Uppsala University
Department of Information Technology

### Technical Report 2004-001

# Maximum Likelihood Modeling of Orbits of Nonlinear ODEs

### Torsten Söderström, Torbjörn Wigren, and Emad Abd-Elrady

January 2004

**Abstract:**This report treats a new approach to the problem of periodic signal estimation. The idea is to model the periodic signal as a function of the state of a second order nonlinear ordinary differential equation (ODE). This is motivated by Poincare theory which is useful for proving the existence of periodic orbits for second order ODEs. The functions of the right hand side of the nonlinear ODE are then parameterized, and a maximum likelihood algorithm is developed for estimation of the parameters of these unknown functions from the measured periodic signal. The approach is analyzed by derivation and solution of a system of ODEs that describes the evolution of the Cramer-Rao bound over time. The proposed methodology reduces the number of estimated unknowns at least in cases where the actual signal generation resembles that of the imposed model. This in turn is expected to result in an improved accuracy of the estimated parameters.

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