Robust H∞ Deep Neural Network-Based Filter Design of Nonlinear Stochastic Signal Systems

Recently, deep neural network (DNN) schemes based on big data-driven methods have been successfully applied to image classification, communication, translation of language, speech recognition, etc. However, more efforts are still needed to apply them to complex robust nonlinear filter design in signal processing, especially for the robust nonlinear $H_{\infty }$ filter design for robust state estimation of nonlinear stochastic signal system under uncertain external disturbance and output measurement noise. In general, the design problem of robust nonlinear $H_{\infty }$ filter needs to solve a complex Hamilton-Jacobi-Isaacs equation (HJIE), which is not easily solved analytically or numerically. Further, the robust nonlinear $H_{\infty }$ filter is not easily designed by training DNN directly via conventional big data schemes. In this paper, a novel robust $H_{\infty }$ HJIE-embedded DNN-based filter design is proposed as a co-design of $H_{\infty }$ filtering algorithm and DNN learning algorithm for the robust state estimation of nonlinear stochastic signal systems with external disturbance and output measurement noise. In the proposed robust $H_{\infty }$ DNN-based filter design, we have proven that when the approximation error of HJIE by the trained DNN through Adam learning algorithm approaches to 0, the HJIE-embedded DNN-based filter will approach the robust nonlinear $H_{\infty }$ filter of nonlinear stochastic signal system with uncertain external disturbance and output measurement noise. Finally, a trajectory estimation problem of 3-D geometry incoming nonlinear stochastic missile system by the proposed robust $H_{\infty }$ HJIE-embedded DNN-based filter scheme through the measurement by the sensor of radar system with external disturbance and measurement noise is given to illustrate the design procedure and validate its robust $H_{\infty }$ filtering performance when compared with the extended Kalman filter and particle filter.