For example, I need to estimate the state (position, speed, acceleration) of a vehicle, but I can only observe the its position inaccurately. The problem is, without knowing the speed, I cannot update the state for the next time point. If I calculate the speed using the position difference between 2 time points i.e. v_t=(p_t-p_(t-1))/(delta t), the estimation becomes very unstable: if one estimation has a large deviation, the speed would also deviate from the real one, so the forecast of the next time point is even more inaccurate.
Does any one have any good ideas? Thanks in advance!
I am not sure if it is a good idea to use Kalman filtering techniques to estimate high order derivatives. I propose to check for Super-Twisting Second Ordre Sliding Mode techniques to estimate velocity and acceleration from the (noisy) measurement of position.
Have a look at the following link in estimating both the state and parameters in your system.
http://dataminingtheworld.blogspot.com.au/2014/04/koalas-vs-possums-modelling-biological.html
Here is how I would make a model for Kalman filtering out of this situation.
State X = (p,v,a)^T
State model with no process noise and assuming constant (perhaps uncertain) acceleration a:
X(t) = Phi(tau) X(t - tau) where Phi = [1 tau tau^2/2 ; 0 1 tau ; 0 0 1]
Measurement z(t) = HX(t) + n(t) where H = [1 0 0], n(t) is measurement noise of position p(t) with E n(t)^2 = r(t) (known or guessed)
You have to make an initial estimate X^(t0) with error covariance matrix
P(t0) = E (X(t0)-X^(t0)) (X(t0)-X^(t0)^T with your guess of uncertainties of (p,v,a) at time t0.
Now the Kalman algorithm will propagate (X^(t),P(t)) for t = t1, t2, ... where tau(n) = tn - t(n-1).
Thank you very much! If we assume a constant acceleration, the model indeed works in your way. However, it is not necessarily that case.
I asked this question more than one year ago and now I've found another solution to that problem with the grid based Bayesian filter (my states are discrete). Anyway, thank you for your answer :-)
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