Robust Adaptive Sliding Mode Consensus of Multiagent Systems with Perturbed Communications and Actuators

This paper deals with the asymptotic consensus problem for a class of multiagent systems with time-varying additive actuator faults and perturbed communications. The L 2 performance of systems is also considered in the consensus controller designs. The upper and lower bounds of faults and perturbations in actuators and communications and controller gains are assumed to be unknown but can be estimated by designing some indirect adaptive laws. Based on the information from the adaptive estimation mechanism, the distributed robust adaptive sliding mode controllers are constructed to automatically compensate for the effects of faults and perturbations and to achieve any given level of L 2 gain attenuation from external disturbance to consensus errors. Through Lyapunov functions and adaptive schemes, the asymptotic consensus of resulting adaptive multiagent system can be achieved with a specified performance criterion in the presence of perturbed communications and actuators. The effectiveness of the proposed design is illustrated via a decoupled longitudinal model of F-18 aircraft.

As we know, state consensus may not be guaranteed under the influence of network perturbations.In [7], the bounded tracking results were obtained for multiagent consensus with an active leader in the presence of bounded perturbations.A stochastic model for distributed average consensus with zero mean noise was presented in [8].Recently, the authors of [9] utilized an adaptive method to deal with a class of known upper bound disturbances.In addition, from the Laplacian eigenvalue standpoint, paper [10] focused on maximizing the second smallest eigenvalue of a state-dependent graph Laplacian to improve the robustness of the dynamic systems.The recent paper [11] gave a lazy consensus protocol against unknown but bounded disturbances and achieved bounded consensus results.Note that the aforementioned systems cannot guarantee asymptotic consensus when the disturbance always exists in the systems.Therefore, the capability of disturbance rejection for the previous systems seems weak.Recently, an asymptotic consensus result has been obtained in [12] under a matching condition in the presence of timedelays and disturbances in communications.But the design is complicated and the performances of systems are not considered in the paper.
It is well known that actuators play an important role in the consensus of multiagent systems.Many researchers were devoted to the study of fault-tolerant control against actuator/sensor faults to ensure the stability and performance optimization of systems (e.g., see [19][20][21]).Paper [17] considered consensus problem with missing data in actuators and Markovian communication failure, and the communication failure process was reduced to a Bernoulli process.The recent paper [22] studied fault diagnosis for a class of discrete time-delayed complex interconnected networks with linear Mathematical Problems in Engineering coupling in the case of actuator faults.The authors of [23] analyzed the performances of a team of unmanned vehicles with some actuator fault types, such that loss of effectiveness and lock-in-place.In [18], the bias actuators have been considered in synchronization of master-slave systems using an indirect adaptive method.Here, the similar additive actuator faults which can also be considered as perturbed actuators are treated by using an adaptive sidling mode method.
It should be noted that the system performances, such as  ∞ ,  2 ,  2 performances, are rarely considered in the consensus designs.In [13], the  ∞ consensus control problem was considered in directed networks of delayed and nondelayed agents using a matrix inequality method.The  ∞ consensus filtering problem was dealt with in [14] by a difference linear matrix inequalities over a finite-horizon for sensor networks with multiple missing measurements.The authors of [15] utilized a model transformation approach and matrix theory to solve the  ∞ consensus for secondorder multiagent systems with multiple asymmetric timevarying delays.Different from those papers using matrix inequality methods, this paper mainly considers the studies of  2 performance of multiagent systems by an adaptive method.
In this paper, we consider the consensus problem of multiagent systems in the presence of perturbed communications and actuators with a specified performance criterion.Here, the bounds of additive faults and the size of perturbations in communications are not necessary to be known.Based on the Lyapunov stability theory, a novel adaptive sliding mode control strategy is developed to achieve asymptotic consensus of the multiagent systems.On the basis of this proposed method, an integral sliding manifold is developed for average consensus and varying consensus.Some adaptive schemes are proposed to estimate the bounds of faults and perturbations and controller gains.Then, adaptive sliding mode controllers are constructed relying on the updated gains.By using the designed controller, the faulty and perturbed factors effects can be completely compensated and the asymptotic consensus can be achieved in the finite time with any given level of  2 gain attenuation.Besides, it should be noted that the new proposed adaptive design method is not necessary for the estimations to give the exact information.
The asymptotic consensus problem formulation is described in Section 2. In Section 3, the adaptive sidling mode state feedback controllers are developed.Section 4 gives an example and simulation.Finally, conclusions are given in Section 5.

Preliminaries and Problem Statement
In this paper, we consider a multiagent system  composed of  interconnected linear time-invariant continuous time agents   ,  = 1, 2, . . ., , which can be illustrated as an undirected graph.Each edge (V  , V  ) corresponds to an available information link from agent  to agent .Then, the  nodes constitute a network as the following state-space equation: where   () ∈   is the state of node , and   () ∈   is the control input;   ∈  represents the topological structure of the network, which is an element of Laplacian matrix [24] satisfying Then, by the Kronecker product, the multiagent system (1) can be rewritten as where  = ( Similar to [12], to ensure the achievement of average consensus objective, the following assumption in consensus design is also assumed to be valid.Assumption 1.For multiagent system (2) and any appropriate dimension matrix  1 , there exist matrix functions ,  of appropriate dimensions such that respectively.Define where   (0),  = 1, 2, . . ., , are the known initial values of system states.Then, for the sake of solving the consensus problems of multiagent systems in the presence of external faults, the local consensus protocol is considered as follows: where ,  = 1, 2, . . ., ,   is defined as in (1),   is a positive constant,  1 is the control gain, and V  () is an adaptive control function which will be designed in later.
Then, the objective of this paper is to make sure that system (12) is asymptotically stable; namely, lim that is, lim with any given  2 performance index under the influence of perturbed communications and actuators.

Main Results
In this section, we develop the adaptive laws to estimate unknown controller gains for designing robust adaptive sliding mode controllers to eliminate the effects of communication perturbations and actuator faults and, simultaneously, to achieve any given  2 performance criterion of the closedloop system (12).The composite sliding surface for the closed-loop system ( 12) is chosen as with (()) ≡: ( 1 ( 1 ()),  2 ( 2 ()), . . .,   (  ()))  and where  1 is the controller gain proposed in (6) which is obtained by solving the following linear matrix inequality: where  is a positive definite matrix.Note that the matrix  1 is designed such that the nominal fault-free system ( 12) is stable and some prescribed specifications would also be satisfied via this nominal state feedback control.Here, the term of   ( 0 ) achieves the nice property that   (  ( 0 )) = 0 such that the reaching phase is eliminated.Now, consider the controller model ( 6) with controller gain  1 solved by (17).We design control function V  () as follows: where   is an existed but unknown large enough positive constant satisfying where   is any given  2 performance index, and φ is the estimation of   updated by the following adaptive laws: where   > 0 is the weight of adaptive law φ ().The sign function sgn(    2 )  = [sgn( 1 ), . . ., sgn(  )]  , where   ,  = 1, 2, . . ., , is the  element of the vector     2 and sgn(  ) defined by sgn and  is a positive symmetric matrix designed in (17);   is the switching factor defined between constants 0 and 1 defined by ê () and ê () are the estimations of   and   , respectively, updated by the following adaptive laws: where   > 0 are the adaptive law gains to be designed according to practical application.Let where   = [ 1 ,  2 , . . .,   ]  ,   = [ 1 ,  2 , . . .,   ]  ,  = 1, 2, . . ., .
Because to   ,   ,   are constants, the error system can be written as the following equations: Thus, for the multiagent system described by (12), we propose the adaptive robust local control scheme (7) with the control gain function  2 given by Hence, the following theorem can be obtained, which shows the uniform ultimate boundedness of the closed-loop system (12) and the error system (25).
Theorem 2. Consider the closed-loop multiagent system described by (12) satisfying Assumption 1.By using the control scheme () described in (7) with adaptive laws (20) and (23) and control gain functions (26), one can guarantee that all closed-loop system signals are bounded and lim  → ∞   () =  with any given  2 performance index   for any initial value ( 0 ), if there exists a symmetric matrix  > 0 in (17).
Proof.For the adaptive robust closed-loop system described by (12), we first define a Lyapunov functional candidate as From ( 12) and ( 16), the derivative of (()) with respect to time can be calculated as follows: Then, following Assumption 1, the time derivative of () for  > 0 can be described as where   is the th column of  ⊗  2 ,  = 1, 2, . . ., .Note that where   are denoted in (22).By the adaptive laws chosen in (23) When inequality (32) is integrated over the interval [ 0 , ], we obtain Then, we have where If the initial conditions are chosen to be zero, then the  2 gain becomes clear such that Hence, it is easy to see that V() < 0 for any  ̸ = 0. Thus, the solutions of closed-loop system are uniformly bounded, and the error () converges asymptotically to zero.Moreover, from (35) and the results of [25], the  2 gain level of the disturbance attenuation can be guaranteed to be a given small value by adjusting   .
Remark 3. The first part of control function V  () in ( 18) is designed for formulating the  2 performance index   .The second part of V  () is constructed for eliminating the effects of the matched perturbations of actuators and communications.It should be noted that the method largely simplifies the designs in [12].
The average consensus problem has been solved in Theorem 2. Actually, the system can also track the timevarying object by the proposed method.Here, we define that () ∈   is the given command of states in time  for the agents.Without loss of generality, we assume that the command is a differentiable continuous signal which satisfies where () is bounded by an unknown positive constant  0 such that ‖()‖ ≤  0 , and  is a real constant matrix with appropriate dimensions and satisfies the condition  =  2 .Without considering   (), then the closed-loop system (12) can be rewritten as where  = (  1 , . . .,    )  ,   () =   ()+  ()+()+().Since   (),   (), (), and () are bounded signals, we know that   is still a bounded signal.

Numerical Example
In this section, an example of robust consensus control system design is given to demonstrate the proposed method.A multiagent system is composed of four dynamical agents, which have the same system matrices as follows: and the topological structure matrix  in  To verify the effectiveness of the proposed adaptive sliding mode method, the simulations are given with the following parameters and initial conditions: Then, following the system state initial values, we get the average value  = [1, −  Figures 1 and 2 are, respectively, the response curves of the system's states and consensus errors with adaptive robust feedback control scheme in aforementioned case.Figure 3 describes the estimate of upper and lower bounds of perturbations ê and ê ,  = 1, 2, 3, 4, respectively.Figure 4 is the response curves of the estimate of controller gain   ,  = 1, 2, 3, 4. It is easy to see that the estimations can convergence but not converge to theirs true values.In our adaptive robust fault-tolerant control design, there is no need for the estimate of ê , ê , φ to converge to their true values.
Furthermore, the proposed method can also deal with the time-varying consensus problem.Here, we consider the previous system with the same initial conditions and perturbations and faults, but without considering the effects of external disturbances.The consensus target is described by α () = 0.5 2 [sin() cos()]  with initial values [1, −1.5]  .Then, Figure 5 illustrates the response curves of the consensus errors with adaptive sidling mode control scheme (38).The results indicated that the proposed method can also solve the time-varying consensus problem with some matched condition.

Conclusions
In this paper, we have shown an adaptive design method to solve the robust asymptotic consensus problem for a class of multiagent systems with perturbed communications and faulted actuators.For the sake of automatically compensating for the effects of faults and networked perturbations and specifying  2 performance criterion, the consensus protocol is constructed with the adaptive schemes, which are based on the updated adaptation laws to estimate the unknown bounds of perturbations and controller gains online.On the basis of Lyapunov stability theory, it has been shown that the resulting adaptive closed-loop multiagent system can be guaranteed to be asymptotic average and varying consensus with any given  2 performance index even in the presence of imperfect communications and actuators.A numerical example has been given to illustrate the effectiveness of the proposed method.

𝜙 1
The estimate of  2The estimate of  3 The estimate of  4
and  12 ∈  × is the inner coupling matrix describing the interconnections among components;   () ∈   is the network perturbation in communication between the  agent and the  agent, which is satisfied by   ≤   ≤   , where   and   denote the unknown amplitude size of   ;   () ∈   denotes the additive actuator faults.Here, we assume that   () can be described by a nonlinear function and bounded by unknown lower and upper bounds   and   ;   () ∈   is a continuous vector function denoting external disturbance and possible variations with respect to the nominal parameter values for the system; ,  1 , and  2 are real constant matrices with appropriate dimensions.