I try to use a Kalman filter on the sensor output data. The measurement is a differential value that represents a difference between two states at time t and t-\tau, y(t) = x(t) - x(t-\tau).
In that case, how to use Kalman filter to optimize the state x(t)?
If tau is a multiple of your sampling time, you can simply augment the state vector, e.g. if y(kk)=x(k)-x(k-1). you can define a new state vector z(k)=[x(k);x(k-1)], write the state transition equation for it, y(k) is then [1 -1] z(k) and you can use the KF for the augmented system.
Your filter has a sample rate \Delta t = t_n - t_(n-1). Divide \tau by \Delta t to see how many lags separate x(t) from x(t-\tau). Assume that is a whole number L. Now make your x state vector have L+1 states so it includes x(t) as first element and x(t-\tau) as last element.
If tau is not a multiple of your sampling time or Asynchronous, you may look at this paper and the paper cited this: "Modeling and Estimation of Asynchronous Multirate Multisensor System With Unreliable Measurements"
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