Nonlinear projection filter with parallel algorithm and parallel sensors

Over the past few decades, the computational power has been increasing rapidly. With advances of the parallel computation architectures it provides new opportunities for solving the optimal estimation problem in real-time. In addition, sensor miniaturization technology enables us to acquire multiple measurements at low cost. Kolmogorov´s forward equation is the governing equation of the nonlinear estimation problem. The nonlinear projection filter presented in the late 90´s is an almost exact solution of the nonlinear estimation problem, which solves the governing equation using Galerkin´s method. The filter requires high-dimensional integration in several steps and the complexity of the filter increases exponentially with the dimension of systems. The current parallel computation speed with the usage of many sensors at the same time make it feasible to implement the filter efficiently for practical systems with some mild dimension sizes. On-line or off-line multi-dimensional integration is to be performed over the parallel computation using the Monte-Carlo integration method and random samples for the state update are obtained more efficiently based on the multiple sensor measurements. A few simplifications of the filter are also derived to reduce the computational cost. The methods are verified with two numerical examples and one experimental example.