Robust Sequential Fusion Estimation Based on Adaptive Innovation Event-Triggered Mechanism for Uncertain Networked Systems

In order to reduce the transmission pressure of the networked system and improve its robust performance, an adaptive innovation event-triggered mechanism is designed for the first time, and based on this mechanism, the robust local filtering algorithm for the multi-sensor networked system with uncertain noise variances and correlated noises is presented. To avoid calculating the complex error cross-covariance matrices, applying the sequential fusion idea, the robust sequential covariance intersection (SCI) and sequential inverse covariance intersection (SICI) fusion estimation algorithms are proposed, and their robustness is analyzed. Finally, it is verified in the simulation example that the proposed adaptive innovation event-triggered mechanism can reduce the communication burden, the robust local filtering algorithm is effective for the uncertainty generated by the unknown noise variances, and two robust sequential fusion estimators show good robustness, respectively.

and fused robust time-varying Kalman predictors, flters, and smoothers were proposed for uncertain systems with uncertain noise variances. Te robust Kalman estimators were designed for sensor systems with mixed uncertainties in [11], which was achieved by converting the original system into a new system with uncertain noise variances and applying the augmented state, derandomization, and fctitious noise technologies.
In the networked systems, some factors, such as the discretization of the continuous system [12,13], the normalization of the generalized system [14], the environment involving the same noise source, and so on, will lead to the correlation between the process noise and the observation noise from the single sensor or that between the observation noises from two diferent sensors. In [15], an optimal sequential fltering algorithm was proposed for linear timevarying discrete systems, where the observation noises are cross-correlated. In [16], a distributed fusion flter was presented for the multi-sensor system with fnite-step autocorrelation of process noises and cross-correlation among observation noises, using the matrix-weighted fusion estimation algorithm.
Some research works focus on the sensor networked system with both uncertainties and correlated noises. A distributed weighted fusion robust Kalman fltering algorithm was proposed for uncertain networked systems with one-step autocorrelated and two-step cross-correlated noises in [17]. In [18], a robust extended Kalman flter was proposed for nonlinear discrete-time systems with unknown inputs and correlated noises.
Te above-used fusion methods require known estimation error cross-covariances among the local sensors exactly, which are not easily available in practice, while covariance intersection (CI) fusion algorithm can overcome this weakness and reduce the computational burden signifcantly [19]. Inverse covariance intersection (ICI) fusion algorithm was proposed in [20,21], which was proved to be less conservative than CI fusion algorithm and has tight consistency. To extend the above two fusion algorithms to the multi-sensor systems and reduce the computational burden and the complexity of the batch algorithm [22,23], based on the sequential fusion idea [24], the sequential inverse covariance intersection (SICI) fusion algorithm was proposed in [25,26], which is equivalent to the multiple twosensor ICI fusion estimation, inherits the advantages of ICI fusion algorithm, and is more suitable for the actual applications.
Te sampling period is usually set in the traditional multi-sensor networked systems, which means that the time-triggered mechanism is used for data transmission. But due to the limited bandwidth, the transmission process may generate packet loss, while using an event-triggered mechanism can reduce the transmission pressure of the networked system and save the cost of communication resources. So, more researchers are concerned with the study of the event-trigger mechanisms. For example, the mode-dependent event-triggered mechanism was designed for the discrete-time fuzzy Markov jump singularly perturbed systems in [27], and the adaptive event-triggered mechanism was designed for the semi-Markovian switching cyber-physical systems in [28]. Based on the innovation triggering mechanism [29], an optimal state estimation algorithm for a multi-sensor system with correlated noises was presented by an iterative white-noise estimator, which can signifcantly reduce the communication requirements compared with the traditional timetriggered scheme, although the estimation performance is slightly degraded. In [30], based on an innovation eventtriggered mechanism, a distributed fusion estimation algorithm was proposed for the multi-sensor nonlinear networked system with stochastic transmission delays, which reduced the measurement transmission data from each communication channel and ensured the estimation performance. Under the linear minimum variance criterion, based on the innovation event-triggered mechanism, an event-triggered optimal sequential fusion flter for the multi-sensor correlated-noise systems was proposed in [31] to reduce the communication rate from the local sensors to the fusion center. However, none of those mentioned research studies consider the situation that the networked system has both uncertainties and limited communication resources simultaneously.
In order to reduce the network transmission pressure and the computational burden on the fusion center and to improve the robustness of the estimators, an adaptive innovation event-triggered mechanism is presented, and for the multi-sensor networked systems with unknown noise variances and correlated noises, based on the arriving order of the local robust estimators at the fusion center, the sequential covariance intersection and sequential inverse covariance intersection fusion robust estimation algorithms are presented. Te above sequential fusion algorithms can avoid the calculation of the estimation error cross-covariance matrices, solve the networked uncertainties brought by the unknown actual noise variances and actual error crosscovariances, and reduce the risk of data transmission failure. Tey have good stability and robustness and are more suitable for real-time application.

System Model.
Considering the linear discrete timevarying multi-sensor networked systems with uncertain noise variances and correlated noises, where x(k) ∈ R n is the state, w(k) ∈ R r is the process noise, y i (k) ∈ R m i is the observation of the ith sensor, v i (k) ∈ R m i is the observation noise of the ith sensor, and η i (k) ∈ R m i is the white noise uncorrelated with w(k). Φ(k), H i (k), and β i (k) are the time-varying matrices with n × n, m i × n, and m i × r dimensions, respectively, and L is the number of sensors.
2 Computational Intelligence and Neuroscience Assumption 1. w(k) and η i (k) are uncorrelated white noises with zero mean and uncertain actual variance matrices � Q(k) and � σ 2 η i (k), respectively, whose conservative upper bounds Q, σ 2 η i (k) are known and satisfy the relations From (3) and Te uncertain actual one-step correlation matrix be- Assumption 2. Suppose that the state x(0) has the mean μ 0 and the uncertain actual variance matrix � P(0|0) and is uncorrelated with w(k) and v i (k), i.e., where exists the relation where P i (0|0) is the known conservative upper bound of � P i (0|0).

Adaptive Innovation Event-Triggered Mechanism.
In order to reduce the pressure of network communication transmission, an adaptive innovation event-triggered mechanism is designed, introducing Bernoulli random variable c i (k) to describe the transmission state of the ith sensor and defning the innovation ε i (k) as the diference between the observation and its estimation, i.e., . When the product ε T i (k)ε i (k) of the ith sensor exceeds the threshold value θ i , the current observation will be transmitted; otherwise, no transmission will be performed. It can be described as with the threshold value where a, b, c ≥ 0 are the appropriate parameters and l i (k) denotes the last triggered moment, updated with time If the product ε T i (k)ε i (k) does not exceed the threshold value of the last transmission moment, the threshold value will remain unchanged; otherwise, since the threshold value is a monotonically decreasing function on [0, +∞), the threshold value will decrease with the increase of the innovation, and thus it will perform the adaptive regulation. When a � 0, the adaptive innovation event-triggered mechanism will degenerate to the conventional time-triggered mechanism, and when c � 0, it will deteriorate to the event-triggered mechanism with a fxed threshold value. Te structure diagram of the adaptive innovation event-triggered mechanism is shown in Figure 1.
Tis paper aims to propose the robust SCI and SICI fusion estimation algorithms for multi-sensor networked systems with uncertain noise variances and correlated noises, based on the adaptive innovation event-triggered mechanism.

Robust Local Filtering Algorithm
For a multi-sensor networked system with known conservative upper bounds of noise variances and correlated noises, based on the adaptive innovation event-triggered mechanism, the classical Kalman fltering theory is applied to the following lemma.

Lemma 1.
(see [32]). Under Assumptions 1 and 2, based on the adaptive innovation event-triggered mechanism, the local conservative recursive Kalman flter for the uncertain networked systems (1)-(3) iŝ with one-step prediction error variance P i (k + 1|k) which satisfes the Riccati equation Computational Intelligence and Neuroscience 3 When the local sensor does not meet the threshold condition at some moment, the estimator cannot receive the observation, and then the estimator will perform a prediction at that moment. Remark 1. Te conservative observations are unknown and not available, since they can only be theoretically generated by the worst-case systems with known conservative upper bounds Q(k), R ii (k), and S i (k) of noise variances, while the actual observations generated by the actual system (2) with actual unknown noise variances � Q(k), � R ii (k), and � S i (k) are available. Terefore, the local Kalman flter only can be obtained using the actual observations in Lemma 1, which is called the conservative Kalman flter.
Defning the augmented noise ] T , the conservative and actual covariance matrices between λ i (k) and λ j (k) are Te local fltering errors can be derived as where From (16), according to Assumptions 1 and 2 and the projection theory, it is easily known that w(k − 1), v i (k) and When c i (k) � 1, the local conservative fltering error variance matrix where When c i (k) � 0, the local conservative fltering error variance matrix P i (k|k) satisfes the Lyapunov equation where Sensor i Estimator i Figure 1: Adaptive innovation event-triggered mechanism structure diagram. 4 Computational Intelligence and Neuroscience To sum up, the local conservative fltering error variance matrix P i (k|k) satisfes the universal Lyapunov equation with the initial value P i (0|0) � P 0 . Similarly, according to Assumption 1, the actual local fltering error variance matrix � P i (k|k) satisfes the Lyapunov equation with the initial value � P i (0|0) � P 0 . When c i (k) � 1 and c j (k) � 1, the actual local fltering error cross-covariance matrix � When c i (k) � 1 and c j (k) � 0, the actual local fltering error cross-covariance matrix � P ij (k|k) is When c i (k) � 0 and c j (k) � 1, the actual local fltering errors cross-covariance matrix � P ij (k|k) is When c i (k) � 0 and c j (k) � 0, the actual local fltering error cross-covariance matrix � P ij (k|k) is In a word, the actual local fltering error cross-covariance matrix � P ij (k | k) can be universally described as Lemma 2 (see [9]). If a matrix Λ ∈ R r×r is a semi-positive defnite matrix, i.e., Λ ≥ 0, a matrix Λ δ ∈ R rL×rL is also a semipositive defnite matrix with form as Lemma 3 (see [9]). If the matrix where m � m 1 + · · · + m L and di ag(·) denotes the block diagonal matrix.

Theorem 1. Under Assumptions 1 and 2, for the uncertain networked systems (1)-(3) with known conservative upper bounds and correlated noises, based on the adaptive innovation event-triggered mechanism, the local conservative
where P i (k|k) is the minimal upper bound of � P i (k|k). Tis local conservative Kalman flter is called the robust local event-triggered Kalman flter.

Robust SCI Fusion Estimation Algorithm
Based on the SCI fusion estimation algorithm [24] and the adaptive innovation event-triggered mechanism, the sequential fusion processing of a networked system which meets the triggering conditions can reduce the computational burden on the fusion center and maintain good realtime performance. Te structure block diagram of the eventtriggered SCI fusion estimation algorithm is shown in Figure 2. Te structure block diagram of the event-triggered SICI fusion estimation algorithm is similar to that of the event-triggered SCI fusion estimation algorithm, which is omitted.

Theorem 2.
Based on the adaptive innovation event-triggered mechanism, the robust SCI fusion estimator for the multisensor networked systems (1)-(3) with known conservative upper bounds Q(k), R i (k), and S i (k) of noise variances iŝ x SCI (k|k) �x CI L (k|k), P SCI (k|k) � P CI L (k|k), x SCI (0| − 1) � μ 0 , Te minimization performance index of the optimal weighting factor ω i (k) is Proof. Te SCI fusion prediction at the current moment is used as the estimation of the 0th sensor for CI fusion with the estimation of the 1st sensor. When c i (k) � 1, the ith fusion fuses the fusion estimation of the (i − 1)th fusion with the estimation of the ith estimator for CI fusion and transmits it to the (i + 1)th fusion. When c i (k) � 0, the CI fusion cannot be performed by the ith fusion, so at this time the fusion estimation is the (i − 1)th fusion result and is transmitted to the (i + 1)th fusion. Terefore, the adaptive innovation event-triggered mechanism can be added on the basis of the original SCI fusion estimation algorithm. Te proof is completed.

Corollary 1. Te conservative and the actual fusion fltering error variance matrices of the event-triggered SCI fusion estimation algorithm, respectively, are
where l(k) is the number of the triggered sensors at moment k (including the 0th sensor), 1 ≤ l(k) ≤ L + 1, and renumbering these sensors, J i is the serial number of the triggered sensors at moment k, 1 ≤ J 1 < · · · < J l(k) ≤ L + 1, denoting P Ji (k|k) as the conservative local fltering error variance matrix of the J i sensor at moment k and � P JiJj (k|k) as the actual local fltering error cross-variance matrix of J i sensor and J j sensor at moment k. Te weight coefcient θ (r) Ji can be calculated recursively: with initial value Proof. Te proof is similar to Teorem 4.12 and Teorem 4.13 in [33]. Note that due to the adaptive innovation eventtriggered mechanism, the number of sensors at moment k is not L but l(k). Te proof is completed. Estimator L  Proof. Te conclusion of Teorem 1 is available for all l(k) sensors triggered at time k. Applying the mathematical induction on this basis and combining it with Teorem 4.10 in [33], the conclusion (46) can be directly introduced. Te proof is completed.

Robust SICI Fusion Estimation Algorithm
It is known that CI fusion algorithm is too conservative, while ICI fusion algorithm is a new method to deal with the unknown correlation between local estimations, which is less conservative than CI fusion algorithm, so the following theorem is proposed by using ICI fusion algorithm.

Theorem . Te actual SICI fusion fltering error variance matrix based on the adaptive innovation event-triggered mechanism is
Also, the weight coefcient θ (r) Ji can be calculated as with the initial values Proof. By [23], when l(k) sensors are triggered at k moment, based on the adaptive innovation event-triggered mechanism, the SICI fusion estimation algorithm can be rewritten in the form of a batch process: 8 Computational Intelligence and Neuroscience (58) Also, the weight coefcient θ (r) Ji can be calculated as with the initial values Te actual fltering error is Terefore, the actual fusion error variance matrix is Te proof is completed.
□ Remark 2. Unlike the event-triggered SCI fusion estimation algorithm, Teorem 4 does not require the use of the robust fusion fltering error variance matrix P SICI (k|k) in the computation of � P SICI (k | k). Terefore, the batch expression form of � P SICI (k|k) is not given.

Theorem 5.
Under Assumptions 1 and 2, the actual SICI fusion estimation algorithm ((47)-(53)) based on the adaptive innovation event-triggered mechanism is robust, i.e., Proof. According to [20], if the local estimation to be fused is robust, i.e., � P i (k|k) ≤ P i (k|k), the two-sensor ICI fusion is also robust, i.e., � P ICI (k|k) ≤ P ICI (k|k). For the actual event-triggered SICI fusion estimation algorithm, l(k) sensors triggered at the moment k have the conclusions of Teorem 1, i.e., � P Ji (k|k) ≤ P Ji (k|k). Te robustness of the two-sensor ICI fusion can be induced by the mathematical induction as � P ICI Ji (k|k) ≤ P ICI Ji (k|k), J i � 1, · · · , l(k). In particular, there exists that � P ICI l(k)−1 (k|k) ≤ P ICI l(k)−1 (k|k), and according to the structure of the robust SICI fusion estimation algorithm based on the event-triggered mechanism, it yields P ICI l(k)−1 (k|k) � P ICI L (k|k) � P SICI (k|k), and � P ICI l(k)−1 (k|k) � � P SICI (k|k), so � P SICI (k|k) ≤ P SICI (k|k). Te proof is completed.
□ Remark 3. Compared with the SCI fusion algorithm, the SICI fusion algorithm is computationally intensive due to its multiple inverse operations, but it has higher accuracy, better tightness, and better consistency [20]. Terefore, the robust event-triggered SICI fusion algorithm has similar properties.

Simulation Example
Consider a three-sensor target tracking system with uncertain noise variances and correlated noises: In simulation, we take σ 2 w � 0.16, σ 2 η 1 � 0.64, σ 2 η 2 � 0.36, σ 2 η 3 � 0.49, β 1 � 0.2, β 2 � 0.4, β 3 � 0.1, and each conservative upper bound of noise variances Q � Γσ 2 In order to clearly demonstrate the adaptive infuence of the adaptive innovation event-triggered mechanism threshold, the thresholds of three sensor changes at the step k � 300 − 400 are selected here, as shown in Figure 3. It can be seen that the thresholds will adaptively adjust according to the changes of the innovation, where the thresholds will produce a change when the sensor is not triggered at that moment. Moreover, the communication rates of the three sensors are calculated to be 60.67%, 48.5%, and 70.67%, respectively. Te efectiveness of the presented mechanism is confrmed.   True Value and Valuation  Te simulation results of the robust SCI and SICI fusion estimation algorithms based on the adaptive innovation event-triggered mechanism are shown in Figure 4, which shows that both robust event-triggered fusion estimation algorithms track well and have efectiveness. True Value and Valuation   Te comparison results of 100 Monte Carlo simulations formed from robust event-triggered SCI and SICI fusion estimation algorithms at the step k � 400 − 600 are plotted in Figure 6, which shows that the robust event-triggered SICI fusion estimation algorithm is more accurate than the robust event-triggered SCI fusion estimation algorithm, and they are consistent as mentioned above.
To validate the infuences of diferent actual noise variances satisfying Assumption 1, we select three sets of noise variances as Te curves of three sets of the actual fltering errors, the robust and actual standard deviation bounds ±3σ(k|k) and ±3� σ (m) (k|k)(m � 1, 2, 3) are shown in Figures 7 and 8, where the solid curve indicates the actual fusion fltering errors, and the dashed and dotted lines indicate the robust and actual ±3-standard deviation bounds, respectively. It can be seen that more than 99% of the actual fusion fltering error values of two robust event-triggered fusion estimation algorithms lie between ±3� σ (m) (k|k), and all three groups have the relation 3� σ (m) (k|k) ≤ 3σ(k|k), which verifes the correctness of � P SCI (k|k), � P SICI (k|k) and the robustness of two robust event-triggered fusion estimation algorithms.
Te conservative and actual error covariance ellipses of robust event-triggered SCI and SICI fusion estimation algorithms are shown in Figure 9, where the local conservative Actual Filter Error 100 200 300 400 500 600 0 k/step error covariance ellipses of each sensor contain the local actual error covariance ellipses, and three local conservative error covariance ellipses contain the conservative error covariance ellipses of two robust event-triggered estimation fusion algorithms. It also illustrates that the robust accuracies of two robust event-triggered fusion estimation algorithms are higher than those of each sensor, which verifes the robustness and efectiveness of the robust local fltering estimation algorithm and two robust event-triggered fusion estimation algorithms.

. Conclusion
An adaptive innovation event-triggered mechanism is designed in this paper, which can adaptively adjust the threshold value according to the innovation and reduce the communication burden of the networked system. Under this mechanism, the robust local flter is proposed for the uncertain networked systems with correlated noises, and its robustness is demonstrated under diferent triggered cases using the Lyapunov equation. In order to avoid the calculation of the cross-covariances in multisensor networked systems, two robust event-triggered sequential fusion estimation algorithms are proposed using SCI and SICI fusion ideas, respectively, and their actual fusion error variances are obtained by converting the two robust fusion estimation algorithms into a batch form, whose robustness is proved under the adaptive innovation event-triggered mechanism. Te simulation example illustrates that the proposed robust event-triggered sequential fusion estimation algorithms work well with the unknown actual noise variances, and they have robustness in the case of only knowing the noise variance upper bounds and can reduce communication rate and noise correlations. Computational Intelligence and Neuroscience 13

Data Availability
Te data that support the fndings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
Te authors declare that there are no conficts of interest regarding the publication of this paper.