1. Introduction
In recent years, as a special kind of random mixed system, the Markovian jump system has shown an advantage of better describing physical systems with sudden change. It has been widely used in industrial control systems, network control systems, fault detection systems, and other fields [
1,
2,
3,
4,
5,
6,
7]. In [
1], the authors based their research on the method of combining the auxiliary function-based double integral inequality and the extended Wirtinger inequality and Jensen inequality and studied the delay-dependent stability of a Markovian jump network delay system with partial information on transition probabilities. In [
2], the authors studied the design of a reduced-order unknown input observer for a one-sided Lipschitz nonlinear continuous-time descriptor Markovian jump system. In [
3], the authors studied the sliding mode control problem of continuous-time Markovian jump systems with partially unknown and uncertain transition rate matrices. It has attracted many scholars to devote themselves to the stability analysis and controller research of Markovian jump systems. In [
4], the authors, based on the properties of Markovian jump systems, considered packet loss as a Bernoulli stochastic process with constant probability, independence, and identical distribution. A method to design
filters for networked control systems with network delay and packet loss was proposed. In [
5], the author proposed a state feedback controller for solving the discrete-time model reference tracking control problem of external disturbances and Markov jump systems under input constraints. At the same time, a state feedback controller was proposed to ensure that the system state vector accurately tracks the given state reference model vector. In [
6], the authors introduced a logarithmic quantizer into the design of a state feedback controller and input the uncertainty into the quantization error matrix represented by an interval matrix. The quantization control problem of a continuous-time uncertain Markovian jump system with mixed delays and partial known transition probabilities was studied. In [
7], the authors established a sufficient condition for exponential almost sure stability by applying a method to its stochastic transfer matrix and studied exponential almost sure stability of continuous-time Markovian jump systems with additional stochastic switching performance and stability problems. In [
7], the author established sufficient conditions for exponential almost certain stability by applying a method to its random transition matrix and studied the exponential almost certain stability of continuous-time Markovian jump systems with additional random switching performance and stability problems.
With the rapid development of computer and network technology, a network was introduced into the traditional control system to form a networked control system. The network control system has the advantages of high cost-effectiveness, simple installation and maintenance, and high reliability, and it is is widely used in aerospace, defense industry, telemedicine, etc. In a practical network control system, the components of the control system (controller, controlled object, actuator et al.) lie in different positions and the signals of each part are transmitted through the communication network. The signal is in the transmission process, and due to the limitation of network bandwidth, the network control system inevitably has problems such as network delay, packet loss, and timing disorder [
8,
9,
10,
11,
12]. In order to overcome the latency of communication networks and solve the problem of limited network resources, signal quantization technology has emerged in recent years [
13,
14,
15,
16]. For example, in [
13], the author considered the joint design of quantizers and estimators to study the state estimation problem of linear dynamic systems. In [
14], the stabilization condition of a Markovian jump system with partial transition probability and input quantization was studied. In [
15], the authors studied the stabilization of switched linear systems with sampling and quantized output feedback. In [
16], the author assumed that each mode of the system was stable and observable, and each mode had feedback gain. The stabilization problem of switched linear systems with sampled data and quantized output feedback was studied.
Recently, a new event-triggered scheme based on different traditional time-triggered schemes has been proposed in the literature, which can be used as an alternative method to minimize the use of communication resources. Compared with a traditional time-triggered scheme, an event-triggered scheme can significantly reduce the sampling frequency of samplers and unnecessary data transmission while maintaining control system performance. In the case of limited communication bandwidth, reducing unnecessary data transmission can improve the efficiency of data transmission and reduce the problem of data transmission delay in communication networks. In recent years, the event-triggered scheme has attracted the attention of domestic scholars [
17,
18,
19,
20,
21,
22,
23,
24,
25]. For example, in [
17], the authors studied the design of an event-triggered controller for networked control systems with network delay. In [
18], the author studied event-triggered controller design for networked control systems via a new integral inequality of a Lyapunov–Krasovskii functional. In [
19], the authors studied the event-triggered output feedback
control problem of a networked control system with non-uniform sampling. In [
20], the authors studied the control design problem of the event-triggered linear network system of state quantification and control input quantification. In [
21], the authors studied the event-triggered static output feedback control of the network control system. In order to reduce the utilization of limited network bandwidth, in [
22], the authors studied the
control problem of the networked Markovian jump system with an event-triggered scheme. In [
23], the authors proposed a centralized/distributed hybrid event-triggering scheme for nonlinear multi-agent systems and studied the leader–follower consistency problem of nonlinear stochastic multi-agent systems. In [
24], in order to save the limited network communication bandwidth of multi-agent systems, the author proposed a novel event-triggered network consensus mechanism and studied the event-triggered guarantee cost consensus of discrete-time singular multi-agent systems with switching topology. In [
25], the authors studied consensus problems for a class of uncertain nonlinear multi-agent systems by using neural networks and event-driven mechanisms.
Through the analysis of the literature above, it was found that there are still some shortcomings in the research on network delay in networked control systems. Due to the limitation of network bandwidth resources, the existing research often adopts time-triggered and event-triggered schemes to improve the utilization of bandwidth resources and to reduce network delay. At the same time, when estimating the upper bound of Lyapunov–Krasovskii functional derivatives, there may be situations where too many matrix variables are introduced and the network delay is amplified. Inspired by the above discussion, this paper focuses on the output feedback control problem of event-triggered Markov-type network control systems. The main contributions are as follows: (1) In order to solve the problem of network delay caused by limited network bandwidth resources, by introducing an event-triggered mechanism and signal quantization, the transmission of redundant information in the network can be reduced, the transmission efficiency of information in the network can be improved, and the network delay can be reduced. (2) When constructing the Lyapunov–Krasovskii function, considering the influence of network delay on the stability of the system and the influence of different modes on the Markov switching strategy in a small switching time region, the Lyapunov–Krasovskii function contains Markov jump information, network delay information, and the coupling relationship between network delay and the upper bound of network delay. (3) When estimating the upper bound of the weak infinitesimal generator operator of the Lyapunov–Krasovskii function, the combination of Wirtinger’s-based integral inequality and the reciprocally convex inequality is used to estimate the weak infinitesimal generator operator of the Lyapunov–Krasovskii function. It can reduce the matrix variable and the computational complexity, and it can avoid the network delay being amplified to reduce the conservatism of the system.
The structure of this paper is as follows. In
Section 2, a new Markov-type networked control systems model with external disturbances is established, considering the effects of event triggering and quantization. In
Section 3, the output feedback
control problem of event-triggered Markov-type networked control systems is studied. In
Section 4, two numerical examples are given to verify the effectiveness and superiority of the method. In
Section 5, the conclusion of this article is summarized, and future research directions and work are emphasized.
Notation: In this paper, and denote the n-dimensional Euclidean space and the set of real matrices; the notation means that P is a real symmetric and positive-definite (semi-positive-definite) matrix; the notation stands for the mathematical expectation; the superscripts ‘−1’ and ‘T’ stand for the inverse and transpose of a matrix, respectively; symmetric terms in a symmetric matrix are denoted by ‘∗’. For any square matrices A and B, it is defined that . The notation I stands for the identity matrix.
4. Numerical Example and Case Study
In this section, we give three examples to illustrate the efficiency and advantage of the obtained results in this paper.
Example 1. Considering the following 2-mode Markov-type networked control system (9), the system matrix parameters are as follows: It is assumed that the response Markov process transition rate matrix is The initial state and external disturbance are given as , and Case 1. Other parameter settings include , , , , , , , and . The comparison results of the upper bound of the maximum allowable time delay and the number of decision variables obtained by Theorem 2 in the literature [35] and Theorem 2 in this paper. The comparison results are shown in
Table 1, and the corresponding controller gain matrices are
and
. The simulation result for the responses of
is shown in
Figure 3. Through a comparison in
Figure 3, it can be found that by introducing an event-triggered scheme and signal quantization separately in the network control systems, compared with the four cases of simultaneously introducing an event-triggered scheme and signal quantization and not introducing an event-triggered scheme and signal quantization, this can indicate that the method adopted in this paper is reasonable and effective.
It can be seen from
Table 1 that Theorem 2 in this paper gives a better result than Theorem 2 in [
35]. On the one hand, because the combination of Wirtinger’s-based integral inequality and the reciprocally convex inequality is used to estimate the upper bound of the weak infinitesimal generator operator of the Lyapunov–Krasovskii function, while retaining some important information of delay as much as possible, the delay is avoided to be amplified and the network transmission delay is reduced, so the upper bound of the maximum allowable transmission network delay is obtained as 1.0200 s. On the other hand, compared with the free-weight matrix method to estimate the upper bound of the weak infinitesimal generating operator of the Lyapunov–Krasovskii function by using the combination of Wirtinger’s-based integral inequality and reciprocally convex inequality. Wirtinger’s-based integral inequality and reciprocally convex inequality can reduce the number of matrix variables; thus, this reduce the computational complexity. The computational complexity of the matrix variable obtained by this method is
.
Case 2. Other parameter settings include , , , , , and . When and to different values, we observe the impact of different values on network delay.
From the results of
Table 2, we can see that by introducing the event-triggered scheme in networked control systems, when
, the event-triggered weakening process is time-triggered, which is the largest delay in the network; when the value of
increases, the delay in networked control systems gradually decreases. The simulation result for the responses of
is shown in
Figure 4. It can be seen from
Figure 4a–d that the networked control system becomes larger with the value of the event-triggered parameters
and
. Although the system is stable, there are some differences in the system state trajectories, indicating that the introduction of the event-triggered mechanism can reduce the delay problem in the networked control system.
Case 3. Other parameter settings include , , , , , , and . When to different values, we observe the impact of different values on network delay.
From the results of
Table 3, we can see that when
, the delay in the network is the largest by introducing the quantization scheme in the networked control system. When the value of
increases, the delay in the network gradually decreases. The simulation result for the responses of
is shown in
Figure 5. From
Figure 5a–d’s system state response curves, it can be seen that under the action of event-triggered mechanism, the state response time changes with the signal quantization parameter
, and the response time of the system to the steady state also changes, indicating that the signal quantization can effectively reduce the delay problem in the network.
From the state response curves of
Figure 3,
Figure 4 and
Figure 5, it can be seen that although there are some differences in the state trajectories, the system is stable and meets the expected performance index. From
Table 1,
Table 2 and
Table 3, it can be seen that the introduction of an event-triggered scheme and quantization scheme can improve the utilization of network bandwidth resources and reduce network delay.
Example 2. Considering the following 4-mode Markov-type networked control systems (9), the system matrix parameters are as follows: It is assumed that the response Markov process transition rate matrix is Considering the influence of time-delay on the stability of the Markov-type networked control systems (
9) and applying Theorem 2, the upper bound of the maximum allowable time delay is 1.0947s, and the corresponding controller gain matrices are
,
,
, and
. It can be seen from
Figure 6 that the system can quickly reach the steady state.
Example 3. Considering the structure of the distributed photovoltaic power generation network control system shown in Figure 7, taking photovoltaic array 1 and photovoltaic array 5 as examples, analyze the distributed photovoltaic power generation network control system. This system considers the different operating conditions and control situations of photovoltaic power generation, power consumption, and energy storage between different photovoltaic arrays. The distributed photovoltaic power generation network control system transmits information through a communication network, and the controller sends control switch K1, K3 information through the network to determine the operation mode of the photovoltaic power generation system. , , , , , and represent the impedance of the line, and is the impedance of the load. Based on the Markov switching strategy, establish a 4-mode Markov-type network control system for network communication between photovoltaic array 1 and photovoltaic array 5: represents the total current of photovoltaic generation in photovoltaic array 1; represents the total current of photovoltaic generation in photovoltaic array 5; represents the total demand current for power supply; and represents the current of the public power grid, where . Among them, mode 1: when , photovoltaic array 1 and photovoltaic array 5 simultaneously generate photovoltaic power; mode 2: when , photovoltaic array 1 generates photovoltaic power, while photovoltaic array 5 generates photovoltaic power for energy storage; mode 2: when , photovoltaic array 5 generates photovoltaic power, while photovoltaic array 1 generates photovoltaic power for energy storage; mode 2: when , both photovoltaic array 1 and photovoltaic array 5 are storing energy for photovoltaic power generation.
The state vector
of the system is
, which is the generation current, and
is the current flowing through the load in the circuit.
and
, respectively, represent the measurable output power and control output power in the circuit. The matrix parameters of the distributed photovoltaic power generation network control system are as follows:
It is assumed that the response Markov process transition rate matrix is
Considering the impact of network delay on the stability of distributed photovoltaic power generation network control systems, co-design is used to implement the design and implementation form of the controller. In the controller design process, the problem of variable coupling can be addressed through the idea of variable transformations, which can avoid matrix inversion and achieve variable decoupling. Provide a design method for the controller through analysis. For the delay problem in the distributed photovoltaic power generation network control system, using Theorem 2, the maximum allowable delay upper bound for ensuring the stable and smooth operation of the entire system is 1.1190 s, and the corresponding control gain matrix is
,
,
, and
. The state response curve of the distributed photovoltaic power generation network control system is shown in
Figure 8. It can be seen from
Figure 8 that by applying Theorem 2, the distributed photovoltaic power generation network control system can quickly respond to reach a steady state.