Xk+1 = Ak Xk + Bk uk
yk = sk Ck Xk-m + vk
where sk is a random variable to demonstrate the probability of packet loss through the network, and m is a variable random transportation delay.
A z-domain representation might be tidier. Then you could use (1/z)^m where m is the (integer) delay (in samples or revisits).
Hello. Thanks for your answer.
Is it practical to use this model as a base to design a tracking algorithm or modifying the existing ones like JPDAF?
Without the delay part (i.e., when m=0), the model y_k=sC_kx_k+v_k is what is already used in target tracking where the binary variable s represents target existence or non-existence. In most tracking filters s is a deterministic constant (either 0 or 1). In some others, e.g. Integrated PDA, s is a random variable (taking values 0 or 1) assuming Markov behavior. I must say though that the measurement model used in tracking are almost never defined in this form. The same model is usually characterized with cases (or hypotheses) as follows.
y_k=C_kx_k+v_k If there is a target
y_k=v_k If there is no target
which is essentially the same thing.
As for the delay m, if it is a constant, deterministic and known, it is a straightforward issue to account for it using standard estimation tools. If not, it is more problematic and would probably require some research to handle.
Hello, Consider for example the situation where targets exist and the correct measurements are available at the sensor side. But when transferring these measurements to the tracking processor through a network, they are lost. Do the following hypotheses show this situation?
y_k=C_kx_k+v_k If there is a target
y_k=v_k If there is no target
In the case you mention when there is an unreliable communication network between the sensor and the tracker, the network is considered a part of the sensor in a target tracking application. Hence there is a so-called meta-sensor which includes both the original sensor and the communication network. In such a case, the model you mention is meaningful and if the variable s is considered to be a Markov chain, it is a classical example of a so-called jump Markov linear system (JMLS). State estimation is usually done using IMM filter.
Hello. I would like to add to the answers already given that you should firstly reformulate your state space model into a more standard form by defining a vector state X(k)=[x(k), x(k-1), ..., x(k-m)]^T to bring the delay into the state vector. The only non-standard item in your model is then the term s(k).
If s(k) is an independent IID random variable then, as suggested by others, this can be viewed as a so-called detection probability, as used in conventional target tracking.
If the variable s(k) is not independent, for instance, it is a Markov process, and in the discrete state case, a Markov chain, then your system is a hybrid system and as Umut Orguner suggested, you can apply techniques like the IMM filter or others, like generalised pseudo-Bayesian filters to estimate x based on y without knowing s.
Please refer to the paper below for more information on this type of problem, which has been extensively studied in manoeuvring target tracking.
Actually I just noticed that your delay m is also variable. You will have to develop a set of conditional filtering problems in which m is known by conditioning on its value. I have not seen this type of problem before but I think it can be handled by a combination of branching filters with an IMM along each branch.
Technical Report A Survey of Manoeuvring Target Tracking Methods
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