please help me regarding stochastic process problem

[Siddhu]

Hello Friends,

I am not able to solve the following problem, which was given for me as an assignment. Could any one please solve this problem for me. please reply me as soon as possible, because the last date of submission is tomarrow.



Consider the system x(with dot) = a x + w(t); a is a real scalar

Z(t) = x(t) + v(t)

{w(t), t >= -∞} is a zero mean scalar gaussian white noise with unit variance
{v(t), t >= -∞} is also a zero mean scalar gaussian white noise with unit variance;
The two noise processes are independent of each other. Find the fixed point smoothing filter error variance assuming
(1). a < 0 but |a| <<1.
(2). a > 0 but |a| >>1.


regards,

Siddhu

 

[steve10]

There are fixed steps to follow when you apply the continuous kalman filtering technique. This process is essentially an optimization one, the aim of which is to minimize the "error variance". which is defined by

M=E[(x-x0)^2],
where x0 is the approximation of x. The next step is to take the derivative of M and take account of all the conditions that are available to you. You end up with an Riccati ordinary equation. Usually, this equation cannot be solved in a closed form. Fortunately, the variable t does not appear in the equation explicitly, so you can integrate it. This process is a tedious process (but it's standard), and you should get a reference (books or course notes). Here is the final equation that you end up with:

M(with dot)=-M^2 +1 +2aM

After you solve this equation, you consider the two extreme cases listed in your question. There will be some limiting process you need to take. Good luck.

 

[Siddhu]

Thank you Steve,

I could manage the problem. I got result more similar to that you have given. any how, thank you for kind response.

Siddhu