OK.  This has come up a bazillion times, and I even have a fairly good 
idea of how I want to go about it.  (It involves drinking more high-
caffeine tea and filling in a blank spot labeled "something clever" in my 
design document).

But before I go reinvent the wheel, here's my problem statement:

Consider a system

x_k = A * x_{k-1} + B * u_k + w_k

y_k = C * x_k + D * u_k + r_k

We want a best estimate of x_k from y_k, and A, B, C and D are known.

So far, so Kalman.

But what's missing from the usual Kalman formulation is the variances of 
w_k and r_k -- these are unknown.  I want to estimate these variances -- 
preferably independently of my estimate of x_k -- and use them in the 
Kalman filter that estimates x_k.

(and yes, I could just use a H-infinity filter.  And I might.  But when 
you get right down to it, an H-infinity filter is just a really good, 
mathematically structured way to insert a wild-ass guess into a Kalman 
filter and get the best possible outcome _given_ that you've made a wild-
ass guess).

So -- has anyone seen literature that addresses this?  I have to admit 
that I haven't gone and done a comprehensive Google search -- my Google-Fu 
for finding papers that actually _mean_ something isn't terribly good, 
and it seems that there are always really old papers, papers loaded with 
enough mathematical oddities so that some recycled idea will be "new 
research", and conference presentation trash that get in the way of the 
nuggets of true wisdom out there.

-- 
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

On Friday, July 27, 2012 5:38:49 AM UTC+12, Tim Wescott wrote:
> OK.  This has come up a bazillion times, and I even have a fairly good 
> idea of how I want to go about it.  (It involves drinking more high-
> caffeine tea and filling in a blank spot labeled "something clever" in my

> design document).
> 
> But before I go reinvent the wheel, here's my problem statement:
> 
> Consider a system
> 
> x_k = A * x_{k-1} + B * u_k + w_k
> 
> y_k = C * x_k + D * u_k + r_k
> 
> We want a best estimate of x_k from y_k, and A, B, C and D are known.
> 
> So far, so Kalman.
> 
> But what's missing from the usual Kalman formulation is the variances of 
> w_k and r_k -- these are unknown.  I want to estimate these variances -- 
> preferably independently of my estimate of x_k -- and use them in the 
> Kalman filter that estimates x_k.
> 
> (and yes, I could just use a H-infinity filter.  And I might.  But when 
> you get right down to it, an H-infinity filter is just a really good, 
> mathematically structured way to insert a wild-ass guess into a Kalman 
> filter and get the best possible outcome _given_ that you've made a wild-
> ass guess).
> 
> So -- has anyone seen literature that addresses this?  I have to admit 
> that I haven't gone and done a comprehensive Google search -- my Google-Fu 
> for finding papers that actually _mean_ something isn't terribly good, 
> and it seems that there are always really old papers, papers loaded with 
> enough mathematical oddities so that some recycled idea will be "new 
> research", and conference presentation trash that get in the way of the 
> nuggets of true wisdom out there.
> 
> -- 
> My liberal friends think I'm a conservative kook.
> My conservative friends think I'm a liberal kook.
> Why am I not happy that they have found common ground?
> 
> Tim Wescott, Communications, Control, Circuits & Software
> http://www.wescottdesign.com

Not just that, an H infinity filter will give you a smaller mean-square error (even though you
are not theoretically minimising the mse in an H inf filter).
The trouble with H inf is it is more computationally intense however.

I believe you could also use an extended Kalman filter though the convergence is not assured.
There are many such approaches in the literature but it may take a lifetime to sort it all out!
For example you can use recursive least squares and a canonical state space description perhaps.
Extended least squares will estimate the innovations model direct including the noise and then
use self-tuning controllers etc