Distributed extended Kalman filtering for state-saturated nonlinear systems subject to randomly occurring cyberattacks with uncertain probabilities

In this paper, the extended Kalman filtering scheme in a distributed manner is presented for state-saturated nonlinear systems (SSNSs), where the randomly occurring cyberattacks (ROCAs) with uncertain occurring probabilities (UOPs) are taken into account. In particular, a novel cyberattack model is constructed by the consideration of false data-injection attacks (FDIAs) and denial-of-service attacks (DoSAs) simultaneously. The ROCAs are described by a series of Bernoulli distributed stochastic variables, where the so-called UOPs are considered and described by the nominal mathematical expectations and error bounds. The major effort is to develop a novel DEKF strategy for SSNSs with consideration of state delay and ROCAs with UOPs. In what follows, an upper bound with respect to the filtering error covariance is derived and minimized by selecting the suitable filter parameter. Besides, the concrete expression of the filter parameter is formed by solving matrix difference equations (MDEs). Meanwhile, a sufficient condition under certain constraints is proposed to testify the boundedness regarding the given upper bound. Finally, we use the experiments and corresponding comparisons to verify the feasibility of the designed extended Kalman filtering approach in a distributed way.

distributed filtering (DF) whose core idea is to estimate the target's state on account of the local information performs better to some extent. To name a few works in this area, the H ∞ filtering problem in a distributed setting has been tackled in [9] for Markovian jump systems by solving some matrix inequalities. In addition, the DF schemes with fading measurements and redundant channels have been presented in [10] and [11], respectively. And in [12], the DF issue has been addressed for nonlinear systems with switching topology and multiplicative link noises, where the filter parameters have been obtained by means of handling convex optimization problems.
It should be noted that the state saturation occurs frequently in the practical engineering, which can reflect the limitation of systems' state or the protection of the machinery equipment. Generally speaking, the state variables of the physical target are constrained in a bounded set [13,14]. If the state saturation is not aptly addressed, the desired performance of dynamic systems cannot be always ensured. So far, many efforts have been devoted to the studies of synthesis and analysis for state saturated systems [15,16]. For instance, the state-saturated filtering issue has been solved in [17] via certain mathematical computations. In [18], a set-membership filtering algorithm has been studied for statesaturated systems in light of dealing with some linear matrix inequalities. The DF strategy has been proposed in [19] for state-saturated systems under fading measurements and quantization in terms of the feasibility of certain matrix difference equations (MDEs). Moreover, the nonlinearities are ubiquitous in actual world, which should be tackled in a proper way [20][21][22][23][24]. For instance, a new filtering algorithm by considering the event-based communication criterion has been proposed in [25] for nonlinear systems. Nevertheless, it should be noted that few approaches are available to tackle the DF problem for timevarying saturated nonlinear systems, which is one motivation of this paper.
Accompanied with the broad utilizations of open but unprotected shared networks, the sensor networks are extremely vulnerable to attacks, see, e.g., [26,27]. In fact, the fundamental goal of these attacks is to break data information through the shared communication network so as to drive the physical plant to become oscillating or even instable. In [28], the recursive filtering method has been proposed for state-saturated systems under deception attacks, where the attacks have been characterized by a series of Bernoulli stochastic variables with a known occurrence probability. It is worthwhile to notice that the behaviors of attacks might be irregularly intercepted by the protection equipment, namely the attacks generally occur in a random manner. In [29], the extended Kalman filtering issue has been addressed for stochastic systems under randomly occurring cyberattacks (ROCAs). Unfortunately, it should be pointed out that the occurrence probability of the attacks is commonly supposed to be deterministic among the existing results, where such a situation cannot be guaranteed always and restricts its applications. Recently, the recursive filtering method under the variance-constraint criterion has been developed in [30] to attenuate the impacts caused by multiple missing measurements with uncertain missing probabilities. However, there are few studies about the cyberattacks with uncertain occurring probabilities (UOPs) and the state saturation in the existing literature, which deserves further investigations in order to develop the corresponding filtering algorithms with potential robustness. Moreover, time delay unavoidably exists in practical systems, see, e.g., [31][32][33]. Accordingly, the performance of the system might be destroyed if the time delay cannot be effectively handled [34,35]. In [36], the variance-constrained filtering method has been established for stochastic genetic regulatory networks subject to state delay. Nevertheless, it should be noticed that few methods can be found that handle the delayed time-varying stochastic systems with state saturation, not to mention the presence of ROCAs with UOPs. Hence, the goal of this paper is to fill such a gap by designing a novel distributed extended Kalman filtering (DEKF) strategy for the addressed problem.
Inspired by the aforementioned discussions, we attempt to provide a DEKF algorithm for state-saturated nonlinear systems (SSNSs) subject to state delay and ROCAs with UOPs. The phenomena of cyberattacks are adequately considered and characterized by a sequence of nominal means, error upper bounds, as well as the bound of the attack signal.
The key motivations/challenges can be mentioned from three aspects: (i) How to comprehensively take the state saturation, time-delay and ROCAs with UOPs into account when conducting the design problem of DEKF? (ii) How to design the proper filter gain with an easy-to-implement form to realize the expected DEKF performance requirement? (iii) How to evaluate the estimation method performance and ensure the boundedness requirement regarding the upper bound matrix of the filtering error covariance. In order to better answer the above questions, an effective optimal DEKF method is established, where the accurate expression of the filter gain is presented by solving two MDEs. The main contributions can be summarized as follows: (1) the constructed dynamic system model is fairly comprehensive and contains nonlinearity disturbance, state saturation, time-delay, and ROCAs with UOPs, thus better reflecting the practical engineering situations; (2) a matrix simplification technology is exploited to cope with the design difficulty of filter gain caused by the sparseness of the fixed topology; (3) a sufficient condition is devised to guarantee the boundedness of the proposed upper bound matrix; and (4) the newly proposed DEKF scheme can be applied for online implementations. Finally, the usefulness of the main results is certified by a few simulations. Now, we outline the rest of the paper. The DEKF problem is formulated in Sect. 2 for state-saturated systems under a given network topology. In Sect. 3, the upper bound for the state estimation error covariance is obtained firstly and then the proper filter parameter is chosen to minimize such an upper bound. Moreover, the boundedness is analyzed and the proof is provided under an assumption. Two numerical simulations are utilized in Sect. 4 to demonstrate the usefulness of our main theoretical results. Finally, we summarize the conclusions in Sect. 5.

Notations
The symbols utilized throughout the paper are standard. The n-dimensional Euclidean space is denoted by R n . The symbol C ≥ D, where C and D are both symmetric matrices, implies that C -D is positive semi-definite. For a matrix W , W , ρ(W ), W T , σ max (W ) and tr{W } stand for the spectral norm of matrix W , the spectral radius of matrix W , the transpose of matrix W , the largest singular value of matrix W , and the trace of matrix W , respectively; col{W 1 , W 2 , . . . , W N } represents [W T 1 W T 2 · · · W T N ] T ; • denotes the Hadamard product operator, with [P •Q] ij = P ij Q ij ; diag{·} is a block-diagonal matrix; I represents an identity matrix with proper dimension. Furthermore, matrices are assumed to have suitable dimensions for the mathematical operations if they are not explicitly pointed out.

Problem formulation
The DEKF problem with attacks is drawn in Fig. 1. From the diagram, it is easy to see that the outputs are measured by some sensor nodes. Meanwhile, the information can be The topological structure of the WSN is modeled by a graph G = (U , E, A), where the set of nodes, the set of edges, as well as the weighted adjacency matrix are denoted by U = {1, 2, . . . , N}, E ⊆ U × U , and A = [α ij ] N×N with α ij > 0, respectively. An edge of G is given by (i, j). Also α ij > 0 ⇐⇒ (i, j) ∈ E which shows that there is an edge from j to i. Besides, N i = {j ∈ U : α ij > 0} represents the set of adjacent nodes for node i, including itself.
In this paper, the considered SSNSs are described by the following dynamical equations: where x k ∈ R n depicts the state vector that cannot be directly measured, the initial value of x k has meanx 0 and covariance P 0 , and y i,k ∈ R m y (i = 1, 2, . . . , N ) is the output signal collected by sensors. The random parameters ω k ∈ R n w and ν i,k ∈ R m y are the process and measurement noises with zero mean and known covariances, R k > 0 and Q i,k > 0, respectively. The known positive integer d depicts the time delay. The system matrices A k and B k are known with proper dimensions. The nonlinearities f (·) and h i (·) are both supposed to be continuously differentiable. The nonlinearity h i (x k ) : R n → R m y satisfies the following condition: where δ 1 and δ 2 are nonnegative scalars. The saturation function μ(·) : R n → R n can be defined as where ς i,max stands for the ith element of the saturation level ς max , ς i denotes the real value, and μ i (ς i ) represents the value under the saturation constraint. The real measured output of the ith node, which might be subject to ROCAs, is modeled by where y i,k ∈ R m y stands for the output signal with potential attacks; ξ i,k ∈ R m y is the attack signal transmitted by malicious attackers satisfying ξ T i,k ξ i,k ≤ξ k . The mutually independent Bernoulli distributed variables γ i,k and λ i,k are utilized to describe the switch behavior of different cyberattacks, where the statistical features are given as: and ρ i,k being known scalars. Furthermore, we assume that γ i,k , λ i,k , ω k , and ν i,k are mutually uncorrelated for any i and k.
Remark 1 In accordance with (6), the different values of λ i,k and γ i,k can describe different cases. To be more specific, it is easy to see that the measurement output model in (6) mainly includes the following three cases: (1) if γ i,k = 1, the measurement signals can be transmitted successfully, i.e., the behaviors of attacks cannot occur; (2) when γ i,k = 0 and λ i,k = 1, the systems encounter the false data-injection attacks (FDIAs); and (3) when γ i,k = 0 and λ i,k = 0, the systems suffer from the denial-of-service attacks (DoSAs).
Remark 2 On the one hand, it is worth noticing that the UOPs are characterized by (7) and (8), where γ i,k and λ i,k denote the UOPs subject to the certain probability error bounds with a hope to reflect the practical engineering environments. On the other hand, the fuzzy uncertainty method can be considered. In fact, the choice of membership functions is crucial for the realization of purpose, which is commonly dependent on the experience for different systems. Compared with the modeling of fuzzy uncertainty, we adopt another method to model the UOPs described by the nominal means and error upper bounds. Moreover, the new modeling way contains the existing ROCAs as a special one when setting λ i,k = 0 and γ i,k = 0.
The following distributed filter is designed for addressed SSNSs: wherex i,k+1|k is the one-step prediction,x i,k+1|k+1 is the filter. In the sequel, the matrix G ij,k+1 is the desirable time-varying filter parameter. Now, we devote to state the major objectives of this paper. Firstly, a new DEKF method is proposed for SSNSs under ROCAs with UOPs. Secondly, the boundedness is analyzed for the newly designed DEKF scheme.
Before proceeding, we provide the following lemmas, which are helpful for the further theoretical derivations.

Lemma 2 ([19])
Let N = diag{n 1 , n 2 , . . . , n p } be a random diagonal matrix and M = [m ij ] p×p be a real-valued matrix. Then, it can shown that the following relationship holds:

DEKF algorithm design and error boundedness discussion
In this section, a DEKF algorithm will be given for SSNSs subject to state delay and RO-CAs with UOPs. Firstly, we analyze the nonlinearities f (x k-d ) and h j (x k+1 ) by employing the Taylor series expansion, where the high-order terms induced by the linearization process are approximately estimated. Secondly, a major effort is made on looking for a minimal upper bound matrix with respect to the state estimation error covariance, where the matrix simplification technique (MST) is adopted to obtain the proper filter parameter with an easy-to-implement manner. Finally, the performance evaluation issue is addressed, where a new sufficient criterion under certain constraint conditions is given in order to characterize the boundedness of the estimation error covariance.

Lemma 4
The prediction error covariance matrix P k+1|k can be shown to be where

Lemma 5
The filtering error covariance matrix P k+1|k+1 obeys P k+1|k+1 where Note that the uncertain terms exist in (18)- (19), and hence it is impossible to get their precise values. In the sequel, an alternative method is utilized, where a recursive method will be given to obtain the optimized upper bound matrix by designing a suitable filter gain.
Remark 3 In Theorem 1, the upper bound has been constructed based on the given G k+1 .
Subsequently, we aim to choose a suitable filter parameter G k+1 in order to optimize such an upper bound. However, it should be noted that the sensors only transmit information with their neighbors, which causes the matrix T i to be noninvertible. Hence, it induces some difficulties for the later filtering developments. Here, an MST is adopted to cope with this problem.
To proceed, some notations are introduced, which are necessary to the subsequent derivations: In what follows, we aim to provide the design approach of filter parameter by using the MST and optimize the given upper bound accordingly. (9) and (10). Then tr{Σ k+1|k+1 } is minimized by utilizing the following filter parameter:

Theorem 2 Consider the filter
Proof In light of (21) and the following equality then we obtain Subsequently, we aim to derive the expression form of the filter parameter G k+1 . In particular, take the partial derivative of tr{Σ k+1|k+1 } with respect to G k+1 . By setting the result be zero, the following equation can be obtained: On the basis of (49), we have Noting T i =L iL T i , we arrive at We know thatL T i has full row rank andL T i L k+1Li is invertible, thus it can be derived that According to (45) and (52), one has and then the filter parameter can be computed by: which ends the proof of Theorem 2.

Error boundedness discussion
In the subsection, the boundedness analysis regarding the trace of the upper bound for filtering error covariance is conducted. Before further developments, the following assumption is essential to be introduced.

Theorem 3 For the SSNSs
Proof Based on (45) and (48), one gets For the first term of (55), one has Substituting (56) into (55) leads to In view of (20), we have In particular, by considering the properties of the trace and Assumption 1, one gets and then tr{Σ k+1|k } ≤ 4nN 2 + Nϑ 2 tr{R}.
On the other hand, it is not difficult to check that and further, one has Next, in view of the matrix operations, the following inequality can be written: For the second term of (57), combining (60) with (63), we arrive at Finally, it can be observed that the second term of (57) is bounded and the first term is nonpositive. Thereby, it can be shown that tr{Σ k+1|k+1 } is bounded, which finishes the proof of Theorem 3.
Remark 4 So far, the DEKF issue has been handled for SSNSs subject to ROCAs with UOPs. In particular, a novel DEKF algorithm has been established to analyze the impact of ROCAs with UOPs, and a recursive filtering scheme has been given to choose the filter parameter matrix. Furthermore, the boundedness of the estimation performance has been discussed, which further clarifies the performance analysis of the proposed filtering method from the theoretical viewpoint. It should be noticed that the proposed filtering method has the time-varying distributed manner, which has a potential advantage in the online implementation/application. Moreover, it is worthwhile to mention that some interesting and effective methods have been given in [37][38][39][40][41][42] for fractional systems, which motivate the further investigation on the DF problem for fractional nonlinear systems under ROCAs.

Two illustrative simulations
In this section, we will present two examples and discuss the effectiveness of the proposed DEKF scheme. In particular, comparative experiments for different cases are provided.
Example 1 Consider the SSNSs (1) with state delay d = 1 and real measurement output (6). The related system parameters are given as: The nonlinear function f (x k-d ) with state delay is expressed as The nonlinear functions h i (x k ) are given by:  21) and (46), the optimal upper bound can be obtained at each sampling time.
In order to evaluate the algorithm accuracy of the new DEKF strategy, the mean square error (MSE) is utilized, which can be calculated by Here, M represents the experimental number of runs. Accordingly, the main results are proposed in Figs. 3-7 with M = 200. In particular, Figs. 3-4 show the trajectories of the estimates and the actual state x k . We discover that the MSE is indeed below the upper bound in Fig. 5. Figure 6 exhibits the successful time of cyberattacks in detail. Figure 7 describes the curves of the upper bound affected by the different occurrence probabilities of cyberattacks, and it can be easily seen that the upper bound will increase when the occurring probabilities of cyberattacks increase. To sum up, those above simulation results show that the designed filtering strategy possesses good performance for addressed SSNSs under ROCAs with UOPs.
Next, we consider a 3-dimensional system in order to propose a comparative experiment and further testify the validity of the proposed DEKF algorithm.     Again, the corresponding simulations can be obtained by implementing the proposed DEKF algorithm. Accordingly, Figs. 8-10 plot the trajectories of the real state and their estimates, respectively. It is clear that the newly presented estimation algorithm performs well. Moreover, in order to illustrate the differences for system with different dimensions, it is observed that the CPU time in Example 1 is 182 s and the CPU time in Example 2 is 193 s. Then, it is not difficult to find that the processing time will increase when the system dimension increases. The major reason is that the complex coupling relationships of nodes are involved and additional computations should be made.

Conclusions
In this paper, the problems of DEKF scheme and its algorithm performance evaluation have been discussed for SSNSs subject to state delay and ROCAs with UOPs. In particular, the attacks including FDIAs and DoSAs have been considered and the case of UOPs has been depicted by the combination of nominal means and error upper bounds. Subsequently, the major effort has been made to design a filter in a distributed manner and  The actual state x 2,k and the corresponding estimates Figure 10 The actual state x 3,k and the corresponding estimates look for an optimized upper bound matrix regarding the filtering error covariance. Furthermore, a suitable filter parameter has been designed at each time step to minimize the obtained upper bound matrix in the sense of matrix trace. Besides, the boundedness analysis has been certified for the DEKF algorithm by exploiting the elaborate mathematical methods. Finally, we have utilized two illustrative examples and comparative results to demonstrate the usefulness and advantages of the new DEKF approach.