Event-Triggered H-Infinity Controller Design of Nonlinear Networked Cascade Control System

Nonlinearity is ubiquitous in practical industrial production, so this paper investigates the problems of modeling, H∞ control and stabilization for a class of nonlinear networked cascade control systems (NCCSs) under an event-triggered scheme, which makes the system have more practical application value. Firstly, aiming at a kind of typical nonlinear system in industrial production, a new mathematical model is established by adding networked cascade control system and event-triggered scheme. The collaborative design method of primary and secondary controller parameters and event-triggered parameters is completed utilizing linear matrix inequality (LMI). Secondly, based on ensuring its stability, combined with H∞ control, the system’s disturbance is suppressed, making the system more universal. Finally, the feasibility and effectiveness of the proposed method in the nonlinear networked cascade control system has verified by a marine boiler liquid level cascade control system.


I. INTRODUCTION
Due to the continuous development and maturity of networked control systems (NCSs) [1] in recent decades, it has received widespread attention from scholars. It uses the network signal to realize the control. Modularization, realtime control, and the low cost of the system are its advantages. Therefore, networked control systems have been applied widely in actual industrial production. However, many papers ignore some nonlinear terms when modeling based on the actual system, such as: [1], [2]. The linear system constructed in this way cannot be applied to most practical systems, and the application field is narrow. Therefore, when designing the control system, it will be more practical to add the possible nonlinear system in production.
Nonlinear factors are inevitable in production. Therefore, the research on nonlinear networked Control systems has important practical significance. Many outstanding scholars have made many achievements in theory. On the one hand, the system model and stability are studied for the delay of network control. For example, in [2], the network delay is defined as nonlinear to establish the model. In contrast, in [3], the network delay is described by the probability density function. In addition, some scholars have studied the generalized systems [4]. On the other hand, in order to deal with different control objects, different control methods are added, such as PID control [5][6], sliding mode control [7][8], adaptive control [9][10], robust control [11][12] and predictive control [13]. According to the other maintenance, or performance requirements, observers [14] and filters [15] are designed. However, the above papers are all aimed at the single closed-loop nonlinear networked control system. However, the cascade structure is rarely mentioned, so the cascade structure is added to the nonlinear system in this paper.
Cascade control System (CCS) [16] is a kind of control structure with unique advantages. When the primary loop has a strong disturbance, the secondary loop can quickly suppress the disturbance. With the primary loop, the system can achieve the desired output. Networked cascade control systems [17] combines most of the advantages of CCSs and NCSs, which can eliminate the interference in the system in time and quickly, and at the same time, improve the efficiency of the system. However, the single closed-loop structure of the networked control system is easier to study, and the addition of the cascade structure further complicates the model. That makes theoretical research very difficult. However, the addition of the cascade structure makes the model more complicated, making theoretical research very difficult.
Regarding the nonlinear term in the system, scholars have studied a lot. Among them, the Lurie system is a widely existing nonlinear system. In Lurie system, the nonlinear system is divided into two parts, one part is a linear timeinvariant system, and the other part is a nonlinear system that satisfies the sector constraint, and its sector boundary can be selected freely. In actual industrial production, the controlled index changes nonlinearly due to the actual influence of various factors. As long as the appropriate fan-shaped boundary is selected, the Lurie system can be used for simulation. Since 1944, Lurie has proposed the system. Its research has been in progress. For example, in [18], the state of the chaotic Lurie system is estimated, and a new communication channel is established and verified by Chua's circuit. In [19], the absolute stability of Lurie nonlinear systems with both sector and slope constraints is studied. The paper [20] combines Lurie system with networked control system, and gives the stability condition. Furthermore, the paper [21] adds robust control, and its stability is verified by simulation. However, most scholars focus on the stability of the Lurie system structure itself. Few people combine it with the networked cascade control system.
On the other hand, over-saturated signals are more likely to cause packet loss and disorder, which can lead to instability of the control system. In order to reduce the transmission frequency of the signal in the network, the event-triggered control method [22][23][24] is proposed. When the signal is filtered by the event-triggered controller, the signal that is meaningful to the system control will be sent, which reduces the transmission pressure of the network. However, under the conditions of some trigger controls, when the time interval for detecting the system state is minimal, it is easy to happen the unreasonable situation of triggering an infinite number of times in a limited time, that is, the Zeno phenomenon [25][26]. To this end, many scholars have proposed a discrete state event-triggered condition for defining sampling time [27][28], thereby avoiding this phenomenon. In addition, some scholars have proposed different event-triggered scheme to deal with some special situations [29][30][31][32]. In order to allow the control system to operate stably, and avoid failures caused by network pressure, the event-triggered scheme is also added. The selected eventtriggered conditions in this paper can avoid the Zeno phenomenon.
In the process of industrial production, there will be control links for specific indicators. Let this index stabilize at the expected value, and the control system should have the advantages of anti-disturbance, optimized resource utilization, and easy maintenance as much as possible. For unavoidable non-linear factors, idealized considerations sometimes cannot be applied to actual production. This paper is based on a class of nonlinear NCCSs, add an event-triggered scheme and H∞ control, completing the collaborative design of the primary state feedback controller and secondary one, and obtain event-triggered parameters. The main contributions of this paper are summarized as follows: 1) To the best of our knowledge, few scholars combine nonlinear term and event-triggered scheme with networked cascade control systems simultaneously. This paper is the first time to combine Lurie nonlinear cascade control system and event-triggered scheme with networked control.
2) A new model which takes nonlinear terms and eventtriggered control is constructed. This model can describe the actual industrial production process more closely and practically. And the system has all the advantages of eventtriggered control and cascade structure.
3) In the presence or absence of disturbances, the codesign methods of the primary and secondary controller parameters and event-triggered parameters are obtained simultaneously. It can be applied to industrial process control systems with the same structure.
4) The event-triggered scheme and H∞ control of the system is considered. The obtained results can be further applied to the industrial production of this kind of nonlinear NCCSs model.
The rest of this paper mainly describes the following contents: In section II, the model of nonlinear NCCS is established. In section III, the sufficient conditions of the system without disturbance are obtained, and the co-design method of controller parameters and event-triggered parameters is given. In section IV, based on the previous section, we complete the H∞ control with disturbance. In section V, a simulation example of marine level boiler with nonlinear condition is selected to verify the effectiveness of the proposed method. Section VI is the conclusion.

II. MODELING OF NONLINEAR NCCS
Event generator x 1 (t) Network FIGURE 1. Configuration diagram of event-triggered controller of nonlinear network cascade control system.
A control system configuration diagram is shown in Figure  1. The event-triggered scheme is introduced into nonlinear NCCSs. In the networked cascade system, the response speed of the inner loop is fast and the communication channel is short, so the network delay can be ignored. Therefore, we construct a model in which the network is in the outer loop. The secondary plant in the outer loop is regarded as nonlinear. An event generator is added between the primary sensor and the primary controller to decide whether to transmit the new sampling signal to the primary controller.
P1 and P2 are two plants. C1 and C2 are the primary controller and secondary controller. S1 is the primary sensor and S2 is the secondary one. A is actuator. It can be seen from This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2021.3127226, IEEE Access VOLUME XX, 2017 3 the figure that the output of P2 is the input of P1, the output of C1 is the set value of C2, and the states of P1 and P2 are respectively transmitted to C1 and C2 through S1 and S2. First, consider the primary controlled object, as follows xtis the state vector of the primary plant, 1 () ytis its output, () t  is a finite disturbance that satisfies the 2 [0, ) L  . A1, B1, C1 and C3 are known matrices.
Considering the following secondary plant, which is nonlinearity and can be described as xtis the state vector of the secondary plant, and 2 () yt is its output, 12 ( ( ), ( )) f x t x t satisfies Lurie system and can be expressed as a linear system part and a nonlinear part, so the secondary plant can be described as In this formula, 2 () utis the output of the secondary controller, A2, B2, B3, B4, C2 and C4 are constant matrices with appropriate dimensions, and the nonlinear term w(t) can be expressed as (5) Considering that in practical application, the network generally exists in the primary loop. It is assumed that there is no time delay in the secondary loop, and the time delay only exists in the primary loop. The delay is time-varying and may be larger than one sampling period.
In this paper, primary controller 1 () ut and secondary controller 2 () ut are state feedback controllers. Considering the network transmission delay of the primary loop, it can be expressed as where k  is the time-varying delay and bounded in the network communication.
Define a function where M  denotes the upper delay bounds. It means for any Combine the event-triggered scheme with cascade networked cascade control system, which will select the signal according to the following conditions and send it to the controller.
, and 1 () k x t h is the last signal sent at the time of triggering k th.
Remark 1: Only when the transmitted signal 1 () k x i h satisfies inequality (8), the data will be sent to the controller. Therefore, this can reduce the times of signal transmission in the network. And the trigger condition (8) can effectively avoid unlimited triggers in a short time, which means the Zeno phenomenon can be avoided.
Remark 2: It is assumed that the system is completely observable, and there is no data packet loss and mis-sequence. Define () k et as the difference between the current sampled signal and the last successfully sent signal, that is, . Therefore, the event-triggered scheme in this paper can be obtained When the event-triggered scheme controller is applied to the primary loop of the cascade control system connected through the network, the closed-loop system corresponding to system (1) and (2) can be expressed In order to design the corresponding controller, the following lemma is needed.
Lemma 1 [33]: For constant matrix 3 X and symmetric matrix 12 , , if one of the following two conditions is true

III. DESIGN OF NONLINEAR NCCS EVENT-TRIGGED CONTROLLER WITHOUT DISTURBANCE
This paper analyzes the stability of the networked cascade control system and designs the corresponding primary controller and secondary one based on event-triggered control. It is assumed that there is no disturbance, and we will give the sufficient condition of stability for nonlinear NCCS (10) in this section.
Taking the derivative of () Vt , we can obtain: From the formulas (6), we can define From the formulas (5), we can define Using (13), (14), (15), we can obtain This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2021.3127226, IEEE Access VOLUME XX, 2017 5 x v e t w t  =− Separate this matrix in formula (16), and define  as a symmetric matrix, then we can obtain 11 12 According to Lemma 1, 0  in Equation (17) (10) is asymptotically stable in a given sector interval, and the theorem is proved.
According to Theorem 1, we give the design method of primary controller and secondary one of the system (10). , such that the following inequality (19) holds, then the system (10) is asymptotically stable, and the expected gain matrix of the primary controller can be obtained:      If there are matrices ,   , , , , , with appropriate dimensions, so that the inequality (19) holds, then the system (10) is asymptotically stable, and the expected gain matrices of the primary controller and secondary controller are Remark 3: Scholars have completed the work of only considering event-triggered control or only considering nonlinear systems, but these two problems are considered in networked cascade control system in this paper. The above part considers the stability of the system without disturbance. But zero disturbance is a special case of disturbance. Therefore, when the disturbance is not constant zero, the specific influence of the disturbance on the system should be considered.

IV. DESIGN OF NONLINEAR NCCS EVENT-TRIGGED CONTROLLER WITH DISTURBANCE
In the previous section, we studied the sufficient condition of stability for nonlinear NCCS (10) without disturbance. We presented sufficient conditions for system stability and the controller design method through Theorem 1 and Theorem 2. This section will study the stability problem and H∞ control of NCCS (10) with disturbances.
Under the zero-initial condition, if the constant 0   , the output of the primary controller y1(t) and the disturbance Δ(t) satisfy the H∞ norm bounded constraint 12 2 ( ) ( ) y t t   ‖ ‖ , and Δ(t) boundary satisfies the 2 L norm bounded, that is,  . It ensures that the system (10) is stable under the H∞ performance index  .
Using calculation methods such as Theorem 1 and Theorem 2, we can obtain sufficient conditions and controller design methods for system stability in the presence of disturbances. The corresponding theorem proposed is as follows.  (14) can be rewritten as Select the same Lyapunov function as in Theorem 1 and derive it, where , , , P Z R Q are symmetric matrices and are positive definite matrices.
The inequality 0  , and can be described as  , J < 0 is verified by LMI. This means that the system (9) has an H∞ performance index  . As a result, if the inequality (25) is true, then the system (9) is asymptotically stable in the known sector interval.
According to Theorem 3, we give the design method of the primary controller and secondary one of the system (9) with disturbance. , such that the following inequality (26) holds, then the system (10) is asymptotically stable, and the expected gain matrix of the primary controller can be obtained   Remark 4: The consideration of nonlinear terms will make the system more universal. The addition of event-triggered scheme will significantly reduce the burden of the network. The cascade structure of the system can quickly restrain the disturbance in the loop. Finally, adding H∞ control to make the system more suitable for practical production. Next, the effectiveness of the above design method will be verified by a simulation example of a marine boiler liquid level cascade control system.

V. SIMULATION EXAMPLE
Nonlinear systems are ubiquitous in industrial production with cascade structure, which is of practical significance. And nowadays, many cascade systems are connected through the network. In this paper, we choose a marine boiler liquid level cascade control system model, use the linear matrix inequality method to design the primary and secondary controllers, and verify the feasibility of the design method. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. As shown in fig. 2, C1 and C2 are primary and secondary controllers, and S1 and S2 are corresponding sensors. The primary loop adjusts the steam valve's water supply pressure difference control loop according to the pressure difference before and after the water supply valve. According to the water level deviation, the secondary loop adjusts the water level regulation loop of the water supply valve. The given value of the water level is input. In order to facilitate the calculation, the given value is set to 0. The steam regulating valve is used as the primary plant P1, and the water pump unit is the secondary plant P2.
In this paper, the nonlinear function is selected as:

A. FOR NONLINEAR NCCS WITHOUT DISTURBANCE
When there is no disturbance, the matrices 4 B , 3 C and . According to Theorem 2, the trigger matrix  is obtained. Out of 200 sampled signals, only 54 sampled signals are sent to the primary controller. In addition, the average signal sending interval after adding event-triggered can be calculated to be 0.7370s. It can be seen that the addition of event-triggered controller can filter the signals and save the network resources. Figure 3 show the state response of the system's primary and secondary loop, and Figure 4 shows the signal transmission interval.
It can be seen that the system reaches a stable state at 40s. It indicates that the primary loop reaches the steady-state at 40s, and the secondary loop reaches the steady-state at 25s. Therefore, it can be proved that the design method proposed in this paper is feasible.
The above-described controller design methods in such cascade control systems can be obtained directly to avoid the disciplined solutions, thereby avoiding the traditional methods of experimentation. If the literature [34] selects the controller gain as a range and repeatedly tests, it is more troublesome. If using the design method in this paper, the result can be obtained directly, and the controller gain can be accurate to 4 decimal places.

B. FOR NONLINEAR NCCS WITH DISTURBANCE
The disturbance of the system mainly comes from the external disturbance caused by the sudden change of the controlled value and the change of the working environment during the operation of the system. Irregular changes in the controlled value of the system itself can be included in the nonlinear term. So we assumed an external disturbance to impact the system to verify the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.  . According to Theorem 2, the trigger matrix  is obtained. Out of 200 sampled signals, only 40 sampled signals are sent to the primary controller. In addition, the average signal sending interval after adding event-triggered scheme can be calculated to be 0.9950s. Figure 5 show the state response of the system, and Figure 6 shows the signal transmission interval.
The nonlinear system model with disturbance is more suitable for practical industrial production. If it is a simple nonlinear system, the system will be fragile when applied to practice. For example, the reference [35] does not consider the influence of disturbance, so it has certain limitations.

Remark 5:
We can see that the system can obtain sufficient conditions for stability after considering nonlinearity. Moreover, the simultaneous addition of event-triggered mechanism and H∞ control can make the system resist disturbance to a certain extent while saving network resources.

VI. CONCLUSION
Considering the needs of practical industrial production, based on a class of nonlinear systems and event-triggered scheme, the primary and secondary controllers of NCCSs have designed in this paper. The method is applied to a liquid level of marine boiler control system, which is connected through the network. This is the first time that the eventtriggered scheme is added to the Lurie nonlinear networked cascade control system, and the event-triggered scheme can avoid Zeno phenomenon based on reducing the burden of network bandwidth. A new model of the networked cascade control system is established, and the collaborative design method of primary controller and secondary one parameters and event-triggered parameters is completed through Lyapunov theorem and linear matrix inequality. Then add H∞ control to effectively suppress the extra disturbance in the nonlinear system. Finally, an example of a marine boiler liquid level cascade control system is used to verify the stability of the system and the optimization of the utilization rate of network resources. Explains the feasibility of this method.