Kalman filtering with finite-step autocorrelated measurement noise☆
Introduction
State estimation problem for discrete-time linear systems has been extensively studied due to its importance in various fields, including communications, industrial electronics, speech recognition, information fusion, etc. The Kalman filter [1] is the most famous method for state estimation of discrete-time linear systems, which describes a recursive solution to the optimal estimate of system state. The Kalman filter has numerous practical applications in many areas, and is also used as a mathematical tool to solve many estimation and control problems [2], [3], [4], [5], [6], [7]. However, to ensure the optimality, the Kalman filter requires the condition that both the process noise and the measurement noise are white. When the above condition is unsatisfied, the Kalman filter becomes suboptimal, and often yields unacceptable performance in applications [8].
To relax the constraint condition mentioned above, the state estimation problem for dynamic systems with finite-step autocorrelated noise has attracted increasing interests [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] because the performance can be significantly improved compared with the case of white noise in many practical applications in a variety of domains such as target tracking, communication network, signal processing, etc. [14], [17], [18]. In [9], a discrete-time linear system is considered in which the process and measurement noises are one-step correlated, and the optimal Kalman filter, smoother, and predictor were proposed. In [10], the filtering problem with multi-step correlated noise was reported where the covariance functions of correlated noises are known. The authors considered a linear time-variant dynamic system in which the process and measurement noises are multi-step correlated, and a Kalman type recursive state estimator was presented. Almost at the same time, a Kalman type recursive filter for discrete-time linear systems with finite-step autocorrelated process noise was designed in [11]. In [12], the Kalman filtering problem is studied for discrete-time linear systems with (i) finite-step autocorrelated process noises, (ii) finite-step autocorrelated measurement noises, as well as (iii) finite-step correlated process and measurement noises, and recursive algorithms of the globally optimal state estimate were developed. Until now, for the state estimation problem with finite-step correlated noise, many research findings involving various systems have been reported. In [13], [14], [15], the optimal linear estimation problem with finite-step correlated noise was investigated where the systems considered in the literatures are different. In [16], a recursive filter for discrete time-varying systems with finite-step correlated noises, random parameter matrices and multiple fading measurements was presented. In [17], [18], robust Kalman filtering problem with finite-step correlated noises was dealt with where an uncertain system with multiple packet dropouts was discussed in [17] and an uncertain system with stochastic nonlinearities and autocorrelated missing measurements was considered in [18]. However, the convergence problem has not been studied in all the aforementioned references about finite-step correlated noises. Also, to the best of our knowledge, the Kalman filtering problem with finite-step autocorrelated measurement noise described by a linear function of several mutually uncorrelated random vectors has not been studied including the optimal Kalman filter design and convergence analysis of the corresponding Kalman filter, which motivates our current work.
In this paper, the Kalman filtering problem of discrete-time linear systems with finite-step autocorrelated measurement noise is considered where the measurement noise is modeled as a linear function of several mutually uncorrelated random vectors. We first present an optimal Kalman filter for the system under consideration using state augment approach. Then, we analyze the asymptotic convergence properties of the presented Kalman filter, which is the core content of this paper. In state estimation, convergence analysis is a fundamental issue, which has received a great deal of attention (see, for example, [19], [20], [21], [22] and the references therein). In [19], by introducing a Lyapunov function, convergence conditions for the Kalman filter were established. The convergence problem for Kalman filtering with stochastic parameters was studied in [20], and the convergence problem for Kalman filtering with intermittent measurements was investigated in [21]. More recently, the convergence conditions for the optimal linear estimator of discrete-time linear systems with multiplicative and time-correlated additive measurement noises have been established in [22]. More results about convergence analysis can be found in [23], [24], [25], [26], [27], [28]. However, the results presented in [19], [20], [21], [22], [23], [24], [25], [26], [27], [28] do not consider the case of finite-step correlated noise. The convergence conditions of the optimal Kalman filter presented in this paper are established based on equivalently studying the convergence of the prediction state error covariance of an augmented system. However, in the matrix difference equation of the prediction augmented-state error covariance (PASEC), there is neither measurement noise covariance nor positive definite constant matrix with the same dimension as the PASEC where the PASEC is defined in (15) and the corresponding matrix difference equation is given in (16). Hence, the existing results are no longer suitable to study the convergence of the PASEC. To study the convergence of the PASEC, which is a critical point toward establishing the convergence conditions of the optimal Kalman filter with finite-step autocorrelated measurement noise, we prove some properties about the PASEC. More precisely, the following properties about the PASEC are obtained:
- (1)
Using an auxiliary measurement and the property of linear minimum mean-square error estimate, we prove that the PASEC is bounded if is detectable where is the state transition matrix and is the measurement matrix of the system under consideration.
- (2)
We prove that if is detectable, the PASEC is convergent when its initial value is . Here, different from the existing results, we cannot obtain the PASEC is convergent with zero initial value because the inverse of the innovation covariance does not exist when the PASEC is equal to zero.
- (3)
We show that, under proper conditions, there exists a stable matrix in the matrix difference equation of the PASEC when the time step approaches infinity.
It is worth mentioning that, in the process of proving these results, the most important technique developed in this paper is that we prove that there exists a non-zero vector under appropriate conditions where the non-zero vector is a subvector of an eigenvector (See Remark 4 for details). It is also addressed that the convergence conditions of the PASEC can be relaxed if the matrices in the measurement noise model are equal.
The rest of this paper is organized as follows. In Section 2, the problem under consideration is formulated and the optimal Kalman filter with finite-step autocorrelated measurement noise is presented. The convergence conditions of the optimal Kalman filter with finite-step autocorrelated measurement noise are established in Section 3. A discussion is provided in Section 4. A numerical example is given in Section 5 to confirm the validity of the presented results. Concluding remarks are drawn in Section 6.
Notation The -dimensional real Euclidean space is denoted by . The positive semidefinite (definite) matrix is denoted by
(). For symmetric matrices and , we say that if . The transpose and inverse of a matrix are denoted by and , respectively. We use to stand for the expectation operation, and utilize to represent the identity matrix of dimension . The covariance operation is denoted by , and the cross-covariance operation is denoted by . For two random vectors and , is used to represent the linear minimum mean square error estimate of given .
Section snippets
Problem formulations and optimal filter
Consider the following dynamical system where is the unknown state; is the process noise; is the measurement; is the measurement noise; is a random vector; in ; , and are matrices of appropriate dimensions; and the initial state is a random vector with mean and covariance matrix .
Throughout the paper, we introduce the following two assumptions.
- (1)
and are zero-mean white noise sequences with
Convergence analysis
In this section, we establish the convergence conditions for the Kalman filter with finite-step autocorrelated measurement noise.
In order to establish the convergence condition, we first propose the following two theorems.
Theorem 1 All the following statements about are true: (a) can be given by where denotes the symmetric part of a matrix, and
Discussion
In this section, we discuss the different definition for new state vector, and address that the convergence is hard to be proved when the new state vector defined in any other way does not include all the elements in .
We define a new state vector that does not contain all the elements in . Without loss of generality, let , namely, the new state vector does not contain . Then, it follows from (1)–(3) that
Numerical example
In this section, the effectiveness of the presented results in this paper is verified by a numerical example.
Consider the system described by (1), (2) where , , and where , and are mutually uncorrelated zero-mean random variables with the same variance , and are uncorrelated with , and . In this example, we take , , , . We easily see that the measurement
Conclusion
Based on equivalently studying the convergence of a PASEC, the convergent conditions for the optimal Kalman filter of discrete-time linear systems influenced by finite-step autocorrelated measurement noise have been established. The difficulty in studying the convergence of the PASEC is that the matrix difference equation of the PASEC contains neither the measurement noise covariance nor positive definite constant matrix with the same dimension as the PASEC. To conquer this difficulty, several
Acknowledgments
The authors would like to thank the Principal Editor, and the anonymous reviewers for their constructive comments and suggestions, which helped to improve the quality of the manuscript.
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This work was partially supported by the National Nature Science Foundation of China (62073125, 61773131), the Australian Research Council (DP170102644), the Key-area Research and Development Program of Guangdong Province, China (2020B0909020001), and the Dongguan Innovative Research Team Program (2020607202006).