Event-triggered control for network-based switched linear systems with transmission delay☆
Introduction
A switched system comprises a finite number of sub-systems and a switching law that determines transitions of the sub-systems. Since switched system can describe many physical plants (e.g., robotic systems [1] and chemical systems [2]), it has received increasing attention in the recent decades [3], [4], [5]. Stability is always the basic but important problem of switched system, which has been widely studied in literatures, such as the review [6] and some recent progresses [7], [8], [9]. In addition, to implement a control design in digital platform, traditionally the controllers sample feedback signals periodically, and it is referred to as time-triggered control. However, time-triggered control is performed at a fixed sampling frequency regardless of whether the sampling is really necessary. As a consequence, it will lead to unreasonable allocation of system resources. Thus, an alternative control method, event-triggered control (ETC), has been proposed [10], [11], [12].
In ETC, the signal is sampled and transmitted depending on a designed event-triggering condition. Compared with time-triggered control, the inter-execution intervals under ETC are no longer fixed. The aperiodic sampling is able to guarantee the desired system performance as well as reduce the sampling rate [13], which further saves the resources of computation and communication. Due to the potential advantages, many related issues have been investigated and event-triggering scheme is the key of the control design. In [14], a state-based event-triggering scheme is presented, and the event is triggered if and only if the error between the current system state and the predicted state exceeds a constant threshold. Under time-varying communication delay, an event-triggering scheme with guaranteed stability is designed in [15]. When the full system state is unavailable [16], the observed system state is utilized for developing the event-triggering scheme. In [17], [18], the system output instead of state is used to determine the samplings. Moreover, for further reducing the sampling rate, a dynamic triggering mechanism based on Lyapunov function and a transition method between two kinds of event-triggering schemes are proposed in [19] and [20], respectively.
On the other hand, to implement the ETC in practical engineering, the Zeno behavior (i.e., continuous triggering) has been an important problem which should be well addressed. Meanwhile, the exogenous disturbance commonly exists in practical systems and it is also one of the main reasons to induce the Zeno phenomenon. In [21], a typical event-triggering scheme is developed and the system state is sampled and transmitted when a condition in the relative error type is satisfied. A lower bound on the inter-execution intervals is further presented to avoid the Zeno behavior. However, when the event-triggering scheme is extended to the system subject to exogenous disturbance, the continuous triggering may occur. In [22], an event-triggering condition based on the passivity theory is adopted and the stability is studied. But as the discussion in the paper, the continuous triggering may appear when the exogenous disturbance is considered in the system. In [23], [24], a kind of sampled-data based event-triggering scheme is investigated. The prior periodic sampling guarantees that any inter-execution interval is not less than a sampling period, which allows the Zeno problem to be excluded whether or not there is a disturbance.
Since ETC has remarkable superiority in saving system resources, it is widely applied to various systems, such as power systems [25], multi-agent systems [26], [27] and networked control systems [28], [29]. However, the (networked) switched systems with event-triggered control are still less concerned [30], [31], [32]. And in the event-triggered switched system, the event-triggered instants and switching instants both exist and likely interlace with each other. To our best knowledge, no reference has thoroughly discussed the effect of them on the ETC design. In addition, Refs. [23], [24] not only study the periodic event-triggering schemes but also present a method to transform the event-triggered closed-loop system into a time-delay model. Nevertheless, the model may no longer hold when the transmission delay in the network is larger than the sampling period such that the stagger of the data updating sequence occurs in the actuator. As far as we know, this issue still has not been noticed and considered.
In this paper, the event-triggered control problem for network-based switched linear systems subject to exogenous disturbance and transmission delay is investigated. Motivated by the event-triggering schemes in [23], [24], the improved event-triggering schemes with extra parameters are first proposed. Then, a time-delay closed-loop switched system is developed under a comprehensive analysis of the relationship between the transmission delay and sampling period. Furthermore, the piecewise Lyapunov functional method and average dwell time technique are utilized to construct sufficient functions for achieving GUUB of the system state and performance of the developed time-delay closed-loop switched system. Finally, the co-design of state feedback gains and event-triggering parameters is provided.
The contributions of this paper can be summarized. First, an improved periodic sampling based event-triggering strategy is proposed, which involves a set of extra parameters and can further reduce the release rate and redundant signal transmissions and updates. Second, in the actuator, since large transmission delays may result in the stagger of data updating sequence and cause difficulties in closed-loop modeling and further derivations, an active packet loss method is introduced to address this issue. And third, the coupling effects from the switching instants and the event-triggered data updating instants are thoroughly clarified. This provides an effective framework to analyze the stability of switched systems with the improved event-triggering strategy. The reminder of this paper is organized as follows. Section 2 provides the improved event-triggering schemes and system modeling. The stability analysis and control design are followed in Section 3. An illustrative example and conclusions are respectively given in Sections 4 Illustrative example, 5 Conclusions.
Notation. Throughout the paper, is the -dimensional Euclidean space; denotes the Euclidean vector norm; is an identity matrix with appropriate dimension; and respectively denote the set of natural numbers and positive natural numbers; is the set of nonnegative real numbers; and are respectively the transpose and inverse of matrix is the left limit of scalar stands for the symmetric blocks of a partitioned matrix; and are respectively the smallest and largest eigenvalues of matrix denotes a block-diagonal matrix; is defined as the set of signals with a finite -norm, i.e., , where .
Section snippets
The improved event-triggering schemes and system modeling
Consider a switched linear system with exogenous disturbance as where is the system state, is the control input, is the controlled output, is an exogenous disturbance belonging to , and is the switching signal. and are constant matrices with appropriate dimensions.
As shown in Fig. 1, the system state is first sampled as with a
Stability analysis and control synthesis
To simplify the analysis, the condition is first considered. Some lemmas and definitions are given here.
Lemma 3.1 For any real matrices , symmetric and positive definite matrix with appropriate dimensions and scalar , it holds that [36]
Lemma 3.2 For any constant matrices and with appropriate dimensions, then, holds for any if and only if [37]
Lemma 3.3 For any symmetric and positive definite matrix, symmetric matrix with[38]
Illustrative example
In this section, an example is provided to demonstrate the effectiveness of the proposed method. Consider the switched linear system (1) with two sub-systems as . The exogenous disturbance is given by if , otherwise . The initial system state is set as and other parameters are set as and . By solving the LMIs (30)–(32)
Conclusions
By implementing the improved periodic event-triggeringschemes with extra parameters in networked switched linear systems, we are interested in the control problem of the systems subject to exogenous disturbance and transmission delay. In particular, an active packet loss strategy has been proposed to fully cope with the influence of the relationship between the sampling period and transmission delay on the time-delay modeling. Besides, the coupling effect of the switching instants and the
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.
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This work was supported in part by the National Natural Science Foundation of China (grant numbers 61873172, 61811530036, 61403261), the Liaoning Revitalization Talents Program, China (grant number XLYC1807101), the Aeronautical Science Foundation of China (grant number 2016ZC54008), the Natural Science Foundation of Liaoning Province, China (grant number 20180550517), the Scientific Research Fund of Liaoning Provincial Education Department, China (grant number L201709), the Liaoning BaiQianWan Talents Program, China (grant number 2018-B-20), the Young and Middle-aged Science and Technology Innovation Talent Support Program of Shenyang, China (grant number RC170445), the Open Fund of Science and Technology on Thermal Energy and Power Laboratory, China (grant number TPL2017CA005), and the Young and Middle-aged Top-notch Talent Support Program of SAU, China (grant number 17091600105).