sorry i may not clearly represent my question. In my point of view, KF
is a sort of estimator.

y(t) = a1(t)+a2(t)*exp(-a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-
a5(t)*y(t-1)^2)*y(t-2) + e(t);

during the estimation process

y_hat(t) = a1(t)+a2(t)*exp(-a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-
a5(t)*y(t-1)^2)*y(t-2);

the measurement error e(t) = y(t) - y_hat(t);

the measurement covariance Q(t) = cov(e(t));

then predicted covariance P(t|t-1) = F*P(t|t)*F+Q(t) where F is the
state process model.


If i want to apply EKF to y(t) = a1(t)+a2(t)*exp(-
a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-a5(t)*y(t-1)^2)*y(t-2) + e(t), how
can I the State-space mode?

Terry <terryzz24@gmail.com> writes:

> the measurement equation is:
> y(t) = a1(t)+a2(t)*exp(-a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-
> a5(t)*y(t-1)^2)*y(t-2);
> 
> and the state equations are
> a1(t) = a1(t-1) + v1;
> a2(t) = a2(t-1) + v2;
> a3(t) = a3(t-1) + v3;
> a4(t) = a4(t-1) + v4;
> a5(t) = a5(t-1) + v5;
> 
> there is no measurement noise.

Then you still haven't read enough about Kalman filters.

Ciao,

Peter K.

-- 
"And he sees the vision splendid 
of the sunlit plains extended
And at night the wondrous glory of the everlasting stars."

 

I don't need y(t-1) or y(t-2) to be a part of state, both of them are
observerable

the measurement equation is:
y(t) = a1(t)+a2(t)*exp(-a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-
a5(t)*y(t-1)^2)*y(t-2);

and the state equations are
a1(t) = a1(t-1) + v1;
a2(t) = a2(t-1) + v2;
a3(t) = a3(t-1) + v3;
a4(t) = a4(t-1) + v4;
a5(t) = a5(t-1) + v5;

there is no measurement noise.

Terry <terryzz24@gmail.com> writes:

> sorry sir
> 
> there is a example of adaptive kalman filter applied to time-varying
> AR model
> 
> http://www.cs.ubc.ca/~emtiyaz/papers/TBME-00664-2005.R2-preprint.pdf
> 
> AR model is represented as Y(t) = a_1*Y(t-1) + ... + a_p*Y(t-p); so in
> that paper, the state space model is given as
> 
>  X(t+1) = AX(t) + V(t);  Y(t) = H(t)X(t) + W(t);
> 
> X(t) = [a_1, a_2, ... , a_p]';       H(t) = [Y(t-1), Y(t-2), ... , Y(t-
> p)];
> 
> so if that paper were wrong?

Quit spouting other people's formulae (badly) and think about what you
want.  I say "badly" because

Y(t) = a_1*Y(t-1) + ... + a_p*Y(t-p);

doesn't make a whole lot of sense, and is certainly not the same as:

X(t+1) = AX(t) + V(t);  Y(t) = H(t)X(t) + W(t);

Your problem:

>  y(t) = a1(t)+a2(t)*exp(-a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-a5(t)*y(t-1)^2)*y(t-2)+randn*v1;
> 
> a1(t) = a1(t-1) + randn*v2;
> a2(t) = a2(t-1) + randn*v3;
> a3(t) = a3(t-1) + randn*v4;
> a4(t) = a4(t-1) + randn*v5;
> a5(t) = a5(t-1) + randn*v6;

Seems poorly defined for the following reasons:

* It's not clear that there is process noise and measurement noise.
  The way it's written, there is only process noise (the v1, ...,
  v6). If there was a measurement noise, I would expect it to be
  written as u1, ... , un.

* It's not clear what a good choice of state is.  As Tim said, a1 ->
  a5 is part of the state, but it also looks like you need y(t-1) and
  y(t-2).

As Tim said, more reading is required... in books.

Ciao,

Peter "Books are an information technology" K.

-- 
"And he sees the vision splendid 
of the sunlit plains extended
And at night the wondrous glory of the everlasting stars."

 

thanks a lot, Tim. I found the problem where it is. :-)

sorry the model is
y(t) = a1(t)+a2(t)*exp(-a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-
a5(t)*y(t-1)^2)*y(t-2);


a1(t) = a1(t-1) + v1;
a2(t) = a2(t-1) + v2;
a3(t) = a3(t-1) + v3;
a4(t) = a4(t-1) + v4;
a5(t) = a5(t-1) + v5;

sorry sir

there is a example of adaptive kalman filter applied to time-varying
AR model

http://www.cs.ubc.ca/~emtiyaz/papers/TBME-00664-2005.R2-preprint.pdf

AR model is represented as Y(t) = a_1*Y(t-1) + ... + a_p*Y(t-p); so in
that paper, the state space model is given as

 X(t+1) = AX(t) + V(t);  Y(t) = H(t)X(t) + W(t);

X(t) = [a_1, a_2, ... , a_p]';       H(t) = [Y(t-1), Y(t-2), ... , Y(t-
p)];

so if that paper were wrong?

On Mon, 21 Jan 2008 11:26:50 -0800, Terry wrote:

> On 21 Jan, 16:46, Tim Wescott <t...@seemywebsite.com> wrote:
>> Tim Wescott
> 
> thanks a lot Tim,
> 
> I write this state space according to the paper "Demonstration of
> Adaptive Extended Kalman Filter for. Low Earth Orbit Formation
> Estimation Using CDGPS" http://acl.mit.edu/papers/Busse_ION02.pdf
> 
> so I write state space model as y(t) = a1(t)+a2(t)*exp(-
> a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-a5(t)*y(t-1)^2)*y(t-2)+randn*v1;
> 
>  y(t) = a1(t)+a2(t)*exp(-a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-
> a5(t)*y(t-1)^2)*y(t-2)+randn*v1;
> 
> a1(t) = a1(t-1) + randn*v2;
> a2(t) = a2(t-1) + randn*v3;
> a3(t) = a3(t-1) + randn*v4;
> a4(t) = a4(t-1) + randn*v5;
> a5(t) = a5(t-1) + randn*v6;
> 
> a1 a2 a3 a4 a5 follows a random walk.
> 
> and I applied the state space model into Extended Kalman filter, the
> model works. According to that paper, the model should be fine however
> the model performance seems not very well. Is it possible that could you
> give me any suggestions?
> 
> many many thanks

I did already: study up on Kalman filters, using a good book (like the 
one I recommended), then come back with specific questions.

-- 
Tim Wescott
Control systems and communications consulting
http://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

On 21 Jan, 16:46, Tim Wescott <t...@seemywebsite.com> wrote:
> Tim Wescott

thanks a lot Tim,

I write this state space according to the paper "Demonstration of
Adaptive Extended Kalman Filter for. Low Earth Orbit Formation
Estimation Using CDGPS" http://acl.mit.edu/papers/Busse_ION02.pdf

so I write state space model as y(t) = a1(t)+a2(t)*exp(-
a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-a5(t)*y(t-1)^2)*y(t-2)+randn*v1;

 y(t) = a1(t)+a2(t)*exp(-a3(t)*y(t-1)^2)*y(t-1)+a4(t)*exp(-
a5(t)*y(t-1)^2)*y(t-2)+randn*v1;

a1(t) = a1(t-1) + randn*v2;
a2(t) = a2(t-1) + randn*v3;
a3(t) = a3(t-1) + randn*v4;
a4(t) = a4(t-1) + randn*v5;
a5(t) = a5(t-1) + randn*v6;

a1 a2 a3 a4 a5 follows a random walk.

and I applied the state space model into Extended Kalman filter, the
model works. According to that paper, the model should be fine however
the model performance seems not very well. Is it possible that could
you give me any suggestions?

many many thanks