Prediction of Dynamical Systems by Constraining the Dynamics on the Observational Manifold
Evolution models of dynamical systems posed as differential equations generally do not include all the factors affecting the system. This leads to a mismatch between the model prediction and the observations. In this work, an approach is developed where the observations are treated as constraints on the system dynamics. This problem is posed as a Differential-Algebraic Equation (DAE) system and the algebraic constraints are obtained from the system observation data. This leads to evaluation of extra forcing terms for keeping the dynamics on the observation constraints manifold by using Lagrange multipliers. The Lagrange multipliers generate the instances of the unmodelled forcing terms which, in turn, are used to modify the evolution rule to improve the prediction capability of the original nominal model. Orbit prediction of Global Navigation Satellite System (GNSS) satellites which is essential for positioning has been studied to elucidate the above approach. In this context, this work studies the GNSS data and some techniques for mitigating the errors and biases in the raw measurements. Using the first principles, the constrained dynamical system equations are derived for the dynamics of the satellites in orbit. The DAE approach outlined in first paragraph is applied to predict orbit of a geostationary satellite of the BeiDou satellite constellation using historical International GNSS Services' precise ephemeris data. In our example, we observe a reduction in the predicted orbital position error by roughly 88% as compared to the nominal gravitational model. In another illustrative application, temperature predictions are done for a one-dimensional heat conduction system. This problem presents itself with a challenge of limited sampling in the temperature domain. The observed data is used to generate instances of extra terms in the evolution model. These instances are fitted to a regression model which is used during prediction. As a further elucidation, estimation of the coefficients of a nominal differential equation model from observed data of the system is briefly studied.