H ∞ Filtering of 2D Linear FM II Model Delayed Systems

. The aim of this study is to address the challenge of H ∞ filtering for 2-D systems with time delays. We will employ the well-known Fornasini-Marchesini state model with delay to mathematically describe these 2-D systems. Our objective is to develop a full-order filter that not only ensures the asymptotic stability of the expanded system but also maintains H ∞ performance. Our approach will rely on Linear Matrix Inequalities (LMIs) to establish the conditions for the filter’s existence and convert the filter design problem into a convex optimization problem. Finally, we conclude with an illustrative example to demonstrate the effectiveness of the proposed filter design.


Introduction
Researchers have increasingly focused on discrete-time 2-D systems, driven by their wideranging applications across diverse fields including image processing, seismographic data analysis, thermal process control, and the optimization of water flow heating, as highlighted by the work of [1].This trend reflects the recognition of the versatility and adaptability of such systems in tackling real-world challenges across multiple domains.Nonetheless, the literature has documented a number of significant findings, including the examination of 2-D system stability [2], which stands out as a noteworthy example.Moreover, the controller and filter design challenges have been tackled in the estimation problem for 2-D digital filters in [3].Recently, The use of the Generalized Kalman Yakobovich Popov (GKYP) lemma has been employed to investigate the design of a Static Output Feedback (SOF) controller, specifically focusing on finite-frequency (FF) specifications, within the context of Two-Dimensional (2D) Fornasini-Marchesini (FM-II) Discrete Systems [4].
Within the realm of signal processing, the H ∞ filtering problem has garnered significant interest among researchers in recent times.The core objective of the H ∞ filtering problem is to identify a full-order filter (or reduced order) capable of ensuring that the associated augmented system adheres to the constraints imposed by the H ∞ norm.This intriguing problem has motivated a series of investigations, as evidenced by the works of [5,8,9], reflecting the active exploration of H ∞ filtering solutions in the field.
Furthermore, it is widely acknowledged that time delays are a common occurrence in dynamic systems and can frequently lead to issues such as instability and suboptimal performance.Consequently, over the past decade, significant progress has been made in the analysis and design of systems with time delays [10].On the other hand, considering both time delays and the problem of H ∞ filtering for discrete-time 2-D systems has unfortunately not received sufficient attention from researchers especially using FMII model.This situation is primarily due to the complexity of stability analysis, despite the evident potential of this approach in engineering applications.For instance, [11] considers the problem of H ∞ filtering for uncertain two-dimensional discrete systems with state-varying delays.Since this work, many researchers have focused on the Roesser model, to mention a few [12] and [13] tackled the problem of filtering for 2D systems with delay described by the Roesser Model.On the other hand, findings concerning filtering of FMII second model systems with time delay still deserve much intentions.
It is in this context that our current research finds its motivation.We are striving to fill this gap by exploring new approaches and developing innovative techniques to solve the H ∞ filtering problem for discrete-time 2-D systems, considering both time delays.We believe that these efforts will contribute to enhancing our understanding of complex 2-D systems based on FMII model and will open new perspectives for practical applications in various engineering domains.The aim of this work is to tackle the problem of H ∞ filtering for 2-D FM systems with delay.Our objective is to design a full-order filter, which will simultaneously guarantee the asymptotic stability of such systems and guarantee a level of the H ∞ performance.Using the LMI (linear matrix inequality) tool, and introducing free-weighting matrices through which sufficient conditions of existence of the desired filter will be derived.Lately, a numerical example from the literature will be used to highlight the correctness of our approach.This work is organized as follows: section 2 is reserved for formulating the problem, while the 3 rd section presents the H ∞ performance analysis condition of the studied system.Afterward, section 4 will be dedicated to the design condition of the filter.Finally, sections 5 and 6 present the numerical example and some conclusions, respectively.

problem formulation
Consider the following 2-D F-M system with state delay: Where χ (i, j) ∈ R n is the state vector; ω(i, j) ∈ l 2 [0, ∞) is the disturbance input; Y (i, j) ∈ R m is the measured output; Z (i, j) ∈ R p is the output signal to be estimated where i, j are integers, d 1 and d 2 are positive integer constants representing respectively the delay in the horizontal and vertical directions.
The objective of the filtering problem is to estimate the output signal Z(i, j) through a full-order linear dynamic filter described by : and C f are constant matrices of appropriate dimension to be determined.Now let's augment the model in (1) to include the filter state (3), and we get the augmented system as follows: with such that the corresponding transfer function in the frequency domain is under the following form such that: Z 1 = e jω 1 and Z 2 = e jω 2 such that ω 1 and ω 2 bolong to [−π, π].

Definition 2
The FM filter in (4) is said to be an H ∞ filter if the augmented FM system in (5) is asymptotically stable and satisfies the H ∞ performance in (9) under the boundary conditions in (2) .
Objective: The target is to determine the matrices A for the 2-D FM delay system in (1) .Knowing that for any non-zero ω(i, j)∈ l 2 [0, ∞), the augmented FM system in (5) is asymptotically stable that satisfies (9).
Lemma 1 [6] Considering matrices Ξ = Ξ T ∈ R n×n , Ω ∈ R k×n and F ∈ R m×n the following problem: is solvable by considering the variable X if and only if the following statements are equivalents: Next, we present some primary results related to the same problem.The following Lemma handling a sufficient analysis condition for the issue of linear filtering for 2D FMII system: Lemma 2 [7] The FM augmented system in (5) is asymptotically stable with a level of disturbance attenuation γ if there exist positive definite matrices P >0 , Q >0 , Q 1 >0 and Q 2 >0 such as the following inequality hold:

H ∞ Filter Analysis
In this section, we will deal with the problem of filtering analysis.In particular, we'll assume that the filter matrices in (4) are known, and we'll study the conditions under which the augmented system is asymptotically stable, taking into account the H ∞ boundedγ norm.
E3S Web of Conferences 469, 00097 (2023) ICEGC'2023 https://doi.org/10.1051/e3sconf/202346900097 Theorem 1 Given the 2-D of F-M system in (1) and the filter in (4), the augmented system in (5) is asymptotically stable and satisfies (9) with an H ∞ performance level γ, if there exist positive definite matrices: Proof 1 Based on the inequality (11) given in Theorem 1, it can be seen that this inequality can be expressed with the following inequality: such that: and F=I.
Using the result provided through Lemma 2, from the inequality in (12), the following statement is held: . and F ⊥ =0.Then, using Schur-complement and including the given parameters in Theorem 1 with their forms, the inequality in Lemma 2 is verified.This proves the correctness of our proposed results in Theorem 1.This completes the proof.

Remark 1
The condition presented via Theorem 1 provides sufficient conditions for the existence of H ∞ filter (4) that guarantee the asymptotic stability of the augmented systems in (5).This result is provided through the use of Lemma 1 which helps to introduce slack variables.This technique allows a separation between system matrices and Lyapunov variables which leads to less conservative results and more flexibility in the solution space.

Synthesis of the H ∞ filter
This part will be devoted to establishing a design for the filter H ∞ in (4), which is equivalent to determining the filter matrices in (4) in such a way that the system is asymptotically stable with respect to the H ∞ norm.
Theorem 2 Consider the 2D FM state delay system given in (1), the augmented filtering systems in (5) is asymptotically stable respecting the performance inequality in (9), if there exist positive definite matrices P > 0, Q > 0, Q 1 > 0 and Q 2 > 0, the free weighting matrices E, K, Q, F and H (The structure of these matrices is given in the proof), the filter parameters Ā1 f , Ā2 f , B1 f , B2 f and Cf such that : Where with the filter design is given by: Proof 2 Based on the conditions in Theorem (1), and by assigning the following variable structures such that: P= let's substitute every matrix and variable by its own structure in the inequality (11), then we adopt the following change of variable Thereafter, by using this change of variable the inequality in (13) is verified.This ends the proof.
Remark 2 Based on the result proposed in Theorem 2, the existence of the filter parameters is related to the guarantee of the invertibility of the matrix K.For this reason, from the inequality (13) we can say that P 3 − λ 1 ( K + KT )<0.Then, while P 3 >0 we can conclude that −λ 1 ( K + KT )<0 which means that K is invertible and has an non-singular matrix K−1 .This fact allows a new design methodology to filter parameters which is totally different from the approach provided in the work presented in [7], and which also leads to less conservative results in comparison to the same work.

Remark 3
The slack structure used in this paper helps to provide better results than what existed in the literature.The proposed structure is based on the use of some tuning parameters λ i (i = 1...5) which play an important role in improving the obtained results.Consequently, by providing these tuning parameters even by choice or by using some algorithms such as fminsearch from Matlab Toolbox [14], the inequality in (13) becomes an LMI.

Simulation example
In order to highlight the effectiveness of our approach, we consider the following example from the literature describing certain dynamic phenomena involving gas absorption, heating of water streams, and air drying processes that can be effectively characterized using the Darboux equation, which incorporates time delays [15].These systems are modeled under the FMII discrete model such that the system matrices are given in the following form [7]: Using the provided result by solving the proposed conditions in Theorem 2, the corresponding H ∞ performance is γ=0.8351, and the corresponding filter parameters are presented by the following matrices: For the simulation result, and based on the filter parameters and system matrices above, let's consider the perturbation signal ω(i, j) under the following form: To verify the obtained H ∞ performance value γ=0.8351, we present Figure 1 that illustrates the correctness of the obtained value of γ in contrast to maximum value of the transfer function given in equation (6).
From Figure 1, the obtained value of H ∞ performance γ under the condition given in Theorem 2 is verified by the maximum value of the transfer function G in equation (6).The idea is to verify the correctness of the obtained value of γ in order to highlight the correctness of the obtained filter values with respect to the transfer function of the augmented systems in (5).
On the other hand, to show the superiority of our approach we provide Table 1 which presents a comparison study based on the H ∞ performance values obtained through our approach and the work presented in [7].  1, it is clear that the proposed approach in this paper provides better results than the work presented in [7].This advantage is illustrated by the mean of H ∞ performance index, which represents in a different way the reduction of the conservatism based on the proposed condition in Theorem 2. Additionally, this superiority is a result of introducing slack variables via Lemma 1, which leads to better results on both sides, H ∞ performances, and conservatism reduction.
A 1 , A 2 , A d 1 , A d 2 , B 1 , B 2 , C, D and L are constant matrices of compatible dimensions.compatible.The boundary conditions are given by