Recurrence in the observation equation

Usually a Kalman filter can be applied to a system described by the
following equations:
X[k] = A*X[k-1] + W[k] ... (1)
Y[k] = C*X[k] + V[k] ... (2),
where (1) is the dynamic equation with the state X[k] and process noise
W[k], and (2) is the observation equation with the observation noise V[k].
The only dynamic is in the difference equation (1). Now, I'm reading a
paper on the Kalman filter application, where the observation Y[k] in a
specific application is subject to a recurrence relation too. In
particular, the observation equation reads
Y[k] = C*X[k-1] + V[k] ... (3).

I'm wondering if the classic Kalman filter still applies to the system
described by (1) and (3). If not, hot to adapt the Kalman filter? Does the
adapted Kalman filter have a specific name? Any hints of literature on this
issue are highly appreciated. In the paper that I'm reading these points
are not treated but only a classic Kalman filter is used.

On Jan 20, 8:30&#4294967295;pm, "lanbaba" <lanbaba@n_o_s_p_a_m.gmx.ch> wrote:
> Usually a Kalman filter can be applied to a system described by the
> following equations:
> X[k] = A*X[k-1] + W[k] ... (1)
> Y[k] = C*X[k] + V[k] ... (2),
> where (1) is the dynamic equation with the state X[k] and process noise
> W[k], and (2) is the observation equation with the observation noise V[k].
> The only dynamic is in the difference equation (1). Now, I'm reading a
> paper on the Kalman filter application, where the observation Y[k] in a
> specific application is subject to a recurrence relation too. In
> particular, the observation equation reads
> Y[k] = C*X[k-1] + V[k] ... (3).
>
> I'm wondering if the classic Kalman filter still applies to the system
> described by (1) and (3). If not, hot to adapt the Kalman filter? Does the
> adapted Kalman filter have a specific name? Any hints of literature on this
> issue are highly appreciated. In the paper that I'm reading these points
> are not treated but only a classic Kalman filter is used.

It's the same. the original equation is X(k)=AX(k-1) whereas it is
usually quoted
as X(k+1)=AX(k). So you estimate the delayed state instead, so what?

Hardy

On 01/19/2011 11:30 PM, lanbaba wrote:
> Usually a Kalman filter can be applied to a system described by the
> following equations:
> X[k] = A*X[k-1] + W[k] ... (1)
> Y[k] = C*X[k] + V[k] ... (2),
> where (1) is the dynamic equation with the state X[k] and process noise
> W[k], and (2) is the observation equation with the observation noise V[k].
> The only dynamic is in the difference equation (1). Now, I'm reading a
> paper on the Kalman filter application, where the observation Y[k] in a
> specific application is subject to a recurrence relation too. In
> particular, the observation equation reads
> Y[k] = C*X[k-1] + V[k] ... (3).
>
> I'm wondering if the classic Kalman filter still applies to the system
> described by (1) and (3). If not, hot to adapt the Kalman filter? Does the
> adapted Kalman filter have a specific name? Any hints of literature on this
> issue are highly appreciated. In the paper that I'm reading these points
> are not treated but only a classic Kalman filter is used.

Let

       [ A   0 ]
A_n = [       ]
       [ I   0 ]


C_n = [ 0   C ]

Then you'll find that you're back to a regular old Kalman filter.

Since you're only seeing one delay you should be able to rearrange the 
normal Kalman order of operations to get the second case for free: 
instead of doing the Kalman as prediction -> correction -> take X[k], do 
the Kalman as correction (from last time) -> prediction -> take X[k].

-- 

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

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