Recursive Estimation for Dynamical Systems with Different Delay Rates Sensor Network and Autocorrelated Process Noises

The recursive estimation problem is studied for a class of uncertain dynamical systems with different delay rates sensor network and autocorrelated process noises. The process noises are assumed to be autocorrelated across time and the autocorrelation property is described by the covariances between different time instants. The system model under consideration is subject to multiplicative noises or stochastic uncertainties. The sensor delay phenomenon occurs in a random way and each sensor in the sensor network has an individual delay rate which is characterized by a binary switching sequence obeying a conditional probability distribution. By using the orthogonal projection theorem and an innovation analysis approach, the desired recursive robust estimators including recursive robust filter, predictor, and smoother are obtained. Simulation results are provided to demonstrate the effectiveness of the proposed approaches.


Introduction
The Kalman filter is very popular for estimating the system states of a class of linear systems which are characterized by state-space models.Since its inception in the early 1960s, it has played an important role in the research fields of target tracking, communication, control engineering, and signal processing.An implied assumption of traditional Kalman filter is that the system model and measurement model are exactly known.Unfortunately, this assumption does not always hold due to the constrained knowledge and the variation of the system and environment.When the system model and measurement model under consideration are not exactly known, the performance of traditional Kalman filter can deteriorate appreciably [1][2][3].Therefore, in the past decades, the recursive robust state-space estimation problem has become a hot topic of the estimation theory.There are many different ways to describe the model uncertainty.Multiplicative noise is an important stochastic uncertainty which is commonly encountered in aerospace systems [4], communication systems [5], and image processing systems [6,7].Different from the additive noise, the second-order statistics of the multiplicative noise are usually unknown and this property leads to more difficulties in the research.Up to now, there are several solutions to treat with the estimation and control problems for systems with multiplicative noises, including linear matrix inequality approach [8], Riccati equation approach [9,10], and game-theoretic method [11], to name just a few.
In traditional state estimation theory, the process noises are usually assumed to be Gaussian and uncorrelated with each other.However, this assumption is not always realistic, correlated noises are commonly encountered in practical applications.For example, in a target tracking system, the system state is usually consecutive (i.e., the system state at time k is correlated with its neighbors); thus, when the process noises are dependent on the system state, the process noises are usually autocorrelated across time.So far, there have been several approaches to deal with the estimation problem for systems with correlated noises [12][13][14][15][16].The optimal Kalman filtering fusion problem for dynamic systems with crosscorrelated measurement noises has been dealt with in [13][14][15].In [16], the state estimation for discrete-time systems with cross-correlated noises has been treated based on an optimal weighted matrix sequence, where the process noises and measurement noises are cross correlated.It should be pointed out that the estimators mentioned previously are only suited for the correlated noises at the same time instant.In [17,18], a Kalman-type recursive filter has been proposed for dynamic systems with finite-step autocorrelated process noises, where the autocorrelation property is described by the covariances between different time instants.The filtering problem with finite-step cross-correlated process noises and measurement noises has been investigated in [19].In [20], the optimal robust nonfragile Kalman-type recursive filter has been designed for a class of uncertain systems with finite-step autocorrelated noises.
On another research frontier, with the development of network technologies, the sensor network has attracted increasing attention from many researchers in different fields due to their wide scope applications in surveillance, environment monitoring, information collection, wireless networks, robotics, and so on.In the sensor network, the network-induced time-delay or/and packet dropouts cannot be avoided due to limited single-sensor energy and communication capability and these have brought us new challenges in the design of the desired state estimators.The binary switching sequence is a popular way to describe the networkinduced time-delay or/and packet dropouts since the timedelay or/and packet dropouts in the sensor network are inherently random [21][22][23][24].The least-mean-square filtering problem for one-step random sampling delay has been studied in [25,26].Unfortunately, the filters designed in [25,26] are suboptimal since a colored noise due to augmentation has been treated as a white noise.The filtering problem for systems with random measurement delays and multiple packet dropouts has also been discussed in [24].In [27], the problem of robust filtering for uncertain systems with missing measurements and finite-step correlated process noises has been investigated.It should be noted that, in all the aforementioned literature, sensors involved in the sensor network have the same delay characteristics.Recently, Hounkpevi and Yaz [28,29] present minimum variance state estimators for multiple sensors with different delay or failure rates.The leastsquare filtering problem for systems with one-or two-step random delay has been studied in [30], where the algorithms are derived without requiring the knowledge of the state space model but only the means and covariance functions of the processes involved in the observation equations.The optimal unbiased filtering problem for uncertain systems with different delay rates sensor network and autocorrelated process noises has also been discussed in [31].However, the estimator obtained in [31] is nonrecursive and a colored noise due to augmentation has been treated as white noise.Up to now, to the best of the authors' knowledge, the recursive robust estimation problem has not yet been addressed for uncertain systems with different delay rates sensor network and autocorrelated noises, and this situation motivates our current study.
Motivated by the above analysis, in this paper, we aim to investigate the recursive robust estimation problem for uncertain systems with different delay rates sensor network and autocorrelated noises.The system model and measurement model under consideration are both subject to stochastic uncertainties or multiplicative noises.Different sensors in the sensor network have different delay rates and different delay rates are described by different binary switching sequences.The process noises are assumed to be one-step autocorrelated across time and the autocorrelation property is described by the covariances between different time instants.Based on an innovation analysis approach (IAA) and the orthogonal projection theorem (OPT), recursive robust estimators including filter, predictor, and smoother are obtained.This paper extends the results in [31], in two directions: (1) the autocorrelated measurement noise due to augmentation leads to more difficulties in the design of the recursive robust estimators; however, in [31], the measurement noise is treated as a white noise; and (2) the filter obtained in [31] is actually a nonrecursive filter; however, in our current work, we do not only derive a recursive robust filter, but also derive a recursive robust predictor and a recursive robust smoother.Also, the current paper differs from [28,30] for the model uncertainties considered and for the autocorrelated process noises considered to derive the desired recursive robust estimators.
The remainder of the paper is organized as follows.In Section 2, the recursive robust estimation problem is formulated for a class of uncertain systems with autocorrelated noises and different delay rates sensor network.The recursive robust estimators including filter, predictor, and smoother are derived in Section 3. In Section 4, a simulation example is provided to illustrate the usefulness of the theory developed in this paper.We end the paper with some concluding remarks in Section 5.
Notation 1.The notation used in the paper is fairly standard.The superscript "" stands for matrix transposition, the notation R  denotes the -dimensional Euclidean space, the notation R × is the set of all real matrices of dimension  × , and  and 0 represent the identity matrix and zero matrix, respectively.The notation  > 0 means that  is real symmetric and positive definite, and diag(⋅ ⋅ ⋅ ) stands for block-diagonal matrix.The notation  − is the Kronecker delta function, which is equal to unity for  =  and zero for  ̸ = .In addition, E{} means mathematical expectation of  and Prob{⋅} represents the occurrence probability of the event "⋅".Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

Problem Formulation
Consider the following system model and measurement model: where x  ∈ R  is the state to be estimated, the vector y   ∈ R is the actual output vector of the th sensor, the vector    ∈ R is the measured output vector of the th sensor, the vector   ∈ R  is the process noise, the vectors   ∈ R and    ∈ R are multiplicative noises, the vector V   ∈ R is the measurement noise of the th sensor, the matrices Ȃ , Ȃ, , B  , C   , and C  , are known real time-varying matrices of appropriate dimensions, and the variable    ∈ R is a mutually uncorrelated binary switching sequence (and uncorrelated with other random variables) taking values on 1 and 0 with ( Remark 1.The measurement model ( 1) is a popular way to model the random sensor delay.It can be seen that if ; that is to say, the measurement of the th sensor is up to date.
The noise signals   , V   , and    are all zero-mean Gaussian white noises.They, together with the initial state x 0 and the process noise   , have the following statistical properties: where where a compact representation of (1) can be expressed as follows: where   is the measurement noise of the newly obtained auxiliary system ( 6) and (7).It follows readily from (4) that   has the statistic properties as follows: Remark 2. It can be seen from ( 3) and ( 8) that the process noise   and the measurement noise   are both one-step autocorrelated across time.For example, the process noise at time  is correlated with the process noises at times  − 1 and  + 1 with covariances  ,−1 and  ,+1 , respectively.
The measurement noise at time  is correlated with the measurement noises at times  − 1 and  + 1 with covariances  ,−1 and  ,+1 , respectively.
Remark 3. Observe that the system model and measurement model of system ( 6) and ( 7) are both subject to stochastic uncertainties and   ,  , , and   involve the stochastic variable    .Thus, system ( 6) and ( 7) is actually a stochastic uncertain system.On the other hand, the process noise   and the measurement noise   are both one-step autocorrelated across time.Therefore, the traditional recursive robust estimation approaches may not satisfy the performance requirements here.Remark 4. A seemingly natural way of handling the autocorrelated noises is the augmentation of the system states.However, such a state augmentation approach gives rise to significant increase in the system dimension, which would inevitably lead to computational burden.In addition, in the state augmentation method, the noises are treated as components of the auxiliary system state, generally, it is difficult for an estimator to track noise signals, and this will affect the estimation of other components of the auxiliary system state.Without resorting to state augmentation, in our current work, we treat system ( 6) and (7) directly by using an IAA and the OPT.

The Main Results
For convenience of later development, let us introduce the following lemmas, which are very useful in establishing our main results.Lemma 5.For stochastic matrices   ,   ,   , and  , , one has the following results: Proof.Lemma 5 follows directly from (2), (4), and ( 5) and the fact that   is zero mean.Lemma 6.For system state   and the process noise   , one has the following result: Proof.Lemma 6 follows directly from (3) and ( 6).

𝑘+1
x   }, one has from (4) and Lemma 7 the following: Lemma 8 (see [32]).If  ∈ R × is a real matrix and  = diag( 1 , . . .,   ) is a diagonal stochastic matrix, then where ⊗ is the Hadamard product (this product is defined as

Recursive Robust Filter
Theorem 9.For the addressed system (6) and (7), one has the following recursive robust filter: where   is the innovation with covariance Π  , the matrix ϝ , is the covariance between  Proof.Please see Appendix A.
Remark 10.In the traditional recursive estimation problem, the innovation is calculated as   =   −   x|−1 .However, due to possible sensor delay which occurs in a random way, this is not true for the problem at hand; thus, we have to recalculate the innovation as in (16).Furthermore, it can be seen that the second term on the right-hand side of ( 14), the last four terms on the right-hand side of ( 15), the second term of the right-hand side of ( 17), and the last ten terms on the right-hand side of ( 18) are caused by the random delays, the stochastic uncertainties, and autocorrelated noises.These terms constitute the main differences between our work and the traditional Kalman filter.
Next, we will derive the recursive robust predictor and recursive robust smoother based on Theorem 9.

Recursive Robust Predictor
Theorem 11.For the addressed system (6) and (7), one has the following -step ( ≥ 2) recursive robust predictor: where the initial values x+1| and  +1| can be calculated as in Theorem 9.
Proof.Please see Appendix B.
Proof.Please see Appendix C.

An Illustrative Example
Consider the following uncertain system with different delay rates sensor network and autocorrelated process noises: where x  ∈ R 2 is the state to be estimated.The vectors   ∈ R,   ∈ R,    ∈ R, and V   ∈ R,  = 1, 2 are zero-mean Gaussian white noises with covariances 0.5, 1, 1, and 1, respectively.Without loss of generality, the process noise   is chosen to be as defined in (23).
In the simulation, the initial value x 0 has mean E{ x where  is the number of the samples.Similarly, MSE2 denotes the mean-square error for estimation of the second component of x  ; that is, 1 and 2, we can see that the smoother has the best performance and the predictor has the worst performance.This is due to the fact that smoother uses the most measurement information and the predictor uses the least measurement information.
From Figures 3 and 4, we can see that the filter developed in this work has better performance than the filter of Zeng et al. [31].This is due to the fact that the autocorrelated measurement noise   was treated as zero-mean Gaussian white noise in the filter of Zeng et al. [31].

Conclusions
In this paper, we have studied the recursive robust estimation problem for a class of uncertain systems with autocorrelated process noises and different delay rates sensor network.The system model and measurement model are both subject to stochastic uncertainties.The process noises are one-step autocorrelated across time.Each sensor in the sensor network has a different delay rate and the delay rate has been described by an individual binary switching sequence obeying a conditional probability distributed.Based on an IAA and the OPT, recursive robust estimators including filter, predictor, and smoother have been obtained.Simulation results have indicated that the smoother has the best performance and the predictor has the worst performance, and the filter obtained in this work has better performance than the filter of Zeng et al. [31].

A. The Proof of Theorem 9
Proof.Using the OPT, the one-step measurement prediction ŷ|−1 can be calculated as follows: Taking into account the fact that   is one-step autocorrelated, we have from ( 4), (8), and ( 9) the following: Substituting (A.2) into (A.1),we have Therefore, the innovation   can be calculated as follows: Again, according to the OPT, the state prediction x|−1 can be obtained as follows: Taking (3) into consideration, the expectation E{ −1    } can be calculated as follows: (A.6) Substituting (A.6) into (A.5),we have Therefore, the one-step prediction error x|−1 can be calculated as follows: where x−1|−1 is the filter error at time instant  − 1. Taking into account the fact that   is one-step autocorrelated across time and   is uncorrelated with other signals, the onestep prediction error covariance  |−1 can be calculated as follows: where the expectation E{ −1   −1 } can be calculated as in (A.6) and expectation E{ x−1|−1   −1 } can be obtained as follows: where the third equality in (A.10) holds since   is onestep autocorrelated across time.Substituting (A.10) into (A.9)yields (15).
Noting the fact that x|−1 is orthogonal to x|−1 , we have from ( 9) and (A.4) the following: It implies from ( 9), (A.4), and Lemmas 5 and 8 that the expectation Π  can be obtained as follows: where the remaining expectations can be obtained as follows: where Lemmas 5-8 have been used.Substituting (A.13) into (A.12)yields to (18).Again, by using the OPT, the state estimation x| can be calculated as follows: Therefore, the estimation error x| can be obtained as follows: From (A.15), the estimation error covariance  | can be calculated as follows: where the remaining expectation E{ x|−1    } can be calculated as follows: (A.17) Substituting (A.17) into (A.16),we have which completes the proof of Theorem 9.

B. The Proof of Theorem 11
Proof.Taking into account the fact that the process noise   is one-step autocorrelated across time, the -step prediction x+| can be calculated as follows: Therefore, the -step prediction error x+| can be obtained as follows: Thus, the -step prediction error covariance  +−1| can be calculated as follows:

C. The Proof of Theorem 12
Proof.According to the OPT, the -step fixed-lag smoother x|+ can be calculated as follows:  Similarly, when  ≥ 2, the expectation Ψ + can be calculated as follows: and   , the vectors x| and x|−1 are the filter and one-step predictor, and the matrices  | and  |−1 are the filter error covariance and one-step prediction error covariance.The initial values are x0|0 = [ x  0 0]  ,  0|0 = diag( P 0 , 0), and  1 =  1 −  1 x1|0 .