Event-triggered dissipative control for 2-D switched systems
Introduction
In recent decades, due to the contribution of modeling many practical systems, two-dimensional (2-D) systems have been gathering an increasing interests and thus gained numerous achievements [1], [2], [3], [4], [5], [6], [7]. With regard to the study of 2-D systems, a simple and intuitive research method has been provided, namely 2-D state-space model. It is generally known that the frequently-used models of 2-D systems are FM model and Roesser model. Notably, the Roesser model can be modeled as a special case of the FM model. Furtheromre, 2-D FM model has been successfully applied in various engineering fields, such as metal rolling process, power grid sensor network, signal processing and thermal process [8], [9], [10]. However, many practical systems are often subject to sudden environmental disturbances, abrupt variations in system structures due to random failures. Consequently, the switching signal is introduced to 2-D systems for the practical requirements in description of these physical processes with abrupt changes. Many practical dynamic processes can be represented by 2-D switched systems, such as bipolar devices [11], transistors [12] and thermal processes in chemical reactors [15]. Nevertheless, compared with 1-D systems [13], [14], only some preliminary achievements are established for 2-D switched systems due to the complexity of analyzing and modeling such systems. To mention a few, the problems of stability analysis and stabilization were considered for discrete-time (D-T) 2-D switched systems in [3]; control and filtering problems for 2-D switched systems were discussed, respectively, in [15], [16]; The dissipative filtering and control for D-T 2-D switched systems were studied in [17].
Dissipativity theory was introduced originally by Willems in [18]. Thereafter, Dissipativity was described by supply rates and storage functions, which represent separately the energy supplied from outside the system and the energy store inside. The actual physical meaning of dissipativity is that the increasing of energy store in the system is no more than the supplied energy from outside the system. More recently, dissipativity has become one of the most important properties to be studied, and played a vital part in various problems for the analysis and synthesis of control systems. It serves as a powerful theory in control community for the following reasons: 1) Based on the consideration of abstract energy, dissipativity provides not only a unified framework for the analysis and design of composite systems, but also a natural candidate for the Lyapunov function; 2) In view of the practical needs, many engineering systems need to satisfy dissipativivity to attain prospective noise attenuation. Furthermore, the theory of dissipativity also involves extensive engineering applications, for instance, robotic control, mechatronic system control, nonlinear control and adaptive control [19], [20], [21]. Therefore, a great amount of efforts has been paid on dissipativity and its applications, and some results about dissipative filtering and control issues for different dynamical systems have been proposed [1], [17], [22], [23], [24]. In brief, it is with far-reaching meanings to explore the dissipativity of systems.
So far, significant progresses have been made on analysis and synthesis of network control systems (NCSs), including the time-triggered mechanism (TTM) [25], the event-triggered mechanism (ETM) [26], the quantized feedback [27] and the Lebesgue sampling [28]. Unlike the traditional TTM, the ETM can effectively reduce the unnecessary waste of communication resources and save the bandwidth resources. The essence of ETM is to transmit the data only when the pre-designed event-triggered condition (ETC) is met. Based on the advantages of improving the efficient usage of network resources, quite a few theoretical results combining ETM have emerged for 1-D systems in [29], [30], [31], [32], and the references therein. In contrast to 1-D systems, the states of 2-D systems propagate along two independent directions, thereby increasing the amount of information. Furthermore, to ensure a more efficient data transmission, it is crucial for 2-D systems to design appropriate ETM. Up to present, event-triggered technique has been extended to the research of 2-D systems. To name a few, the issue of event-triggered sliding mode control for 2-D Roesser systems was discussed in [33]; By aid of genetic algorithm and ETM, the sliding mode controller was designed for 2-D systems in [34]. Taking these into consideration, it makes practical sense to study the event-triggered control problem of 2-D switched systems.
Following the aforementioned considerations, we are motivated to deal with the dissipativity-based event-triggered control problem for 2-D switched systems described by FM model. Particularly, there are indeed technical challenges for the addressed problem: 1) how to define the event-triggered sequence and further implement the ETC for 2-D switched FM model, which is more reasonable and easily carried out aiming at the characteristic of 2-D systems? 2) How to ensure the dissipativity and stability of the considered system by the proposed? 3) How to make a balance between the dissipativity performance and communication rate reduction? In response to these problems, the following contributions are summarized:
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[1] A novel notion of event-triggered sequence is presented by utilizing the line of cross-cut technology, and an event-triggered rule is formulated for the concerned 2-D switched systems.
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[2] An ETM and dissipative controller are developed for 2-D switched systems. Sufficient conditions that ensure the dissipative stability for the closed-loop system under the ETM are obtained.
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[3] The specific relationship among triggered parameter, data transmission rate (DTR) and optimal dissipativity performance index (DPI) is discussed.
Notations. The superscript “” refers to matrix transposition; and , denote, respectively, the set of non-negative integers, real numbers, all real matrices, and the n-dimensional Euclidean space; means that matrix P is real symmetric and positive definite; denotes rounding a real number a to the nearest integer , with and indicates a block-diagonal matrix; stands for the Euclidean vector norm; Symbol “” is used to represent the symmetric term of matrix.
Section snippets
Discrete-time (D-T) 2-D switched systems
Consider the following D-T 2-D switched system, which is described by the FM model:where and represent, respectively, the local state, the input, the disturbance input and the control output; expresses the switching signal; And and
Main results
In this section, we will address dissipative control problem under ETM for 2-D switched FM systems. The main issues to be solved can be stated as:
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[1] Dissipative stability analysis. For closed-loop system (8) under the ETM, sufficient condition satisfying the exponentially stable and -dissipative is obtained;
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[2] Dissipativity-based event-triggered control. Based on the ETM, the control law in (7) is designed such that the closed-loop system in (8) is exponentially stable with the 2-D
Illustrative example
This section will demonstrate the effectiveness of the developed theoretical results by employing a simulation concerning the Darboux equation [35]:which can be applied to model numerous practical dynamical processes. As discussed in [36], the Darboux equation can be converted into the D-T switched 2-D FM system (1) with the following parameter matrices ():
Conclusion
In this work, the issue of ETM and dissipative controller design has been studied for D-T 2-D switched FM systems. First, to reduce the number of data transmission, the new event-triggered sequence and the ETM have been established. Then, based on the proposed ETM, the dissipative stability analysis and dissipative controller design have been conducted. Meanwhile, sufficient conditions have been deduced to guarantee that the closed-loop system is exponentially stable with the specified
CRediT authorship contribution statement
Lingling Li: Writing – original draft. Rongni Yang: Validation. Zhiguang Feng: Writing – review & editing. Ligang Wu: Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This work was supported by the National Key R&D Program of China (No. 2019YFB1312001), National Natural Science Foundation of China (62033005, 62022030, 62003114,61873147), the Natural Science Foundation of Heilongjiang Province (ZD2021F001), and the Youth Innovation Group Project of Shandong University, China (2020QNQT016).
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