You can always do the computation iteratively, processing one measurement at a time. The matrix then reduces to a scalar. This procedure allows you to compute the update state *but not* the gain matrix (but then again, you probably do not care about it).

You can always do the computation iteratively, processing one measurement at a time. The matrix then reduces to a scalar. This procedure allows you to compute the update state *but not* the gain matrix (but then again, you probably do not care about it).

Square Root Filter does not require any matrix inversion operation for the Kalman gain calculation

I was going to say what Hector Rotstein said

In general Kalman filter or any linear update based filters make use of prediction error covariance and innovation covariance matrices in computation, and subsequently make use of inverse calculations. I fear that, if any solution is proposed, I would like to recommend to validate such solution (may be an offline analysis) with the solution obtained in conventional manner (with usage of inverse matrices).

Going with individual measurements, fine, but you may need to go for multiple times predictions. This may not hamper the solution, provided the systems dynamics should strictly follows a linear model.

Please read nelles 's book (page 66)

If you are using a regular Kalman filter for an LTI system, then you can just use the steady-state Kalman filter, which means that you precompute a constant Kalman gain. The loss here is the accuracy in the initial transient, but an accurate estimate of the initial state is usually difficult to obtain in practice, so this should not matter much. This is thus the same as a Luenberger observer, but computing the gain using the Riccati equation.

This help-page from MATLAB pretty much sums it up:

http://www.mathworks.com.au/help/control/ref/kalman.html

Square Root Filter does not require any matrix inversion operation

Check the paper "The Kalman Filter Revisited Using Maximum Relative Entropy" out. We tried to avoid matrix manipulation, because it has an artifact of producing instability of the filter at certain cases. MrE is especially good for loosely coupled systems.

I am agree with Hector Rotstein. Except that you actually can compute the gain matrix. Have a look at the book of Dan Simon “Optimal State Estimation” 2006. There the so called “sequential Kalman filter” is described (sec. 6.1, page 150ff).

Brilliant! Thanks for all your input! My experience with Kalman-Filters up to now was for stationary signals (and rather simple cases), so there's going to be a lot of reading in my near future!

Check out the sequential Kalman Filter by Chiu and Chen, "Kalman Filtering: with Real-Time Applications". It was designed to work in computationally challenged environments.

There is something called the matrix inversion lemma which avoids the problem of inversion. Do you know about that?

I agree with F. Barret. The matrix inversion lemma is commonly used in recursive identification and associated adaptive control problems to be able to proceed in real time with the identification and controller synthesis algorithms. The measurements are processed one- by -one, at each new sample, so as to update the gain and the estimator values based on their previous values. This is based in making mathematically equivalent the batch and the recursive versions of the standard (or generalized least ) squares method provided that a batch procedure is used for the initialization of the recursive counterpart for the first few " n " measurements ( n depending on the number of parmeters to be estimated ). It is not necessary a matrix inversion process procceding in this way.

In practice , it is not necessary even this " batch initialization" of the recursive algorithm , since both the batch and the recursive least- suares estimation methods can be made ( not for all samples- but asymptotically ) equivalent almost for any arbitrary initialization of the recursive least-squares estimation algorithm.

I do not know if the filtering problem propsed here is similar but probably it is since the kinds of batch and updating problems in both cases are typically very close.

Search for “sequential Kalman filter” in different available references, e.g., “Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches” by Dan Simon (2006), or Kalman Filtering: With Real-Time Applications by Charles K. Chui, and Guanrong Chen (2009) (http://books.google.it/books/about/Kalman_Filtering.html?id=zxWlHqLnlL0C&redir_esc=y)

For linear time-invariant systems the Kalman gain is constant. Other systems are relatively or mostly constant. In those cases, it may be acceptable to compute the gain offline and just use it to update the states all the time.

I worked out an iterative formula for the overall Kalman gain K of a sequence of scalar measurements after a temporal update. Let the state vector be of length n. Suppose there are m>1 scalar measurements using the row vectors H1,...,Hm of length n. Calculate the Kalman gains K1,...,Km (column vectors) for the m measurements. Then the following iteration gives K.

Let K = K1. For i = 2 to m; calculate Ki; let K = [ [I - Ki*Hi]K  Ki].

John: to be more precise, for LTI systems, under the right assumptions, the Kalman gain converges to a constant matrix.