Decentralized Output Sliding-Mode Fault-Tolerant Control for Heterogeneous Multiagent Systems

This paper proposes a novel decentralized output sliding-mode fault-tolerant control (FTC) design for heterogeneous multiagent systems (MASs) with matched disturbances, unmatched nonlinear interactions, and actuator faults. The respective iteration and iteration-free algorithms in the sliding-mode FTC scheme are designed with adaptive upper bounding laws to automatically compensate the matched and unmatched components. Then, a continuous fault-tolerant protocol in the observer-based integral sliding-mode design is developed to guarantee the asymptotic stability of MASs and the ultimate boundedness of the estimation errors. Simulation results validate the efficiency of the proposed FTC algorithm.


I. INTRODUCTION
M ULTIAGENT systems (MASs) have attracted considerable attention in diverse fields, such as aerospace, transportation, wireless networks, and power systems [1], [2]. Heterogeneous MASs (HMASs) are MASs composed of a large number of nonidentical agents that are connected by mechanical interconnections or communication networks [3]- [6]. A decentralized control is equipped with a simpler architecture with local information than a single centralized control and, thus, is more practical to realize on physical HMASs [7]- [9]. One essential issue of the decentralized control for HMASs involves the interaction of different agents [10]. Thus, remarkable benefits can be obtained by specifying the decentralized control concept for HMASs with interactions and subsequently achieve the satisfactory local performance of each agent and the global property of the overall HMASs.
Faults may occur more frequently in HMASs than in single agents due to the existence of a number of controllers, sensors, and interconnected equipment [11], [12]. Therefore, HMASs are required to operate safely, and the fault-tolerant control (FTC) is regarded as an effective approach to guarantee the stability and desired performance of HMASs with unpredicted faults. An active FTC scheme was presented with a high-gain observer for high-order HMASs in the presence of actuator faults and communication disconnections [13]. An FTC design with fault detection and recovery mechanism was developed for stabilization and navigation of heterogeneous multiagent formations of autonomous aerial and ground robots [14]. However, most existing studies of HMASs in dealing with faults have centralized or distributed configurations. The centralized and distributed schemes are easy to implement, but both rely on information sharing and transmission among agents [15]- [19]. A decentralized FTC (DFTC) design should be able to automatically compensate the effects of faults without necessarily exchanging information between individual agents [20]. Various DFTC approaches, such as adaptive DFTC [21], [22]; fuzzy DFTC [23], [24]; and neural-networkbased DFTC [25], [26], have been introduced by research in recent years. Most of the results are based on either the state [7], [23], [27] or output feedbacks [4], [8], [13], [25]. The literature indicates that the research on local output DFTC for HMASs has received minimal attention thus far. Furthermore, the DFTC protocol in [20] is usually effective for HMASs with weak interactions and small couplings, and a more general assumption is needed in this paper compared with the strong interconnections [26], [28]. In addition, the existing disturbances and actuator faults in single agents are easy to spread in HMASs. Sliding-mode control (SMC) has been widely applied in handling uncertainties and improving the robustness of HMASs [29]- [31]. A continuous SMC method with an adaptive strategy was designed for second-order nonlinear MASs with actuator faults and disturbances to realize the consensus-tracking objective [30]. An SMC design with a heterogeneous finite-time disturbance observer was proposed for high-order HMASs to attenuate the effect of uncertainties [31]. On the one hand, most results on SMC for HMASs focused on matched disturbances in the input channel and did not consider the unmatched one, such as flexible joint manipulators and multimachine power systems [9], which may not always satisfy the matching condition [24], [32]. On the other hand, the results on the decentralized output sliding-mode FTC (SM-FTC) of coupled nonlinear HMASs with matched disturbances, unmatched nonlinear interactions, and actuator faults are dearth, hence the motivation of this investigation.
To tackle the aforementioned difficulty, the decentralized output SM-FTC scheme for HMASs is designed with adaptive upper bounding laws. The matched disturbances, unmatched interactions, and actuator faults are effectively compensated, and then the robust performance of the SM-FTC strategy is further achieved by rigorously performing a stability analysis of HMASs and conducting a reachability analysis of the SMC motion. To the best of our knowledge, integral SMC can be adopted in FTC schemes for various complex MASs [33]- [35]. Compared with the conventional SMC method that partitions the state space into matched and unmatched parts [29], [36], the matched disturbances and actuator faults are compensated in the integral SMC strategy [37]. An integral SMC scheme was presented for HMASs composed of quadrotors and two-wheeled mobile robots associated with model uncertainties and external disturbances to reach a consensus [34]. Actuator faults and external disturbances/model uncertainties in a group of nonlinear systems were tolerated by using the integral SMC technique [35]. More important, there is little literature on the subject of observer-based consensus for MASs [38], [39]. The distributed pinning observer [40] and the finite-time observer [41] in the consensus control strategy are designed to estimate the state information of each single agent. In this case, further introducing the observer-based integral SMC into the decentralized output SM-FTC strategy for HMASs can lead to complex research and analysis.
The major contributions of this paper can be summarized as follows.
1) This paper is the first trial to consider the output DFTC problem for HMASs with actuator faults and nonlinear interactions, which contain unmatched components, in contrast with the existing work that focuses mostly on matched model uncertainties and external disturbances [26], [30], [34]. To overcome this difficulty, an augmented dynamic configuration is constructed with the dynamic compensator to satisfy the Kimura-Davison condition [42], and the decentralized output SM-FTC technique is introduced. 2) In comparison with the adaptive bounds [21], prescribed performance bounds [22], and Lipschitz boundedness of the disturbances, model uncertainties, and nonlinearities [13], [19], [23], [35], the adaptive law is used in this paper to relax the known upper bound assumptions of the matched disturbances and actuator faults. 3) Two multistep algorithms based on the extended linear matrix inequality (LMI) characterization are given to reduce the conservativeness of the solution of the nonlinear matrix inequality with nonconvex algebraic constraints by implementing iteration and iteration-free strategies. The remainder of this paper is organized as follows. Section II introduces the system formulation. Section III is devoted to the SMC design with stability analysis of HMASs and reachability analysis of the SMC motion. The iteration and iteration-free algorithms are further illustrated. The observerbased integral SMC design is proposed in Section IV to guarantee the ultimately uniformly boundedness of the estimation errors. Simulations in Section V validate the efficiency of the proposed algorithm. Finally, the conclusions follow in Section VI. The symbol sgn(·) denotes the sign function, the vector x = col( He(X) = X+X T and denotes the symmetric part of the specific matrix.

II. SYSTEM FORMULATION
Consider a group of N agents in the presence of matched disturbances, unmatched nonlinear interactions, and actuator faults. The ith agent of HMASs (i = 1, . . . , N) is given aṡ where x i ∈ R n i , u i ∈ R m i , and y i ∈ R p i are the system state, and input and output vectors, respectively; ≤d i with the unknown upper boundd i . Assumption 2: The nonlinear interaction term ξ i (x, t) in HMASs is satisfied with the quadratic constraint, that is, where E i is a known constant matrix and α i is a known upper bound scalar.
Remark 1: Assumption 1 provides the controllable and observable conditions for the described HMASs and guarantees that the actuator fault f i (t) will be constrained in a given compensation range. The interaction term ξ i (x, t) in Assumption 2 can be described as the physical interconnections, for example, the transmission links of smart grids [1] and the interconnections of multimachine power systems [9], [36]. Define ξ(x, t) = col(ξ i (x, t)), and it follows that the overall interaction term satisfies

Remark 2:
In comparison with the known smooth and bounded interactions [28], the known upper bounds of the actuator faults [24], [26] and external disturbances [16], and the unknown but locally Lipschitz nonlinearities [13], [19], [23], the proposed assumptions in this paper are more general by adaptively approximating a surrogate of the upper bounds. By contrast, the previous works on actuator faults have been combined with interconnection delays [21], [22]; unmeasured states [23]; and unstructured uncertainties [25], [26].
Lemma 1 [43]: Consider the following inequality with a symmetric matrix ∈ R l×l and matrices S and H of the column dimension l:

III. SMC DESIGN AND STABILITY ANALYSIS
A dynamic compensator of the appropriate dimension is used in the output feedback SMC design to satisfy the Kimura-Davison condition [42]. On the basis of the available output information, the dynamic compensator is given bẏ and the decentralized control input is given associated with the dynamic compensator (3) as follows: wherex i ∈ R r i denotes the state of the compensator and v i ∈ R m i denotes the nonlinear SM-FTC switching function to compensate for the effects of matched disturbances and actuator faults. MatricesĀ i ,B i ,C i , andD i are system gains with compatible dimensions to be designed.
The ith augmented dynamics of the combined HMASs and the dynamic compensator are described aṡ wherẽ Then, the augmented HMASs can be written aṡ The SMC function for the ith augmented HMASs on the basis of the newly available output Remark 3: If the so-called Kimura-Davison condition [42] is not satisfied, it is verified that it is difficult to propose a static output feedback SMC scheme. In this case, a sufficient condition to solve this problem is to add an extra compensator (3), (4) (i.e., a particular subsystem with the relative dimensionsx i ∈ R r i ), thus providing additional degrees of freedom.
Here, the following theorem is given to achieve the stability of the augmented HMASs and the insensitivity to the matched disturbances and actuator faults.
Theorem 1: Consider the augmented HMASs (6) and the decentralized control input (4). The overall dynamic system is quadratically stable and the matched disturbances and actuator faults can be compensated with the adaptive algorithms (9), (10) where = 0 is satisfied, which implies that each SMC function has been reached. Then, the gain matrices of the nonlinear SM-FTC switching function (13) are derived as Proof: (14) is a symmetric positive-definite matrix, a Lyapunov function Vx for the augmented HMASs (6) is considered as follows: The time derivative of Vx in (15) is obtained aṡ Since Then, it follows that: It is obtained thatVx < 0 when the inequality in (12) and η i > 0 are satisfied. This completes the proof.
Notably, the inequality in (12) is not convex and, thus, cannot be solved with the LMI toolbox. Here, the following corollary is given with iteration to solve the nonconvex constraints.
Corollary 1: Given the predesigned symmetric matrixP, scalars α i , positive scalars μC and μ W , and matrices where 1 = He(PÃ) − μCC TC and 2 = He(P(Ã +BK)) with the derived matrixK = −B TP . Proof: According to Lemma 1, it is easy to check that the inequality in (12) is in the form of that in (2). Define W = B T P, and the inequality in (12) is solvable for matrix K if and only if there exists a symmetric positive-definite matrix P such that where NC and N W denote the column form bases of the null spaces ofC and W, respectively. Then, the inequalities in (20) are solvable by using Finsler's lemma if the following forms are satisfied with positive scalars μC and μ W : The existing condition, that is, μ W PBB T P > 0, makes the inequality in (22) nonconvex, which needs to be further dealt with. Furthermore, the inequality in (19) is equivalent to the following form withK = −B TP : Thus, the inequality in (19) is equivalent to that in (22). Meanwhile, the formulation in (18) is equivalent to that in (21). The solvable LMI formulations in (18) and (19) imply that there exists a solution of matrix P satisfying the forms in (21) and (22). Hence, the inequality in (12) is solvable by using Lemma 1 and this completes the proof.
The following algorithm is given with the iteration strategy. Furthermore, the matrices P and K in (12) can be decoupled without iteration and the following corollary is given to reduce the space complexity of the LMI formulation.
3) Solve the LMI minimization problem in (18), (19) and derive the symmetric matrix P. 4) Fix P =B 0 W 1B T 0 +C T W 2C and derive the matrices W 1 and W 2 . Solve the following LMI minimization problem and derive the matrix K. ⎡ where 3 = He(P(Ã +BKC)). Then, the dynamic compensator matrix K = diag(K i ) and the matrix in the SMC function (8) is derived as Proof: Define the symmetric positive-definite matrices P = diag(P i ) and Q = diag(Q i ) = P −1 . According to Lemma 1, the necessary and sufficient condition of the feasible solution in (12) is that matrices P and Q are satisfied with Partition every P i and its inverse matrix Q i as Meanwhile, the following compact is derived on the basis of the matrix inversion lemma: and Algorithm 2 Iteration-Free Algorithm 1) Solve the LMI minimization problem in (26)- (28) and derive the matrices P, Q, W 1 and W 2 . 2) Define P i22 = I r i and M i = P i12 P T i12 . Then, P i11 − Q −1 i11 = M i is obtained with the derived matrices P and Q in step 1. Then, the following form is obtained with the diagonal decomposition method.
where λ i = diag(λ i,1 , . . . , λ i,n i ) and V i denote the respective eigenvalues diagonal matrix and eigenvector matrix of matrix M i . Furthermore, defineλ i = diag( λ i,1 , . . . , λ i,r i ). Then, P i12 = V iλi and the corresponding matrices P i and Q i are given as Hence, matrices P and Q are obtained. (25) with the derived matrix P in step 2 and derive the dynamic compensator matrix K = diag(K i ). (8) are derived with W 2i in step 1 and P i12 in step 2.

4) The matrices T i1 and T i2 in the SMC function
The following form is then obtained and is equivalent to the formulation in (28): Select the first row and column in (29), and it is derived as where P = diag(P i11 ) and Q = diag(Q i11 ).
The inequalities in (33) are equivalent to those in (26) and (27). The solvable LMI formulations in (26)- (28) imply that there exists a solution of matrices P, Q, W 1 , and W 2 , satisfying the forms in (32) and (33). Thus, the inequality in (12) is solvable by using the following algorithm with an iteration-free strategy and this completes the proof.
Remark 4: 1) The orders of the LMI formulations (26)- (28) in Corollary 2 are less than the formulations (18), (19) in Corollary 1. Hence, compared with the multistep iterative algorithm, the iteration-free algorithm has the advantage of the lower space dimension and computational complexity, which is more conducive to solving the nonconvex constraints in (12). 2) Compared with the FTC based on the fault detection and isolation approach [11], [15], the nonlinear SM-FTC switching compensation (13) does not need a fault diagnosis mechanism, and the DFTC protocol is designed by the local output feedbacks without the threshold setting and fault isolation. 3) Unlike the adaptive approximation algorithm of the unknown disturbance effects and actuator fault function [21], [22], [33], the unknown functions in this paper are compensated through adaptively approximating a surrogate of the estimated upper bound parameters. Furthermore, the known boundedness of the disturbances/faults [13], [16], [24], [26] and their first-order derivatives [35] are not required in this paper. Consider the SMC function for the ith agent in (8). The time derivative of the SMC function σ i (x i , y i , t) is given aṡ It is necessary to prove the reachability of the sliding surface σ i (x i , y i , t) and the following theorem is given.
Theorem 2: Consider the precalculated matrix P = diag(P i ) satisfying (12) in Theorem 1. The SMC function (8) for each agent (1) can be reached with the adaptive algorithms (9), (10) for the unknown upper bounds and the SM-FTC algorithm (13) in the decentralized control input (4).
Proof: Consider a Lyapunov function V σ i for the SMC function The time derivative of V σ i in (38) is derived in the following form with a positive scalar η i : It follows thatV σ i < 0 when the inequality in (12) and η i > 0 is satisfied, and each SMC function σ i (x i , y i , t) = 0 can be reached and subsequently remains there.
Remark 5: From Theorems 1 and 2, both the stability of HMASs and the reachability of the SMC motion depend on the nonlinear inequality constraint in (12): PÃ+Ã T P+PBKC+ C T K TBT P + PP +Ẽ TẼ < 0 and η i > 0. The positive scalar η i is related to the estimated upper bounds of the matched disturbances and actuator faults (d i andf i ) in the SM-FTC function (13). In fact, the SMC function σ i (x i , y i , t) is essential to limit the updates ind i andf i . Furthermore, the parameters A i ,B i ,C i , andD i in the dynamic compensator (3), (4) and the matrix T i in the SMC function (8) can be derived in the two multistep algorithms with iteration and iteration-free strategies.

IV. OBSERVER-BASED INTEGRAL SMC DESIGN
In this section, both the matched disturbances and actuator faults occur in the control input. The observer-based integral SMC design can effectively reject the matched perturbations and improve the robustness of the unmatched interactions.
First, the linear Luenberger observer is designed aṡ wherex i andŷ i are the respective estimations of x i and y i . Matrix L i ∈ R n i ×p i is the observer gain and u i0 = K ixi is the linear component to make the SMC motion stable. Define the state error e xi = x i −x i and the decentralized control input u i = u i0 +u i1 with the nonlinear FTC component u i1 to compensate for the effects of disturbances and actuator faults and to guarantee the system trajectories to remain within the SMC surface. Then, the state error dynamics is given aṡ The SMC function here is modified in the following compact with an integral component: The SMC function in (42) implies that the system trajectories start from the SMC manifold, that is, σ i (x i , y i , t 0 ) = 0. Then, the time derivative of the SMC function (42) is derived aṡ ). (43) Here, an extra bound condition of the nonlinear interaction term is given as  , y i , t , i = 1, . . . , N. (44) Theorem 3: Given a matrix R i = B T i C † i , which satisfies the invertibility of matrix R i C i B i and a large enough scalar η i , the SMC function (42) can be reached with the adaptive algorithm (44) and the nonlinear FTC input as follows: Furthermore, given diagonal matrices A and J with elements of the respective the HMASs are asymptotically stable and the overall state estimation errors e x = col(e xi ) are ultimately bounded if there exist symmetric positive-definite diagonal matrices S 1 and S 2 with elements of matrices S i1 ∈ R n i ×n i and S i2 ∈ R n i ×n i , and diagonal matrices L and K with elements of matrices L i ∈ R n i ×p i and K i ∈ R m i ×n i such that ⎡ ⎢ ⎢ ⎣ 11 12 where matrices A, B, and C are of diagonal forms with elements of A i , B i , and C i , respectively. Proof: Consider a Lyapunov function Vσ i for the SMC function σ i (x i , y i , t) in (42) as It is easy to check that the system trajectory will remain within the SMC function if the Euclidean norm of the state estimation error e xi is bounded and decreasing. Furthermore, This implies that the SMC function can be reached with the nonlinear FTC strategy (45), and the reachability analysis is completed. Hence, the dynamic trajectory of HMASs can remain within it on the basis of the stability of the state estimation error e xi , which needs to be further dealt with.
Note that the dynamic trajectory can remain within the SMC surface, that is, Hence, the equivalent FTC input u i1 can be derived as Then, the dynamics of the ith agent can be modified aṡ Note that the matched disturbances d i and actuator faults f i are completely compensated with the equivalent FTC input (50). Then, the state estimation error dynamics are also modified aṡ Furthermore, consider a Lyapunov function V(x i , e xi ) for the dynamics (51) and the state estimation error dynamics (52) with symmetric positive-definite matrices S i1 and S i2 The time derivative of V(x i , e xi ) in (53) is obtained aṡ x e x (54) The inequality in (54) is equivalent to that in (46) with the Schur lemma. Hence,V(x i , e xi ) < 0 is obtained, which implies The conventional separation principle cannot be used in HMASs because of the existing interactions among HMASs. The bidirectional interaction term ξ i (x, t) appears in both the state dynamics (51) and the estimation error dynamics (52). In comparison with the work that the interactions are satisfied with the matching condition [24], [32], the unmatched interactions in this paper are further handled with additional freedoms provided in the integral SMC design.
Remark 8: 1) The decentralized controller in the integral SMC contains two parts, namely, the linear and nonlinear FTC compensating components. The nominal linear controller serves to guarantee the stability of the SMC motion, whereas the FTC compensating controller aims to attenuate the effect caused by disturbances and actuator faults. 2) As opposed to the discontinuous dynamics in the SMC protocol [33] and the sign function sgn(·) in (13), the nonlinear FTC input in the integral SMC protocol is continuous within a boundary layer of the hyperbolic tangent tanh(·) and prevents system chattering [30], [31]. More important, with the help of an adaptive mechanism, the corresponding protocol can work effectively even without prior knowledge of the actuator faults and interactions [13], [21], [22], [25].

V. SIMULATION RESULTS
In this section, a numerical simulation of three-machine power systems with nonlinear interactions and actuator faults is put forward to validate the effectiveness of the proposed control designs, that is, the decentralized output SM-FTC scheme and the observer-based integral SMC scheme.
The HMASs model of the three-machine power systems with steam valve control is given in the form of (1) with the state vector , and X ei denote the rotor angle, relative speed, per-unit mechanical power, and steam valve aperture of the ith machine (i = 1, 2, 3), respectively. δ i0 , P mi0 , and X ei0 denote the respective nominal values. Furthermore, the three-machine power systems are characterized by the following matrices [36]: The structure of the three-machine power systems is shown in Fig. 1, and the physical meanings and values of the parameters are illustrated in Table I [36], [37]. Furthermore, the nonlinear interaction term ξ i (x, t) is given as where p ij is a weight coefficient and p ii = 0, p ij = 1 if there is a connection between the ith and jth machines; otherwise, p ij = 0, q i , and q j are the per-unit internal transient voltages and B ij is the per-unit nodal susceptance between the ith and jth machines. It follows that the overall interaction term is To demonstrate the efficiency of the proposed algorithms in Theorems 1-3, the actuator faults in the power systems are considered in the steam valve control inputs and the fault distribution matrices are satisfied with F i = B i f 1 = |0.1sin(0.5t)|, t ≤ 8 0.05sat(0.1sin(0.5t)), t > 8 , The matched disturbances d i , i = 1, 2, 3 are considered as d 1 = 0.1sin(0.5t), d 2 = 0.1cos(0.5t), and d 3 = |0.1sin(0.5t)|.
Simulation parameters are designed as ε = 0.01 and μC = μ W = 0.1, and the gains in the augmented dynamics (5) and the SMC function (8)  Furthermore, the following gains, that is, the symmetric positive-definite matrices P i in (12), the SMC matrices T i in (8), and the dynamic compensator matrices K i in (5) are derived by solving Algorithm 2: In the presence of the time-varying additive faults both in the first and third machines and the time-invariant additive faults in the second machine, the results in Figs. 2 and 3 indicate the effectiveness of the decentralized output SM-FTC design in Theorems 1 and 2. The respective deviations of the rotor angles δ i in the three machines are depicted in Fig. 2 with the application of Algorithms 1 and 2. The respective deviations of the relative speed ω i , the per-unit mechanical power P mi , and the steam valve aperture X ei in the three machines in Fig. 3 show the robust stability of the threemachine power systems. Note that the first machine fails at t = 8 s, the second one fails at t = 40 s, and the third one suffers a failure at t = 20 s. These figures show that with the proposed SM-FTC strategy, the HMASs are insensitive to the considered actuator faults, and the matched disturbances can be compensated in finite-time simulations.
In the observer-based integral SMC scheme, the following gains, that is, the linear feedback gains K i and the observer      Hence, the numerical simulation case demonstrates the effectiveness of the proposed decentralized output SM-FTC and the observer-based integral SMC schemes for the threemachine power systems.

VI. CONCLUSION
In this paper, a decentralized output SMC-FTC design is developed for a class of nonlinear HMASs in the presence of matched disturbances, unmatched interactions, and actuator faults. The nonlinear interaction term is treated as a quadratic constraint. Meanwhile, the disturbances and faults are compensated by adaptively estimating the unknown upper bounds. Subsequently, two LMI minimization algorithms, namely, iteration and iteration-free protocols, are integrated into the SM-FTC design to solve the nonlinear matrix inequality. The observer-based integral SMC is further introduced into the proposed SM-FTC scheme to guarantee the asymptotic stability of HMASs and ultimately realize the boundedness of the estimation errors. Current investigations focus on the extension of the proposed method to nonlinear HMASs with model uncertainties, unmatched disturbances, and simultaneous actuator/sensor faults under the circumstance of network disconnections and cyber attacks.