Optimal Risk-sensitive Filtering and Control for Linear Stochastic Systems

The optimal exponential-quadratic control problem and exponential mean-square filtering problems are considered for stochastic Gaussian systems with polynomial first degree drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form optimal control and filtering algorithms are obtained using quadratic value functions as solutions to the corresponding Hamilton-Jacobi-Bellman equations. The performance of the obtained risk-sensitive regulator and filter for stochastic first degree polynomial systems is verified in a numerical example against the conventional linear-quadratic regulator and Kalman-Bucy filter, through comparing the exponential-quadratic and exponential mean-square criteria values. The simulation results reveal strong advantages in favor of the designed risk-sensitive algorithms in regard to the final criteria values.