"Tim Wescott" <tim@seemywebsite.com> wrote in message
news:118kveet9k0024e@corp.supernews.com...
> susheemg@yahoo.com wrote:
>
> > Has there been work performed on the case in Kalman Filtering where you
> > are attempting to design a controller for a plant for which you have an
> > estimate of the state transition matrix?  Then you would want to adapt
> > the actual state transition matrix as well as predicting the value of
> > the state.  How do you go about setting up this problem?  I am more
> > than a little lost.
> >
> > thanks,
> > Susheem
> >
> So you want to have a filter that not only estimates the state vector
> but also estimates the state transition matrix?
>
> You can use adaptive techniques to estimate the system transfer
> function, but for a SISO system with n states that only gives you 2n-1
> parameters and your state transition matrix has n^2 elements in it.  So
> for n > 1...
>
Not if you write your parametric model found from RLS in controllable
canonical form ie pahse variable form.

0 1 0...
0 0 1...
0 0 0 1
--an -an-1 -an-2 ... -a1

and so on.

Rimmer

robert bristow-johnson wrote:

> in article 118kveet9k0024e@corp.supernews.com, Tim Wescott at
> tim@seemywebsite.com wrote on 05/17/2005 19:27:
> 
> 
>>So you want to have a filter that not only estimates the state vector
>>but also estimates the state transition matrix?
> 
> 
> now i'm a little lost.  isn't the Kalman filtering problem one of estimating
> the state vector, x, by knowing something about the plant, the A, B, C and D
> matrices, and observing the output, y, that has a little noise added to it?
> 
> you know:
> 
>     x[n+1] = A*x[n] + B*u[n]
>     y[n]   = C*x[n] + D*u[n]
> 
> and the observer can measure y[n] + e[n] where e[n] is uncorrelated random
> white noise.  from measuring y[n] + e[n], the state x[n] is estimated (and
> then y[n] can be determined from that).  IIRC, A, B, C are known and the D
> matrix is 0 and the u[n] input is also white, random, and uncorrelated to
> anything else.
> 
> isn't that what a Kalman filter problem is?
> 
> the state transition matrix is the inverse Z transform of z*I - A, where I
> is the square identity matrix of the same size as A.
> 
> this is the extent that i can remember anything else about Kalman filters.
> 
In adaptive control when you construct an optimum RLS system identifier 
it turns out to be a Kalman filter for a time-varying system who's 
modulation matrix is (IIRC) the system stimulation.  Combine that with 
an observer and you are, in effect, constructing an extended Kalman filter.

And that's most of what _I_ know on the subject (at least that part of 
that subject); I've been needing to get a good book on Kalman filtering 
for years.

-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

in article 118kveet9k0024e@corp.supernews.com, Tim Wescott at
tim@seemywebsite.com wrote on 05/17/2005 19:27:

> So you want to have a filter that not only estimates the state vector
> but also estimates the state transition matrix?

now i'm a little lost.  isn't the Kalman filtering problem one of estimating
the state vector, x, by knowing something about the plant, the A, B, C and D
matrices, and observing the output, y, that has a little noise added to it?

you know:

    x[n+1] = A*x[n] + B*u[n]
    y[n]   = C*x[n] + D*u[n]

and the observer can measure y[n] + e[n] where e[n] is uncorrelated random
white noise.  from measuring y[n] + e[n], the state x[n] is estimated (and
then y[n] can be determined from that).  IIRC, A, B, C are known and the D
matrix is 0 and the u[n] input is also white, random, and uncorrelated to
anything else.

isn't that what a Kalman filter problem is?

the state transition matrix is the inverse Z transform of z*I - A, where I
is the square identity matrix of the same size as A.

this is the extent that i can remember anything else about Kalman filters.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

susheemg@yahoo.com wrote:

> Has there been work performed on the case in Kalman Filtering where you
> are attempting to design a controller for a plant for which you have an
> estimate of the state transition matrix?  Then you would want to adapt
> the actual state transition matrix as well as predicting the value of
> the state.  How do you go about setting up this problem?  I am more
> than a little lost.
> 
> thanks,
> Susheem
> 
So you want to have a filter that not only estimates the state vector 
but also estimates the state transition matrix?

You can use adaptive techniques to estimate the system transfer 
function, but for a SISO system with n states that only gives you 2n-1 
parameters and your state transition matrix has n^2 elements in it.  So 
for n > 1...

You _could_ start with a system model that has 2n-1 adjustable 
parameters, make sure that there is a 1:1 mapping between those 
parameters and your transfer function, then do a running least-squares 
estimate of them.  It's kind of indirect, but if you're starting with a 
solid model of the system it may be a good way to go.

-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

in article 1116368267.627430.121030@f14g2000cwb.googlegroups.com,
susheemg@yahoo.com at susheemg@yahoo.com wrote on 05/17/2005 18:17:

> Has there been work performed on the case in Kalman Filtering where you
> are attempting to design a controller for a plant for which you have an
> estimate of the state transition matrix?  Then you would want to adapt
> the actual state transition matrix as well as predicting the value of
> the state.  How do you go about setting up this problem?  I am more
> than a little lost.

haven't done KF since grad school, but...

i thought you already knew the "A" matrix of the plant (and it has to be a
"completely observable system"), when you design a Kalman filter to estimate
the states of the system (by looking at the system output corrupted by white
noise).  if you know the A matrix, then you know the state transition
matrix, no?  i think the converse is also true, no?

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Has there been work performed on the case in Kalman Filtering where you
are attempting to design a controller for a plant for which you have an
estimate of the state transition matrix?  Then you would want to adapt
the actual state transition matrix as well as predicting the value of
the state.  How do you go about setting up this problem?  I am more
than a little lost.

thanks,
Susheem