Path: news.mathworks.com!newsfeed-00.mathworks.com!nlpi057.nbdc.sbc.com!prodigy.net!news.glorb.com!Xl.tags.giganews.com!border1.nntp.dca.giganews.com!nntp.giganews.com!local02.nntp.dca.giganews.com!nntp.web-ster.com!news.web-ster.com.POSTED!not-for-mail
NNTP-Posting-Date: Mon, 21 Jul 2008 11:20:27 -0500
From: Tim Wescott <tim@seemywebsite.com>
Subject: Re: Kalman filtering with multiplicative noise
Newsgroups: sci.stat.math,comp.soft-sys.matlab,sci.engr.control
References: <1309743c-3d5c-4165-9cdb-1ab51892727d@z16g2000prn.googlegroups.com>
User-Agent: Pan/0.132 (Waxed in Black)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit
Message-ID: <v_WdnRJ79rDWKhnVnZ2dnUVZ_j-dnZ2d@web-ster.com>
Date: Mon, 21 Jul 2008 11:20:27 -0500
Lines: 45
X-Usenet-Provider: http://www.giganews.com
X-Trace: sv3-Cq0pAW/xAVwsLxqiueK6qOe3OhGkU2ioY3mwutVVpFQMwiOGLqjK6PXgQIP9oqiNjqRquyiLfQ4M687!SE2ZVd3VVdmDcPWa5h4uslMMCqEaxpvhQ41jxYYSKhse3NnPrbheR8PfwanOe2V0PTbK4SUpz5aq!rEb+ArrYlXGUxrI=
X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers
X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly
X-Postfilter: 1.3.39
Xref: news.mathworks.com sci.stat.math:83491 comp.soft-sys.matlab:480702 sci.engr.control:50814



On Sun, 20 Jul 2008 19:51:02 -0700, dsp wrote:

> I'm trying to implement a Kalman filter in MATLAB that will use two
> types of measurements: volume and in/out flow rate. For the flow rate,
> the measurement error is additive Gaussian, but for the volume the
> measurement error is expressed as a percentage of the volume, so that
> the volume measurement is less accurate when its value is higher. I
> think the measurement model should therefore be:
> 
> Flow rate measurement model:
> z1 = x1 + v1 where v1 ~ N(0,e1)
> 
> Volume measurement model:
> z2 = x2*v2 where v2 ~ N(1,e2)
> 
> I assumed the volume filtering should be done in the log domain to make
> the noise additive but how do I deal with a noise mean of one when the
> Kalman filter assumes a mean of zero? And how can I have a Kalman filter
> using both the measurements if one is in the log domain and the other
> one isn't?
> 
> I am also dealing with a system where measurements will usually be
> missing (they are arriving sequentially) and at an uneven sampling rate,
> any other pointers on these too would be appreciated.

As Ray pointed out these seem to be completely independent measurements, 
unless V1 and V2 are correlated.

With any affine system, such as the log-domain volume system, you can 
subtract out the offset to make the system linear, do your linear 
processing, then add the offset back in.  I.O.W., calculate log z2 = log 
x2 + log v2 + 1, then define dlz2 = log z2 - 1, then find your estimated 
dlz2, then back up to your estimated z2.  (Note that your estimate of z2 
probably won't be strictly optimal in the mean-squared sense, only your 
estimate of dlz2 will be.  But you'll certainly be closer than you would 
by a lot of other means).

-- 
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html