1. Introduction
The Euler–Lagrange system serves as a paradigmatic model for unraveling complex dynamics and finds extensive application in various engineering domains, including robotic manipulators [
1,
2], spacecrafts [
3,
4] and unmanned aerial vehicle systems [
5,
6]. Over the past few years, there has been a notable focus on cooperative control in multiagent systems, which involve multiple Euler–Lagrange systems and a leader system. The main difficulty in multiagent cooperative control lies in devising a distributed control law for each agent to accomplish particular global goals, such as consensus [
7], synchronization [
8] and formation [
9].
Various efforts have been devoted to addressing this issue [
10]. Networked cooperative systems offer numerous advantages over single subsystems, including enhanced efficiency and capabilities. Over recent years, there has been significant attention toward the cooperative control of multiple Euler–Lagrange systems. The fundamental challenge is to devise a distributed control scheme for each entity so that particular overarching goals (e.g., consensus [
1,
7], tracking [
11,
12], synchronization [
8,
13] and formation [
9,
14]) can be accomplished via local maneuvers. The challenge arises from the incomplete knowledge of the parameters associated with multiple Euler–Lagrange systems, and the existence of uncertainty will diminish the controllability performance of the system. Currently, the adaptive method [
11,
13], neural networks method [
12,
14] and backstepping design procedure [
8] have become effective methods for addressing the uncertainties associated with multiple Euler–Lagrange systems. More specifically, Ref. [
11] investigated the distributed adaptive control of a group of underactuated flexible spacecraft operating under a leader–follower framework, relying solely on the measurements of the rigid bodies. Ref. [
13] developed strategies to facilitate synchronization among networked uncertain Lagrange systems through adaptive coordination control protocols. In [
12], a distributed finite-time tracking control algorithm was formulated, leveraging neural networks for estimating model uncertainties. In [
14], a design of an adaptive neural network multilayer formation controller was proposed to handle model uncertainties. In [
8], a fixed-time robust controller was developed by converting uncertain Euler–Lagrange systems into second-order systems and introduced the backstepping design methodology. However, these systems also exhibit increased complexity and a higher frequency of faults, attributed to their extensive array of sensors, controllers and communication equipment. At times, a fault within a system can lead to a decline in performance or even result in catastrophic accidents. Therefore, ensuring reliability is a crucial objective in engineering system design. It is imperative to maintain stable and appropriate performance in systems regardless of whether components are functioning properly or have failed. Fault-tolerant control is seen as one of the most promising control technologies because it maintains specified performance of dynamic systems even if component faults occur. Fault tolerance control is divided into passive fault tolerance and active fault tolerance according to design methods. As the complexity of the system deepens, the fixed control gain designed by passive fault tolerance and the limited conservatism of fault tolerance are amplified. Active fault tolerance can adjust the control gain or reconstruct the controller online, thereby compensating for the limitations of passive fault tolerance and having stronger fault tolerance. However, it can also lead to time delays caused by fault detection, isolation and controller reconstruction, as well as computational resource waste caused by time estimation of fault information [
15].
When dealing with Euler–Lagrange systems, controlling and executing tasks becomes essential, particularly in the face of environmental disruptions, model uncertainties and actuator malfunctions. This issue has garnered significant attention in recent years due to mission failures caused by these uncertainties. To achieve the desired control performance, various studies have been carried out and put into practice, e.g., sliding mode control [
16], back-stepping control [
17], adaptive control [
18,
19], etc. In current research, the extensive use of sliding mode control for fault-tolerant control demonstrates strong robustness to parameter uncertainties and external disturbances. Nonetheless, it is vital to deal with the inherent chattering issues in sliding mode control when utilizing this control approach. In [
20], a resilient fault-tolerant controller based on sliding mode was introduced to lessen chattering. Moreover, a quasi-continuous second-order sliding mode control law was developed to address system malfunctions and alleviate chattering. Terminal sliding mode has delivered satisfactory outcomes in fault-tolerant control. In [
21], a decentralized fault-tolerant controller for robot manipulators was proposed by combining terminal sliding modes and a super-twist algorithm to minimize chattering and expedite convergence. In [
22], a sliding mode surface for integral terminal was carefully developed, resulting in the development of a model-free finite-time resilient adaptive fault-tolerant control for robot manipulators, demonstrating advantages in global finite-time convergence. Fault-tolerant cooperative control stands out as a leading control technology ensuring consistent and optimal performance of dynamic systems despite component faults [
18,
19,
23]. Ref. [
18] devised an adaptive fault-tolerant cooperative controller to attain the coordinated tracking error to be uniformly ultimately bounded for networked uncertain Lagrange systems. Ref. [
23] proposed a distributed coordinated control scheme to make the leader-following tracking error uniformly ultimately bounded. Ref. [
19] introduced an adaptive neural networks-based fixed-time fault-tolerant controller to make leader-following tracking errors converge into compact sets.
In the domain of output regulation, as described in [
24], it is common to reference an exosystem that produces both the reference signal and external disturbances. Furthermore, in the realm of cooperative output regulation, as discussed in [
25], the exosystem is regarded as the leader system, distinct from the follower system. To design a distributed control law, a distributed observer was proposed as a dynamic compensator following the certainty equivalence principle [
26]. This observer possessed the capability to assess the leader’s state and transmit it to all followers via the communication network. The observer presented in [
27] initially addressed the cooperative output regulation issue in linear multiagent systems and was subsequently extended to multiple Euler–Lagrange systems [
28]. Nevertheless, it was contingent on the assumption of possessing knowledge regarding the system matrix of the leader system. To loosen the above assumption, an adaptive distributed observer was introduced in [
28,
29], capable of estimating both the state and matrix of the leader system. A self-tuning distributed observer was designed in [
30] to address the challenge of calculating the observer gain online when dealing with numerous agents. These observers assume that certain followers have knowledge of the leader system’s dynamics. Nonetheless, in practice, the dynamics of the leader system might remain unknown to any follower, meaning that none of the followers possesses precise knowledge of the leader system’s dynamics. In light of this, a distributed dynamic compensator was introduced in [
10].
Through the above investigation, we have learned how to save resources and achieve fault-tolerant performance in multiagent collaborative control without using fault detection. The designed controller can handle actuator faults, system uncertainties and external disturbances while tracking uncertain leader signals, which is a challenging research topic. In this paper, we synthesize a controller based on an adaptive distributed observer and nonsingular fast terminal sliding mode surface to solve the fault-tolerant cooperative control problem of multiple uncertain Euler–Lagrange systems, which are subjected to actuator faults. In existing research, designing a distributed observer requires assuming that some followers know the dynamics of the leader system [
27,
28]. However, in many practical applications, followers may not have precise knowledge of the leader system’s dynamics. The main contributions of this work can be stated as follows. Initially, an adaptive distributed observer is devised for a linear leader system with neutral stability, featuring an uncertain system matrix. This observer can estimate and transmit the leader’s state to each follower via the system’s communication network, even without precise knowledge of the leader’s system matrix. Secondly, an adaptive approach is applied to estimate the aggregate uncertainty, removing the need to compute its upper limit. This method exhibits resilience to actuator faults, uncertain parameters and external disturbances, the upper bounds of which can be arbitrarily large and unidentified. Finally, the controller in this paper has a simple structure without fault diagnosis, and it is also suitable for situations where the actuator is healthy. The complicated and costly fault detection, diagnosis and identification are not required.
This paper is structured as follows.
Section 2 presents the formulation of the fault-tolerant cooperative control problem.
Section 3 describes the development of a self-adjusting observer for dynamic leader systems with unknown parameters. In
Section 4, we design the adaptive fault-tolerant controller. In
Section 5, an example is presented. Finally, we conclude this paper.
In what follows, the following notation will be adopted. For , , . denotes the Euclidean norm of a vector a. For , and is defined as . For , , . For , let be such that .
2. Problem Formulation
Consider the following Euler–Lagrange equations with unknown dynamic uncertainties:
where
,
and
are the joint variable vectors of generalized position and velocity of the
ith agent,
is the symmetric positive definite inertia matrix,
is the Coriolis and Centripetal forces,
is the gravity term,
,
,
,
,
and
are nominal values of inertia, Coriolis and centripetal forces and gravity, respectively,
,
and
are unknown dynamic uncertainties of inertia, Coriolis and centripetal forces and gravity, respectively,
is the control torque and also represents the actuator’s output and
signifies uncertain external disturbances.
The following exosystem produces the reference signal:
where
is the state,
is the system matrix,
is an unspecified parameter vector,
is a known constant matrix and
is the desired trajectory to track.
Inspired by [
31], this paper considers the following actuator faults:
where
is nominal actuator output,
is actuator fault,
is bounded bias fault,
,
is actuator effectiveness,
is the identity matrix.
The multiagent system consists of
N followers as described in (
1) and an uncertain leader as defined in (
2). The communication graph is represented by a directed graph
, where
is the node set and
is the edge set. The edge set
is defined such that, for
,
,
if and only if the node
i can receive information of node
j. The weighted adjacency
,
, where
;
, if
; otherwise
. Define the Laplacian matrix associated with
as
, where
and
for
,
. Define the subgraph
of
where
and
is obtained from
by eliminating all edges between the node 0 and the nodes in
.
Fault-tolerant Cooperative Control Problem: Consider a multiagent system composed of multiple uncertain Euler–Lagrange systems (
1) subject to actuator faults (
3) and a leader system (
2), a fixed graph
; find an adaptive control law such that for any initial conditions
,
and
, the solution of the closed-loop system exists and is bounded for all
and the leader-following cooperative tracking error
is ultimately bounded,
.
Remark 1. The fault-tolerant cooperative control problem is also addressed by [32], ensuring uniform bounds on tracking errors. To solve the above problem, we need the following assumptions.
Assumption 1. The actuator fault is bounded by , where the constant .
Assumption 2. The graph contains a directed spanning tree with the leader as the root and the subgraph is undirected.
Assumption 3. The matrix is neutrally stable.
Remar 2. Assumption 1 is the standard assumption. The fault in this article considers partial loss failure fault and bounded bias fault of the actuator; thus, it can be assumed that is bounded [31]. Assumption 2 is standard in a communication topology graph for multiagent systems and has been widely found in existing results [26,33]. Assumption 3 is also assumed in [10,28] and it is not conservative. Under Assumption 3, without loss of generality, we can presume that , where , represents a diagonal matrix, , . 4. Adaptive Fault-Tolerant Cooperative Controller Design
For the solvability of the fault-tolerant cooperative control problem, we need three properties for system (
1) [
8,
37].
Property 1. There are two positive constants and , such that .
Property 2. There are two positive constants and , such that and , respectively..
Property 3. The matrix is skew-symmetric.
Let
; the dynamics model of (
1) subject to actuator faults (
3) is equivalently rearranged as the following form:
where
and the lumped uncertainty
.
Using the self-adjusting observer
for the leader signal
v in (
2) which is designed in (
5), we further design an adaptive controller to solve the fault-tolerant cooperative control problem for the multiple uncertain Euler–Lagrange systems. The definition of the tracking error
is as follows:
The second error
is defined as follows:
where
is the virtual control,
,
,
is a positive constant and
is an n-dimensional identity matrix. Differentiating
, we obtain
The time derivative of
is
By Theorem 1 in [
38], we can choose a nonsingular fast terminal sliding mode surface
where
and
are positive matrices,
and
are positive odd numbers satisfying the relations
and
. If
, the convergence time
T is defined as
, where
where
is the initial value of
and
is Gauss’ hypergeometric function.
As the quantity of follower agents expands, the task of computing the upper limit becomes more cumbersome and involved. Therefore, we design an adaptive method to estimate the lumped uncertainty , denoted as ; we establish the lumped uncertainty estimation as . Subsequently, the estimation error is expressed as . Afterwards, we offer the subsequent assumption.
Assumption 4. It is assumed that the lumped uncertainty is upper bounded [31] and the rate of change is also upper bounded [39]; i.e., there exist unknown constants and , , such that and , such that and . Remark 3. In [39], the norm of partial derivatives for the unknown component is bounded, which is composed of system state variables, disturbances and actuator faults. In this paper, our lumped uncertainty is also composed of system state variables, disturbances and actuator faults; thus, it is reasonable to make Assumption 4 in the following theoretical analysis. The controller
is designed as
where
and the adaption law is designed as
where
,
,
is the Moore–Penrose pseudoinverse of
, which satisfies
,
and
are positive constants.
Theorem 1. Consider a multiagent system consisting of (27) and (2), a fixed graph and assume that Assumptions 1–4 hold. Then, take the adaptive fault-tolerant controller (34), such that for any initial conditions , , , , , the solution of the closed-loop system exists and is bounded for all and the tracking errors , and the lumped uncertainty estimation error are ultimately bounded. Proof. The Lyapunov function candidate is proposed
Differentiating (
39), we have
Substituting (
34) into (
40), we have
where
Hense, substituting (
42) into (
41), we can have
where
,
are two positive constants provided as
where
and
. In order to guarantee that
, the gains
are chosen to satisfy
By multiplying both sides with
in Equation (
43), we obtain
Define
we have
Thus, we obtain the compact sets
where
with
,
given in (
45) and (
44), respectively. Consequently, we can obtain that
and
and
are ultimately bounded. By (
25), we can obtain
and combine the above
, which is ultimately bounded; thus, we can obtain that
is ultimately bounded; thus, the fault-tolerant cooperative problem can be solved and the tracking error
can be made as small as possible. □
Remark 4. By (48) and (49), we can see that and determine the size of , and ; that is, the smaller the tracking errors, the bigger and the smaller should be. By (44), we can see that the bigger is, the bigger the control parameters , and should be. By (45), we can see that the smaller is, the smaller the parameter should be. Combining (44), (45) and (48), the influence of becoming smaller in (45) is greater than the influence of becoming larger in (44) for , i.e., becoming smaller can make smaller. Thus, the right sides of (50) can be set as small as possible, i.e., the tracking error is guaranteed to be small enough by choosing the small parameter and the big parameters , . Compared with the semi-globally bounded tracking error in [40], the tracking error in this paper is globally bounded and the bound can be sufficiently small from the above analysis.