with a dramatically coarser spatial grid which means a much smaller set of sampling points can be used.
Practical implementation via Fast Fourier Transform (FFT) of the Fourier spectral differentiation operator derived is discussed and used to obtain the said regression form.
For the resulting over-determined system of linear equations, we propose a transformation to an orthogonal space and then propose a structure selection procedure, by which we detect and eliminate redundant parameters based on their insufficient contribution to the reduction of the Mean Squared Error (MSE), decided by an Error Reduction Ratio (ERR) threshold.
When no more redundant parameters are left, the reduced order over-determined system of linear equations is solved for the remaining parameters in a least squares sense, giving the maximum likelihood estimates provided the modeling errors are assumed to be zero-mean white noise.
The method is thoroughly exemplified with numerical experiments for the identification of the Kuramoto-Sivashinsky (KS) and Burgers PDEs, in which we use the time series coming from the direct solution of those PDEs with spectral methods and Exponential Time Differencing (ETD). The superiority of spectral methods over finite differences (for inverse problems) is illustrated with the numerical analysis of prediction errors.
 

 

[Blasts from the past]

TI Programs

Routh-Hurwitz stability matrix generator. Calculates the Routh stability table for the given characteristic polynomial. Handles both basic and special cases, namely when the table has (1) a Zero only in the first column (surmounted by epsilon method) (2) an Entire row of zeros (surmounted by even polynomial detection). Symbolic capability; generates the table even in the case of a parametric polynomial.

Root Locus Data calculator. Calculates three most quintessential features pertaining to a root locus; namely its (1) Behavior at infinity, (2) Real-axis breakaway and break in points, and (3) Angles of arrival and departure. Any combination of positive/negative feedback and gain supported. Handles non-minimum as well as minimum phase systems. Multiplicity of the zeros and poles also taken care of in the program.

Spur, Helical, and Bevel Gear set designer based on AGMA standards. Through hardened, Nitrided, and 2.5% Chrome Steel as well as Nitralloy St-HB utilized. Face width initial estimator capability. Internal surface geometry factor (I) calculation. Capable of solving both design problems through iteration, and analysis problems (like power rating) through symbolic solution.