H ∞ Control for Network-Based 2 D Systems with Missing Measurements

and Applied Analysis 3 3. Main Results In this section, we assume that the system matrices A, B, B 1 and controller gain matrix G are known, and then we study the condition under which the closed-loop 2D system is mean-square asymptotically stable with a guaranteed H ∞ performance. Then, a feasible controller gain matrix can be given based on the condition. From system (1), (3), and (4), we can obtain the following closed-loop system:


Introduction
Two-dimensional (2D) systems have attracted considerable research interest over the past few decades due to their wide applications in the areas such as multidimensional digital filtering, linear image processing, signal processing, process control, and iterative learning control [1][2][3][4][5].Thus the stability and stabilization of 2D systems have attracted a lot of interests; see, for example, [6][7][8][9][10][11][12][13] and the references therein. ∞ optimization is a powerful tool that can be used to design a robust controller or filter, which has been proved to be one of the most important strategies.Recently, such problems on 2D systems have stirred a great deal of research attention.For example, several versions of 2D bounded real lemma have been established in [2,6,14].The problem of  ∞ control for 2D systems with state delays has been considered in [15].The problem of  ∞ filtering for 2D systems has been studied extensively in [16].An  ∞ controller is designed for a class of 2D nonlinear discrete systems with sector nonlinearity in [17,18].
Notice that all the above-mentioned results are based on an implicit assumption that the communication between the physical plant and controller is perfect; that is, the signals transmitted from the plant will arrive at the controller simultaneously and perfectly.However, in many practical situations, there may be a nonzero probability that all the signals can be measured during their transmission.In other words, the systems may have missing measurements.Moreover, networked systems are becoming more and more popular for the reason that they have several advantages over traditional systems, such as low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability [19][20][21][22].If network is introduced to controller design, the data packet dropout phenomenon, which appears in a typical network environment, will naturally induce missing measurements from the plant to the controller.In 1D system, the problems of stability, stabilization, filtering, or state estimation for networked control systems have been widely researched [23][24][25][26][27][28][29][30].However, in the network-based 2D system case, only few results have been available.For instance, the problem of robust  ∞ filtering for 2D systems described by the Fornasini-Marchesini (FM) second model with missing measurements is considered in [31].To the best of the authors' knowledge, the  ∞ control for 2D systems represented by the Roesser model which is structurally quite distinct from FM second model has not been addressed in the literature so far.
Motivated by the aforementioned observations, this paper considers the problem of  ∞ control for networkbased 2D systems presented by the Roesser model with missing measurements.We also notice that some dynamical processes such as gas absorption, water stream heating, and 2 Abstract and Applied Analysis air drying can be described by a 2D Roesser model.In practical, these systems are often implemented by distribute control systems (DCSs) or field-bus control systems (FCSs), where control loops that are closed over a communication network.Hence, the considered topic in this paper is of practical significance.Compared with existing results, this paper proposes a state feedback controller design method for 2D systems in the framework of networked control systems.Meanwhile, the introduction of the random missing measurements renders the 2D system to be stochastic instead of a deterministic one.To analyze the stability, we introduce the stochastic mean-square asymptotically stable and stochastic  ∞ disturbance attenuation level.The controller is also designed under the framework of 2D stochastic systems.
The remainder of this paper is organized as follows.In Section 2, the mathematical description and design objectives of this paper are presented.In Section 3, a sufficient condition of mean-square asymptotic stability with  ∞ performance for such 2D stochastic systems is derived by means of LMI technique, and then formulas can be given for the control law design.In Section 4, the design result is extended to the 2D systems with uncertain parameters.Numerical examples are given in Section 5 and conclusions are drawn in Section 6. Notation 1.The superscript "" denotes the matrix transposition,   denotes the -dimensional Euclidean space,  denotes the identity matrix, 0 denotes the zero vector or matrix with the required dimensions, and diag{⋅} denotes the standard (block) diagonal matrix whose off-diagonal elements are zero.In symmetric block matrices, an asterisk * is used to denote the term that is induced by symmetry.The notation ‖ ⋅ ‖ refers to the Euclidean vector norm and  min (⋅),  max (⋅) denote the minimum and the maximum eigenvalue of the corresponding matrix, respectively.{}, { | } mean the expectation of  and the expectation of  conditional on .Matrices, if the dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

Problem Formulation
Consider the following 2D discrete system given by where  ℎ (, ) ∈ Now, we consider the following state feedback control law: where x(, ) is the measurement of state signals,  is appropriately dimensioned controller matrix to be determined.When the feedback control is implemented via a networked control system, the data  ℎ (, ),  V (, ) are transferred as two separate packets from the remote plant to the controller.
In this process, the data may be missed due to the network transmission failure, resulting in what we call missing measurement.It is assumed that the data packet dropout can be described by a stochastic variable; that is, where (, ) = [  ℎ (,)  V (,) ], the stochastic parameters { , } is a Bernoulli distributed white sequence taking the values of 0 and 1 with Prob { , = 1} =  { , } = , and 0 ≤  ≤ 1 is a known constant.
Notice that the introduction of the stochastic variable { , } renders the 2D system to be stochastic instead of a deterministic one.Before proceeding further, we need to introduce the following definition of stochastic stability for the 2D system, which will be essential for our derivation.
Definition 2 (see [31]).The 2D system (1) is said to be mean-square asymptotically stable if under the zero input and for every bounded initial condition  ℎ (, 0),  V (0, ), the following is satisfied lim Definition 3. Given a scalar  > 0, the 2D system (1) is said to have an  ∞ disturbance attenuation level , if it is mean-square asymptotically stable and under zero initial conditions, ‖‖  < ‖‖ 2 is satisfied for any external disturbance (, ) ∈ ℓ 2 , where To this end, the design objective of this paper can be described as follows.
Problem Statement.For any initial condition satisfying Assumption (1) and missing measurements described as (5), design a state feedback law (3) such that the closed-loop 2D system is mean-square asymptotically stable and has an  ∞ disturbance attenuation level .

Main Results
In this section, we assume that the system matrices , ,  1 and controller gain matrix  are known, and then we study the condition under which the closed-loop 2D system is mean-square asymptotically stable with a guaranteed  ∞ performance.Then, a feasible controller gain matrix can be given based on the condition.
From system (1), (3), and (4), we can obtain the following closed-loop system: Define α, =  , − ; it is obvious that Then, the 2D closed-loop system can be rewritten as Theorem 4. The 2D closed-loop system (10) is mean-square asymptotically stable with a given  ∞ disturbance attenuation level , if there exists a positive define symmetric matrix where Proof.We first prove the mean-square asymptotically stability of 2D system (10) with zero disturbance (, ) = 0.In this case, the system becomes Define where Substituting (13) into the above index, we can obtain where From (11), it is easy to see that Ψ < 0. Hence, for all x ̸ = 0, we have where  = 1 − ( min (−Ψ)/ max ()).
Adding both sides of the inequalities yields Then, under Assumption 1, the right side of this inequality is bounded, which means that lim  → ∞ {  } = 0; that is, lim + → ∞ {‖(, )‖ 2 } = 0. Then the 2D stochastic system (10) is mean-square asymptotically stable.Now, the  ∞ performance for the 2D stochastic system (10) will be established.To this end, assume zero initial boundary conditions; that is,  ℎ (0, ) = 0,  V (, 0) = 0 for all .In this case, the index  becomes Another index is introduced as where  = [x  w ]  .From condition (11), we have Π < 0 for any  ̸ = 0; that is, Taking the expectation of both sides yields Due to the relationship (31), it can be established that . . .
Adding both sides of the inequality system, we have Summing up both sides of this inequality from  = 0 to  = , we have Considering the zero initial boundary conditions, (35) means This completes the proof.
Remark 5. Theorem 4 provides a sufficient condition of the mean-square asymptotic stability and  ∞ disturbance attenuation level  for 2D systems with missing measurements.
If the communication links existing between the plant and the controller are perfect, that is, there is no packet dropout during their transmission, then  = 1 and  = 0.In this case, the condition in Theorem 4 becomes the condition obtained in [6] for 2D deterministic system.From this point of view, Theorem 4 can be seen as an extension of [6] to 2D systems with missing measurement.Theorem 4 gives a mean-square asymptotic stability condition with  ∞ disturbance attenuation level  for system (10) where the controller gain matrix  is known.However, our eventual purpose is to determine a suitable  by system matrices , ,  1 and parameter .
The following well-known lemma is needed in the proof of our main result.
Lemma 6 (Schur complement).Assume , ,  are given matrices with appropriate dimensions, where  and  are positive definite symmetric matrices.Then if and only if or Based on the above lemma, we can give our main result.
Theorem 7.For the 2D closed-loop system (10), if there exist a positive define symmetric matrix  and a matrix  satisfying then the 2D closed-loop system (10) is mean-square asymptotically stability and has an  ∞ disturbance attenuation level .
In this case, a suitable state feedback control law can be given as  =  −1 .
Proof.The condition in Theorem 4 can be rewritten as where By applying Lemma 6, condition (41) is equivalent to the following LMI condition: Define  =  −1 , and pro-and postmultiplying diag(, , , , , ) for the above condition give Set  =  to obtain the LMI of (40) and the proof is complete.
Remark 8. Theorem 7 provides an LMI condition for the mean-square asymptotic stability and  ∞ disturbance attenuation level  of 2D stochastic system, which can be solved by LMI Toolbox.Then by (40), we also can give a suitable state feedback control law.It is noted that LMI (40) only contains few elements; hence, the computation is not big.However, when the system matrices get bigger, the solution of LMI (40) may be time-consuming due to complicated computation.In practical system, the system dimension is not very big (often less than 5).Hence, the computation complexity is acceptable.
Remark 9.For a fixed , the feasibility of (40) is a suboptimal  ∞ controller.When is not fixed, the minimization of  that satisfies (40) can be searched.Hence, an optimal  ∞ controller can be obtained by solving the following optimization problem: min (45)

Robust 𝐻 ∞ Control for Uncertain 2D Systems
In this section, we extend the above design to the case of robust  ∞ control.Consider the following 2D system with uncertain parameter perturbations where Δ, Δ, Δ 1 denote admissible uncertain perturbations of matrices , , and  1 , which can be represented as where ,  1 ,  2 ,  3 are known real constant matrices characterizing the structures of uncertain perturbations and Σ is an uncertain perturbation of the system that satisfies Σ  Σ ≤ .
Lemma 10.Assume ,  are matrices or vectors with appropriate dimensions.For any scalar  > 0 and all matrices Δ with appropriate dimensions satisfying ΔΔ  ≤ , the following inequality holds: The main result of this section is given in the following theorem.
Theorem 11.For the uncertain 2D system (46) used the state feedback control law (3) with missing measurement, if there exist a positive define symmetric matrix , a matrix  , and scalars  > 0 satisfying where That is, Therefore, using Lemma 10, there exists a scalar  > 0 such that Note that the above condition is equivalent to By the Schur complement, LMI (54) implies LMI in Theorem 11 holds.This completes the proof.
In addition, by solving the following optimization problem min ,,  2 s.t (49). ( we can obtain a robust optimal  ∞ controller for the uncertain 2D stochastic systems (46).
Remark 12.This paper considers the problem of  ∞ stabilization for a class of 2D systems with missing measurements.
Here, we describe the missing measurements as a Bernoulli random binary distribution, which renders the 2D systems to be stochastic ones.Under this framework, we give the definition of stochastic mean-square stability and  ∞ performance, and then a state feedback control design approach is addressed.The proposed design approach is systematic for 2D stochastic system.The results in this paper can be extended to solve other problems such as network-induced delay and  ∞ filter design.

Illustrative Examples
In this section, two numerical examples are used to illustrate the effectiveness of the proposed results.
It is assumed that measurements transmitted between the plant and the controller are imperfect; that is, the state signal may be lost during their transmission.Suppose  = 0.8; that is, in the communication link, the probability of the data packet missing is 20%.By applying Theorem 7 and solving the optimization problem (45), the minimum  ∞ disturbance attenuation level is  opt = 0.32.Meanwhile, we can obtain Hence, a feasible state feedback control law can be selected as Assume the disturbance is Simulation results are shown in Figures 1 and 2, where the state response of  ℎ (, ) is plotted in Figure 1 and  V (, ) is plotted in Figure 2. It can be seen from Figures 1 and 2 that the closed-loop 2D system is asymptotically stable.Hence, even though the 2D system is affected by external disturbances and has significant data dropout in the output measurements, the proposed design approach can guarantee the stability and disturbance attenuation ability for the 2D system.(60) It is also assumed that the probability of the data packet missing is 0.2; that is,  = 0.8.In this case, by applying Theorem 11 and solving the optimization problem (55), the The disturbance (, ) is also given as Example 1. Simulation results are shown in Figures 3 and 4, where the state response of  ℎ (, ) is plotted in Figure 3 and  V (, ) is plotted in Figure 4.It is observed that the closed-loop uncertain 2D system is also asymptotically stable.Hence, the proposed design approach is also effective for uncertain 2D system.

Conclusions
In this paper, we have investigated the problem of  ∞ stabilization for a class of 2D systems described with missing measurements.A sufficient condition has been developed in terms of LMIs, which guarantees mean-square asymptotic stability and  ∞ disturbance attenuation level for closedloop 2D system.The result is also extended to more general cases where the system matrices contain uncertain parameters.Numerical examples have been provided to illustrate the effectiveness of proposed approach.In our future work, the control and filtering problems for networked-based 2D system with packet dropouts and network-induced delay will be discussed.

Example 1 .
Let us consider 2D system (1) with the following parameters:

Example 2 .
Let us consider uncertain 2D system (46) with the following parameters: