# Adaptive Fuzzy-Sliding Consensus Control for Euler–Lagrange Systems with Time-Varying Delays

Authors: Yeong-Hwa Chang; Cheng-Yuan Yang; Hung-Wei Lin
Journal: Complexity
 (2022)

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## Abstract

This paper presents an adaptive fuzzy sliding-mode controller for multiple Euler–Lagrange systems communicated with directed topology. Based on the graph theory and Lyapunov–Krasovskii functions, a delay-dependent sufficient condition for the existence of sliding surfaces is given in terms of linear matrix inequalities. The asymptotic stability is analyzed by using the Lyapunov method in the presence of unknown parametric dynamics, actuator faults, and time-varying delays. The usage of adaptive techniques is to adapt the unknown parameters so that the objective of globally asymptotic stability is achieved. Finally, simulation results are provided to illustrate the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

Cooperative control of multiagent systems has attracted much attention in recent years, such as consensus [1–4], formation [5–8], and flocking problems [9–12]. The solving of the consensus problem is an essential and interesting topic of cooperative multiagent research. Basically, the idea of consensus implies that a group of agents can reach an agreement on certain quantities of interest. In graph-based approaches, a directed or indirected graph is popularly applied to describe the communication topology of networked multiagent systems. In the last few years, several methods have been proposed to deal with the consensus problems of the multiagent systems [13–17]. In [13], adaptive synchronization protocols for a heterogeneous multiagent network were investigated, where the interaction between agents is represented by a direct graph. In [14], the leaderless consensus problem over strict-feedback nonlinear multiagent systems with unknown model parameters and control directions was investigated. Also, a robust continuous-time optimization algorithm was presented for multiagent systems with guaranteed fixed-time convergence. Zhang al. presented a robust consensus tracking strategy for multiple unmanned underwater vehicles with switching topology [16]. In [17], the leader-following average consensus problem was addressed for linear multiagent systems.Besides the first-order or second-order linear models, one important class of multiagent control systems is the so-called Euler–Lagrange systems, which generally describes the dynamic properties of robot manipulators, rigid body systems, and so on. In [18], the leader-following consensus problem was studied of multiple Euler–Lagrange (EL) systems subject to an uncertain leader. In [19], an adaptive sliding mode control technique was proposed for the EL systems with actuator faults and system uncertainties. In addition, a distributed optimal consensus strategy based on an event-triggered scheme for EL multiagent systems was investigated [20]. The model-free optimal consensus problem was addressed for networked Euler–Lagrange systems without velocity measurements [21]. Chen et al. proposed a robust adaptive finite-time tracking control scheme for Euler-Lagrange systems subject to nonparametric uncertainties, unknown disturbances, and input saturation [22]. In [23], a robust adaptive finite-time tracking control scheme was proposed for Euler–Lagrange systems subject to nonparametric uncertainties, unknown disturbances, and input saturation.The sliding-mode method has been studied for nonlinear systems because of some attractive features, such as robustness to parameter variations and good transient performance. Recently, a sliding-mode controller has been developed to deal with nonlinearities and uncertainties for multiagent systems [24–28]. In [24], the finite-time consensus tracking of multirobotic systems with disturbances was investigated via utilizing integral sliding mode control. In [25], an optimal sliding mode control approach was presented for the consensus of nonlinear discrete-time high-order multiagent systems. Jina et al. investigated the consensus control problem of Takagi–Sugeno fuzzy multiagent systems by using an observer-based distributed adaptive sliding mode control [26]. In addition, the event-triggered tracking control problem of second-order uncertain multiagent systems was addressed by utilizing the distributed sliding-mode control approach [27]. In [28], the consensus tracking problem of networked control systems with disturbances was discussed, where an integral sliding mode protocol was developed to achieve the consensus in a setting time. Recently, fuzzy sliding-mode control has attracted much attention, where the fuzzy mechanism is useful to decrease the chattering behaviors. In [29], an adaptive back-stepping fuzzy neural network controller using a fuzzy sliding mode controller was designed to suppress the harmonics of a shunt active power filter. In addition, a fuzzy sliding mode control method was proposed to improve the ability of magnetic levitation feed platform subject to external disturbance [30]. In [31], a fuzzy sliding-mode control was developed to deal with unmodeled dynamics and external disturbances in a human-exoskeleton system.On the other hand, in multiagent systems, actuator failures generally result in poor system performance or even cause the instability. In [32], an adaptive fixed-time controller was designed for a class of uncertain nonstrict feedback multiagent systems subject to actuator faults and external disturbances. In [33], a robust consensus control strategy was addressed for nonlinear second-order multiagent systems against actuator faults and uncertainties. Dong et al. presented an augmented control system for a quadrotor unmanned aerial vehicle with parameter uncertainties, external disturbance, and the partial loss of actuator effectiveness [34]. In [35], a cooperative fault tolerant control was presented for linear leader-follower networks subject to actuator faults. Moreover, the fault-tolerant leader-following consensus problem was discussed for multiagent systems with input saturation and actuator faults [36]. In [37], the consensus problem was investigated for a class of nonaffine nonlinear multiagent systems with actuator faults of partial loss of effectiveness.Because of the interactive communication in multiagent systems, the coupling delays between agents become more crucial due to practical considerations. In [38], the consensus problem of discrete-time linear multiagent systems was addressed with unbounded time-varying delays. In [39], the containment control problem of the double-integrator multiagent systems was investigated with time-varying communication delays. Tan et al. discussed the output feedback control problem for a class of nonlinear multiagent systems governed by the high-order strict-feedback model with time delays [40]. Also, the finite-time consensus of leader-following multiagent systems was addressed with multiple time delays over time-varying topology [41]. The leader-following consensus problem was discussed for discrete-time multiagent systems with time-varying delays [42]. In [43], the second-order multiagent networks with time-varying delays were investigated, where a sufficient condition was presented to make all agents asymptotically reach consensus using the linear matrix inequality theory.In this paper, an adaptive fuzzy sliding-mode fault-tolerant controller (AFSFC) is presented for multiple Euler–Lagrange systems. Also, the parametric uncertainties, actuator faults, and time-varying communication delays are considered. The proposed control scheme is based on adaptive sliding-mode techniques combining with the fuzzy logic strategy. An adaptive algorithm is provided to estimate the unknown parametric vector. Moreover, by employing the Lyapunov–Krasovskii function and linear matrix inequalities (LMIs), a sufficient condition is established such that the resulting sliding-mode dynamics is stable. The main contributions of this paper are stated as follows: (1) an adaptive fuzzy sliding-mode controller is proposed for networked Euler–Lagrange systems with communication time-varying delays. The fuzzy sliding mode and adaptive controller are combined to deal with the parametric uncertainties. (2) For the multiagent systems, provided with controller parameters and communication topology, the maximum tolerated delay of all agents can be determined by using the Lyapunov–Krasovskii analysis and LMIs. (3) The proposed control scheme can be applied to a group of agents with a directed communication topology. The tracking errors are shown to be asymptotically convergent with a directed spanning tree of communication topology. (4) The unknown parametric dynamics and actuator faults can be estimated online using adaptive strategies. (5) The overall closed-loop stability can be preserved using Lyapunov stability analysis in both fault-free and faulty situations. Moreover, the allowable communication delays can be obtained and formulated as some LMIs.This paper is organized as follows: in Section 2, the dynamic model of Euler–Lagrange systems is presented with the consideration of partial loss of effectiveness faults. The stability of the sliding motion of multiagent systems is investigated in Section 3. In Section 4, an adaptive fuzzy sliding-mode controller for multiagent systems with time-varying delays and actuator faults is discussed. In Section 5, the simulation results are provided for performance comparisons. Finally, the concluding remarks are given in Section 6.


## 2. Preliminaries

### 2.1. Fundamentals of Graph Theory

A directed graphG=V,E consists of a vertex set V=v1,v2,…,vn and an edge set E⊆V×V, where vj,vi∈E means that the ith node can receive information from the jth node. Ni=vj∈V:vj,vi∈E denotes the neighboring set of vi. The adjacent matrix A=aij∈ℛn×n, where aij=1, if vj,vi∈E, or aij=0, otherwise. The degree matrix D=diagd1,d2,…,dn∈ℛn×n, di=∑j∈Niaij of a digraph G is a diagonal matrix.Lemma 1 (see [44]).
The graphG has a directed spanning tree if and only if there is at least one node with a directed path to all other nodes. If a graph G has a spanning tree, then a right eigenvector L associated with the zero eigenvalue is 1n=1,1,…,1T, i.e. ℒ1n=0.Assumption 1.
The graphG has a directed spanning tree.


### 2.2. Dynamic Models of Multiagent Systems

This paper considers a group of n-agent Euler–Lagrange systems, which can be represented as(1)Miqiq¨i+Ciqi,q˙iq˙i+giqi=τi,where q˙i∈ℛp is the vector of joint positions, Miqi∈ℛp×p is the inertia matrix, Ciqi,q˙i∈ℛp×p is the Coriolis matrix, giqi∈ℛp is the gravitational vector, τi∈ℛp is the vector of input torques, i = 1, 2, …, n. The actuator fault model considered can be described as(2)τif=σiτi+Δτi,where σi is the effectiveness factor, Δτi is an additive fault, 0 < σ¯i ≤ 1, and i = 1, 2, …, n. With fault model (2), the Euler–Lagrangian dynamics (1) of ith agent can be rewritten as follows:(3)Miqiq¨i+Ciqi,q˙iq˙i+giqi=σiτi+Δτi.Letσi be the lower bound of σi, 0<σ¯i≤σi, in which σ¯i is an unknown positive constant. The actuators are fault-free when σ¯i = 1 and Δτi=0, and σ¯i ∈ (0, 1) corresponds to the cases with partial loss of effectiveness (PLOE) faults.Property 1.
The Euler–Lagrangian dynamics (1) is linearly parameterizable as(4)Miqiq¨i+Ciqi,q˙iq˙i+giqi=Yiqi,q˙i,q¨iΘi,where Yi⋅∈ℛp×pθ is the regression matrix and Θi∈ℛpθ is a vector of unknown constant parameters.Property 2.
The matrixMiqiq¨i−2Ciqi,q˙i is skew symmetric.Lemma 2 (see [45]).
Given a positive definite matrixZ, the following inequality holds:(5)2ATB≤ATZA+BTZ−1B,where A and B are two matrices with proper dimensions.Assumption 2.
There exists a positive constantτ¯i such that Δτi≤τ¯i.Lemma 3 (see [46]).
LetO=O11O12O12T−O22 be a matrix with proper dimensions, O22〉0. Then, the following equation holds,(6)O〈0ΔO11+O12O22−1O12T〈0.


## 2.1. Fundamentals of Graph Theory

A directed graphG=V,E consists of a vertex set V=v1,v2,…,vn and an edge set E⊆V×V, where vj,vi∈E means that the ith node can receive information from the jth node. Ni=vj∈V:vj,vi∈E denotes the neighboring set of vi. The adjacent matrix A=aij∈ℛn×n, where aij=1, if vj,vi∈E, or aij=0, otherwise. The degree matrix D=diagd1,d2,…,dn∈ℛn×n, di=∑j∈Niaij of a digraph G is a diagonal matrix.Lemma 1 (see [44]).
The graphG has a directed spanning tree if and only if there is at least one node with a directed path to all other nodes. If a graph G has a spanning tree, then a right eigenvector L associated with the zero eigenvalue is 1n=1,1,…,1T, i.e. ℒ1n=0.Assumption 1.
The graphG has a directed spanning tree.


## 2.2. Dynamic Models of Multiagent Systems

This paper considers a group of n-agent Euler–Lagrange systems, which can be represented as(1)Miqiq¨i+Ciqi,q˙iq˙i+giqi=τi,where q˙i∈ℛp is the vector of joint positions, Miqi∈ℛp×p is the inertia matrix, Ciqi,q˙i∈ℛp×p is the Coriolis matrix, giqi∈ℛp is the gravitational vector, τi∈ℛp is the vector of input torques, i = 1, 2, …, n. The actuator fault model considered can be described as(2)τif=σiτi+Δτi,where σi is the effectiveness factor, Δτi is an additive fault, 0 < σ¯i ≤ 1, and i = 1, 2, …, n. With fault model (2), the Euler–Lagrangian dynamics (1) of ith agent can be rewritten as follows:(3)Miqiq¨i+Ciqi,q˙iq˙i+giqi=σiτi+Δτi.Letσi be the lower bound of σi, 0<σ¯i≤σi, in which σ¯i is an unknown positive constant. The actuators are fault-free when σ¯i = 1 and Δτi=0, and σ¯i ∈ (0, 1) corresponds to the cases with partial loss of effectiveness (PLOE) faults.Property 1.
The Euler–Lagrangian dynamics (1) is linearly parameterizable as(4)Miqiq¨i+Ciqi,q˙iq˙i+giqi=Yiqi,q˙i,q¨iΘi,where Yi⋅∈ℛp×pθ is the regression matrix and Θi∈ℛpθ is a vector of unknown constant parameters.Property 2.
The matrixMiqiq¨i−2Ciqi,q˙i is skew symmetric.Lemma 2 (see [45]).
Given a positive definite matrixZ, the following inequality holds:(5)2ATB≤ATZA+BTZ−1B,where A and B are two matrices with proper dimensions.Assumption 2.
There exists a positive constantτ¯i such that Δτi≤τ¯i.Lemma 3 (see [46]).
LetO=O11O12O12T−O22 be a matrix with proper dimensions, O22〉0. Then, the following equation holds,(6)O〈0ΔO11+O12O22−1O12T〈0.


## 3. Stability of Sliding Motion

For the time-delayed multiagent systems (1), a sliding-mode controller will be designed so that the corresponding sliding motion is asymptotically stable. The sliding surface for the ith agent is defined as(7)si=q˙i+Kiϵi,where Ki∈ℛp×p is a constant positive diagonal matrix, ϵi=∑j∈Niaijqit−dt−qjt−dt, i = 1, 2, …, n.Assumption 3.
The communication time-varying delaydt satisfies that(8)0〈dt≤d¯〈∞,d˙t≤μ〈1,where d¯ and μ are positive constants.
In the following, the notationsqi,d and qj,d stand for qit−dt and qjt−dt, respectively, for simplicity. We denote s=s1T,s2T,…,snTT. The sliding surface of multiagent systems is summarized as(9)s=q˙+Kℒ⊗Ipqd,where q˙=q˙1T,q˙2T,…,q˙nTT, qd=q1,dT,q2,dT,…,qn,dTT∈ℛpn, and K=diagK1,K2,…,Kn∈ℛpn×pn. When the sliding mode is achieved, (9) can be equivalently described as(10)q˙=−Kℒ⊗Ipqd.
Let the error function between the 1st agent and other agents as(11)e1r=q1−qr,r=2,3,…,n.
Moreover, (11) can be rewritten as the following augmented form:(12)e=q1⊗1n−1+Eq,where e=e12T,e13T,…,e1nTT, q=q1T,q2T,…,qnTT, 1n−1=1,1,…,1T, and(13)E=1p−1p0p⋯0p1p0p−1p⋯0p⋮⋮⋮⋱⋮1p0p0p⋯−1p∈ℛpn−1×pn,in which 1p is the p-dimension identity matrix, and 0p is the p-dimensional matrix with all zeros. Thus, (11) can be rewritten as(14)q=q1⊗1n+Fe,where(15)F=0p0p⋯0p−1p0p⋯0p0p−1p⋯0p⋮⋮⋱⋮0p0p⋯−1p∈ℛpn×pn−1.
The derivative of (12) along (10) and (13) is derived as(16)e=−EKℒ⊗Ipq1,d⊗1n−EKℒ⊗IpFed,where the notation ed stands for et−dt. According to Lemma 1, it is obtained as(17)e˙=−Ψed,where Ψ=EKℒ⊗IpF.Lemma 4 (See [47]).
The matrixΨ is Hurwitz if and only if the communication topology G of the multiagent systems has a directed spanning tree.Theorem 1.
Suppose that the communication graph of a multiagent Euler–Lagrange system of (1) has a directed spanning tree. From Assumption 3, the error dynamics of (15) is asymptotically stable, if the following inequalities hold:(18)−PΨ−ΨTP+QPΨΨTP−d¯R−1〈0,−μ˜QΨΨTd¯R−1〈0,where P, Q, and R are symmetric positive definite matrices of ℛpn−1×pn−1 and μ˜=1−μ.Proof.
The Lyapunov–Krasovskii function [48] is chosen as(19)Vs=eTPe+∫t−dtteTsQesds+∫−d¯0∫t+θte˙TsRe˙sdsdθ.
The time derivative ofVs along (15) is derived as(20)V˙s≤−2eTPΨed+eTQe−μ˜eaTQed+d¯eaTΨTRΨed−∫t−dt∈TsRe˙sds.
It is noted thated=e−∫t−dttesds. Then, from Lemma 2 and Assumption 3, it leads to the following inequality:(21)−2eTPΨed=−2eTPΨe+∫t−dtte˙sds≤−2eTPΨe+d¯eTΨTPR−1PΨe+∫t−dtte˙TsRe˙sds.
Substituting (19) into (18), it results in(22)V˙s≤−2eTPΨe+eTQe−μ˜edTQed+d¯eTΨTPR−1PΨe+d¯edTΨTRΨed=eedTΩ100Ω2eed,where Ω1=−PΨ−ΨTP+Q+d¯ΨTPR−1PΨ and Ω2=−μ˜Q+d¯ΨTRΨ. From Lemma 3, V˙s is negative if (16) holds. It implies that the error dynamics of (15) is asymptotically stable. This completes the proof.Remark 1.
It is noted that Theorem1 proposed a delay-dependent sufficient condition of stability for the sliding surfaces (9) based on the LMIs (16). The error function (12) with time-varying communication delay will converge to zero as t⟶∞. Also, the allowable bound of time-varying delays can be obtained. Since it provides only sufficient conditions for stability, the results could be likely conservative.


## 4. Consensus Controller Design

### 4.1. Fault-Free Cases

Note that in the fault-free case, i.e.,σi=1 and Δτi=0, the torques acting on the dynamic system (1) are designed as(23)τi=τeq,i+τsw,i,where τeq,i is the equivalent control action and τsw,i is the switching controller. To obtain the equivalent control action τeq,i, the state trajectory is desired to stay in the sliding surface, i.e., s˙i=0. From (7), it gives that s˙i=q¨i+Kiϵ˙i=0. From (1), with the equivalent control action τi=τeq,i, the equivalent control action τeq,i can be derived as(24)τeq,i=−MiKiϵ˙i−CiKiϵi+gi,where ϵ˙i=1−d˙t∑j∈Niaijq˙i,d−q˙j,d. We consider the unknown parameter vector Θi of (4). Let the estimation of Θi be defined as(25)Θ˜i=Θi−Θ^i,where Θ^i is the estimation of Θi. The adaptive law of Θ^i is designated as follows:(26)Θ^˙i=ΓYiTqi,q˙i,ϵi,ϵ˙isi,where Γ∈Rp×pθ is a constant positive definite matrix.Letsi and τsw,i be the input and output variables of a switching control system, respectively. Therefore, the switching system is represented by a single input-output fuzzy logic system. The fuzzy system is a collection of the fuzzy IF-THEN rules in the form of(27)Rule k:IFsriisMki,THENτsw,riisF6−ki,k=1,2,…,5,where the Mki and F6−ki are the input and output fuzzy sets, si=s1i,s2i,…,spiT, τsw,i=τsw,1i,τsw,2i,…,τsw,piT, r = 1, 2, …, p, respectively. The triangular input and singleton output membership functions are shown in Figure 1. By using the centroid defuzzification technique, the output τsw,ri of the fuzzy system is(28)τsw,ri=∑k=15gkiμkisri∑k=15μkisri=−∑k=15gkisgnsriμkisri,where gki is the value of the corresponding fuzzy output and μkisri is the firing strength of the antecedent membership function, and sgn. is a standard sign function. We rewrite (25) as the following augmented form:(29)τsw,i=−∏isgnsi,where sgnsi=sgns1i,sgns2i,…,sgnspiT and(30)∏i=diag∑k=15gkiμkis1i,∑k=15gkiμkis2i,…,∑k=15gkiμkispi.Figure 1
Membership functions of the fuzzy switching controller.Remark 2.
Triangular and Gaussian membership functions are typical membership functions chosen for the process of fuzzy inference. In this paper, the reason of choosing triangular functions as the input membership functions is to reduce the computation complexity in the calculation of firing strengths in (25).Theorem 2.
We consider a multiagent Euler–Lagrange system of (1) with a directed spanning-tree communication graph. From (7), the state trajectories of (1) will be driven onto the sliding surface si=0 with the adaptive fuzzy sliding-mode controller (AFSC) (21), (22), (26), and the adaptive law (24).Proof.
Let the Lyapunov function be chosen as(31)V=∑i=1n12siTMisi+Θ˜iTΓ−1Θ˜i.
The time derivative of V can be expressed as(32)V˙=∑i=1nsiTMisi+12siTM˙isi+Θ˜iTΓ−1Θ˜˙i=∑i=1nsiTτsw,i+Yiqi,q˙i,ϵi,ϵ˙iΘ˜i−Cisi+12siTM˙isi−Θ˜iΓ−1Θ^˙i.
From Property1, (29) can be rewritten as(33)V˙=∑i=1nsiTτsw,i+Yiqi,q˙i,ϵi,ϵ˙iΘ˜i−Θ˜iTΓ−1Θ^˙i.
Substituting (24) and (26) into (30), it yields(34)V˙=∑i=1nsiTτsw,i=−∑i=1nsiT∏isgnsi<0.
Therefore, it implies thatsi=0 and that the system trajectory is enforced on the sliding surfaces. The proof is completed.


### 4.2. Fault Cases

In this section, we consider the controller design for multiagent systems with PLOE and unknown effectiveness fault, i.e.,0<σi<1 and Δτi≠0. From (7), the dynamical system (3) can be rewritten as(35)Mis˙i+Cisi−MiKiϵ˙i−CiKiϵi+gi=σiτi+Δτi.For the multiagent systems ofn agents, a faulty actuator of the ith agent can be described as(36)τi=τeq,i+τsw,i+τc,i,where the auxiliary controller τc,i is provided to compensate the faulty influences.The auxiliary controller can be expressed as(37)τc,i=−1σ^¯isgnsi‖τi‖+τ¯^isgnsi.The adaptive algorithms of fault are given as(38)σ^¯˙i=ϕi1σ^¯isgnsi‖τi‖,(39)τ¯^i˙=ςi‖si‖,where ϕi and ςi are positive constants. To make the design concepts more concise and clearer, the overall adaptive fuzzy sliding-mode fault-tolerant controller (AFSFC) structure is shown in Figure 2.Figure 2
Structure of adaptive fuzzy-sliding fault-tolerant control.Theorem 3.
We consider a multiagent Euler–Lagrange system of (3) with a directed spanning-tree communication graph. From (7), the state trajectories of the system (32) will be driven onto the surface si=0 with the AFSFC (33), the auxiliary controller (34), and the adaptive laws (24), (35), (36).Proof.
Let the Lyapunov function be chosen as(40)Vf=V+∑i=1n121ϕiσ˜¯i2+1ςiτ¯˜i2,where σ˜¯i=σ¯i−σ^¯i, and τ¯˜i=τ¯i−τ¯^i. The time derivative of Vf can be expressed as(41)V˙f=V˙+∑i=1n−1ϕiσ˜¯iσ^˙¯i−1ςiτ¯˜iτ¯^˙i=∑i=1nsiTMis˙i+12siTM˙isi−Θ˜iTΓ−1Θ^˙i−1ϕiσ˜¯iσ^˙¯i−1ςiτ¯˜iτ¯^˙i.
Substituting (32) and (33) into (38), it yields(42)V˙f=∑i=1nsiTMiKiϵi+CiKiϵi+gi+σiτi+σiτc,i+Δτi−Θ˜iTΓ−1Θ^˙i−1ϕiσ˜¯iσ^˙¯i−1ςiτ¯˜iτ¯^˙i.
From Assumption2 and Theorem 2, the time derivative of V can be described as(43)V˙f≤∑i=1nsiTYiqi,q˙i,ϵi,ϵ˙iΘ˜i+siTτsw,i+‖si‖‖τi‖+σisiTτc,i+τ¯i‖si‖−Θ˜iTΓ−1Θ^˙i−1ϕiσ˜¯iσ^˙¯i−1ςiτ¯˜iτ¯^˙i.
From (43), one has(44)σisiTτc,i=−σiσ^¯i‖si‖‖τi‖−σiτ¯^i‖si‖≤−σ¯iσ^¯i‖si‖‖τi‖−τ¯^i‖si‖=−1+σ˜¯iσ^¯i‖si‖‖τi‖−τ¯^i‖si‖,where ∀−1≤−σi≤−σ¯i<0.
Substituting (41) into (40), it leads to(45)V˙f≤∑i=1nsiTYiqi,q˙i,ϵi,ϵ˙iΘ˜i+siTτsw,i+τ¯˜i‖si‖−σ¯iσ^¯i‖si‖‖τi‖−Θ˜iTΓ−1Θ^˙i−1ϕiσ˜¯iσ^˙¯i−1ςiτ¯˜iτ¯^˙i.
From (24), (35) and (36), one has Θ˜iTΓ−1Θ^˙i=−siTYiqi,q˙i,ϵi,ϵ˙iΘ˜i, 1/ςiτ¯˜iτ¯^˙i=τ¯˜i‖si‖, and 1/ϕiσ˜¯iσ^˙¯i=−σ˜¯iσ^¯i−1‖si‖‖τi‖. Therefore, (42) can be rewritten as(46)V˙f≤∑i=1nsiTτsw,i=−∑i=1nsiT∏isgnsi<0.
Therefore, it implies thatsi=0 so that the system trajectory is enforced on the sliding surfaces. The proof is completed.Remark 3.
In summary, with the sliding-mode control, the switching control actionτsw,i (26) will drive the ith agent to the sliding surface si. As the agents converge to the sliding surface, the equivalent controller τeq,i (22) will ensure the agents stay on the sliding surface. Moreover, the existence of unknown parameters and actuator faults can be resolved by using the adaptive law (24) and auxiliary controller (34).


## 4.1. Fault-Free Cases

Note that in the fault-free case, i.e.,σi=1 and Δτi=0, the torques acting on the dynamic system (1) are designed as(23)τi=τeq,i+τsw,i,where τeq,i is the equivalent control action and τsw,i is the switching controller. To obtain the equivalent control action τeq,i, the state trajectory is desired to stay in the sliding surface, i.e., s˙i=0. From (7), it gives that s˙i=q¨i+Kiϵ˙i=0. From (1), with the equivalent control action τi=τeq,i, the equivalent control action τeq,i can be derived as(24)τeq,i=−MiKiϵ˙i−CiKiϵi+gi,where ϵ˙i=1−d˙t∑j∈Niaijq˙i,d−q˙j,d. We consider the unknown parameter vector Θi of (4). Let the estimation of Θi be defined as(25)Θ˜i=Θi−Θ^i,where Θ^i is the estimation of Θi. The adaptive law of Θ^i is designated as follows:(26)Θ^˙i=ΓYiTqi,q˙i,ϵi,ϵ˙isi,where Γ∈Rp×pθ is a constant positive definite matrix.Letsi and τsw,i be the input and output variables of a switching control system, respectively. Therefore, the switching system is represented by a single input-output fuzzy logic system. The fuzzy system is a collection of the fuzzy IF-THEN rules in the form of(27)Rule k:IFsriisMki,THENτsw,riisF6−ki,k=1,2,…,5,where the Mki and F6−ki are the input and output fuzzy sets, si=s1i,s2i,…,spiT, τsw,i=τsw,1i,τsw,2i,…,τsw,piT, r = 1, 2, …, p, respectively. The triangular input and singleton output membership functions are shown in Figure 1. By using the centroid defuzzification technique, the output τsw,ri of the fuzzy system is(28)τsw,ri=∑k=15gkiμkisri∑k=15μkisri=−∑k=15gkisgnsriμkisri,where gki is the value of the corresponding fuzzy output and μkisri is the firing strength of the antecedent membership function, and sgn. is a standard sign function. We rewrite (25) as the following augmented form:(29)τsw,i=−∏isgnsi,where sgnsi=sgns1i,sgns2i,…,sgnspiT and(30)∏i=diag∑k=15gkiμkis1i,∑k=15gkiμkis2i,…,∑k=15gkiμkispi.Figure 1
Membership functions of the fuzzy switching controller.Remark 2.
Triangular and Gaussian membership functions are typical membership functions chosen for the process of fuzzy inference. In this paper, the reason of choosing triangular functions as the input membership functions is to reduce the computation complexity in the calculation of firing strengths in (25).Theorem 2.
We consider a multiagent Euler–Lagrange system of (1) with a directed spanning-tree communication graph. From (7), the state trajectories of (1) will be driven onto the sliding surface si=0 with the adaptive fuzzy sliding-mode controller (AFSC) (21), (22), (26), and the adaptive law (24).Proof.
Let the Lyapunov function be chosen as(31)V=∑i=1n12siTMisi+Θ˜iTΓ−1Θ˜i.
The time derivative of V can be expressed as(32)V˙=∑i=1nsiTMisi+12siTM˙isi+Θ˜iTΓ−1Θ˜˙i=∑i=1nsiTτsw,i+Yiqi,q˙i,ϵi,ϵ˙iΘ˜i−Cisi+12siTM˙isi−Θ˜iΓ−1Θ^˙i.
From Property1, (29) can be rewritten as(33)V˙=∑i=1nsiTτsw,i+Yiqi,q˙i,ϵi,ϵ˙iΘ˜i−Θ˜iTΓ−1Θ^˙i.
Substituting (24) and (26) into (30), it yields(34)V˙=∑i=1nsiTτsw,i=−∑i=1nsiT∏isgnsi<0.
Therefore, it implies thatsi=0 and that the system trajectory is enforced on the sliding surfaces. The proof is completed.


## 4.2. Fault Cases

In this section, we consider the controller design for multiagent systems with PLOE and unknown effectiveness fault, i.e.,0<σi<1 and Δτi≠0. From (7), the dynamical system (3) can be rewritten as(35)Mis˙i+Cisi−MiKiϵ˙i−CiKiϵi+gi=σiτi+Δτi.For the multiagent systems ofn agents, a faulty actuator of the ith agent can be described as(36)τi=τeq,i+τsw,i+τc,i,where the auxiliary controller τc,i is provided to compensate the faulty influences.The auxiliary controller can be expressed as(37)τc,i=−1σ^¯isgnsi‖τi‖+τ¯^isgnsi.The adaptive algorithms of fault are given as(38)σ^¯˙i=ϕi1σ^¯isgnsi‖τi‖,(39)τ¯^i˙=ςi‖si‖,where ϕi and ςi are positive constants. To make the design concepts more concise and clearer, the overall adaptive fuzzy sliding-mode fault-tolerant controller (AFSFC) structure is shown in Figure 2.Figure 2
Structure of adaptive fuzzy-sliding fault-tolerant control.Theorem 3.
We consider a multiagent Euler–Lagrange system of (3) with a directed spanning-tree communication graph. From (7), the state trajectories of the system (32) will be driven onto the surface si=0 with the AFSFC (33), the auxiliary controller (34), and the adaptive laws (24), (35), (36).Proof.
Let the Lyapunov function be chosen as(40)Vf=V+∑i=1n121ϕiσ˜¯i2+1ςiτ¯˜i2,where σ˜¯i=σ¯i−σ^¯i, and τ¯˜i=τ¯i−τ¯^i. The time derivative of Vf can be expressed as(41)V˙f=V˙+∑i=1n−1ϕiσ˜¯iσ^˙¯i−1ςiτ¯˜iτ¯^˙i=∑i=1nsiTMis˙i+12siTM˙isi−Θ˜iTΓ−1Θ^˙i−1ϕiσ˜¯iσ^˙¯i−1ςiτ¯˜iτ¯^˙i.
Substituting (32) and (33) into (38), it yields(42)V˙f=∑i=1nsiTMiKiϵi+CiKiϵi+gi+σiτi+σiτc,i+Δτi−Θ˜iTΓ−1Θ^˙i−1ϕiσ˜¯iσ^˙¯i−1ςiτ¯˜iτ¯^˙i.
From Assumption2 and Theorem 2, the time derivative of V can be described as(43)V˙f≤∑i=1nsiTYiqi,q˙i,ϵi,ϵ˙iΘ˜i+siTτsw,i+‖si‖‖τi‖+σisiTτc,i+τ¯i‖si‖−Θ˜iTΓ−1Θ^˙i−1ϕiσ˜¯iσ^˙¯i−1ςiτ¯˜iτ¯^˙i.
From (43), one has(44)σisiTτc,i=−σiσ^¯i‖si‖‖τi‖−σiτ¯^i‖si‖≤−σ¯iσ^¯i‖si‖‖τi‖−τ¯^i‖si‖=−1+σ˜¯iσ^¯i‖si‖‖τi‖−τ¯^i‖si‖,where ∀−1≤−σi≤−σ¯i<0.
Substituting (41) into (40), it leads to(45)V˙f≤∑i=1nsiTYiqi,q˙i,ϵi,ϵ˙iΘ˜i+siTτsw,i+τ¯˜i‖si‖−σ¯iσ^¯i‖si‖‖τi‖−Θ˜iTΓ−1Θ^˙i−1ϕiσ˜¯iσ^˙¯i−1ςiτ¯˜iτ¯^˙i.
From (24), (35) and (36), one has Θ˜iTΓ−1Θ^˙i=−siTYiqi,q˙i,ϵi,ϵ˙iΘ˜i, 1/ςiτ¯˜iτ¯^˙i=τ¯˜i‖si‖, and 1/ϕiσ˜¯iσ^˙¯i=−σ˜¯iσ^¯i−1‖si‖‖τi‖. Therefore, (42) can be rewritten as(46)V˙f≤∑i=1nsiTτsw,i=−∑i=1nsiT∏isgnsi<0.
Therefore, it implies thatsi=0 so that the system trajectory is enforced on the sliding surfaces. The proof is completed.Remark 3.
In summary, with the sliding-mode control, the switching control actionτsw,i (26) will drive the ith agent to the sliding surface si. As the agents converge to the sliding surface, the equivalent controller τeq,i (22) will ensure the agents stay on the sliding surface. Moreover, the existence of unknown parameters and actuator faults can be resolved by using the adaptive law (24) and auxiliary controller (34).


## 5. Simulation Results

The simulations are conducted for 2-DOF (degree of freedom) with six robot agents. The Euler–Lagrange model (1) is obtained as [49].(47)Miqi=Mi11Mi12Mi21Mi22,Ciqi,q˙i=Ci11Ci12Ci21Ci22,Gi=0,qi=q1iq2i,where Mi11=α1+2α2cosq2i+2α3sinq2i, Mi12=Mi21=α2+α3cosq2i+α4sinq2i, Mi22=α2, Ci11=−α3sinq2i+α4cosq2iq˙2i, Ci12=−α3sinq2i+α4cosq2iq˙1i+q˙2i, Ci21=α3sinq2i+α4cosq2iq˙1i, Ci22=0, in which α1=I1+m1lc12+Ie+mele2, α2=I1+melc12, α3=melclcecosσe, α4=melclcesinσe. The parameters of the multiagent system are set as m1=1, me=2, lc=1, lc1=0.5, lce=0.8, σe=π/6, Ie=0.25, and I1=0.15.The six-agent connected network is shown in Figure3. Form Figures 3(a)–3(f), the directed graph exists the spanning tree. According to Theorem 1, the agents of the proposed topology in Figure 3(a) are asymptotically stable using the AFSFC scheme with the delay bound d¯=0.16 (sec). The parameters of the AFSFC are set as μ˜=0.5, Ki=diag1,1 , f1i=−f5i=1, f2i=−f4i=0.5, f3i=0, g1i=−g5i=144, g2i=−g4i=12, g3i=1.5, and Γi=diag2,2. The adaptive laws are set as Θ^˙i=α^1,α^2,α^3,α^4T, ϕi=1×10−4, ςi=1, and(48)Yiqi,q˙i,ϵi,ϵ˙i=Yi11Yi12Yi13Yi14Yi21Yi22Yi23Yi24T,where Yi11=k1iϵ1i, Yi12=k2iϵ˙2i, Yi21=0, Yi12=k1iϵ˙1i+k2iϵ˙2i, Yi23=k1icosq2ϵ˙1i+k1iq˙1isinq2iϵ1i, Yi24=k1isinq2ϵ˙1i+k1iq˙1icosq2iϵ1i, and(49)Yi13=2k1icosq2iϵ˙1i+k2isinq2iϵ˙2i−k1iq˙2isinq2iϵ1i−k2iq˙1isinq˙2iϵ2i−k2iq˙2isinq˙2iϵ2i,Yi14=2k1isinq2iϵ˙1i+k2isinq2iϵ˙2i−k1iq˙2icosq2iϵ1i−k2iq˙1isinq˙2iϵ2i−k2iq˙2icosq˙2iϵ2i.Figure 3
Communication topology of a multiagent system. (a)d¯=0.16 sec. (b)d¯=0.16 sec. (c)d¯=0.06 sec. (d)d¯=0.36 sec. (e)d¯=0.15 sec. (f)d¯=0.11 sec.
(a)(b)(c)(d)(e)(f)In this paper, the time delay is chosen asdt=0.08+0.08sint (sec) for simulations. The initial states of the six agents are set as q1=−1,1T, q2=1,0.5T, q3=−2.5,−1T, q4=2,−0.5T, q5=2.5,1.5T, q6=−2,1.5T, and the initial velocities and acceleration are zeros. The following indices are considered for the combined formation errors, the integral absolute error (IAE), the integral time absolute error (ITAE), the integral square error (ISE), and the integral time square error (ITSE) [50].Remark 4.
The indices IAE, ITAE, ISE, and ITSE are considered for the performance comparisons. The IAE and ISE indicate the accumulated consensus error of agents, where the weights in the transient and steady-state stages are equal. On the other hand, ITAE and ITSE indicate the time-weighted consensus errors, and these two indices can be better used to highlight the performance superiority in the steady state.


### 5.1. Fault-Free Cases

The comparisons between the adaptive fuzzy sliding-mode controller (AFSC) and the proposed adaptive fuzzy sliding-mode fault-tolerant controller (AFSFC) are also given in Figs.4–9. Figures 4 and 7 show the responses of multiagent systems with the ASC and proposed AFSC, respectively. Lines A1-A6 are the trajectories of agents 1–6, the symbol “o” is initial state positions, and the symbol “⋆” is final state positions. Figures 5 and 8 show the states q1i and q2i of multiagent systems with the ASC and proposed AFSC, respectively. Figures 6 and 9 show the control inputs τ1i and τ2i of multiagent systems with the AFSC and AFSFC, respectively. The position errors are summarized in Table 1. In this case, the AFSC and AFSFC method can support a certain degree of consistence in position responses.Figure 4
Multiagent trajectories and the fault-free case (AFSC).Figure 5
Simulation results of statesq1i and q2i and the fault-free case (AFSC).Figure 6
Simulation results of control inputsτ1i and τ2i and the fault-free case (AFSC).Figure 7
Multiagent trajectories and the fault-free case (AFSFC).Figure 8
Simulation results of statesq1i and q2i and the fault-free case (AFSFC).Figure 9
Simulation results of control inputsτ1i and τ2i and the fault-free case (AFSFC).Table 1
Performance comparisons (fault-free case).
IAEITAEISEITSEAFSC19454388885535850697AFSFC20697425876181963843


### 5.2. Fault Cases

In this case, the effectiveness faultσi is present randomly in [0.01 0.5], and the additive faults are set to be Δτ=2rand.2rand.T, where rand. is a random number uniformly distributed in 1−1. The comparisons between the AFSC and the proposed AFSFC are also given in Figs. 10–15. Figures 10 and 13 show the response of multiagent systems with the ASC and proposed AFSC. Figures 11 and 14 show the states q1i and q2i of multiagent systems with the AFSC and AFSFC, respectively. All the agents can achieve consensus. The states q1i and q2i are more stable by the proposed AFSFC compared with the AFSC. Figures 12 and 15 show the control inputs τ1i and τ2i of multiagent systems with the AFSC and AFSFC, respectively. From Figures 12 and 15, the amplitude of the control inputs is smaller in the first few seconds with the proposed AFSFC. In summary, from Figures 10–15, the performance of the proposed AFSFC is obviously better than the AFSC. Subsequently, the position errors are summarized in Table 2. It can be observed that the proposed AFSFC method extends the performance improvement from 7.07% to 86.84%.Figure 10
Multiagent trajectories and the fault case (AFSC).Figure 11
The statesq1i and q2i and the fault case (AFSC).Figure 12
The control inputsτ1i and τ2i and the fault case (AFSC).Figure 13
State trajectories and the fault case (AFSFC).Figure 14
The statesq1i and q2i and the fault case (AFSFC).Figure 15
The control inputsτ1i and τ2i and the fault case (AFSFC).Table 2
Performance comparisons (fault case).
IAEITAEISEITSEAFSC3624116331090090222250AFSFC22046487836437869119Remark 5.
From Figs.6, 9, 12, and 15, it can be observed that greater inputs are required to deal with the problem of actuator faults. In practical applications, the control inputs should have prescribed limits. The consensus stability issues of multiagent agents with input constraints open a new theoretical problem, which cannot be solved at the current stage and needs to be investigated separately.


## 5.1. Fault-Free Cases

The comparisons between the adaptive fuzzy sliding-mode controller (AFSC) and the proposed adaptive fuzzy sliding-mode fault-tolerant controller (AFSFC) are also given in Figs.4–9. Figures 4 and 7 show the responses of multiagent systems with the ASC and proposed AFSC, respectively. Lines A1-A6 are the trajectories of agents 1–6, the symbol “o” is initial state positions, and the symbol “⋆” is final state positions. Figures 5 and 8 show the states q1i and q2i of multiagent systems with the ASC and proposed AFSC, respectively. Figures 6 and 9 show the control inputs τ1i and τ2i of multiagent systems with the AFSC and AFSFC, respectively. The position errors are summarized in Table 1. In this case, the AFSC and AFSFC method can support a certain degree of consistence in position responses.Figure 4
Multiagent trajectories and the fault-free case (AFSC).Figure 5
Simulation results of statesq1i and q2i and the fault-free case (AFSC).Figure 6
Simulation results of control inputsτ1i and τ2i and the fault-free case (AFSC).Figure 7
Multiagent trajectories and the fault-free case (AFSFC).Figure 8
Simulation results of statesq1i and q2i and the fault-free case (AFSFC).Figure 9
Simulation results of control inputsτ1i and τ2i and the fault-free case (AFSFC).Table 1
Performance comparisons (fault-free case).
IAEITAEISEITSEAFSC19454388885535850697AFSFC20697425876181963843


## 5.2. Fault Cases

In this case, the effectiveness faultσi is present randomly in [0.01 0.5], and the additive faults are set to be Δτ=2rand.2rand.T, where rand. is a random number uniformly distributed in 1−1. The comparisons between the AFSC and the proposed AFSFC are also given in Figs. 10–15. Figures 10 and 13 show the response of multiagent systems with the ASC and proposed AFSC. Figures 11 and 14 show the states q1i and q2i of multiagent systems with the AFSC and AFSFC, respectively. All the agents can achieve consensus. The states q1i and q2i are more stable by the proposed AFSFC compared with the AFSC. Figures 12 and 15 show the control inputs τ1i and τ2i of multiagent systems with the AFSC and AFSFC, respectively. From Figures 12 and 15, the amplitude of the control inputs is smaller in the first few seconds with the proposed AFSFC. In summary, from Figures 10–15, the performance of the proposed AFSFC is obviously better than the AFSC. Subsequently, the position errors are summarized in Table 2. It can be observed that the proposed AFSFC method extends the performance improvement from 7.07% to 86.84%.Figure 10
Multiagent trajectories and the fault case (AFSC).Figure 11
The statesq1i and q2i and the fault case (AFSC).Figure 12
The control inputsτ1i and τ2i and the fault case (AFSC).Figure 13
State trajectories and the fault case (AFSFC).Figure 14
The statesq1i and q2i and the fault case (AFSFC).Figure 15
The control inputsτ1i and τ2i and the fault case (AFSFC).Table 2
Performance comparisons (fault case).
IAEITAEISEITSEAFSC3624116331090090222250AFSFC22046487836437869119Remark 5.
From Figs.6, 9, 12, and 15, it can be observed that greater inputs are required to deal with the problem of actuator faults. In practical applications, the control inputs should have prescribed limits. The consensus stability issues of multiagent agents with input constraints open a new theoretical problem, which cannot be solved at the current stage and needs to be investigated separately.


## 6. Conclusions

This paper has proposed an adaptive fuzzy sliding-mode fault-tolerant controller for Euler–Lagrange systems in the presence of unknown parametrics, actuator faults, and communication time-varying delays. In the design of the fuzzy sliding-mode fault-tolerant controller, a delay-dependent sufficient condition is derived, and the allowable bound of time delays can be obtained in the form of linear matrix inequalities. Based on the Lyapunov stability theory, the overall stability of the multiagent system is guaranteed such that the desired consensus of agents can be asymptotically attained. Simulation results indicate that the proposed control scheme has superior responses, compared to the AFSC method. Especially, the proposed AFSFC method provides significant improvement in the case of actuator faults. In practical applications, the control inputs should have prescribed limits. Thus, the systematic analysis and synthesis of input-constrained multiagent systems are an interesting topic in the future.



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*Source: 9914940-2022-10-26.xml*