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A robust filtering problem is formulated and investigated for a class of nonlinear systems with correlated noises, packet losses, and multiplicative noises. The packet losses are assumed to be independent Bernoulli random variables. The multiplicative noises are described as random variables with bounded variance. Different from the traditional robust filter based on the assumption that the process noises are uncorrelated with the measurement noises, the objective of the addressed robust filtering problem is to design a recursive filter such that, for packet losses and multiplicative noises, the state prediction and filtering covariance matrices have the optimized upper bounds in the case that there are correlated process and measurement noises. Two examples are used to illustrate the effectiveness of the proposed filter.

In recent years, the state estimation theory has received extensive attention in many fields of application, such as attitude estimation [

In literature mentioned above, however, only additive noises are considered for nonlinear systems. Actually, another important noise called multiplicative noise is often encountered in many engineering systems, such as attitude estimation systems and airborne synthetic aperture radar systems. It is coupled with the state and has an unknown noise variance, which results in a negative impact on the state estimation. Hence, the multiplicative noise is usually viewed as a model uncertainty. Currently, the nonlinear robust filtering problem with multiplicative noises has been much less researched. In [

In addition, the correlation of additive noises is one of the key factors to the filtering algorithm. Disturbed by the complicated environment, the additive noises often show the characteristic of correlation in the practical application. Unluckily, the design procedures of all the above filters for multiplicative noises or packet losses are based on the assumption that there are uncorrelated additive noises in the system. In fact, this assumption does not always come into existence, and the process noise might be correlated with the measurement noise in real applications. In [

Motivated by the above discussion, we present a robust recursive filter for a class of nonlinear systems with correlated additive noises, multiplicative noises, and packet losses. In this paper, multiplicative noises are assumed as zero mean Gaussian white noises and the packet losses are modeled as independent Bernoulli random variables. Based on the structure of the extended Kalman filter with correlated noises, the proposed filter designs an optimal upper bound of the prediction error and the filtering error covariance matrices, respectively. The main contributions of the paper are as follows.

Consider a general class of discrete time-varying systems with multiplicative noises, correlated additive noises, and packet losses:

The deterministic nonlinear functions

Because of existing correlated additive noises, for system (

State prediction:

State correction:

The aim of the paper is to design a recursive filter for the structures (

The addressed filtering problem is that the designed filter parameters

In engineering applications, multiplicative noises constantly existing in the systems depend on the real state value, which results in the unknown noise variance. As discussed in [

Denote the two-step prediction error as

The nonlinear functions

According to (

Substituting (

The one-step prediction error covariance can be obtained as

Denote the estimation error as

From (

Subsequently, in the light of (

Since there are the high-order errors, the matrices

To develop the robust recursive filter, the following lemmas are given.

Let

Given matrices

For

According to these lemmas, the following theorem is given to obtain the main results of the robust recursive filter.

Consider the covariance matrices of the one-step prediction errorand the filtering error in (

According to (

Assume that

From (

According to the literature [

Substituting (

Furthermore, inserting (

Similar to (

Substituting (

From (

Assume that there exist

According to Lemma

Combining (

To minimize the upper bounds, constructingthe optimized prediction gain

Considering (

For the sake of clarity, the robust recursive filter is summarized as follows.

Given

The filtering gain

Repeat Step

The robust recursive filter problem is removed by using Theorem

To show the effectiveness of the proposed robust recursive filter (RRF), it is compared with the finite-horizon extended Kalman filter (FEKF) in the literature [

The discretized maneuvering target tracking example in [

The initial state and covariance are set as

To evaluate the performance of the proposed robust recursive filter, the mean square error (MSE) is employed. And it can be expressed as

Simulation results are shown in Figures

The trajectory of the actual state

The trajectory of the actual state

MSE of the estimated state

MSE of the estimated state

According to the literature [

The simulation conditions are set as follows: the gyro sampling interval is

The simulated results are shown in Figures

The quaternion estimation errors of the proposed filter.

RMSE of attitude angles in the proposed filter.

Due to the fact that existing robust filtering algorithms are difficult in dealing with correlated additive noises, a robust recursive filter is developed in this paper for nonlinear systems with consideration of correlated additive noises, multiplicative noises, and packet losses. The proposed algorithm is designed to minimize the upper bound on the prediction covariance and the filtering covariance. Simulated results demonstrate that the proposed filter provides effective performance for controlling correlated additive noises, multiplicative noises, and packet losses.

The authors declare that there is no conflict of interests regarding the publication of this paper.

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