Note this article below is somewhat technical and is written primarily for the benefit of system and control scholars who want to have a quick introduction to this subtopic in system and control. Other scholars may need to read additional popularized material on statistics and Kalman estimation before attempting to follow what is written below.

 I have written earlier on the crowning achievement of modern system and control theory – the Draper prize winning Kalman filter http://bbs.sciencenet.cn/home.php?mod=space&uid=1565&do=blog&id=16065  . Now the Kalman filter strictly speaking is applicable only to linear systems with Gaussian disturbances and measurement noises. Since many real world problems are nonlinear and non -Gaussian, the question about nonlinear filtering and estimation naturally arise and the related question as to “what shall we do in practice?”

The most popular approach to non linear estimation is what is known as

EXTENDED KALMAN FILTERING - Here you simply start with a guess estimate/trajectory of the nonlinear system. Linearize around this trajectory and apply the usual Kalman filtering formula to this linearized system. You can also recursive update the estimated trajectory with new data and re-linearize.  All very efficient. The only problem is that Kalman filtering is based on a unimodal (Gaussian) distribution. If your problem actually involves/ results in multimodal distribution, then this approach often will breakdown or lead to very bad estimates

The second approach is

NONLINEAR LEAST SQUARE FIT THE DATA – This approach is quite stable and works. The only problem is the fact that it is not real time. For each new measurement you need to solve a new two point boundary value problem. Unless you update very infrequently, the computational load is quite high and not practical in real time.

The third approach is

PARTICLE FILERING –One of the leading practitioner of this approach is Dr. Fred Daum of Raytheon company with over two decade of real experience and success stories to his credit. So what is Particle Filtering  (PF)? In one sentence – you try to propagate and update the multidimensional density function of the system state directly. Of course, you approximate the multidimensional density function by histograms made up by actual samples (i.e., particles) of the state vector.  Propagate a histogram (i.e., the particles making up the histogram) is not that hard computationally (think about it. It is a simple Monte Calro run involving the particles). But the problem of updating a histogram using a new measurement is not simple.  First of all, to adequately represent the histogram you may need an exponentially large number of particles if the state dimension is high. This is a universal dirty secret applied mathematician does not like to talk about (http://bbs.sciencenet.cn/home.php?mod=space&uid=1565&do=blog&id=26889 ). To make matters worse, the Bayes rule used in the updating formula requires you to multiply the prior density by the likelihood function which falls off rapidly around the actual measurement value. Thus, only a small percentage of the particles will get significant updating. Computationally this is rather inefficient. Daum’s expertise and contribution are the tricks he has developed to ameliorate this problem.  These have their limits. Fortunately, many real problems have only a six dimensional state vector (3 position and 3 velocity in aerospace guidance and control applications) which helps in computational load.  In fact, by hindsight, the nonlinear filtering example in my 41 year old text book on page 381 http://www.amazon.com/Applied-Optimal-Control-Optimization-Estimation/dp/0891162283  is an extremely  simple problem of particle filtering (it has only two possible states) except the problem/solution was not labeled as such since the name PF has not been invented yet 41 years ago.

OK, Now in my opinion you know what are worth knowing about practical nonlinear filtering.