Computer-implemented method for controlling a multi-agent system with N autonomous agents, N ≧ 2

Abstract

Computer-implemented method for controlling a multi-agent system (10) with N autonomous agents (12, 12a, 12b, 12c, 12d), N ≥ 2, for n-dimensional navigation of the latter, wherein each agent (12, 12a, 12b, 12c, 12d) comprises an active actuator device (14) and at least one inactive actuator device (22) each for n-dimensional navigation of the agent (12, 12a, 12b, 12c, 12d), wherein the at least one inactive actuator device (22) is configured redundantly to the active actuator device (14), wherein each agent (12, 12a, 12b, 12c, 12d) comprises a control switching device (24) for switching a control (26) from an actuator device (14, 22) to another actuator device (14, 22), wherein the multi-agent system (10) is configured in a leader-follower formation (16) in which each agent (12, 12a, 12b, 12c, 12d) is to perform a respective desired movement using the active actuator device (14),wherein each agent (12, 12a, 12b, 12c, 12d) has a sensor device for tracking a respective actual motion, wherein the desired motion and the actual motion of the i-th agent (12), i = 1, ...,N, are defined by a motion-related quantity x, i , x i ∈ ℜ n be described.

Description

TECHNICAL FIELD

[0001] The invention relates to a computer-implemented method for controlling a multi-agent system with N autonomous agents, N ≥ 2. The invention further relates to a control device, an autonomous agent, a multi-agent system, a computer program and a computer-readable storage medium.

[0002] Robustness and reliability are very important for multi-agent systems, such as a team of robots or a team of autonomous vehicles. Formation tracking control of multi-agent systems has attracted considerable attention in various industrial fields, as it enables agents to follow desired trajectories while maintaining their autonomy. During the control of multi-agent systems, actuator failure, which directly affects the safety and reliability of the system, is a crucial issue that must be addressed. In this context, some researchers have attempted to develop fault-tolerant control concepts to account for various types of actuator failures in multi-agent systems, such as partial loss of effectiveness, stuckness, bias, etc.However, currently existing methods are based on the assumption that the faulty actuator retains some control capability and are therefore not applicable to actuator failure, where the actuator fails without any output signal and completely loses its control capability. Given this, an effective control solution needs to be developed to deal with actuator defects, especially potential actuator failures, using actuator and controller redundancy.

[0003] The following documents are hereby cited and incorporated by reference: [1] Z. Zuo, J. Zhang, and Y. Wang, “Adaptive fault-tolerant tracking control for linear and Lipschitz nonlinear multi-agent systems,” IEEE Transactions on Industrial Electronics, vol. 62, no. 6, pp. 3923-3931, 2014; [2] P. Gong, W. Lan and Q. Han, „Robust adaptive fault-tolerant consensus control for uncertain nonlinear fractional-order multi-agent systems with directed topologies,“ Automatica, vol. 117, p. 109011, 2020; [3] G. Cui, S. Xu, Q. Ma, Z. Li, and Y. Chu, „Command-filter-based distributed containment control of nonlinear multi-agent systems with actuator failures,“ International Journal of Control, vol. 91, no. 7, pp. 1708-1719, 2018; [4] G.-H. Yang, S.-Y. Zhang, J. Lam, and J. Wang, „Reliable control using redundant controllers,“ IEEE Transactions on Automatic Control, vol. 43, no. 11, pp. 1588-1593, 1998; [5] C. P. Bechlioulis and G. A. Rovithakis, „Robust Adaptive Control of Feedback Linearizable MIMO Nonlinear Systems With Prescribed Performance,“ in IEEE Transactions on Automatic Control, vol. 53, no. 9, pp. 2090-2099, Oct. 2008, doi: 10.1109 / TAC.2008.929402; und [6] J. Liu, “Radial Basis Function (RBF) neural network control for mechanical systems: design, analysis and Matlab simulation,” Springer Science & Business Media, 2013. SUMMARY OF THE INVENTION

[0004] The object of the invention is to provide a method for controlling a multi-agent system, wherein the method takes into account possible failures of actuator devices in the multi-agent system.

[0005] The object of the invention is achieved by the subject matter of the independent claims. Advantageous embodiments of the invention are the subject matter of the dependent claims.

[0006] According to one aspect, the invention provides a computer-implemented method for controlling a multi-agent system with N autonomous agents, N ≥ 2, for n-dimensional navigation of the latter, wherein each agent comprises an active actuator device and at least one inactive actuator device, each configured for n-dimensional navigation of the agent, wherein the at least one inactive actuator device is configured to be redundant with the active actuator device, wherein each agent has a control switching device for switching control from one actuator device to another actuator device, wherein the multi-agent system is configured in a leader-follower formation in which each agent is to execute a respective desired movement using the active actuator device, wherein each agent has a sensor device for tracking a respective actual movement,where the desired movement and the actual movement of the i-th agent, i = 1, ...,N are determined by a movement-related quantity x, i , x i ∈ ℜ n described, where the procedure for each agent includes: a) Controlling the active actuator device for navigating the agent by limiting a corresponding tracking error s i,1 , i = 1, ...,N, between the motion-related quantity x i the actual movement and the movement-related quantity x i the desired movement within a specified range, as long as no failure of the active actuator device is detected; and b) if a failure of the active actuator device is detected, ignoring the active actuator device for control and causing the control switching device to switch control of the ignored actuator device to one of the at least one inactive actuator device, and using the latter as the active actuator device for control to navigate by detecting the respective tracking error s i,1 , i = 1, ..., N remains bounded according to step a).

[0007] In multi-agent systems, the agents are typically dynamically decoupled from each other, implying the need for cooperation to achieve collective behavior. The goal of distributed formation control is for multiple agents to work together efficiently to achieve collective group behavior through local interaction. Such multi-agent systems can be used in warehouse automation, dial-a-ride scenarios, parcel delivery, mothership applications, and so on. Actuator defects in multi-agent systems can severely impact system performance and even lead to catastrophic instability of the closed-loop system. Common types of actuator defects in multi-agent systems include partial loss of effectiveness or failure (complete loss of effectiveness).

[0008] In embodiments of the invention, initial conditions are taken into account in the event of actuator failures. The method can enable a control activator based on a fault detection system to handle potential actuator failures. Possible actuator failures can be handled, and the controller's performance can be guaranteed by limiting tracking errors within specified ranges. Since only one functioning actuator operates during control and the redundant channel is only activated when needed, energy consumption can be reduced. In other words, the redundant channels can be inactive or switched off as long as no failure is detected and are only switched on when needed.

[0009] Preferably, the movement-related quantity x i a position, a speed and / or an acceleration of the respective agent.

[0010] Preferably, the tracking error s i,1 of the i-th agent, i = 1, ..., N given by: si,1=ai,0(yi−yh,i−y0)+∑μ∈Niai,μ((yi−yh,i)−(yμ−yh,μ)), where y i = x i,1 , x i,m an m-th state or coordinate of the motion-related quantity x i ∈ ℜ n of the i-th agent, where y0 is a predefined global task reference of the multi-agent system, where y h,i ∈ ℜ is a predefined local reference of the i-th agent, where A^=[aij]∈ℜN×N an adjacency matrix is ​​denoted by a ii = 0, a ij = 1 if there is a communication connection between the i-th agent and the j-th agent, and otherwise a ij = 0, and where Ni is a neighborhood set of the i-agent, and wherein step a) further comprises one or both of the following: a1) Limiting the tracking error s i,1of the i-th agent, i = 1, ...,N, such that: −γ_iϕi <si,1<γ¯iϕi, wenn si,1(t0)> 0, and −γ¯iϕi <si,1<γi_ϕi,wenn si,1(t0)<0, for all t ∈ [t0, ∞), where s i,1 (t0) is the tracking error of the i-th agent, i = 1, ..., N , at a start time t0, 0 < γ i γ i ≤ 1 are predefined parameters, and ϕ i is a positive predetermined power function that decreases with time t and satisfies the condition limt→∞ϕ=ϕ∞>0 fulfilled; and / or a2) Limiting the tracking error s i,1 of the i-th agent, i = 1, ...,N, such that: −κα,i <si,1<κβ,i, for all t ∈ [t0, ∞), where κ α,i and κ β,i are predefined time-varying constraint functions, each of which κ α,i , κ β,i > 0 and the by κ α,i ≤ ĸ i and κ β,i ≤ κ i are limited, where ĸ i and κ itwo limit constants, and where the k-th time derivatives, κα,i(k) and κβ,i(k) k = 1, ..., n + 1 are each limited.

[0011] An advantage of the method may be that two options for initial conditions are available and the corresponding control performance in the event of possible actuator failures can be achieved by the preferred control strategies, so that the task performance can be specified as needed.

[0012] Preferably, step a) further comprises: a3) Controlling the active actuator device of the i-th agent, i = 1, ..., N according to: u^i=τi,n−Disgn(si,n)−θ^iTfi(xi), with adaptive laws: θ^˙=fiT(xi)si,n, where: ϕi=(ϕ0,i−ϕ∞,i)e−cit+ϕ∞,i, fu¨rt∈[0,∞), τi,1=1εi(∑μ∈Niai,μ(y˙h,i−y˙h,μ)+ϕ˙isi,1ϕi+∑μ∈Niai,μxμ,2+ai,0(y˙0+y˙h,i)− ciϕi(∂Γi−1(si,1ϕi)∂(si,1ϕi))−1Γi−1(si,1ϕi)), τi,2=−vi,2si,2+τ˙i,1−εiϕi∂Γi−1(si,1ϕi)∂(si,1ϕi)Γi−1(si,1ϕi), τi,j=−vi,jsi,j+τ˙i,j−1−si,j−1,for rj=3,...,n, εi=∑μ∈Niai,μ+ai,0, si,j=xi,j−τi,j−1,for rj=2,...,n, and the multi-agent system 10 is described as: {x˙i,m=xi,m+1,for m=1,...,n−1,x˙i,n=ui+wi(xi,t),for i=1,...,N,yi=xi,1, where wi(xi,t)=di+θiTfi(xi)∈ℜ a possible and / or permissible constant external disturbance d i with an upper limit D i ≥ |d i | and an unknown uncertainty f i (x i ) with a constant vector θ i , each related to the movement of the i-th agent, where ϕ 0,i , ϕ ∞,i , c i are predefined positive parameters for the maximum overshoot, the maximum stationary error and the lower limit of the convergence speed, where ϕ 0,i> ϕ ∞,i , with v i,j and D i are predefined positive parameters, and u i ∈ ℜ denotes a control signal for the active actuator device of the i-th agent, where Γ i (Δ) of the i-th agent depending on any time-related quantity Δ has the following properties 1) to 3): 1) smooth and steadily rising; 2) −γ_<Γ(Δ)<γ¯,if Δ(t0)>0, and -γ¯<Γ(Δ)<γ_, if Δ(t0)<0; 3) limΔ→−∞Γ(Δ)=−γ_ and limΔ→−∞Γ(Δ)=γ¯, if Δ(t0)>0, and limΔ→−∞Γ(Δ)=−γ¯ andlimΔ→−∞Γ(Δ)=γ_,if Δ(t0)<0; and / or a4) Controlling the active actuator device of the i-th agent, i = 1, ..., N according to: u˜i=αi,n−D˜i sgn(zi,n)−v^i,n(xi), with adaptive laws: v^i(xi)=W^iTh⌣i(xi)and W^˙i=σizi,nh⌣i(xi)−σiσi'W^i, where: αi,1=1εi(∑μ∈Niai,μ(y˙h,i−y˙h,μ)+∑μ∈Niai,μxμ,2+ai,0(y˙0+y˙h,i)−(v˜i,1+ (κ˙α,iκα,i)2+(κ˙β,iκβ,i)2zi,1), αi,2=−v˜i,2zi,2+α˙i,1−εiςizi,12pi−1, αi,j=−v˜i,jzi,j+α˙i,j−1−zi,j−1,for rj=3,...n, εi=∑μ∈Niai,μ+ai,0 zi,1=si,1, zi,j=xi,j−αi,j−1,for rj=2,…,n, and the multi-agent system is described as: {x˙i,m=xi,m+1,for m=1,…,n−1x˙i,n=vi(xi)+ui+di,for i=1,…,N,yi=xi,1 where d i a possible and / or permissible time-varying external disturbance with an upper limit D i ≥ |d i | and vi(xi)=Wi∗Th⌣i(xi)+g⌣i∈ℜ are an unknown uncertainty, each related to the movement of the i-th agent, where Wi∗ is an ideal weight vector that satisfies the condition Wi∗TWi≤w¯i with w i > 0, ǧ i fulfilled, where |ǧ i | ≤ ǧ i with q i > 0, where ȟi (x i ) = [ȟ i,1 (x i ), ...,ȟ i,qi (x i )] T ∈ ℜ qi is a predefined vector of basis functions, q i is a predefined parameter, where ṽ i,j , σ i , σ′i are predefined positive parameters, where D̃ i > D i + q i is a predefined parameter, where p i a predefined parameter with 2p i ≥ n, and where u i ∈ ℜ denotes a control signal for the active actuator device of the i-th agent, and ςi=η(zi,1)Kβ,i2pi−zi,12pi−1−η(zi,1)Kα,i2pi−zi,12pi, with η( * ) = 1, if * > 0, otherwise η = 0.

[0013] An advantage of the method may be that external disturbances and system uncertainties are taken into account in order to further improve the robustness and reliability of the multi-agent system.

[0014] Preferably, the i-th agent, i = 1, ...,N, l i redundant actuator devices, l i ≥ 1, where in step b) the failure of the (r + 1)-th active actuator device in the i-th agent, r = 0, ..., l i - 1, at a time t i,r+1 which is defined as follows: b1) ti,r+1=min{t|Si,1=−γ_i,rϕi,Si,1=γ¯i,rϕi,i=1,…,N}, where: γ_i,r=−Γi(−ψi,r), γ¯i,r=Γi(ψi,r), ψi,r=(εi2(ti,r)+∑j=2nSi,j2(ti,r)+σ¯i+θ¯iT(ti,r)θ¯i(ti,r))−1 / 2, θ¯i(ti,r)=|θ^i(ti,r)|+θ′i, εi=Γi−1(si,1ϕi), where σ i and θ′i≥|θi| are predefined positive parameters, and t i,0 = t0 applies; and / or b2) ti,r+1=min{t||ϑi|=1−e−qi,r22pi),i=1,…,N}, where: qi,r=(ln11−ϑi2pi(ti,r)+pi∑j=2nzi,j2(ti,r)+2piD¯iΛi+2piσiW^iT(ti,r)W^i(ti,r)+Di'+χi)1 / 2, Λi=2min{piv˜i,1,v˜i,j,σiσi'}, ϑi=η(zi,1)ϑβ,i+(1−η(zi,1))ϑα,i, ϑα,i=zi,1κα,iand ϑβ,i=zi,1κβ,i where Di'=2piw¯iσi is a predefined parameter and D¯i=12σi'w¯i applies and χ i is a predefined positive parameter, and t i,0 = t0 applies.

[0015] One advantage of the method is that it allows for any number of redundant channels. This allows the number of redundant channels to be chosen according to practical needs, for example, with regard to the security requirements and the execution environment of the task.

[0016] According to a further aspect, the invention provides a control device comprising means for carrying out the method according to one of the preceding embodiments.

[0017] According to a further aspect, the invention provides an autonomous agent for n-dimensional navigation, wherein the agent has an active actuator device and at least one inactive actuator device, each for n-dimensional navigation of the agent, wherein the at least one inactive actuator device is designed redundantly to the active actuator device, wherein the agent has a control switching device for switching control from one actuator device to another actuator device, wherein the agent is suitable for forming a multi-agent system in a leader-follower formation in which the agent is to execute a desired movement using the active actuator device, wherein the agent has a sensor device for tracking an actual movement of the agent, wherein the desired movement and the actual movement of the agent are determined by a movement-related quantity x i , x i ∈ ℜn described, whereby the agent is characterized by the control device.

[0018] According to a further aspect, the invention provides a multi-agent system with N autonomous agents, N ≥ 2.

[0019] According to a further aspect, the invention provides a computer program containing instructions which, when the program is executed by the control device, cause the control device to carry out the method according to one of the preceding embodiments.

[0020] According to a further aspect, the invention provides a computer-readable storage medium on which the computer program is stored.

[0021] Embodiments of the invention preferably have the following advantages and effects: Preferred embodiments of the invention propose a novel fault-tolerant formation tracking control method for multi-agent systems for handling actuator defects, which ensures the robustness and reliability of the system. Unlike existing methods, the proposed control method can advantageously handle possible actuator failures and guarantee the control performance by limiting tracking errors within predetermined ranges. Since only one functioning actuator operates during control, energy consumption can be reduced.

[0022] In order to expand the scope of the preferred embodiment of the invention, two possibilities of predetermined functions are provided, as shown in (Eq. 1) and (Eq. 4), and the corresponding control performance can be preferably achieved in the event of possible actuator failures by the developed control strategies, so that the task performance can be specified as needed.

[0023] In a preferred embodiment of the invention, external disturbances and system uncertainty are taken into account to further increase the robustness and reliability of the control system. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Embodiments of the invention will now be explained in more detail with reference to the accompanying drawings, in which Fig. 1 shows a first embodiment of a multi-agent system in n = 2 dimensions; Fig. 2 shows a second embodiment of the multi-agent system; Fig. 3 shows a first embodiment of a computer-implemented method for controlling the multi-agent system; Fig. 4 shows a simulation for the multi-agent system according to Fig. 1, in which the procedure according to Fig. 3 is implemented; Fig. Figure 5 shows a third embodiment of the multi-agent system in n = 2 dimensions; Fig. 6 shows a second embodiment of the computer-implemented method for controlling the multi-agent system; and Fig. 7 shows a simulation for the multi-agent system from Fig. 5, in which the procedure according to Fig. 6 is implemented. DETAILED DESCRIPTION OF THE DRAWINGS

[0025] Fig. 1 shows a first embodiment of a multi-agent system 10 in n = 2 dimensions.

[0026] The multi-agent system 10 comprises N = 4 autonomous agents 12, a first agent 12a, a second agent 12b, a third agent 12c, and a fourth agent 12d. In this case, the autonomous agents 12, 12a, 12b, 12c, 12d can each be configured as a robot, vehicle, etc. However, within the scope of the invention, any number N ≥ 2 of agents 12 and any dimension n, for example, n = 1 or n = 3, are possible. In this context, the autonomous agents 12 can each be configured as a train, airplane, drone, etc.

[0027] In Fig. 1, each agent 12 has an active actuator device 14 with which the agent 12 can be navigated two-dimensionally.

[0028] The multi-agent system 10 is configured in a leader-follower formation 16, in which each agent 12, 12a, 12b, 12c, 12d is to execute a respective desired movement using the active actuator device 14, wherein the desired movement of the i-th agent 12, i = 1, ..., N is determined by a movement-related quantity x i , x i ∈ ℜ 2 described.

[0029] In this case, the movement is a translational movement, but within the scope of the invention, it can also be a rotational movement. The movement-related variable in this case is the position of the respective agent 12. Within the scope of the invention, however, it can also be a speed and / or acceleration of the respective agent 12.

[0030] The desired movement is based on the movement of a leader agent 18. In this case, the leader agent 18 is virtual. However, within the scope of the invention, the leader agent 18 can also be any of the N autonomous agents 12 of the multi-agent system 10.

[0031] In the leader-follower formation 16, the agents 12, 12a, 12b, 12c, 12d communicate with each other. Fig. In the example shown in Figure 1, the second agent 12b communicates with the first agent 12a and the third agent 12c. The third agent 12c communicates with the first agent 12a and the second agent 12b. The fourth agent 12d communicates with the first agent 12a. The virtual leader 18 communicates with the third agent 12c and the fourth agent 12d. In other words, the agents 12, 12a, 12b, 12c, 12d and the leader agent 18 form a communication topology 20 that prescribes a collective movement of the multi-agent system 10 with respect to the leader agent 18, corresponding to the respective desired movement of the agents 12, 12a, 12b, 12c, 12d. For two-dimensional navigation according to the respective desired movement, each agent 12, 12a, 12b, 12c, 12d uses its active actuator device 14.

[0032] One idea is to solve the following problem: A defect and / or failure of the active actuator device 14 of a particular agent 12, 12a, 12b, 12c, 12d may cause this agent 12, 12a, 12b, 12c, 12d to deviate from the respective desired movement.

[0033] However, due to the leader-follower formation 16, the deviation of a given agent 12, 12a, 12b, 12c, 12d from the respective desired movement can also affect the movement of all other agents 12, 12a, 12b, 12c, 12d in the multi-agent system 10, even though all other agents 12, 12a, 12b, 12c, 12d may have an intact active actuator device 14. In other words, a failure of the active actuator device 14 of a single agent 12, 12a, 12b, 12c, 12d can lead to incorrect behavior of the entire multi-agent system 10.

[0034] Fig. 2 shows a second embodiment of the multi-agent system 10.

[0035] To solve the above problem, each agent 12 initially comprises at least one inactive actuator device 22, wherein the at least one inactive actuator device 22 is configured to be redundant with the active actuator device 14.

[0036] Secondly, each agent 12 has a control switching device 24 for switching a control 26 from one actuator device 14, 22 to another actuator device 14, 22. In Fig. 2, each actuator device 14 includes a corresponding controller 26 that is separate from the control switching device 24. However, within the scope of the invention, the corresponding controllers 26 may also be included in the control switching device 24.

[0037] Thirdly, each agent 12 has a sensor device for tracking a respective actual movement, where the actual movement of the i-th agent 12, i = 1, ..., N, is determined by the movement-related quantity xi , x i ∈ ℜ n , is described.

[0038] Fig. 3 shows a first embodiment of a computer-implemented method for controlling the multi-agent system 10 with the four autonomous agents 12, 12a, 12b, 12c, 12d.

[0039] In a step S11, the method comprises: - for each agent 12, 12a, 12b, 12c, 12d, controlling the active actuator device 14 to navigate the agent 12, 12a, 12b, 12c, 12d by limiting a respective tracking error s i,1 , i = 1, ...,N, between the motion-related quantity x i the actual movement and the movement-related quantity x i the desired movement within a prescribed range, as long as no failure of the active actuator device is detected.

[0040] In a step S12, the method comprises: - for each agent 12, 12a, 12b, 12c, 12d, if a failure of the active actuator device 14 is detected, ignoring the active actuator device 14 for control and causing the control switching device 24 to switch the control 26 from the ignored actuator device 14 to one of the at least one inactive actuator devices 22, and using the latter as the active actuator device 14 for control to navigate by detecting the respective tracking error s i,1 , i = 1, ..., N remains limited according to step S11.

[0041] According to the procedure, the respective tracking error remains s i,1 , i = 1, ..., N, between the motion-related quantity x i the actual movement and the movement-related quantity x ithe desired movement within the specified range, even if a failure of the active actuator device 14 is detected. In other words, limiting the respective tracking error s i,1 , i = 1, ..., N is not interrupted.

[0042] If a failure of the active actuator device 14 is detected, the active actuator device 14 is ignored for control and the control switching device 24 is caused to switch the control 26 from the ignored actuator device 14 to one of the at least one inactive actuator devices 22 and to use this as the active actuator device 14 for control for navigation. According to the method, the ignoring, switching, and use of the at least one inactive actuator device 22 as the active actuator device 14 is carried out such that the respective tracking error s i,1, i = 1, ..., N remains within the specified range at all times. Thus, a failure of the active actuator device 14 of a single agent 12, 12a, 12b, 12c, 12d can prevent malfunction of the entire multi-agent system 10.

[0043] The tracking error s i,1 of the i-th agent 12, i = 1, ..., N, can be given as follows: si,1=ai,0(yi−yh,i−y0)+∑μ∈Niai,μ((yi−yh,i)−(yμ−yh,μ)), where y i = x i,1 , x i,m an m-th state or a coordinate of the motion-related quantity x i ∈ ℜ n of the i-th agent 12, y0 is a predefined global task reference of the multi-agent system 10, y h,i ∈ ℜ is a predefined local reference of the i-th agent 12, A^=[aij]∈ℜN×N an adjacency matrix is ​​denoted by a ii = 0, a ij= 1, if there is a communication connection between the i-th agent 12 and the j-th agent 12, and otherwise a ij = 0, and Ni is a neighborhood set of the i-th agent 12. y0 can represent the movement of the virtual leader 18. The adjacency matrix can be calculated according to the communication topology 20 of Fig. 1 can be defined.

[0044] In a step S13, the method comprises the following: - Limiting the tracking error s i,1 of the i-th agent, i = 1, ...,N, such that: −γ_iϕi <si,1<γ¯iϕi, wenn si,1(t0)> 0, and −γ¯iϕi <si,1<γ_iϕi, wennsi,1(t0)<0, for all t ∈ [t0, ∞), where s i,1 (t0) is the tracking error of the i-th agent 12, i = 1, ...,N, at a start time t0, 0 < γ i , γ i ≤ 1 are predefined parameters, and ϕ i of the i-th agent 12 is a positive predetermined performance function that decreases with time t and limt→∞ϕ=ϕ∞>0 fulfilled.

[0045] According to a preferred embodiment of the invention, any positive function ϕ , which limt→∞ϕ=ϕ∞>0 fulfilled, can be chosen. ϕ can also be defined differently for different agents 12. In the present case, the positive prescribed performance function ϕ is defined as: ϕ=(ϕ0−ϕ∞)e−ct+ϕ∞,fu¨rt∈[0,∞), where ϕ0, ϕ ∞ , c are predefined positive parameters for the maximum overshoot, the maximum stationary error and the lower limit of the convergence speed, where ϕ0 > ϕ ∞ applies.

[0046] In a step S14, the method comprises: - Controlling the active actuator device 14 of the i-th agent 12, i = 1, ..., N according to: u^i=τi,n−Disgn(si,n)−θ^iTfi(xi) with adaptive laws: θ^˙i=fiT(xi)si,n, where: τi,1=1ϵi(∑μ∈Niai,μ(y˙h,i−y˙h,μ)+ϕ˙isi,1ϕi+∑μ∈Niai,μxμ,2+ai ,0(y˙0+y˙h,i)−ciϕi(∂Γi−1(si,1ϕi)∂(si,1ϕi))−1Γi−1(si,1ϕi)), τi,2=−vi,2si,2+τ˙i,1−ϵiϕi∂Γi−1(si,1ϕi)∂(si,1ϕi)Γi−1(si,1ϕi), τi,j=−vi,jsi,j+τ˙i,j−1−si,j−1, for rj=3, …, n, ϵi=∑μ∈Niai,μ+ai,0, si,j=xi,j−τi,j−1, for rj=2, …, n, and the multi-agent system 10 as: {x˙i,m=xi,m+1, for m=1, …, n−1,x˙i,n=ui+wi(xi, t), for i=1, …, N,yi=xi,1, is described, where wi(xi, t)=di+θiTfi(xi)∈ℜ a possible and / or permissible constant external disturbance d i with an upper limit D i ≥ |d i | and an unknown uncertainty f i (x i ) with a constant vector θ i which are each related to the movement of the i-th agent 12, where v i,j and D i predefined positive parameters and u i∈ ℜ denotes a control signal for the active actuator device 14 of the i-th agent 12, where Γ i (Δ) of the i-th agent 12 as a function of any time-related quantity Δ has the following properties 1) to 3): 1) smooth and steadily rising; 2) −γ_<Γ(Δ)<γ¯, if Δ(t0)>0, and −γ¯<Γ(Δ)<γ_ if Δ(t0)<0; 3) limΔ→−∞Γ(Δ)=−γ_ and limΔ→∞Γ(Δ)=γ¯ if Δ(t0)<0, and limΔ→−∞Γ(Δ)=−γ¯ and limΔ→∞Γ(Δ)=γ_ if Δ(t0)<0; and / or

[0047] According to a preferred embodiment of the invention, any function Γ that satisfies properties 1), 2), and 3) can be chosen. Γ can also be defined differently for different agents 12, 12a, 12b, 12c, 12d. An example of Γ is given in reference [5]: Γ(Δ)={γ¯eΔ−γ_e−ΔeΔ+e−Δ, if Δ(t0)≥0,γ_eΔ−γ¯e−ΔeΔ+e−Δ, if Δ(t0)<0, where γ, γ are predefined parameters:

[0048] Suppose the i-th agent 12, i = 1, ..., N, comprises l i redundant actuator devices, l i ≥ 1. In a step S15, the method comprises: - detecting the failure of the (r + 1)-th active actuator device 14 in the i-th agent 12, r = 0, ..., l i - 1, at a time t i,r+1 , which is defined as follows: ti,r+1=min{t|si,1=−γ_i,rϕi, si,1=γ¯i,rϕi, i=1, …, N}, where: γ_i,r=−Γi(−ψi,r). γ¯i,r=Γi(ψi,r), ψi,r=(εi2(ti,r)+∑j=2nsi,j2(ti,r)+σ¯i+θ¯iT(ti,r))−1 / 2, θ¯i(ti,r)=|θ^i(ti,r)|+θ′i, εi=Γi−1(si,1ϕi), where σ i and θ′i≥|θi| are predefined positive parameters, and t i,0 = t0 applies.

[0049] Fig. 4 shows a simulation for the multi-agent system 10 according to Fig. 1, in which the procedure according to Fig. 3 is implemented.

[0050] The multi-agent system 10 is described as follows: {x˙i=vi,for i=1,…,4,v˙i=1m(ui−wi),for i=1,…,4, where x i ∈ ℜ 2 and v i ∈ ℜ 2 are the position and velocity of the i-agent 12, respectively, and m is the mass of each agent 12, 12a, 12b, 12c, 12d, which is defined as m = 1 kg.

[0051] For the purposes of the simulation, w i , which represents the possible and / or permissible constant external disturbance d i with the upper limit D i ≥ |d i | and the unknown uncertainty f i (x i ) with the constant vector θ i , each related to the movement of the i-th agent 12, are taken into account, as follows: wi(xi,t)=μ˜(tanh(a˜xi)−tanh(b˜xi)),fu¨ri=1,…,4, where µ̃ = 0.01, ã = 1, and b̃ = 10.

[0052] For each agent 12, 12a, 12b, 12c, 12d, the positive prescribed performance function ϕ is defined according to (Eq. 2), where ϕ 0,i = [6, 10] T , ϕ ∞,i = [0.4, 0.3] T , and c i = [0.2, 0.2] T , for i = 1, ...,4. For each agent 12, 12a, 12b, 12c, 12d, the function Γ according to (Eq. 3) is with γ i = γ i = 1, for i = 1, ...,4.

[0053] The remaining parameters v i,j , D i , and σ i, i = 1, ...,4, are defined as follows: vi,j=

[20005] ,fu¨rj=1,2,Di=0.01,θ′j=0.01 and σ¯i=0.01.

[0054] Fig. 4 shows an x-dimension of the tracking error s i,1 (thin line) based on given initial positions of the agents 12, 12a, 12b, 12c, 12d, the given predefined global task reference y0 of the multi-agent system 10 and the given predefined local reference y h,i∈ ℜ of the i-th agent 12, i = 1, ...,4. The defect of the active actuator device 14 is designed such that the first agent 12a and its active actuator device 14 fail at t = 6 s. In other words, the output signal of the active actuator device 14, which controls the position along the x-axis, drops to zero. The thick line in Fig. 4 shows the restriction according to (Eq. 1).

[0055] The inventors realized that the tracking error s i,1 continuously increases and approaches the constraint condition in the event of a defect in the active actuator device 14. With the aid of preferred embodiments of the invention, it can i,1Despite the defect of the active actuator device 14, the performance can be maintained within the specified performance as expected. The failure of the active actuator device 14 is successfully detected by the proposed method, and the activation of the redundant channel, i.e., the inactive actuator device 22, is triggered at t = 12.69 s.

[0056] Fig. 5 shows a second embodiment of the multi-agent system 10 in n = 2 dimensions.

[0057] The second embodiment of the multi-agent system 10 includes the features of the first embodiment of the multi-agent system 10 as described in connection with Fig. 1. However, the second embodiment of the multi-agent system 10 differs from the first embodiment of the multi-agent system 10 with regard to the communication topology 20.

[0058] As in Fig. As shown in Figure 5, in the leader-follower formation 16, the first agent 12a communicates with the third agent 12c and the fourth agent 12d. The second agent 12b communicates with the first agent 12a and the third agent 12c. The third agent 12c communicates with the second agent 12b. The fourth agent 12d does not communicate with any of the other agents 12a, 12b, or 12c. The virtual leader 18 communicates with the third agent 12c and the fourth agent 12d.

[0059] Fig. 6 shows a second embodiment of the computer-implemented method for controlling the multi-agent system 10 with the four autonomous agents 12, 12a, 12b, 12c, 12d.

[0060] The second embodiment includes step S11 as step S21 and step S12 as step S22.

[0061] In a step S23, the method comprises: - Limiting the tracking error s i,1 of the i-th agent 12, i = 1, ...,N, such that: −Kα,i <Si,1<Kβ,i, for all t ∈ [t0, ∞), where κ α,i and κ β,i are predefined time-varying constraint functions, each of which κ α,i , κ β,i > 0 and by κ α,i ≤ ĸ i and κ β,i ≤ κ i are limited, where ĸ i and κ i two limit constants, and where the k-th time derivatives, Kα.i(k) and Kβ,i(k) κ = 1, ..., n + 1, are each limited.

[0062] Compared to the function ϕ according to the first embodiment of the method, the limitations κ α,i and κ β,i According to the second embodiment, they are more flexible and can be freely designed according to different needs.

[0063] In a step S24, the method comprises: - Controlling the active actuator device 14 of the i-th agent 12, i = 1, ..., N according to: u˜i=αi,n−D˜isgn(zi,n)−v^i,n(xi), with adaptive laws: v^i(xi)=W^iTh⌣i(xi)and W^i=σizi,nh⌣i(xi)−σiσ′iW^i, where: αi,1=1εi(∑μ∈Niαi,μ(y˙h,i−y˙h,μ)+∑μ∈Niαi,μxμ,2+αi,0(y˙0+y˙h,i)−(v˜i,1+(K˙α,iKα,i)2+(K˙β,iKβ,i)2)Zi,1), αi,2=−v˜i,2zi,2+α˙i,1−∈iςizi,12pi−1, αi,j=−v˜i,jzi,j+α˙i,j−1−zi,j−1, for rj=3, …, n, ϵi=∑μ∈Niai,μ+ai,0, zi,1=si,1, zi,j=xi,j−αi,j−1, for rj=2, …, n, and the multi-agent system 10 as {x˙i,m=xi,m+1, for m=1, …, n−1x˙i,n=vi(xi)+ui+di, for i=1, …, Nyi=xi,1 is described, where d i is a possible and / or permissible time-varying external disturbance with an upper limit, D i ≥ |d i |, and vi(xi)=Wi*Th⌣i(xi)+g⌣i∈ℜ is an unknown uncertainty, each related to the movement of the i-th agent 12, where Wi* W i* is an ideal weight vector that satisfies the condition Wi*TWi≤w¯i with w i > 0, ǧ i |ǧ i | ≤ q̌ i with q i > 0, where ȟ i (x i ) = [h i,1 (x i ), ..., ȟ i,qi (x i )] T ∈ ℜ qi is a predefined vector of basis functions, q i is a predefined parameter, ṽ i,j , σ i , σi' are predefined positive parameters, D̃ i > D i + q i is a predefined parameter, p i a predefined parameter with 2p i ≥ n, and u i ∈ ℜ denotes the control signal for the active actuator device 14 of the i-th agent 12, and ςi=η(zi,1)κβ,i2pi−1−η(zi,1)κα,i2pi−zi,12pi, where η( * ) = 1, if * > 0, otherwise η = 0.

[0064] In step S24, vi (x i ) is approximated using a neural network with radial basis functions according to [6]. The basis functions ȟ i,m̃i (x i ), m i ; = 1, ..., q i can be chosen as Gaussian functions, for example: h⌣i,m˜i(xi)=exp(‖xi−h⌣i,m˜i‖22c⌣i,m˜i2), where ℏ̌ i,m̃i ∈ ℜ n and č i,m̃i ∈ ℜ are predefined parameters.

[0065] Suppose that the i-th agent 12, i = 1, N, again comprises l i redundant actuator devices l i ≥ 1. In a step S25, the method comprises: - Detecting the failure of the (r + 1)-th active actuator device 14 in the i-th agent 12, r = 0, ...,l i - 1, at a time t i,r+1 , which is defined as follows: ti,r+1=min{t||ϑi|=2pi1−e−qi,r2), i=1, …, N}, where: qi,r=(ln11−ϑi2pi(ti,r)+pi∑j=2nzi,j2(ti,r)+2piD¯iΛi+2piσiW^iT(ti,r)W^i(ti,r)+Di'+χi)1 / 2, Λi=2min{piv˜i,1, v˜i,j, σiσi'}, ϑi=η(zi,1)ϑβ,i+(1−η(zi,1))ϑα,i, ϑα,i=zi,1κα,iand ϑβ.i=zi,1κβ,i, where Di'=2piw¯iσi is a predefined parameter, D¯i=12σi'w¯i, and where χ i is a predefined positive parameter, and t i,0 = t0 applies.

[0066] Fig. Figure 7 shows a simulation for the multi-agent system 10 according to Fig. 5, in which the procedure according to Fig. 6 is implemented.

[0067] The multi-agent system 10 is again described as: {x˙i=vi,for i=1,...,4,v˙i=1m(ui−wi),for i=1,...,4, defined, where x i ∈ ℜ 2 and v i ∈ ℜ 1are the position and velocity of the i-th agent 12, respectively, and m is the mass of each agent 12, 12a, 12b, 12c, 12d, which is set as m = 1 kg.

[0068] For the purposes of the simulation, w i , which describes the possible and / or permissible time-varying external disturbances d i with the upper limit D i ≥ |d i | and the unknown uncertainty of i (x i ), taken into account, set as: wi(xi,vi)=μ˜(tanh(a˜vi)−tanh(b˜vi))+di,fu¨ri=1,...,4, with µ̃ = 0.01, ã = 1, b̃ = 10, and d i = 0.01 sin(πt) .

[0069] For each agent 12, 12a, 12b, 12c, 12d, the time-varying constraint functions κ α,i and κ β,i , i = 1, ...,4 according to κα,i=ϕi+κα1,i+κα2,icos(3×10−3πt+π4) and κ β,i = ϕ i + κ β1,i + κ β2,i COS (3 × 10 -3 πt), where ϕ 0,i= [2,2] T , ϕ ∞,i = [0,0] T , c i = [1.5,1.5] T , κ α1,i = [0.8,0.8] T , κ α2,i = [0.1,0.1] T , κ β1,i = [0.5,0.5] T , and κ β2,i = [0.1,0.1] T , for i = 1, ...,4.

[0070] The parameters ṽ i,j , σ i , σi' D i , pi, Di' and χi , i = 1, ...,4 are as: v˜i,1=v˜i,2=

[5005] ,σi=σi'=1,D˜i=0.1, pi=4,Di'=0.2. and χi=10−3 defined.

[0071] The vector of basis functions ȟ i (x i ) is defined according to (Eq. 5). The parameters are: qi=5,c^i,mi=1, mi=1,...,5, ℏ⌣1=[−2,−2]T,ℏ⌣2=[−1,−1]T,ℏ⌣3=[0,0]T,ℏ⌣4=[1,1]T.and ℏ⌣1=[2,2]T.

[0072] Fig. 7 shows the x-dimension of the tracking error s i,1(thin line) based on given initial positions of the agents 12, 12a, 12b, 12c, 12d, the given predefined global task reference y0 of the multi-agent system 10 and the given predefined local reference y h,i ∈ ℜ of the i-th agent 12, i = 1, ..., 4, is simulated. The defect of the active actuator device 14 is designed to occur successively at the first agent 12a and the fourth agent 12d, and at their active actuator device 14, the failure occurs at t = 1 s and t = 2 s. In other words, the output signal of the respective active actuator device 14, which controls the position along the x-axis, drops to zero. The thick line in Fig. 4 shows the restriction according to (Eq. 4).

[0073] The inventors realized that the tracking error s i,1 always the time-varying constraint -κ α,i < s i,1 < κ β,iin the presence of failures of the active actuator device 14 by timely activation of the redundant channel, ie the inactive actuator device 22 of the first agent 12a and the fourth agent 12d at t = 2.45 s and t = 3.05 s.

[0074] In addition to the methods, the invention provides a control device (not shown) comprising means for carrying out the described methods. The invention further provides the agent 12, 12a, 12b, 12c, 12d including the control device and the multi-agent system 10. The invention further provides a computer program (not shown) containing instructions which, when executed by the control device, cause the control device to carry out the described methods. The invention further provides a computer-readable medium (not shown) on which the computer program is stored. LIST OF REFERENCE SYMBOLS 10 Multi-agent system 12 agents 12a first agent 12b second agent 12c third agent 12d fourth agent 14 active actuator device 16 Leader-Follower Formation 18 (virtual) leaders 20 Communication topology 22 inactive actuator device 24 Control switching device 26 Control QUOTES CONTAINED IN THE DESCRIPTION

[0000] This list of documents submitted by the applicant was generated automatically and is included solely for the convenience of the reader. This list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions. Cited non-patent literature

[0000] Z. Zuo, J. Zhang, and Y. Wang, „Adaptive fault-tolerant tracking control for linear and Lipschitz nonlinear multi-agent systems,“ IEEE Transactions on Industrial Electronics, vol. 62, no. 6, pp. 3923-3931, 2014

[0003] P. Gong, W. Lan and Q. Han, „Robust adaptive fault-tolerant consensus control for uncertain nonlinear fractional-order multi-agent systems with directed topologies,“ Automatica, vol. 117, p. 109011, 2020

[0003] G. Cui, S. Xu, Q. Ma, Z. Li, and Y. Chu, „Command-filter-based distributed containment control of nonlinear multi-agent systems with actuator failures,“ International Journal of Control, vol. 91, no. 7, pp. 1708-1719, 2018

[0003] G.-H. Yang, S.-Y. Zhang, J. Lam, and J. Wang, „Reliable control using redundant controllers,“ IEEE Transactions on Automatic Control, vol. 43, no. 11, pp. 1588-1593, 1998

[0003] C. P. Bechlioulis and G. A. Rovithakis, „Robust Adaptive Control of Feedback Linearizable MIMO Nonlinear Systems With Prescribed Performance,“ in IEEE Transactions on Automatic Control, vol. 53, no. 9, pp. 2090-2099, Oct. 2008, doi: 10.1109 / TAC.2008.929402

[0003] J. Liu, „Radial Basis Function (RBF) neural network control for mechanical systems: design, analysis and Matlab simulation“, Springer Science & Business Media, 2013

[0003]

Claims

[1] Computer-implemented method for controlling a multi-agent system (10) with N autonomous agents (12, 12a, 12b, 12c, 12d), N ≥ 2, for n-dimensional navigation of the latter, wherein each agent (12, 12a, 12b, 12c, 12d) comprises an active actuator device (14) and at least one inactive actuator device (22), each designed for n-dimensional navigation of the agent (12, 12a, 12b, 12c, 12d), wherein the at least one inactive actuator device (22) is designed redundantly to the active actuator device (14), wherein each agent (12, 12a, 12b, 12c, 12d) comprises a control switching device (24) for switching a control (26) from a Actuator device (14, 22) to another actuator device (14, 22), wherein the multi-agent system (10) is configured in a leader-follower formation (16) in which each agent (12, 12a, 12b, 12c,12d) is to execute a respective desired movement using the active actuator device (14), wherein each agent (12, 12a, 12b, 12c, 12d) has a sensor device for tracking a respective actual movement, wherein the desired movement and the actual movement of the i-agent (12), i = 1, ...,N, are determined by a movement-related quantity x, i , x i ∈ ℜ n described, wherein the method comprises for each agent (12, 12a, 12b, 12c, 12d): a) controlling the active actuator device (14) to navigate the agent (12, 12a, 12b, 12c, 12d) by limiting a respective tracking error s i,1 , i = 1, ...,N, between the motion-related quantity x i the actual movement and the movement-related quantity x i the desired movement within a prescribed range, as long as no failure of the active actuator device (14) is detected; and b) if a failure of the active actuator device (14) is detected, ignoring the active actuator device (14) for control and causing the control switching device (24) to switch control (26) from the ignored actuator device (14) to one of the at least one inactive actuator devices (22), and using the latter as the active actuator device (14) for control to navigate by the respective tracking error s i,1 , i = 1, ..., N remains bounded according to step a). [2] Method according to claim 1, characterized by that the motion-related quantity x i a position, a speed and / or an acceleration of the respective agent (12, 12a, 12b, 12c, 12d). [3] Method according to one of the preceding claims, characterized by that the tracking error s i,1 of the i-th agent (12), i = 1, ..., N, by: si,1=ai,0(yi−yh,i−y0)+∑μ∈Niai,μ((yi−yh,i)−(yμ−yh,μ)) where y i = x i,1 , x i,m an m-th state or coordinate of the motion-related quantity x i ∈ ℜ n of the i-th agent (12), where y0 is a predefined global task reference of the multi-agent system (10), where y h,i ∈ ℜ is a predefined local reference of the i-th agent (12), where A˜=[aij]∈ℜN×N an adjacency matrix with a ii = 0, a ij = 1 if there is a communication connection between the i-th agent (12) and the j-th agent (12), and otherwise a ij = 0, and where Ni is a neighborhood set of the i-th agent (12), and that step a) further comprises one or both of the following: a1) Limiting the tracking error s i,1 of the i-th agent (12), i = 1, ..., N, such that: −γi_ϕi <si,1<γ¯iϕi,wenn si,1(t0)> 0, and −γ¯iϕi <si,1<γi_ϕi, wennsi,1(t0)<0, for all t ∈ [t0, ∞), where s i,1 (t0) is the tracking error of the i-th agent (12), i = 1, ..., N, at a start time t0, 0 < γ1, γ i ≤ 1 are predefined parameters, and ϕ i is a positive predetermined power function that decreases with time t and satisfies the condition limt→∞ϕ=ϕ∞>0 fulfilled; and / or a2) Limiting the tracking error s i,1 of the i-th agent (12), i = 1, ...,N, such that: −κα,i <si,1<κβ,i, for all t ∈ [t0, ∞), where κ α,i and κ β,i are predefined time-varying constraint functions, each of which κ α,i , κ β,i > 0 and the by κ α,i ≤ ĸ i and κ β,i ≤ κ i are limited, where ĸ i and κ i two limit constants, and where the k-th time derivatives, κα,i(k) and κβ,i(k) , κ = 1, ..., n + 1, are each bounded. [4] Method according to claim 3, characterized by that step a) continues to include: a3) controlling the active actuator device (14) of the i-th agent (12), i = 1, ..., N according to: u^i=τi,n−Di sgn(si,n)−θ^iTfi(xi), with adaptive laws: θ^˙=fiT(xi)si,n, where: ϕi=(ϕ0,i−ϕ∞,i)e−cit+ϕ∞,i, fu¨rt∈[0, ∞), τi,1=1ϵi(∑μ∈Niai,μ(y˙h,i−y˙h,μ)+ϕ˙isi,1ϕi+∑μ∈Niai,μxμ,2+ai ,0(y˙0+y˙h,i)−ciϕi(∂Γi−1(si,1ϕi)∂(si,1ϕi))−1Γi−1(si,1ϕi)), τi,2=−vi,2si,2+τ˙i,1−ϵiϕi∂Γi−1(si,1ϕi)∂(si,1ϕi)Γi−1(si,1ϕi), τi,j=−vi,jsi,j+τ˙i,j−1−si,j−1, for rj=3, …, n, ϵi=∑μ∈Niai,μ+ai,0, si,j=xi,j−τi,j−1, for rj=2, …, n, and the multi-agent system (10) is described as: {x˙i,m=xi,m+1, for m=1, …, n−1,x˙i,n=ui+wi(xi, t), for i=1, …, N,yi=xi,1, where wi(xi, t)=di+θiTfi(xi)∈ℜ a possible and / or permissible constant external disturbance d i with an upper limit D i ≥ |d i | and an unknown uncertainty f i (x i ) with a constant vector θ i which are each related to the movement of the i-th agent (12), where ϕ 0,i , ϕ ∞,i , c i are predefined positive parameters for the maximum overshoot, the maximum stationary error and the lower limit of the convergence speed, where ϕ 0,i > ϕ ∞,i , with v i,j and D i are predefined positive parameters, and u i ∈ ℜ denotes a control signal for the active actuator device (14) of the i-th agent (12), where Γ i (Δ) of the i-th agent (12) as a function of any time-related quantity Δ has the following properties 1) to 3): 1) smooth and steadily rising; 2) −γ_<Γ(Δ)<γ¯, if Δ(t0)>0, and −γ¯<Γ(Δ)<γ_,if Δ(t0)<0; 3) limΔ→−∞Γ(Δ)=−γ_andlimΔ→∞Γ(Δ)=γ¯,if Δ(t0)>0, and limΔ→−∞Γ(Δ)=−γ_andlimΔ→∞Γ(Δ)γ_,if Δ(t0)<0; and / or a4) controlling the active actuator device (14) of the i-th agent (12), i = 1, ..., N according to: u˜i=αi,n−D˜isgn(zi,n)−v^i,n(xi), with adaptive laws: v^i(xi)=W^iTh⌣i(xi)and W^˙i=σizi,nh⌣i(xi)−σiσ′iW^i, where: αi,1=1∈i(∑μ∈Niai,μ(y˙h,i−y˙h,μ)+∑μ∈Niai,μxμ,2+ai,0(y˙0+y˙h,i)−(v˜i,1+(K˙α,iKα,i)2+(K˙β,iKβ,i)2)zi,1), αi,2=−v˜i,2zi,2+α˙i,1−∈iςizi,12pi−1, αi,j=−v˜i,jzi,j+α˙i,j−1−zi,j−1fu¨rj=3,…n, ∈i=∑μ∈Niai,μ+ai,0, zi,1=si,1, zi,j=xi,j−αi,j−1,for rj=2,…,n, and the multi-agent system (10) will be described as: {x˙i,m=xi,m+1,for m=1,…,n−1x˙i,n=vi(xi)+ui+di,for i=1,…,N,yi=xi,1 where d i a possible and / or permissible time-varying external disturbance with an upper limit D i ≥ |d i | and vi(xi)=Wi∗Th⌣i(xi)+g⌣i∈ℜ are an unknown uncertainty, each related to the movement of the i-th agent (12), where Wi∗ W i * is an ideal weight vector that satisfies the condition Wi*TWi≤w¯i with w i > 0, ǧ i |ǧ i | ≤ q̌ i with q i > 0, where ȟ i (x i ) = [ȟ i,1 (x i ), ..., ȟ i,qi (x i )] T ∈ ℜ qi is a predefined vector of basis functions, q i is a predefined parameter, where ṽ i,j , σ i , σi' are predefined positive parameters, where D̃ i > Di + q i is a predefined parameter, where p i a predefined parameter with 2p i ≥ n and where u i ∈ ℜ denotes a control signal for the active actuator device (14) of the i-th agent (12), and ςi=η(zi,1)κβ,i2pi−zi,12pi−1−η(zi,1)κα,i2pi−zi,12pi, where η( * ) = 1, if * > 0, otherwise η = 0. [5] Method according to claim 4, characterized by , that the i-th agent (12), i = 1, ..., N, l i comprises redundant actuator devices (22), l i ≥ 1, and wherein in step b) the failure of the (r + 1)-th active actuator device (14) in the i-th agent (12), r = 0, ..., l i - 1, to a t i,r+1 time is detected, which is defined as follows: b1) ti,r+1=min{t|si,1=−γ_i,rϕi, si,1=y¯i,rϕi, i=1,...,N}, where: γ_i,r=−Γi(−ψi,r), γ¯i,r=−Γi(−ψi,r), ψi,r=(εi2(ti,r)+∑j=2nsi,j2(ti,r)+σ¯i+θ¯iT(ti,r)θ¯i(ti,r))−1 / 2, θ¯i(ti,r)=|θ^i(ti,r)|+θi', εi=Γi−1(si,1ϕi), where σ i and θi'≥|θi| are predefined positive parameters, and t i,0 = t0 applies; and / or b2) ti,r+1=min{t||ϑi|=1−e−qi,r22pi),i=1,...,N}, where: qi,r=(ln11−ϑi2pi(ti,r)+pi∑j=2nzi,j2(ti,r)+2piD¯iΛi+2piσiW^iT(ti,r)W^i(ti,r)+Di'+χi)1 / 2, Λi=2min{piv˜i,1,v˜i,j,σiσi'}, ϑi=η(zi,1)ϑβ,i+(1−η(zi,1))ϑα,i, ϑα,i=zi,1κα,i and ϑβ,i=zi,1κβ,i, where Di'=2piw¯iσi is a predefined parameter and D¯i=12σi'w¯i applies, and where χi is a predefined positive parameter, and t i,0 = t0 applies. [6] Control device with means for carrying out the method according to one of the preceding claims. [7] Autonomous agent (12) for n-dimensional navigation, wherein the agent (12) comprises an active actuator device (14) and at least one inactive actuator device (22), each for n-dimensional navigation of the agent (12), wherein the at least one inactive actuator device (22) is configured to be redundant to the active actuator device (14), wherein the agent (12) comprises a control switching device (24) for switching a control (26) from one actuator device (14, 22) to another actuator device (14, 22), wherein the agent (12) is suitable for forming a multi-agent system (10) in a leader-follower formation (16) in which the agent (12) is to execute a desired movement using the active actuator device (14), wherein the agent (12) comprises a sensor device for tracking an actual movement of the agent (12),wherein the desired movement and the actual movement of the agent (12) are determined by a movement-related quantity x, i , x i ∈ ℜ n described, wherein the agent (12) is characterized by a controller according to claim 6. [8] A multi-agent system (10) with N autonomous agents (12), N ≥ 2, according to claim 7. [9] A computer program comprising instructions which, when the program is executed by a control device, cause the control device to carry out the method according to any one of claims 1 to 5. [10] A computer-readable storage medium on which the computer program according to claim 9 is stored.

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