Nonlinear Filtering

Optimal state estimation for linear system with Gaussian noise as discussed in Chaps. 9 – 11 was possible by finding the minimum mean square estimate and its associated error covariance matrix. The minimum mean square estimate is the expected state given the data and its derivation and analysis are straightforward for linear systems. However, linearity is an idealization that is only approximately valid under limited conditions and most physical systems are nonlinear. For nonlinear systems, the expected value of the state given the estimate and its error covariance can only be obtained approximately and the estimators are suboptimal. This chapter discusses suboptimal state estimation for nonlinear systems. The simplest and most commonly used approach to nonlinear filtering is based on linearization. This works well if the nonlinearities of the system are not severe. For highly nonlinear systems, linearization results in large estimation errors and other approaches are recommended. Several approaches require selecting a number of points and using them to approximate the conditional probability density function (pdf) of the state given the data. The unscented Kalman filter selects the points using a deterministic rule. The ensemble Kalman filter selects points from a known pdf. These approaches are suitable for the case of Gaussian noise. Particle filters, which are also based on a discrete approximation of the pdf can handle nonlinear systems with non-Gaussian distribution. This comes at a high computational cost because of the phenomenon of degeneracy, which causes the number of selected points to drop to a level that does not provide a good approximation of the pdf. This requires resampling to replenish the number of points used to approximate the pdf.