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This paper proposes a new distributed Kalman filtering fusion with random state transition and measurement matrices, i.e., random parameter matrices Kalman filtering. It is proved that under a mild condition the fused state estimate is equivalent to the centralized Kalman filtering using all sensor measurements; therefore, it achieves the best performance. More importantly, this result can be applied to Kalman filtering with uncertain observations including the measurement with a false alarm probability as a special case, as well as, randomly variant dynamic systems with multiple models. Numerical examples are given which support our analysis and show significant performance loss of ignoring the randomness of the parameter matrices.

Linear discrete time system with random state transition and observation matrices arise in many areas such as radar control, missile track estimation, satellite navigation, digital control of chemical processes, economic systems. Koning [

Many advanced systems now make use of large number of sensors in practical applications ranging from aerospace and defense, robotics and automation systems, to the monitoring and control of a process generation plants. An important practical problem in the above systems is to find an optimal state estimator given the observations.

When the processing center can receive all measurements from the local sensors in time, centralized Kalman filtering can be carried out, and the resulting state estimates are optimal in the Mean Square Error (MSE) sense. Unfortunately, due to limited communication bandwidth, or to increase survivability of the system in a poor environment, such as a war situation, every local sensor has to carry on Kalman filtering upon its own observations first for local requirement, and then transmits the processed data-local state estimate to a fusion center. Therefore, the fusion center now needs to fuse all received local estimates to yield a globally optimal state estimate.

Under some regularity conditions, in particular, the assumption of independent sensor noises, an optimal Kalman filtering fusion was proposed in [

In the multisensor random parameter matrices case, sometimes, even if the original sensor noises are mutually independent, the sensor noises of the converted system are still cross-correlated. Hence, such multisensor system seems not satisfying the conditions for the distributed Kalman filtering fusion given in [

The remainder of this paper is organized as follows. In Section 2, we present the concept of random parameter matrices Kalman filtering. In Section 3, we present an optimal Kalman filtering fusion with random parameter matrices and show that under a mild condition the fused state estimate is equivalent to the centralized Kalman filtering with all sensor measurements. In Section 4, we show that the result can be applied to Kalman filtering with uncertain observations as well as randomly variant dynamic systems with multiple models. More importantly, we will see that the Kalman filtering with false alarm probability is a special case of Kalman with random parameter matrices. A simulation example is given in Section 5. And finally, in Section 6, we present our conclusions.

Consider a discrete time dynamic system:
_{k}^{r}_{k}^{N}_{k}^{r}_{k}^{N}_{k}^{r}^{×}^{r}_{k}^{N}^{×}^{r}

We assume the system has the following statistical properties: {_{k}_{k}_{k}_{k}_{0}. Moreover, we assume _{k}_{k}_{k}_{0}, the noises _{k}_{k}_{k}_{k}

_{k}_{k}

Rewriting _{k}_{k}

System

In the

Theorem 1. The Linear Minimum Variance recursive state estimation of system _{k}_{+1|}_{k}_{k}_{+1}, _{k}_{+1|}_{k}_{k}_{+1|}_{k}_{k}_{+1|}_{k}_{+1} denotes the update of _{k}_{+1} and _{k}_{+1|}_{k}_{+1} denotes the covariance of _{k}_{+1|}_{k}_{+1}.

Compared with the standard Kalman filtering and noting the notations in

In this section, a new distributed Kalman filtering fusion with random parameter matrices is proposed. The framework of the distributed tracking system is the same as those considered in [

The _{k}^{r}_{k}^{r}_{k}

Convert system

The stacked measurement equation is written as:
_{k}

Consider the covariance of the measurement noise of single sensor in new system. By the assumption above, we have:

As shown in the last part of Section 2, every entry of the last matrix term of the above equation is a linear combination of
_{k} is non-diagonal block matrix.

Luckily, when sensor noises are cross-correlated, in [

According to Theorem 1 and the Kalman filtering formulae given in [

We assume that the system has the following properties: the row dimensions of all sensor measurement matrices

According to [

Using

Using

To express the centralized filtering _{k}_{|}_{k}

Thus, substituting

That is to say that the centralized filtering

In this paper it is assumed that all sensor observations are synchronous. In practice, this may be very rarely true. However, in the past 20 years, such assumption was used very often in the track fusion community (for example, see [

The distributed systems here and in [

In this section, we will see that the results in the last two sections can be applied to the Kalman filtering with uncertain observations as well as randomly variant dynamic systems with multiple models.

The Kalman filtering with uncertain observation attracted extensive attention [

Consider a system:
_{k}_{k}_{k}_{k}_{k}

Consider that in _{k}_{k}_{k}

All that remains in order to apply the random measurement matrix Kalman filtering is just to calculate:

Substituting (

In the classical Kalman filtering problem, the observation is always assumed to contain the signal to be estimated. However, in practice, when the exterior interference is strong, i.e., total covariance of the measurement noise is large; the estimator will mistake the noise as the observation sometimes. In radar terminology, this is called a false alarm. Usually, the estimator cannot know whether this happens or not, only the probability of a false alarm is known. In the following, we will show that the Kalman Filtering with a false alarm probability is a special case of the uncertain observations of the above model

Consider a discrete dynamic process:
_{k}_{k}_{k}_{k}_{k}_{k}_{k}

Then,we can rewrite the measurement equations as follows:
_{k}

Due to

In the false alarm case, the state transition matrix is still deterministic, but the measurement matrix is random, by

Thus, the Kalman filtering with false alarm probability in this case is given by:

In this section, we consider the application to a general uncertain observation for one sensor case. In a manner analogous to the derivation of Section 4.1, we can also give an application to a general uncertain observation for multisensor case using Section 3. The procedure is omitted here.

The multiple-model (MM) dynamic process has been considered by many researchers. Although the possible models considered in those papers are quite general and can depend on the state, but no optimal algorithm in the mean square error (MSE) sense was proposed in the past a few decades. On the other hand, when some of the MM systems satisfy the assumptions in this paper, they can be reduced to dynamic models with random transition matrix and thus the optimal real-time filter can be given directly according to the random transition matrix Kalman filtering proposed in Theorem 1.

Consider a system:
_{k}_{k}_{k}_{k}_{k}

A necessary step for implementing the random Kalman filtering is to calculate:

Thus, all the recursive formulas of random Kalman filtering can be given by:

In this section, three simulations will be done for a dynamic system with random parameter matrices modeled as an object movement with process noise and measurement noise on the plane. The simulations give the special applications of results in the last section and show that fused random parameter matrices Kalman filtering algorithms can track the object satisfactorily.

Remember that we have rigorously proved in Section 3 that the centralized algorithm using all sensor observations

Firstly, we consider a three-sensor distributed Kalman filtering fusion problem with false alarm probabilities.

The object dynamics and measurement equations are modeled as follows:
_{k}

The false alarm probability of the

The initial state _{0} = (50, 0),
_{v}_{ωi}

In Example 1, both the sensor noises and the random measure matrices of the original system are mutually independent, so the sensor noise of the converted system are mutually independent. Now, we consider another example that both the noises and the random measure matrices of the original system are cross-correlated.

The object dynamics and measurement equations are modeled as follows:
_{k}_{k}_{k}_{k}_{v}_{ω}_{ωi}_{0} = (50, 0),

In this example, both the measurement noises and the random measure matrices of the original system are cross-correlated. Hence, the sensor noises of the converted system are cross-correlated.

In this simulation, there are three dynamic models with the corresponding probabilities of occurrence available. The object dynamics and measurement matrix in

The covariance of the noises are diagonal, given by _{v}_{ω}

In the multisensor random parameter matrices case, it was proven in this paper that when the sensor noises, or the measurement matrices of the original system are correlated across sensors, the sensor noises of the converted system are cross-correlated. Hence, such multisensor system seems not to satisfy the conditions for the standard distributed Kalman filtering fusion. This paper propose a new distributed Kalman filtering fusion with random parameter matrices Kalman filtering and proves that under a mild condition the fused state estimate is equivalent to the centralized Kalman filtering using all sensor measurements, therefore, it achieves the best performance. More importantly, this result can be applied to Kalman filtering with uncertain observations as well as randomly variant dynamic systems with multiple models. The Kalman filtering with false alarm is a special case of Kalman filtering with uncertain observations. Numerical examples are given which support our analysis and show significant performance loss of ignoring the randomness of the parameter matrices.

Suppose random matrix

By the properties of conditional expectation, we have that:

By the assumptions on the model

Since {_{k}_{k}_{0},

Similarly,

Without loss of generality, we consider the case of

For _{l}_{l}_{−1}_{l}_{−}_{2}_{1}_{0}_{0} ,_{l}_{−1}_{l}_{−1,}…_{l}_{−}_{i}_{+1}_{l}_{−}_{i,}i_{k,}v_{k}_{0},
_{k}

Hence,

Also consider the case of _{l}_{l}_{−1}_{l}_{−2}…_{1}_{0}_{0},_{l}_{−1},_{l}_{−1}_{l}_{−2}…_{l}_{−}_{i}_{+1}_{l}_{−i},_{k}, H_{k}, v_{k}_{k}_{0}, and {_{k}_{k}_{l}

By Lemma 2, system

Supported in part by NSF of China (#60874107, 10826101) and Project 863 through grant 2006AA12A104.

Comparison of standard Kalman filtering fusion and random Kalman filtering fusion.

Comparison of standard Kalman filtering fusion and random Kalman filtering fusion.

Comparison of IMM and random Kalman filtering.