Fuzzy reduced-order filtering for nonlinear parabolic PDE systems with limited communication

This paper investigates a fuzzy reduced-order filter design for a class of nonlinear partial differential equation (PDE) systems. First, a Takagi-Sugeno (T-S) fuzzy model is considered to reconstruct the nonlinear PDE system. Then, the employment of an event-triggered mechanism (ETM) can effectively avoid signal redundancy and improve network resource utilization. Furthermore, based on the advantages of the fuzzy model and ETM, several Lyapunov functions are designed and the proposed filter parameters are obtained by adopting linear matrix inequality methods to satisfy the asymptotic stability condition with 𝐻 1 performance. Finally, a simulation example is presented to demonstrate the practicality and effectiveness of the proposed filter design method.


INTRODUCTION
Numerous processes in industry are related not only to time but also to spatial location [1][2][3] , such as nuclear reaction processes, fluid heat exchange processes and biological systems [4][5][6] . These systems are called distributed parameter systems, usually described by partial differential equations (PDE) [7][8][9] . According to the different characteristics of the spatial differential operators, the PDE systems can be further divided into three categories, namely, hyperbolic [7] , parabolic [8] and elliptic [9] . In particular, parabolic PDEs can be applied to express the dynamic of industrial processes involve diffusion-convection-reaction processes, such as crystal growth processes, semiconductor thermal processes and wavy behavior in chemistry [10,11] . Therefore, control/filtering studies for parabolic PDE systems have attracted extensive attention [12][13][14][15][16][17] . For example, Wang et al. [12] introducted the estimator-based ∞ sampled-data fuzzy control for nonlinear parabolic PDE systems; Zhang et al. [16] addressed the controller design under mobile collocated actuators and sensors; Song et al. [17] discussed the reliable ∞ filter design for PDE system with Markovian jumping sensor faults, which stimulated the author' s interest in PDE systems.
In another research field, numerous feasible methods have been developed to solve the analysis and synthesis problem of nonlinear systems. Among them, Takagi-Sugeno (T-S) fuzzy model is widely adopted as an effective method for stability analysis of nonlinear systems [18] , which can express any smooth nonlinear function with arbitrary accuracy in any convex compact region [19,20] . In addition, the stability analysis problem of fuzzy control/filtering can be solved by employing the linear matrix inequality (LMI) methods [21] . Based on these advantages of the fuzzy model, many studies have been conducted to apply fuzzy models in controller/filter design for parabolic PDE systems [22,23] . Kerschbaum et al. [22] addressed the backstepping control for parabolic PDE systems; Qiu et al. [23] addressed the distributed adaptive output feedback consensus problem for parabolic PDE systems.
Meanwhile, the above study mainly used the traditional time-triggered (periodical sampling) network transmission method. The periodic sampling will lead to waste of the network resources and redundancy of transmission signals [24] . Therefore, to better save communication resources, Tabuada [25] proposed an eventtriggered mechanism (ETM) that can effectively save network transmission resources, which can determine whether to transmit data according to different trigger conditions [26][27][28][29][30][31][32] . For example, Wang et al. [26] developed an output-feedback backstepping control method for PDE systems with ETM; based on ETM, Ji et al. [27] considered the filtering control for PDE systems. These results prove that, compared to the traditional periodic sampling, ETM can effectively reduce the burden of network bandwidth and improve resource utilization.
Many nonlinear systems are multi-input and multi-output systems [33][34][35][36][37] , which the system models are often complex and diverse, how to design simple controllers/filters to meet the corresponding needs. Recently, Su et al. [38] proposed a reduced-order filter (ROF) design method, namely, the order of the ROF is lower than the original plant, which will be more favorable for the real-time filtering procedure because some redundancy and extra calculation can be effectively avoided by this filter [39,40] . However, as far as the author knows, there is little work on the design of fuzzy ROF for PDE systems, which arouses the author' s interest.
Based on the above discussion, this article intends to design a fuzzy ROF with ∞ performance for the PDE systems, which its main contributions are as follows: (1) Different from the fuzzy ROF design [38] , the influence of space position is fully considered in the system model and filter design process, the considerations are more comprehensive. Meanwhile, the ROF is designed on the basis of the early work [17] , which makes the complex engineering mathematical model more simplified and flexible.
(2) Compared with time-triggered fuzzy filter design [41] , the ETM is applied to determine whether to send a sampled signal, which can effectively solve the problem of signal resource transmission in the network.
Organization: Section 2 provides a problem statement and gives relevant definitions and lemmas, which include fuzzy system models, ROF structures and ETM. Section 3 is the main result of this paper, which include stability analysis and ROF design. A numerical example to prove the practicability of the obtained results in Section 4. Section 5 gives conclusions and future research directions.
Notations: R represents -dimensional Euclidean space; H indicates Hilbert space; The notation and −1 stand for the transpose and inverse matrix of the matrix , respectively; > 0 (≥ 0) means the matrix is positive-definite (semi-definite); for ease of expression, define = ( , ), = ( , ), = ( , ), Matrices not explicitly stated in the text are assumed to have the appropriate dimensionality.

Structure of reduced-order fuzzy filter
Consider the T-S fuzzy model, the following fuzzy ROF is obtained: Filter Rule : IF 1 is 1 ... and is , THEN whereˆ∈ R stands for the filter state with < ; the practical filter input signal isˆ;ˆexpresses the estimate signal of ; , and are the coefficient matrix of the filter with suitable dimensions. Thus, the overall T-S fuzzy filter can be expressed as follows:

Event-triggered mechanism
To improve the utilization of communication resources in the network, this paper introduces samplers and zero-order holder (ZOH) in the network. Consider an ETM, which meets the threshold condition as follows: where ∈ [0, 1), Ω is a symmetric positive definite matrix, ( ) and ( ) denote the current sampling signal (CSS) and latest transmitted signal (LTS), respectively. Definition 1. The transmission error ( ) between the CSS and the LTS as: where = + , which implies that the sampling time. Therefore, the transmission error can be expressed as: Based on the ETM, the system (4) can be transformed into a new system with time-delay. This network delay defined as ℎ( ) = − , 0 ≤ ℎ( ) ≤ ℎ, where ∈ [ + , +1 + +1 ). Therefore, the ETM inferred as: By taking into account the influence of the ZOH, the filter input expressed as: Therefore, the above-mentioned ETM is used to convert the fuzzy filter in (6) into a time-delay system. Remark 1. It is worth noting that the ETM will be reduced to time-triggered communication Mechanism (TTCM) when ≡ 0, which means that ETM is more practical in engineering applications.

Remark 2.
Because the order of the ROF is lower than the order of the plant. H = is introduced to expand the order of the filter, which will be explained in detail in Theorem 2.

Definition 2. The error system (13) with
≡ 0 is considered to be asymptotically stable with boundary conditions (2), if satisfies lim (13) is considered to be asymptotically stable with ∞ disturbance attention performance, if there exists a scalar > 0 such that following inequality holds:

Definition 3. System
Lemma 1. [42] (Jensens inequality) Let ∈ H be a vector function. Then, for any matrix > 0, the following inequality holds: ) . Lemma 2. [40] Let 1 , 2 , · · · , : R ↦ → R have positive values in the open subset C of R. Then, the mutually convex combination of over C satisfies Consider the following main problem: for the known systems (1), design a fuzzy ROF such that the error system (13) satisfies the asymptotically stable condition with ∞ performance.

RESULTS
In this section, the asymptotically stable condition of the error system (13) is obtained in Theorem 1. The ROF parameters will be obtained by the LMI method in Theorem 2.

Proof. The Lyapunov function is selected as follows
where The derivative of ( ) is as follows: According to Lemmas 1 and 2, the following inequality can be obtained Furthermore, from (13), we can get: Under the boundary conditions (2), by partial integration we can obtain: Consider the ETM (10), define: and Combining (18)- (21) and schur complement, we have: When ( , ) ≡ 0 and according to Theorem 1 and Definition 1, with the processing method [17] , we can get the error system (13) is asymptotically stable. Under zero initial condition, one can obtain: which indicates that which completes the proof.□ Next, solving several LMIs to obtain the parameters of the designed ROF, the main results are as follows: Theorem 2. For given scalars ∈ [0, 1), > 0, ℎ > 0, if there exist real matrices > 0, > 0, > 0, Ω > 0, 1 > 0, 1 > 0,˜,˜,˜satisfy the following matrix inequations for ∈ {1, 2, · · · , }, ∈ {1, 2, · · · , }: then the designed ROF (6) can be obtained by the following relation: Proof. If the conditions (14)- (16) in Theorem 1 are satisfied, then the non-singular matrix can be divided into: where For the purpose of proving that 2 is non-singular, define: Since > 0, it is easily to obtain > 0 for > 0. Consequently, it is convenient to verify that 2 is nonsingular for arbitrarily > 0 and the above expression is feasible with . In general, it is assumed that 2 is non-singular subject to 2 . Based on the above discussion the following definitions can be obtained: and [˜˜0 ] = Then, we can get: Pre-and post-multiply both sides of (16) with diag{ 1×6 } and its transpose, respectively. If (32)-(34) is considered, then the inequality (26)- (28) can hold. Therefore, the error system (13) can be guaranteed to be asymptotically stable with ∞ performance. In addition, (33) is equivalent to Thus, the , , in (6) can be obtained by (35). In case of general, let − 2 3 = , we obtain (29), which can be adopted to construct the ROF in (6). This completes the proof.□ Remark 3. It is noted that the matrix H presented in Theorem 2 plays a pivotal role in the filter design problem because it is used as a reduced-order factor. When H is a unit matrix, the designed filter is a full-order filter (FOF). The presence of H allows for efficient conversion between ROF and FOF.

SIMULATION EXAMPLE
In this section, to illustrate the effectiveness of the proposed method, the ROF design problem is studied for the FHN equation, which is a widely adopted model of excitable medium fluctuations in chemistry and can be described as follows: with boundary (2) and initial conditions The following output and measurement signals are given: The systems (36) can be represented by the following fuzzy rule [43] : Plant Rule 1: IF ( 1 ) is "Big", THEN where Thus, the following T-S fuzzy model is written as follows: Finally, to observe the ∞ performance more conveniently, define Simulation result: the trajectory of the error system (13) is shown in Figures 1 and 2. Figure 3 shows the release moment under the ETM, the trajectory of ( ) defined in (37) is shown in Figure 4. It can be observed that the filtering error system is asymptotically stable with ∞ performance. Moreover, the designed ETM (7) can effectively reduce the amount of network signal transmissions and improve the efficiency of network resources utilization.

Remark 4.
Inspired by ETM [25] , we introduce an ETM in the signal transmission process. Compared with the original TTCM [41] , it can effectively reduce the number of signal transmission and improve the network resource utilization. On the other hand, Song et al. [17] considered filtering for parabolic PDE systems, but did not consider the problem of ROF design. Compared with Example 2 [17] , the method proposed in this paper can achieve the same filtering function and the order of the filter is lower than the order of the plant, which simplifies the filter design.

CONCLUSIONS
This paper investigates the design method of fuzzy ROF based on ETM for nonlinear parabolic PDE systems. First, a T-S fuzzy model has been considered to reconstruct the nonlinear parabolic PDE systems. In addition, an ETM has been employed to reduce the amount of network transmission data to improve the network resource utilization. Then, the parameters of the designed ROF have been obtained by solving several LMIs based on Lyapunov direct method. Finally, the effectiveness of the proposed method is illustrated by simulation experiments. However, due to the time-delay phenomenon, the systems output signal is difficulty to keep synchronized through the network transmission between the filter and the plant. Therefore, in future studies, we will further consider asynchronous ROF for fuzzy PDE systems, which will be a more interesting topic.