Optimal Filtering and Control for First Degree
MarAracelia Alcorta Garc ́ ia · 2007
The algorithms for the optimal filter and control have been obtained for systems with polynomial first degree drift term in the state and observations equations. Two cases are presented: systems with disturbances in L 2 and systems with Brownian motion and parameter e in the state and observation equations. The algorithms of the optimal risk- sensitive filter are obtained in each case and their performance is verified and compared with the algorithms of the optimal Kalman-Bucy filter through an example. Besides the solution to the optimal control risk-sensitive problem for stochastic system as in the filter, and quadratic cost function to be minimized is obtained. The algorithms for the optimal control are obtained using PDE HJB. These algorithms are compared with the traditional control algorithms through numerical example. The optimal risk-sensitive filter and control show better performance for large values of the parameter e: Since the linear optimal filter was obtained by Kalman and Bucy (60's), numerous works are based on it. I could mention some as (M. V. Basin et al. (2003)), (M. V. Basin et al. (2003)),(F. L. Lewis (1992)), (V. S. Pugachev et al. (2001)), (S. S.-T. Yau(1994)), of the variety of all those. More than thirty years ago, Mortensen (R. E. Mortensen(1968)) introduced a deterministic filter model which provides an alternative to stochastic filtering theory. In this model, errors in the state dynamics and the observations are modeled as deterministic functions, and a mean-square disturbance error criterion is to be minimized. In this case, special conditions for the existence, continuity and boundedness of f(xt) in the state equation, which is considered nonlinear, and for the linear function h(xt) in the observation equation, are given. A concept of the deterministic estimator, which is introduced more recently by McEneaney (W. M. McEneaney(1998)), is reviewed and applied to system with disturbances in L 2 ; where f(x) has a nonlinear form in the dynamics of the system and linear