Dynamic event-triggered-based human-in-the-loop formation control for stochastic nonlinear MASs

The dynamic event-triggered (DET) formation control problem of a class of stochastic nonlinear multi-agent systems (MASs) with full state constraints is investigated in this article. Supposing that the human operator sends commands to the leader as control input signals, all followers keep formation through network topology communication. Under the command-ﬁlter-based backstepping technique, the radial basis function neural networks (RBF NNs) and the barrier Lyapunov function (BLF) are utilized to resolve the problems of unknown nonlinear terms and full state constraints, respectively. Furthermore, a DET control mechanism is proposed to reduce the occupation of communication bandwidth. The presented distributed formation control strategy guarantees that all signals of the MASs are semi-globally uniformly ultimately bounded (SGUUB) in probability. Finally, the feasibility of the theoretical research result is demonstrated by a simulation example


Introduction
In the past decade, with the development of artificial intelligence technology, many effective control strategies have been proposed in the area of cooperative control, such as neural network control, sliding mode control, and fuzzy control [1][2][3][4][5][6].In addition, the formation problem of multi-agent systems (MASs) is a significant branch of cooperative control, that has received extensive attention [7].Formation control in MASs pertains to the coordinated motion of numerous agents in accordance with a predetermined formation pattern, which has found extensive utility in diverse domains such as aerial vehicles, land vehicles, and ships [8][9][10][11][12][13].Based on the position of agents, a collaborative control algorithm for auxiliary systems was proposed, which did not require direct measurement of linear velocity to achieve aerial vehicle formation tracking control [9].It is noteworthy that perturbations in the external environment can substantially impact the accuracy of formation control.For a class of MASs subject to interference, a distributed controller in [10] was formulated to eliminate the influence of the bounded disturbances and optimize the system to reach the intended formation.A robust adaptive controller was developed to effectively minimize the tracking errors of multiple vehicle formation systems with perturbations [11].
Facing the challenge of obstacle avoidance within formation control, a light transmission model was introduced in [12], which improved the traditional artificial potential method.In [13], a dynamic formation tracking control strategy was presented for a class of unmanned aerial vehicles equipped with switching topology to make the drone swarm reach the surrounding area of a designated target.
In recent years, there has been a swift evolution of intelligent control technology for unmanned autonomous systems, which has also garnered the interest of numerous scholars [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].However, there were numerous reported incidents concerning the autonomous control of unmanned systems, which have caused significant negative impacts, prompting concerns and further investigation.In complex environments, enhancing the security of autonomous agent systems is of utmost importance.To improve this, the human-in-the-loop (HiTL) control approaches were analyzed in [29][30][31][32][33][34], which were proven to be highly effective in enhancing the reliability and security of the system.In light of this background, HiTL control technology has risen rapidly in the field of cooperative control.For the HiTL MASs with time delay, Koru et al. [31] proposed an inner loop distributed control method, but it was unsuitable for nonlinear systems.For the HiTL MASs with uncertainty disturbances, a distributed adaptive containment control strategy was presented in [32] to achieve containment control of MASs.In [33], a novel adaptive inverse optimal control method using only state measurement was designed to solve the problem of online adaptive learning human behavior.Lin et al. [34] constructed a dynamic model for an unmanned aerial vehicle attitude system and proposed a finite-time command-filtered HiTL control method to achieve a faster convergence rate.
The aforementioned research works have successfully addressed the HiTL control problems.However, the issue of communication network bandwidth is seldom taken into account, which leads to excessive consumption of communication resources.In order to reduce the communication burden, many sampling schemes were proposed [35][36][37][38][39][40][41][42][43][44][45].For the second-order integral MASs with communication noise, a distributed sampling-data protocol was studied in [46], which solved the average consensus problem.To enable stability for nonlinear systems with sampled-data inputs, Bernuau et al. [47] designed a system satisfying a homogeneous condition.It is worth mentioning that the event-triggered-based mechanisms perform with higher resource utilization efficiency than the time-triggered-based mechanisms.In [48], An adaptive control approach was proposed to reduce the communication burden by implementing a fixed threshold event triggering mechanism.In [49,50], two event-triggered control strategies with relative threshold parameters were considered to further optimize control performance and obtain higher resource utilization efficiency, respectively.It was noteworthy that a novel dynamic event-triggered (DET) strategy was proposed in [51] to solve the network bandwidth problem for a stochastic nonlinear system, which enabled the controller to attain a lower triggering frequency than the traditional sampling control strategy.In addition, Cao et al. [52] designed a novel switching DET control mechanism to reduce communication costs.
System states, such as aircraft flight altitude, motor speed, and vehicle turning angle, are often subject to constraints arising from the actual.A system with state constraints is more secure and reliable [53][54][55].In [53], a nonlinear state space model of synchronous motors was established to introduce state constraints into the model and guarantee that the motor current was constrained within a certain range.Zhang et al. [54] proposed a one-to-one nonlinear mapping strategy, whereby the constrained system is transformed into an unconstrained system, thereby alleviating the burden of state constraints.It is noteworthy that the introduction of the barrier Lyapunov function (BLF) effectively addresses the issue of full-state constraints.For a category of multiple constraint problems, a BLF was designed to handle state constraint problem [55].
Inspired by the above works, a DET formation control strategy is researched for a category of HiTL stochastic nonlinear MASs with full state constraints.The major contributions are outlined below.
• Unlike the formation control of [7][8][9][10][11][12][13] where the leader's output trajectory is given as a reference signal, this paper considers that the leader's control signal is given by a human operator, which is conducive to realizing the variability of the formation trajectories, with better utility value and safety performance.
• By comparing several time-triggering control and event -triggering control mechanisms [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52], a DET control mechanism is developed to reduce waste of communication resources between controller and actuator in this paper, which obtains higher resource utilization efficiency.• The distributed formation controller is developed under the framework of the command filtering backstepping method to achieve formation control, and a filter compensation system is used to handle the filtering error problem.Furthermore, a BLF is constructed to constrain the states within a certain range, thereby improving the control performance.
2 Preliminaries and problem statement

Problem statement
The ith(i = 1, 2, . . ., N ) follower dynamic model is described by where x i, denotes the systems state with = 1, 2, . . ., n, xi, = [x i,1 , x i,2 , . . ., x i, ] T ∈ R , the f i, (x i, ) ∈ R and g i, (x i, ) ∈ R represent the unknown nonlinear functions.Assume g T i, (x i, ) is bounded.u i and y i denote the control input and output of the ith agent, respectively.ω represents an independent qdimensional Wiener process which is included in a complete probability space (Ξ, ae, {ae t } t≥0 , ), where Ξ represents a sample space, ae represents a σ-filed, ae t represents a filtration,  represents a probability measure.
The leader's input u 0 is supplied by a human operator, which indirectly controls all followers through the network topology, followers can't directly access the information of u 0 .The dynamic model of the leader is considered as where x 0,1 expresses the leader state, y 0 is the leader output.
The main objective of this article is to design a distributed controller where all the agents can maintain a desired formation.All signals in the closed-loop system are bounded in probability and do not violate the full state constraints.To ensure completeness in the design process, the following assumptions and lemmas are made.
Assumption 1.The leader has at least one directional path to any other node, and the leader output y 0 and its derivative ẏ0 are bounded.
To enhance the efficiency of communication resources, defining u i (t) = u i , the DET control mechanism is described as where v i ( tκ ) is the sampled controller input, e i (t) = v i (t) − u i (t), v i (t) expresses the intermediate continuous control signal, µ i and β i are positive parameters.Furthermore, the initial value of λ i (t) satisfies inequality 0 < λ i (0) < 1. according to (3), it yields where £ i (t) and £i (t) are design parameters within the range of [−1, 1].
Assumption 2. The compensation error , where E i, is a designed constraint parameter for compensation error z i, .

Lemma 2. [56]
The radial basis function neural networks (RBF NNs) w * i, φ i, (η i, ) are given to approximate the unknown smooth nonlinear term Ji, (η where η i, indicates the independent variable of the function, i, is the bounded approximation error, = 1, 2, . . ., n.The ideal weight vector w * T i, ∈ R is expressed as where Γ i and ℘ i express compact sets of w T i, and η i, , respectively. The basis function φ i, (η i, ) ∈ R is selected as the following function: where ζi, and Ψ i, represent the center and width of φ i, (η i, ), respectively.

Lemma 3. [57]
For real variables A and F , the following Young's inequality holds where C > 0, O > 0, U > 0, and
The compensating error z i, ( = 1, 2, . . ., n) is described as where γ i, represents the error compensating signal.The error compensation system is described as where , c i, and k i, ( = 1, 2, . . ., n) represent positive parameters.
The virtual controller α i, and adaptive law ˙ i, are designed as where H i, is a positive parameter.Based on (31)-( 35) and ( 23), we can know Step n.Similar to previous steps, according to ( 1) and ( 7), the derivative of s i,n is given as According to ( 9), ( 10) and ( 37), we can know where g T i,n (x i,n ) has a bounded range.There is a positive parameter i,n such that ||g T i,n (x i,n )|| ≤ i,n .Then, the BLF is defined as where | with E i,n being the constraint parameter of the compensating error z i,n , τ i,n is a positive parameter.Combining (38), (39) and Lemma 1, one has Substituting ( 4) into (40), it yields v i (t) is constructed as where δ i and ϕ i,n are positive parameters with ϕ i,n > µi 1−λi(t) .Because £ i (t) and £i (t) are design parameters within the range of [−1, 1], we have Substituting ( 42)-( 44) into (41), we can get From Lemma 4, we have Then, combining ( 45)- (47), it yields By using Lemma 3, we have Substituting ( 49)-( 51) into (48), one has A nonlinear function Ji,n (η i,n ) is defined as where η i,n = [x i,1 , x i,2 , . . ., x i,n ] T , By using Lemmas 2 and 3, there is a positive parameter ¯ i,n such that where i,n represents a design parameter.
The virtual controller α i,n and adaptive law ˙ i,n are designed as Security and Safety, Vol. 2, 2023024 where H i,n is a positive parameter.
According to (3), we can know 0 < λ i (0) < 1 and λi (t) = −β i λ 2 i (t), β i is a positive parameter, so there has a positive parameter μi with 52)-( 56) and (36), defining 2 , then we have According to the above description, the block diagram of a distributed formation controller based on DET for HiTL stochastic MASs is shown in Figure 1.
The following demonstrates that the full state constraints of MASs do not be violated.According to (9), we can obtain According to Assumption 2, one has |y 0 | < y 0 , where y 0 is a positive constant.Substituting ( 7) into (66), one has where = 1, 2, . . ., n, i = N ı=1 ρ i,ı i,ı + i i,0 .Remark 2. The design parameters adhere to several guidelines.The appropriate design parameters are selected such that C 1 > 0. Therefore, c i, > 0 and H i, > 0 are selected to ensure system stability.Then, another parameter E i, satisfies |z i, | < |E i, | to ensure that all states of the closed-loop system are constrained.It is worth noting that the larger the value of c i, , the better the system performance, but the computation time also increases.Therefore, it is important to choose appropriate parameters to meet the requirements of the system.

Simulation
To illustrate the feasibility of the proposed control strategy, a numerical simulation example is provided.In Figure 2, the communication topology diagram with a leader and four followers is displayed.

Conclusion
The formation control problem for a category of HiTL stochastic nonlinear MASs with full state constraints has been addressed.A DET mechanism has been presented to effectively reduce the communication burden.A BLF has been constructed to address the full state constraints problem.Under the backstepping control framework, the RBF NNs and command filter have been introduced to address the challenges with unknown nonlinear terms and "explosion of complexity", respectively.Moreover, a filtering error compensation system has been designed to compensate for errors caused by the command filter.Through the provision of the control signal to the leader, the designed distributed formation controller has guaranteed that all followers can achieve the desired formation trajectory and all signals in the MASs have been SGUUB in probability.Finally, the simulation example has validated the effectiveness of the designed control method.

Figure 1 .
Figure 1.The block diagram of the distributed formation control

Table 1 .
Triggering times in 50 seconds of different event-triggering control strategies , ∂ i, + γ i, + |E i, | < H i, , respectively.We can get |x i,1 | < H i,1 and |x i, | < H i, , which indicates that the full state constraints are not transgressed.