YANG Pu,HU Xukai,WANG Zixin,and ZHANG Zhiqing

College of Automation,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

Abstract:The fault-tolerant consensus problem for leader-following nonlinear multi-agent systems with actuator faults is mainly investigated.A new super-twisting sliding mode observer is constructed to estimate the velocity and undetectable fault information simultaneously.The time-varying gain is introduced to solve the initial error problem and peak value problem,which makes the observation more accurate and faster.Then,based on the estimated results,an improved sliding mode fault-tolerant consensus control algorithm is designed to compensate the actuator faults.The protocol can guarantee the finite-time consensus control of multi-agent systems and suppress chattering.Finally,the effectiveness and the superiority of the observer and control algorithm are proved by some simulation examples of the multi-aircraft system.

Keywords:multi-agent system,sliding mode control,fault-tolerant consensus control,super-twisting sliding mode observer.

1.Introduction

With the continuous development of modern control technology,in recent years,researchers have made many innovative achievements and progress in the field of automatic control.However,smart devices will inevitably fail due to a long time of frequent use.If there is no measure to compensate for the failure in advance,once the failure occurs,it may affect the progress of the whole task,and even lead to property loss or casualties.Therefore,the fault-tolerant control algorithm has gradually become a significant direction of the contemporary control field[1-4].Generally speaking,methods of fault-tolerant control can be divided into passive fault-tolerant control and active fault-tolerant control according to different ways of a control system using redundancy to handle faults.Passive fault-tolerant control makes use of the pre-set system redundancy to achieve system insensitivity to defects,which does not require the help of fault diagnosis modules.Active fault-tolerant control is generally achieved by adjusting controller parameters or structure online after a fault occurs.It often needs to adapt or reconstruct the control law with the help of fault information obtained from fault diagnosis.

Since the concept of multi-agent systems (MASs) came into being,the distributed collaborative control method of MASs has attracted more and more attention due to its better robustness,practicability and flexibility compared with traditional centralized control [5-7].For example,multi-robot system and multi-aircraft system have been widely used in reality.As one of the most fundamental and essential issues,the consensus of MASs has attracted a large number of researchers [8-10].The so-called consensus means that with appropriate consensus control strategies,each agent can make its final state tend to the same expectation through the information interaction between each other,which is mainly reflected in the formation control and cluster control of MASs.Many papers have studied the consensus control of MASs.In [11],Yu et al.designed a distributed linear consensus protocol with second-order dynamics to achieve consensus in multi-agent dynamical systems with sampled position data.In [12],Gong et al.investigated the output feedback consensus control problem for a class of nonlinear fractional-order MASs with general directed topologies.Focusing on the problem of finite-time consensus control for MASs,Tian et al.[13] constructed a robust finite-time consensus control by using the backstepping method,Zhou et al.[14] proposed a new nonsingular terminal sliding mode control method based on the pinning error.Based on the above background,the consensus control problem of MASs is studied in this paper.

However,for complex and large systems such as MASs,the occurrence of faults appears more frequent and dangerous.Take the multi-aircraft system composed of quad-rotor unmanned aerial vehicles as an example.Affected by ageing,wear,or unexpected conditions,the quad-rotors may experience various types of faults such as actuator fault [15,16],sensor fault [17],and communication fault [18].These faults can affect not only the fault aircraft itself,but also other aircraft,which hinder the consensus control of the entire system.Therefore,a suitable fault-tolerant controller is needed to compensate the influence of faults.Sliding mode control (SMC) has been widely used for fault-tolerant systems due to its simple algorithm and strong robustness.The application of SMC realizes the stability of the systems and the suppression of matched disturbance and actuator fault [19,20].The consensus tracking problem of MAS with disturbance and actuator fault was solved by using the SMC protocol in[21].The formation control problem was investigated for MASs with different terminal sliding mode methods to overcome the effect of actuator fault in [22] and [23]respectively.Qin et al.[24] proposed a fault-tolerant control method based on an integrated sliding mode surface for MASs with multiplicative actuator fault.

The above achievements assume that the information of multiple agents is measurable in the study.In practice,however,considering the cost of detection and load on agents,a fully equipped sensor device may not be configured for the system,so in some cases there are states which cannot be measured.In response to this practical problem,current research results often adopt the method of combining observers to estimate the unmeasured states.For example,Liu et al.[25] designed an observer in solving unmeasurable states of agents.For MASs with unknown external disturbances,a finite time disturbance observer was designed in [26] to realize the fast estimation of unknown disturbances in each following agent.In[27],for MASs consisting of second-order nonlinear systems,state observer was used to achieve finite-time consensus convergence in the case that all agents had input constraints and velocities were unmeasurable.In the area of fault estimation,Menon et al.[28] considered sliding mode observers to estimate the actuator faults in MASs.Zhao et al.[29] designed a corresponding unknown input observer with reduced/full order to obtain the fault estimation.And a distributed anti-disturbance fault observer was proposed for MASs to reject the influence of disturbances for fault estimation results in [30].It can be seen that some progress has been made in the research field of MASs to design effective control schemes by using state observers.

Considering the current consensus research results of nonlinear MASs with actuator faults,most of them focus on the design of fault-tolerant control protocols and fail to consider the rapidity and accuracy of fault information estimation,which affects the system and the performance of the controller.Therefore,this paper focuses on the fault-tolerant consensus problem of nonlinear MASs with actuator fault and unmeasurable fault information.The main contributions are as follows: First,aiming at the undetectable problem of the system state of MASs,a super-twisting sliding mode fault observer (STSMFO) is designed in this paper.Compared with the observer in[31],the designed observer uses the velocity states of agents to estimate the fault information of actuators.The improvement in structure makes the observation effect smoother.Secondly,in case of large initial error,by introducing a time-varying gain into the observer,the system can rapidly converge to reduce the peak value.Finally,a sliding mode fault-tolerant control algorithm is designed to solve the consensus problem.Compared with the algorithm designed in [32],the designed algorithm improves the robustness of the control system by adding an integral item and introduces hyperbolic tangent function into the controller,so as to suppress the influence of chattering on the system and enable the controller to play a better control effect.

The remaining part of this paper is as follows: In Section 2,the basic graph theory and the model description of MASs are introduced,and the definition of the problem studied in this paper is given.Section 3 mainly introduces the design and related proof of the STSMFO.A sliding mode fault-tolerant control protocol for MASs with actuator fault and its convergence proof are presented in Section 4.In Section 5,the simulation experiment is given to verify the effectiveness of the observer in Section 3 and the control algorithm in Section 4.In Section 6,the comparison experiments prove the superiority of the observer and control algorithm designed in this paper.

The symbols quoted in this paper have the following definitions.Rmrepresentsm-dimensional vector space.diag{·} is a diagonal matrix.? denotes the Kronecker product of the matrix.‖x‖ represents Euclidean norm and satisfies ‖x‖=

2.Problem statement and preliminaries

2.1 Graph theory

Lemma 1[33] If graphis connected,the matrixL+Bis a reversible matrix.

Lemma 2[34] Consider a nonlinear system described with=f(x) in whichf(0)=0,if there exist a positive definite continuous functionV(x),real numbersc＞0 and φ ∈(0,1) with an open neighborhood ofV(x)at the origin such that

then the origin is a finite-time-stable equilibrium of the nonlinear system.Moreover,ifTis the setting-time function,then

2.2 Problem formulation

Considering the state-space model of leader agent 0 without actuator fault and inherent nonlinear dynamic behavior as shown below.

wherex0(t),v0(t)∈Rmare positio n and velocity state of leader agent,respectively.u0(t)∈Rmdenotes the control input.

For follower agenti(i=1,2,···,n) with actuator fault,the following state-space model can be used to represent:

wherexi(t) arevi(t)∈Rmare the position and the velocity state of follower agentirespectively.ui(t)∈Rmdenotes the control input.The continuous vector functionfi(xi(t),vi(t),t)∈Rmrepresents the inherent nonlinear dynamic behavior of follower agenti.βi(t)∈Rmdenotes the actuator additive fault vector of theith follower.βi(t)=0 represents no fault occurs,and βi(t)≠0 represents that the actuator of follower agentihas failed.

Assumption 1For the nonlinear functionfi(xi(t),vi(t),t),it is assumed that a non-negative real numberfˉsatisfies

Assumption 2For the actuator additive faultβi(t)and its first derivative,it is assumed that a non-negative real numbersatisfies

According to the neighbor information obtained by the follower agenti,the consensus position tracking error variableexi(t) and the velocity error variableevi(t)are respectively defined as follows:

Equation (7) can be written as the following vector form by using the Kronecker product.

Definition 1The consensus goal of MASs is said to be achieved if and only if the state variables of each agent in MASs (3) and (4) satisfy the following relationship under any initial conditions:

3.Super-twisting sliding mode observer design

Since it is difficult to detect the actuator fault information of MASs in the process of motion completely,an STSMFO is designed as shown in (11) to predict the unmeasured fault information of each agent’s actuator in this paper.The estimation variable of failure is added to the control protocol as compensation.

In order to avoid the slow convergence of the observer due to the large initial estimation error,the gain coefficient η(t) is designed in this paper,so that the system can converge to the final state rapidly even when the initial error is large.η(t) is expressed as

Take the first-order derivative of (12).One can get the error dynamics

positive definite matrix according to (15).

Substituting (20) into (22),one can get

Therefore,according to Lemma 2,the observer error can converge to the zero within finite time,the proof is thus completed.□

It shows that the observer can estimate the velocity and fault information of the follower agentiin a finite time,and the finite convergence timeT1satisfies

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4.Sliding mode controller design

Based on the consensus tracking error variablesexi(t) andevi(t),the following integral sliding mode variable is designed as

When the sliding mode variable enters the sliding surface,it should meet thats(t)=(t)=0.

In this subsection,according to MASs (3) and (4),the designed STSMFO (11) and the integral sliding surface constructed as (26) in this paper,a distributed fault-tolerant consensus control strategy designed for follower agentiis written as

whereuai(t) is the control law used by MASs in the failure-free state,i.e.,βi(t)=0,which can be written as

where γ and ε are positive constants,(t) is the estimation of actuator additive fault of agentiobtained by STSMFO (11).uci(t) is used to deal with the inherent nonlinear dynamics of the system,which can be expressed as

Theorem 2Under the premise of Assumption 2,for MASs (3) and (4),choosing (27) as the integral sliding surface of MASs.The system trajectory will reach and remain on the integral sliding surface in finite time.The consensus tracking error variables will converge to zero under the distributed fault-tolerant consensus control strategy (29).Then MASs (3)-(4) can achieve the goal of fault-tolerant consensus.

ProofAccording to the separation principle [36],the design of the observer and controller can be considered and analyzed separately,which means real states can replace the observer states in controller (30)-(32).Integrating (9) and (29)-(32),the fault-tolerant consensus control law can be simplified into the following form:

The following proof is divided into two steps,namely,the reaching phase and the sliding phase.

Step 1(The reaching phase) Consider the following Lyapunov function:

The first term in (35) can be further simplified as

According to Lemma 3 and (33),the second term in(35) can be further simplified as

Combing (35)-(37),one can get

Therefore,according to Lemma 2,it is easy to be known that the consensus tracking errors (9) of MASs can reach the integral sliding-mode surfaces(t)=0 in finite time,and the finite convergence timeT2satisfies

Remark 1In practical applications,the SMC process is affected by inertia,time-delay and spatial-lag,which may lead to chattering phenomena.In order to weaken the impact of chattering,the hyperbolic tangent function tanh(s(t)/ε) is adopted in this paper,where ε ＞0denotes linear width.Since the control accuracy of the system is often interfered in the process of reducing chattering and improving the smoothness of SMC,the value of ε is usually small.

Remark 2Consider the finite convergence timesT1andT2,the upper bounds of both depend on the initial values of the selected Lyapunov functions.Specifically,the initial value ofV1(t) is based on the observer estimation errors,and the initial value ofV2(t) is based on the consensus state tracking errors between agents.Obviously,in practice,the latter has larger initial conditions,which results in a longer convergence time.

Step 2(The sliding phase) After the consensus tracking errors reach the sliding-mode surface,the dynamics of consensus tracking errors decrease into

Sincek1,k2＞0,the system matrix of (40) satisfies the Routh-Hurwitz criterion [37].Therefore,the consensus tracking error variablesexi(t) andevi(t) can converge asymptotically to zero on the sliding surface,and the consensus tracking error system (9) is asymptotically stable along the sliding surface,which completes the proof.□

5.Simulation verification

In this section,we combine the simulation of MAS to verify the theoretical validity according to the results obtained in the previous theoretical parts.A MAS consisting of six aircraft with a leader index of 0 and follower index ofi(i=1,2,···,5) is considered with its communication topology being in Fig.1.The communication topology graph is connected obviously.

Fig.1 MAS communication topology network

Set all the weights of edge to 1.Therefore,the Laplacian matrixLand the adjacency matrixBbetween all of leader and follower agents can be calculated as

In the simulation experiments of this paper,all the aircraft are set up to fly on the same horizontal plane and the velocity direction of the leader agent is set as theY-axis.We carry out simulation analysis for the control of the system in the direction ofX-axis.The leader agent dynamic is described as follows:

wherex0(t) andv0(t) represent the displacement and the linear velocity of the leader in theX-axis direction respectively.Initial values of the leader’s dynamic equation are assigned withx0(0)=0,v0(0)=1.And control input of the leader agent isu0=sin(t/15).

The follower agenti(i=1,2,···,5) by considering inherent nonlinear dynamic behavior and actuator fault can be gotten as follows:

wherexi(t) andvi(t) represent the displacement and the linear velocity of theith follower in theX-axis direction,fi(xi(t),vi(t),t)=sin(xi(t))+sin(vi(t)),and‖fi(xi(t),vi(t),t)‖≤2,so that Assumption 1 is obviously satisfied.Initial values of the followers’ dynamic equations are selected asx1(0)=1,(0)=1.2,x3(0)=0.5,x4(0)=-1,x5(0)=-1.5,v1(0)=1.5,v2(0)=1.2,v3(0)=2,v4(0)=0.5,v5(0)=0.

To verify the effectiveness of the fault-tolerant control algorithm,we assume that follower agents 2 and 4 have actuator additive faults and that the remaining agents normally work without faults.The actuator fault functions of aircraft 2 and 4 are as follows:

Obviously,these faults are bounded and differentiable andcan be chosen as 0.3.Therefore,according to Theorem 1,we can set the parameters of the STSMFO as λ1=5,λ2=13.The coefficient of the time-varying gain is chosen as ?=3,σ=5.

In order to better illustrate that the improved supertwisting sliding mode fault observer designed in this paper has better performance than the observer designed in [31],this paper sets the same parameters for the two observers.It applies them to the estimations of the same actuator fault functions.The fault estimation curves can be obtained in Fig.2 and Fig.3.

Fig.2 Estimations of β2(t) by two observers

Fig.3 Estimations of β4(t) by two observers

From the comparisons of the estimation curves in Fig.2 and Fig.3,we can clearly draw a conclusion.The observer designed in this paper can achieve accurate estimations of actuator additive faults with smoother curves and lower peaks than that in [31].Table 1 shows the time consumption of the two methods for fault function estimation.The results demonstrate that the proposed STSMFO can realize fault estimation more quickly in the face of actuator additive fault.

Table 1 Estimated time of faults in Fig.2 and Fig.3 s

Choose the sliding mode surface coefficients ask1=2,k2=1,the coefficients of the fault-tolerant controller are set as γ=0.5,n=5,ε=0.15.Fig.4 and Fig.5 are theX-axis position and linear velocity trajectories of each agent under the action of control protocol (30).Based on a fixed topology and through timely information interaction,followers can eventually track the leader within finite time.More clearly,it can be seen from the simulation results in Fig.6 and Fig.7 that although different actuator faults are given for different follower aircraft,their position and velocity states are finally consensus in a finite time under the fault-tolerant consensus control law.

Fig.4 Position tracking of leader and followers

Fig.5 Velocity tracking of leader and followers

Fig.6 Position tracking errors of followers

Fig.7 Velocity tracking errors of followers

Fig.8 shows the control input of each agent,and it can be seen clearly that the control signal runs smoothly under the control protocol (30).Besides,in order to increase the persuasiveness of the algorithm proposed in this paper,we have also added some comparison experiments with [32].The control input of each follower aircraft with the method proposed in [32] is shown in Fig.9,and Fig.10 shows the control input of follower agent 2 under different control protocols.From the comparison curve in Fig.10,we can see that compared with [32],since the control law proposed in this paper can better suppress chattering,the control input under the proposed control protocol is smoother than that in [32].

Fig.8 Control input of followers with method in this paper

Fig.9 Control input of followers with method in [32]

Fig.10 Control input of two control protocols

To sum up,through the comprehensive comparison of the above simulation results,we can make the following summary and analysis.Firstly,compared with [31],the super-twisting sliding mode observer designed in this paper is improved to estimate the fault information of actuators in MASs,and its observation curve for the signal is smoother.Secondly,the observer proposed in this paper has apparent advantages in rapidity and robustness.The reason for these advantages is that a time-varying gain is introduced into the observer,which enables the observer to estimate the unknown information quickly in finite time and avoids the problem of peak value when the initial error is too large.Finally,this paper designs a new distributed sliding mode fault-tolerant control protocol for MASs.The algorithm can ensure the finite-time consensus convergence of MASs in the case of failure.At the same time,the introduction of the integral item and hyperbolic tangent function has a good effect on reducing chatter.

6.Conclusions

This paper mainly focuses on the consensus problem of leader-following MASs.For the unmeasurable fault information in the system,a new STSMFO is designed.The observer can achieve fast estimation in finite time without the impact of the initial error problem and the peak problem.Then,an improved sliding mode fault-tolerant control protocol is proposed to achieve the finite-time convergence of consensus tracking error variables,which also means that MASs with actuator fault and inherent nonlinear dynamic behavior can achieve consensus.Subsequently,simulation results verify the advantages of the designed observer compared with the observer in existing literature,and further demonstrate the effectiveness of the proposed control protocol and its better chattering suppression performance.