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Time Domain Inverse Problems in Nonlinear Systems Using Collocation & Radial Basis Functions

Time Domain Inverse Problems in Nonlinear Systems Using Collocation & Radial Basis Functions

by T.A. Elgohary, L. Dong, J.L. Junkins and S.N. Atluri

This paper can be freely downloaded at: http://care.eng.uci.edu/pdf/Dong2014d.pdf

 

In this study, we consider ill-posed time-domain inverse problems

for dynamical systems with various boundary conditions and unknown controllers.

Dynamical systems characterized by a system of second-order nonlinear ordinary

differential equations (ODEs) are recast into a system of nonlinear first order ODEs

in mixed variables. Radial Basis Functions (RBFs) are assumed as trial functions

for the mixed variables in the time domain. A simple collocation method is developed

in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source

points as well as collocation points. The duffing optimal control problem with various

prescribed initial and final conditions, as well as the orbital transfer Lambert’s

problem are solved by the proposed RBF collocation method as examples. It is

shown that this method is very simple, efficient and very accurate in obtaining the

solutions, with an arbitrary solution as the initial guess. Since methods such as the

Shooting Method and the Pseudo-spectral Method can be unstable and require an

accurate initial guess, the proposed method is advantageous and has promising applications

in optimal control and celestial mechanics. The extension of the present

study to other optimal control problems, and other orbital transfer problems with

perturbations, will be pursued in our future studies.

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