High-Order State and Parameter Transition Tensor Calculations
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Most mathematical models governing physical systems are described by first-order ordinary differential equations. The local phase space flows of such nonlinear differential equations are described by state and parameter transition tensors. This paper presents a mixed numeric/symbolic algorithm for automating the derivation, coding, and computation of tensor-based generalizations of such flows. Automatic differentiation algorithms provide the partial derivative models required for assembling the tensor differential equations for the state and parameter transition tensors. The tensor equations are described by implicit differential equations. Complicated analytic models are available through the use of Faá di Bruno's formula for differentiation of composite functions. A simple vector generalization for Faá di Bruno's formula is presented that builds on a recursive integer algorithm. An array-of-arrays data structure is introduced for (1) tracking the derivative terms arising during a derivation of the tensor necessary conditions, and (2) assembling a generalized state space model for integrating the state and tensor differential equation models. Pell exponential numbers model the total number of implicit derivative calculations performed at each expansion order. The integer codes that describe the derivative derivation for the tensors are transformed automatically into FORTRAN by a Fortran-based string manipulation routine. The goal of the analysis is to generate a perturbation model, based on the local flow of solutions for the nonlinear system, for general-purpose applications in uncertainty analysis, quantification of state estimate uncertainty for use in navigation algorithms and construction of neighboring extremal solutions in guidance problems. The new algorithms presented herein are expected to expedite the computation of the perturbation descriptions of such departure motion nonlinear systems by a systematic exploitation of the structure and symmetry of the tensor fields. Copyright © 2008 by James D. Turner Ph.D.
author list (cited authors)
Turner, J., Majji, M., & Junkins, J.