Finite Horizon H-Infinity Control of Nonlinear Time-Varying Systems Under WTOD Protocol

The finite horizon <inline-formula> <tex-math notation="LaTeX">$H_{\infty } $ </tex-math></inline-formula> control problem is investigated for a class of discrete nonlinear time-varying systems subject to Weighted Try-Once-Discard communication protocol. The equations of state and output under investigation involve both deterministic and stochastic nonlinearities. By resorting to the Taylor series expansion formula, Lyapunov stability theory and cross-amplification lemma, sufficient conditions are established for the existence of the desired observer-based <inline-formula> <tex-math notation="LaTeX">$\text{H}\infty $ </tex-math></inline-formula> controller. The gain matrices of the controller and observer are obtained by solving a set of recursive linear matrix inequalities.


I. INTRODUCTION
In recent years, more and more scholars have developed a strong interest in the control and filtering of nonlinear systems [1]- [3]. Literature [4] studied the sliding mode control (SMC) problem for Markovian jump systems with probabilistic denial-of-service (DoS) attacks. In Literature [5], an event-triggered nonfragile distributed H ∞ control problem was considered for uncertain nonlinear networked control systems over sensor networks with random communication packet dropouts and redundant channels. Literature [6] treated the problem of periodic tracking control of fractionalorder IT-2 fuzzy systems with state and external disturbance, where an observer-based fractional-order modified repetitive controller has been designed to make the resulting closedloop fuzzy control system asymptotically stable together with a prescribed H ∞ disturbance attenuation level. In literature [7], the envelope constraint and filtering problem of the system were considered simultaneously. The nonlinearities here include deterministic nonlinearities and stochastic nonlinearities. Literature [8] investigated the recursive filtering problem for nonlinear time-varying stochastic systems with measurement outliers. Literature [9] assumed that the spatial distance between the filter and the sensor is relatively long, and the measured value may be lost in the transmission The associate editor coordinating the review of this manuscript and approving it for publication was Min Wang. process from the sensor to the filter due to insufficient energy stored in the sensor. Under such a condition, the finite horizon filtering problem of a class of nonlinear time-delay systems with energy acquisition sensors is studied.
It is worth noting that the most current literature on the study of nonlinear system is not enough in-depth, only the existence of nonlinear term in the system is considered in the literature. And the parameters of the system is steady, For the tracking control problem of nonlinear continuous time system, literature [15] gave the maximum allowable sampling interval of tracking control under the constraint of WTOD protocol. In literature [16], the influence of WTOD protocol was considered in the study of synovial control algorithm of automobile electronic valve system, and good stability effect was achieved.
Based on the above discussion, this paper investigates the finite horizon control method for networked time-varying nonlinear systems under the weighted try-once-discard communication protocol. Compared with the existing research, the main contributions of this paper are as follows: (1) Based on the Lyapunov stability theory, cross amplification lemma technology and recursive linear matrix inequality technique, an observer-based finite horizon controller is designed.
(2) The influence of WTOD communication protocol is considered in the finite time domain H ∞ control problem for networked time-varying nonlinear systems. (3) The proposed controller can ensure both the stability and the prescribed H ∞ performance index of the closed-loop system over a given finite horizon.

II. NONLINEAR CONTROL SYSTEM MODEL
Take the nonlinear time-varying system in finite horizon as the research object, and the controlled object is described as where x k ∈ R n x , u k ∈ R n u , z k ∈ R n z and y k ∈ R n y respectively represent system state, control input, controlled output and measurement output vector without network transmission. w k ∈ l 2[0,N ] and v k ∈ l 2[0,N ] are the process noise and measurement interference of the system respectively. The matrices B k , D k , E k and L k are time-varying parameter matrices with known appropriate dimensions. g (x k ) and h (x k ) are known nonlinear functions which can be analyzable everywhere in the finite horizon k ∈ [0, N ]. f (x k ) is the random nonlinear function and the first moment and covariance of the random nonlinear function f (x k ) satisfy where q is known as a non-negative integer, n,k and n,k (n = 1, 2, , . . . , q) are known as the appropriate dimensional matrix.
Consider the communication network between the sensor of the controlled object and the remote controller using a WTOD protocol. Dynamic scheduling protocol is a scheduling method in which network nodes obtain communication network access permissions according to some given rules. The Weighted try-once-Discard protocol is a kind of dynamic scheduling protocol which determines the node receiving network communication permissions at instant k based on the difference in size between the data to be sent by each transfer node at time k and the data sent last time. The greater the difference is, the higher the transmission demand of this node is, and the sensor node with the highest transmission demand will have the priority to obtain the permission to use the communication network. The sensor node defined to obtain access to the network at the instant k of sampling is ξ k ∈ S@ 1, 2, , . . . , n y , and the value of ξ k is calculated by the following formula: whereȳ i,k−1 refers to the data last sent by sensor node i before moment k (excluding k ), and Q i refers to the known positive determination matrix, which represents the weight matrix of transfer node i under Weighted try-once-discard protocol schedule. Equation (5) can be further converted into where, under the dispatching action of WTOD protocol, the measured output after transmission from the sensor . . , δ i − n y I , and δ (·) ∈ {0, 1} are Kronecker-delta functions. Consider equations (1) and (3), we can get:ȳ Remark 1: In a networked control system with multiple sensor nodes, the WTOD protocol can solve the problem of distribution of communication authority of each sensor node. In the network where the WTOD protocol exists, sensor nodes obtain communication rights through ''competition''. The rules of this ''competition'' are based on the difference between the amount of data sent by the node at the previous moment and the amount of pre-sent data at the current moment. The greater the difference, the higher the transmission demand, so the sensor with the highest demand will have priority in obtaining the communication authority.
For system (1), an observer-based state feedback controller is designed in the following form: wherex k ∈ R n x is the state estimation of system (1),x 0 = 0, G k ∈ R n x ×n x , H k ∈ R n x ×n y , K k ∈ R n u ×n x are the gain matrix of the observer and controller to be solved. In the finite horizon k ∈ [0, N ], the nonlinear function g (x k ) and h (x k ) can be linearized according to the Taylor series expansion formula, and the Taylor expansion formula of g (x k ) and h (x k ) at the system state estimationx k is: ∂x x=x k , L 1 ∈ R n x ×n l 1 and L 2 ∈ R n y ×n l 2 are known scaling matrices, 1 ∈ R n l 1 ×n x and 2 ∈ R n l 2 ×n x is unknown matrix, which satisfies 1 ≤ 1, 2 ≤ 1. The estimation error e k = x k −x k and the augmented vector Combined with equations (7)- (10), the closed-loop nonlinear system (1) can be transformed into the following closed-loop augmented networked time-varying system: where A k , F, T ,D k ,w k , andL k , as shown at the bottom of the page.
The main research purpose of this paper is on the finite domain k ∈ [0, N ], design a kind of controller based on observer such as type (8), by solving the controller and observer parameter K k , G k , H k for the pre-given disturbance suppression level γ , positive definite matrix S and initial state x 0 of the system, the controlled output z k of the system (11) satisfies the following H ∞ performance: ∀w k = 0 (12) Remark 2: Taylor expansion technique will unavoidably lead to conservatism, cause the higher order terms are ignored. Furthermore, the definition of finite horizon H ∞ performance index (12) of networked time-varying systems refers to literatures [17], [18]. The physical meaning of Equation (12) refers to that considering the influence of the initial state η 0 of the system, on a given finite horizon [0, N ], the energy gain of external interferencew k to the controlled output z k of the system is less than the given disturbance suppression level γ .

III. PERFORMANCE ANALYSIS OF NONLINEAR SYSTEM CONSTRAINED BY WTOD PROTOCOL
Lemma 1: For given the scalar ε > 0, the inequality (13) is always true for any matrix X and Y with appropriate dimensions. Proof: Theorem 1: For given H ∞ performance indexes γ > 0 and positive definite matrix S, the gain matrix K k of the controller and the gain matrix G k , H k of the observer are known. If there is a symmetric positive definite matrix P k , which satisfies matrix inequality (14) and (15), as shown at the bottom of the page, then the closed-loop augmented networked control system (11) satisfies H ∞ performance requirements.
Proof: Construct Lyapunov function as: According to equations (11) and (2)-(4), the difference of the preceding term for can be obtained Add zero term z k 2 −γ 2 w k 2 +γ 2 w k − z k 2 to the right side of equation (17) and defineη k = η T kw T k T , we can get: Whenw k = 0, if matrix inequality (14) is true, then V k < 0; For any non-zero external disturbancew k ∈ l 2 [0, ∞) , if matrix inequality (14) is true, then Add the sum from 0 to N on both sides of inequality (19), and you can get Due to V N +1 > 0, if matrix inequality (15) was set up, then Therefore, the system (11) meets the required H ∞ performance. The proof is completed.
where˜ k ,¯ k ,¯ k ,Ā k ,Ā k ,L,L 1 ,L 2 ,˜, and J, as shown at the bottom of the page. Proof: according to Schur complement lemma, matrix inequality (22) is equivalent to Then by using the property of matrix trace, we can get Similarly, according to Schur's complement lemma, matrix inequality (23) is equivalent to equation (26).
Define k = kDk , it can be obtained from the matrix inequality (14) that According to Schur complement lemma, equation (27) is equivalent to

Further converting equation (28) can be obtained
According to lemma 1, k ≤ 0 is true, that is, theorem 2 is true to ensure that theorem 1 is true, to ensure that the closed-loop augmented system (11) meets the H ∞ performance requirements. According to Theorem 2, the observer gain matrix K k and controller gain matrices G k and H k can be obtained by solving the recursive time-varying matrix inequalities (22)-(23) using LMI toolbox. The proof is completed.

V. THE SIMULATION RESULTS
In order to verify the effectiveness of the method proposed in this paper, the non-stationary growth system model proposed in literature [7] and literature [19] are taken into account, and its specific parameters are as follows: be solved by LMI toolbox. The corresponding simulation results of the system are shown in Figure 1, Figure 2 and Figure 3. Figure 1 depicts the changes of sensor nodes that obtain network permissions under the influence of WTOD communication protocol. FIG. 2 and FIG. 3 show the controlled output images of the open-loop system and closedloop system respectively. It can be seen from Figure 3 that, on a given finite horizon AA, the controlled output curve of the closed-loop system oscillates and converges. It can be calculated from Figure 3 that when there is disturbance, the H ∞ performance index formula (12) in the finite horizon is established. The simulation results demonstrate the effectiveness of the finite horizon H ∞ controller design method.

VI. CONCLUSION
In this paper, under the influence of WTOD protocol, the finite horizon H ∞ control problem with both nonlinear and random nonlinear time-varying systems that can be analyzed everywhere in the state equation and the output equation is studied. First, a nonlinear time-varying system model is constructed. Then, in the sensor-controller transmission network, the WTOD communication protocol is added to avoid data conflicts. Then, the nonlinear terms determined by Taylor expansion technique are processed, the stochastic nonlinear terms are processed by statistical method, and the sufficient conditions for the system to meet the H ∞ performance requirements are obtained by Lyapunov stability theory and cross amplification lemma. On this basis, the controller design algorithm based on observer is given. Finally, simulation shows the effectiveness of the proposed method. In addition, extending the proposed finite horizon H ∞ control method to nonlinear systems under Round-Robin protocol and Random Access protocol is a problem that requires further research. From 2005 to 2018, she was a Research Assistant at the Wuxi Institute of Technology. Since 2018, she has been an Associate Professor of computer science with the School of Internet of Things Technology, Wuxi Institute of Technology. She is the author of one book, more than 30 articles, and one invention. She took charge of many studies at provincial and municipal levels, and enterprise projects. Her research interests include computer network control, networked control system, fault detection, artificial intelligence, and pattern recognition.
Prof. Huang won the ''Excellent Young Teacher'' Award and got the first prize in the teaching competition of the Wuxi Institute of Technology, in 2017. In 2018, her thesis won the first prize of an excellent thesis of vocational and technical education in Jiangsu. Since 2012, she has guided students to participate in international, national, provincial, and municipal competitions, and won the first prize, second prize, and third prize for many times.
FENG PAN received the M.S. degree in industrial automation and the Ph.D. degree in fermentation engineering from Jiangnan University, China. He is currently a Professor and a Ph.D. Candidate Supervisor with the School of Internet of Things, Jiangnan University. His research interests include process control, biochemical process intelligence control, and computer distribution control systems.