Now I want to solve a navigation problem using both MHE and EKF. I found that MHE has a better performance than EKF for highly nonlinear system. Can anyone tell me the reason? What is the relationship between these for estimation method? Are they identical to each other under certain assumptions?
In MHE, at every iteration the new estimate is computed using a previous set of estimates of a fixed size (this set slides in time, hence moving window). Apart from estimating the present state, it also smoothes(makes it more accurate) the previous estimates as well, whereas in KF only the former part is done. MHE merges with KF when window size =1.
Nonlinear MHE, theoretically gives a complete solution. But practical challenges are:
1) in solving a non-convex nonlinear optimization problem numerically
2) selection of weights used in each iteration
Courtesy - Prof. Sachin Patwardhan, Dept. of Chemical engineering, IIT Bombay
Hello!, you may find very useful information in the book of Jazwinski "Stochastic Processes and Filtering Theory". In the very fundamentals, both the Kalman Filter and the MHE are equivalent if the system under consideration is linear and there are no constraints. However, both approaches start to diverge once constraints are considered. In the same way, you may find some similarities between the EKF and the MHE, but again, once constraints are considered differences will come out. EKF is very useful when nonlinearities are not that strong. If this is the case, the UKF appears to be a better choice. MHE is an interesting filtering approach but, as Sajith mentioned above the solution of non-convex nonlinear optimization problems related to the MHE formulation is to be considered.
The MHE arises as a maximum a posteriori (MAP) probability estimator (maximisation of the log posterior probability density), whereas the EKF is a suboptimal solution in the minimum mean square error (MMSE) sense. If the state and measurement processes are linear Gaussian without constraints, the "one-step" solutions coincide.
The MHE can tackle highly nonlinear problems where the state and uncertainty may have constraints, but at a higher computational cost and some practical difficulties such as approximating the arrival cost. The EKF is computationally simple but it requires both the system nonlinearities to be mild and the variances of the noises to be small (in order to keep the linearisation valid).
Depending on the nature of the system, I would rather consider either the UKF or sequential Monte-Carlo samplers (e.g., particle filters).
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