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

Abstract

This paper deals with the asymptotic consensus problem for a class of multiagent systems with time-varying additive actuator faults and perturbed communications. The 𝐿₂ 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 𝐿₂ 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.

1. Introduction

The consensus behavior of multiagent systems has received significant attention over the last few years. It involves in lots of practical control systems and many physical phenomenon, such as synchronization of coupled oscillators [1], rendezvous in space [2], aircraft formation control [3], and flocking theory [4]. Recently, lots of researchers have focused on the development of methodologies to solve consensus problem with some usual issues existed in communications such as time-delays [5, 6], perturbations [7–15], and faulty communication links [16–18].

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 time-delays 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 [19–21]). 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 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 𝐻_∞, 𝐻₂, 𝐿₂ performances, are rarely considered in the consensus designs. In [13], the 𝐻_∞ consensus control problem was considered in directed networks of delayed and nondelayed agents using a matrix inequality method. The 𝐻_∞ 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 𝐻_∞ consensus for second-order multiagent systems with multiple asymmetric time-varying delays. Different from those papers using matrix inequality methods, this paper mainly considers the studies of 𝐿₂ 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 𝐿₂ 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.

2. Preliminaries and Problem Statement

In this paper, we consider a multiagent system 𝐺 composed of 𝑁 interconnected linear time-invariant continuous time agents 𝐺_𝑖, 𝑖=1,2,…,𝑁, which can be illustrated as an undirected graph. Each edge (𝑣_𝑗,𝑣_𝑖) corresponds to an available information link from agent 𝑖 to agent 𝑗. Then, the 𝑁 nodes constitute a network as the following state-space equation: 𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝐴₁₂𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)+𝑑_𝑖𝑗(𝑡)+𝐵₂𝑢_𝑖(𝑡)+𝑓_𝑖(𝑡)+𝐵₁𝑤_𝑖(𝑡), (1) where 𝑥_𝑖(𝑡)∈𝑅^𝑛 is the state of node 𝑖, and 𝑢_𝑖(𝑡)∈𝑅^𝑞 is the control input; 𝑎_𝑖𝑗∈𝑅 represents the topological structure of the network, which is an element of Laplacian matrix [24] satisfying 𝑎_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗, and 𝐴₁₂∈𝑅^𝑛×𝑛 is the inner coupling matrix describing the interconnections among components; 𝑑_𝑖𝑗(𝑡)∈𝑅^𝑛 is the network perturbation in communication between the 𝑖 agent and the 𝑗 agent, which is satisfied by 𝑑_𝑖𝑗≤𝑑_𝑖𝑗≤𝑑_𝑖𝑗, where 𝑑_𝑖𝑗 and 𝑑_𝑖𝑗 denote the unknown amplitude size of 𝑑_𝑖𝑗; 𝑓_𝑖(𝑡)∈𝑅^𝑞 denotes the additive actuator faults. Here, we assume that 𝑓_𝑖(𝑡) can be described by a nonlinear function and bounded by unknown lower and upper bounds 𝑓_𝑖 and 𝑓_𝑖; 𝑤_𝑖(𝑡)∈𝑅^𝑚 is a continuous vector function denoting external disturbance and possible variations with respect to the nominal parameter values for the system; 𝐴, 𝐵₁, and 𝐵₂ are real constant matrices with appropriate dimensions.

Then, by the Kronecker product, the multiagent system (1) can be rewritten as 𝑥̇(𝑡)=𝐼⊗𝐴+𝐿⊗𝐴₁₂𝑥(𝑡)+𝐼⊗𝐴₁₂𝑑(𝑡)+𝐼⊗𝐵₂𝑢(𝑡)+𝐼⊗𝐵₂𝑓(𝑡)+𝐼⊗𝐵₁𝑤(𝑡), (2) where 𝑥=(𝑥^𝑇₁,…,𝑥^𝑇_𝑁)^𝑇, 𝑢=(𝑢^𝑇₁,…,𝑢^𝑇_𝑁)^𝑇, 𝑓=(𝑓^𝑇₁,…,𝑓^𝑇_𝑁)^𝑇, 𝑤=(𝑤^𝑇₁,…,𝑤^𝑇_𝑁)^𝑇, 𝑑=(𝑑^𝑇₁,…,𝑑^𝑇_𝑁)^𝑇, 𝑑_𝑖=∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑑_𝑖𝑗, 𝑖=1,2,…,𝑁, and 𝐿 is called the graph Laplacian induced by the information flow 𝐺 and is defined by 𝑙_𝑖𝑗=−∑^𝑁_{𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘𝑎,𝑗≠𝑖,_𝑖𝑗,𝑗=𝑖. (3)

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 𝐾₁, there exist matrix functions 𝐻, 𝑍 of appropriate dimensions such that 𝐴₁₂+𝐵₂𝐾₁=𝐵₂𝐻,𝐴=𝐵₂𝑍, (4) respectively.
Define 1𝛼=𝑁^𝑁_𝑖=1𝑥_𝑖(0), (5) where 𝑥_𝑖(0), 𝑖=1,2,…,𝑁, 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: 𝑢_𝑖(𝑡)=−𝐾₁^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑑_𝑖𝑗+𝑐_𝑖𝑥_𝑖(𝑡)−𝛼+𝑣_𝑖(𝑡), (6) where 𝑖,𝑗=1,2,…,𝑁, 𝑎_𝑖𝑗 is defined as in (1), 𝑐_𝑖 is a positive constant, 𝐾₁ is the control gain, and 𝑣_𝑖(𝑡) is an adaptive control function which will be designed in later.
Then, by the Kronecker product, (6) can be rewritten as 𝑢(𝑡)=−𝐾₁(𝐿+𝐶)⊗𝐼𝑥(𝑡)+𝐾₁𝐶𝟏_𝑁⊗𝛼+𝐼⊗𝐾₁𝑑+𝑣, (7) where 𝑢=(𝑢^𝑇₁,…,𝑢^𝑇_𝑁)^𝑇, 𝑣=(𝑣^𝑇₁,…,𝑣^𝑇_𝑁)^𝑇, and 𝐶 is an 𝑁×𝑁 diagonal matrix whose 𝑖th diagonal element is 𝑐_𝑖, while 𝟏_𝑁 represents [1,1,…,1]^𝑇_𝑁.
Then, substituting (7) into (2), the closed-loop system model is stated as 𝑥̇(𝑡)=𝐼⊗𝐴+𝐿⊗𝐴₁₂𝑥(𝑡)−𝐼⊗𝐵₂𝐾₁()(𝑡)+𝐿+𝐶⊗𝐼𝑥𝐼⊗𝐵₂𝐾₁𝐶𝟏_𝑁+𝐴⊗𝛼𝐼⊗₁₂+𝐵₂𝐾₁𝑑(𝑡)+𝐼⊗𝐵₂𝑓(𝑡)+𝐼⊗𝐵₂𝑣(𝑡)+𝐼⊗𝐵₁𝑤(𝑡). (8)
Now, let 𝑥(𝑡)=𝑥(𝑡)−𝟏_𝑁𝛼. (9) Due to the characteristic of Laplacian matrix 𝐿, see (3), we have 𝐼⊗𝐵₂𝐾₁(𝐿+𝐶)⊗𝐼𝑥(𝑡)=𝐼⊗𝐵₂𝐾₁()(𝑡)−𝐿+𝐶⊗𝐼𝑥𝐼⊗𝐵₂𝐾₁𝐶𝟏_𝑁⊗𝛼(𝑡). (10)
Then, following (10) and substituting (9) into (8), the closed-loop system can be rewritten as 𝑥̇‌(𝑡)=𝐼⊗𝐴+𝐿⊗𝐴₁₂−𝐼⊗𝐵₂𝐾₁(𝐿+𝐶)⊗𝐼𝑥(𝑡)+𝐼⊗𝐵₂𝑣(𝑡)+𝐼⊗𝐴₁₂+𝐵₂𝐾₁𝑑(𝑡)+𝐼⊗𝐵₂𝑓(𝑡)+(𝐼⊗𝐴)𝛼+𝐼⊗𝐵₁𝑤(𝑡). (11)
According to Assumption 1, we have 𝑥̇‌(𝑡)=𝐼⊗𝐴+𝐿⊗𝐴₁₂−𝐼⊗𝐵₂𝐾₁(𝐿+𝐶)⊗𝐼𝑥(𝑡)+𝐼⊗𝐵₂𝑣(𝑡)+𝐼⊗𝐵₂𝑒(𝑡)+𝐼⊗𝐵₁𝑤(𝑡), (12) where  𝑒=(𝑒^𝑇₁,…,𝑒^𝑇_𝑁)^𝑇, 𝑒_𝑖(𝑡)=𝑓_𝑖(𝑡)+𝐻𝑑_𝑖(𝑡)+𝑍𝛼. Since 𝑓_𝑖(𝑡), 𝑑_𝑖(𝑡), and 𝛼 are bounded signals, we know that 𝑒_𝑖 is also a bounded signal and denote 𝑒_𝑖 and 𝑒_𝑖 as unknown larger constants than lower and upper bounds of 𝑒_𝑖, respectively.
Then, the objective of this paper is to make sure that system (12) is asymptotically stable; namely, lim_𝑡→∞𝑥(𝑡)=0, (13) that is, lim_𝑡→∞𝑥(𝑡)=𝟏_𝑁𝛼, (14) with any given 𝐿₂ performance index under the influence of perturbed communications and actuators.

3. 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 𝐿₂ performance criterion of the closed-loop system (12).

The composite sliding surface for the closed-loop system (12) is chosen as 𝑠(𝑥(𝑡))=0, (15) with 𝑠(𝑥(𝑡))≡∶(𝑠₁(𝑥₁(𝑡)),𝑠₂(𝑥₂(𝑡)),…,𝑠_𝑁(𝑥_𝑁(𝑡)))^𝑇 and 𝑠_𝑖𝑥_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥_𝑖𝑡₀−^𝑡_𝑡0𝐴𝑥_𝑖(𝜏)+^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝐴₁₂𝑥_𝑗(𝜏)−𝑥_𝑖(𝜏)−𝐵₂𝐾₁^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑥_𝑗(𝜏)−𝑥_𝑖(𝜏)+𝑐_𝑖𝑥_𝑖(𝜏)𝑑𝜏, (16) where 𝐾₁ is the controller gain proposed in (6) which is obtained by solving the following linear matrix inequality: 𝐼⊗𝐴+𝐿⊗𝐴₁₂−𝐵₂𝐾₁(𝐿+𝐶)⊗𝐼^𝑇()+()𝐼⊗𝑃𝐼⊗𝑃(𝐼⊗𝐴+𝐿⊗𝐴₁₂−𝐵₂𝐾₁(𝐿+𝐶)⊗𝐼<0, (17) where 𝑃 is a positive definite matrix. Note that the matrix 𝐾₁ 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 𝑥_𝑖(𝑡₀) achieves the nice property that 𝜎_𝑖(𝑥_𝑖(𝑡₀))=0 such that the reaching phase is eliminated.

Now, consider the controller model (6) with controller gain 𝐾₁ solved by (17). We design control function 𝑣_𝑖(𝑡) as follows: 𝑣_𝑖=𝜙_𝑖𝑠^𝑇_𝑖𝑃𝐵₂sgn𝑠^𝑇_𝑖𝑃𝐵₂^𝑇+𝐾₂𝐼−𝜌_𝑖𝑒_𝑖̂(𝑡)+𝜌_𝑖𝑒_𝑖̂(𝑡), (18) where 𝜙_𝑖 is an existed but unknown large enough positive constant satisfying 2𝜙_𝑖𝑠^𝑇_𝑖𝑃𝐵₂²≥𝑠^𝑇_𝑖𝑃𝐵₁²+𝛾_𝑖𝑠_𝑖², (19) where 𝛾_𝑖 is any given 𝐿₂ performance index, and 𝜙_𝑖 is the estimation of 𝜙_𝑖 updated by the following adaptive laws: 𝑑𝜙_𝑖(𝑡)𝑑𝑡=𝑟_𝑖𝑠^𝑇_𝑖𝑃𝐵₂², (20) where 𝑟_𝑖>0 is the weight of adaptive law 𝜙_𝑖(𝑡). The sign function sgn(𝑠^𝑇_𝑖𝑃𝐵₂)^𝑇=[sgn(𝑏_𝑖1),…,sgn(𝑏_𝑖𝑞)]^𝑇, where 𝑏_𝑖𝑙, 𝑙=1,2,…,𝑞, is the 𝑙 element of the vector 𝑠^𝑇_𝑖𝑃𝐵₂ and sgn(𝑏_𝑖𝑙) defined by sgn𝑏_𝑖𝑙=−1,if𝑏_𝑖𝑙>0,1,if𝑏_𝑖𝑙<0,0,if𝑏_𝑖𝑙=0, (21) and 𝑃 is a positive symmetric matrix designed in (17); 𝜌_𝑖 is the switching factor defined between constants 0 and 1 defined by 𝜌_𝑖=diag𝜌_𝑖1,𝜌_𝑖2,…,𝜌_𝑖𝑞,𝜌_𝑖𝑙=0,𝑏_𝑖𝑙≥0,1,𝑏_𝑖𝑙<0. (22)𝑒̂‌_𝑖𝑙(𝑡) and 𝑒̂_𝑖𝑙(𝑡) are the estimations of 𝑒_𝑖𝑙 and 𝑒_𝑖𝑙, respectively, updated by the following adaptive laws: 𝑑𝑒̂_𝑖𝑙(𝑡)𝑑𝑡=𝑡_𝑖𝑙𝑏_𝑖𝑙,𝑑𝑒̂_𝑖𝑙(𝑡)𝑑𝑡=𝑡_𝑖𝑙𝑏_𝑖𝑙, (23) where 𝑡_𝑖𝑙>0 are the adaptive law gains to be designed according to practical application.

Let 𝜙^∼_𝑖(𝑡)=𝜙_𝑖(𝑡)−𝜙_𝑖,_∼𝑒_𝑖(𝑡)=𝑒_𝑖̂(𝑡)−𝑒_𝑖,_∼𝑒_𝑖(𝑡)=𝑒_𝑖̂(𝑡)−𝑒_𝑖, (24) where 𝑒_𝑖=[𝑒_𝑖1,𝑒_𝑖2,…,𝑒_𝑖𝑛]^𝑇, 𝑒_𝑖=[𝑒_𝑖1,𝑒_𝑖2,…,𝑒_𝑖𝑛]^𝑇, 𝑖=1,2,…,𝑁.

Because to 𝜙_𝑖, 𝑒_𝑖, 𝑒_𝑖 are constants, the error system can be written as the following equations: ̇‌^∼𝜙_𝑖(𝑡)=𝜙̇_𝑖(𝑡),^∼̇𝑒_𝑖(𝑡)=𝑒_𝑖̇̂(𝑡),^∼̇𝑒_𝑖(𝑡)=𝑒_𝑖̇̂(𝑡). (25)

Thus, for the multiagent system described by (12), we propose the adaptive robust local control scheme (7) with the control gain function 𝐾₂ given by 𝐾₂=−𝐼. (26)

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 𝑢(𝑡) 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_𝑡→∞𝑥_𝑖(𝑡)=𝛼 with any given 𝐿₂ performance index 𝛾_𝑖 for any initial value 𝑥(𝑡₀), if there exists a symmetric matrix 𝑃>0 in (17).

Proof. For the adaptive robust closed-loop system described by (12), we first define a Lyapunov functional candidate as 𝑉(𝑡)=𝑠^𝑇(𝐼⊗𝑃)𝑠+^𝑁∼_𝑖=1𝜙²_𝑖𝑟_𝑖+^𝑁𝑛_𝑖=1𝑙=11−𝜌𝑒_𝑖𝑙^∼2_𝑖𝑙𝑡_𝑖𝑙+^𝑁𝑛_𝑖=1𝑙=1𝜌^∼_𝑖𝑙𝑒²_𝑖𝑙𝑡_𝑖𝑙. (27)
From (12) and (16), the derivative of 𝑠(𝑥(𝑡)) with respect to time can be calculated as follows: 𝑠̇(𝑥(𝑡=̇))𝑥(𝑡)−𝑥̇𝑡₀−𝐼⊗𝐴+𝐿⊗𝐴₁₂−𝐾₁(𝐿+𝐶)⊗𝐼𝑥(𝑡)=𝐼⊗𝐵₂sgn𝑠^𝑇𝜙_𝑠⊗𝑃𝐵₂^𝑇+𝐼⊗𝐵₂𝐼⊗𝐾₂𝐼⊗𝐼−𝜌𝑒̂(𝑡)+𝐼⊗𝜌𝑒̂(𝑡+)𝐼⊗𝐵₂𝑒(𝑡)+𝐼⊗𝐵₁𝑤(𝑡), (28) where 𝑒̂‌=(𝑒^𝑇₁̂,𝑒^𝑇₂̂,…,𝑒^𝑇_𝑁̂)^𝑇, 𝑒̂=(𝑒^𝑇₁̂,𝑒^𝑇₂̂,…,𝑒^𝑇_𝑁̂)^𝑇, 𝜌=diag[𝜌₁,𝜌₂,…,𝜌_𝑁], 𝜙_𝑠=diag[𝜙₁‖𝑠^𝑇₁𝑃𝐵₂‖,𝜙₂‖𝑠^𝑇₂𝑃𝐵₂‖,…,𝜙_𝑁‖𝑠^𝑇_𝑁𝑃𝐵₂‖].
Then, following Assumption 1, the time derivative of 𝑉(𝑡) for 𝑡>0 can be described as 𝑉̇(𝑡)=+2𝑠^𝑇(𝐼⊗𝑃)𝐼⊗𝐵₂sgn𝑠^𝑇𝜙_𝑠⊗𝑃𝐵₂^𝑇+2𝑠^𝑇(𝐼⊗𝑃)𝐼⊗𝐵₂𝑒+2𝑠^𝑇(𝐼⊗𝑃)𝐼⊗𝐵₁𝑤+2𝑠^𝑇(𝐼⊗𝑃)𝐼⊗𝐵₂𝐼⊗𝐾₂𝑒̂+𝜌𝑒̂−𝜌𝑒̂+^𝑁_𝑖=12𝜙^∼_𝑖̇‌^∼𝜙_𝑖𝑟_𝑖+^𝑁𝑛_𝑖=1𝑙=121−𝜌𝑒_𝑖𝑙^∼_𝑖𝑙̇‌^∼𝑒_𝑖𝑙𝑡_𝑖𝑙+^𝑁𝑞_𝑖=1𝑙=12𝜌^∼_𝑖𝑙𝑒_𝑖𝑙𝑒̇‌^∼_𝑖𝑙𝑡_𝑖𝑙=2^𝑁_𝑖=1𝜙_𝑖𝑠^𝑇_𝑖𝑃𝐵₂𝑠^𝑇_𝑖𝑃𝐵₂sgn𝑠^𝑇_𝑖𝑃𝐵₂^𝑇+2^𝑁_𝑖=1𝑠^𝑇_𝑖𝑃𝐵₁𝑤_𝑖+2𝑠^𝑇(𝐼⊗𝑃)^𝑁𝑞_𝑖=1𝑙=1𝐵_𝑖𝑒_𝑖𝑙+2𝑠^𝑇(𝐼⊗𝑃)^𝑁𝑞_𝑖=1𝑙=1𝐵_𝑖𝑘_2𝑙𝑒̂_𝑖𝑙+𝜌_𝑖𝑙𝑒̂_𝑖𝑙−𝜌_𝑖𝑙𝑒̂‌_𝑖𝑙+^𝑁_𝑖=12𝜙^∼_𝑖̇‌^∼𝜙_𝑖𝑟_𝑖+^𝑁𝑞_𝑖=1𝑙=121−𝜌𝑒_𝑖𝑙^∼_𝑖𝑙̇‌^∼𝑒_𝑖𝑙𝑡_𝑖𝑙+^𝑁𝑞_𝑖=1𝑙=12𝜌^∼_𝑖𝑙𝑒_𝑖𝑙𝑒̇‌^∼_𝑖𝑙𝑡_𝑖𝑙, (29) where 𝐵_𝑖 is the 𝑖th column of 𝐼⊗𝐵₂, 𝑖=1,2,…,𝑁.
Note that 𝑠^𝑇(𝐼⊗𝑃)^𝑁𝑞_𝑖=1𝑙=1𝐵_𝑖𝑒_𝑖𝑙≤𝑠^𝑇(𝐼⊗𝑃)^𝑁𝑞_𝑖=1𝑙=1𝐵_𝑖𝑒_𝑖𝑙+𝜌_𝑖𝑙𝑒_𝑖𝑙−𝜌_𝑖𝑙𝑒_𝑖𝑙, (30) where 𝜌_𝑖𝑙 are denoted in (22). By the adaptive laws chosen in (23) and controller gain function chosen in (26), then (29) can be rewritten as 𝑉̇(𝑡)≤2^𝑁_𝑖=1𝜙_𝑖𝑠^𝑇_𝑖𝑃𝐵₂𝑠^𝑇_𝑖𝑃𝐵₂sgn𝑠^𝑇_𝑖𝑃𝐵₂^𝑇+2^𝑁_𝑖=1𝑠^𝑇_𝑖𝑃𝐵₁𝑤_𝑖+2𝑠^𝑇()×𝐼⊗𝑃_{𝑁𝑞𝑖=1𝑙=1}𝐵_𝑖𝑒_𝑖𝑙+𝜌_𝑖𝑙𝑒_𝑖𝑙−𝜌_𝑖𝑙𝑒_𝑖𝑙+2𝑠^𝑇()×𝐼⊗𝑃_{𝑁𝑞𝑖=1𝑙=1}𝐵_𝑖𝑘_2𝑙𝑒̂_𝑖𝑙+𝜌_𝑖𝑙𝑒̂_𝑖𝑙−𝜌_𝑖𝑙𝑒̂‌_𝑖𝑙+^𝑁_𝑖=12𝜙^∼_𝑖̇‌^∼𝜙_𝑖𝑟_𝑖+^𝑁𝑞_𝑖=1𝑙=121−𝜌𝑒_𝑖𝑙^∼_𝑖𝑙̇‌^∼𝑒_𝑖𝑙𝑡_𝑖𝑙+^𝑁𝑞_𝑖=1𝑙=12𝜌^∼_𝑖𝑙𝑒_𝑖𝑙𝑒̇‌^∼_𝑖𝑙𝑡_𝑖𝑙≤2^𝑁_𝑖=1𝜙_𝑖𝑠^𝑇_𝑖𝑃𝐵₂𝑠^𝑇_𝑖𝑃𝐵₂sgn𝑠^𝑇_𝑖𝑃𝐵₂^𝑇+2^𝑁_𝑖=1𝑠^𝑇_𝑖𝑃𝐵₁𝑤_𝑖+^𝑁_𝑖=12𝜙^∼_𝑖̇‌^∼𝜙_𝑖𝑟_𝑖. (31)
From the fact that 𝑠^𝑇_𝑖𝑃𝐵₂sgn(𝑠^𝑇_𝑖𝑃𝐵₂)^𝑇≤−‖𝑠^𝑇_𝑖𝑃𝐵₂‖ and the adaptive law proposed in (20) with inequality (19), then we have 𝑉̇(𝑡)≤−2^𝑁_𝑖=1𝜙_𝑖𝑠^𝑇_𝑖𝑃𝐵₂²+2^𝑁_𝑖=1𝑠^𝑇_𝑖𝑃𝐵₁𝑤_𝑖+^𝑁_𝑖=12𝜙^∼_𝑖̇‌^∼𝜙_𝑖𝑟_𝑖≤−^𝑁_𝑖=12𝜙_𝑖𝑠^𝑇_𝑖𝑃𝐵₂²−𝑠^𝑇_𝑖𝑃𝐵₁²−𝛾_𝑖𝑠_𝑖²+^𝑁_𝑖=1𝑤_𝑖²−𝛾_𝑖𝑠_𝑖²+^𝑁_𝑖=12𝜙^∼_𝑖̇‌^∼𝜙_𝑖𝑟_𝑖=^𝑁_𝑖=1𝑤_𝑖²−𝛾_𝑖𝑠_𝑖². (32)
Obviously, we have 𝑉̇(𝑡)<0 if ‖𝑠_𝑖‖>‖𝑤_𝑖‖/√𝛾_𝑖. It means that the sliding manifold signal is uniformly ultimately bounded by lim_𝑡→∞‖𝑠_𝑖‖∈{𝑠_𝑖∈𝑅^𝑛∣‖𝑠_𝑖‖≥‖𝑤_𝑖‖/√𝛾_𝑖}.
When inequality (32) is integrated over the interval [𝑡₀,𝑡], we obtain 𝑉(𝑡)−𝑉𝑡₀≤^𝑁_𝑖=1−^𝑡_𝑡0𝛾_𝑖𝑠_𝑖(𝜏)²𝑑(𝜏)+^𝑡_𝑡0𝑤_𝑖(𝜏)²𝑑(𝜏). (33)
Then, we have ^𝑁_𝑖=1^𝑡_𝑡0𝛾_𝑖𝑠_𝑖(𝜏)²𝑑(𝜏)≤𝑉𝑡₀+^𝑁_𝑖=1^𝑡_𝑡0𝑤_𝑖(𝜏)²𝑑(𝜏), (34) where 𝑉(𝑡₀)=𝑠(𝑡₀)^𝑇(𝐼⊗𝑃)𝑠(𝑡₀)+∑^𝑁_𝑖=1(𝜙^∼2_𝑖(𝑡₀)/𝑟_𝑖)+∑^𝑁_𝑖=1∑^𝑛_𝑙=1((1−𝜌_𝑖𝑙)𝑒^∼2_𝑖𝑙(𝑡₀)/𝑡_𝑖𝑙)+∑^𝑁_𝑖=1∑^𝑛_𝑙=1(𝜌^∼_𝑖𝑙𝑒²_𝑖𝑙(𝑡₀)/𝑡_𝑖𝑙).
If the initial conditions are chosen to be zero, then the 𝐿₂ gain becomes clear such that 𝛾_𝑖^𝑡_𝑡0𝑠_𝑖(𝜏)²𝑑(𝜏)≤^𝑡_𝑡0𝑤_𝑖(𝜏)²𝑑(𝜏)+^∼𝜙²_𝑖𝑡₀𝑟_𝑖+^𝑛_𝑙=11−𝜌𝑒_𝑖𝑙^∼2_𝑖𝑙𝑡₀𝑡_𝑖𝑙+^𝑛_𝑙=1𝜌^∼_𝑖𝑙𝑒²_𝑖𝑙𝑡₀𝑡_𝑖𝑙. (35)
Hence, it is easy to see that 𝑉̇(𝑡)<0 for any 𝑥≠0. Thus, the solutions of closed-loop system are uniformly bounded, and the error 𝑥(𝑡) converges asymptotically to zero. Moreover, from (35) and the results of [25], the 𝐿₂ gain level of the disturbance attenuation can be guaranteed to be a given small value by adjusting 𝛾_𝑖.

Remark 3. The first part of control function 𝑣_𝑖(𝑡) in (18) is designed for formulating the 𝐿₂ performance index 𝛾_𝑖. The second part of 𝑣_𝑖(𝑡) 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 time-varying object by the proposed method. Here, we define that 𝛼(𝑡)∈𝑅^𝑛 is the given command of states in time 𝑡 for the agents. Without loss of generality, we assume that the command is a differentiable continuous signal which satisfies 𝛼̇(𝑡)=𝐶𝑔(𝑡), (36) where 𝑔(𝑡) is bounded by an unknown positive constant 𝑔₀ such that ‖𝑔(𝑡)‖≤𝑔₀, and 𝐶 is a real constant matrix with appropriate dimensions and satisfies the condition 𝐶=𝐵₂𝑀.
Without considering 𝑤_𝑖(𝑡), then the closed-loop system (12) can be rewritten as 𝑥̇‌(𝑡)=𝐼⊗𝐴+𝐿⊗𝐴₁₂−𝐼⊗𝐵₂𝐾₁(𝐿+𝐶)⊗𝐼𝑥(𝑡)+𝐼⊗𝐵₂𝑣(𝑡)+𝐼⊗𝐵₂𝑒(𝑡), (37) where 𝑒=(𝑒^𝑇₁,…,𝑒^𝑇_𝑁)^𝑇, 𝑒_𝑖(𝑡)=𝑓_𝑖(𝑡)+𝐻𝑑_𝑖(𝑡)+𝑀𝑔(𝑡)+𝑍𝛼(𝑡). Since 𝑓_𝑖(𝑡), 𝑑_𝑖(𝑡), 𝑔(𝑡), and 𝛼(𝑡) are bounded signals, we know that 𝑒_𝑖 is still a bounded signal.
Then, the following corollary is proposed to obtain asymptotic consensus results of perturbed communications and actuators.

Corollary 4. Suppose that Assumption 1 holds. Consider the closed-loop multiagent system described by (37). Then, by using the control scheme 𝑢(𝑡) described by 𝑢_𝑖(𝑡)=−𝐾₁^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑑_𝑖𝑗+𝑐_𝑖𝑥_𝑖(𝑡)−𝛼+𝐾₂𝐼−𝜌_𝑖𝑒_𝑖̂(𝑡)+𝜌_𝑖𝑒_𝑖̂(𝑡) (38) with adaptive laws (23) and control gain functions (26), one can guarantee that all closed-loop multiagent system signals are bounded and lim_𝑡→∞𝑥_𝑖(𝑡)=𝛼(𝑡) for any initial value 𝑥(0), if there exists a symmetric matrix 𝑃>0 in (17).

Proof. Similar to the proof of Theorem 2 and according to (30), we have 𝑠^𝑇(𝐼⊗𝑃)∑^𝑁_𝑖=1∑^𝑞_𝑙=1𝐵_𝑖ℎ_𝑙𝑒_𝑖𝑙=𝑠^𝑇(𝐼⊗𝑃)∑^𝑁_𝑖=1∑^𝑞_𝑙=1𝐵_𝑖ℎ_𝑙(𝑒_𝑖𝑙+𝜌_𝑖𝑙𝑒_𝑖𝑙−𝜌_𝑖𝑙𝑒_𝑖𝑙) if and only if 𝑠_𝑖=0. Thus, it is obvious that 𝑉̇=0 if and only if 𝑠_𝑖=0, 𝑖=1,2,…,𝑁, and a set 𝐸 can be found as 𝐸={(𝑠,𝑒,̂𝑒)̂∶𝑉̇=0}={(𝑠,𝑒,̂𝑒)̂∶𝑠_𝑖=0}, where 𝑠=(𝑠₁,𝑠₂,…,𝑠_𝑁)^𝑇, 𝑒̂=(𝑒₁̂,𝑒₂̂,…,𝑒_𝑁̂)^𝑇, and 𝑒̂‌=(𝑒₁̂,𝑒₂̂,…,𝑒_𝑁̂)^𝑇. Starting with arbitrary initial values 𝑠_𝑖(0), 𝑒_𝑖̂(0), and 𝑒_𝑖̂‌(0), the orbit converges asymptotically to 𝑠_𝑖=0, 𝑒_𝑖̂=𝑒_𝑖, and 𝑒_𝑖̂‌=𝑒_𝑖, where 𝑒_𝑖, 𝑒_𝑖 are constants. Based on LaSalle invariance principle, the system trajectories converge to the largest positively invariant subset 𝑀={(𝑠,𝑒,̂𝑒)̂∶𝑠=0,𝑒̇̂=0,𝑒̇̂=0}. Moreover, according to (12) and (28), we know that the error 𝑠_𝑖(𝑡) converges asymptotically to zero if there exist 𝐾₁ and 𝑃 making the inequality (17) hold true.

4. 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: 𝐴=−10.210.1,𝐵₂=,𝐴2−1.51−1₁₂=0.4000.4,𝐵₁=,1.51−2−1 (39) and the topological structure matrix 𝐿 in (3) described by 𝐿=−10.20.50.30.1−10.60.30.20.5−10.30.10.30.6−1. (40)

To verify the effectiveness of the proposed adaptive sliding mode method, the simulations are given with the following parameters and initial conditions: 𝑟_𝑖=100,𝑡_𝑖𝑙=100,𝑒̂_𝑖𝑙(0)𝑒̂=10,_𝑖𝑙(0)=−10,𝑐_𝑖𝑥=1,₁(0)=[2,−1]^𝑇,𝑥₂(0)=[1,−0.5]^𝑇,𝑥₃(0)=[0.5,−1.5]^𝑇,𝑥₄(0)=[0.5,−3]^𝑇,𝑖=1,2,3,4,𝑙=1,2. (41)

Then, following the system state initial values, we get the average value 𝛼=[1,−1.5]^𝑇.

Then, solving LMI (17), we can obtain 𝑃=0.1260−0.0156−0.01560.0262,𝐾₁=1.8286−2.74291.8286−3.6571. (42)

The following case is considered in the simulations; that is, the communication perturbations 𝑑₁₂(𝑡)=[−0.5,0.2sin(2𝑡)]^𝑇, 𝑑₂₃(𝑡)=[0.4sin(0.2𝑡),−0.5]^𝑇, 𝑑₃₄(𝑡)=[0.5,0.5cos(2𝑡)]^𝑇, and 𝑑₄₁(𝑡)=[−0.2,0.3sin(20𝑡)]^𝑇 and the additive actuator faults 𝑓₁(𝑡)=[0.5,0.5cos(0.5𝑡)]^𝑇, 𝑓₂(𝑡)=[−0.5,0.4sin(20𝑡)]^𝑇, 𝑓₃(𝑡)=[0.5,−0.4sin(2𝑡)]^𝑇, and 𝑓₄(𝑡)=[0.5sin(𝑡),−1.5]^𝑇 enter into the systems at the beginning (𝑡≥0). Moreover, the external disturbances 𝑤₁(𝑡)=[−5,2sin(2𝑡)]^𝑇, 𝑤2(𝑡)=[4sin(0.2𝑡),−5]^𝑇, 𝑤₃(𝑡)=[5,5cos(2𝑡)]^𝑇, and 𝑤₄(𝑡)=[−2,3sin(20𝑡)]^𝑇 enter into the systems in 𝑡∈[0.8,0.9].

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 𝑒_𝑖̂‌ and 𝑒_𝑖̂, 𝑖=1,2,3,4, respectively. Figure 4 is the response curves of the estimate of controller gain 𝜙_𝑖, 𝑖=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  𝑒_𝑖̂‌, 𝑒_𝑖̂, 𝜙_𝑖 to converge to their true values.

Figure 1 

Response curves of the system states 𝑥𝑖(𝑡), 𝑖=1,2,3,4.

Figure 2 

Response curves of the consensus errors 𝑥𝑖(𝑡), 𝑖=1,2,3,4.

(a)

(b)

Figure 3 

Response curves of the estimate of upper bound and lower bound of 𝑒𝑖(𝑡), 𝑖=1,2,3,4.

Figure 4 

Response curves of the estimate of 𝜙𝑖, 𝑖=1,2,3,4.

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 𝛼̇(𝑡)=0.5𝐵₂[sin(𝑡)cos(𝑡)]^𝑇 with initial values [1,−1.5]^𝑇. 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.

Figure 5 

Response curves of the consensus errors 𝑥𝑖(𝑡) with varying target; 𝑖=1,2,3,4.

5. 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 𝐿₂ 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 𝐿₂ 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.

Acknowledgments

This work is supported by the National Nature Science Foundation (Grant nos. 61104029, 61273155, and 61203087), the Science and Technology Plan of Liaoning province (2011219011), the Natural Science Foundation of Liaoning Province (Grant no. 201202156), and the Scientific Research Foundation for Doctor of Liaoning Province (Grant no. 20121040).