Security and Safety
Volume 2, 2023
Security and Safety in Unmanned Systems
Article Number 2023024
Number of page(s) 17
Section Industrial Control
Published online 11 September 2023

© The Author(s) 2023. Published by EDP Sciences and China Science Publishing & Media Ltd.

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. 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 [16]. 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 [813]. 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 [1428]. 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 [2934], 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 [3545]. 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 [5355]. 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 [713] 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 [3552], 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

2.1. Graph theory

MASs exchange information through directed communication topology . ℵ = {ℵ1, ℵ2, …, ℵN} stands for the point set, represents the edge set, and means the adjacency matrix, where ℵi is the node in the directed communication topology , i = 1, 2, …, N, ı = 1, 2, …, N. If ρi, ı = 1, , the follower ı is able to send information to follower i, and if ρi, ı = 0, or i = ı. Define ℌ = diag{d1, d2, …, dN} as the diagonal in-degree matrix and . is described as the graph Laplacian matrix. Define the expansion diagram as , where , ℵ0 indicates the leader node. ℏi = 1 when the ith follower can receive the leader information, else, ℏi = 0. An expansion diagram adjacency matrix is .

2.2. Problem statement

The ith(i = 1, 2, …, N) follower dynamic model is described by


where xi, ϶ denotes the systems state with ϶  =  1, 2, …, n, , the and represent the unknown nonlinear functions. Assume is bounded. ui and yi denote the control input and output of the ith agent, respectively. ω represents an independent q-dimensional Wiener process which is included in a complete probability space , where Ξ represents a sample space, represents a σ-filed, represents a filtration, ȷ represents a probability measure.

The leader’s input u0 is supplied by a human operator, which indirectly controls all followers through the network topology, followers can’t directly access the information of u0. The dynamic model of the leader is considered as


where x0, 1 expresses the leader state, y0 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 y0 and its derivative are bounded.

To enhance the efficiency of communication resources, defining ui(t)=ui, the DET control mechanism is described as


where is the sampled controller input, ei(t)=vi(t)−ui(t), vi(t) expresses the intermediate continuous control signal, μi and βi are positive parameters. Furthermore, the initial value of λi(t) satisfies inequality 0 <  λi(0)< 1. according to (3), it yields


where and are design parameters within the range of [ − 1, 1].

Lemma 1

[51] Suppose that dð=𝔈(ð) dt + 𝔇(ð) dω expresses a stochastic nonlinear differential equation, where ð ∈ ℝn represents the system state, 𝔈(ð) and 𝔇(ð) represent the unknown nonlinear functions, ω represents the ȷ-dimensional Wiener process. The corresponding Lyapunov function V(ð)∈C2, the differential operator L of V(ð) is given as


where Tr{ ⋅ } expresses a matrix trace.

Assumption 2

The compensation error zi, ϶(϶  =  1, 2, …, n) satisfies |zi, ϶|< |Ei, ϶|, where Ei, ϶ is a designed constraint parameter for compensation error zi, ϶.

Lemma 2

[56] The radial basis function neural networks (RBF NNs) wi, ϶*ϕi, ϶ (ηi, ϶) are given to approximate the unknown smooth nonlinear term


where ηi, ϶ indicates the independent variable of the function, 𝜚i, ϶ is the bounded approximation error, ϶  =  1, 2, …, n. The ideal weight vector is expressed as

where Γi and ℘i express compact sets of wi, ϶T and ηi, ϶, respectively. Define ℜi, ϶* = ||wi, ϶*||2, with being the estimate of ℜi, ϶*. The basis function is selected as the following function:

where and Ψi, ϶ represent the center and width of ϕi, ϶ (ηi, ϶), respectively.

Lemma 3

[57] For real variables A and F, the following Young’s inequality holds

where ℂ >  0, 𝕆 >  0, 𝕌 >  0, and (𝕆 − 1)(𝕌 − 1)=1.

Lemma 4

[58] The hyperbolic tangent function tanh(𝔨) satisfies that , where 𝔨 ∈ ℝ and δ >  0.

3. Controller design

The formation error si, 1(i = 1, 2, …, N) and error surfaces si, ϶ (϶  =  2, 3, …, n) are defined as follows:


where ℑi, ı and ℑi, 0 represent the follow i formation distance with follow ı and leader, respectively, ∂i, ϶ − 1 represents the command filter output signal, which is constructed as follows:


where αi, ϶ is the virtual controller, ∂i, ϶ = σi, ϶1, Li, ϶ and Ri, ϶ are positive parameters, ϶  =  1, 2, …, n − 1.

The compensating error zi, ϶ (϶  =  1, 2, …, n) is described as


where γi, ϶ represents the error compensating signal.

The error compensation system is described as


where , ci, ϶ and ki, ϶(϶  =  1, 2, …, n) represent positive parameters.

Step 1. According to (1), (2) and (7), the derivative of formation error si, 1 is written as


Defining , Υi, 1 = [xi, 1, xı, 1]T, and considering (9)–(11), one has


The BLF is constructed as


where |zi, 1|< |Ei, 1| with Ei, 1 being the constraint parameter of the compensating error zi, 1, τi, 1 represents a positive parameter. Combining (12), (13) and Lemma 1, one has


where 𝔗i, 1i, 1) has a bounded range. There is a positive parameter ⅁i, 1 such that ||𝔗i, 1i, 1)||≤⅁i, 1. By using Lemma 3, the following inequalities hold:

(15) (16) (17)

Substituting (15)–(17) into (14), LVi, 1 can be rewritten as


A nonlinear function is described as


where ηi, 1 = [xi, 1, xı, 1, xı, 2]T. By using Lemmas 2 and 3, there is a positive parameter such that , one has


where ϵi, 1 is a design parameter.

The virtual controller αi, 1 and adaptive law are designed as

(21) (22)

where Hi, 1 is a positive parameter. Combining (18)–(22), we can obtain that


Step ϶ (϶  =  2,3,…,n1). From (1) and (7), one has


Using (9), (10) and (24), it yields


where has a bounded range. There is a positive parameter ⅁i, ϶ such that .

The BLF is constructed as


where |zi, ϶|< |Ei, ϶| with Ei, ϶ being the constraint parameter of the compensating error zi, ϶, τi, ϶ is a positive parameter. Combining (25), (26) and Lemma 1, one has


From Lemma 3, the following inequalities hold:

(28) (29) (30)

Substituting (28)–(30) into (27), it yields


A nonlinear function is defined as


where ηi, ϶ = [xi, 1, xi, 2, …, xi, ϶]T. By using Lemmas 2 and 3, there is a positive constant such that , one has


where ϵi, ϶ expresses a design parameter.

The virtual controller αi, ϶ and adaptive law are designed as

(34) (35)

where Hi, ϶ 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 si, n is given as


According to (9), (10) and (37), we can know


where has a bounded range. There is a positive parameter ⅁i, n such that .

Then, the BLF is defined as


where |zi, n|< |Ei, n| with Ei, n being the constraint parameter of the compensating error zi, n, τi, n is a positive parameter. Combining (38), (39) and Lemma 1, one has


Substituting (4) into (40), it yields


vi(t) is constructed as


where δi and φi, n are positive parameters with . Because and are design parameters within the range of [ − 1, 1], we have

(43) (44)

Substituting (42)–(44) into (41), we can get


From Lemma 4, we have

(46) (47)

Then, combining (45)–(47), it yields


By using Lemma 3, we have

(49) (50) (51)

Substituting (49)–(51) into (48), one has


A nonlinear function is defined as


where ηi, n = [xi, 1, xi, 2, …, xi, n]T, By using Lemmas 2 and 3, there is a positive parameter such that , it yields


where ϵi, n represents a design parameter.

The virtual controller αi, n and adaptive law are designed as

(55) (56)

where Hi, n is a positive parameter.

According to (3), we can know 0 <  λi(0)< 1 and , βi is a positive parameter, so there has a positive parameter with . Combining (52)–(56) and (36), defining , 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.

thumbnail Figure 1.

The block diagram of the distributed formation control

4. Stability analysis

Theorem 1 For the MASs (1) with full state constraints, based on (5), (6), incorporating the DET mechanism (3), the virtual controllers (21), (34) and (55), the adaptive laws (22), (35) and (56), and the intermediate control signal (45) are constructed, then the DET formation controller guarantees that all signals in the MASs are semi-globally uniformly ultimately bounded (SGUUB) in probability.

Proof A Lyapunov function is constructed as


According to Lemmas 2 and 3, one has


where ϶  =  1, 2, …, n. Substituting (57) and (59) into (58), one has


According to the definition of Vi, ϶, yields , one has


Then, one has


where C1 = min{4ci, ϶, Hi, ϶} and . Finding the integral of (62), we can know


From (63), by designing appropriate initial values and parameters, all signals in MASs are SGUUB in probability.

Zeno behavior will occur if the event is triggered infinitely in a limited time. The following proof demonstrates the exclusion of Zeno behavior. The derivative of |ei(t)| is given as


where . According to (3) and (42), it yields


Based on (65), we know that represents a continuous function. There has a positive parameter ℚ with , and . Therefore, the minimum triggering interval tmin satisfies , which proves the exclusion of Zeno behavior. ▫

Remark 1. From [59], the command filter (8) is stable, and the command filter output signal ∂i, ϶ is bounded such that . It is worth noting that we have made modifications to the error compensation system (10) to make the algorithm simpler. The system remains stable, and the error compensation signal γi, ϶ is bounded such that , where and are positive constants, ϶  =  1, 2, …, n.

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 , where is a positive constant. Substituting (7) into (66), one has


where ϶  =  1, 2, …, n, .

From (67), we define functions and satisfy , , respectively. We can get and , which indicates that the full state constraints are not transgressed.

Remark 2. The design parameters adhere to several guidelines. The appropriate design parameters are selected such that C1 >  0. Therefore, ci, ϶ >  0 and Hi, ϶ >  0 are selected to ensure system stability. Then, another parameter Ei, ϶ satisfies |zi, ϶|< |Ei, ϶| to ensure that all states of the closed-loop system are constrained. It is worth noting that the larger the value of ci, ϶, 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.

5. 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.

thumbnail Figure 2.

Communication topology

From Figure 2, we know that the expansion adjacency matrix is and the adjacency matrix is given as

The follower dynamic model is given as

where , , i = 1, 2, 3, 4.

The control input u0(t) of leader (2) is designed as follows:

Table 1.

Triggering times in 50 seconds of different event-triggering control strategies

thumbnail Figure 3.

Trajectories of followers yi(i = 1, 2, 3, 4) and Leader y0

thumbnail Figure 4.

Curves of formation errors si, 1(i = 1, 2, 3, 4)

thumbnail Figure 5.

Curves of filtering errors ∂i, 1 − αi, 1(i = 1, 2, 3, 4)

The main design parameters are selected as βi = 0.03, μi = 0.01, Li, 1 = 75, Ri, 1 = 3, ci, 1 = 75, ci, 2 = 85, ki, 1 = 0.3, ki, 2 = 0.12, τi, 1 = τi, 2 = 2, Hi, 1 = Hi, 2 = 1.2, φi, 2 = 2, ϵi, 1 = ϵi, 2 = 1.5 and the constraint parameters of the compensating error are Ei, 1 = 0.8 and Ei, 2 = 0.5, the initial values of MASs are designed as x1, 1(0)=0.6, x2, 1(0)= − 0.8, x3, 1(0)= − 1, x4, 1(0)= − 1.2 and xi, 2(0)=0. According to the communication topology diagram. The formation distances are designed such that ℑ1, 0 = 0.6, ℑ1, 3 = 1.8, ℑ2, 0 = −0.6, ℑ2, 3 = 0.6, ℑ3, 2 = −0.6, ℑ4, 1 = 0.6. Figure 3 shows the formation trajectory of the MASs under state constraints. The formation error is displayed in Figure 4. Figure 5 displays the error curves of filter output signals and virtual control signals. Figure 6 displays the time intervals of triggering events with the followers i(i = 1, 2, 3, 4) from top to bottom. In Table 1, compared with two different event-triggered control methods, the results show that the DET control scheme obtains fewer trigger times, which proves that the proposed control method can reduce the consumption of communication resources more efficiently. The input signals of followers are depicted in Figure 7.

thumbnail Figure 6.

Interevent times of control signals

thumbnail Figure 7.

Curves of control inputs ui(i = 1, 2, 3, 4)

6. 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.

Conflict of Interest

The authors declare that they have no conflict of interest.

Data Availability

No data are associated with this article.

Authors’ Contributions

Yonghua Peng constructed and wrote the article, Guohuai Lin participated in writing and refined the language, and Guangdeng Chen and Hongyi Li sorted out and analyzed the research results and discussed the latest developments.


We thank all editors and reviewers who helped us improve this paper.


This work was supported in part by the National Natural Science Foundation of China (62121004, 62033003, 61973091, 62203119), the Local Innovative and Research Teams Project of Guangdong Special Support Program (2019BT02X353), the Natural Science Foundation of Guangdong Province (2023A1515011527, 2022A1515011506), and the China National Postdoctoral Program (BX20220095, 2022M710826).


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Yonghua Peng

Yonghua Peng received a B.S. degree in Information Engineering from Henan University of Science and Technology, Luoyang, China, in 2022. He is currently pursuing an M.S. degree in control science and engineering at Guangdong University of Technology, Guangzhou, China. His research interests include human-in-the-loop control, formation control, and event-triggered control for nonlinear multi-agent systems.

Guohuai Lin

Guohuai Lin received a B.S. degree in Electrical Engineering and Automation from the Guangdong University of Petrochemical Technology, Maoming, China, in 2018. He is currently pursuing his Ph.D. degree in Control Science and Engineering at Guangdong University of Technology, Guangzhou, China. His current research interests include adaptive control and human-in-the-loop control for multi-agent systems.

Guangdeng Chen

Guangdeng Chen received a B.S. degree from the Guangdong University of Petrochemical Technology, Maoming, China, in 2018, and a M.S. degree in control science and engineering from the Guangdong University of Technology, Guangzhou, China, in 2021, where he is currently pursuing the Ph.D. degree in control science and engineering. His research interests include event-triggered control, nonlinear systems, and cyber-physical systems.

Hongyi Li

Hongyi Li (Senior Member, IEEE) received a Ph.D. degree in intelligent control from the University of Portsmouth, Portsmouth, U.K., in 2012. He is currently a professor at the Guangdong University of Technology, Guangdong, China. His research interests include intelligent control, cooperative control, sliding mode control, and their applications.

All Tables

Table 1.

Triggering times in 50 seconds of different event-triggering control strategies

All Figures

thumbnail Figure 1.

The block diagram of the distributed formation control

In the text
thumbnail Figure 2.

Communication topology

In the text
thumbnail Figure 3.

Trajectories of followers yi(i = 1, 2, 3, 4) and Leader y0

In the text
thumbnail Figure 4.

Curves of formation errors si, 1(i = 1, 2, 3, 4)

In the text
thumbnail Figure 5.

Curves of filtering errors ∂i, 1 − αi, 1(i = 1, 2, 3, 4)

In the text
thumbnail Figure 6.

Interevent times of control signals

In the text
thumbnail Figure 7.

Curves of control inputs ui(i = 1, 2, 3, 4)

In the text

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