Event-Triggered Resilient Filtering for 2-D Systems with Asynchronous-Delay: Handling Binary Encoding Decoding with Probabilistic Bit Flips
Yu Chen Wei Wang



Event-Triggered Resilient Filtering for 2-D Systems with Asynchronous-Delay: Handling Binary Encoding Decoding with Probabilistic Bit Flips
This paragraph of the first footnote will contain the date on which you submitted your paper for review. This paper was produced by the IEEE Publication Technology Group. This work was supported by the National Natural Science Foundation of China (No. 62250056), the Natural Science Foundation of Shandong Province (No.ZR2021MF069, ZR2021JQ24).

    Yu Chen and Wei Wang
Y. Chen and W. Wang are with the School of Control Science and Engineering, Shandong University, Jinan 250061, P. R. China (Email: ychen_2000@163.com, w.wang@sdu.edu.cn).

    
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    In this paper, the event-triggered resilient filtering problem is investigated for a class of two-dimensional systems with asynchronous-delay under binary encoding-decoding schemes with probabilistic bit flips. To reduce unnecessary communications and computations in complex network systems, alleviate network energy consumption, and optimize the use of network resources, a new event-triggered mechanism is proposed, which focuses on broadcasting necessary measurement information to update innovation only when the event generator function is satisfied. A binary encoding-decoding scheme is used in the communication process to quantify the measurement information into a bit stream, transmit it through a memoryless binary symmetric channel with a certain probability of bit flipping, and restore it at the receiver. In order to utilize the delayed decoded measurement information that a measurement reconstruction approach is proposed. Through generating space equivalence verification, it is found that the reconstructed delay-free decoded measurement sequence contains the same information as the original delayed decoded measurement sequence. In addition, resilient filter is utilized to accommodate possible estimation gain perturbations. Then, a recursive estimator framework is presented based on the reconstructed decoded measurement sequence. By means of the mathematical induction technique, the unbiased property of the proposed estimator is proved. The estimation gain is obtained by minimizing an upper bound on the filtering error covariance. Subsequently, through rigorous mathematical analysis, the monotonicity of filtering performance with respect to triggering parameters is discussed.



    Two-dimensional system, event-triggered mechanism, binary encoding-decoding scheme, measurement reconstruction, recursive estimator.




§ INTRODUCTION

    In recent years, two-dimensional (2-D) systems have stimulated an ever-increasing research interest due to their application prospects in different scenarios, including but not limited to image analysis <cit.>, environmental monitoring systems <cit.>, chromatography detection <cit.>, and wave propagation <cit.>. Unlike the well-known one-dimensional (1-D) systems, information propagation in 2-D systems evolves independently in both horizontal and vertical directions. 2-D systems are usually represented by two types of models, namely the Fornasini-Marchesini state space model <cit.> and the Roesser state space model <cit.>. To date, stability/performance analysis and control/filtering synthesis issues of various 2-D systems have been extensively studied. In <cit.>, under unknown but bounded external source interference and measurement noise, the zonotope-based interval estimation of a 2-D systems based on the event-triggered mechanism has been studied. In <cit.>, the co-design scheme of asynchronous H_∞ control for a class of 2-D Markov jump systems under adaptive event-triggered mechanism has been studied, by constructing nonlinear matrix inequalities, the asymptotic mean square stability of closed-loop system has been guaranteed. In <cit.>, the design problem of cost-guaranteed controller for linear repetitive process event-triggered with multiplicative noises have been solved by utilizing chi-square distribution, complementary square technique, and actual information of triggering instants. In <cit.>, the dissipative control and filtering of a class of 2-D switching systems have been studied. By using the switching quadratic Lyapunov function technique, the sufficient conditions for the asymptotic stability of the system under arbitrary switching signals and the exponential stability of the 2-D TSR-ϑ-dissipation rate under restricted switching signals has been given.
	
    For communication strategies transmitted over the network, event-triggered mechanisms (ETMs), as an effective method to save energy and bandwidth resources, are often used to control signal transmission from sensors to filter/controller. Compared with traditional time-triggered mechanism <cit.> where the communication interval is known a priori, ETMs is an aperiodic communication protocol in which the transmission instant is determined by a defined event generator. Intuitively speaking, system components (such as sensors or actuators) are authorized to access the network to send signals only when they meet the triggering conditions. Otherwise, the signal will be held by the zero-order holder (ZOH) to maintain the signal value at the last triggering instant. So far, a large number of results have been reported on filter design issues under ETMs <cit.>. However, how to use event-triggered mechanism to achieve effective transmission and utilization of measurement information while considering multi-channel delay has not been fully studied. This is the main motivation of this article.
	
    With the rapid development of network communication technology and the widespread deployment of network devices, communication channels have brought multiple advantages to modern communication systems, including flexibility of architecture, ease of implementation, and low-cost maintenance. Nevertheless, due to limited communication capacity, transmitted signals are inevitably affected by network-induced phenomena. In order to improve the efficiency and reliability of data transmission, quantization-based encoding-decoding schemes (EDSs) have been widely used in the signal transmission process <cit.>. Currently, because binary data is easy to implement and has transmission robustness, binary EDSs has become one of the widely used signal transmission strategies. Under binary EDSs, the transmission signal has been first encoded into a set of binary bit streams and transmitted through a memoryless binary symmetric channel. The final bit stream is restored to the transmission signal at the receiving end. During the transmission process, due to channel noise or malicious network attacks, bit flips may occur randomly with a certain crossover probability. If not handled properly, it will cause distortion of the transmission signal and lead to communication errors. Relevant results have appeared for 1-D systems <cit.>, but there have been no relevant in-depth discussions on 2-D systems due to the difficulty of bidirectional evolution.
	
    In control systems that depend on network communication, delay is an inevitable problem. The causes of network delay is manifold, mainly including the distance of the transmission medium, network bandwidth, the processing and queuing of data packets, and hardware performance. In <cit.>, the distributed H_∞ state estimation problem of a class of 2-D systems with one-way stochastic delays have been studied. In <cit.>, the problem of reliable estimation of 2-D discrete linear systems with one-way delays and incomplete observations have been studied. In <cit.>, the problem of robust filter design for a class of 2-D systems with norm-bounded parameter uncertainty and incomplete measurements (including random sensor delays and missing measurements) has been studied. The one-way random sensor delays have been presented in a unified form through the stochastic Kronecker function. Existing research on estimation problems of 2-D delays systems is all about one-way delays and is highly conservative. In <cit.>, the design scheme of a 2-D systems filter under two-way multi-step observation delays have been given for the first time. This paper will consider the more general state estimation of 2-D systems with asynchronous-delays.
	
    To tackle the aforementioned emerging issues, this paper investigates the estimation problem of 2-D systems with ETMs and asynchronous-delays under probabilistic bit flips binary EDSs. Some urgent challenges are outlined as follows: 1) How to propose a new approach to deal with the asynchronous-delays caused by the sensor transmission? 2) How to define an event-triggered shift sequence in view of the bidirectional signal propagations? 3) How to mathematically quantify the signal distortion during communication under binary EDSs? 4) How to design a recursive filter and present estimation gain? The main contributions of this paper can be highlighted as follows: 1) In view of the asynchronous-delay, a decoded measurement reconstruction approach is proposed to achieve effective utilization of measurement information. 2) In order to reduce the communication burden of the network system, a new definition of triggering sequence and event generator function are proposed, based on which the occurrence of certain event is determined. 3) A new estimator framework is established for event-triggered filtering with asynchronous-delay under binary EDSs. 4) The impact of ETMs on filtering performance is discussed. In particular, a rigorous mathematical analysis of the monotonicity of filtering performance with respect to triggering parameters is performed.



§ PROBLEM FORMULATION

    

 §.§ System description

        Notations: ℰ{·} represents the mathematical expectation. The scrip “T” represents the transpose of vectors or matrices. ℝ^n stands the n-dimensional Euclidean space. δ(i,j) is the Kronecker delta function with δ(i,j) being unity for i=j but zero elsewhere. col_0≤ c ≤ s{a_c} represents a column vector with a_0,a_1,⋯,a_s as an element. diag_0≤ c ≤ s{a_c} stands for a diagonal matrix with a_0,a_1,⋯,a_s as the diagonal elements. [0,k] represents the set {0,1,⋯,k}. sym(A)=A+A^T.

		Consider the following 2-D systems are described by FM-II model:
		
    x(i,j)   =f_1((i,j-1),x(i,j-1))
        +f_2((i-1,j),x(i-1,j))
        +B_1(i,j-1)w(i,j-1)
        +B_2(i-1,j)w(i-1,j)

        where i,j on a finite horizon [0,℘] with ℘>0 being a given integer, x(i,j)∈ℝ^n is the system state vector and w(i,j)∈ℝ^n is the process noise with zero-mean and covariance Q(i,j)>0. For ι∈[1,2], B_ι(i,j) are known shift-varying matrices with appropriate dimensions, the nonlinear functions f_ι((i,j),x(i,j)) satisfying
		
    {   f_ι((i,j),0)=0
       ‖ f_ι((i,j),x_1)-f_ι((i,j),x_2)-A_ι(i,j)(x_1-x_2)‖
         ≤ a_ι(i,j)‖ x_1-x_2 ‖  ∀ x_1, x_2∈ℝ^n
    				.

		where A_ι(i,j) are known shift-varying matrices and a_ι(i,j)≥ 0 are given scalars.
		
        In this study, the signal transmission between sensors and the remote estimator is conducted across multiple channels simultaneously, with the aim of enhancing the system fault tolerance. If a single channel fail or a signal be lost, other channels can serve as backups to ensure the complete transmission of the signal. However, given the restrictions on network bandwidth, this transmission method often leads to transmission delays. To accurately describe the process of signal transmission, the measurement model with asynchronous delays can be represented as:
        
    {   y_(0)(i,j)=C_(0)(i,j)x(i-_0,j-_0)+v_(0)(i,j)
       y_(1)(i,j)=C_(1)(i,j)x(i-_1,j-_1)+v_(1)(i,j)
                ⋮
       y_(N)(i,j)=C_(N)(i,j)x(i-_N,j-_N)+v_(N)(i,j)
                    .

        where y_(s)(i,j)∈ℝ^m(s∈[0,N]) is the s+1-th delayed measurement information and v_(s)(i,j)∈ℝ^m is measurement noise with zero-mean and covariance R_(s)(i,j)>0, C_(s)(i,j) is known shift-varying matrices with appropriate dimensions.

        
            Without loss of generality, the delay _s and _s are assumed to be in a increasing order: 0=_0=_0<_1≤_1⋯ <_N≤_N.
        
		
            For i,j∈[0,℘], s∈[0,N], the initial state x(i,0) and x(0,j), random noises w(i,j) and v_(s)(i,j) are mutually uncorrelated with each other. The system (<ref>) initial condition satisfy the following statistical constraints:
			
    ℰ{x(i,0)}=x_u(i,0)   ℰ{x(0,j)}=x_u(0,j)
       ℰ{[x^T(i,0) x^T(0,j)]^T [x^T(r,0) x^T(0,c)]}
        =diag{Ξ(i,0)δ(i,r),Ξ(0,j)δ(j,c)}

            where x_u(i,0), x_u(0,j), Ξ(i,0), and Ξ(0,j) are known shift-varying matrices with appropriate dimensions.
		
        
            In actual systems, due to factors such as equipment aging, working environment effects or cognitive limitations, it may lead to problems such as unknown model parameters or inaccurate descriptions, especially in the modeling of artificial systems and maneuverable targets. For example, In the dynamic evolution process of wave equation describing string vibration problem and heat equation describing water heating process, due to the existence of damped oscillation, elastic deformation and friction, a nonlinear model in which the displacement change of the underlying system is coupled with the system state will appear.
		
		
            The framework of the estimation problem with the ETMs and EDSs for 2-D systems are presented in Fig. <ref>. Sensors are only authorized to access the network and send measurement information when triggering conditions are met. Otherwise, the signal will be held by the ZOH to hold the measurement information at the last triggering instant. Then, the triggering measurement information is encoded into a set of binary bit streams and transmitted through a memoryless binary symmetric channel. Finally, the bit stream is restored to decoded measurement information at the receiver. All decoded measurement information is sent to the remote estimator for reconstruction.
		
        

	

 §.§ Dynamic event-triggered mechanisms

        Considering that periodic or continuous data transmission in the communication network, the shared network is occupied, resulting in a large number of redundant measurement outputs and causing network congestion. In addition, when there are only small fluctuations between adjacent measurement outputs, the filter does not need to consume resources again to update. Based on the above considerations, in order to reduce the data transmission frequency and improve network utilization efficiency, ETMs are used to determine whether to transmit the current measurement output to the remote filter.
		
		Define the event generator function as follows:
		
    φ_(s)^γ(i,j)≜ς_(s) h_(s)^γ(i,j)+ξ_(s)^γ(i,j)

		where
		
    h_(s)^γ(i,j)=   σ_(s)^γ(y_(s)^γ(i,j))^2-ρ_(s)^γ(e_(s)^γ(i,j))^2
    
    			e_(s)^γ(i,j)=   y_(s)^γ(i,j)-y_(s)^γ(i_t,j_t)

        for any γ∈[1,m], y_(s)^γ(i,j) and y_(s)^γ(i_t,j_t) represents the γ-th component of the current measurement and the measurement received at the latest triggering instant (i_t,j_t), respectively. e_(s)^γ(i,j) represent the γ-th component of the dynamic event-triggered error e_(s)(i,j). ς_(s),ρ_(s)^γ, and σ_(s)^γ represent triggering parameters. ξ_(s)^γ(i,j) represent the γ-th component of the internal dynamic variable ξ_(s)(i,j) governed by the following dynamics:
		
    ξ_(s)^γ(i,j)=   α_(s),1ξ_(s)^γ(i,j-1)+α_(s),2ξ_(s)^γ(i-1,j)
       +h_(s)^γ(i,j-1)+h_(s)^γ(i-1,j)

        where α_(s),1 and α_(s),2 represent the pre-assigned scalars, and ξ_(s)^γ(i,0)=ξ_(s)^γ(0,j)=ξ_(s)^0≥0 are given initial conditions of the internal dynamic variable ξ_(s)^γ(i,j).

        It is worth noting that two indices are used in (<ref>), and the sorting problem of generating index sequences will be explained below, prior to this, introduce a definition of partial order on integer pairs
		
    {   (i_t+1,j_t+1)>(i_t,j_t),   i_t+1=i_t, j_t+1>j_t 
                         or  i_t+1>i_t,  j_t∈[0,℘],
       (i_t+1,j_t+1)=(i_t,j_t),   i_t+1=i_t,j_t+1=j_t.			
    				.

		Based on the definition of partial order mentioned above, the following triggering sequences are proposed:
		
    (0,0)≤(i_1,j_1)<(i_2,j_2)<⋯<(i_t,j_t)<⋯

		The forthcoming triggering instant (i_t+1,j_t+1) is determined by
		
    (i_t+1,j_t+1)=inf   {(i,j) | (i,j)>(i_t,j_t),
        ∃ s, γ, s.t. φ_(s)^γ(i,j)≤ 0}

		
            In view of the two-way propagation characteristics of the 2-D systems,This paper proposes a more general dynamic ETMs for 2-D systems, The event generator function φ_(s)^ι(i,j) has two indicators to represent the triggering instant (i_t, j_t). Then, the trigger threshold is closely connected with system information and can be dynamically adjusted according to the operation of the system. When the measurement in the recent period fluctuates greatly, the internal dynamic variable ξ_(s)^ι(i,j) will be dynamically adjusted to reduce the trigger threshold to ensure that the current measurement satisfies the event trigger generator to implement trigger transmission. For the convenience of subsequent analysis, define y̌_(s)(i,j)=y_(s)(i_t,j_t),(i_t,j_t)≤(i,j)<(i_t+1,j_t+1).
		
		
            It should be noted that in the ETMs, the forthcoming triggering instant is not set independently in each channel. Instead, when there is a channel that meets the trigger execution conditions, all channels update the triggering instant and the current measurement is transmitted to the remote filter. This is different from the traditional ETMs. Considering the impact of delays on measurement, the ETMs adopted in this paper is conducive to the reconstruction of subsequent delayed measurement.
		
        

 §.§ Encoding-decoding schemes

            Consider using probabilistic bit flips EDSs to transmit the triggered measurement y̌_(s)(i,j). More specifically, first utilize an encoder to preprocess the triggered measurement y̌_(s)(i,j), i.e.,
			
    𝒬(·):y̌_(s)^γ(i,j)⟶𝒬(y̌_(s)^γ(i,j),L_(s))

            where y̌_(s)^γ(i,j) represents the γ-th component of y̌_(s)(i,j), γ∈[1,m] . Let the value range of the triggered measurement y̌_(s)^γ(i,j) at the horizon (i,j) be [-Z_(s),Z_(s)], among them, Z_(s)∈ℝ>0 is a known scalar determined by the range of the sensor s. By dividing [-Z_(s),Z_(s)] into uniform length intervals Δ_(s), the triggered measurement y̌_(s)^γ(i,j) is encoded as a binary bit string of length L_(s),
			
    Δ_(s)=(2Z_(s))/(2^L_(s)-1)

		   when cΔ_(s)≤y̌_(s)^γ(i,j)≤ (c+1)Δ_(s),
            𝒬(y̌_(s)^γ(i,j),L_(s))=cΔ_(s) or 𝒬(y̌_(s)^γ(i,j),L_(s))=(c+1)Δ_(s). The encoded output 𝒬(y̌_(s)^γ(i,j),L_(s)) is generated in the following probabilistic manner:
			
    {   Pr{𝒬(y̌_(s)^γ(i,j),L_(s))=cΔ_(s)}=1-p_(s)^γ(i,j)
       Pr{𝒬(y̌_(s)^γ(i,j),L_(s))=(c+1)Δ_(s)}=p_(s)^γ(i,j)
    					.

            where p_(s)^γ≜ (y̌_(s)^γ(i,j)-cΔ_(s))/Δ_(s) and 0≤ p_(s)^γ(i,j)≤ 1.
 Furthermore, the encoded output 𝒬(y̌_(s)^γ(i,j),L_(s)) can be represented in binary bits,
			
    𝒬(y̌_(s)^γ(i,j),L_(s))=-Z_(s)+∑_ν=1^L_(s)b_(s),ν^γ(i,j)2^ν-1Δ_(s)

            where y̌_(s)^γ(i,j) is encoded as a binary bit string {b_(s),1^γ(i,j),
 b_(s),2^γ(i,j),⋯,b_(s),L_(s)^γ(i,j)} and b_(s),ν^γ(i,j)∈{0,1} is an encoded bit stream. Such flow is transmitted through the memoryless binary symmetric channel, and bit flipping may occur due to channel noise. Correspondingly, the received bit stream written is {b⃗_(s),1^γ(i,j), b⃗_(s),2^γ(i,j),⋯,b⃗_(s),L_(s)^γ(i,j)} and b⃗_(s),ν^γ(i,j)∈{0,1} where
			
    b⃗_(s),ν^γ(i,j)=   ϱ_(s),ν(i,j)(1-b_(s),ν^γ(i,j))
       +(1-ϱ_(s),ν(i,j))b_(s),ν^γ(i,j)

			where ϱ_(s),μ(i,j) is a random variable obeying the Bernoulli distribution,
			
    ϱ_(s),μ(i,j)={   1, the ν-th bit is flipped
       0, the ν-th bit is not flipped.

            For the convenience of subsequent discussion, assuming that ϱ_(s),ν(i,j) is white and mutually uncorrelated, and have the same crossover probabilities. Pr{ϱ_(s),ν(i,j)=1}=ϱ_(s). Decode the received bit string and restore the triggered measurements according to the following rules:
			
    𝒬^∘(y̌_(s)^γ(i,j),L_(s))=-Z_(s)+∑_ν=1^L_(s)b⃗_(s),ν^γ(i,j)2^ν-1Δ_(s)

		
			In light of bit encoding rules, it can be seen that the encoded output can be rewritten as
			
    𝒬(y̌_(s)^γ(i,j),L_(s))=y̌_(s)^γ(i,j)+q_(s)^γ(i,j)

            where q_(s)^γ(i,j)=𝒬(y̌_(s)^γ(i,j),L_(s))-y̌_(s)^γ(i,j) represents the encoding truncation error. On the basis of (<ref>), q_(s)^γ(i,j) obeying the Bernoulli distribution taking values at p_(s)^γ(i,j)Δ_(s) or (p_(s)^γ(i,j)-1)Δ_(s), i.e.,
			
    {   Pr{q_(s)^γ(i,j)=-p_(s)^γ(i,j)Δ_(s)}=1-p_(s)^γ(i,j)
       Pr{q_(s)^γ(i,j)=(1-p_(s)^γ(i,j))Δ_(s)}=p_(s)^γ(i,j)
    					.

			The proof that the encoding truncation error is unbiased and the variance is bounded will be given later.
			
                Under the consideration of probabilistic bit flips EDSs, transmission errors (including quantization encoding errors and flipping bit errors) are inevitably generated during the transmission process. However, the quantization encoding error can be reduced by extending the length of the binary encoding stream, and the bit flipping error can be mitigated by utilizing more reliable communication channels and reducing the crossover probability.
			


§ PRELIMINARIES

    

 §.§ The internal dynamic variable properties

        Below, we provide a lemma to illustrate that the interval dynamic variable ξ^ι(i,j)≥0 is always hold.
		
            Let the parameters satisfies ρ_(s)≥1/α_(s),1, ρ_(s)≥1/α_(s),2, 0<α_(s),1,α_(s),2,σ_(s)^γ<1 for any γ∈[1,m]. Then, we have ξ_(s)^γ(i,j)≥0 for i,j∈[0,℘].
		
		
			1) For (i,j)=(i_t,j_t),  retrospecting the event generator function, it is easy to obtain that
			
    φ_(s)^γ(i,j)=   ς_(s)σ_(s)^γ(y_(s)^γ(i_t,j_t))^2+ξ_(s)^γ(i_t,j_t)
    
    				h_(s)^γ(i,j)=   σ_(s)^γ(y_(s)^γ(i_t,j_t))^2

            Due to the initial conditions of the interval dynamic variable and h_(s)^γ(i,j)≥0, so ξ_(s)^γ(i,j)=ξ_(s)^γ(i_t,j_t)≥0 is valid. Then, it indicates that for (i_t,j_t)≤ (i,j)<(i_t+1,j_t+1), φ_(s)^γ
 (i,j)>0, it is easy to know that
			
    h_(s)^γ(i,j)>-1/ς_(s)ξ_(s)^γ(i,j)

			Next, combining with (<ref>), we conclude that
            
			
    ξ_(s)^γ(i,j)>   (α_(s),1-1/ς_(s))ξ_(s)^γ(i,j-1)
       +(α_(s),2-1/ς_(s))ξ_(s)^γ(i-1,j)

            
		
			2) For (i_t,j_t)≤ (i,j) <(i_t+1,j_t+1), according to the above analysis, we have
            
			
    ξ_(s)^γ(i,j)   >(α_(s),1-1/ϱ_(s))^2ξ_(s)^γ(i,j-2)
       +2(α_(s),1-1/ς_(s))(α_(s),2-1/ς_(s))ξ_(s)^γ(i-1,j-1)
       +(α_(s),2-1/ς_(s))^2ξ_(s)^γ(i-2,j)
       >⋯
       >(α_(s),1-1/ς_(s))^j_tξ_(s)^γ(i,j_t)
       +∑_t=j_t+1^j-1i(j-t)(α_(s),1-1/ς_(s))^j-t(α_(s),2-1/ς_(s))^iξ_(s)^γ(i_t,t)
       +∑_t=i_t+1^i-1j(i-t)(α_(s),1-1/ς_(s))^j(α_(s),2-1/ς_(s))^i-tξ_(s)^γ(t,j_t)
       +(α_(s),2-1/ς_(s))^i_tξ_(s)^γ(i_t,j)
       ≥0.

            
			To this end, by the induction, it is easy to get that ξ_(s)^γ(i,j)≥0 for i,j∈[0,℘].
		

        Then, constrain the dynamic event-triggered error δ_(s)(i,j) by constructing an auxiliary variable ξ̌_(s)^γ(i,j). For (i_t,j_t)≤ (i,j) <(i_t+1,j_t+1), the auxiliary variable governed by the following dynamics:
        
		
    {ξ̌_(s)^γ(i+1,j+1)   =α_(s),1ξ̌_(s)^γ(i+1,j)+α_(s),2ξ̌_(s)^γ(i,j+1)
       +σ_(s)^ι(y_(s)^γ(i+1,j))^2+σ_(s)^γ(y_(s)^γ(i,j+1))^2
    ξ̌_(s)^γ(i_t,j)   =ξ_(s)^γ(i_t,j), ξ̌_(s)^γ(i,j_t)=ξ_(s)^γ(i,j_t)
    				.

        
        ξ̌_(s)^γ(i,0)=ξ̌_(s)^γ(0,j)=ξ_(s)^0≥0 is given initial conditions of the auxiliary variable ξ̌_(s)^γ(i,j). The following lemma will illustrate that ξ̌_(s)^γ(i,j)≥ξ_(s)^γ(i,j).
		
            For i,j∈[0,℘], the interval dynamic variable ξ_(s)^γ(i,j) is not greater than the auxiliary variable ξ̌_(s)^γ(i,j).
		
		
			Prove this lemma via using mathematical induction, for (i_t,j_t)≤ (i,j) <(i_t+1,j_t+1).
			
			 Initial Step: In light of (<ref>) and (<ref>), it is easy to obtain
            
			
    ξ_(s)^γ(i_t+1,j_t+1)   =α_(s),1ξ^γ_(s)(i_t+1,j_t)+α_(s),2ξ^γ_(s)(i_t,j_t+1)
       +h^γ_(s)(i_t+1,j_t)+h^γ_(s)(i_t,j_t+1)
    ξ̌_(s)(i_t+1,j_t+1)   =α_(s),1ξ̌^γ_(s)(i_t+1,j_t)+α_(s),2ξ̌^γ_(s)(i_t,j_t+1)
       +σ_(s)^γ(y_(s)^γ(i_t+1,j_t))^2+σ_(s)^γ(y_(s)^γ(i_t,j_t+1))^2

            
            Due to ξ̌_(s)^γ(i_t,j_t+1)=ξ_(s)^γ(i_t,j_t+1), ξ̌_(s)^γ(i_t+1,j_t)=ξ_(s)^γ(i_t
 +1,j_t) and the definition of h_(s)^γ(i,j). It is obvious that
			
    ξ̌_(s)^γ(i_t+1,j_t+1)≥ξ_(s)^γ(i_t+1,j_t+1)

		
            Inductive Step: Let ξ̌_(s)^γ(i,j)≥ξ_(s)^γ(i,j) is hold for (i,j)∈{(l,k)|l∈[i_t+1,i_t+1),k∈[j_t+1,j_t+1),l+k=L} with L∈[i_t+j_t+2,i_t+1+j_t+1-3), then ,for (i,j)∈{(l,k)|l∈[i_t+1,i_t+1),k∈[j_t+1,j_t+1),l+k=L+1}, we has
            
			
    ξ_(s)^γ(i,j)=   α_(s),1ξ_(s)^γ(i,j-1)+α_(s),2ξ_(s)^γ(i-1,j)
       +h_(s)^γ(i,j-1)+h_(s)^γ(i-1,j)
    ξ̌_(s)^γ(i,j)=   α_(s),1ξ̌_(s)^γ(i,j-1)+α_(s),2ξ̌_(s)^γ(i-1,j)
       +σ_(s)^γ(y_(s)^γ(i,j-1))^2+σ_(s)^γ(y_(s)^γ(i-1,j))^2.

            
            It can be known from the hypothesis that ξ̌_(s)^γ(i,j)≥ξ_(s)^γ(i,j). Therefor, along the same lines, we can conclude ξ̌_(s)^γ(i,j)≥ξ_(s)^γ(i,j) for i,j∈[0,℘].
		

        Based on the Lemma <ref>, a constrained upper bound for dynamic triggered error is given.
        
		
    (e_(s)^γ(i,j))^2   ≤σ_(s)^γ/ρ_(s)^γ(y_(s)^γ(i,j))^2+1/ς_(s)ρ_(s)^γξ_(s)^γ(i,j)
       ≤σ_(s)^γ/ρ_(s)^γ(y_(s)^γ(i,j))^2+1/ς_(s)ρ_(s)^γξ̌_(s)^γ(i,j)
       ≜(e̅_(s)^γ(i,j))^2.

        
		Thereby, the local upper bound of the dynamic triggering error covariance can be obtained
		
    𝔼{e_(s)(i,j)(e_(s)(i,j))^T}
         ≤diag_1≤γ≤ m{(e̅_(s)^γ(i,j))^2}≜δ_(s)(i,j).


	

 §.§ Effects of binary encoding-decoding schemes

		
            In the EDSs process, the encoding truncation error q_(s)^γ(i,j) is of zero-mean and bounded variance, i.e.
			
    ℰ{q_(s)^γ(i,j)}=0,
       ℰ{(q_(s)^γ(i,j))^2}≤1/4Δ_(s)^2.

		
		
			 Retrospecting the probability distribution of encoding truncation error, it is easy to conclude that
			 
    ℰ{q_(s)^γ(i,j)}=   -p_(s)^γ(i,j)Δ_(s)(1-p_(s)^γ(i,j))
       +(1-p_(s)^γ(i,j))Δ_(s)p_(s)^γ(i,j)
    
    			 	=   0.

		 	Similarly, it is easy to know
		 	
    Pr{(q_(s)^γ(i,j))^2=(p_(s)^γ(i,j))^2Δ_(s)^2}=1-p_(s)^γ(i,j)
       Pr{(q_(s)^γ(i,j))^2=(1-p_(s)^γ(i,j))^2Δ_(s)^2}=p_(s)^γ(i,j)

	 		which means that
	 		
    ℰ{(q_(s)^γ(i,j))^2}=   (p_(s)^γ(i,j))^2Δ_(s)^2(1-p_(s)^γ(i,j))
       +(1-p_(s)^γ(i,j))^2Δ_(s)^2p_(s)^γ(i,j)
    ≤   1/4Δ_(s)^2.

 			This completes the proof.
		
		
            In the EDSs process, the encoding output 𝒬(y̌_(s)^γ(i,j),L_(s)) generated by the uniformly randomized quantizer is unbiased.
		
		
            According to (<ref>) and the truncation error q_(s)^γ(i,
 j) is of zero-mean, it is easy to know that 𝒬(y̌_(s)^γ(i,j),L_(s)) is unbiased.
		
	
		An important property of memoryless binary symmetric channel is given below.
		
            Suppose that the encoded output 𝒬(y̌_(s)^γ(i,j),
L_(s)) is transmitted through a memoryless binary symmetric channel with a crossover probability of ϱ_(s). Then, the mean and variance of the decoded output 𝒬^∘(y̌_(s)^γ(i,j),L_(s)) are
            
    ℰ{𝒬^∘(y̌_(s)^γ(i,j),L_(s))}=(1-2ϱ_(s))𝒬(y̌_(s)^γ(i,j),L_(s))
       Var{𝒬^∘(y̌_(s)^γ(i,j),L_(s))}=ϱ_(s)(1-ϱ_(s))4Z_(s)^2(2^2L_(s)-1)/3(2^L_(s)-1)^2

		    where the expectation is about the random variable ϱ_(s),ν(i,j).
		
        
			According to (<ref>), one has
            
			
    ℰ{𝒬^∘(y̌_(s)^γ(i,j),L_(s))}=-Z_(s)+∑_ν=1^L_(s)ℰ{b^γ_(s),ν(i,j)}2^ν-1Δ_(s)
        =-Z_(s)+∑_ν=1^L_(s)(ϱ_(s)(1-b_(s),ν^γ(i,j))
         +(1-ϱ_(s))b_(s),ν^γ(i,j))2^ν-1Δ_(s)
       =𝒬(y_(s)^γ(i,j),L_(s))+ϱ_(s)∑_ν=1^L_(s)(1-2b_(s),ν^γ(i,j))2^ν-1Δ_(s).

            
			The last term of the above equation can be rewritten as
			
            
    ϱ_(s)∑_ν=1^L_(s)(1-2b_(s),ν^γ(i,j))2^ν-1Δ_(s)
         =ϱ_(s)(∑_ν=1^L_(s)2^ν-1Δ_(s)-2∑_ν=1^L_(s)b_(s),ν^γ(i,j)2^ν-1Δ_(s))
         =2ϱ_(s)(Z_(s)-∑_ν=1^L_(s)b_(s),ν^γ(i,j)2^ν-1Δ_(s))

            
            In view of (<ref>), ℰ{𝒬^∘(y̌_(s)^γ(i, j),L_(s))}=(1-2ϱ_(s)) 𝒬(y̌_(s)^γ(i,
 j),L_(s)) can be verified. Moreover, the variance of 𝒬^∘(y̌_(s)^γ(i,
 j),L_(s)) is:
			
            
    Var{𝒬^∘(y̌_(s)^γ(i,j),L_(s))}=ℰ{(-Z_(s)+∑_ν=1^L_(s)b_(s),ν^γ(i,j)
         ×
                    2^ν-1Δ_(s))^2}-(ℰ{𝒬^∘(y_(s)^γ(i,j),L_(s))})^2
       =ℰ{(∑_ν=1^L_(s)(b_(s),ν^γ(i,j)-ℰ{b_(s),ν^γ(i,j)})2^ν-1Δ_(s)
         +ℰ{𝒬^∘(y_(s)^γ(i,j),L_(s))})^2}+(ℰ{𝒬^∘(y_(s)^γ(i,j),L_(s))})^2
       =ℰ{(∑_ν=1^L_(s)(b_(s),ν^γ(i,j)-ℰ{b_(s),ν^γ(i,j)})2^ν-1Δ_(s))^2}

            
            It is worth noting that b_(s),μ^γ(i,j)(μ∈[1,L_(s)]) are mutually independent with b_(s),μ^γ(i,j)∈{0,1}. Therefore, the above equation can be further expressed as
			
            
    ℰ{(∑_ν=1^L_(s)(b_(s),ν^γ(i,j)-ℰ{b_(s),ν^γ(i,j)})2^ν-1Δ_(s))^2}
       =∑_ν=1^L_(s)(ℰ{(b_(s),ν^γ(i,j))^2}-(ℰ{b_(s),ν^γ(i,j)})^2)2^2ν-2Δ_(s)^2
       =ϱ_(s)(1-ϱ_(s))4Z_(s)^2(2^2L_(s)-1)/3(2^L_(s)-1)^2

            
			This completes the proof.
		

        Based on Lemma <ref>, the decoded output 𝒬^∘(y̌_(s)^γ(i,j),L_(s)) received through the memoryless binary symmetric channel can be restated as
		
    𝒬^∘(y̌_(s)^γ(i,j),L_(s))
        =(1-2ϱ_(s))𝒬(y̌_(s)^γ(i,j),L_(s))+ϰ_(s)^γ(i,j)

        where ϰ_(s)^γ(i,j)∈ℝ is random variable with ℰ{ϰ_(s)^γ(i,j)}=0 and Var{ϰ_(s)^γ(i,j)}=Var{𝒬^∘(y̌_(s)^γ(i,j),L_(s))}≜ℶ_(s)^γ.
	

 §.§ Original decoded measurement description

        Let 𝒬_o^∘(y_s(l,k),L_s) denotes the original decoded measurement received at the (l,k) horizon of the 2-D systems (<ref>) and (<ref>). For the sake of clarity, define the following sets
		
    𝒯_0^+={(l,k)|_0≤ l< _1,_1≤ k≤ j},
       𝒯_0^∘ ={(l,k)|_0≤ l< _1,d_0≤ k< _1},
       𝒯_0^- ={(l,k)|_1≤ l≤ i,d_0≤ k< _1},
                           ⋮
       𝒯_N-1^+={(l,k)|_N-1≤ l< _N,_N≤ k≤ j},
       𝒯_N-1^∘={(l,k)|_N-1≤ l< _N,_N-1≤ k< _N},
       𝒯_N-1^-={(l,k)|_N≤ l≤ i,_N-1≤ k< _N},
       𝒯_N^∘   ={(l,k)|_N≤ l≤ i,_N≤ k≤ j},
       𝒯_s=𝒯_s^+∪𝒯_s^∘∪𝒯_s^-(s∈[0,N-1]),
                𝒯_N=𝒯_N^∘.

		The original decoded measurement can be describe as
		
    𝒬_o^∘(y_s(l,k),L_s)≜[ 𝒬^∘(y_(0)(l,k),L_(0)); 𝒬^∘(y_(1)(l,k),L_(1));                     ⋮; 𝒬^∘(y_(s)(l,k),L_(s)) ]

        where (i_t,j_t)≤(i,j)<(i_t+1,j_t+1), s∈{0,1,⋯,N} for (l,k)∈{𝒯_0,𝒯_1,⋯,𝒯_N}.
        
            We will utilize the original decoded output sequence to estimate the system state. In light of (<ref>), the original decoded measurement 𝒬_o^∘(y_s(l,k),L_s) contain asynchronous-delay, the filter developed in <cit.> and <cit.> cannot be used to solve the proposed estimation problem in this paper. Thus, it is necessary to develop a new approach to address asynchronous-delay and propose a novel recursive filter.
		
    

 §.§ Reconstruction of the decoded measurement

        In order to detailed investigate the estimation problem with measurement delays, we only consider the case of i≥_N, j≥_N. Other cases can be handled in the same way. For brevity, denote i_s=i-_N-s,j_s=j-_N-s and define the following sets
		
    𝒮_0^∘={(l,k)|0≤ l≤ i_0,0≤ k≤ j_0},
       𝒮_1^-={(l,k)|0≤ l≤ i_0,j_0<k≤ j_1},
       𝒮_1^∘={(l,k)|i_0<l≤ i_1,j_0< k≤ j_1},
       𝒮_1^+={(l,k)|i_0<l≤ i_1,0≤ k≤ j_0},
                     ⋮
       𝒮_N^-={(l,k)|0≤ l≤ i_N-1,j_N-1<k≤ j_N},
       𝒮_N^∘={(l,k)|i_N-1<l≤ i_N,j_N-1< k≤ j_N},
       𝒮_N^+={(l,k)|i_N-1<l≤ i_N,0≤ k≤ j_N-1},
       𝒮_0=𝒮_0^∘,𝒮_s=𝒮_s^-∪𝒮_s^∘∪𝒮_s^+(s∈[1,N]),


		The original decoded measurements can be reconstructed as follows:
		
    𝒬_r^∘(y_s(l,k),L_s)≜[       𝒬^∘(y_(0)(l,k),L_(0)); 𝒬^∘(y_(1)(l+_1,k+_1),L_(1));                           ⋮; 𝒬^∘(y_(s)(l+_s,k+_s),L_(s)) ]

        where (i_t,j_t)≤(i,j)<(i_t+1,j_t+1), s∈{0,1,⋯,N} for (l,k)∈{𝒮_N,𝒮_N-1,⋯,𝒮_0}.
		
        In view of the definition of the original measurement and the original encoded measurement, it is easy to conclude that the corresponding reconstruction are y_s(l,k)≜col_0≤ c ≤ s{y_(c)(l+_c,k+_c)},𝒬_r(y_s(l,k),L_s)≜col_0≤ c ≤ s{𝒬(y_(c)(l+_c,k+_c),L_(c))}.
	
		It is easy to know that y_s(l,k) satisfy
		
    y_s(l,k)=   C_s(l,k)x(l,k)+v_s(l,k)

        where C_s(l,k)=col_0≤ c ≤ s{C_(c)(l+_c,k+_c)} and v_s(l,k)=col_0≤ c ≤ s{v_(c)(l+_c,k+_c)}. v_s(l,k) is zero-mean white noise with covariance matrices R_s(l,k)=diag_0≤ c≤ s{R_(c)(l+_c,k+_c)}. Obviously, there is delay-free existed in (<ref>). The following lemma will show that the reconstructed decoded measurement sequence and the original decoded measurement sequence have the same information.
		
            The linear space spanned by the reconstructed measurement sequences and the linear space spanned by the original measurement sequences are equivalent, i.e.
			
    ℒ{𝒬_r^∘(y_s(l,k),L_s)| (l,k)∈𝒮_N∪𝒮_N-1∪⋯∪𝒮_0}=
    ℒ{𝒬_o^∘(y_s(l,k),L_s)| (l,k)∈𝒯_0∪𝒯_1∪⋯∪𝒯_N}

		
        
            The proof resembles that in <cit.>, due to space limitations, it is unmentioned.
		
		
            The new decoded measurement sequence, {𝒬_r^∘
 (y_s(l,k),L_s)|(l,k)∈𝒮_N∪𝒮_N-1∪⋯∪𝒮_1} is named as reconstructed measurement sequence of {𝒬_o^∘(y_s(l,k),L_s)|(l,k)∈𝒯_0∪𝒯_1∪⋯∪𝒯_N}. The reconstructed decoded measurement sequence is all about the state information at the same horizon. And it is obvious from Lemma <ref> that the reconstructed decoding measurement sequence contains the same information as the original decoding measurement sequence.
		

        The next section will give a filter design framework based on the reconstructed decoding measurement sequence. In order to facilitate analysis, for (i_t,j_t)≤(i,j)<(i_t+1,j_t+1), define y̌_s(l,k)≜ y_s(l_t,k_t),
			𝒬_r(y̌_s(l,k),L_s)≜𝒬_r(y_s(l_t,k_t),L_s),
𝒬_r^∘(y̌_s(l,k),L_s)≜𝒬_r^∘(y_s(l_t,k_t),L_s)



§ MAIN RESULTS

	

 §.§ Filter development

			In this section, a N+1-step recursive filter is proposed as follows:
			
			
				𝐒𝐭𝐞𝐩  1: For (l,k)∈𝒮_0
				
    x̂_p^1(l,k)=   f_1((l,k-1),x̂_u^1(l,k-1))
       +f_2((l-1,k),x̂_u^1(l-1,k))
    x̂_u^1(l,k)=   x̂_p^1(l,k)+(K_1(l,k)+α_1(l,k))
       ×(𝒬_r^∘(y̌_N(l,k),L_N)-C_N(l,k)x̂_p^1(l,k))

			
            where x̂_p^1(l,k) is the one-step prediction of the state x(l, k), x̂_u^1(l,k) is the corresponding updated estimate with the initial
			conditions x̂_u^1(l,0)=x_u(l,0) and x̂_u^1(0,k)=x_u(0,k).
			
			
				𝐒𝐭𝐞𝐩  2: For (l,k)∈𝒮_1
				
    x̂_p^2(l,k)=   f_1((l,k-1),x̂_u^1(l,k-1))
       +f_2((l-1,k),x̂_u^1(l-1,k))
    x̂_u^2(l,k)=   x̂_p^2(l,k)+(K_2(l,k)+α_2(l,k))
       ×(𝒬_r^∘(y̌_N-1(l,k),L_N-1)-C_N-1(l,k)x̂_p^2(l,k))

			
            where x̂_p^2(l,k) is the one-step prediction of the state x(l,k), x̂_u^2(l,k) is the corresponding updated estimate with the initial conditions x̂_u^2(l,0)=x_u(l,0)(i_0+1≤ l ≤ i_1),x̂_u^2(l,j_0)=x̂_u^1(l,j_0)(0 ≤ l ≤ i_0) and x̂_u^2(0,k)=x_u(0,k)(j_0+1 ≤ k ≤ j_1), x̂_u^2(i_0,k)=x̂_u^1(i_0,k)(0 ≤ k ≤ j_0).
			
			⋮
			
			
				𝐒𝐭𝐞𝐩  𝐍+1: For (l,k)∈𝒮_N
				
    x̂_p^N+1(l,k)=   f_1((l,k-1),x̂_u^N+1(l,k-1))
       +f_2((l-1,k),x̂_u^N+1(l-1,k))
    x̂_u^N+1(l,k)=   x̂_p^N+1(l,k)+(K_N+1(l,k)+α_N+1(l,k))
       ×(𝒬_r^∘(y̌_0(l,k),L_0)-C_0(l,k)x̂_p^N+1(l,k))

			
            where x̂_p^N+1(l,k) is the one-step prediction of the state x(l,k), x̂_u^N+1(l,k) is the corresponding updated estimate with the initial conditions x̂_u^N+1(l,0)=x_u(l,0)(i_N-1+1≤ l ≤ i_N), x̂_u^N+1(l,j_N-1)=x̂_u^N(l,j_N-1)(0 ≤ l ≤ i_N-1) and x̂_u^N+1(0,k)=x_u(0,k) (j_N-1+1 ≤ k ≤ j_N), x̂_u^N+1(i_N-1,k)
= x̂_u^N(i_N-1,k)(0 ≤ k ≤ j_N-1).
			
            In (<ref>)-(<ref>), K_1(l,k), K_2(l,k), ⋯, K_N+1(l,k) are estimation gain parameters to be determined. α_1, α_2, ⋯, α_N+1 are the estimation gain variation. α_τ(l,k)=∑_ν=1^r_τβ_τ,ν (i,j)H_τ,ν, where
            H_τ,ν∈ℝ^n × (N+1-τ)m is a set of known matrices and β_τ,μ(i,j) is scalar-valued white noise with ℰ{β_τ,ν(i,j)}=0 and Var{β_τ,ν(i,j)}=β̌_τ,ν.
			
                In control systems, accurate calculation of estimation gain is one of the key factors in ensuring system performance. Nevertheless, due to various reasons such as the non-ideal characteristics of electronic components, external interferences, and temperature changes, errors or uncertainties may arise in the calculation of estimation gain during the hardware implementation process. Such error or uncertainty can lead to a decrease in system performance or even cause system instability. Therefore, introducing the resilient characteristic in the estimation gain design is an effective strategy <cit.>. Specifically, the introduction of the matrix α_τ(l,k) to simulate gain perturbations allows for the consideration of possible changes and uncertainty in estimation gain during the design process, thereby enhancing the robustness of the control system.
			
			
			For the N+1-step recursive filter, for the sake of simplicity,
			we define
			
    x̃_p^τ(l,k)   ≜ x(l,k)-x̂_p^τ(l,k)
    x̃_u^τ(l,k)   ≜ x(l,k)-x̂_u^τ(l,k)
    
                    P_p^τ(l,k)   ≜ℰ{x̃_p^τ(l,k)(x̃_p^τ(l,k))^T}
    
                    P_u^τ(l,k)   ≜ℰ{x̃_u^τ(l,k)(x̃_u^τ(l,k))^T}
    
    				X(l,k)   ≜ℰ{x(l,k)x^T(l,k)}

			Then, it follows from (<ref>)-(<ref>) and (<ref>)-(<ref>) that,
			
			
				𝐒𝐭𝐞𝐩  1: For (l,k)∈𝒮_0
				
                
    x̃_p^1(l,k)=   f̃_1((l,k-1),x̃_u^1(l,k-1))
       +f̃_2((l-1,k),x̃_u^1(l-1,k))+B_1(l,k-1)
       × w(l,k-1)+B_2(l-1,k)w(l-1,k)
    x̃_u^1(l,k)=   [I-(K_1(l,k)+α_1(l,k))C_N(l,k)]x̃_p^1(l,k)
       -[K_1(l,k)+α_1(l,k)]{-2ϱ_N C_N(l,k)x(l,k)
       +(I-2ϱ_N)[q_N(l,k)-e_N(l,k)]+ϰ_N(l,k)
       +(I-2ϱ_N)v_N(l,k)}

                
			

            
				𝐒𝐭𝐞𝐩  2: For (l,k)∈𝒮_1
				
                
    x̃_p^2(l,k)=   f̃_1((l,k-1),x̃_u^2(l,k-1))
       +f̃_2((l-1,k),x̃_u^2(l-1,k))+B_1(l,k-1)
       × w(l,k-1)+B_2(l-1,k)w(l-1,k)
    x̃_u^2(l,k)=   [I-(K_2(l,k)+α_2(l,k))C_N-1(l,k)]x̃_p^2(l,k)
       -[K_2(l,k)+α_2(l,k)]{-2ϱ_N-1 C_N-1(l,k)x(l,k)
       +(I-2ϱ_N-1)[q_N-1(l,k)-e_N-1(l,k)]+ϰ_N-1(l,k)
       +(I-2ϱ_N-1)v_N-1(l,k)}

                
			
		
			⋮
			
			
				𝐒𝐭𝐞𝐩  𝐍+1: For (l,k)∈𝒮_N
				
                
    x̃_p^N+1(l,k)=   f̃_1((l,k-1),x̃_u^N+1(l,k-1))
       +f̃_2((l-1,k),x̃_u^N+1(l-1,k))+B_1(l,k-1)
       × w(l,k-1)+B_2(l-1,k)w(l-1,k)
    x̃_u^N+1(l,k)=   [I-(K_N+1(l,k)+α_N+1(l,k))C_0(l,k)]x̃_p^N+1(l,k)
       -[K_N+1(l,k)+α_N+1(l,k)]{-2ϱ_0 C_0(l,k)x(l,k)
       +(I-2ϱ_0)[q_0(l,k)-e_0(l,k)]+ϰ_0(l,k)
       +(I-2ϱ_0)v_0(l,k)}

                
			
			for ι∈[1,2],τ∈[1,N+1],s∈[0,N], where
			
    f̃_ι((l,k),x̃_u^τ(l,k))=f_ι((l,k),x(l,k))-f_ι((l,k),x̂_u^τ(l,k))
       ϱ_s=diag_0≤ c ≤ s{ϱ_(c)⊗ I_m}
       e_s(l,k)=col_0 ≤ c ≤ s{e_(c)(l+_c,k+_c)}
       e_(c)(l+_c,k+_c)=col_1≤γ≤
                    m{e_(c)^γ(l+_c,k+_c)}
       q_s(l,k)=col_0 ≤ c ≤ s{q_(c)(l+_c,k+_c)}
       q_(c)(l+_c,k+_c)=col_1≤γ≤
                    m{q_(c)^γ(l+_c,k+_c)}
       ϰ_s(l,k)=col_0 ≤ c ≤ s{ϰ_(c)(l+_c,k+_c)}
       ϰ_(c)(l+_c,k+_c)=col_1≤γ≤
                    m{ϰ_(c)^γ(l+_c,k+_c)}

            
                It is worth noting that due to the existence of event triggering errors and encoding truncation errors, it is difficult to accurately calculate the error covariance dynamics in analytical form. Therefore, the target of this paper is to design an upper bound for the actual filtering error covariance, and to minimize this upper bound by appropriately designing estimation gains.
            	
    

 §.§ Design of estimation gain parameters

        In this subsection, we investigate estimation gains design problem for the resilient filter. Before presenting the main results, first, give the upper bound of the second-order moment of system state x(l,k).
			
                Let ǎ_1,ǎ_2>0 are given scalars, the upper bound of the second-order moment of the state x(l,k) is the solution of the following recursive evolution:
				
    X̅(l,k)=   (1+ǎ_1)X̂_1(l,k-1)+(1+ǎ_1^-1)X̂_2(l-1,k)
       +Q_1(l,k-1)+Q_2(l-1,k)

                with initial condition X̅(l,0)=X(l,0) and X̅(0,k)= X(0,k
), where X̂_ι(l,k)≜ (1+ǎ_2)a_ι^2tr{X̅(l,k)}I+(1+ǎ_2^-1)A_ι(l,k)
X̅(l,k)A_ι^T(l,k) and Q_ι(l,k)=B_ι(l,k)Q(l,k)B_ι^T(l,k), ι=1,2.
			
			
				According to (<ref>)-(<ref>) and the statistical property of w(l,k)
				
    X(l,k)≤    (1+ǎ_1)ℰ{f_1((l,k-1),x(l,k-1))
       × f_1^T((l,k-1),x(l,k-1))}+Q_1(l,k-1)
       +(1+ǎ_1^-1) ℰ{f_2((l-1,k),x(l-1,k))
       × f_2^T((l-1,k),x(l-1,k))}+Q_2(l-1,k)

                Next, we will conduct a detailed analysis of ℰ{f_ι((l,k),
 x(l,k))f_ι^T((l,k),x(l,k))},
				
                
    ℰ{f_ι((l,k),x(l,k))f_ι^T((l,k),x(l,k))}
         =ℰ{(f_ι((l,k),x(l,k))-f_ι((l,k),0)-A_ι(l,k)x(l,k)
            +A_ι(l,k)x(l,k))(f_ι((l,k),x(l,k))-f_ι((l,k),0)
            -A_ι(l,k)x(l,k)+A_ι(l,k)x(l,k))^T}
         ≤ (1+ǎ_2)ℰ{(f_ι((l,k),x(l,k))-f_ι((l,k),0)
           -A_ι(l,k)x(l,k))(f_ι((l,k),x(l,k))-f_ι((l,k),0)
           -A_ι(l,k)x(l,k))}+(1+ǎ_2^-1)A_ι(l,k)X(l,k)A_ι^T(l,k)
         ≤
                        (1+ǎ_2)a_ι^2tr{X(l,k)}I+(1+ǎ_2^-1)A_ι(l,k)X(l,k)A_ι^T(l,k)

                
				Based on the above derivation, it can be inferred that
				
    X(l,k)≤    (1+ǎ_1)X_1(l,k-1)+(1+ǎ_2)X_2(l-1,k)
       +Q_1(l,k-1)+Q_2(l-1,k)

                where X_ι(l,k)≜ (1+ǎ_2)tr{X(l,k)}I+(1+ǎ_2^-1)A_ι(l,k)
X(l,k)A_ι^T(l,k).
                Then, using mathematical induction to prove that X̅(l,k)≥ X(l,k). To begin with, retrospecting the initial condition and (<ref>) that X̅(1,1)≥ X(1,1). Assume that X̅(l,k)≥ X(l,k) hold for (l,k)∈{(l_0,k_0)|l_0,k_0∈[1,℘]; l_0
 +k_0=♭} with ♭∈[2,2℘-1], then, for (l,k)∈{(l_0,k_0)|l_0,
 k_0∈[1,℘];l_0+k_0=♭+1}, one has
				
    X̅(l,k)-X(l,k)
         ≥ (1+ǎ_1)(X̂_1(l,k-1)-X_1(l,k-1))
            +(1+ǎ_1^-1)(X̂_2(l-1,k)-X_2(l-1,k))
         ≥0

				This completes the proof.
			
            
                Let b̌_1, b̌_2, b̌_3, b̌_4, b̌_5, and b̌_6 are given positive scalar, the upper bound of the one-step prediction error covariance and the filtering error covariance are the solution of the following recursive evolution of the difference equations:
			
			
				𝐒𝐭𝐞𝐩  1: For (l,k)∈𝒮_0
				
                
    Ξ_p^1(l,k)=   (1+b̌_1)Ξ_1p^1(l,k-1)+(1+b̌_1^-1)Ξ_2p^1(l-1,k)
       +Q_1(l,k-1)+Q_2(l-1,k)
    Ξ_u^1(l,k)=   (1+b̌_3+b̌_4)(Ξ_p^1(l,k)-K_1(l,k)C_N(l,k)
       ×Ξ_p^1(l,k)-Ξ_p^1(l,k)C_ħ^T(l,k)K_1^T(l,k))
       +K_1(l,k)ϝ̌_N(Ξ_p^1(l,k))K_1^T(l,k)
       +∑_ν=1^r_1β̌_1,μH_1,μϝ̌_N(Ξ_p^1(l,k))H_1,μ^T

                
			
			with the initial condition Ξ_u^1(l,0)=P_u^1(l,0) and Ξ_u^1(0,k)=P_u^1(0,k).
			
			
				𝐒𝐭𝐞𝐩  2: For (l,k)∈𝒮_1
                
				
    Ξ_p^2(l,k)=   (1+b̌_1)Ξ_1p^2(l,k-1)+(1+b̌_1^-1)Ξ_2p^2(l-1,k)
       +Q_1(l,k-1)+Q_2(l-1,k)
    Ξ_u^2(l,k)=   (1+b̌_3+b̌_4)(Ξ_p^2(l,k)-K_2(l,k)C_ħ-1(l,k)
       ×Ξ_p^2(l,k)-Ξ_p^2(l,k)C_N-1^T(l,k)K_2^T(l,k))
       +K_2(l,k)ϝ̌_N-1(Ξ_p^2(l,k))K_2^T(l,k)
       +∑_ν=1^r_2β̌_2,μH_2,μϝ̌_N-1(Ξ_p^2(l,k))H_2,μ^T

                
			
            with the initial condition Ξ_u^2(l,0)=P_u^2(l,0)(i_0+1≤ l ≤ i_1), Ξ_u^2(l,j_0)=Ξ_u^1(l,j_0)(0≤ l ≤ i_0) and Ξ_u^2(0,k)=P_u^2(0,k)(j_0+1≤ k ≤ j_1), Ξ_u^2(i_0,k)=Ξ_u^1(i_0,k)(0≤ k ≤ j_0).
			
			⋮
			
			
				𝐒𝐭𝐞𝐩  𝐍+1: For (l,k)∈𝒮_N
				
                
    Ξ_p^N+1(l,k)=   (1+b̌_1)Ξ_1p^N+1(l,k-1)+(1+b̌_1^-1)Ξ_2p^N+1(l-1,k)
       +Q_1(l,k-1)+Q_2(l-1,k)
    Ξ_u^N+1(l,k)=   (1+b̌_3+b̌_4)(Ξ_p^N+1(l,k)-K_N+1(l,k)C_0(l,k)
       ×Ξ_p^N+1(l,k)-Ξ_p^N+1(l,k)C_0^T(l,k)K_N+1^T(l,k))
       +K_N+1(l,k)ϝ̌_0(Ξ_p^N+1(l,k))K_N+1^T(l,k)
       +∑_ν=1^r_N+1β̌_N+1,μH_N+1,μϝ̌_0(Ξ_p^N+1(l,k))H_N+1,μ^T

                
			
            with the initial condition Ξ_u^N+1(l,0)=P_u^N+1(l,0)(i_N-1+1≤ l ≤ i_N), Ξ_u^N+1(l, j_N-1)=Ξ_u^N(l,j_N-1)(0≤ l ≤ i_N-1) and Ξ_u^N+1(0,k)=P_u^N+1(0,k)(j_N-1+1≤ k ≤ j_N), Ξ_u^N+1(i_N-1,k)=Ξ_u^N(i_N-1,k)(0≤ k ≤ j_N-1).
			
			where
			
            
    Ξ_ι p^τ(l,k)   =(1+b̌_2)a_ι^2(l,k)tr{Ξ_u^τ(l,k)}I
        +(1+b̌_2^-1)A_ι(l,k)Ξ_u^τ(l,k)A_ι^T(l,k)
    Θ̌_N+1-τ(X̅(l,k))   =(1+b̌_3^-1+b̌_5)4ϱ_N+1-τC_N+1-τ(l,k)
       ×X̅(l,k)C_N+1-τ^T(l,k)ϱ_N+1-τ
       +(I-2ϱ_N+1-τ)ϑ_N+1-τ(I-2ϱ_N+1-τ)
       +(1+b̌_4^-1+b̌_5^-1+b̌_6)(I-2ϱ_N+1-τ)
       ×δ_N+1-τ(l,k)(I-2ϱ_N+1-τ)+(1+b̌_6^-1)
       ×(I-2ϱ_N+1-τ)R_N+1-τ(I-2ϱ_N+1-τ)
       +ℶ_N+1-τ
    ϝ̌_N+1-τ(Ξ_p^τ(l,k))   =Θ̌_N+1-τ(X̅(l,k))+(1+b̌_3+b̌_4)
       × C_N+1-τ(l,k)Ξ_p^τ(l,k)C_N+1-τ^T(l,k)

            
		
        
			According to (<ref>) and (<ref>), it is easy to know that
			
            
    P_p^1(l,k)=   ℰ{f̃_1((l,k-1),x̃_u^1(l,k-1))f̃_1^T((l,k-1),
       x̃_u^1(l,k-1))}+ℰ{f̃_2((l-1,k),x̃_u^1(l-1,k))
       ×f̃_2^T((l-1,k),x̃_u^1(l-1,k))}+Q_1(l,k-1)
       +Q_2(l-1,k)+sym(ℰ{f̃_1((l,k-1),x̃_u^1(l,k-1))
       ×f̃_2^T((l-1,k),x̃_u^1(l-1,k))})

            
            Then, we will conduct a detailed analysis of ℰ{f̃_ι((l,k),
x̃_u^1(l,k))f̃_ι^T((l,k),x̃_u^1(l,k))},
			
            
    ℰ{f̃_ι((l,k),x̃_u^1(l,k))f̃_ι^T((l,k),x̃_u^1(l,k))}
          =ℰ{(f̃_ι((l,k),x̃_u^1(l,k))-(A_ι(l,k)-A_ι(l,k))x̃_u^1(l,k))
           ×(f̃_ι((l,k),x̃_u^1(l,k)) -(A_ι(l,k)-A_ι(l,k))x̃_u^1(l,k))^T}
         ≤(1+b̌_2)ℰ{(f̃_ι((l,k),x̃_u^1(l,k))-A_ι(l,k)x̃_u^1(l,k))
           ×(f̃_ι((l,k),x̃_u^1(l,k))-A_ι(l,k)x̃_u^1(l,k))^T}
            +(1+b̌_2^-1)A_ι(l,k)P_u^1(l,k)A_ι^T(l,k)
         ≤ (1+b̌_2)a_ι^2(l,k)tr{P_u^1(l,k)}I
            +(1+b̌_2^-1)A_ι(l,k)P_u^1(l,k)A_ι^T(l,k)≜ P_ι p^1(l,k)

            
            By using inequality ab^T+ba^T≤b̌_1aa^T+b̌_1^-1bb^T, (<ref>) can be rewritten as
			
    P_p^1(l,k)≤    (1+b̌_1)P_1p^1(l,k-1)+(1+b̌_1^-1)P_2p^1(l-1,k)
       +Q_1(l,k-1)+Q_2(l-1,k)

			According to (<ref>) and (<ref>), it is easy to know that
			
            
    P_u^1(l,k)=   ℰ{[I-(K_1(l,k)+α_1(l,k))C_N(l,k)]P_p^1(l,k)
       ×[I-(K_1(l,k)+α_1(l,k))C_N(l,k)]^T
       +[K_1(l,k)+α_1(l,k)]{4ϱ_N C_N(l,k)X(l,k)C_N^T(l,k)ϱ_N
        +(I-2ϱ_N)(ℰ{q_N(l,k)q_N^T(l,k)}
       +ℰ{e_N(l,k)e_N^T(l,k)}+R_N(l,k))(I-2ϱ_N)
       +ℰ{ϰ_N(l,k)ϰ_N^T(l,k)}}[K_1(l,k)+α_1(l,k)]^T
       +sym{ℸ_11+ℸ_21+ℸ_31+ℸ_41+ℸ_51}}

            
			where
			
            
    ℸ_11=   -[I-(K_1(l,k)+α_1(l,k))C_N(l,k)]
       ×ℰ{x̃_p^1(l,k)(-2ϱ_N C_N(l,k)x(l,k)+(I-2ϱ_N)
       ×[q_N(l,k)-e_N(l,k)+v_N(l,k)]+ϰ_N(l,k))^T}
       ×[K_1(l,k)+α_1(l,k)]^T
    ℸ_21=   [K_1(l,k)+α_1(l,k)]ℰ{-2ϱ_N C_N(l,k)x(l,k)((I-2ϱ_N)
       ×[q_N(l,k)-e_N(l,k)+v_N(l,k)]+ϰ_N(l,k))^T}
       ×[K_1(l,k)+α_1(l,k)]^T
    ℸ_31=   [K_1(l,k)+α_1(l,k)]ℰ{(I-2ϱ_N)q_N(l,k)((I-2ϱ_N)
       ×[-e_N(l,k)+v_N(l,k)]+ϰ_N(l,k))^T}
       ×[K_1(l,k)+α_1(l,k)]^T
    ℸ_41=   [K_1(l,k)+α_1(l,k)]ℰ{(I-2ϱ_N)e_N(l,k)((I-2ϱ_N)
       × v_N(l,k)+ϰ_N(l,k))^T}[K_1(l,k)+α_1(l,k)]^T
    ℸ_51=   [K_1(l,k)+α_1(l,k)]ℰ{(I-2ϱ_N)v_N(l,k)ϰ_N^T(l,k)}
       ×[K_1(l,k)+α_1(l,k)]^T

            
            It is worth noting that x̃_p^1(l,k) is independent of q_N(l,k),
 v_N(l,k), and ϰ_N(l,k). Can be derived that
			
            
    sym{ℸ_11}≤    (b̌_3+b̌_4)[I-(K_1(l,k)+α_1(l,k))C_N(l,k)]
       × P_p^1(l,k)[I-(K_1(l,k)+α_1(l,k))C_N(l,k)]^T
       +b̌_3^-1[K_1(l,k)+α_1(l,k)]4ϱ_NC_N(l,k)
       × X(l,k)C_N^T(l,k)ϱ_N[K_1(l,k)+α_1(l,k)]^T
       +b̌_4^-1[K_1(l,k)+α_1(l,k)](I-2ϱ_N)
       ×ℰ{e_N(l,k)e_N^T(l,k)}(I-2ϱ_N)[K_1(l,k)+α_1(l,k)]^T.

            
			Furthermore, x(l,k) is independent of q_N(l,k), v_N(l,k), and ϰ_N(l,k).
			
    sym{ℸ_21}≤   b̌_5[K_1(l,k)+α_1(l,k)]4ϱ_NC_N(l,k)
       × X(l,k)C_N^T(l,k)ϱ_N[K_1(l,k)+α_1(l,k)]^T
       +b̌_5^-1[K_1(l,k)+α_1(l,k)](I-2ϱ_N)ℰ{e_N(l,k)
       × e_N^T(l,k)}(I-2ϱ_N)[K_1(l,k)+α_1(l,k)]^T

            It is known that q_N(l,k), v_N(l,k), and ϰ_N(l,k) are mutually independent, thus, sym{ℸ_31+ℸ_51}=0,
			
    sym{ℸ_41}≤   b̌_6 [K_1(l,k)+α_1(l,k)](I-2ϱ_N)ℰ{e_N(l,k)
       × e_N^T(l,k)}(I-2ϱ_N)[K_1(l,k)+α_1(l,k)]^T
       +b̌_6^-1[K_1(l,k)+α_1(l,k)](1-2ϱ_N)
       × R_N(l,k)(I-2ϱ_N)[K_1(l,k)+α_1(l,k)]^T

			Substituting all cross terms into (<ref>) to obtain
			
            
    P_u^1(l,k)≤   ℰ{(1+b̌_3+b̌_4)[I-(K_1(l,k)+α_1(l,k))C_N(l,k)]
       × P_p^1(l,k) [I-(K_1(l,k)+α_1(l,k))C_N(l,k)]^T
       +(1+b̌_3^-1+b̌_5)[K_1(l,k)+α_1(l,k)]4ϱ_N
       × C_N(l,k)X(l,k)C_N^T(l,k)ϱ_N[K_1(l,k)+α_1(l,k)]^T
       +[K_1(l,k)+α_1(l,k)](I-2ϱ_N)ℰ{q_N(l,k)
       × q_N^T(l,k)}(I-2ϱ_N)[K_1(l,k)+α_1(l,k)]^T
       +(1+b̌_4^-1+b̌_5^-1+b̌_6)[K_1(l,k)+α_1(l,k)](I-2ϱ_N)
       ×ℰ{e_N(l,k)e_N^T(l,k)}(I-2ϱ_N)[K_1(l,k)+α_1(l,k)]^T
       +(1+b̌_6^-1)[K_1(l,k)+α_1(l,k)](I-2ϱ_N)R_N(l,k)
       × (I-2ϱ_N)[K_1(l,k)+α_1(l,k)]^T+[K_1(l,k)
       +α_1(l,k)] ℰ{ϰ_N(l,k)ϰ_N^T(l,k)}
       ×[K_1(l,k)+α_1(l,k)]^T}

            
			The first term in the above equation can be decomposed to obtain,
			
            
    ℰ{[I-(K_1(l,k)+α_1(l,k))C_N(l,k)]P_p^1(l,k)
         ×[I-(K_1(l,k)+α_1(l,k))C_N(l,k)]^T}
        =P_p^1(l,k)-K_1(l,k)C_N(l,k)P_p^1(l,k)
          -P_p^1(l,k)C_N^T(l,k)K_1^T(l,k)+ℰ{(K_1(l,k)+α_1(l,k))
         × C_N(l,k)P_p^1(l,k)C_N^T(l,k)(K_1(l,k)+α_1(l,k))^T}

            
			and
			
            
    ℰ{(K_1(l,k)+α_1(l,k))C_N(l,k)P_p^1(l,k)
           × C_N^T(l,k)(K_1(l,k)+α_1(l,k))^T}
         =K_1(l,k)C_N(l,k)P_p^1(l,k)C_N^T(l,k)K_1^T(l,k)
           +∑_ν=1^r_1β̌_1,νH_1,νC_N(l,k)P_p^1(l,k)C_N^T(l,k)H_1,ν^T

            
			In view of the ETMs,
			
    ℰ{e_N(l,k)e_N^T(l,k)}   ≤diag_0≤ c ≤ N{δ_(c)(l+_c,k+_c)}≜δ_N(l,k).

			According to the statistical property of the truncation error,
			
    ℰ{q_N(l,k)q_N^T(l,k)}
        = diag_0≤ c ≤ N{ℰ{q_(c)(l+_c,k+_c)q_(c)^T(l+_c,k+_c)}}
       ℰ{q_(c)(l+_c,k+_c)q_(c)^T(l+_c,k+_c)}
        = diag_1 ≤γ≤ m{ℰ{(q_(c)^γ(l+_c,k+_c))^2}}
        ≤diag_1 ≤γ≤ m{1/4Δ_(c)^2}.

			In light of (<ref>), it is easy to conclude that
			
    ℰ{ϰ_N(l,k)ϰ_N^T(l,k)}
        =diag_0 ≤ c ≤ N{ℰ{ϰ_(c)(l+_c,k+_c)ϰ_(c)^T(l+_c,k+_c)}}
       ℰ{ϰ_(c)(l+_c,k+_c)ϰ_(c)^T(l+_c,k+_c)}
        =diag_1≤γ≤ m{ℰ{(ϰ_(c)^γ(l+_c,k+_c))^2}}
        ≤diag_1≤γ≤ m{ℶ_(c)^γ}

			so, we can record briefly
			
    ℰ{q_N(l,k)q_N^T(l,k)}≤ϑ_N,  ℰ{ϰ_N(l,k)ϰ_N^T(l,k)}≤ℶ_N.

			Denote
			
            
    Θ_N(X(l,k))   =(1+b̌_3^-1+b̌_5)4ϱ_NC_N(l,k)X(l,k)C_N^T(l,k)ϱ_N
       +(I-2ϱ_N)ϑ_N(I-2ϱ_N)+(1+b̌_4^-1+b̌_5^-1+b̌_6)
       ×(I-2ϱ_N)δ_N(l,k)(I-2ϱ_N)+(1+b̌_6^-1)
       × (I-2ϱ_N)R_N(l,k)(I-2ϱ_N)+ℶ_N
    ϝ_N(P_p^1(l,k))   =Θ_N(X(l,k))+(1+b̌_3+b̌_4)
       × C_N(l,k)P_p^1(l,k)C_N^T(l,k).

            
			It is easy to know that
			
    P_u^1(l,k)≤    (1+b̌_3+b̌_4)(P_p^1(l,k)-K_1(l,k)C_N(l,k)
       × P_p^1(l,k)-P_p^1(l,k)C_N^T(l,k)K_1^T(l,k))
       +K_1(l,k)ϝ_N(P_p^1(l,k))K_1^T(l,k)
       +∑_ν=1^r_1β̌_1,νH_1,νϝ_N(P_p^1(l,k))H_1,ν^T.

            Then, by using mathematical induction to prove that Ξ_p^1(l,k)
≥ P_p^1(l,k) and Ξ_u^1(l,k)≥ P_u^1(l,k). According to the initial condition and (<ref>), it is easy to know that Ξ_p^1(1,1)≥ P_p^1(1,1), which imply Ξ_u^1(1,1)≥ P_u^1(1,1). Next, assume Ξ_u^1(l,k)≥ P_u^1(l,k) is valid for (l,k)∈{(l_0,k_0)|l_0,k_0∈[1,℘]; l_0+k_0=♭} with ♭∈[2,2℘-1], then, for (l,k)∈{(l_0,k_0)|l_0,k_0∈[1,℘];l_0+k_0=♭+1}, one has
			
            
    Ξ_p^1(l,k)-P_p^1(l,k)
        ≥ (1+b̌_1)(Ξ_1p^1(l,k-1)-P_1p^1(l,k-1))
          +(1+b̌_1^-1)(Ξ_2p^1(l-1,k)-P_2p^1(l-1,k))
        ≥ 0
       Ξ_u^1(l,k)-P_u^1(l,k)
        ≥ (1+b̌_3+b̌_4)[I-(K_1(l,k)+α_1(l,k))C_N(l,k)]
         ×(Ξ_p^1(l,k)-P_p^1(l,k))[I-(K_1(l,k)+α_1(l,k))C_N(l,k)]^T
          +K_1(l,k)(Θ̌_N(X̅(l,k))-Θ_N(X(l,k)))K_1^T(l,k)
         +∑_μ^r_1β̌_1,μH_1,μ(ϝ̌_N(Ξ̅_p^1(l,k))-ϝ_N(Ξ_p^1(l,k)))H_1,μ^T
        ≥ 0

            
			Similar methods can be used to prove other areas. This completes the proof of the lemma.
		

        The following theorem gives the estimation gain parameters by minimizing the local upper bound on the estimation error covariance.
		
            For the 2-D systems (<ref>) and (<ref>), the recursive evolution of the upper bound of filtering error covariance and the estimation gains are given as follows,
			
			
				𝐒𝐭𝐞𝐩  1: For (l,k)∈𝒮_0
				
                
    K_1(l,k)=   (1+b̌_3+b̌_4)Ξ_p^1(l,k)C_N^T(l,k)(ϝ̌_N(Ξ_p^1(l,k)))^-1
    Ξ_u^1(l,k)=   (1+b̌_3+b̌_4)Ξ_p^1(l,k)-K_1(l,k)ϝ̌_N(Ξ_p^1(l,k))
       × K_1^T(l,k)+∑_ν=1^r_1β̌_1,νH_1,νϝ̌_N(Ξ_p^1(l,k))H^T_1,ν

                
			
		
			
				𝐒𝐭𝐞𝐩  2: For (l,k)∈𝒮_1
				
                
    K_2(l,k)=   (1+b̌_3+b̌_4)Ξ_p^2(l,k)C_N-1^T(l,k)(ϝ̌_N-1(Ξ_p^2(l,k)))^-1
    Ξ_u^2(l,k)=   (1+b̌_3+b̌_4)Ξ_p^2(l,k)-K_2(l,k)ϝ̌_N-1(Ξ_p^2(l,k))
       × K_2^T(l,k)+∑_ν=1^r_2β̌_2,νH_2,νϝ̌_N-1(Ξ_p^2(l,k))H^T_2,ν

                
			
			⋮
			
			
				𝐒𝐭𝐞𝐩  𝐍+1: For (l,k)∈𝒮_N
                
                
    K_N+1(l,k)=   (1+b̌_3+b̌_4)Ξ_p^N+1(l,k)
       × C_0^T(l,k)(ϝ̌_0(Ξ_p^N+1(l,k)))^-1
    Ξ_u^N+1(l,k)=   (1+b̌_3+b̌_4)Ξ_p^N+1(l,k)-K_N+1(l,k)
       ×ϝ̌_0(Ξ_p^N+1(l,k))K_N+1^T(l,k)+∑_ν=1^r_N+1β̌_N+1,ν
       × H_N+1,νϝ̌_0(Ξ_p^N+1(l,k))H^T_N+1,ν

                
			
		
		
            Since the proof of each step is similar, for simplicity, only the proof of 𝐒𝐭𝐞𝐩 1 is given here. By using the square completion method for (<ref>), it can be obtained that
			
    Ξ_u^1(l,k)=   (K_1(l,k)-K_1^*(l,k))ϝ̌_N(Ξ_p^1(l,k))
       ×(K_1(l,k)-K_1^*(l,k))^T+(1+b̌_3+b̌_4)Ξ_p^1(l,k)
       -K_1^*(l,k)ϝ̌_N(Ξ_p^1(l,k))(K_1^*(l,k))^T
       +∑_ν=1^r_1β̌_1,νH_1,νϝ̌_N(Ξ_p^1(l,k))H_1,ν^T

            where K_1^*(l,k)=(1+b̌_3+b̌_4)Ξ_p^1(l,k)C_N^T(l,k) (ϝ̌_N(Ξ_p^1(l,
k)))^-1 is minimized if and only if K_1(l,k)=K_1^*(l,k), so the assert (<ref>) is valid. Inserting (<ref>) into (<ref>), one has
			
    Ξ_u^1(l,k)=   (1+b̌_3+b̌_4)Ξ_p^1(l,k)-K_1(l,k)ϝ̌_N(Ξ_p^1(l,k))
       × K_1^T(l,k)+∑_ν=1^r_1β̌_1,νH_1,νϝ̌_N(Ξ_p^1(l,k))H^T_1,ν

			This completes the proof.
		
        
            So far, the event-triggered filtering problem of 2-D systems with asynchronous-delay under binary encoding-decoding schemes have been solved. Due to the existence of asynchronous-delay, the measurements are divided into N+1 regions for reconstruction, and the corresponding N+1-step recursive filter is given. Considering the impact of triggering error and encoding truncation error on the filter, the local upper bounds of prediction error covariance and filtering error covariance are given. The estimation gain parameters are obtained by minimizing the local upper bound of filtering error covariance. It is obvious from (<ref>)-(<ref>) that all information about the system model (triggering parameter, encoding truncation error, and relevant decoding information) are reflected in the filter design and has a direct impact on filtering performance.
		
	

 §.§ Monotonicity of triggering parameters

        As shown in Theorem <ref>, the evolution of the minimum upper bound of prediction error covariance and filtering error covariance depends on the triggering parameters, which is essentially reflected in the impact of the trigger error on the minimum upper bound of the estimated error covariance. Therefore, in this subsection, strict mathematical derivation will be used to discuss the relationship between the triggering error δ_τ(l,k) and the local minimum upper bound of filtering error covariance Ξ_u^N+1-τ(l,k).
		
        For the sake of simplicity, only discuss (l,k)∈𝒮_0. The notations Ξ_p^1δ_N(l,k), Ξ_u^1δ_N(l,k), and K_1^δ_N represent the local upper bound of error covariance and the estimation gain parameter with the triggering error δ_N(l,k). The following defines the operators 𝒢((l,k),δ(l,k),K(l,k),Ξ(l,k)) and ℱ((l,k),δ(l,k),K(l,k),Ξ(l,k)),
		
    𝒢((l,k),δ(l,k),K(l,k),Ξ(l,k))
         ≜(1+b̌_3+b̌_4)[I-K(l,k)C_N(l,k)]
           ×Ξ(l,k)[I-K(l,k)C_N(l,k)]^T
            +∑_ν=1^r_1β̌_1,νH_1,νϝ̌_N(Ξ(l,k))H_1,ν^T
            +K(l,k)Θ̌(X̅(l,k))K^T(l,k)
       ℱ((l,k),δ(l,k),K(l,k),Ξ(l,k))
         ≜(1+b̌_1)Ξ_1(l,k-1)+(1+b̌_1^-1)Ξ_2(l-1,k)
            +Q_1(l,k-1)+Q_2(l-1,k)

        where Ξ_ι(l,k)=(1+b̌_2)a^2_ι(l,k)tr{Ξ(l,k)}I+(1+b̌_2^-1)A_ι(l,
 k)Ξ(l,k)A_ι^T(l,k). Obviously, it can be seen that Ξ_u^1δ_N(l,k)=
𝒢((l,k),δ_N(l,k),K_1^δ_N(l,k),Ξ_p^1δ_N(l,k)).
		
        From the definition of ϝ̌_N(Ξ_p^1δ_N(l,k)), it is easy to see that ϝ̌_N(Ξ_p^1δ_N(l,k)) is monotonic about δ_N(l,k). Then, 𝒢((l,k),
δ_N(l,k),K_1^δ_N(l,k),Ξ_p^1δ_N(l,k)) is monotonic about δ_N(l,k) and Ξ_p^1δ_N(l,k).
		
            For given scalars σ_(s)^γ,1, σ_(s)^γ,1, ρ_(s)^γ,1,ρ_(s)^γ,2,ς_s^1, and ς_s^2, if the conditions σ_(s)^γ,1≤σ_(s)^γ,1,ρ_(s)^γ,1≥ρ_(s)^γ,2, and ς_s^1≤ς_s^2, i.e. 0<δ_N^1(l,k)≤δ_N^2(l,k), then the following inequality holds
			
    Ξ_u^1δ_N^1(l,k)≤Ξ_u^1δ_N^2(l,k)

			for every k∈[0,i_0], k∈[0,j_0].
		
        
            In view of Theorem 1, Ξ_u^1δ_N(l,k) can be minimized if and only if K_1^δ_N(l,k)=K_1^*δ_N(l,k)=(1+b̌_3+b̌_4)Ξ_p^1δ_N(l,k)C_N^T(l,k)(ϝ̌(Ξ_p^1δ_N(l,k)))^-1, so
			
    𝒢((l,k),δ_N(l,k),K_1^*δ_N(l,k),Ξ_p^1δ_N(l,k))
               ≤𝒢((l,k),δ_N(l,k),K_1^δ_N(l,k),Ξ_p^1δ_N(l,k))

            Next, by using mathematical induction to prove this theorem.
            Initial Step:According to the initial condition Ξ_u^1δ_N(l,0) and Ξ_u^1δ_N(0,k) are independent of the triggering error δ_N(l,k), so Ξ_p^1δ_N^1(1,1)=Ξ_p^1δ_N^2(1,1), due to (<ref>), it is easy to know that
			
    Ξ_u^1,δ_N^1(1,1) ≤Ξ_u^1,δ_N^2(1,1).

            Inductive Step: Assume Ξ_u^1δ_N^1(l,k)≤Ξ_u^1δ_N^2(l,k) is valid for (l,k)∈{(l_0,k_0)|l_0∈[1,i_0],k_0∈[1,j_0]; l_0+k_0=♭} with ♭∈[2,i_0+j_0-1], then, for (l,k)∈{(l_0,k_0)|l_0∈[1,i_0],k_0∈[1,j_0];l_0+k_0=♭+1}, one has
			 
    Ξ_u^1,δ_N^1(l,k)   =𝒢((l,k),δ_N^1(l,k),K_1^*δ_N^1(l,k),Ξ_p^1δ_N^1(l,k))
       ≤𝒢((l,k),δ_N^1(l,k),K_1^*δ_N^2(l,k),Ξ_p^1δ_N^1(l,k))
       ≤𝒢((l,k),δ_N^2(l,k),K_1^*δ_N^2(l,k),Ξ_p^1δ_N^2(l,k))
       =Ξ_u^1,δ_N^2(l,k).

		 	This completes the proof.
		
		
            It is noteworthy that, according to Theorem <ref>, the upper bound of the filtering error covariance does not decrease as the triggering error increases. Intuitively speaking, a larger triggering error leads to a higher frequency of update triggers, resulting in increased utilization of network resources by each channel, thereby exacerbating the tension on network resource utilization. At the same time, an increase in triggering error implies that more signals will be transmitted for estimation, which to a certain extent enhances the accuracy of the estimation. Therefore, setting the triggering parameters reasonably can achieve a satisfactory compromise between energy saving and estimation performance.
		



§ CONCLUSION

    To summarize, this paper has contributed some of the first efforts to the problem of recursive filtering with asynchronous-delay. First, an ETMs have been introduced under the premise of asynchronous-delay to reduce the number of signal transmissions, improve communication quality, and enable measurement reconstruction after triggering. Then, the mathematical quantification of signal distortion during communication under binary EDSs has been given. Then, through mathematical induction, a local upper bound on the estimation error covariance has been found, and the estimation gain parameters have been given by minimizing the upper bound on the estimation error covariance. In addition, the monotonicity of the filtering performance with respect to the triggering parameters has been further discussed. Future research topics include extending existing filter design results to have more complex dynamics and communication mechanisms, such as dynamic ETMs with random communication delay, and unknown input estimation problems with asynchronous-delay <cit.>.


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