Estimation of states and parameters in continuous non-linear systems with discrete observations |
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Authors: | Connie Jean Weeks Alan Schumitzky |
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Institution: | Department of Mathematics, University of Southern California, Los Angeles, California 90007 U.S.A. |
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Abstract: | We consider a class of continuous non-linear systems defined by the ordinary differential equation x = f(x, t) + g(x, t)u, where u is an unknown input representing noise or disturbances. The object is to estimate states and parameters in these systems by means of a fixed number of discrete observations yi = h(x(ti), ti) + vi, 1 ? i ? m, where the vi represents unknown errors in the measurements yi. No statistical assumptions are made concerning the nature of the unknown input u or the unknown measurement errors vi. A weighted least squares criterion is defined as a measure of the optimal estimate. A result concerning the existence of solutions of the differential equation which minimize the criterion is presented. The necessary conditions for an optimal estimate, a set of Euler-Lagrange equations and multi-point discontinuous non-linear boundary conditions, are given. The multi-point problem is converted to an equivalent continuous two-point boundary value problem of larger dimension in the case in which the observations are assumed to be linear functions of the state. A pair of equivalent quasilinearization algorithms is defined for the two-point system and the multi-point system. Quadratic convergence for these algorithms is proved. |
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