Adaptive Fault-Tolerant Control for Second-Order Multiagent Systems with Unknown Control Directions via a Self-Tuning Distributed Observer

Abstract

In this paper, we first design a self-tuning distributed observer for second-order multi-agent systems which is capable of providing the estimation of the leader’s signal to various followers. We then further develop an adaptive sliding-mode controller to solve the cooperative tracking problem between leader and followers for second-order multi-agent systems subject to time-varying actuator faults and unknown external disturbances, which can ensure that the leader-following cooperative tracking errors converge to zero asymptotically. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed controller. This control law offers three advantages: first, the problem of communication barriers among the leader and followers can be solved by the self-tuning distributed observer, which can calculate the observer gain online; second, a new type of adaptive sliding-mode controller is proposed by introducing a Nussbaum function; and lastly, the bounds of unknown actuator faults and unknown external disturbances can be adaptively estimated.

1. Introduction

During practical operation involving complex work, actuators are most likely to suffer from faults in the control system. Partial loss of actuator effectiveness (PLOAE) is a common actuator fault. In the past past decade, there have been different controllers developed for PLOAE faults, among which adaptive fault-tolerant controllers are the most common and effective. Different types of fault-tolerant controllers have been designed according to the changing conditions and control objectives of the faults. PLOAE faults are commonly divided into two types: those in which the change of actuator partial failure failure suddenly becomes an unknown bounded constant at an unknown moment [1,2,3], and those in which partial actuator failure is time-varying and the degree of failure remains bounded [4,5]. The adaptive controller in [1] and the distributed adaptive controller in [3] are designed for PLOAE faults, and can realize the tracking error between followers and leader asymptotically converging to zero; yet, they do not consider how to deal with unknown external disturbances in the follower system. Controllers have been designed for continuous time-varying PLOAE faults [4,5], where consensus errors were steered into small residual sets. Similarly, the controller in [4] was not designed to deal with unknown external disturbances in the followers system. In [3], an adaptive fault-tolerant control scheme was proposed for a category of multiagent systems, where the unknown fault parameters were limited to be constants only. A more practical (yet challenging) case is that the fault parameters, including the actuation effectiveness and the additive fault, vary with time, resulting in time-varying actuator faults. The controller in [1,2,3,4,5] required the control directions of agents to be known. A fault-tolerant controller for nonlinear multiagent systems with Unknown Control Directions was designed in [6,7].

On the other hand, the design of controllers for multi-agent systems is challenging because of communication constraints. The distributed observer launched in [8] represents a key technique for generating a distributed controller, which is a dynamic compensator capable of providing the estimation of the leader’s signal to various followers. However, a shortcoming of this distributed observer is that it assumes each follower knows the system matrix of the leader, which is not desirable in certain applications. For those followers who are not the children of the leader, this assumption is not desirable. An adaptive distributed observer was further proposed in [9,10], which only requires those followers who are the children of the leader to know the system matrix of the leader. A key parameter in a distributed observer or adaptive distributed observer is the observer gain; it is quite difficult to find the lower bound of this parameter in order to guarantee the stability of the closed-loop system. The gain of the observer needs to be adequately large in [11], and it is difficult to obtain an appropriate value of the gain when N is very large. Thus, the design for the gain of an adaptive observer is an interesting area of research, in the sense that it can avoid the need to calculate the observer gain offline.

Therefore, for second-order multi-agent systems subject to time-varying actuator faults with unknown control direction, designing a controller able to make the tracking error between followers and the leader converge to zero asymptotically and reject unknown disturbances from external signals is challenging. In this paper, the principal contributions can be formally summarized as follows: (1) The leader signal is produced by a nonlinear autonomous system, requiring further improvements to the distributed observers in [12], where the leader signal was provided by a linear autonomous system. The problem of communication barriers between the leader and the followers is solved by a self-tuning distributed observer, which can calculate the observer gain online. (2) The actuator suffers from PLOAE faults and additive faults with unknown external disturbances, which are considered in this article. Many unknown time-varying parameters are caused by the occurrence of faults together with an unknown control direction. To handle this problem, a new type of adaptive sliding-mode controller is introduced. To establish closed-loop stability, new Lyapunov functions are introduced. It is proved that all signals in the closed-loop system are bounded. (3) By directly estimating the bounds of the loss of actuator effectiveness, additive faults, and unknown external disturbances instead of errors themselves, the proposed controller can ensure that leader-following cooperative tracking errors approach to zero asymptotically by adjusting control parameters. To the best of our knowledge, this is the first adaptive sliding-mode controller designed for the category of second-order multi-agent systems suffering from time-varying actuator faults and unknown external disturbances via a self-tuning distributed observer.

The rest of this paper is structured as follows. In Section 2, the control problem is formulated. In Section 3, a self-tuning distributed observer for a nonlinear leader system is established. In Section 4, an adaptive sliding-mode controller is designed to solve the leader-following cooperative tracking problem for a category of second-order multi-agent systems subject to actuator faults and unknown external disturbances. In Section 5, a simulation is provided to demonstrate the effectiveness of the controller. Finally, we conclude the paper in Section 6.

In this paper, the following notations are adopted: $\left|\right|x\left|\right|$ indicates the Euclidean norm of a vector x; furthermore, for $x_i\in R^{n_i\times p}$, $i=1,\dots ,q$, $col\left(x_1,\dots ,x_q\right)={\left[x_1^T,\dots ,x_q^T\right]}^T$.

2. Problem Formulation

In this paper, we consider a category of second-order multi-agent systems of the following form:

$$\begin{array}{cc}\hfill {\dot{x}}_{i1}& =x_{i2}\hfill \\ \hfill {\dot{x}}_{i2}& =b_i{u}_i+d_i(x_i,t)\hfill \\ \hfill y_i& =x_{i1}\hfill \\ \hfill e_i& =y_i-y_0,\hfill \end{array}$$

(1)

where, for $i=1,\dots ,N$, $x_i=col(x_{i1},x_{i2})\in {\mathbb{R}}^{2n}$ is the state, $u_i\in {\mathbb{R}}^n$ is the ith input (i.e., the output of the ith actuator), the signs of $b_i$ are a constant scalar the signs and magnitudes of which are unknown and which represents the control directions of the ith agent, $d_i(x_i,t)\in {\mathbb{R}}^n$ is an unknown external disturbance, $y_i\in {\mathbb{R}}^n$ is the measurement output of the ith subsystem, $e_i\in {\mathbb{R}}^n$ is the error output of the ith subsystem, $y_0\in {\mathbb{R}}^n$ is the output of exogenous signal v, and v is an exogenous signal indicating the reference input and/or the external disturbance.

It is assumed that actuators in the multi-agent system suffer from faults of the same form as in [13], described by

$$u_i=h_i\left(t\right)u_{ci}+{\varphi }_i\left(t\right)$$

(2)

where $u_{ci}\in {\mathbb{R}}^n$ is defined as the input of the ith actuator, while $h_i\left(t\right)\in \mathbb{R}$ and ${\varphi }_i\left(t\right)\in \mathbb{R}$ are unknown and denote the actuation effectiveness and the additive fault of the ith subsystem, respectively

Then, we can consider system (1) with actuator faults described by

$$\begin{array}{cc}\hfill {\dot{x}}_{i1}& =x_{i2}\hfill \\ \hfill {\dot{x}}_{i2}& ={\sigma }_i\left(t\right)u_{ci}+{\varphi }_i\left(t\right)+d_i(x_i,t)\hfill \\ \hfill y_i& =x_{i1}\hfill \\ \hfill e_i& =y_i-y_0,i=1,\dots ,N\hfill \end{array}$$

(3)

where ${\sigma }_i\left(t\right)=b_i{h}_i\left(t\right)$.

It is assumed that the exogenous signal $v\left(t\right)$ is generated by a nonlinear autonomous system with the same form as in [11]:

$$\begin{array}{cc}\hfill \dot{v}& =g\left(v\right)\hfill \\ \hfill y_0& =a\left(v\right)\hfill \end{array}$$

(4)

where $v\in {\mathbb{R}}^m$, $g(·){\mathbb{R}}^m\mapsto {\mathbb{R}}^m$, and $a(·):{\mathbb{R}}^m\mapsto {\mathbb{R}}^n$ are globally defined and sufficiently smooth functions vanishing severally at the origin of ${\mathbb{R}}^m$, ${\mathbb{R}}^m$ and ${\mathbb{R}}^{m+n}$.

We now define a digraph $\overline{\mathcal{G}}=(\overline{\mathcal{V}},\overline{\mathcal{E}})$, where $\overline{\mathcal{V}}=\{0,1,\dots ,N\}$ is the node set with node 0 associated with the leader system (4) and node i, $i=1,\dots ,N$, associated with the ith follower system of (3) and $\overline{\mathcal{E}}\subseteq \overline{\mathcal{V}}\times \overline{\mathcal{V}}$, is the edge set. The edge set $\overline{\mathcal{E}}$ is defined such that, for $i=1,\dots ,N$, $j=0,1,\dots ,N$, $i\ne j,(j,i)\in \overline{\mathcal{E}}$, if and only if the control $u_{ci}$ of the ith follower can approach the data of agent j. In addition, we define a subgraph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ of $\overline{\mathcal{G}}$ where $\mathcal{V}=\{1,\dots ,N\}$ and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is obtained from $\overline{\mathcal{E}}$ by eliminating all edges between the nodes in $\mathcal{V}$ and node 0. Here, let ${\overline{\mathcal{N}}}_i=\left\{j,(j,i)\in \overline{\mathcal{E}}\right\}$ indicate the neighbor set of agent i.

Problem 1. 

Considering systems (3) and (4), a fixed graph $\overline{\mathcal{G}}$ and any compact subset ${\mathbb{V}}_0\subset {\mathbb{R}}^m$ contain the origin. We can consider a multi-agent system subject to actuator faults and unknown external disturbances and find an adaptive sliding-mode control law such that for any initial conditions $v\left(0\right)\in {\mathbb{V}}_0$, $x_i\left(0\right)\in {\mathbb{R}}^n$, the solution of the closed-loop system exists and is bounded for all $t\ge 0$ and the leader-following cooperative tracking errors satisfy ${lim}_{t\to \infty }{e}_i\left(t\right)=0$ and ${lim}_{t\to \infty }{\dot{e}}_i\left(t\right)=0$, $i=1,\dots ,N$.

To solve Problem 1, we list several assumptions.

Assumption 1. 

The graph $\overline{\mathcal{G}}$ contains a spanning tree with the leader as the root, and the subgraph $\mathcal{G}$ associated with the followers is undirected.

Assumption 2. 

For any $v_0\left(0\right)\in R^m$, the solution of system (4) exists and is bounded for all $t\ge 0$.

Assumption 3. 

For each follower i, there exists an unknown and bounded constant ${\omega }_i$ such that $\left|\right|d_i(x_i,t)\left|\right|\le {\omega }_i$, $i=1,\dots ,N$.

Assumption 4. 

For the time-varying effectiveness factor indicator of the actuator $h_i\left(t\right)$ in the ith subsystem, there exist unknown constants ${\underline{h}}_i$ and ${\overline{h}}_i$ such that the inequality ${\underline{h}}_i\le \left|h_i\right|\le {\overline{h}}_i$ is always satisfied.

Assumption 5. 

For each follower i, there exists an unknown and bounded constant ${ϵ}_i$ such that $\left|\right|{\varphi }_i\left(t\right)\left|\right|\le {ϵ}_i$, $i=1,\dots ,N$.

Assumption 6. 

Let the function $g\left(v\right)=G_{1}v+G_2\left(v\right)v$, $G_1\in {\mathbb{R}}^{m\times m}$ be the Jacobian matrix of the function $g\left(v\right)$ at the origin, and let $G_2\left(v\right)=diag\left\{d_1\left(v\right),\dots ,d_m\left(v\right)\right\}$ with $d_j\left(v\right)\le 0$, $j=1,\cdots ,m$.

Remark 1. 

Assumption 1 is standard in systems subject to static networks, and can be widely found in existing results (see, e.g., [11,14]). In many cases, the communication among agents is bidirectional, and as such can be described by an undirected graph. Assumption 2 is used to make the problem reasonably well-posed, otherwise the solution of the closed-loop system may not be bounded [11]. In the actual system, the information of Assumption 3 can be easily obtained [15]. Assumption 4 indicates that there is no singularity problem for the control system [13]. Per Assumption 4 and ${\sigma }_i\left(t\right)=b_i{h}_i\left(t\right)$, $b_i$ is constant scalar the signs and magnitudes of which are unknown; thus, there exist unknown constants ${\underline{\sigma }}_i$ and ${\overline{\sigma }}_i$ such that the inequality ${\underline{\sigma }}_i\le \left|{\sigma }_i\right|\le {\overline{\sigma }}_i$ is satisfied. Assumption 5 is standard and indicates that the degree of executor fault is bounded. Assumption 6 is used to guarantee the stability of system (10), and holds for many nonlinear systems, including the Van der Pol system.

3. Self-Tuning Distributed Observer

The observer for the leader system (4) for $i=1,\cdots ,N$ is designed as follows. Let $\overline{\mathcal{A}}={\left[a_{ij}\right]}_{i,j=0}^N\in {\mathbb{R}}^{(N+1)\times (N+1)}$ be the weighted adjacency matrix of the graph $\overline{\mathcal{G}}$. We first recall the concept of a distributed observer developed in [11]:

$$\begin{array}{ccc}\hfill {\dot{\eta }}_i& =& g\left({\eta }_i\right)-\gamma \sum _{j=0}^N{a}_{ij}\left({\eta }_i-{\eta }_j\right)\hfill \end{array}$$

(5)

For $i=1,\cdots ,N$, $j=0,1,\cdots ,N$, $a_{ii}=0,a_{ij}=1$ if $(j,i)\in \overline{\mathcal{E}}$ and $a_{ij}=0$; otherwise, let ${\eta }_0=v$.

The drawback of the observer (5) is that the observer gain $\gamma$ has to be sufficiently large in order to guarantee the solvability of the problem, and it is difficult to obtain an appropriate value of $\gamma$ when N is large [12]. To overcome the difficulty, we propose the following self-tuning distributed observer:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\eta }}_i& =g\left({\eta }_i\right)-\sum _{j=0}^N{\gamma }_{ij}{a}_{ij}\left({\eta }_i-{\eta }_j\right),\hfill \\ \hfill {\dot{\gamma }}_{ij}& =k_{ij}{a}_{ij}{\left({\eta }_i-{\eta }_j\right)}^T\left({\eta }_i-{\eta }_j\right)\hfill \\ & i=1,\dots ,N,j=0,1,\dots ,N;\hfill \end{array}\end{array}$$

(6)

here, let ${\eta }_{vi}={\sum }_{j=0}^N{a}_{ij}\left({\eta }_i-{\eta }_j\right)$, $k_{ij}=k_{ji}>0$, $i,j=1,\dots ,N$, $k_{i0}>0$, $i=1,\dots ,N$, ${\gamma }_{ij}={\gamma }_{ji}$ for $i,j=1,\dots ,N$.

To construct an energy function for the system (6), let $\eta =col\left({\eta }_1,{\eta }_2,\dots ,{\eta }_N\right)$. We define a matrix $F_1$ as follows:

$$\begin{array}{c}\hfill F_1={\left[f_{ij}\right]}_{i,j=1}^N\end{array}$$

(7)

where $f_{ii}={\sum }_{j=0}^N{\gamma }_{ij}{a}_{ij}$ and $f_{ij}=-{\gamma }_{ij}{a}_{ij}$ for any $i\ne j$, $i,j=1,\cdots ,N$. We define a matrix $F_2$ as follows:

$$\begin{array}{c}\hfill F_2=diag({\gamma }_{i0}{a}_{i0},\cdots ,{\gamma }_{N0}{a}_{N0})\end{array}$$

(8)

Let $\mathbf{g}\left(\eta \right):=col(g\left({\eta }_1\right),\dots ,g\left({\eta }_N\right))$; then,

$$\begin{array}{c}\hfill \dot{\eta }=\mathbf{g}\left(\eta \right)-(F_1\otimes I_m)\eta +(F_2\otimes I_m)({\mathbf{1}}_N\otimes v).\end{array}$$

(9)

Supposing that $v\left(t\right)\equiv 0$ for all $t\ge 0$, the system (9) decreases to

$$\begin{array}{c}\hfill \dot{\eta }=\mathbf{g}\left(\eta \right)-(F_1\otimes I_m)\eta \end{array}$$

(10)

Lemma 1. 

Considering system (10), under Assumptions 1 and 6 there exist ${\gamma }_{ij}>0$ that can change adaptively such that the solution of system (9) is bounded for any initial conditions $v\left(0\right),{\eta }_i\left(0\right)\in \mathbb{V}\left(0\right)$, $i=1,\cdots ,N$.

Proof. 

By definition, ${\gamma }_{ij}>0$; then, per Part (iv) of Theorem 2.5.2 of [16], $F_1$ is a Metzler-matrix if and only if Assumption 1 is satisfied. Thus, per Corollary 2.5.6 of [16], there exists a diagonal $S=diag\{s_1,\cdots ,s_N\}$ with $s_i>0$, $i=1,\cdots ,N$ such that the matrix

$$\begin{array}{c}\hfill T=S{F}_1+F_1^{T}S\end{array}$$

(11)

is symmetric and positive definite. Then, the our energy function can be presented as follows:

$$\begin{array}{c}\hfill V\left(\eta \right)={\eta }^T(S\otimes I_m)\eta \end{array}$$

(12)

where the construction of the matrix S proceeds as in [17]. Under Assumption 5,

$$\begin{array}{c}\hfill \mathbf{g}\left(\eta \right)=\left(I_N\otimes G_1+{\mathbf{G}}_2\left(\eta \right)\right),\eta \end{array}$$

(13)

where ${\mathbf{G}}_2\left(\eta \right)=blockdiag\left\{G_2\left({\eta }_1\right),\cdots ,G_2\left({\eta }_N\right)\right\}$. Then, we can calculate the derivative of the energy function (12) along system (10):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left.\dot{V}\left(\eta \right)\right|}_{\left(10\right)}& =-{\eta }^T(T\otimes I_m)\eta +2{\eta }^T(S\otimes I_m)(I_N\otimes G_1+{\mathbf{G}}_2\left(\eta \right))\eta \hfill \\ & \le -{\eta }^T\left(T\otimes I_m\right)\eta +2{\eta }^T\left(S\otimes G_1\right)\eta \hfill \\ & \le (2{\lambda }_{max}(S\otimes G_1)-{\lambda }_{min}\left(T\right)){\parallel \eta \parallel }^2\hfill \end{array}\end{array}$$

(14)

Because ${\gamma }_{ij}$ in matrix T can adaptively change such that ${\lambda }_{min}\left(T\right)>2{\lambda }_{max}(S\otimes G_1)$, the origin of system (10) is globally exponentially stable. Furthermore, because the Lyapunov function $V\left(\eta \right)$ for system (10) as defined in (12) is positive definite and quadratic, and its derivative along system (10) is negative definite and quadratic when ${\gamma }_{ij}$ in matrix T adaptively changes such that ${\lambda }_{min}\left(T\right)>2{\lambda }_{max}(S\otimes G_1)$, there exist positive constants $d_1$, $d_2$, $d_3$, $d_4$ such that the following inequalities are satisfied globally:

$$\begin{array}{c}\hfill \begin{array}{c}{d}_1{\parallel \eta \parallel }^2\le V\left(\eta \right)\le d_2{\parallel \eta \parallel }^2\hfill \\ \frac{\partial V\left(\eta \right)}{\partial \eta }(\mathbf{g}\left(\eta \right)-(F_1\otimes I_m)\eta )\le -d_3{\parallel \eta \parallel }^2\hfill \\ ∥\frac{\partial V\left(\eta \right)}{\partial \eta }∥\le d_4\parallel \eta \parallel \hfill \end{array}\end{array}$$

(15)

Then, along the trajectory of system (9) and with $0<d_5<d_3$, we can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left.\dot{V}\left(\eta \right)\right|}_{\left(9\right)}=& \frac{\partial V\left(\eta \right)}{\partial \eta }\left(\mathbf{g}\left(\eta \right)-\left(F_1\otimes I_m\right)\eta \right)+\frac{\partial V\left(\eta \right)}{\partial \eta }\left(F_2{\mathbf{1}}_N\otimes v\right)\hfill \\ \hfill \le & -d_3{\parallel \eta \parallel }^2+d_4\parallel \eta \parallel ∥F_2{\mathbf{1}}_N\otimes v∥\hfill \\ \hfill \le & -d_3{\parallel \eta \parallel }^2+\frac{d_4^2}{4d_5}{∥F_2{\mathbf{1}}_N\otimes v∥}^2+d_5{\parallel \eta \parallel }^2\hfill \\ \hfill \le & -(d_3-d_5){\parallel \eta \parallel }^2+\frac{d_4^2}{4d_5}{∥F_2{\mathbf{1}}_N\otimes v∥}^2\hfill \\ \hfill \le & -d_{6}V\left(\eta \right)+\epsilon \left(v\right)\hfill \end{array}\end{array}$$

(16)

where $d_6:=\frac{d_3-d_5}{d_2}$, $\epsilon \left(v\right):=\frac{d_4^2}{4d_5}{∥F_2{\mathbf{1}}_N\otimes v∥}^2$. By Comparison with Lemma [18],

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V\left(\eta \right(t\left)\right)& \le e^{-d_{6}t}V\left(\eta \left(0\right)\right)+{\int }_0^t{e}^{-d_6(t-\tau )}\epsilon \left(v\left(\tau \right)\right)d\tau \hfill \\ & \le c_2{e}^{-d_{6}t}{\parallel \eta \left(0\right)\parallel }^2+{\int }_0^t{e}^{-d_6(t-\tau )}\epsilon \left(v\left(\tau \right)\right)d\tau .\hfill \end{array}\end{array}$$

(17)

Under Assumption 1, $v\left(t\right)$ is bounded over $[0,\infty )$ for any initial condition $v\left(0\right)\in \mathbb{V}\left(0\right)$, $V\left(\eta \right(t\left)\right)$, and hence $\eta \left(t\right)$ of system (9) is bounded over $[0,\infty )$ for any initial conditions ${\eta }_i\left(0\right)\in \mathbb{V}\left(0\right)$, $i=1,\cdots ,N$. □

Lemma 2. 

Given Systems (4) and (6), under Assumptions 1, 2, and 6, for all ${\eta }_0\in {\mathbb{V}}_0$, with ${\mathbb{V}}_0$ being a compact set of ${\mathbb{R}}^m$, there exist ${\gamma }_{ij}>0$, $k_{ij}>0$, $i=1,\dots ,N$, $j=0,\dots ,N$ such that for any ${\eta }_0\in {\mathbb{V}}_0$ the solution of system (9) exists for all $t\ge 0$ and satisfies ${lim}_{t\to \infty }(\eta \left(t\right)-{\mathbf{1}}_N\otimes v\left(t\right))=0$ asymptotically.

Proof. 

For $i=1,\dots ,N$, $j=0,1,\dots ,N$, let ${\overline{\gamma }}_{ij}={\gamma }_{ij}-\gamma$, with some unknown $\gamma >0$, ${\overline{\eta }}_i={\eta }_i-{\eta }_0$. Then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\overline{\eta }}}_i& ={\overline{g}}_i\left({\overline{\eta }}_i,{\eta }_0\right)-\sum _{j=0}^N{\gamma }_{ij}{a}_{ij}({\overline{\eta }}_i-{\overline{\eta }}_j)\hfill \\ \hfill {\dot{\overline{\gamma }}}_{ij}& =k_{ij}{a}_{ij}{\left({\overline{\eta }}_i-{\overline{\eta }}_j\right)}^T\left({\overline{\eta }}_i-{\overline{\eta }}_j\right),\hfill \\ & i=1,\dots ,N,j=0,1,\dots ,N\hfill \end{array}\end{array}$$

(18)

where ${\overline{g}}_i({\overline{\eta }}_i,{\eta }_0)=g\left({\eta }_i\right)-g\left({\eta }_0\right)=g({\overline{\eta }}_i+{\eta }_0)-g\left({\eta }_0\right)$. To construct an energy function for system (18), let $\overline{\eta }=col\left({\overline{\eta }}_1,{\overline{\eta }}_2,\dots ,{\overline{\eta }}_N\right)$, $\overline{\gamma }=col({\gamma }_{10},\dots ,{\gamma }_{N0},{\gamma }_{12},\dots ,{\gamma }_{1N},\dots ,{\gamma }_{(N-1)1},$$\dots ,{\gamma }_{(N-1)N})$.

We are now ready to present our energy function for system (18), as follows:

$$\begin{array}{c}\hfill V(\overline{\eta },\overline{\gamma },t)=V_1(\overline{\eta },t)+V_2(\overline{\gamma },t)\end{array}$$

(19)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_1(\overline{\eta },t)& =\frac{1}{2}\sum _{i=1}^N{\overline{\eta }}_i^T{\overline{\eta }}_i,V_2(\overline{\gamma },t)=\frac{1}{2}\sum _{i=1}^N\left(\sum _{j\in {\mathcal{N}}_i\left(t\right)}\frac{{\overline{\gamma }}_{ij}^2}{2k_{ij}}+\sum _{0\in {\overline{N}}_i\left(t\right)}\frac{{\overline{\gamma }}_{i0}^2}{k_{i0}}\right)\hfill \end{array}\end{array}$$

(20)

To calculate the derivative of the energy function (19) along system (18), we recall the Laplacian matrix of $\overline{\mathcal{G}}$ defined in [12], as follows:

$$\begin{array}{c}\hfill H={\left[h_{ij}\right]}_{i,j=1}^N\end{array}$$

(21)

where $h_{ii}={\sum }_{j=0}^N{a}_{ij}$ and $h_{ij}=-a_{ij}$ for any $i\ne j$, $i,j=1,\cdots ,N$.

The derivative of the functions $V_1$ along system (18) is as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_1(\overline{\eta },t)& =\sum _{i=1}^N{\overline{\eta }}_i^T({\overline{g}}_i\left({\overline{\eta }}_i,{\eta }_0\right)-\sum _{j=0}^N{\gamma }_{ij}{a}_{ij}({\overline{\eta }}_i-{\overline{\eta }}_j))\hfill \\ & =\sum _{i=1}^N{\overline{\eta }}_i^T{\overline{g}}_i\left({\overline{\eta }}_i,{\eta }_0\right)-\sum _{i=1}^N{\overline{\eta }}_i^T\sum _{j\in {\overline{N}}_i}{\gamma }_{ij}({\overline{\eta }}_i-{\overline{\eta }}_j)\hfill \end{array}\end{array}$$

(22)

Because ${\overline{g}}_i({\overline{\eta }}_i,{\eta }_0)=g({\overline{\eta }}_i+{\eta }_0)-g\left({\eta }_0\right)$ and $\overline{\eta }=col({\overline{\eta }}_1,{\overline{\eta }}_2,\cdots ,{\overline{\eta }}_N)$, we have the following compact form:

$$\begin{array}{c}\hfill \overline{\mathbf{g}}(\overline{\eta },{\eta }_0)=\mathbf{g}\left(\eta \right)-{\mathbf{1}}_N\otimes \mathbf{g}\left({\eta }_0\right)\end{array}$$

(23)

Let $\mathbf{g}\left({\eta }_0\right):=col(g_1\left({\eta }_0\right),\dots ,g_m\left({\eta }_0\right))$ where $g_j(·):{\mathbb{R}}^m\mapsto \mathbb{R}$, $j=1,\dots ,m$, then

$$\begin{array}{c}\hfill {\dot{V}}_1(\overline{\eta },t)={\overline{\eta }}^T\overline{\mathbf{g}}(\overline{\eta },{\eta }_0)-\sum _{i=1}^N{\overline{\eta }}_i^T\sum _{j\in {\overline{N}}_i}{\gamma }_{ij}({\overline{\eta }}_i-{\overline{\eta }}_j)\end{array}$$

(24)

The derivative of the functions $V_2$ along system (18) is as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_2(\overline{\gamma },t)& =\sum _{i=1}^N\left(\sum _{j\in {\mathcal{N}}_i}\frac{{\overline{\gamma }}_{ij}{\dot{\overline{\gamma }}}_{ij}}{2k_{ij}}+a_{i0}\frac{{\overline{\gamma }}_{i0}{\dot{\overline{\gamma }}}_{i0}}{k_{i0}}\right)\hfill \\ & =\sum _{i=1}^N\left(\sum _{j=1}^N\frac{{\overline{\gamma }}_{ij}}{2k_{ij}}{k}_{ij}{a}_{ij}{({\overline{\eta }}_i-{\overline{\eta }}_j)}^T({\overline{\eta }}_i-{\overline{\eta }}_j)\right.+\left.a_{i0}\frac{{\overline{\gamma }}_{i0}}{k_{i0}}{k}_{i0}{\overline{\eta }}_i^T{\overline{\eta }}_i\right)\hfill \\ & =\sum _{i=1}^N\frac{1}{2}\left(\sum _{j=1}^N{a}_{ij}{\overline{\gamma }}_{ij}{\overline{\eta }}_i^T\left({\overline{\eta }}_i-{\overline{\eta }}_j\right)\right.+\left.\sum _{j=1}^N{a}_{ij}{\overline{\gamma }}_{ij}{\overline{\eta }}_j^T\left({\overline{\eta }}_j-{\overline{\eta }}_i\right)\right)+\sum _{i=1}^N{a}_{i0}{\overline{\gamma }}_{i0}{\overline{\eta }}_i^T{\overline{\eta }}_i\hfill \end{array}\end{array}$$

(25)

Because $a_{ij}{\overline{\gamma }}_{ij}=a_{ji}{\overline{\gamma }}_{ji}$, $i,j=1,\dots ,N$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\sum _{i=1}^N\sum _{j=1}^N{a}_{ij}{\overline{\gamma }}_{ij}{\overline{\eta }}_j^T\left({\overline{\eta }}_j-{\overline{\eta }}_i\right)=\sum _{i=1}^N\sum _{j=1}^N{a}_{ji}{\overline{\gamma }}_{ji}{\overline{\eta }}_i^T\left({\overline{\eta }}_i-{\overline{\eta }}_j\right)\hfill \\ =\sum _{i=1}^N\sum _{j=1}^N{a}_{ij}{\overline{\gamma }}_{ij}{\overline{\eta }}_i^T\left({\overline{\eta }}_i-{\overline{\eta }}_j\right)=\sum _{i=1}^N\sum _{j\in {\mathcal{N}}_i\left(t\right)}{\overline{\gamma }}_{ij}{\overline{\eta }}_i^T\left({\overline{\eta }}_i-{\overline{\eta }}_j\right)\hfill \end{array}\end{array}$$

(26)

Thus,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_2(\overline{\gamma },t)& =\sum _{i=1}^N(\sum _{j\in {\mathcal{N}}_i}{\overline{\gamma }}_{ij}{\overline{\eta }}_i^T\left({\overline{\eta }}_i-{\overline{\eta }}_j\right)+\sum _{0\in {\overline{N}}_i\left(t\right)}{\overline{\gamma }}_{i0}{\overline{\eta }}_i^T{\overline{\eta }}_i)\hfill \\ & =\sum _{i=1}^N\sum _{j\in {\overline{N}}_i}{\overline{\gamma }}_{ij}{\overline{\eta }}_i^T\left({\overline{\eta }}_i-{\overline{\eta }}_j\right)\hfill \end{array}\end{array}$$

(27)

Combining (24) and (27) provides

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{V}(\overline{\eta },\overline{\gamma },t)& ={\overline{\eta }}^T\overline{\mathbf{g}}(\overline{\eta },{\eta }_0)-\sum _{i=1}^N{\overline{\eta }}_i^T\sum _{j\in {\overline{N}}_i}{\gamma }_{ij}({\overline{\eta }}_i-{\overline{\eta }}_j)+\sum _{i=1}^N\sum _{j\in {\overline{N}}_i}{\overline{\gamma }}_{ij}{\overline{\eta }}_i^T\left({\overline{\eta }}_i-{\overline{\eta }}_j\right)\hfill \\ & ={\overline{\eta }}^T\overline{\mathbf{g}}(\overline{\eta },{\eta }_0)-\sum _{i=1}^N\sum _{j\in {\overline{N}}_i}{\gamma }_{ij}{\overline{\eta }}_i^T({\overline{\eta }}_i-{\overline{\eta }}_j)+\sum _{i=1}^N\sum _{j\in {\overline{N}}_i}({\gamma }_{ij}-\gamma ){\overline{\eta }}_i^T\left({\overline{\eta }}_i-{\overline{\eta }}_j\right)\hfill \\ & ={\overline{\eta }}^T\overline{\mathbf{g}}(\overline{\eta },{\eta }_0)-\sum _{i=1}^N\sum _{j\in {\overline{N}}_i}\gamma {\overline{\eta }}_i^T({\overline{\eta }}_i-{\overline{\eta }}_j)\hfill \end{array}\end{array}$$

(28)

Then, per Remark A.1 in [19] and following Assumption 6, H is positive definite and symmetric; thus, the eigenvalue of H is positive real number. We then have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{V}(\overline{\eta },\overline{\gamma },t)=& {\overline{\eta }}^T\overline{\mathbf{g}}(\overline{\eta },{\eta }_0)-\gamma {\overline{\eta }}^T\left(H\otimes I_m\right)\overline{\eta }\hfill \\ \hfill \le & \parallel \overline{\eta }\parallel \parallel \overline{\mathbf{g}}(\overline{\eta },{\eta }_0)\parallel -\gamma {\lambda }_{min}\left(H\right){\parallel \overline{\eta }\parallel }^2\hfill \end{array}\end{array}$$

(29)

Because ${\overline{g}}_i({\overline{\eta }}_i,{\eta }_0)=g({\overline{\eta }}_i+{\eta }_0)-g\left({\eta }_0\right)=g\left({\eta }_i\right)-g\left({\eta }_0\right)$, let $g\left({\eta }_i\right):=col(g_1\left({\eta }_i\right),\dots ,g_m\left({\eta }_i\right))$, $i=1,\cdots ,N$, $g\left({\eta }_0\right):=col(g_1\left({\eta }_0\right),\dots ,g_m\left({\eta }_0\right))$, where $g_j(·):{\mathbb{R}}^m\mapsto \mathbb{R}$, $j=1,\dots ,m$. Following Appendix A in [20], we can use the inequality of the norm of vector $\parallel x\parallel \le \sqrt{m}{\parallel x\parallel }_{\infty }$ with all vectors $x\in {\mathbb{R}}^n$; then,

$$\begin{array}{c}\hfill \parallel {\overline{g}}_i({\overline{\eta }}_i,{\eta }_0)\parallel \le \sqrt{m}\underset{j=1,\dots ,m}{max}\left|g_j\left({\eta }_i\right)-g_j\left({\eta }_0\right)\right|\end{array}$$

(30)

Thus,

$$\begin{array}{c}\hfill \parallel \overline{\mathbf{g}}(\overline{\eta },{\eta }_0)\parallel \le \sqrt{Nm}\underset{\genfrac{}{}{0pt}{}{i=1,\dots ,N}{j=1,\dots ,m}}{max}\left|g_j\left({\eta }_i\right)-g_j\left({\eta }_0\right)\right|\end{array}$$

(31)

Because ${\overline{g}}_j\left({\overline{\eta }}_i,{\eta }_0\right):=g_j\left({\overline{\eta }}_i+{\eta }_0\right)-g_j\left({\eta }_0\right)=g_j\left({\eta }_i\right)-g_j\left({\eta }_0\right)$ is a sufficiently smooth function satisfying ${\overline{g}}_j\left(0,{\eta }_0\right)=0$ for all ${\eta }_0\in {\mathbb{V}}_0$, with ${\mathbb{V}}_0$ being a compact set of ${\mathbb{R}}^m$, then by Lemma 7.8 in [20] there exists some smooth function ${\pi }_j\left({\overline{\eta }}_i,{\eta }_0\right)\ge 1$, $i=1,\cdots ,N$ such that

$$\begin{array}{c}\hfill {\overline{g}}_j\left({\overline{\eta }}_i,{\eta }_0\right)=g_j\left({\eta }_i\right)-g_j\left({\eta }_0\right)\le {\pi }_j\left({\overline{\eta }}_i,{\eta }_0\right)∥{\overline{\eta }}_i∥\end{array}$$

(32)

Then, we have

$$\begin{array}{c}\hfill \parallel {\overline{g}}_i({\overline{\eta }}_i,{\eta }_0)\parallel \le \sqrt{m}\underset{j=1,\dots ,m}{max}{\pi }_j\left({\overline{\eta }}_i,{\eta }_0\right)∥{\overline{\eta }}_i∥\end{array}$$

(33)

and

$$\begin{array}{c}\hfill \parallel \overline{\mathbf{g}}(\overline{\eta },{\eta }_0)\parallel \le \sqrt{Nm}\underset{\genfrac{}{}{0pt}{}{i=1,\dots ,N}{j=1,\dots ,m}}{max}{\pi }_j\left({\overline{\eta }}_i,{\eta }_0\right)∥{\overline{\eta }}_i∥\end{array}$$

(34)

Because $\parallel {\overline{\eta }}_i\parallel \le \parallel \overline{\eta }\parallel$ for $i=1,\cdots ,N$, we have

$$\begin{array}{c}\hfill \parallel \overline{\mathbf{g}}(\overline{\eta },{\eta }_0)\parallel \le \sqrt{Nm}\underset{\genfrac{}{}{0pt}{}{i=1,\dots ,N}{j=1,\dots ,m}}{max}{\pi }_j\left({\overline{\eta }}_i,{\eta }_0\right)∥\overline{\eta }∥.\end{array}$$

(35)

Combining (29) and (35) provides

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{V}(\overline{\eta },\overline{\gamma },t)\le & \parallel \overline{\eta }\parallel \sqrt{Nm}\underset{\genfrac{}{}{0pt}{}{i=1,\dots ,N}{j=1,\dots ,m}}{max}{\pi }_j\left({\overline{\eta }}_i,{\eta }_0\right)∥\overline{\eta }∥-\gamma {\lambda }_{min}\left(H\right){\parallel \overline{\eta }\parallel }^2\hfill \\ \hfill \le & (\sqrt{Nm}\underset{\genfrac{}{}{0pt}{}{i=1,\dots ,N}{j=1,\dots ,m}}{max}{\pi }_j\left({\overline{\eta }}_i,{\eta }_0\right)-\gamma {\lambda }_{min}\left(H\right)){\parallel \overline{\eta }\parallel }^2\hfill \end{array}\end{array}$$

(36)

Let $\pi (\overline{\eta },{\eta }_0):=\sqrt{Nm}{max}_{\begin{array}{c}i=1,\dots ,N\\ j=1,\dots ,m\end{array}}{\pi }_j\left({\overline{\eta }}_i,{\eta }_0\right)$, and let $\overline{\pi }={sup}_{t\ge 0}\pi (\overline{\eta }\left(t\right),{\eta }_0\left(t\right))$; according to Lemma 1, ${\eta }_0\left(t\right)$, $\eta \left(t\right)$, and hence $\overline{\eta }\left(t\right)$ are bounded over [0,∞), $\pi (\overline{\eta },{\eta }_0)$ is finite, and as a result, $\overline{\pi }$ is finite. Then,

$$\begin{array}{cc}\hfill \dot{V}(\overline{\eta },\overline{\gamma },t)\le & (\overline{\pi }-\gamma {\lambda }_{min}\left(H\right)){\parallel \overline{\eta }\parallel }^2\end{array}$$

(37)

Thus, there exists a positive real number $\gamma$ satisfying

$$\begin{array}{c}\hfill \gamma >\frac{{\pi }^{\ast }}{{\lambda }_{\phantom{(}min}\left(H\right)}\end{array}$$

(38)

such that for $i=1,\cdots ,N$, $\dot{V}(\overline{\eta },\overline{\gamma },t)\le 0$. Then,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \underset{t\to \infty }{lim}{\int }_0^t\dot{V}(\overline{\eta },\overline{\gamma },\tau )d\tau \le 0\end{array}\end{array}$$

(39)

where, $\gamma$ satisfies (38). Thus, ${lim}_{t\to \infty }{\int }_0^t\dot{V}(\overline{\eta },\overline{\gamma },\tau )d\tau$ exists and is finite. Then, ${lim}_{t\to \infty }{\int }_0^t\dot{V}(\overline{\eta },\overline{\gamma },\tau )d\tau ={lim}_{t\to \infty }(V(\overline{\eta },\overline{\gamma },t)-V(\overline{\eta },\overline{\gamma },0))$ is finite as well. Thus, $V(\overline{\eta },\overline{\gamma },t)$ is bounded for $t\ge 0$, and by (19), ${\overline{\eta }}_i$ and ${\overline{\gamma }}_{ij}$ are bounded for all $t\ge 0$ as well. Then, ${\eta }_i$ is bounded for all $t\ge 0$, as ${\eta }_0$ is bounded under the Assumption 2. Then, ${\eta }_{vi}$ is bounded for all $t\ge 0$ and ${\gamma }_{ij}$ is bounded for all $t\ge 0$. Because $g(·)$ is a sufficiently smooth function and ${\overline{\eta }}_i$, ${\eta }_0$ is bounded for all $t\ge 0$, it follows that ${\overline{g}}_i({\overline{\eta }}_i,{\eta }_0)$ is bounded for all $t\ge 0$. Because $g(·)$ is a sufficiently smooth function and is bounded for all $t\ge 0$, then ${\dot{\overline{g}}}_i({\overline{\eta }}_i,{\eta }_0)$ is bounded for all $t\ge 0$. Thus, from (18), $\dot{\overline{\eta }},\dot{\overline{\gamma }}$ are all bounded on $t\ge 0$. From (28), $\ddot{V}(\overline{\eta },\overline{\gamma },t)={\sum }_{i=1}^N{\dot{\overline{\eta }}}_i^T({\overline{g}}_i\left({\overline{\eta }}_i,{\eta }_0\right)-\gamma {\eta }_{vi})+{\sum }_{i=1}^N{\overline{\eta }}_i^T({\dot{\overline{g}}}_i({\overline{\eta }}_i,{\eta }_0)-\gamma {\dot{\eta }}_{vi})$. Then, $\ddot{V}(\overline{\eta },\overline{\gamma },t)$ is bounded for all $t\ge 0$. Thus, $\dot{V}(\overline{\eta },\overline{\gamma },t)$ is uniformly continuous for $t\ge 0$. Then, by Lemma 8.2 in [18], ${lim}_{t\to \infty }\dot{V}(\overline{\eta },\overline{\gamma },\tau )=0$. Thus,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}({\eta }_i\left(t\right)-{\eta }_0\left(t\right))=0\end{array}$$

(40)

asymptotically. □

Remark 2. 

Because $a(·)$ is a globally defined and sufficiently smooth function vanishing at the origin of ${\mathbb{R}}^{m+n}$, and because ${\eta }_0\left(t\right)=v\left(t\right)$, $y_0\left(t\right)=a\left(v\left(t\right)\right)=a\left({\eta }_0\left(t\right)\right)$, by (40) we have

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}(a\left({\eta }_i\left(t\right)\right)-y_0\left(t\right))=0\end{array}$$

(41)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{a}\left({\eta }_i\left(t\right)\right)& =\frac{\partial a\left({\eta }_i\left(t\right)\right)}{\partial {\eta }_i\left(t\right)}\frac{\partial {\eta }_i\left(t\right)}{\partial t}=\frac{\partial a\left({\eta }_i\left(t\right)\right)}{\partial {\eta }_i\left(t\right)}(g\left({\eta }_i\left(t\right)\right)-\sum _{j=0}^N{\gamma }_{ij}{a}_{ij}({\eta }_i\left(t\right)-{\eta }_j\left(t\right)))\hfill \end{array}\end{array}$$

(42)

By (40), we have

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\sum _{j=0}^N{\gamma }_{ij}{a}_{ij}\left({\eta }_i\left(t\right)-{\eta }_j\left(t\right)\right))=0\end{array}$$

(43)

By (42) and (43), we have

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\dot{a}\left({\eta }_i\left(t\right)\right)=\frac{\partial a\left({\eta }_i\left(t\right)\right)}{\partial {\eta }_i\left(t\right)}g\left({\eta }_i\left(t\right)\right)\end{array}$$

(44)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{a}\left({\eta }_0\left(t\right)\right)& =\frac{\partial a\left({\eta }_0\left(t\right)\right)}{\partial {\eta }_0\left(t\right)}\frac{\partial {\eta }_0\left(t\right)}{\partial t}=\frac{\partial a\left({\eta }_0\left(t\right)\right)}{\partial {\eta }_0\left(t\right)}g\left({\eta }_0\left(t\right)\right)\hfill \end{array}\end{array}$$

(45)

Finally, by (40), (44), and (45), we have

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}(\dot{a}\left({\eta }_i\left(t\right)\right)-{\dot{y}}_0\left(t\right))& =0.\end{array}$$

(46)

4. Adaptive Sliding-Mode Controller Design

Because the sign of $b_i$ is unknown, the control direction is unknown. In the literature, Nussbaum functions $N\left(k\right)$ have been introduced to deal with the absence of information on the control direction; these functions are smooth, and have the following oscillation properties [21,22]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\underset{k\to \infty }{lim\; inf}\frac{1}{k}{\int }_0^k\mathrm{N}\left(\tau \right)d\tau =-\infty \hfill \\ \underset{k\to \infty }{lim\; sup}\frac{1}{k}{\int }_0^k\mathrm{N}\left(\tau \right)d\tau =\infty \hfill \end{array}\right.\end{array}$$

(47)

Inspired by the design of the sliding-mode function in [23], the filtered tracking error can be defined according to

$$\begin{array}{c}\hfill s_i=c_i{e}_i+{\dot{e}}_i\end{array}$$

(48)

where $s_i\in {\mathbb{R}}^n$, $c_i$ is a positive constant, $i=1,\cdots ,N$, and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill e_i=y_i-y_0=x_{i1}-a\left({\eta }_0\right),{\dot{e}}_i=x_{i2}-\dot{a}\left({\eta }_0\right).\end{array}\end{array}$$

(49)

Using the parameters obtained by the observer, we can perform the following coordinate transformation on the tracking error:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\overline{e}}_i=x_{i1}-a\left({\eta }_i\right),{\dot{\overline{e}}}_i=x_{i2}-\dot{a}\left({\eta }_i\right)\end{array}\end{array}$$

(50)

where, ${\eta }_i\in {\mathbb{R}}^m$, $a\left({\eta }_i\right)\in {\mathbb{R}}^n$, $\dot{a}\left({\eta }_i\right)\in {\mathbb{R}}^n$, ${\overline{e}}_i\in {\mathbb{R}}^n$, ${\dot{\overline{e}}}_i\in {\mathbb{R}}^n$. Correspondingly, the filtered tracking error is converted to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\overline{s}}_i& =c_i{\overline{e}}_i+{\dot{\overline{e}}}_i=c_i{x}_{i1}-c_{i}a\left({\eta }_i\right)+x_{i2}-\dot{a}\left({\eta }_i\right)\hfill \\ \hfill {\dot{\overline{s}}}_i& =c_i{\dot{\overline{e}}}_i+{\ddot{\overline{e}}}_i=c_i{x}_{i2}-c_i\dot{a}\left({\eta }_i\right)+{\sigma }_i{u}_{ci}+{\varphi }_i\left(t\right)+d_i(x_i,t)-\ddot{a}\left({\eta }_i\right)\hfill \end{array}\end{array}$$

(51)

where ${\overline{s}}_i\in {\mathbb{R}}^n$, ${\dot{\overline{s}}}_i\in {\mathbb{R}}^n$.

We now design the adaptive sliding-mode cooperative tracking control law as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_{ci}& =N\left(k_i\right){\overline{u}}_i\hfill \\ \hfill {\overline{u}}_i& =-k_{2i}{\overline{s}}_i+{\overline{\alpha }}_i\hfill \\ \hfill {\dot{\overline{k}}}_i& ={\beta }_i{\overline{s}}_i^T{\overline{u}}_i,{\dot{\widehat{ϵ}}}_i={\delta }_i\left|\right|{\overline{s}}_i\left|\right|,{\dot{\widehat{\omega }}}_i={\varsigma }_i\left|\right|{\overline{s}}_i\left|\right|\hfill \end{array}\end{array}$$

(52)

where $k_{2i}$, ${\beta }_i$, ${\delta }_i$, ${\varsigma }_i$ are positive constants and $u_i\in {\mathbb{R}}^n$, $sgn(·)$ is the signum function. The function ${\overline{\alpha }}_i$ is designed as follows:

$$\begin{array}{c}\hfill {\overline{\alpha }}_i=k_{1i}{\overline{s}}_i+c_i{\dot{\overline{e}}}_i-\ddot{a}\left({\eta }_i\right)+{\widehat{\omega }}_i\mathit{sgn}\left({\overline{s}}_i^T\right)+{\widehat{ϵ}}_i\mathit{sgn}\left({\overline{s}}_i^T\right)\end{array}$$

(53)

where, $k_{1i}$ is a positive constant.

Let $\mathit{sgn}\left({\overline{s}}_i^T\right)={\left[sgn({\overline{s}}_i^T:1),\cdots ,sgn({\overline{s}}_i^T:l),\cdots ,sgn({\overline{s}}_i^T:n)\right]}^T$, where, ${\overline{s}}_i^T:l$ is the element in column l of ${\overline{s}}_i^T$, $1\le l\le n$.

Under the control law (52), the closed-loop system becomes

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\overline{e}}}_i& =x_{i2}-\dot{a}\left({\eta }_i\right)\hfill \\ \hfill {\ddot{\overline{e}}}_i& ={\sigma }_i\left(t\right)N\left(k_i\right)(-k_{2i}{\overline{s}}_i+{\overline{\alpha }}_i)+{\varphi }_i\left(t\right)+d_i(x_i,t)-\ddot{a}\left({\eta }_i\right)\hfill \\ \hfill {\dot{\overline{k}}}_i& ={\beta }_i{\overline{s}}_i^{T}N\left(k_i\right)(-k_{2i}{\overline{s}}_i+{\overline{\alpha }}_i))\hfill \\ \hfill {\dot{\widehat{ϵ}}}_i& ={\delta }_i\left|\right|{\overline{s}}_i\left|\right|,{\dot{\widehat{\omega }}}_i={\varsigma }_i\left|\right|{\overline{s}}_i\left|\right|\hfill \end{array}\end{array}$$

(54)

Theorem 1. 

Consider a multi-agent system (1) subject to time-varying actuator faults and unknown external disturbances along with system (4), a fixed graph $\overline{\mathcal{G}}$, any compact subset ${\mathbb{V}}_0\subset {\mathbb{R}}^m$ containing the origin, and that Assumptions 1–6 hold. Then, take the adaptive sliding-mode control law in Equations (52), such that for $i=1,\dots ,N$, for any initial conditions $v\left(0\right)\in {\mathbb{V}}_0$, $x_i\left(0\right)\in {\mathbb{R}}^n$ the solution of the closed-loop system exists and is bounded for all $t\ge 0$ and the global tracking errors satisfy ${lim}_{t\to \infty }{\overline{e}}_i\left(t\right)=0,{lim}_{t\to \infty }{\dot{\overline{e}}}_i\left(t\right)=0$ asymptotically.

Proof. 

We now present our energy function for the closed-loop system (54) as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V& =\sum _{i=1}^N\frac{1}{2}{\overline{s}}_i^T\overline{s_i}+\sum _{i=1}^N\frac{1}{2{\delta }_i}{({ϵ}_i-{\widehat{ϵ}}_i{\vartheta }_i)}^2+\sum _{i=1}^N\frac{1}{2{\varsigma }_i}{({\omega }_i-{\widehat{\omega }}_i{\varpi }_i)}^2\hfill \end{array}\end{array}$$

(55)

where ${\vartheta }_i$ and ${\varpi }_i$ are positive constants.

Then, along the the trajectory of (54), the time derivative of (55) satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{V}& =\sum _{i=1}^N{\overline{s}}_i^T{\dot{\overline{s}}}_i+\sum _{i=1}^N\frac{1}{{\delta }_i}({ϵ}_i-{\widehat{ϵ}}_i{\vartheta }_i)(-{\dot{\widehat{ϵ}}}_i)+\sum _{i=1}^N\frac{1}{{\varsigma }_i}({\omega }_i-{\widehat{\omega }}_i{\varpi }_i)(-{\dot{\widehat{\omega }}}_i)\hfill \\ & =\sum _{i=1}^N{\overline{s}}_i^T({\overline{\alpha }}_i-k_{1i}{\overline{s}}_i-{\widehat{\omega }}_i\mathit{sgn}\left({\overline{s}}_i^T\right)-{\widehat{ϵ}}_i\mathit{sgn}\left({\overline{s}}_i^T\right)+{\sigma }_i\left(t\right)N\left(k_i\right){\overline{u}}_i+{\varphi }_i\left(t\right)+d_i(x_i,t))\hfill \\ & -\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|({ϵ}_i-{\widehat{ϵ}}_i{\vartheta }_i)-\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|({\omega }_i-{\widehat{\omega }}_i{\varpi }_i)+\frac{{\dot{\overline{k}}}_i}{{\beta }_i}-\frac{{\dot{\overline{k}}}_i}{{\beta }_i}\hfill \\ & =\sum _{i=1}^N({\overline{s}}_i^T{\sigma }_i\left(t\right)N\left(k_i\right){\overline{u}}_i+\frac{{\dot{\overline{k}}}_i}{{\beta }_i})-\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|({ϵ}_i-{\widehat{ϵ}}_i{\vartheta }_i)-\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|({\omega }_i-{\widehat{\omega }}_i{\varpi }_i)\hfill \\ & +\sum _{i=1}^N{\overline{s}}_i^T({\overline{\alpha }}_i-k_{1i}{\overline{s}}_i-{\widehat{\omega }}_i\mathit{sgn}\left({\overline{s}}_i^T\right)-{\widehat{ϵ}}_i\mathit{sgn}\left({\overline{s}}_i^T\right)+{\varphi }_i\left(t\right)+d_i(x_i,t))-{\overline{s}}_i^T(-k_{2i}{\overline{s}}_i+{\overline{\alpha }}_i)\hfill \\ & =\sum _{i=1}^N\frac{1}{{\beta }_i}({\sigma }_i\left(t\right)N\left(k_i\right)+1){\dot{\overline{k}}}_i-\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|({ϵ}_i-{\widehat{ϵ}}_i{\vartheta }_i)-\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|({\omega }_i-{\widehat{\omega }}_i{\varpi }_i)\hfill \\ & +\sum _{i=1}^N{\overline{s}}_i^T(k_{2i}{\overline{s}}_i-k_{1i}{\overline{s}}_i-{\widehat{ϵ}}_i\mathit{sgn}\left({\overline{s}}_i^T\right)-{\widehat{\omega }}_i\mathit{sgn}\left({\overline{s}}_i^T\right)+{\varphi }_i\left(t\right)+d_i(x_i,t))\hfill \\ & =\sum _{i=1}^N\frac{1}{{\beta }_i}({\sigma }_i\left(t\right)N\left(k_i\right)+1){\dot{\overline{k}}}_i+\sum _{i=1}^N{\overline{s}}_i^T(k_{2i}{\overline{s}}_i-k_{1i}{\overline{s}}_i)\hfill \\ & +\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|(-{\widehat{ϵ}}_i\mathit{sgn}\left({\overline{s}}_i^T\right)+{\varphi }_i\left(t\right)-{ϵ}_i+{\widehat{ϵ}}_i{\vartheta }_i)\hfill \\ & +\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|(-{\widehat{\omega }}_i\mathit{sgn}\left({\overline{s}}_i^T\right)+d_i(x_i,t)-{\omega }_i+{\widehat{\omega }}_i{\varpi }_i)\hfill \end{array}\end{array}$$

(56)

Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{V}& \le \sum _{i=1}^N\frac{1}{{\beta }_i}({\sigma }_i\left(t\right)N\left(k_i\right)+1){\dot{\overline{k}}}_i-\sum _{i=1}^N{\overline{s}}_i^T(k_{1i}-k_{2i}){\overline{s}}_i+\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|(-{\widehat{ϵ}}_i+{ϵ}_i-{ϵ}_i+{\widehat{ϵ}}_i{\vartheta }_i)\hfill \\ & +\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|(-{\widehat{\omega }}_i+{\omega }_i-{\omega }_i+{\widehat{\omega }}_i{\varpi }_i)\hfill \\ & \le \sum _{i=1}^N\frac{1}{{\beta }_i}({\sigma }_i\left(t\right)N\left(k_i\right)+1){\dot{\overline{k}}}_i-\sum _{i=1}^N{\overline{s}}_i^T(k_{1i}-k_{2i}){\overline{s}}_i-\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|(1-{\vartheta }_i){\widehat{ϵ}}_i\hfill \\ & -\sum _{i=1}^N\left|\right|{\overline{s}}_i\left|\right|(1-{\varpi }_i){\widehat{\omega }}_i\hfill \end{array}\end{array}$$

(57)

Let the positive constants $k_{1i}$, $k_{2i}$, ${\vartheta }_i$ and ${\varpi }_i$ satisfy $k_{2i}\le k_{1i}$, ${\vartheta }_i\le 1$ and ${\varpi }_i\le 1$; then,

$$\begin{array}{c}\hfill \dot{V}\le \sum _{i=1}^N\frac{1}{{\beta }_i}({\sigma }_i\left(t\right)N\left(k_i\right)+1){\dot{\overline{k}}}_i\end{array}$$

(58)

Now, we can take the integral of both sides of (58) as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill V\left(t\right)\le V\left(0\right)+\sum _{i=1}^N{\int }_0^t\frac{1}{{\beta }_i}({\sigma }_i\left(\tau \right)N\left(k_i\right)\left(\tau \right)+1){\dot{\overline{k}}}_i\left(\tau \right)d\tau \end{array}\end{array}$$

(59)

the boundedness of ${\sum }_{i=1}^N{\int }_0^t\frac{1}{{\beta }_i}({\sigma }_i\left(\tau \right)N\left(k_i\right)\left(\tau \right)+1){\dot{\overline{k}}}_i\left(\tau \right)d\tau$ can be proved from the properties of a Nussbaum function by seeking a contradiction, which is similar to the proof of Theorem 1 in [24].

First, we define $V_M$ on $(t_y,t_z)$ as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill V_M\left(t_y,t_z\right)=\sum _{i=1}^N{\int }_{t_y}^{t_z}\frac{1}{{\beta }_i}(\varphi N\left(k_i\right)+1){\dot{k}}_{i}d\tau \end{array}\end{array}$$

(60)

For brevity, we define $V_M(t_y,t_z)=V_M(k_{iy},k_{iz})$ on $0<t_y<t_z$. Then, $|V_M\left(k_{iy},k_{iz}\right)|\le {\sum }_{i=1}^N{\int }_{k_{iy}}^{k_{iz}}\frac{1}{{\beta }_i}\left(\right|{\overline{\sigma }}_{i}N\left(k_i\right)|+1)d{k}_i\le {\sum }_{i=1}^N\frac{1}{{\beta }_i}\left(k_{iz}-k_{iy}\right)\left({sup}_{k_i\in \left[k_{iy},k_{iz}\right]}\left|{\overline{\sigma }}_{i}N\left(k_i\right)\right|+1\right)$. In this paper, the Nussbaum function $N\left(k_i\right)=k_i^{2}cos(\pi k_i/2)$ is considered. The Nussbaum function is positive in $k_i\in (4m_i-1,4m_i+1)$ and negative in $k_i\in (4m_i+1,4m_i+3)$ for any positive integer m. Then, we consider the two time intervals $[k_{i0},k_{i1}]=[k_{i0},4m_i+1]$ and $[k_{i1},k_{i2}]=[4m_i+1,4m_i+3]$ for $k_{i0}>0$, where $m_i$ is a sufficiently large positive integer. For $[k_{i0},k_{i1}]=[k_{i0},4m_i+1]$, we can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_M\left(k_{i0},k_{i1}\right)& \le \sum _{i=1}^N\frac{1}{{\beta }_i}\left(k_{i1}-k_{i0}\right)(\underset{k_i\in \left[k_{i0},k_{i1}\right]}{sup}|{\overline{\sigma }}_{i}N\left(k_i\right)+1\left|\right)\hfill \\ & =\sum _{i=1}^N\frac{1}{{\beta }_i}(l_{1i}{(4m_i+1)}^2+l_{1i})\hfill \end{array}\end{array}$$

(61)

where, $l_{1i}=4m_i+1-k_{i0}>0$. It should be noted that $\forall k_i\in [k_{i1},k_{i2}]=[4m_i+1,4m_i+3]$, $N\left(k_i\right)\le 0$, and

$$\begin{array}{c}\hfill V_M(k_{i1},k_{i2})\le \sum _{i=1}^N\frac{1}{{\beta }_i}{\int }_{4m_i+2-{\mathsf{\Psi }}_i}^{4m_i+2+{\mathsf{\Psi }}_i}({\overline{\sigma }}_{i}N\left(k_i\right)+1){\dot{k}}_{i}d\tau \end{array}$$

(62)

where, ${\mathsf{\Psi }}_i\in (0,1)$. Then, we can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_M(K_{i1},k_{i2})& \le \sum _{i=1}^N\frac{1}{{\beta }_i}2{\mathsf{\Psi }}_i({\overline{\sigma }}_i\underset{k_i\in [k_{i1},k_{i2}]}{inf}N\left(k_i\right)+1)\hfill \\ & =\sum _{i=1}^N\frac{1}{{\beta }_i}(-l_{2i}{(4m_i+2-{\mathsf{\Psi }}_i)}^2+l_{3i})\hfill \end{array}\end{array}$$

(63)

where $l_{2i}=2{\mathsf{\Psi }}_{i}cos(\pi {\mathsf{\Psi }}_i/2)>0$, $l_{2i}=2{\mathsf{\Psi }}_i>0$. Thus,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_M(k_{i0},k_{i2})& =V_M\left(k_{i0},k_{i1}\right)+V_M(k_{i1},k_{i2})\hfill \\ & \le \sum _{i=1}^N\frac{1}{{\beta }_i}{(4m_i+1)}^2(-l_{2i}(8m_i+2-(8m_i+2){\mathsf{\Psi }}_i+{(1-{\mathsf{\Psi }}_i)}^2)+4m_i\hfill \\ & +1-k_{i0}+\frac{l_{1i}+l_{3i}}{{(4m_i+1)}^2})\hfill \end{array}\end{array}$$

(64)

Now, we can establish the boundedness of $k_i$ on $(0,\infty ]$ by searching for a contradiction. Suppose that $k_i$ is not bounded on $(0,\infty ]$; then, two cases must be considered, namely, ${lim}_{t\to \infty }{k}_i=+\infty$ and ${lim}_{t\to \infty }{k}_i=-\infty$, as follows.

(1) If $k_i\left(t\right)$ has no upper bound on the interval $(0,\infty ]$, there exists a time period $[t_s,t_f]$ where $t_f\to \infty$. However, from (64), $V_M(K_{i0},K_{i2})\to -\infty$ as $m_i\to \infty$, which leads to a contradiction. Therefore, $k_i\left(t\right)$ has an upper bound.

(2) If $k_i\left(t\right)$ has no lower bound on the interval $(T,t_f]$, by defining $k_i=-{\mathsf{\Gamma }}_i$ we know that ${\mathsf{\Gamma }}_i$ does not have an upper bound. Because $N\left(k_i\right)$ is an even function, we know that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V\left(t\right)& \le V\left(0\right)+\sum _{i=1}^N{\int }_0^t\frac{1}{{\beta }_i}({\sigma }_{i}N\left(k_i\right)\left(\tau \right)+1){\dot{\overline{k}}}_i\left(\tau \right)d\tau \hfill \\ & \le V\left(0\right)+\sum _{i=1}^N{\int }_0^t\frac{1}{{\beta }_i}({\sigma }_{i}N\left({\mathsf{\Gamma }}_i\right)\left(\tau \right)+1){\dot{\overline{\mathsf{\Gamma }}}}_i\left(\tau \right)d\tau \hfill \\ & =V\left(0\right)+V_M({\mathsf{\Gamma }}_i\left(0\right),{\mathsf{\Gamma }}_i\left(t\right))\hfill \end{array}\end{array}$$

(65)

which means there exists a time interval $[t_s,t_f]$ such that ${\mathsf{\Gamma }}_i\left(t\right)$ is monotonically increasing and ${lim}_{t\to t_f}{\mathsf{\Gamma }}_i\left(t\right)=\infty$ with ${\mathsf{\Gamma }}_i\left(t_s\right)>0$. Similarly, we can define an period that can result in a contradiction. Thus, $k_i$ is lower-bounded on the interval $[t_s,t_f]$. From (61), we can obtain the conclusion that $V_M$ is bounded. Thus, $k_i$ and ${\sum }_{i=1}^N{\int }_0^{\infty }\frac{1}{{\beta }_i}({\sigma }_{i}N\left(k_i\right)\left(\tau \right)+1){\dot{\overline{k}}}_i\left(\tau \right)d\tau$ are bounded, meaning that ${\sum }_{i=1}^N{\int }_0^t\frac{1}{{\beta }_i}({\sigma }_{i}N\left(k_i\right)\left(\tau \right)+1){\dot{\overline{k}}}_i\left(\tau \right)d\tau$ is bounded, and V(0) is bounded, meaning that $V\left(t\right)$ is bounded. Thus, ${lim}_{t\to \infty }{\int }_0^t\dot{V}d\tau$ exists and is finite, and in turn ${lim}_{t\to \infty }V\left(t\right)$ is bounded for all $t\ge 0$. Following (57), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \ddot{V}& =\sum _{i=1}^N({\dot{\overline{s}}}_i^T{\dot{\overline{s}}}_i+{\overline{s}}_i^T{\ddot{\overline{s}}}_i)+\sum _{i=1}^N\frac{1}{{\delta }_i}({{\dot{\widehat{ϵ}}}_i}^2-({ϵ}_i-{\widehat{ϵ}}_i{\vartheta }_i){\ddot{\widehat{ϵ}}}_i)+\sum _{i=1}^N\frac{1}{{\varsigma }_i}({{\dot{\widehat{\omega }}}_i}^2-({\omega }_i-{\widehat{\omega }}_i{\varpi }_i){\ddot{\widehat{\omega }}}_i)\hfill \end{array}\end{array}$$

(66)

Because it has been proven that $li{m}_{t\to \infty }{\int }_0^{t}V\left(\tau \right)d\tau$ is bounded for all $t\ge 0$, per (55), ${lim}_{t\to \infty }{\overline{s}}_i\left(t\right)$, ${\widehat{ϵ}}_i$ and ${\widehat{\omega }}_i$ are bounded for all $t\ge 0$. Because ${\overline{s}}_i$ is bounded and ${\overline{s}}_i$ is a sufficiently smooth function, ${\dot{\overline{s}}}_i$ is sufficiently smooth and bounded, meaning that ${\ddot{\overline{s}}}_i$ is bounded. Because ${\overline{s}}_i$ is bounded and ${\overline{s}}_i$ is a sufficiently smooth function, ${\overline{s}}_i^T$ is bounded and ${\overline{s}}_i^T$ i as sufficiently smooth function, meaning that ${\dot{\overline{s}}}^T$ is bounded. Thus, ${\dot{\overline{s}}}_i^T{\dot{\overline{s}}}_i+{\overline{s}}_i^T{\ddot{\overline{s}}}_i$ is bounded as well. Because it has been proven that ${lim}_{t\to \infty }{\int }_0^{t}V\left(t\right)$ is bounded for all $t\ge 0$, per (55), ${lim}_{t\to \infty }{\widehat{ϵ}}_i\left(t\right)$ is bounded for all $t\ge 0$. Because ${\widehat{ϵ}}_i$ is bounded and ${\widehat{ϵ}}_i$ is a sufficiently smooth function, ${\dot{\widehat{ϵ}}}_i$ is sufficiently smooth and bounded, meaning that ${\ddot{\widehat{ϵ}}}_i$ is bounded. Thus, $\frac{1}{{\delta }_i}({{\dot{\widehat{ϵ}}}_i}^2-({ϵ}_i-{\widehat{ϵ}}_i{\underline{\sigma }}_i){\ddot{\widehat{ϵ}}}_i)$ is bounded as well. Similarly, $\frac{1}{{\varsigma }_i}({{\dot{\widehat{\omega }}}_i}^2-({\omega }_i-{\widehat{\omega }}_i{\varpi }_i){\ddot{\widehat{\omega }}}_i)$. Thus, $\ddot{V}$ is bounded. Furthermore, $\dot{V}$ is uniformly continuous for all $t\ge 0$. Following Lemma 8.2 in [18], ${lim}_{t\to \infty }\dot{V}=0$. Thus,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\overline{s}}_i\left(t\right)=0\end{array}$$

(67)

asymptotically, i.e., ${lim}_{t\to \infty }{c}_i{\overline{e}}_i\left(t\right)+{\dot{\overline{e}}}_i\left(t\right)=0$ asymptotically. Thus, when $t\to \infty$, ${\dot{\overline{e}}}_i\left(t\right)=-c_i{\overline{e}}_i\left(t\right)$, ${\overline{e}}_i\left(t\right)=c{e}^{-c_{i}t}$, where c is a constant. Thus, ${lim}_{t\to \infty }{\overline{e}}_i\left(t\right)=0$ asymptotically, because ${lim}_{t\to \infty }{c}_i{\overline{e}}_i\left(t\right)+{\dot{\overline{e}}}_i\left(t\right)=0$ asymptotically; thus, ${lim}_{t\to \infty }{\dot{\overline{e}}}_i\left(t\right)=0$ asymptotically as well. □

We know that ${lim}_{t\to \infty }{\overline{e}}_i\left(t\right)=0$, ${lim}_{t\to \infty }{\dot{\overline{e}}}_i\left(t\right)=0$ asymptotically, i.e.

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \underset{t\to \infty }{lim}(x_{i1}\left(t\right)-a\left({\eta }_i\right)\left(t\right))=0,\underset{t\to \infty }{lim}(x_{i2}\left(t\right)-\dot{a}\left({\eta }_i\left(t\right)\right))=0\end{array}\end{array}$$

(68)

Combining (41), (46), and (68) from Remark 2 of Lemma 2 provides ${lim}_{t\to \infty }(x_{i1}\left(t\right)-a\left({\eta }_i\right)\left(t\right))+{lim}_{t\to \infty }(a\left({\eta }_i\left(t\right)\right)-y_0\left(t\right))={lim}_{t\to \infty }(x_{i1}\left(t\right)-y_0\left(t\right))=0$ and ${lim}_{t\to \infty }(x_{i2}\left(t\right)-\dot{a}\left({\eta }_i\left(t\right)\right))+{lim}_{t\to \infty }(\dot{a}\left({\eta }_i\left(t\right)\right)-{\dot{y}}_0\left(t\right))={lim}_{t\to \infty }(x_{i2}\left(t\right)-{\dot{y}}_0\left(t\right))=0$, i.e., ${lim}_{t\to \infty }{e}_i\left(t\right)=0$, ${lim}_{t\to \infty }{\dot{e}}_i\left(t\right)=0$ asymptotically. Thus, Problem 1 is solved.

Remark 3. 

The use of a signum term makes system (54) discontinuous and causes chattering. This design method is feasible in theory. In simulation, the saturate function can be used to eliminate chattering, and system (54) can be considered as continuous when ${\overline{s}}_i\left(t\right)$ trends to 0 with an unknown direction by the Fillipov solution in [25].

5. Simulation Studies

The reference signal is generated by a Van der Pol system with the following form:

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\dot{v}}_1\\ {\dot{v}}_2\end{array}\right]& =\left[\begin{array}{c}{v}_2\\ -a{v}_1+b\left(1-v_1^2\right)v_2\end{array}\right]\hfill \\ \hfill y_0& =v_1\hfill \end{array}$$

(69)

where, $a>0$, $b>0$, and $v=col(v_1,v_2)$. Clearly, Assumption 2 is satisfied.

Consider a multi-agent system (3) with actuator faults described by

$$\begin{array}{cc}\hfill {\dot{x}}_{i1}& =x_{i2}\hfill \\ \hfill {\dot{x}}_{i2}& ={\sigma }_i{u}_{ci}+{\varphi }_i\left(t\right)+d_i(x_i,t)\hfill \\ \hfill y_i& =x_{i1}\hfill \\ \hfill e_i& =y_i-y_0,i=1,\dots ,4\hfill \end{array}$$

(70)

The communication network among the five agents is described by Figure 1, and hence Assumption 1 is satisfied. The leader-following cooperative tracking problem of systems (3) and (69) can be solved by an adaptive sliding-mode control law of the form (52).

Simulation is performed with $b_i=1$, $h_i\left(t\right)=1+0.1sin\left(t\right)$; then, ${\sigma }_i\left(t\right)=b_i{h}_i=1+0.1sin\left(t\right)$, ${\varphi }_i\left(t\right)=10\ast sin\left(x_{i2}\right)$, $d_i(x_i,t)=10\ast sin\left(x_{i2}\right)$, $a=b=1$, ${\beta }_i=10$, $c_i=10$, $k_{1i}=1$, $k_{2i}=0.5$, ${\delta }_i=10$, ${\varsigma }_i=10$, $k_{ij}=1$, the initial value of ${\widehat{ϵ}}_i$ satisfies ${\widehat{ϵ}}_i\left(0\right)\ge 0$ and is stochastically emanated, and ${\mathbb{V}}_0=\{v\left(0\right)\mid \parallel v\left(0\right)\parallel \le 2\sqrt{2}\}$ and stochastically emanates other initial conditions.

Figure 2 shows the errors between the observers and leader; the errors can converge to 0 asymptotically. Figure 3 shows the dynamic gain ${\gamma }_{ij}$ of the adaptive distributed observer, $i=1,2,3,4$, $j\in {\overline{\mathcal{N}}}_i$. Figure 4 shows the tracking errors between the followers and leader; the tracking errors can converge to 0 asymptotically. Figure 5 shows the $k_i$ and Nussbaum function $N\left(k_i\right)$, $i=1,2,3,4$. Figure 6 shows the parameter estimates ${\widehat{ϵ}}_i\left(t\right)$ and ${\widehat{\omega }}_i$ of the controller $i=1,2,3,4$.

6. Conclusions

This paper has studied the leader-following cooperative tracking problem for a category of second-order multi-agent systems subject to time-varying actuator faults and unknown external disturbances via an adaptive sliding-mode controller based on a self-tuning distributed observer. The controller uses the information of the self-tuning distributed observer, which can solve the problem of communication barriers among the leader and the followers and calculate the observer gain while remaining online. The unknown sign of the state feedback gain can be tolerated using a Nussbaum function. In addition, the gain of the controller can adaptively change according to the change of the actuator faults and unknown external disturbances; thus, the leader-following cooperative tracking errors can approach zero asymptotically, and the controller structure does not need to be redesigned. The extension to sliding-mode observer based consensus, disturbance-observer based consensus in high-order systems suffering from disturbance rejection, and PLOAE faults are interesting topics for future research [26,27,28].