Abstract
The state feedback control problem for a class of nonlinear networked control systems with data packet loss is studied using an event-triggered scheme. The data packet loss is described as an independent and homogeneous Bernoulli process. Under an event-triggered scheme, the nonlinear networked control system with packet loss is modeled as a Takagi-Sugeno (T-S) fuzzy system, based on which sufficient conditions on the existence of event-triggered state feedback controllers are derived such that the closed-loop system is mean-square stable with a desired performance index. The simulation results show that the presented event-triggered scheme can not only ensure the closed-loop performance but also effectively reduce the data transmission rate.
1. Introduction
Networked control systems (NCSs) are control systems in which the control loop is closed via a wired or wireless communication network [1–3]. However, since the bandwidth of the communication network is limited, only a finite number of executions can be performed over the communication network, so that the actuator cannot communicate with the controller timely during the transmission of the control signal due to uncertain external disturbances and packet loss phenomenon [4–7]. Therefore, studying NCSs with packet loss and resource constraints is of much significance both in theory and in practice.
As is known, a Takagi-Sugeno (T-S) fuzzy model can generate more complex nonlinear functions with a small number of fuzzy rules. Therefore, it is a very useful tool for dealing with nonlinear systems. In [8], the problem of an event-triggered nonparallel distribution compensation (PDC) control is considered for the networked T-S fuzzy systems with the limited data transmission bandwidth and the imperfect premise matching membership functions. In [9–11], the control problem for a class of T-S fuzzy Markov jump systems with time-varying delay under unreliable communication relatives has been studied. In [12], it is assumed that the data transmissions between the plant and the controller are subject matter to random packet loss which satisfies Bernoulli distribution. In [13, 14], the controller is designed by the PDC technique in which the controller shares the same membership functions and fuzzy premise variables with the T-S fuzzy model. An event-triggered fuzzy controller design method for a class of discrete-time nonlinear NCSs with time-varying communication delays has been studied in [15].
Up to now, event-triggered control has gained increasing interest in the last several years, which can effectively shrink feedback data flow while retaining a certain level of control performance compared with a time-triggered scheme. In [16], a class of continuous linear NCSs are considered, where some event-triggered schemes have been devised based on the system states. Concerning distributed energy management and control issues of both generators and loads, papers [17, 18] aim to maximize the total social welfare that balances generation-side expanses, user-side payments, and transmission line costs and provide an overview of recent advances on event-triggered consensus of MASs. A pulse system method has been proposed in [19], which simulates and analyzes the scattered control system with an event-triggered scheme. In [20–22], a time delay system method has been proposed, which can be used to analyze the stability of a continuous NCS under an event-triggered transmission scheme. In [23], a codesign method of corresponding event-triggered transmission with quantizer and controller has been proposed. The state observation has been used to describe the system model. Then the NCSs have been modeled as a discrete-time switching system with uncertain parameters.
It is well known that packet loss often arises in the network transmission. To solve this problem, in [24], the data measurements processes from the plant to the filter are subject to random packet loss which satisfies Bernoulli distribution with bounded nonlinearity. In [25, 26], the network-based output tracking control is investigated for a T-S fuzzy system and it is concluded that the system cannot be stabilized by a nondelayed fuzzy static output feedback controller but can be stabilized by a delayed fuzzy static output feedback controller. In [27], jump neural NCSs with transmission delay and packet loss have been considered. However, there is no further consideration for the nonlinear problem. The resource constraints and packet loss problems in [28] also have similar discussions, using the channel constraints to reduce resource consumption, but there is only one actuator connected to the communication network at a time. In [29], the packet loss and channel constraints processes have been modeled as , from which it can be seen that the study is for single-packet transmission strategy, and the input strategy is applicable to different NCSs. Because the above-mentioned packet loss model is a probability event and sometimes does not match the real situation. In order to address two issues mentioned above, in [30, 31], an event-triggered mechanism is proposed, which can effectively reduce feedback data flow and reduce the broadband occupancy rate such that network resources can be saved. In [32], packet loss in both sensor-controller (S/C) and controller-actuator (C/A) channels is considered. The Markovian chain principle is used in modeling the packet loss in S/C and C/A channels. The time scale adopted in these two independent homogeneous Markov chains is linear with the physical time. However, most of the above studies are based on linear systems. In fact, practical systems are usually nonlinear, while few results have focused on nonlinear NCSs by taking event-triggered mechanisms and data packet loss into account, which motivates the study of this paper.
In this paper, in order to deal with the resource constraints and packet loss, an event-triggered mechanism strategy based on a relative error is proposed. Then the state feedback control problem is investigated for a class of nonlinear NCSs using an event-triggered mechanism. First, the packet loss under multichannel strategy is described as an independent and homogeneous Bernoulli process. Then a new model of nonlinear NCSs is established by employing T-S fuzzy model method. Based on this model and by employing the Lyapunov stability theory and linear matrix inequality technique, a sufficient condition for the existence of controller is presented to ensure the mean-square stability of NCSs and an improved performance. Moreover, the corresponding controller can be designed using the PDC technique. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.
The remaining of this paper is organized as follows. Section 1 introduces the background and significance of the research. Section 2 formulates the problem consideration. performance analysis and state feedback controller design are presented in Section 3. An illustrative example is given in Section 4 to demonstrate the effectiveness of the presented method. The conclusion is drawn in Section 5.
2. Problem Description and Modeling
In this paper, the NCS with packet loss and transmission data generator is shown in Figure 1. The sensors in the communication network are time-triggered with a constant sampling period , while the controller and actuators are event-triggered. The TD generator in Figure 2 is event-triggered, which is used to generate sampled data and transmission data. The zero-order holder (ZOH) is used to hold the sampling signal until the new sampling signal reaches the controller. Consider the nonlinear plant model as follows: where is the state vector of plant; is the input vector and is the external interference signal satisfying . , , , , and are continuous functions of , and . Since the transmission process belongs to the discrete process, the nonlinear system on the compact set is considered to the following discrete-time T-S model with plant rules:
Plant rule : IF is and is , THEN where ( is the number of IF-THEN rules); , , and are the state vector, input vector, and regulated output, respectively. , , , , and are system matrices with appropriate dimensions. are fuzzy sets, via the membership function to , and are the premise variables, which are defined on , and indicates the number of rules on the S plane, and indicates the number of states on the S plane. Then the T-S fuzzy system can be described as where ; ; ) is the grade of membership of in , and .
For each transmission of measurement data and control signal, due to the uncertainty of the communication network, a packet loss usually occurs; these measurement data and control signal packets may not be able to reach the actuator. Suppose that the two-valued function and express the packet loss of the th control signal. Then the corresponding control signal will be transferred to the actuator when the th actuator accesses the channel. and express that the th control signal is lost during transmission; and express that the th control signal is successfully transmitted. Notice that data packet loss process often occurs randomly. Thus, in this paper, it is described as Bernoulli process. Assume that the channel packet loss processes are independent of each other, and the measurement data packet loss rate is ; that is, ; . Similarly, the loss of the control signal has the same process; the packet loss rate is ; that is, ; .
In this paper, the design of the event-triggered transmission is based on a relative error threshold: where are error threshold parameters and , , and are the last transmission signal, output signal, and input signal, respectively.
Set and . Then, from the event-triggered transmission scheme (4), the input-output relationship of ETM can be transformed to where .
From the above analysis, we can see that, in the NCS with an event-triggered mechanism, since each sampling period signal does not need to be transmitted, the purpose of reducing the data transmission rate can be achieved. Suppose that there are channels in the event trigger to send the data. We set ;
Considering packet loss, the final measurement output is given asSimilarly, setting and , the transmission of the control signal can be described by With formulas (5), (6), and (7) we can get the following closed-loop model of NNCSs:where In this section, a T-S fuzzy-model controller will be designed via PDC technique to stabilize the T-S fuzzy system (1) with th controller rule:
Controller rule : IF is and is , THEN where ( is the number of IF-THEN rules); and are the controller output and input data, and is the controller gain to be designed. Thus, the T-S fuzzy controller reads as From (5), (6), (7), and (8), the closed-loop system can be obtained as follows: where
Definition 1. If there exist scalars , , and such that then the stochastic state of the discrete-time stochastic system is mean-square stable.
The goal of our research is to design the gain matrix of the state feedback controller (10) based on the event-triggered strategy (4) within a tolerable threshold such that system (12) is mean-square stable and satisfies a given performance parameter .
In the case of external disturbances , the nonlinear system (12) is mean-square stable.
Under zero initial conditions, for any nonzero , the closed-loop system (12) satisfies the following performance index, where is a given constant.
3. Stability Analysis and Controller Design
In the beginning, we introduce two lemmas.
Lemma 2 (see [33]). If there exist scalars , , and such that then sequence satisfies
Lemma 3 (see [34]). Let , , , and be real matrices of appropriate dimensions with and ; then holds, if and only if there exists a real scalar , satisfying Regarding the random sequence , , we suppose that
Theorem 4. For given scalars and , under the communication scheme (4), the nonlinear closed-loop system (12) is mean-square stable with performance index if there exist symmetric positive-definite matrices , , and with appropriate dimensions such that where
Proof. For the nonlinear closed-loop system (12), we construct the Lyapunov function where is a symmetric positive definite matrix. Then we have where When , we have From (25), (27), and the Schur complement, we can get that To derive the expression for the controller parameters, set and as follows: where , , , and
From (28), , , and , one has Set Applying Lemma 3, it is clear that (30) is equivalent to Employing Lemma 3, we can get that Applying Lemma 2, (27) is equivalent to where When and , from (34), we can get that the nonlinear system (12) is mean-square stable. When , we have that where When , by Schur complement, we can get that Similarly, substituting into (38) gives Likewise, employing Lemma 3, we have that which is equivalent to (21). Furthermore, by the use of Schur complement, we can get that and (15) is obtained. Hence, it is clear that which leads to Due to the initial state , we get , where is a given positive constant. Thus holds for any nonzero . So the nonlinear system (12) is mean-square stable and has an performance level . This completes the proof.
Remark 5. If the initial state , , and inequalities (21) and (22) hold, then the nonlinear system (12) also has an performance level such that where and are given constants.
The following theorem provides an algorithm to design suitable controller gains.
Theorem 6. For given scalars and , the nonlinear system described by (12) under the event-triggered condition (4) is mean-square stable with performance index , if there exist real symmetric positive definite matrices , , and and with appropriate dimensions such that where Moreover, the desired controller gains are given by
Proof. From Theorem 4, if (21) and (22) hold, the nonlinear closed-loop system (12) is mean-square stable with performance index . Firstly, the matrix inequality (21) is multiplied by and its transpose, respectively. Setting and performing the congruent transformation on the left side of inequality (21), we can get (45). Similarly, the matrix inequality (22) is multiplied by and its transpose, respectively. Performing the congruent transformation on the left side of inequality (22), we can get (46). This completes the proof.
Remark 7. Although the conditions in Theorem 6 are not convex, using the cone complementarity linearization (CCL) method [35], we can change it to the nonlinear minimization problem with LMI constraints which are equivalent to the nonlinear minimization problem with linear matrix inequality constraints as follows: The above nonlinear minimization problem can be solved by an iterative algorithm as presented as follows.
Algorithm 8.
Step 1. Find a set of feasible solutions such that (51), and set the iterative number .
Step 2. Use LMI toolbox of mincx solver to solve the following linear objective function minimization problem: The solution is set to be .
Step 3. Substitute the solution into the matrix inequalities (45) and (46) in Theorem 6. If (45) and (46) are satisfied, then is the state feedback gain matrix and stop; otherwise, go to Step 4.
Step 4. If the iterative number satisfies (L is a predetermined iterative number upper bound), set and and go to Step 2 for the next iteration. Otherwise, go to Step 1 and reselect a set of feasible solutions for calculation.
4. Simulation Example
In this section, we will give a simulation example to demonstrate the effectiveness of the proposed approach. The discrete-time T-S model fuzzy systems are expressed as follows. Plant rules: IF is , THEN IF is , THENwhereWe suppose that the sampling period and the initial condition and the external disturbance is The membership functions and are described, respectively, as
Let the event-triggered parameter , the packet loss rate and , and performance level . Applying Theorem 6, we can obtain controller feedback gains as Figures 3 and 4 illustrate that the state responses of the system are mean-square stable. Compared with these two pictures, packet loss may have some influence on the stability of the system at the same transmission conditions and states. Figure 5 shows the Bernoulli communication sequence about packet loss. Figure 6 shows the intervals of different times between event-triggered transmitters. IAE denotes the control performance of the system. Under four different thresholds , the performance of transfer rate and is shown for the system with the above three error threshold values in Figure 7, where case 1 is , case 2 is , and case 3 is .
It can be seen that although only a small section is triggered to be transmitted to the controller, the networked control system is mean-square stable with the prescribed performance level. Comparing Table 1 with Figure 7, we find that when the value of gets larger, the transmission rate is lower, but the error increases, so the appropriate value can be determined. Only 20.8% of the measurement data is transmitted if , which means that the required transmission can save limited network resources. From Figure 6, we find that the event-triggered mechanism can not only relieve the problem of resource constraints but also make the data in the transmission process faster and more stable, so the method proposed in this paper is effective.
5. Conclusion
This paper focuses on the packet loss and limited resources problems for the nonlinear NCSs. A discrete-time model of nonlinear NCSs is established by the T-S fuzzy approach. The packet loss process of each channel is described as an independent and distributed Bernoulli process. An event-triggered scheme is put forward for the nonlinear NCSs based on a relative error-triggered mechanism. Using the Lyapunov stability theory and linear matrix inequality method, the feedback controller is given via PDC technique, which guarantees the mean-square stability of nonlinear NCSs with the desired performance. Since the principles and algorithms of the trigger in this paper are relatively simple, the smart sensor is easy to implement. In addition, the simulation results show that the designed communication scheme can effectively decrease the data transmission rate, so it is very suitable for application to NCSs with limited bandwidth resources.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.