Model of time-varying linear systems and Kolmogorov equations
<jats:title>Abstract</jats:title> <jats:p>In the paper an approximate model of time-varying linear systems using a sequence of time-invariant systems is suggested. The conditions for validity of the approximation are proven with a theorem. Examples comparing the numerical solution of the original system and the analytical solution of the model are given. For the system under the consideration a new criterion giving sufficient conditions for robust Lagrange stability is suggested. The criterion is proven with a theorem. Examples are given showing stable and non stable solutions of a time-varying system and the results are compared with the numerical Runge-Kutta solution of the system. In the paper an important application of the described method of solution of linear systems with time-varying coefficients, namely analytical solution of the Kolmogorov equations is shown.</jats:p>
Chungand (1990), Parallel solution of sparse linear systems on a vector multiprocessor computer on Circuits and Systems New Orleans, Proc IEEE Int Symp USA, 1577. ; Roychowdhury (1999), Reduced - order modelling of time - varying systems on Circuits and Systems - II : Analog and Digital Signal processing, IEEE Trans, 46, 1273. ; Krumov (2005), Model and stability of a class of time - varying linear systems of the IEEE Conf on Computer as a tool, Proc, 1216. ; Enright (1979), On the efficient and reliable numerical solution of large linear systems of ODE s on Automatic Control, IEEE Trans, 24, 905. ; Joubertand (1994), PCG : a software package for the iterative solution of linear systems on scalar vector and parallel computers of the Scalable High - Performance Computing Conf USA, Proc, 811. ; Helton (1998), Computer simplification of formulas in linear systems theory on Automatic Control, IEEE Trans, 4, 302. ; Ascherand (1990), Projected implicit Runge - Kutta methods for differential - algebraic boundary value problems of the th IEEE Conf on Decision and Control Honolulu HI USA, Proc, 448. ; Davariand (1994), Short - time stability analysis of time - varying linear systems of the th Southeastern Symp on System Theory, Proc, 302.