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 consultinghttp://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

d... at (no spam) myallit.com wrote in message

You might consider Extended Kalman filtering (EKF). Be aware
about the eventual non-stability of the scheme.

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.

On Jul 22, 2:20 am, Tim Wescott <t... at (no spam) seemywebsite.com> wrote:
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 consultinghttp://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

My state transition for these two measurements is given by:

F = [dt 1
1 0]

where dt is the time step, so as far as I understand it these
measurements are correlated - if z1 is positive, then z2 will be
increasing and vice versa.

Doing this in the log domain makes sense, but how can I have one
measurement in the log domain and the other not? The state transition
won't make sense.

Equation 2 will actually reach zero quite often, so maybe I need
multiplicative and additive noise in my measurement model for z2, not
too sure about that.

On Jul 21, 5:35 pm, "Bruno Luong" <b.lu... at (no spam) fogale.fr> wrote:
d... at (no spam) myallit.com wrote in message

You might consider Extended Kalman filtering (EKF). Be aware
about the eventual non-stability of the scheme.


What do you mean by the eventual non-stability? I did look at the EKF,
there is some simple sample MATLAB code here:
http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=18189

But in the first few lines of this script it says:

% for nonlinear dynamic system:
% x_k+1 = f(x_k) + w_k
% z_k = h(x_k) + v_k
% where w ~ N(0,Q) meaning w is gaussian noise with covariance Q
% v ~ N(0,R) meaning v is gaussian noise with covariance R

so the EKF looks appropriate for non-linear process models and
measurement models that can be represented by any arbitrary functions
f(x) and h(x), but the noise is still assumed to be additive.

On Jul 21, 5:35 pm, "Bruno Luong" <b.lu... at (no spam) fogale.fr> wrote:

d... at (no spam) myallit.com wrote in message

You might consider Extended Kalman filtering (EKF). Be aware
about the eventual non-stability of the scheme.

What do you mean by the eventual non-stability? I did look at the EKF,
there is some simple sample MATLAB code here:http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objec...

But in the first few lines of this script it says:

% for nonlinear dynamic system:
% x_k+1 = f(x_k) + w_k
% z_k = h(x_k) + v_k
% where w ~ N(0,Q) meaning w is gaussian noise with covariance Q
% v ~ N(0,R) meaning v is gaussian noise with covariance R

so the EKF looks appropriate for non-linear process models and
measurement models that can be represented by any arbitrary functions
f(x) and h(x), but the noise is still assumed to be additive.

On Jul 22, 2:20 am, Tim Wescott <t... at (no spam) seemywebsite.com> wrote:



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 consultinghttp://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

My state transition for these two measurements is given by:

F = [dt 1
1 0]

where dt is the time step, so as far as I understand it these
measurements are correlated - if z1 is positive, then z2 will be
increasing and vice versa.

Doing this in the log domain makes sense, but how can I have one
measurement in the log domain and the other not? The state transition
won't make sense.

Equation 2 will actually reach zero quite often, so maybe I need
multiplicative and additive noise in my measurement model for z2, not
too sure about that.

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.

On Jul 21, 5:35 pm, "Bruno Luong" <b.lu... at (no spam) fogale.fr> wrote:

d... at (no spam) myallit.com wrote in message

You might consider Extended Kalman filtering (EKF). Be aware
about the eventual non-stability of the scheme.

What do you mean by the eventual non-stability? I did look at the EKF,
there is some simple sample MATLAB code here:http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objec...

But in the first few lines of this script it says:

% for nonlinear dynamic system:
% x_k+1 = f(x_k) + w_k
% z_k = h(x_k) + v_k
% where w ~ N(0,Q) meaning w is gaussian noise with covariance Q
% v ~ N(0,R) meaning v is gaussian noise with covariance R

so the EKF looks appropriate for non-linear process models and
measurement models that can be represented by any arbitrary functions
f(x) and h(x), but the noise is still assumed to be additive.

Contrary to the beliefs of some, the MathWorks isn't on high, and MatLab
isn't an extension of the Bible. I would trust them for examples, but
don't take their documentation for anything but a means of selling you
copies of MatLab.

If this is a student project then why not assume there will be a set
point and the controller will maintain that set point with little
variation. Now the volume is fixed and the variance is therefore
fixed. Now you can assume the system is linear around that set point.

I really suggest that you make up a graphical model of the system,
Simulink or Scicos ( I can deal with Scicos), both the physical system
and then the proposed filtering.

So your multiplicative noise can be negative? This seems strange.

In any case, why not simply derive the appropriate filtering from
scratch instead of trying to find off the shelf solutions? That way
you can incorporate all the specifics of your problem.

The EKF propagates first and second moments which is essentially
treating the problem like it can be approximated like a linear
Gaussian.

Particle filters come in various flavors and there are versions that
can handle multiplicative noise.

Contrary to the beliefs of some, the MathWorks isn't on high, and MatLab
isn't an extension of the Bible. I would trust them for examples, but
don't take their documentation for anything but a means of selling you
copies of MatLab.

The link I posted is a user contribution, not an example from the
Mathworks, so I don't trust it entirely. However the Wikipedia article
on EKF's seems to show the same formulation with the noise being
assumed to be additive, only the process and measurement models are
allowed to be non-linear:

http://en.wikipedia.org/wiki/Extended_Kalman_filter#Formulation

Does this mean the EKF is not appropriate for my problem or am I
misunderstanding how it works?

If this is a student project then why not assume there will be a set
point and the controller will maintain that set point with little
variation. Now the volume is fixed and the variance is therefore
fixed. Now you can assume the system is linear around that set point.

I have already implemented a regular Kalman filter simply by
approximating both of the noise terms as additive noise and the
results are okay but not great (another idea I might try is
approximating both of them as multiplicative noise, and then I can do
all the calculations in the log domain). To improve it I want to try
incorporating the multiplicative noise. When z2 terms have smaller
values, we have more confidence in them (less variance), so I want to
explicitly incorporate this into the filter. In particular, when z2
reaches zero, we have almost 100% confidence it is correct.

Contrary to the beliefs of some, the MathWorks isn't on high, and MatLab
isn't an extension of the Bible. I would trust them for examples, but
don't take their documentation for anything but a means of selling you
copies of MatLab.

The link I posted is a user contribution, not an example from the
Mathworks, so I don't trust it entirely. However the Wikipedia article
on EKF's seems to show the same formulation with the noise being
assumed to be additive, only the process and measurement models are
allowed to be non-linear:

http://en.wikipedia.org/wiki/Extended_Kalman_filter#Formulation

Does this mean the EKF is not appropriate for my problem or am I
misunderstanding how it works?

If this is a student project then why not assume there will be a set
point and the controller will maintain that set point with little
variation. Now the volume is fixed and the variance is therefore
fixed. Now you can assume the system is linear around that set point.

I have already implemented a regular Kalman filter simply by
approximating both of the noise terms as additive noise and the
results are okay but not great (another idea I might try is
approximating both of them as multiplicative noise, and then I can do
all the calculations in the log domain). To improve it I want to try
incorporating the multiplicative noise. When z2 terms have smaller
values, we have more confidence in them (less variance), so I want to
explicitly incorporate this into the filter. In particular, when z2
reaches zero, we have almost 100% confidence it is correct.

I really suggest that you make up a graphical model of the system,
Simulink or Scicos ( I can deal with Scicos), both the physical system
and then the proposed filtering.

I'm not too familiar with Simulink or Scicos but here's a small
diagram:

z2
+---------+
| |
======>| |
z1 | |
====== | |
| |
+---------+

The measurement z1 is measuring the amount going in and out of the
container (flow rate), the measurement z2 is measuring the total
currently in the container (volume). I am trying to find the true
value of x2 (the volume) by combining the noisy measurements of z1 and
z2. Actually there are multiple sensors measuring flow rate and
volume, but I've only shown one of each to simplify the problem. The
z1 measurement has additive Gaussian noise, the z2 measurement has
multiplicative (and possibly additive?) Gaussian noise.

So your multiplicative noise can be negative? This seems strange.
In any case, why not simply derive the appropriate filtering from
scratch instead of trying to find off the shelf solutions? That way
you can incorporate all the specifics of your problem.

No I suppose it should always should be positive, I have uploaded an
example image to help clarify:

http://picasaweb.google.com.au/dspmyallit/ExampleVolumeMeasurements/p...

The EKF propagates first and second moments which is essentially
treating the problem like it can be approximated like a linear
Gaussian.
Particle filters come in various flavors and there are versions that
can handle multiplicative noise.

I will look into particle filters. Are you aware of any good C++
implementations? (For the Kalman filter I have been using MATLAB to
design and test it, OpenCV to implement it.)
So your multiplicative noise can be negative? This seems strange.
In any case, why not simply derive the appropriate filtering from
scratch instead of trying to find off the shelf solutions? That way
you can incorporate all the specifics of your problem.

No I suppose it should always should be positive, I have uploaded an
example image to help clarify:

http://picasaweb.google.com.au/dspmyallit/ExampleVolumeMeasurements/p...