H infinity error covariance matrix

Hello,

I was hoping that someone might know how to compute an error
covariance matrix for an H infinity filter when the noise terms are
known to be uncorrelated white and Gaussian.   I suspect that P
(defined below) is the error covariance matrix since it becomes that
when configured to be equivalent to a Kalman filter.  However, based
on my understanding of H infinity theory P is not an error covariance
matrix...

P(k+1) = F(k) P(k)[ I - theta S(k) P(k) + H'(k)R(k)^(-1)H(k)P(k)]^(-1)F
(k)' + Q(k)

Thanks,
- Peter

On Jan 27, 12:37�pm, dihydro...@gmail.com wrote:
> Hello,
>
> I was hoping that someone might know how to compute an error
> covariance matrix for an H infinity filter when the noise terms are
> known to be uncorrelated white and Gaussian. � I suspect that P
> (defined below) is the error covariance matrix since it becomes that
> when configured to be equivalent to a Kalman filter. �However, based
> on my understanding of H infinity theory P is not an error covariance
> matrix...
>
> P(k+1) = F(k) P(k)[ I - theta S(k) P(k) + H'(k)R(k)^(-1)H(k)P(k)]^(-1)F
> (k)' + Q(k)
>
> Thanks,
> - Peter

Doesn't mean much without explaining the symbols. For H infinity you
have a lambda term somewhere from teh cost function and the noise
covariance matrices don't come into it.

Hardy

On Jan 29, 4:21&#4294967295;am, HardySpicer <gyansor...@gmail.com> wrote:
> On Jan 27, 12:37&#4294967295;pm, dihydro...@gmail.com wrote:
>
> > Hello,
>
> > I was hoping that someone might know how to compute an error
> > covariance matrix for anHinfinityfilter when the noise terms are
> > known to be uncorrelated white and Gaussian. &#4294967295; I suspect that P
> > (defined below) is the error covariance matrix since it becomes that
> > when configured to be equivalent to a Kalman filter. &#4294967295;However, based
> > on my understanding ofHinfinitytheory P is not an error covariance
> > matrix...
>
> > P(k+1) = F(k) P(k)[ I - theta S(k) P(k) +H'(k)R(k)^(-1)H(k)P(k)]^(-1)F
> > (k)' + Q(k)
>
> > Thanks,
> > - Peter
>
> Doesn't mean much without explaining the symbols. For H infinity you
> have a lambda term somewhere from teh cost function and the noise
> covariance matrices don't come into it.
>
> Hardy

Just noticed that someone had replied.  Here is a more detailed
explanation:

System equations:

x(k+1) = F(k) x(k) + w(k)
y(k) = H(k) x(k) + v(k)
z(k) = L(k) x(k)

In this specific case w(k) and v(k) are independent white Gaussian
processes, which is not true for h infinity filters in general.  The
cost function is the following:

J_1 =          sum[k=0...N-1 , ||z(k)-\hat{z}(k)||^2_{S(k)} ]
 
--------------------------------------------------------------------------------------------------
       ||x(0) - \hat{x}(0)||^2 _(P(0)^{-1}) + sum[k=0...N-1 , (||w(k)||
^2_{Q(k)^-1} + ||v(k)||^2_{R(k)^-1})]

and J_1 < 1 / theta

The notation ||a||^2_{B} = a'Ba.

In this cost function the noise terms do come into play.  In my
situation L and S(k) are set to identity matrices and the noise terms
are all assumed to be Gaussian.  I'm wondering if anyone has already
shown that P(k+1), defined in the first message, is an error
covariance matrix in this situation.  If in addition theta is set to
zero, then it becomes equivalent to a Kalman filter and P(k+1) is an
error covariance matrix.

Thanks,
- Peter