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Error bounds for nonlinear filtering are very important for performance evaluation and sensor management. This paper presents a comparative study of three error bounds for tracking filtering, when the detection probability is less than unity. One of these bounds is the random finite set (RFS) bound, which is deduced within the framework of finite set statistics. The others, which are the information reduction factor (IRF) posterior Cramer-Rao lower bound (PCRLB) and enumeration method (ENUM) PCRLB are introduced within the framework of finite vector statistics. In this paper, we deduce two propositions and prove that the RFS bound is equal to the ENUM PCRLB, while it is tighter than the IRF PCRLB, when the target exists from the beginning to the end. Considering the disappearance of existing targets and the appearance of new targets, the RFS bound is tighter than both IRF PCRLB and ENUM PCRLB with time, by introducing the uncertainty of target existence. The theory is illustrated by two nonlinear tracking applications: ballistic object tracking and bearings-only tracking. The simulation studies confirm the theory and reveal the relationship among the three bounds.

In the Bayesian framework, the complete posterior density of the state is necessary, in order to obtain the optimal recursive random state estimate for a classical nonlinear filtering problem by using various sensors [

Error bounds for nonlinear filtering can be applied in many fields. Firstly, error bounds can be used as a performance evaluation of suboptimal nonlinear filters and as a judgment of the effects of introduced approximations. For example, error bounds were applied in the cases of bearings-only tracking by a moving platform carrying sensor [

There is a long history of the development of the error bounds for nonlinear filtering. Initially, the Cramér-Rao lower bound (CRLB) was introduced as a bound of the estimation when the state dynamics are deterministic, and a comprehensive review of pre-1989 attempts are presented in [_{D}_{FA}

The assumptions of the sensors where _{D}_{FA}_{D}_{FA}_{D}_{FA}

All the bounds mentioned above are based on the assumption that the target exists from the beginning to the end. However, the appearance of the target varies with time in many practical situations. Moreover, we cannot determine whether the target exists or not from the measurements, because it is unlikely to know whether there is missing detection or there are false measurements, especially in defense and surveillance [

Because both target states and estimates are modeled as RFS, traditional Euclidean distance could not be applied for calculating the error. Therefore, a meaningful distance named Optimal Sub-pattern Assignment (OSPA) distance in [

For the problem of the error bound in the framework of finite set statistics, [

The paper presents a comparative study of the RFS bound in [_{D}

It is noted that the result in this paper is for the condition of sensors where _{D}_{FA}_{d}_{FA}^{−6}, as indicated in [_{D}_{FA}

In this paper, Section 2 introduces some background knowledge about the dynamic and sensor models, the PCRLB and the main theoretical results of ENUM PRRLB and IRF PCRLB. Section 3 reviews the basic knowledge of random set statistics, the random set dynamic and measurement models and the RFS bound. Section 4 compares these three bounds in two cases: when the target exists from the beginning to the end; and when targets might appear or disappear. Section 5 is devoted to the application examples: the tracking of ballistic missiles in the re-entry phase and bearings only tracking. Conclusions are given in Section 6.

For a discrete-time nonlinear filtering problem, the target state is modeled as a random vector, and the state dynamic equation is given by:
_{k}^{m}_{k}_{k}_{k}

The sensor measurement model is a function of the target state, in which there is a single sensor and a single measurement at each time step. There are no false alarms _{FA}_{D}_{k}^{r}_{k}_{k}_{k}

The covariance of the estimate of _{k}_{k}_{1},⋯,_{k}_{k}_{k}_{k}^{−1} is the PCRLB. The inequality in (3) means that the difference _{k}_{k}^{−1} is a positive semi-definite matrix.

As in [

The matrices _{k}_{k}_{k}_{k}

As defined in [

The initial FIM is calculated from the prior probability function _{0}(_{0}):

For the case where _{D}_{D}

Then, the PCRLB calculated by IRF method is denoted by _{k}_{k}_{k}^{−1}.

In [_{D}_{FA}_{i}

The uncertain target dynamics is as:
_{k}_{+ 1}(_{i}

Then, as introduced in [_{k}

At time step k, there are 2^{k}_{i}_{k}_{k}_{i}^{−1}.

Random finite set (RFS) is a random variable which takes value as a finite set [

The filter derived from Bernoulli RFS attracts substantial interest and is used widely recently [

For the function

The expectation of the function

For the error between the set

For the cardinality of

The state dynamics and measurement model are as similar as what in [

For the dynamical model, the Markov transition density is defined by:
_{k}_{k}_{k}_{k}_{−1}) is the probability density of a transition from _{k}_{k}_{+ 1}, which can be calculated by

The prior probability function of the state set is also Bernoulli RFS:

The probability of detection is _{D}_{k}_{k}

In [_{D}_{FA}_{k,n}^{k}

When _{k}_{+1,}_{n}

For certain time-sequence of observation-sets Θ_{k,n}, the error bound Σ_{k,n} is as follows:

_{k,n}_{k,n}_{k,n}.

Let us denote the error bound between two sets _{k}_{k}_{1} ⋯ _{k}_{k}

As indicated in _{k}_{k,n}

When time-step ^{k}^{+ 1}, the bounds sequence _{k}_{+ 1,n} obeys the recursion:

As defined in _{k}_{+1} = ∅ corresponds to the sequence number 1 ≤ ^{k}_{k}_{+1} = ∅, which are:
_{k}_{+1} is the prediction, whose bound is calculated by _{k}_{+1,}_{n}

Recursive relationships of all above auxiliary elements are derived rigorously in [^{k}^{+ 1}, the FIM _{k}_{+ 1,n} obey the recursion:

Second, when

When _{k,n}_{k}_{+ 1}_{,n}_{k}_{+1,}_{n}^{k}^{+ 1}:

When

Finally, from _{k}_{+ 1,n} and Pr(Θ_{k+1,n}) is the recursion of _{k+1,n} = ∅|Θ_{k,n}), 1 ≤ ^{k}

When

Then the recursive form of bound recursions is turned up, and we obtain the recursive _{k,n}

This section presents a comparison of the RFS bound, the IRF PCRLB and the ENUM PCRLB, as introduced in the previous sections. This problem is discussed in two cases. Section 4.1 introduces the first case, where the target exists from the beginning to the end. Section 4.2 is devoted to the other case, where the new targets might appear and existing targets could disappear. In Section 4.1, we firstly deduce a comparable form of the RFS bound. Then, this form of bound is applied to compare with ENUM PCRLB in Section 4.1.2, while it is also used to compare with IRF PCRLB in Section 4.1.3. In Section 4.2, these three bounds are compared directly both quantitatively and qualitatively.

All PCRLBs are based on the assumption that the target always exists. In order to compare the bound in [

Condition1:

Condition2:

Proof:

Assume that:

As the number of the scans of measurements increases, Condition 2 should be met easily. Then from

Combining the definition of _{k}_{+ 1}_{,n}

Because of Condition 1 that _{1} = _{0} = [_{1,1} _{2,1} ⋯ _{m}_{,1}]^{T}

As defined in

Then, the bound for a time-sequence of observation-sets Θ_{k,n}

Proposition I denotes that, when the probability of not empty state set is more than which of empty, the bound is in the form that

The calculation of PCRLB is based on the assumption that the target exists from the beginning to the end. Moreover, there is no false alarm. For the RFS bound, as defined in

Therefore, the bound calculated by

Then, the calculation of _{k}_{+ 1}(RFS) in

As indicated in _{k}_{+1,}_{n}

Although we have the recursion of the conditional probability _{k}_{+1,}_{n}_{k,n}

At time step ^{k}^{− 1}, if it is satisfied that:

Condition 3: the maintenance probability is the unity as:

Condition 4: the initial probability is the unity as:

Then the conditional probability _{k}_{+1,}_{n}_{k,n}

Proof:

For the case where the false alarm probability _{FA}_{D}

Additionally, when _{k}_{+1,n} = ∅|Θ_{k,n}_{D}_{k}_{+ 1,n} = 0,1 ≤ ^{k}

From

As in [_{k,n}_{k,n}_{k,n}

Rewrite the EUNM PCRLB in

Therefore, we can now show the relationship between _{k}_{+ 1}(_{k}_{+ 1}(

_{D}_{FA}

_{k}_{+1,}_{n}_{k,n}_{D}_{k}_{k}

In [_{D}_{FA}

Condition 5:

Condition 6:

It is obvious that Condition 5 and Condition 6 are satisfied when we get the recursive bound for random finite set in [

Therefore, for the case where _{D}_{FA}

All above proofs are in the case of _{D}_{FA}

When the probability of detection is zero or unity, such as _{D}_{D}

All PCRLBs are based on the assumption that the target exists from the beginning to the end. However, the existence of the target is difficult to be deterministic, with phenomena such as target spawning, new-born targets or disappearing ones. This problem finds lots of applications in defense and surveillance [

In fact, there are many tracking algorithms, such as [_{D}_{FA}

Although the RFS bound _{k}_{k}_{k}_{k}_{k}_{k}_{k}_{k}_{k}

The quantitative comparison is difficult. The reason is that the RFS bound and ENUM PCRLB are calculated by different models. The model for calculating _{k}_{k}_{k}_{k}_{k,n}_{k}_{k,n}_{D}_{k}_{−1,}_{n}_{k−2,n}). But, when _{k}_{k,n}_{D}

The qualitative comparison analysis is also difficult. If we only considered the uncertainty brought by new-born and disappearing targets, the RFS bound _{k}_{k}_{k}

Although it is hard to determine that the bound _{k}_{k}

In this section, two examples are used to illustrate previous results. In the first case, we give an example to show how the four conditions in Section 4 influence the relationship between the RFS bound _{k}_{k}

Online estimation of the kinematic state of a ballistic object re-entering the atmosphere is an important problem. This section used a simple motion model as in [_{l}_{l}v_{l}β_{l}_{l}_{l}_{l}

The exponentially decaying model of air density is adopted as _{l}_{l}^{−4}. The covariance matrix _{l}

The radar measures the height of target with _{k}

The prior distribution is assumed Gaussian with covariance, and the initial FIM is calculated as in

The Jacobian defined by

Following parameters are applied in this example. The initial target state vector _{0} = [55,000 m 300 m/s, 22,500 kg/ms^{2}]. The integration time _{r}_{β}^{2}.

Here we set _{1} = _{0}, and thus the Condition 1 is met. In this example, we always set

_{k}_{k}_{1} ⋯ _{k}_{k}_{k}_{k}_{D}_{D}

In _{k}

When we compare the two solid lines, _{1}_{1}^{T}_{0}_{0}^{T}_{0}. Therefore, as the scan number increases and measurements become more, the Condition 2 is satisfied. In this condition, we can see that _{k}_{k}_{1}_{1}^{T}_{0}_{0}^{T}_{0}, the Condition 2 is not met, as shown by the purple solid line. At the beginning, the purple line is below the green one, because its cardinality mismatches _{0} and _{1} are smaller. But after these scans, the purple line is more the green one. In conclusion, by the influence of wrong settings _{0} and _{1}, _{k}_{k}_{k}_{+1,}_{n}_{D}_{k,n−2k}, _{k}_{+1} ≠ ∅) is much more bigger than the probability Pr(Θ_{k,n−2k}, _{k}_{+1} = ∅) in _{k}_{+1,}_{n}^{−1}*Pr(Θ_{k,n−2k}, _{k}_{+1} ≠ ∅) provides more contribution than _{k}_{+1,}_{n}_{k}_{0} and _{1} becomes less important with time, and eventually, both _{k}

In _{k}_{1}_{1}^{T}_{0}_{0}^{T}_{0}. Thus the Condition 2 is satisfied as the scan number increases. When we compare the three solid lines, _{k}_{k}_{k}_{k}_{k}

This example is similar to the bearings-only tracking case in [

The observer, named ownship, is a moving platform carrying sensor. Its state vector is denoted as
_{k}_{k}_{k}_{k}_{k}

The sensor measurement equation is:

The measurement noise _{k}_{k}_{k}

The initial FIM is
_{r}_{v}

The errors in cardinality mismatches are:

Ownship is moving as a uniform circular motion. The dynamic equation of the observer is given by:

The initial target state vector
_{r}_{v}

_{k}_{k}_{1} ⋯ _{k}

In _{k}_{D}_{k}_{k}

There are many lines in _{k}_{D}_{D}_{k}_{k}_{D}_{k}_{k}_{k}

In _{k}_{k}_{k}_{k}

To the RFS bound _{k}_{k}_{k}_{k}_{k}_{k}_{k}_{k}

The influence of the initial probability _{k}_{k}_{k}_{k}_{k}_{k}

Admittedly, it is impossible to prove all the intersections of the lines in _{k}_{k}_{k}_{k}_{k}

In _{FA}

As in _{D}

In conclusion, allowing the disappearance of existing targets and the appearance of new targets, RFS bound _{k}_{k}_{k}

The paper is devoted to comparison of the recently reported bounds within the framework of finite set statistics and the traditional PCRLB bounds, which are applicable to the case where the detection probability _{D}_{FA}

This work was supported in part by the National Natural Science Foundation of China (No. 60901057) and in part by the National Natural Science Foundation of China (No. 61201356).

Comparisons of bounds when the target exists from the beginning to the end with different setting cardinality mismatches: (

Comparisons of bounds when the target disappears with different maintenance probability r: (

Bearing-only tracking scenario.

Comparisons of bounds with different detection probability P_{D}: (

Comparisons of bounds with different maintenance probability r: (

Comparisons of bounds with different initial existence probability b: (

Comparisons of the performance of PHD filter and the bounds with different maintenance probability r: (