Human-in-the-Loop Consensus Control for Nonlinear Multi-Agent Systems With Actuator Faults

Guohuai Lin, Hongyi Li,, Hui Ma, Deyin Yao, and Renquan Lu,

Abstract—This paper considers the human-in-the-loop leaderfollowing consensus control problem of multi-agent systems(MASs) with unknown matched nonlinear functions and actuator faults. It is assumed that a human operator controls the MASs via sending the command signal to a non-autonomous leader which generates the desired trajectory. Moreover, the leader’s input is nonzero and not available to all followers. By using neural networks and fault estimators to approximate unknown nonlinear dynamics and identify the actuator faults, respectively, the neighborhood observer-based neural fault-tolerant controller with dynamic coupling gains is designed. It is proved that the state of each follower can synchronize with the leader’s state under a directed graph and all signals in the closed-loop system are guaranteed to be cooperatively uniformly ultimately bounded.Finally, simulation results are presented for verifying the effectiveness of the proposed control method.

I. INTRODUCTION

IN the past years, the consensus control problem of multiagent systems (MASs) has attracted considerable attention due to its widespread applications such as synchronization of autonomous underwater vehicles (AUVs), formation control of unmanned air vehicles (UAVs), and cooperative control of traffic vehicles [1]–[4]. Generally, the consensus control problem can be roughly divided into two categories: the leaderless consensus [5]–[8] and the leader-following consensus [9]–[13]. The leaderless consensus was studied in [5] for a class of heterogeneous linear MASs with a predictive event-triggered mechanism. Two control methods for the leader-following consensus were first given in [9],including static state feedback and dynamic output feedback control methods. In [10], a robust adaptive controller with event-driven mechanism was established for the state consensus problem of linear MASs subject to external disturbances. However, nonlinearity is ubiquitous in the vast majority of practical control systems. In response to this problem, neural networks and fuzzy logic systems were introduced in [14]–[24]. For example, in [14], by using neural networks to approximate the nonlinear dynamics, an adaptive finite-time control protocol was presented to achieve output consensus for nonstrict feedback nonlinear MASs with unknown control coefficients and output dead zones. In [15]and [16], a fuzzy event-triggered bipartite containment controller and a distributed fuzzy containment controller were designed for nonlinear MASs.

In the era of artificial intelligence, MASs are expected to be autonomous in practical applications. However, plenty of accidents about autonomous systems have been reported. The reliability and safety of real systems are improved under a human operator’s judgement and decision, which indicates that the presence of the human element is important for some safety-critical autonomous systems [25]. For instance, in order to ensure that UAVs can achieve leader-following consensus safely and successfully, the human operator can supervise the entire team by utilizing the visual equipment connected to the leader and all followers, and manipulate them to avoid certain barriers through broadcasting commands to control the leader directly. Therefore, the leader is required to be nonautonomous. Under the communication graph of MASs, all followers autonomously synchronize with the trajectory of the leader via the distributed protocol. Recently, studies on human-in-the-loop control for MASs have been rapidly increasing [26]–[30]. In [26], for a class of human-in-the-loop MASs with a non-autonomous leader, a novel distributed output feedback control strategy was developed to ensure that the states of all followers can synchronize with the leader’s state using only the relative input-output measurements.Under the remote operation, the information transmission between systems and humans was time-delayed, then authors in [27] investigated the stability of a class of linear timeinvariant human-in-the-loop systems in time-domain and frequency-domain. Similarly, a cooperative controller was designed in [28] to achieve the goal that an ensemble of kinematic robots asymptotically synchronize with the reference inputs through the interaction with human operator.

On the other hand, as the scale of control systems grow more and more complex, the probability of actuator faults increases, which may lead to system instability. Therefore, it is of great significance to investigate the fault-tolerant control(FTC) problem for MASs subject to actuator faults [31]–[36].In order to stabilize the closed-loop system, a fault estimation module was constructed in [31] for compensating the effect of actuator faults. In [32], an adaptive FTC protocol based on an event-triggered mechanism was studied for a class of general linear MASs with multiplicative actuator faults. By designing a continuous robust adaptive FTC method for heterogeneous fractional-order MASs in [33], the problems of coupling nonlinearities, actuator faults, and external disturbances were solved at the same time.

In addition, the design of an observer-based controller for the consensus control problem is of great importance in real scenarios [37]–[43]. In [37], based on the local and neighborhood observers, two different kinds of neighborhood controllers were constructed to achieve the state synchronization of MASs. For ann-dimensional stochastic time-delay system subject to unknown control direction, a novel observer-based neural controller was put forward in[38]. Furthermore, the objectives of [39] were to design a reduced-order observer-based fuzzy adaptive bipartite tracking control protocol for nonlinear MASs and achieve that the consensus tracking errors converge to a small neighborhood of the origin. However, the neighborhood observer-based human-in-the-loop consensus control has not been fully studied for general nonlinear MASs with actuator faults.

Motivated by the aforementioned observations, in this paper, we design the neighborhood observer and the neural fault-tolerant controller with adaptive coupling gains to investigate the human-in-the-loop consensus control problem for nonlinear MASs with actuator faults. The main contributions are given as follows:

1) The leader of the nonlinear MASs is considered as nonautonomous, whose control input is time-varying and is provided by a human operator. Furthermore, the restriction assumption that a subset of followers can access the leader’s input in [44]–[46] is removed in this paper.

2) In [37] and [47], the control laws with static coupling strengths were designed. Different from [37] and [47], the control protocol proposed in this paper achieves the leaderfollowing consensus via adaptive coupling strengths for adjusting online.

3) Compared with [9]–[11], the observer-based protocol design of nonlinear MASs is more challenging, because the separation principle is noneffective for nonlinear MASs generally. By using the relative information of neighboring nodes, the neighborhood observer is designed for nonlinear MASs to obtain the consensus in this paper.

Notations:INdenotes an identity matrix with dimensionN.‖·‖ is the Euclidean norm of a vector and ‖·‖F(xiàn)is the Frobenius norm of a matrix.andare the maximum singular value and the minimum singular value of a matrix,respectively. Let diag{J1,...,JN} be a diagonal matrix andJiis theith diagonal element, wherei=1,...,N. t r(·) is the trace of a matrix. The Kronecker product is denoted by ?. There are some properties of the Kronecker product, such as(Q?M)T=QT?MT,s(Q?M)=(sQ)?M=Q?(sM), and(Q?M)(I?X)=(QI)?(MX) , where Q,M,X,I are matrices andsis a scalar. Let=[1,1,...,1]T∈RN×1.

II. PROBLEM FORMULATION AND PRELIMINARIES

A. Basic Graph Theory

B. System Formulation

Consider the nonlinear MASs consisting ofNidentical agents and one leader, and theith agent’s dynamics are described by

wherexi=[xi1,...,xin]T∈Rnis the state,ui=[ui1,...,uim]T∈Rmis the control input,yi=[yi1,...,yim]T∈Rmis the measurement output of theith agent.fi(xi) = [fi1(xi),...,fim(xi)]T∈Rm, and ωi(t)=[ωi,1(t),...,ωi,ι(t)]T∈Rιrepresent the unknown matched nonlinear functions and the actuator faults of theith follower, respectively. In the context of consensus tracking control of nonlinear MASs,S∈Rn×n,W∈Rn×m,J∈Rm×nandD∈Rn×ιare known constant matrices.

The dynamics of the leader are given by

wherex0=[x01,...,x0n]T∈Rnandy0=[y01,...,y0m]T∈Rmare the state and the measurement output of the leader.u0∈Rmis an unknown bounded control input.

Remark 1:According to [26], the human-in-the-loop control method requires the human operator to control the leader directly through control inputu0and influences each follower indirectly. More precisely, all followers are led by the leader through the designed controller and the connectivity of network topology for synchronization. In addition, unlike previous works [9], [48] on the synchronization problem for linear MASs, the control inputu0of the leader is nonzero and not available to all followers in this paper.

The following assumptions are given to facilitate analyzing and designing.

Assumption 1 [31]:Actuator faults satisfy ‖ω(t)‖≤ω1M,, where ω1M>0, ω2M>0.

Assumption 2 [31]:(S,W) is controllable and (S,J) is observable.

Remark 2:From [31], Assumption 2 implies rank(W,D) =rank(W). Therefore, the actuator faults considered in this paper belong to the actuation spaceIm(W). In other words,actuation spaceIm(D) is contained in actuation spaceIm(W),which meansIm(D)?Im(W), and there existsW? ∈Rm×nsuch that (In?WW?)D=0.

Assumption 3 [49]:The unknown matched nonlinear functionsfi(xi) can be approximated by neural networksas

The control objective of this paper is to design the distributed adaptive FTC protocoluifor the synchronization errors converging to a small neighborhood of the origin.

III. NEIGHBORHOOD OBSERVER AND CONTROLLER DESIGN

The neighborhood observer is constructed to estimate the unmeasurable states of the dynamics (1) by only using the neighborhood information in this section. Based on the designed neighborhood observer, the adaptive neural FTC protocol is also proposed by using the neighborhood information.

The following neural networks can be utilized to approximate the unknown nonlinear functions of the followers dynamics:

By using the relative states of neighboring nodes, the distributed neural adaptive neighborhood control lawuiis designed as

whereuiαis the nominal linear controller,uiβis the robust term,uiΞa(chǎn)nduiDare used to compensate the nonlinear functionsfi(xi) and actuator faults ωi(t), respectively. By using (5), they are designed as

where ξ is a positive constant, adaptive parametersandare specified by the following update laws:

where Γαi, Γβi,aαandaβare positive constants,K∈Rm×nis the controller gain matrix withK=?WTP2, and P2is the unique positive definite solution to the following Riccati equation:

whereQ2is a positive definite matrix.

Remark 3:In [44]–[46], the leader’s control inputu0is contained in the controller design; hence, an assumption that a subset of followers are aware of the leader’s input is made,which is impractical in real applications. In this work, we remove the above restricted condition by the fact that the control input of the leader is neither zero nor available for any follower. For this case, the control termuiβis introduced to improve the robustness of the proposed scheme.

By using (6), the dynamics of theith node are rewritten as

Substituting (6) into (3), the observer dynamics ofare expressed as

According to (12), the global state estimation error dynamics are rewritten as

The dynamics of the neighborhood estimation error are written as

In the light of the above description, the block diagram of the neighborhood observer-based neural fault-tolerant distributed controller for agentiis shown in Fig. 1.

Fig. 1. Block diagram of the control technique.

IV. STABILITY ANALYSIS

The stability analysis made in this section follows from the aforementioned observer and controller design.

Theorem 1:Under Assumptions 1–3 and the considered directed graph contains a spanning tree, for MASs (1) and (2),if the neighborhood state observer (3), the controller (6) with the adaptive coupling strengths (7), (8) are designed well, and the adaptive law is given by

V. SIMULATION EXAMPLE

A numerical simulation is given to demonstrate the effectiveness of the designed controller by considering a harmonic oscillator. We consider a group of five followers indexed by 1 to 5 and one leader indexed by 0 with agent dynamics described by

fi(xi)=xi1sin2(xi2)i=1,2,3,4,5

where , , and the actuator faults are given as

The directed communication graph containing five followers and one leader is shown in Fig. 2. Next, the leader’s inputu0(t) is considered as

Fig. 2. Communication graph for the followers and leader.

Moreover, it needs to be emphasized that the leader’s inputu0(t) is not available to any follower and is provided by the human operator. According to Fig. 2, the leader adjacency matrix isG=diag{1,0,0,0,0}, the adjacency matrix Λ and the Laplacian matrix L are given as

From (4), (9) and (21), we obtainamax=0.2912 ,=3,R=?0.1400 and

The correlative design parameters areaΞ=0.1,aα=0.0015,aβ=0.5,aω=0.001, ΓΞ1=ΓΞ2=ΓΞ3=ΓΞ4=ΓΞ5=100,Γα1=Γα2=Γα3=Γα4=Γα5=1000, Γβ1=Γβ2=Γβ3=Γβ4=Γβ5=500, and Γω1=Γω2=Γω3=Γω4=Γω5=300. Fig. 3 (a)and Fig. 4 (a) imply the state tracking trajectories of followers under the adaptive output feedback fault-tolerant controller.Besides, in order to show the effectiveness of the fault-tolerant controller, the state tracking trajectories of followers without FTC are given in Fig. 3 (b) and Fig. 4 (b) for comparison. The trajectories of consensus errors are shown in Fig. 5, which depicts the tracking result more clearly. Fig. 6 shows the observing errors for revealing that the unmeasurable states are efficiently observed by the neighborhood state observer. The coupling strengthsandare shown in Fig. 7, which illustrates thatandwill finally converge to some bound values. In addition, the effectiveness of fault estimations is shown in Fig. 8. The control inputs of all followers are depicted in Fig. 9. The simulation results for nonlinear MASs are shown in Figs. 3–9, from which we can clearly find that the neighborhood observer-based neural fault-tolerant control protocol with adaptive gains can simultaneously guarantee the leader-following consensus of the MASs and all signals in the closed-loop system are CUUB.

Fig. 3. States xi1 and x01 of multi-agent systems. (a) States of all followers xi1 and state of the leader x01 with FTC; (b) States of all followers xi1 and state of the leader x01 without FTC.

Fig. 4. States xi2 and x02 of multi-agent systems. (a) States of all followers xi2 and state of the leader x02 with FTC; (b) States of all followers xi2 and state of the leader x02 without FTC.

Fig. 5. Trajectories of consensus errors δi with i =1,2,3,4,5.

Fig. 6. Observer errors with i =1,2,3,4,5.

Fig. 7. Coupling strengths and with i =1,2,3,4,5.

Fig. 8. Actuator faults ω i(t) and the fault estimations with i =1,3,5.

Fig. 9. Control inputs ui with i =1,2,3,4,5.

Remark 4:In practical applications, for example, the leaderfollowing consensus control of UAVs, the safety problem and tracking performance should be considered at the same time.The adaptive distributed controller is developed to ensure that the states of all followers can track the leader state. The human operator can adjust the trajectory of the entire team through changing the leader’s control inputu0to fulfill some complex tasks like multi-obstacles avoidance and collision avoidance withu0being a time-varying piecewise function given in (41). From Figs. 3 and 4, we can see that the state trajectories of all agents are changed in 15 s and 30 s due to the differentu0.

VI. CONCLUSION

The human-in-the-loop leader-following cooperative tracking control problem has been investigated for a class of nonlinear MASs subject to actuator faults. It has been assumed that the human operator manipulates the entire team through sending the control signal to the non-autonomous leader whose input is not available to all followers. Each follower has synchronized with the leader autonomously based on the directed connected graph. The unknown nonlinear functions have been approximated by neural networks and the actuator faults have been identified by the fault estimators. By designing the state observer only using the neighborhood information among all agents, the neighborhood observer-based adaptive fault-tolerant distributed controller with dynamic coupling gains has been established. It has been shown that all signals in the closedloop system are guaranteed to be CUUB. The numerical example has been given to verify the effectiveness of the developed control method. Future research efforts will be devoted to considering the event-triggered mechanism and extending the results to the formation control of UAVs with time delays [50]–[54].