Robust Filtering for 2-D Systems With Uncertain-Variance Noises and Weighted Try-Once-Discard Protocols

The robust filtering problem is tackled for a class of shift-varying two-dimensional systems with uncertain-variance noises under the scheduling of the weighted try-once-discard (WTOD) protocol. The measurements collected from the sensors are transmitted to a remote filter via a shared network. To alleviate the communication burden and obviate the network congestion, the WTOD protocol is adopted to orchestrate the data transmission, where a sensor node is solely permitted to broadcast its information to the remote filter at every transmission step. Moreover, the resilient filter is exploited to regulate the possible gain perturbation. The objective of the addressed problem is to design a robust filter in a recursive structure such that, in the simultaneous presence of the uncertain-variance noises and the WTOD protocol, the minimal upper bounds (UBs) on the filtering error variances (EVs) are developed for the considered system. First, by means of induction and stochastic analysis technique, certain UBs in terms of coupled recursive difference equations are derived for the actual EVs. Then, a proper filter is carefully designed which achieves the minimization of the obtained UBs at each step. Finally, an illustrative example is presented to verify the usefulness of the proposed protocol-based filtering method.


I. INTRODUCTION
W ITH the rapid advance of industrial processes, there is an increasing adoption of two-dimensional (2-D) models which effectively describe many real-world systems with multiple variables in various engineering fields. Compared with the traditional state-space model whose signals evolve along a single direction, the 2-D system possesses its own distinctive feature that the system states transmit along two independent directions. To date, 2-D systems have already found successful applications in varieties of research fields ranging from image processing, chemical process, and industrial automation to multivariable network visualization [2], [9]. Owing to its promising application prospects, the 2-D system has been garnering a sizeable amount of research interest, and various issues in the 2-D case have been studied (see [4], [8], [10], [15], [29], [31], [38], [39], [44] and the references cited therein). It is worth mentioning that the state estimation or filtering problems for 2-D systems, which intend to extract the true signals based on the available measurements, have drawn particular research attention.
Pertaining to 2-D systems, the filter design issue has served as one of the critical research topics, where many efficient approaches have been proposed in the literature including the H ∞ filtering schemes, the Kalman filtering algorithms, and the robust filtering strategies [1], [7], [37], [40]. Among others, the H ∞ filtering is a useful method in handling the deterministic yet bounded-in-energy disturbances, which aims at insuring a prescribed disturbance attenuation level on the filtering error [1], [7]. Kalman filtering is an effective tool to tackle linear systems subject to Gaussian noises with known statistics in hope of acquiring the optimal filtering error variance (EV) at every iteration [40].
Note that many types of noises are neither deterministic nor stochastic with exactly known statistics. A typical example is the so-called uncertain-variance noises which are commonly encountered in the real world owing to the difficulty in acquiring the precise statistics. Accordingly, an accurate solution of the EV is barely impossible to be obtained. In this case, robust filtering is hence adopted as an especially suitable approach to dealing with systems subjected to uncertain-variance noises or other types of uncertainties [6]. The core idea of robust filtering is to first construct a proper filter with desirable gains and then recursively calculate the minimal upper bounds (UBs) as an alternative yet practical performance replacing the estimation EVs [32]. On account of their engineering insight, the robust filtering problems have hitherto been investigated to accomplish expected performance specifications [14], [26], [27], [41]. Despite the fruitful robust filtering results on the classical one-dimensional (1-D) systems, the 2-D robust filtering issues have not received adequate research attention, which motivates the current investigation.
Concerning the filtering issue in practice, sensor nodes are likely to undergo limited sensing/communication capacities due to the restriction of network resource and power storage [12], [18]. To be more specific, the communication channel inevitably suffers from limited bandwidth if all sensors simultaneously have access to the shared network. In addition, data transmission executed at each instant may give rise to redundant/unnecessary signals, which further leads to the energy waste. In this case, the so-called energyefficient filtering schemes have been a focus of research and a wealth of literature has been published [20], [35], [43]. For instance, the event-based mechanism has been applied to reduce the data transmission frequency for energy-saving purposes. Moreover, the information scheduling policies have been widely exploited to relieve data congestions, thereby better allocating the limited network resource. Typical scheduling policies include the weighted try-once-discard (WTOD) protocol, periodic protocol, and random access protocol [22], [25], [30], [33], [34], [45]. Particularly, the WTOD protocol has been utilized in [19] where a robust filter has been designed for a type of Markovian jump systems with the aim of guaranteeing a prescribed H ∞ disturbance attenuation level. The periodic protocol has been applied in [3] to address the distributed filtering problem over sensor networks with randomly switching topologies. Very recently, the random access protocol has been adopted in [46] to tackle the moving horizon estimation issue for a class of networked nonlinear systems, where the desired estimator has been dexterously designed to ensure the exponential ultimate boundedness of the estimation error.
Among various scheduling policies, the WTOD scheduling determines the sensor to deliver its measurement based on a quadratic selection principle that enables to reflect the significance of certain demands/missions, and such a scheme is more reasonable as compared to the random access protocol where the selected node at each transmission step is determined by a given probability distribution. Moreover, in contrast with the periodic protocol acting as a static scheduling rule, the WTOD protocol accommodates the transmission order of sensors dynamically, which would ensure a higher reliability of the expected performance. In light of its assignment property, the WTOD protocol has been preferred in both academia and industry, and many representative results on the WTOD protocol-based filtering strategies have been reported in the last decade [5], [16]. However, the robust filter design for 2-D systems under the WTOD protocol has not been adequately investigated, despite its engineering significance.
On another research front, fluctuations in the filter realization are a common phenomenon arising primarily from the round-off errors in computation and the imprecision of instruments [13]. As a result, the resilient filter is constructed to reflect the possible gain fluctuation, which usually occurs in a random fashion because of the abrupt/changeable working conditions. It is noteworthy that the design of a resilient filter is of benefit to the system robustness, and the relevant filtering problem has begun to draw some initial research attention [11], [21], [23], [36]. From a practical perspective, almost all systems in reality possess time/shift-varying parameters due probably to temperature change and component aging. As such, it is meaningful to investigate the filter design of the targeted plant with time/shift-varying parameters and further consider the transient performance instead of the steady-state one [17].
Inspired by the foregoing discussions, we intend to launch a systematic study on the robust filtering scheme for 2-D shift-varying systems undergoing uncertain-variance noises and WTOD transmission scheme. The stochastic noises enter into both the target plant and the measurement model, whose variances are unknown but bounded within certain ranges. The resilience issue is also considered in the design of the filter for characterizing the random gain variation. To deal with the addressed issue, great effort shall be devoted to answering the following nontrivial questions: 1) how to develop appropriate scheduling and updating rule of the data transmission to effectively extend the conventional WTOD protocol suitable for the 2-D framework? 2) how to seek an applicable robust filtering method to cope with the introduced noises and the gain variation? 3) how to devise a proper filter for solving the 2-D robust filtering problem with an acceptable performance? and 4) how to reveal the impact of the uncertain-variance noises, the WTOD scheduling, and the gain variation on the estimation performance?
In response to the aforementioned questions, the novelties of this article are emphasized as threefold.
1) The class of systems under investigation is rather general which includes shift-varying parameters, uncertaincovariance noises, communication protocol, as well as the filter gain perturbation. 2) The WTOD scheduling is, for the first time, introduced for 2-D shift-varying models with uncertain-variance noises. 3) A recursive method is established to design the filter gain and derive the minimal UBs on the EVs by using mathematical induction, stochastic analysis, and difference equation techniques. The outline of this article is stated as follows. Section II gives the underlying system, presents a description of a new WTOD protocol, and introduces the 2-D recursive filter. Section III develops the main results, in which sufficient conditions are given to look for the minimal UBs and the desirable filter gains. The validity of the proposed filter design approach is clarified via an illustrated example in Section IV, and conclusions are drawn in Section V.
Notations: R n means the n-dimensional Euclidean space. For a square matrix A, A T and A −1 indicate the transpose and the inverse matrix of A, respectively. diag 1≤l≤n {A l } is the block-diagonal matrix with elements being matrices A 1 , A 2 , . . . , A n . For a real and symmetric matrix Z, the symbol Z > 0 infers that Z is positive definite, while Z is positive semidefinite if Z ≥ 0. I and 0 are, respectively, the identity and zero matrices of compatible dimensions. For integers a 1 and a 2 with a 1 ≤ a 2 , [a 1 a 2 ] is used to signify the set {a 1 , a 1 + 1, . . . a 2 }. E{α} stands for the mathematical expectation of certain random variable α.

II. PROBLEM FORMULATION
Consider a discrete shift-varying 2-D system over a finite horizon s, r ∈ [1 ] described as where x(s, r) ∈ R n x is the state vector and y(s, r) ∈ R n y is the measurement output.
, and D(s, r) are known shift-varying matrices of appropriate dimensions. w(s, r) ∈ R p is the process noise while v(s, r) ∈ R q is the measurement noise, which are two white sequences with zero means and uncertain variances confined to the following inequalities: where Q 1 (s, r) and Q 2 (s, r) are given positive-definite matrix sequences.
The initial conditions of system (1) are two white-noise sequences satisfying E{x(s, where u 1 (s), u 2 (r), 1 (s), and 2 (r) are known vectors with u 1 (0) = u 2 (0) and 1 (0) = 2 (0). Remark 1: Nowadays, 2-D systems have played an indispensable role in reality due to their profound application insights in many industrial areas. Notice that many chemical processes are depicted by specific partial differential/difference equations (PDEs), which enable some physical variables of interest to evolve along the space and time directions. By means of proper sampling techniques, most of the PDEs can be converted into the 2-D system (1) that is one of the most general 2-D models. A case in point is the thermal/heating process, where the dynamic behavior of the reactor temperature varying with different spatial coordinates is presented by (1) as mentioned in [2] and [36].
Remark 2: In practical engineering, the precisely statistical information of the noises is usually unobtainable in view of the artificial electromagnetic interference and/or other disturbances. For the underlying system, the process and measurement noises are white-noise sequences with zero mean, while their respective variances are unknown but bounded by certain given matrices Q i (s, r) (i = 1, 2), which are recognized as the UBs on the noise amplitudes. In this case, the renowned Kalman filtering is no longer applicable, and the robust filtering is then adopted for the considered 2-D systems with uncertain-variance noises.

A. Weighted Try-Once-Discard Protocol
Let us consider the effect induced by the communication protocol. As is well known, the measurements transmitted from the sensors to the remote filter through a shared network of limited bandwidth ineluctably undergo certain communication constraints. For purpose of avoiding data congestions, the sensor node is exclusively allowed to have access to the network at every transmission step, and the WTOD protocol is exploited to schedule the communication between the sensors and the filter. The structure diagram of the filtering problem with the WTOD protocol is presented in Fig. 1.
Without loss of generality, sensors are grouped into m nodes in accordance with their spatial distribution, and all these nodes would unify together to measure the output information of the target plant. Let us denote with y j (s, r) ∈ R n y j and m j=1 n y j = n y for j ∈ [1 m], where y j (s, r) signifies the measurement of the jth sensor node before transmission. Let ξ(s, r) be the selected node taking the network access at the transmission instant (s, r). The determination of ξ(s, r) is specified as follows on the basis of a selection principle according to the WTOD protocol: whereỹ j (s, r) = y j (s, r)−(1/2)(y j (s, r−1)+y j (s−1, r)) infers the difference between the measurement output at instant (s, r) and the nearest measurement outputs at instants (s, r − 1) and (s−1, r) about node j, and W j (j ∈ [1 m]) is a known positivedefinite matrix signifying the weight matrix for the jth node. Apparently, the index ξ(s, r) is chosen as the minimal value of j that maximizesỹ T j (s, r)W jỹj (s, r), which is solely determined, thereby ensuring the uniqueness of the selected sensor node. r), then the selected sensor node ξ(s, r) is exclusively set as j 1 .

Remark 3:
The received measurement affected by the communication protocol is given as (4) [or (5)]. Generally, two popular strategies have been proposed in the existing literature to undertake the measurements after being transmitted via a shared network, one is the zero input policy and the other is the zero-order hold (ZOH) policy (see [42] for the 1-D case). Unlike the former one, the ZOH policy is employed as a compensation mechanism to offset the measurements that have no network access at each transmission instant. In this research, the measured output after transmission abides by a generalized 2-D ZOH strategy (3), (4). In particular, the solely selected sensor node is determined to access the communication channel based on the selection principle (3). The measurement of the selected node is successfully delivered via the network, while the others take the value stored in the corresponding zero-order holder.
Remark 4: When it comes to the WTOD scheduling proposed for 1-D systems, the selected node obtaining the network access relies on the last transmitted signal. Such a conventional WTOD protocol, however, cannot be used directly for 2-D systems due to the two-directional broadcast. To circumvent such an obstacle, in this article, the selection principle of the WTOD protocol in the 1-D case has been successfully extended to the 2-D case. In comparison with the WTOD scheduling in 1-D systems, the proposed scheduling in the 2-D setting takes into account the measurement difference between the signal at the current instant and those along two directions at the immediate preceding instants.
Remark 5: Different from the periodic and random access protocols whose transmission schemes are uncorrelated with the measurements, the proposed WTOD scheduling designates the transmission order for the sensors in terms of a quadratic selection principle, which depends on the measurement outputs. This, unfortunately, would ineluctably lead to additional challenges when analyzing the addressed problem because of the measurement-dependent selection principle of the WTOD scheduling. Furthermore, due to the mathematical analysis complexity, such a dynamical scheduling of the WTOD protocol also poses extra difficulties in establishing certain UBs on the EVs and designing the robust filter. In view of these substantial challenges, it is a nontrivial task to develop a robust filtering strategy for the underlying 2-D system.

B. Augmented System and Resilient Filter
For notational brevity, let us definē Then, the considered system (1) with the proposed WTOD protocol is rewritten as follows: For the augmented system in (6), the following robust filter in a recursive form is adopted: wherex u (s, r) andx p (s, r) ∈ R n x +2n y are the estimate and the prediction of statex(s, r), respectively, G(s, r) is the filter gain to be devised, whereas G(s, r) is the random parameter perturbation with zero mean and the following statistics: where ρ(s, r) > 0 is a known scalar. The initial states of (7) are given asx u (s, 0) = E{x(s, 0)} andx u (0, r) = E{x(0, r)}. Remark 6: In this article, the relationship (8) imposes a statistical constraint on the second-order moment of G(s, r), which is competent in characterizing the variation range of the random gain perturbation. The constructed filter involving stochastic gain variation (on account of the implementation uncertainty or computation inaccuracy) gives rise to the resilient estimation issue and, accordingly, the desired filter would be insensitive to gain variations in its implementation.
For ease of representation, we definē Then, the following error system is derived from (6) and (7):

s, r) = I −Ḡ(s, r)C(s, r) e p (s, r) −Ḡ(s, r)D(s, r)w(s, r). (9b)
The presence of uncertain-variance noises and gain perturbation makes an accurate filtering EV unobtainable. To enhance the reliability of the filter design, a natural and alternative way is to derive the UBs on the filtering EVs as an acceptable filtering performance. Consequently, this article focuses on devising the robust filter (7) in order to give the tightest UB on the EV for all allowed modeling uncertainties.

III. MAIN RESULTS
This section investigates the robust filtering problem for the underlying system, where the minimization of the UBs on the EVs and determination of the filter parameter are established in terms of two recursive difference equations. Denote P p (s, r) E e p (s, r)e T p (s, r) P u (s, r) E e u (s, r)e T u (s, r) . Before proceeding with the main results, the following lemmas are presented to show the basic matrix inequality technique as well as the dynamics of matrices P p (s, r) and P u (s, r). Lemma 1: Let be an arbitrary positive scalar and X and Y be the real-valued matrices of suitable dimensions. One has the following inequality: Lemma 2: For the error system (9), recursions of the EVs are given as follows:

P u (s, r) = E I −Ḡ(s, r)C(s, r) P p (s, r)
× I −Ḡ(s, r)C(s, r) Proof: According to the statistical properties of random variables w(s, r) and v(s, r), the variablew(s, r) is uncorrelated with not only e p (s, r) but also e u (s 0 , The validity of the assertion in this lemma is not difficult to be obtained from (9) and the expressions of P p (s, r) and P u (s, r).
Remark 7: Due primarily to the existence of 1) the uncertain-variance noises; 2) the random gain perturbation; and 3) the cross terms in the recursion of P p (s, r), it is technically impossible to calculate the exact filtering EV, much less design the expected filter. More specifically, 1) the secondorder moment of noisew(s, r) is inaccessible; 2) the gain perturbation G(s, r) is subject to the statistical constraint (8); and 3) the cross term E{e u (s, r − 1)e T u (s − 1, r)} is hard to be computed because of the complicated system dynamics. To this end, we shall cope with the above terms in (10) and (11) and turn to derive the local minimum UB for the EV by elegantly devising the desired filter gain. Such an alternative strategy is well suited to ensure a satisfactory filtering performance at the cost of sacrificing a bit acceptable conservatism.
The following theorem presents a sufficient condition that provides the UBs of the EVs.
Theorem 1: Let μ, α, β, and λ be given positive scalars. For the considered system (6) with filter (7), assume that there are two series of positive-definite matrices M p (s, r) and M u (s, r) such that the following two recursions hold: Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
To devise the 2-D robust filter, the following theorem presents the filter gain that optimizes the UBs on the actual EVs.
Theorem 2: Let μ, α, β, and λ be given positive scalars. For the considered system (6) with filter (7), the desirable filter gain is determined by The proof of this theorem is hence complete. Remark 8: On the basis of the induction, recursive method, and stochastic matrix theory, certain UBs have been established in Theorem 1 to suppress the actual EVs. Furthermore, the gain parameter has been devised in Theorem 2,under which the solutions to the coupled difference equations (12) and (24) provide the locally minimal bounds on the EVs. It is observed that the expressions of the minimal UBs developed in the main results embrace all the information about the system parameters, the noise constraints, the scheduling of the data transmission, and the variation range of the gain perturbation.
Remark 9: In this article, a resilient filter in a recursive form has been first proposed to handle the robust filtering for 2-D shift-varying systems subject to uncertain-variance noises and a WTOD protocol. Distinguished from the other existing literature, the novelties of this article lie in the following aspects: 1) the addressed filtering problem is new in the sense that the considered 2-D system is shift varying, the involved noises suffer from uncertain covariances, and the filter implementation undergoes the stochastic gain perturbation; 2) a novel WTOD scheduling is proposed, where the selection principle of the data transmission order is presented in the 2-D setting; and 3) a theoretical framework of the robust filtering algorithm is developed for the underlying 2-D system to guarantee the desired estimation performance.

IV. NUMERICAL EXAMPLE
In this section, a thermal process is given to justify the applicability and validity of the developed filtering scheme. In engineering practice, a heating process based on vessels is formulated by the following equation [36]: where H(k, l) is the temperature of interest at the space coordinate k ∈ [0, K] and time l ∈ [0, L], f (k, l) stands for a given force function, and φ and ψ are known scalars inferring the heat transfer parameters. The above equation can be approximated to a 2-D discrete-time system by using the discretization approach. Let k and l be the sampling periods regarding the indices k and l, respectively. For k ∈ [s k, (s + 1) k) and l ∈ [r l, (r + 1) l), the following approximate equations are obtained:

Denote x(s, r)
[H T ((s − 1) k, r l) H T (s k, r l)] T and set ψ = 0. Then, the PDE in (25) can be transformed to the following discrete-time counterpart in the 2-D setting: In consideration of the possibly contaminated reagents or vessels, the external disturbances or noises should be taken into account for better describing the real situation. In addition, the system parameters may be changeably induced by the period oscillations of the fluid/liquid or gas in the practical vessels. For simulation purposes, the system parameters are given as follows:

A. Effectiveness of the Developed Filtering Scheme
In this example, the process and measurement noises are given by w(s, r) = 0.5 sin(s)w 0 (s, r) and v(s, r) = 0.6 cos(s)v 0 (s, r), where w 0 (s, r) and v 0 (s, r) are uncorrelated Gaussian white sequences with mean zero and variance I. In this regard, the matrices Q 1 (s, r) and Q 2 (s, r) that bound the amplitudes of noises w(s, r) and v(s, r) are taken as Q 1 (s, r) = 0.25I and Q 2 (s, r) = 0.36I, respectively. Moreover, the weighted matrices of the adopted WTOD scheduling are set to be W 1 = 0.5 and W 2 = 1.2. The parameter regarding the gain perturbation is assumed to be ρ(s, r) = 0.05. The initial states of the 2-D system are given by u 1 (s) = u 2 (r) = [0 0] T and 1 (s) = 2 (r) = 0.1I. Moreover, the scaling scalars are chosen to be α = β = λ = 1 and μ = 0.01.
For simplicity, let us denote the th element ofx(s, r) and x u (s, r) asx ( ) (s, r) andx ( ) u (s, r), respectively, and further denote M ( ) u (s, r) and P ( ) u (s, r) as the th diagonal element of matrices M u (s, r) and P u (s, r), respectively. By virtue of the filter design strategy obtained in Theorem 2, the 2-D robust filtering problem under the WTOD scheduling can be iteratively calculated using MATLAB. The filter parameters are   listed in Table I, where only part of the gain matrices is given for saving space. The corresponding experimental results are presented in Figs. 2-5. Among them, Figs. 2 and 3 plot the trajectories of the system statex (1) (s, r) and its estimate. Fig. 4 depicts the trajectory of the minimal UB M (11) u (s, r). Fig. 5 displays the trajectory of the actual EV P (11) u (s, r) averaged from 1000 independent runs. Obviously, these simulations validate that the developed protocol-based filtering method is indeed effective and the filtering performance is well achieved.

B. Comparison of the Results
In what follows, some comparison simulations are conducted to assess the impacts of the random gain perturbation,  the noises as well as the WTOD scheduling on the filtering performance.
Case I: To begin with, let us choose ρ(s, r) = 0.1 without changing all the other parameters presented in the above experiment. In this case, the simulation result of the minimal UB M (11) u (s, r) is given in Fig. 6. It is observed from Figs. 4 and 6 that a smaller gain perturbation generally results in a tighter UB, namely, a better performance.
Case II: To clarify influence of the noises, we reset w(s, r) = sin(s)w 0 (s, r) and v(s, r) = 1.2 cos(s)v 0 (s, r), where their noise variances are confined by Q 1 (s, r) = I and Q 2 (s, r) = 1.44I, respectively. In this scenario, the trajectory of the minimal UB M (11) u (s, r) is shown in Fig. 7. Comparing with Figs. 4 and 7, we can easily see that the expected UB becomes larger as the noise amplitudes increase.  In other words, the filtering performance is degraded if the noise amplitudes become larger.
Case III: A comparison is carried out to further examine the effectiveness of the filtering strategy under different WTOD protocols. Consider the case that the zero-input policy is applied to the WTOD scheduling. In this case, the update of the measurementȳ j (s, r) is expressed bȳ y j (s, r) = y j (s, r), ξ(s, r) = j 0, ξ(s, r) = j and thus we haveȳ(s, r) = ξ(s,r) y(s, r). Based on such measurements, the corresponding theoretical and simulation results can be attained. Denotex (1) u,0 (s, r) and M (11) u,0 (s, r) as the respective counterparts of the state estimate and the minimal UB subjected to the WTOD scheduling with zero-input policy. Fig. 8 plots the state estimatex (1) u,0 (s, r), and Figs. 9 and 10 show the distinctions of the minimal UB and the actual EV under different updating rules. Figs. 3 and 8-10 reveal that 1) the filtering strategies under the WTOD scheduling perform well for both of the updating rules and 2) the protocol-based filtering method with the ZOH policy is superior to that with the zero-input policy because special attention has been paid to the compensation mechanism.     (11) u,0 (s, r) and P (11) u (s, r).

V. CONCLUSION
This article has dealt with the robust filtering issue for the 2-D shift-varying system subjected to uncertain-variance noises and communication protocol. The WTOD scheme has been implemented to schedule the transmission order for sending measurement collected by a single sensor at each transmission instance. The proposed filter with a recursive structure contains the stochastic gain parameter. Based on the stochastic analysis together with the induction approach, the sufficient criterion has been provided to derive the UBs on the EVs in terms of recursive difference equations. Moreover, the desired filter gain has been determined under which the optimal UB is obtained. Finally, an illustrated example has been presented to show the usefulness of the robust filtering strategy. Further research directions would include some potential extensions of the main results to more complicated systems under communication constraints, such as 2-D nonlinear systems under coding-decoding mechanisms and eavesdropping attacks [24], [28].