1 Introduction

In the research field of multi-agent systems, due to the greater efficiency and operational capability, the distributed cooperative control has attracted extensive attention from a large number of domestic and foreign researchers in the past two decades [1, 17]. Many results have been available for the consensus control problem of multi-agents systems [8, 16, 22, 27]. The fundamental research on cooperative control of multi-agent systems mainly included cooperative regulator problems and cooperative tracking problems. As the basis of distributed cooperative control of multi-agent systems, the consensus problem has attracted researchers’ interests [8]. In [1], a method was proposed for the stability of the Vicsek model and proved that the first-order agent system can achieve consensus under the undirected communication topology. In addition, Jadbabaie et al. changed the Vicsek model and studied a model with leader–follower form in [9]. In [16], the consensus problem was investigated for single integrator multi-agent systems and proposed a basic framework for multi-agent system consensus algorithm. It turned out that balanced digraphs played a critical role in addressing consensus problems. Based on the convexity theory with the Lyapunov method, the authors solved the consensus problem for discrete multi-agent systems. It pointed out that as long as the coupling relationship between the agents meet certain convexity conditions and the communication structure is connected, the behavior of the multi-agent system will eventually become consensus [15]. In [18], the form of communication network was extended to weighted directed communication topology, and the authors proposed the sufficient condition for consensus. In [27], the synchronization problem was studied for networked higher-order nonlinear systems with an active leader, and a robust adaptive sliding mode control scheme was proposed. In [24], a distributed control method was proposed to solve the distributed consensus tracking problems of multi-agent systems. In [20], the consensus problem was investigated for uncertain second-order nonlinear multi-agent systems with unknown nonlinear dead zone. In [29], the finite-time consensus tracking control problem was studied for uncertain nonlinear multi-agent systems. However, the above studies do not take into account the situation, in which the system becomes faulty.

In general, actuators, sensors and components may become faulty in practical application and these faults may cause system instability, which can lead to catastrophic consequences. Therefore, many experts have proposed effective fault-tolerant control methods to improve the system reliability and ensure the stability of the controlled system in all situations [2,3,4,5, 7, 19, 25, 26, 28]. In [21], a novel cooperative adaptive fuzzy tracking control scheme was proposed to guarantee that all followers asymptotically synchronize a leader in spite of actuator faults. The leader-following consensus problem was studied for multi-agent systems in [23], where it was assumed that there exist gain and bias transmission nonlinearities in the links in the communication network. Two distributed adaptive control schemes were designed to compensate for the faults. In [13], the finite-time fault-tolerant control problem was investigated for multiple-input multiple-output nonlinear systems. However, only actuator faults are considered in [3,4,5, 12,13,14, 21, 30].

In this paper, we investigate the cooperative adaptive fault tolerant tracking control problem of high-order multi-agent systems and propose an active fault-tolerant control scheme against signal transmission faults. Compared with the results in [8, 15, 20, 24, 27, 29], signal transmission faults in communication network are considered, and a novel distributed adaptive control method is designed.

The rest of paper is organized as follows. In Sect. 2, the basic graph theory and radial basis function neural networks are introduced. The problem under study is formulated. In Sect 3, the distributed adaptive control scheme is designed. Based on Lyapunov theory, the closed-loop system stability analysis is developed in Sect 4. Simulation results and discussion are reported in Sect. 5. The conclusion is drawn in Sect. 6.

Notations Throughout this paper, \(R,{R^N}\) denote the real numbers and the real n vectors, respectively; \(| \cdot |\) is the absolute value of a real number; \(|| \cdot ||\) is the Euclidean norm of a vector; \(tr\{ \cdot \}\) is the trace of a matrix; \(\sigma ( \cdot )\) is the set of singular values of a matrix; \({\bar{\sigma }} ( \cdot )\) is the maximum singular value of a matrix; \(( \cdot )\) is the minimum singular value of a matrix; matrix \(P > 0\) means P is positive definite.

2 Preliminaries

2.1 Basic Graph Theory and Notations

Let \(G = (\upsilon ,E)\) be a weighted digraph, \(\upsilon = ({\upsilon _1}, \ldots ,{\upsilon _N})\) is the nonempty set of nodes/agents, \( E \subseteq \upsilon \times \upsilon \) is the set of edges, \(({\upsilon _k},{\upsilon _j}) \in E\) means \({\upsilon _j}\) can obtain information from \({\upsilon _k}\). Define an adjacency matrix \(A = [{a_{kj}}] \in {R^{N \times N}}\) with \({a_{kj}} > 0\) if \(({\upsilon _k},{\upsilon _j}) \in E\) ; otherwise, \({a_{kj}} = 0\). In this paper, it is assumed that \({a_{kk}} = 0\) and the topology is fixed, i.e., A is time invariant. Define \({d_k} = \sum \nolimits _{j = 1}^N {{a_{kj}}}\) as the weighted in-degree of node k and \(D = \hbox {diag}({d_1}, \ldots ,{d_N}) \in {R^{N \times N}}\) as in-degree matrix. The graph Laplacian matrix is \(L = [{l_{kj}}] = D - A \in {R^{N \times N}}\). Let \( \underline{1}= {[1, \ldots ,1]^\mathrm{T}} \in {R^{N \times 1}}\) with appropriate dimension; then, \(L\underline{1} = 0\) . We use the set \({N_k}\) to describe all neighboring agents of \({\upsilon _k}\), i.e.,\({N_k} = \{ j|({\upsilon _j},{\upsilon _k}) \in E\}\).

2.2 Problem Formulation

In this paper, we consider a team of \(N + 1\) agents consisting of N followers and one leader. The dynamics of the kth follower agent is described as

$$\begin{aligned} \left\{ \begin{aligned}&{{\dot{x}}_{_{k,i}}}(t) = {x_{k,i + 1}}(t),~~i= 1, \ldots ,{n_k} - 1\\&{{\dot{x}}_{k,{n_k}}}(t) = {f_k}({{{\bar{x}}}_k}) + {u_k}(t) + {h_k}(t) \end{aligned} \right. \end{aligned}$$
(1)

where \(k = 1, \ldots ,N\), \({n_k}\) denotes the order number, \({x_{k,i}} \in R\) denotes the ith state and \({{\bar{x}}_k} = {[{x_{k,1}}, \ldots ,{x_{k,{n_k}}}]^\mathrm{T}} \in {R^{{n_k}}}\) denotes the state vector of node k; \({f_k}({{\bar{x}}_k}) \in R\) is an unknown continuous function; \({u_k} \in R\) is the control input; \({h_k}(t) \in R\) is an external disturbance, which is unknown but bounded.

The dynamics of the leader is given by

$$\begin{aligned} \left\{ \begin{aligned}&{{\dot{x}}_{0,i}}(t) = {x_{0,i + 1}}(t)\\&{{\dot{x}}_{0,{n_0}}}(t) = {f_0}({{{\bar{x}}}_0},t) \end{aligned} \right. ,~~\;i = 1, \ldots ,{n_0} - 1 \end{aligned}$$
(2)

where \({n_0}\) denotes the order number, \({x_{0,i}} \in R\) denotes the ith state and \({{\bar{x}}_0}= {[{x_{0,1}}, \ldots ,{x_{0,n}}]^\mathrm{T}}\in {R^{{n_0}}}\) is the state vector of the leader; \({f_0}({{\bar{x}}_0},t) \in R\) is piecewise continuous in time t and locally Lipschitz in \({{\bar{x}}_0}\) with \({f_0}({\mathrm{0}},t)= 0\) for all \(\forall t \ge 0\) and \({{\bar{x}}_0} \in {R^n}\).

In this paper, we assumed \({n_0} = {n_1} = \cdots = {n_k} = n\).

Define \({x_i} = {[{x_{1,i}}, \ldots ,{x_{N,i}}]^\mathrm{T}} \in {R^N}\), (1) can be written in the following compact form:

$$\begin{aligned} \left\{ \begin{aligned}&{{\dot{x}}_i}(t) = {x_{i + 1}}(t)\\&{{\dot{x}}_n}(t) = f({\bar{x}}) + u(t) + h(t) \end{aligned} \right. ,~~i = 1, \ldots ,n - 1 \end{aligned}$$
(3)

where \(f = {[{f_1}, \ldots ,{f_N}]^\mathrm{T}} = {[{f_1}({{\bar{x}}_1}), \ldots ,{f_N}({{\bar{x}}_N})]^\mathrm{T}}\), \(u = [{u_1}, \ldots ,{u_N}{]^\mathrm{T}}\), \({\bar{x}} = {[{\bar{x}}_1^\mathrm{T}, \ldots ,{\bar{x}}_N^\mathrm{T}]^\mathrm{T}}\), \(h = {[{h_1}, \ldots ,{h_N}]^\mathrm{T}} = {[{h_1}(t), \ldots ,{h_N}(t)]^\mathrm{T}}\).

Under normal condition (no fault), define the ith tracking error for follower k as follows:

$$\begin{aligned} {\delta _{k,i}} = {x_{k,i}} - {x_{0,i}},~~i = 1, \ldots ,n,k = 1, \ldots , N \end{aligned}$$
(4)

Let \({\delta _i} = {[{\delta _{1,i}}, \ldots ,{\delta _{N,i}}]^\mathrm{T}} \in {R^N}\), then

$$\begin{aligned} {\delta _i} = {x_i} - {\underline{x}_{0,i}} \end{aligned}$$
(5)

where \({\underline{x}_{0,i}} = {[{x_{0,i}}, \ldots ,{x_{0,i}}]^\mathrm{T}} \in {R^N}\).

The control objective of this paper is to design a distributed controller u for each follower such that each follower tracks the leader and the tracking error \({\delta _i}(i = 1, \ldots ,n)\) converges to the small neighborhoods of the origin.

Define the neighborhood synchronization error

$$\begin{aligned} {e_{k,i}}(t) = \sum \limits _{j \in {N_k}} {{a_{kj}}({x_{j,i}} - {x_{k,i}}) + {b_k}({x_{0,i}} - {x_{k,i}})} \end{aligned}$$
(6)

Note that if the node k can obtain the leader information, then \({b_k} > 0\); otherwise, \({b_k} = 0\);

Define the following notations: \({e_i} = {[{e_{1,i}}, \ldots ,{e_{N,i}}]^\mathrm{T}} \in {R^N}\), \({\underline{f}_0} = {[{f_0}({x_0},t), \ldots ,}{{f_0}({x_0},t)]^\mathrm{T}} \in {R^N}\), \(B= \hbox {diag}\{ {b_1}, \ldots ,{b_N}\} \in {R^{N \times N}}\).

The above tracking error can be written as:

$$\begin{aligned} \left\{ \begin{aligned}&{{{\dot{e}}}_i}(t) = {e_{i + 1}}(t)\;,~~i = 1, \ldots ,n - 1\\&{{{\dot{e}}}_n}(t) = - (L + B)(f + u(t) + h - {\underline{f}_0}) \end{aligned} \right. \end{aligned}$$
(7)

Define the new augmented graph as \({\bar{G}} = \{ {\bar{\upsilon }} ,{\bar{E}}\}\), \({\bar{\upsilon }} = \{ {\upsilon _0},{\upsilon _1}, \ldots ,{\upsilon _N}\}\) and \({\bar{E}} \subseteq {\bar{\upsilon }} \times {\bar{\upsilon }}\).

It is well known that information transmission in multi-agent systems is via communication network. There is a communication link between each two agents. For example, there are three agents k, j and h. Agents h and j are the neighbors of agent k, which can obtain the information from agent k, shown in Fig. 1. From Fig. 1, it is easily seen that there exist two communication links. One is used for the signal transmission between agents k and j, and the other one is used for the signal transmission between agents k and h.

Fig. 1
figure 1

Information transmission with faults

Full size image

In practical applications, communication links in the network may become faulty. From Fig. 1, when the state of node k is transmitted to node j, the polluted state obtained by node j is

$$\begin{aligned} {x_{f,k,i}} = {x_{k,i}} + {g_{j,k,i}} \end{aligned}$$
(8)

where \({g_{j,k,i}}\) is an unknown real constant, which denotes a bounded signal transmission fault. When the state of node k is transmitted to node h, the polluted state obtained by node h is

$$\begin{aligned} {x_{f,k,i}} = {x_{k,i}} + {g_{h,k,i}} \end{aligned}$$
(9)

where \({g_{h,k,i}}\) is an unknown real constant, which denotes bounded a signal transmission fault.

In this paper, for convenience, let \({g_{j,k,i}} = {g_{h,k,i}}\). The following fault model is considered:

$$\begin{aligned} {x_{f,k,i}} = {x_{k,i}} + {g_{k,i}}, ~~{x_{f,0,i}} = {x_{0,i}} + {g_{0,i}} \end{aligned}$$

where \({g_{k,i}}\) and \({g_{0,i}}\) are unknown bounded constant, which, respectively, denote follower and leader communication link faults.

Define the following symbols:

$$\begin{aligned} {x_{f,k}} = {{\bar{x}}_k} + {g_k},~~{x_f} = {\bar{x}} + g \end{aligned}$$

where \({x_{f,k}} = {[{x_{f,k,1}}, \ldots , {x_{f,k,{n_k}}}]^\mathrm{T}}\), \({g_k} = {[{g_{k,1}}, \ldots ,{g_{k,{n_k}}}]^\mathrm{T}}\), \({x_f} = {[x_{f,1}^\mathrm{T}, \ldots ,x_{f,N}^\mathrm{T}]^\mathrm{T}}\), \(g = {[g_1^\mathrm{T}, \ldots ,g_N^\mathrm{T}]^\mathrm{T}}\). If signal transmission becomes faulty, the neighborhood synchronization error \({e_{a,k,i}}\) is

$$\begin{aligned} \begin{aligned} {e_{a,k,i}}&=\sum \limits _{j \in {N_i}} {{a_{kj}}({x_{f,j,i}} - {x_{f,k,i}}} ) + {b_k}({x_{f,0,i}} - {x_{f,k,i}})\\&= \sum \limits _{j \in {N_i}} {{a_{kj}}[({x_{j,i}} + {g_{j,i}}) - ({x_{k,i}} + {g_{k,i}})]} \\&\quad + {b_k}[({x_{0,i}} + {g_{0,i}}) - ({x_{k,i}} + {g_{k,i}})]\\&= {e_{k,i}}+ \sum \limits _{j \in {N_i}} {{a_{kj}}({g_{j,i}} - {g_{k,i}}) + {b_k}({g_{0,i}} - {g_{k,i}}} ) \end{aligned} \end{aligned}$$
(10)

Let \({\omega _k} = \sum \limits _{j \in {N_1}} {{a_{kj}}({g_{j,1}} - {g_{k,1}}) + {b_k}({g_{0,1}} - {g_{k,1}}} )\), then

$$\begin{aligned} {e_{a,k,1}}= {e_{k,1}}+ {\omega _k} \end{aligned}$$
(11)

The following assumptions are made for cooperative tracking problems.

Assumption 1

The augmented graph \({\bar{G}}\) contains a spanning tree with the root node being the leader node 0.

Assumption 2

There exists a positive constant \({M_0} \in R\) and \({M_{f0}} \in R\) such that \(||{{\bar{x}}_0}(t)|| \le {M_0}\), \(|{f_0}({{\bar{x}}_0},t)| \le {M_{f0}}\).

Assumption 3

There exists a positive constant \({M_{h ,k}} > 0\in R\; ~~ k = 1, \ldots ,N\) such that \(|{h_k}({{\bar{x}}_k},t)| \le {M_{h,k}}\).

Assumption 4

There exists a positive constant \({M_{\omega ,k}} > 0 \in R\; ~~ k = 1, \ldots ,N\) such that \(|{\omega _k}| \le {M_{\omega ,k}}\).

Lemma 1

[27] Define \(q = [{q_1}, \ldots ,{q_N}] = {(L + B)^{ - 1}}\underline{1}, \underline{1}= {[1, \ldots ,1]^\mathrm{T}} \in {R^N}\), \(P = \hbox {diag}\{ {p_i}\} = \hbox {diag}\{ 1/{q_i}\} \), \(Q = P(L + B) + {(L + B)^\mathrm{T}}P\), then \(P > 0\) and \(Q > 0\).

Lemma 2

[27] \(||{\varphi _i}|| \le ||{e_{a,i}}||/\underline{\sigma }(L + B),i = 1, \ldots ,n,\) where \(\underline{\sigma }(L + B)\) is a minimum singular value of matrix \(L + B\).

2.3 Neural Networks

Fig. 2
figure 2

RBF neural networks (in normal)

Full size image

Neural networks have been widely used in modeling and controlling of nonlinear systems. The feasibility of applying neural networks to unknown dynamic systems control has been demonstrated in many studies [10, 11]. As can be seen from Fig. 2, radial basis function (RBF) neural networks (NNs) is presented in

$$\begin{aligned} {f_k}({{\bar{x}}_k}) = \theta _k^\mathrm{T}{\xi _k}({{\bar{Z}}_k}) + {\varepsilon _k}({{\bar{Z}}_k}) \end{aligned}$$

where \({\varepsilon _k}({{\bar{Z}}_k})\) denotes the optimal approximation error, \({{\bar{Z}}_k} = {[{z_{k,1}}, \ldots , {z_{k,{p_k}}}]^\mathrm{T}} = {[{{\bar{x}}^\mathrm{T}}_k,1]^\mathrm{T}}\), \({\xi _k}({{\bar{Z}}_k}) = {\left[ {{\zeta _{k,1}}({{{\bar{Z}}}_k}), \ldots {\zeta _{k,{N_W}}}({{{\bar{Z}}}_k})} \right] ^\mathrm{T}}\), \({N_W}\) is the number of the NNs.

\({\zeta _{k,i}}({{\bar{Z}}_k}) = \exp ( - \sum \nolimits _{j = 1}^{{\psi _k}} {\frac{{{{({z_{k,j}} - {q_{k,i,j}})}^2}}}{{{c_{k,i}}^2}}} )\), \({\psi _k}\) is the dimension of \({Z_k}\), where \({c_{k,i}} > 0\) is the center of the receptive field, and \({q_{k,i,j}}\) is the width of the Gaussian function. Let

$$\begin{aligned} \theta _k^{^*}= & {} \arg \mathop {\min }\limits _{\theta \in {\varOmega _{\;\theta }}} \left[ \mathop {\sup }\limits _{z \in {\varOmega _z}} \left| {\theta _k^\mathrm{T}{\xi _k}({{{\bar{Z}}}_k}) - {f_k}({{{\bar{x}}}_k})} \right| \right] \\ {\varOmega _\theta }= & {} \{ {\theta _k}| {||} {\theta _k}|| \le {\beta _\theta }\} \end{aligned}$$

with a constant \({\beta _\theta } > 0\), \({\varOmega _z}\) denotes an enough large compact set.

From [6], we know NNs can approximate any continuous function to any accuracy on a compact set.

Fig. 3
figure 3

RBF neural networks (with faults)

Full size image

From Fig. 2, we know the input of NNs contain the state. In this paper, signal transmission faults are considered, the state \({{\bar{x}}_k}\) cannot taken as the input of NNs, and only \({x_{f,k}}\) is accessed. Therefore, we use the neural network to approximate the unknown smooth function \({f_k}({{\bar{x}}_k})\) in system (1). From Fig. 3, we know

$$\begin{aligned} \begin{aligned} {f_k}({{{\bar{x}}}_k})&= \theta _k^{*\mathrm{T}}{\xi _k}({{{\bar{Z}}}_k}) + {\varepsilon _k}\\&=\theta _k^{*\mathrm{T}}{\xi _k}({{{\bar{Z}}}_{f,k}}) - \theta _k^{*\mathrm{T}}{\xi _k}({{{\bar{Z}}}_{f,k}}) + \theta _k^{*\mathrm{T}}{\xi _k}({{{\bar{Z}}}_k}) + {\varepsilon _k}\\&=\theta _k^{*\mathrm{T}}{\xi _k}({{{\bar{Z}}}_{f,k}}) - \theta _k^{*\mathrm{T}}[{\xi _k}({{{\bar{Z}}}_{f,k}}) - {\xi _k}({{{\bar{Z}}}_k})] + {\varepsilon _k} \end{aligned} \end{aligned}$$
(12)

Assumption 5

There exists a positive constant \({M_{\varepsilon ,k}} > 0\in R{\mathrm{}} ~~ k = 1, \ldots ,N\) such that\(|{\varepsilon _k}| \le {M_{\varepsilon ,k}}\) .

3 Main Results

Define the sliding mode error for the kth follower as follows:

$$\begin{aligned} {s_k} = {\left( \frac{\mathrm{d}}{{\mathrm{d}t}} + \lambda \right) ^{n - 1}}{e_{k,1}}(t) = \sum \limits _{i = 1}^{n - 1} {{c_{k,i}}{e_{k,i}}(t) + {e_{k,n}}(t)} , k = 1, \ldots ,N \end{aligned}$$
(13)

where \({c_{k,i}} = C_{n - 1}^{i - 1}{\lambda ^{n - i}}\) , \({\lambda _k} > 0\). Let \({{\bar{e}}_k}(t) = {[{e_{k,1}}(t), \ldots ,{e_{k,n}}(t)]^\mathrm{T}}\).

Lemma 3

[21] Let \(s_k\) be defined by (13), and then

  1. (1)

    if \({s_k} = 0\), then \({\lim _{t \rightarrow \infty }}{{\bar{e}}_k}(t) = 0\);

  2. (2)

    if \(|{s_k}| \le {\alpha _k},{{\bar{e}}_k}(0) \in {\varOmega _{{\alpha _k}}}\), then \({e_k}(t) \in {\varOmega _{{\alpha _k}}},\forall t \ge 0\);

  3. (3)

    if \(|{s_k}| \le {\alpha _k},{{\bar{e}}_k}(0) \notin {\varOmega _{{\alpha _k}}}\), then \(\exists {T_k} = ({m_k} - 1)/{\lambda _k},\forall t \ge {T_k}\), \({{\bar{e}}_k}(t) \in {\varOmega _{{\alpha _k}}}\); where \({\varOmega _{{\alpha _k}}} = \{{{{{\bar{e}}}_k}(t)} ||{e_{k,i}}| \le {2^{(j - 1)}}\lambda _{^k}^{j - {m_k}} {\alpha _k},i = 1,2, \ldots ,n,j = 1,2, \ldots {m_k}\}\).

Let \({c_{1,i}} = \cdots = {c_{N,i}} = {\lambda _i},{\lambda _n} = 1\), then

$$\begin{aligned} {s_k} = {\lambda _1}{e_{k,1}} + \cdots + {\lambda _n}{e_{k,n}} \end{aligned}$$
(14)

Define the global sliding mode error \(s = {[{s_1}, \ldots ,{s_N}]^\mathrm{T}}\), then

$$\begin{aligned} s = {\lambda _1}{e_1} + \cdots + {\lambda _n}{e_n} \end{aligned}$$
(15)

Differentiating s, we have

$$\begin{aligned} \begin{aligned} {\dot{s}}&= {\lambda _1}{{{\dot{e}}}_1} + \cdots + {\lambda _n}{{{\dot{e}}}_n} = \sum \nolimits _{i = 1}^{n - 1} {{\lambda _i}{e_i}} + {{{\dot{e}}}_n}\\&= \gamma - (L + B)(f({\bar{x}}) + u + h - {\underline{f}_0}) \end{aligned} \end{aligned}$$
(16)

where \(\gamma = \sum \nolimits _{i = 1}^{n - 1} {{\lambda _i}{e_{i + 1}}}\).

Let \({\xi _{f,k}} = \theta _k^{*\mathrm{T}}[\xi ({{\bar{Z}}_{f,k}}) - \xi ({{\bar{Z}}_k})],{\xi _f} = {[{\xi _{f,1}}, \ldots ,{\xi _{f,N}}]^\mathrm{T}}\), \(\varepsilon = {[{\varepsilon _1}, \ldots ,{\varepsilon _N}]^\mathrm{T}}\), \({\theta ^{*\mathrm{T}}} = \hbox {diag}(\theta _1^{*\mathrm{T}}, \ldots ,\theta _N^{*\mathrm{T}}),\xi = {[\xi _1^\mathrm{T}({{\bar{Z}}_{f,1}}), \ldots ,\xi _N^\mathrm{T}({{\bar{Z}}_{f,N}})]^\mathrm{T}}\), then

$$\begin{aligned} f({\bar{x}}) = {\theta ^{*\mathrm{T}}}\xi - {\xi _f} + \varepsilon \end{aligned}$$

where \({\xi _f}\) is a bounded vector function.

If the signal transmission becomes faulty, \({s_k}\) is not obtained and only \({s_{f,k}}\) is accessed which is defined as follows:

$$\begin{aligned} \begin{aligned} {s_{f,k}}&= {\left( \frac{\mathrm{d}}{{\mathrm{d}t}} + \lambda \right) ^{n - 1}}{e_{a,k,1}}(t) = \sum \limits _{i = 1}^{n - 1} {{c_{k,i}}{e_{k,i}}(t)} + {e_{k,n}}(t)\\&\quad + {\lambda _1}\left[ \sum \limits _{j \in {N_i}} {{a_{kj}}({g_{f,j,1}} - {g_{f,k,1}})} + {b_k}({g_{f,0,1}} - {g_{f,k,1}})\right] \\&= {s_k} + {\lambda _1}{\omega _k} \end{aligned} \end{aligned}$$
(17)

Let \({s_f} = {[{s_{f,1}}, \ldots ,{s_{f,n}}]^\mathrm{T}},\omega = {[{\lambda _1}{\omega _1}, \ldots ,{\lambda _1}{\omega _n}]^\mathrm{T}}\) , then

$$\begin{aligned} {s_f} = s + \omega \end{aligned}$$
(18)

Differentiating \({s_f}\) with respect to time t , we have

$$\begin{aligned} {{\dot{s}}_f} = {\dot{s}} + {\dot{\omega }} = \gamma - (L + B)(f({{\bar{x}}}) + u + h - {\underline{f}_0}) \end{aligned}$$
(19)

Define the following Lyapunov function

$$\begin{aligned} {V_s} = {s^\mathrm{T}}Ps/2 \end{aligned}$$
(20)

where \(P = {P^\mathrm{T}} > 0 \in {R^{n \times n}}\).

Differentiating \({V_s}\) with respect to time t, we have

$$\begin{aligned} \begin{aligned} {{{\dot{V}}}_s}&={s^\mathrm{T}}P\left[ {\gamma - (L + B)({\theta ^{*\mathrm{T}}}\xi - {\xi _f} + \varepsilon + u + h - {\underline{f}_0}}) \right] \\&={s^\mathrm{T}}P\gamma - {s^\mathrm{T}}P(L + B)u - {s^\mathrm{T}}P(D + B){\theta ^{*\mathrm{T}}}\xi \\&\quad +{s^\mathrm{T}}PA{\theta ^{*\mathrm{T}}}\xi + {s^\mathrm{T}}P(D + B){\xi _f} - {s^\mathrm{T}}PA{\xi _f}\\&\quad - {s^\mathrm{T}}P(D + B)(\varepsilon + h - {\underline{f}_0}) + {s^\mathrm{T}}PA(\varepsilon + h - {\underline{f}_0}) \end{aligned} \end{aligned}$$
(21)

Define the following control law:

$$\begin{aligned} u = {(D + B)^{ - 1}}\gamma - \hat{f} - \mathop {\mathrm{sgn}} ({({s_f} - \hat{\omega } )^\mathrm{T}}P(D + B)){{\hat{{\bar{M}}}}_{\varepsilon hf}} + m({s_f} - \hat{\omega } ) \end{aligned}$$
(22)

where \(\hat{f} = {[{\hat{f}_1}, \ldots ,{\hat{f}_N}]^\mathrm{T}}\), \({\hat{f}_k} = \hat{\theta } _k^\mathrm{T}{\xi _k}({{\bar{Z}}_{f,k}})\) is the estimate of \({f_k}({{\bar{Z}}_{f,k}})\). \({P_{s,k}}\) is the kth element of \({({s_f} - \hat{\omega } )^\mathrm{T}}P(D + B), {\hat{{\bar{M}}}_{\varepsilon hf}} = {[{\hat{{\bar{M}}}_{\varepsilon hf,1}}, \ldots ,{\hat{{\bar{M}}}_{\varepsilon hf,N}}]^\mathrm{T}}\), \({\hat{M}_{\varepsilon hf,k}}\) is the estimate of \({M_{\varepsilon hf}} = {M_{\varepsilon ,k}} + {M_{h,k}} + {M_{f0}}\). \(m > 0 \in R\) is a design parameter, which satisfies

$$\begin{aligned} m\underline{\sigma }(Q)/2 - (5r + 2rm + {{\bar{\lambda }} ^2}/4r{\underline{\lambda }^2}){\bar{\sigma }} (A){\bar{\sigma }} (P) - r{\bar{\sigma }} (P) > 0 \end{aligned}$$

Let \({\hat{\theta } ^\mathrm{T}} = \hbox {diag}(\hat{\theta } _1^\mathrm{T}, \ldots ,\hat{\theta } _N^\mathrm{T})\), which \(\hat{\theta } _k^\mathrm{T}\) is the estimates of \(\theta _k^{*\mathrm{T}}\).

Define the notations as follows:

$$\begin{aligned} {\tilde{\theta }} = {\theta ^*} - \hat{\theta } ,{\tilde{{\bar{M}}}_{\varepsilon hf}} = {{\bar{M}}_{\varepsilon hf}} - {\hat{{\bar{M}}}_{\varepsilon hf}},{\tilde{\omega }} = \omega - \hat{\omega } \end{aligned}$$

Substituting control law (22) in (21), one has

$$\begin{aligned} \begin{aligned} {{{\dot{V}}}_s} =&- m{s^\mathrm{T}}P(D + B)s + m{s^\mathrm{T}}PAs - {s^\mathrm{T}}P(D + B){{{\tilde{\theta }} }^\mathrm{T}}\xi \\&- {s^\mathrm{T}}P(D + B)\mathop {\mathrm{sgn}} ({s^\mathrm{T}}P(D + B)){{\tilde{{\bar{M}}}}_{\varepsilon hf}} + {s^\mathrm{T}}PA{{{\bar{M}}}_{\varepsilon hf}}\\&+ m{s^\mathrm{T}}PA\omega - m{s^\mathrm{T}}PA\hat{\omega } + {s^\mathrm{T}}PA{{{\tilde{\theta }} }^\mathrm{T}}\xi - {s^\mathrm{T}}PA{\xi _f}\\&- m{s^\mathrm{T}}P(D + B){\tilde{\omega }} - {s^\mathrm{T}}PA\mathop {\mathrm{sgn}} ({s^\mathrm{T}}P(D + B)){{\hat{{\bar{M}}}}_{\varepsilon hf}}\\&+ {s^\mathrm{T}}PA{(D + B)^{ - 1}}\gamma + {s^\mathrm{T}}P(D + B){\xi _f} \end{aligned} \end{aligned}$$
(23)

Since

$$\begin{aligned}&- {s^\mathrm{T}}PA{\xi _f} \le {\bar{\sigma }} (P){\bar{\sigma }} (A)r{s^\mathrm{T}}s + {\bar{\sigma }} (P){\bar{\sigma }} (A)/4r\xi _f^\mathrm{T}{\xi _f}\\&{s^\mathrm{T}}PA{{\tilde{\theta }} ^\mathrm{T}}\xi \le {\bar{\sigma }} (P){\bar{\sigma }} (A)r{s^\mathrm{T}}s + {\bar{\sigma }} (P){\bar{\sigma }} (A)/4r{\xi ^\mathrm{T}}{\tilde{\theta }} {{\tilde{\theta }} ^\mathrm{T}}\xi \\&m{s^\mathrm{T}}PA\omega \le {\bar{\sigma }} (P){\bar{\sigma }} (A)rm{s^\mathrm{T}}s + {\bar{\sigma }} (P){\bar{\sigma }} (A)/4rm{\omega ^\mathrm{T}}\omega \\&- m{s^\mathrm{T}}PA\hat{\omega } \le {\bar{\sigma }} (P){\bar{\sigma }} (A)rm{s^\mathrm{T}}s + {\bar{\sigma }} (P){\bar{\sigma }} (A)/4rm\beta _\omega ^2\\&{s^\mathrm{T}}PA{{{\bar{M}}}_{\varepsilon hf}} \le {\bar{\sigma }} (P){\bar{\sigma }} (A)r{s^\mathrm{T}}s + {\bar{\sigma }} (P){\bar{\sigma }} (A)/4r{\bar{M}}_{\varepsilon hf}^\mathrm{T}{{{\bar{M}}}_{\varepsilon hf}}\\&{s^\mathrm{T}}PA{(D + B)^{ - 1}}\gamma \le {\bar{\sigma }} (P){\bar{\sigma }} (A)r{s^\mathrm{T}}s + {\bar{\sigma }} (P){\bar{\sigma }} (A)/4r{\gamma ^\mathrm{T}}\gamma \\&{s^\mathrm{T}}P(D + B){\xi _f} \le {\bar{\sigma }} (P)r{s^\mathrm{T}}s + {\bar{\sigma }} (P)/4r\xi _f^\mathrm{T}{(D + B)^\mathrm{T}}(D + B){\xi _f}\\&- {s^\mathrm{T}}PA{\mathrm{sgn}} ({s^\mathrm{T}}P(D + B)){\hat{{\bar{M}}}_{\varepsilon hf}} \le {\bar{\sigma }} (P){\bar{\sigma }} (A)r{s^\mathrm{T}}s + {\bar{\sigma }} (P){\bar{\sigma }} (A)/4r\beta _M^2 \end{aligned}$$

where \(r > 0 \in R\) is a design parameter, and \({{\bar{M}}_{\varepsilon hf}} ={[{M_{\varepsilon hf,1}}, \ldots ,{M_{\varepsilon hf,N}}]^\mathrm{T}}\), one has

$$\begin{aligned} \begin{aligned} {{{\dot{V}}}_s}&\le - m{s^\mathrm{T}}P(L + B)s - {s^\mathrm{T}}P(D + B){{{\tilde{\theta }} }^\mathrm{T}}\xi - m{s^\mathrm{T}}P(D + B){\tilde{\omega }}\\&\quad - {s^\mathrm{T}}P(D+ B)\mathop {\mathrm{sgn}} ({s^\mathrm{T}}P(D + B)){{\tilde{{\bar{M}}}}_{\varepsilon hf}}+ {\bar{\sigma }} (P){\bar{\sigma }} (A)/4r{\gamma ^\mathrm{T}}\gamma \\&\quad + r{\bar{\sigma }} (P)(5{\bar{\sigma }} (A) + 2m{\bar{\sigma }} (A) + 1){s^\mathrm{T}}s+ {\bar{\sigma }} (P)/4r\xi _f^\mathrm{T}{(D + B)^\mathrm{T}}(D + B){\xi _f}\\&\quad + {\bar{\sigma }} (P){\bar{\sigma }} (A)/4r({\xi ^\mathrm{T}}{\tilde{\theta }} {{{\tilde{\theta }} }^\mathrm{T}}\xi + \xi _f^\mathrm{T}{\xi _f}+ \beta _M^2 + m\beta _\omega ^2 + m{{ \omega }^\mathrm{T}} \omega +{\bar{M}}_{\varepsilon hf}^\mathrm{T}{{{\bar{M}}}_{\varepsilon hf}} )\\ \end{aligned} \end{aligned}$$
(24)

Since \(\theta _k^*\) and \({\hat{\theta } _k}\) are bounded, \({{\tilde{\theta }} ^\mathrm{T}}\xi \) is bounded. From Assumptions 2, 3 and 5, \({{\bar{M}}_{h\varepsilon f}}\) is bounded. Adaptive law (31) ensures that \(||{\hat{{\bar{M}}}_{h\varepsilon f}}|| \le {\beta _M}\). Further, \({\tilde{{\bar{M}}}_{h\varepsilon f}}\) is bounded as well.

Since \({{\tilde{\theta }} ^\mathrm{T}}\xi ,{{\bar{M}}_{h\varepsilon f}}\) and \({\omega }\) are bounded, there exists an appropriate parameter r , then

$$\begin{aligned}&{\bar{\sigma }} (P){\bar{\sigma }} (A)/4r({\xi ^\mathrm{T}}{\tilde{\theta }} {{{\tilde{\theta }} }^\mathrm{T}}\xi + \xi _f^\mathrm{T}{\xi _f}+ \beta _M^2 + m\beta _\omega ^2 + m{{ \omega }^\mathrm{T}} \omega +{\bar{M}}_{\varepsilon hf}^\mathrm{T}{{{\bar{M}}}_{\varepsilon hf}} )\le {\mu _{s1}}\\&{\bar{\sigma }} (P)/4r\xi _f^\mathrm{T}{(D + B)^\mathrm{T}}(D + B){\xi _f} \le {\mu _{s2}} \end{aligned}$$

Since

$$\begin{aligned} \gamma _k^2= & {} \sum \limits _{i = 1}^{n - 1} {\lambda _i^2} e_{k,i + 1}^2 \le {{\bar{\lambda }} ^2}s_k^2/{\underline{\lambda }^2}\\ {\gamma ^\mathrm{T}}\gamma= & {} \sum \limits _{k = 1}^N {\gamma _k^2} \le \sum \limits _{k = 1}^N {\frac{{{{{\bar{\lambda }} }^2}}}{{{{\underline{\lambda }}^2}}}s_k^2} = \frac{{{{{\bar{\lambda }} }^2}}}{{{{\underline{\lambda }}^2}}}\sum \limits _{k = 1}^N {s_k^2} = \frac{{{{{\bar{\lambda }} }^2}}}{{{{\underline{\lambda }}^2}}}{s^\mathrm{T}}s \end{aligned}$$

where \({\bar{\lambda }} = \max \{ {\lambda _1}, \ldots ,{\lambda _n}\}\), \(\underline{\lambda } = \min \{ {\lambda _1}, \ldots ,{\lambda _n}\} \). Further, one has

$$\begin{aligned} \begin{aligned} {{{\dot{V}}}_s} \le&- m{s^\mathrm{T}}P(L + B)s + ((5r + 2rm + {{{\bar{\lambda }} }^2}/4r{{\underline{\lambda }}^2}){\bar{\sigma }} (A) + r){\bar{\sigma }} (P){s^\mathrm{T}}s\\&- {s^\mathrm{T}}P(D + B){{{\tilde{\theta }} }^\mathrm{T}}\xi - \;{s^\mathrm{T}}P(D + B)\mathop {\mathrm{sgn}} ({s^\mathrm{T}}P(D + B)){{\tilde{{\bar{M}}}}_{\varepsilon hf}}\\&- m{s^\mathrm{T}}P(D + B){\tilde{\omega }} + {\mu _s} \end{aligned} \end{aligned}$$
(25)

where \({\mu _s} = {\mu _{s1}} + {\mu _{s2}}\), \({\mu _s}\) is a design parameter.

Define the Lyapunov function

$$\begin{aligned} {V_\theta } = tr\{ {{\tilde{\theta }} ^\mathrm{T}}{\tilde{\theta }} \} /2{\eta _1} + {\tilde{{\bar{M}}}}_{\varepsilon hf}^\mathrm{T}{\tilde{{\bar{M}}}_{\varepsilon hf}}/2{\eta _2} + {{\tilde{\omega }} ^\mathrm{T}}{\tilde{\omega }} /2{\eta _3} \end{aligned}$$
(26)

where \({\eta _1} > 0 \in R\), \({\eta _2} > 0 \in R\), \({\eta _3} > 0 \in R\) are design parameters.

Differentiating \({V_\theta }\) with respect to time t, we have

$$\begin{aligned} {{\dot{V}}_\theta } = - tr\{ {{\tilde{\theta }} ^\mathrm{T}}\dot{\hat{\theta }} \} /{\eta _1} - {\tilde{{\bar{M}}}}_{\varepsilon hf}^\mathrm{T}{\dot{\hat{{\bar{M}}}}_{\varepsilon hf}}/{\eta _2} - {{\tilde{\omega }} ^\mathrm{T}}\dot{\hat{\omega }} /{\eta _3} \end{aligned}$$
(27)

Define Lyapunov function

$$\begin{aligned} {V_0} = {V_s} + {V_\theta } \end{aligned}$$
(28)

Differentiating \({V_{\mathrm{0}}}\) with respect to time t, we have

$$\begin{aligned} \begin{aligned} {{{\dot{V}}}_0} \le&- m{s^\mathrm{T}}P(L + B)s + ((5r + 2rm + {{{\bar{\lambda }} }^2}/4r{{\underline{\lambda }}^2}){\bar{\sigma }} (A)+ r){\bar{\sigma }} (P){s^\mathrm{T}}s\\&- {s^\mathrm{T}}P(D + B){{{\tilde{\theta }} }^\mathrm{T}}\xi - {s^\mathrm{T}}P(D+ B)\mathop {\mathrm{sgn}} ({s^\mathrm{T}}P(D + B)){{\tilde{{\bar{M}}}}_{\varepsilon hf}}\\&- m{s^\mathrm{T}}P(D + B){\tilde{\omega }} - {{{\tilde{\theta }} }^\mathrm{T}}\dot{\hat{\theta }} /{\eta _1} - {{\tilde{{\bar{M}}}}}_{\varepsilon hf}^\mathrm{T}{{\dot{\hat{{\bar{M}}}}}_{\varepsilon hf}}/{\eta _2} - {{{\tilde{\omega }} }^\mathrm{T}}\dot{\hat{\omega }} /{\eta _3} + {\mu _s} \end{aligned} \end{aligned}$$
(29)

Define the following adaptive laws:

$$\begin{aligned} \dot{\hat{\theta }}= & {} \left\{ \begin{array}{ll} {\tau _\theta },&{}\quad \hbox {if }||\hat{\theta } || < {\beta _\theta }\hbox { or }||\hat{\theta } || = {\beta _\theta }\hbox { and }{{\hat{\theta } }^\mathrm{T}}{\tau _\theta } \le 0\\ {\tau _\theta } - \frac{{\hat{\theta } {{\hat{\theta } }^\mathrm{T}}}}{{||\hat{\theta } |{|^2}}}{\tau _\theta },&{}\quad \hbox {if }||\hat{\theta } || = {\beta _\theta }\hbox { and }{{\hat{\theta } }^\mathrm{T}}{\tau _\theta } > 0 \end{array} \right. \end{aligned}$$
(30)
$$\begin{aligned} {\dot{\hat{{\bar{M}}}}_{\varepsilon hf}}= & {} \left\{ \begin{array}{ll} {\tau _M},&{}\quad \hbox {if }||{{\hat{{\bar{M}}}}_{\varepsilon hf}}|| < {\beta _M}\hbox { or }||{{ {\hat{{\bar{M}}}}}_{\varepsilon hf}}|| = {\beta _M}\hbox { and }{{\hat{{\bar{M}}}}}_{\varepsilon hf}^\mathrm{T}{\tau _M} \le 0\\ {\tau _M} - \frac{{{{\hat{{\bar{M}}}}_{\varepsilon hf}}{{\hat{{\bar{M}}}}}_{\varepsilon hf}^\mathrm{T}}}{{||{{\hat{{\bar{M}}}}_{\varepsilon hf}}|{|^2}}}{\tau _M},&{}\quad \hbox {if } ||{{\hat{{\bar{M}}}}_{\varepsilon hf}}|| = {\beta _M} \hbox { and }{{\hat{{\bar{M}}}}}_{\varepsilon hf}^\mathrm{T}{\tau _M} > 0 \end{array} \right. \end{aligned}$$
(31)
$$\begin{aligned} \dot{\hat{\omega }}= & {} \left\{ \begin{array}{ll} {\tau _\omega },&{}\quad \hbox {if }||\hat{\omega } || < {\beta _\omega }\hbox { or }||\hat{\omega } || = {\beta _\omega }\hbox { and }{{\hat{\omega } }^\mathrm{T}}{\tau _\omega } \le 0\\ {\tau _\omega } - \frac{{\hat{\omega } {{\hat{\omega } }^\mathrm{T}}}}{{||\hat{\omega } |{|^2}}}{\tau _\omega },&{} \quad \hbox {if }||\hat{\omega } || = {\beta _\omega }\hbox { and }{{\hat{\omega } }^\mathrm{T}}{\tau _\omega } > 0 \end{array} \right. \end{aligned}$$
(32)

where \({\tau _\theta } = - {\eta _1}\xi {({s_f} - \hat{\omega } )^\mathrm{T}}P(D + B) + {\eta _\theta }\hat{\theta }\), \({\tau _M} = - {\eta _2}\mathop {\mathrm{sgn}} ({({s_f} - \hat{\omega } )^\mathrm{T}}P(D + B)){({s_f} - \hat{\omega } )^\mathrm{T}}P(D + B) \; + {\eta _M}{{\hat{{\bar{M}}}}_{\varepsilon hf}}\), \({\tau _\omega } = - {\eta _3}m{\lambda _1}{({s_f} - \hat{\omega } )^\mathrm{T}}P(D + B) + {\eta _\omega }\hat{\omega }\). \({\eta _\theta } > 0 \in R\), \({\eta _M} > 0 \in R\), \({\eta _\omega } > 0 \in R\) are design parameters. Note that project operators are adopted to ensure that \(\hat{\theta } \), \({\hat{{\bar{M}}}_{\varepsilon hf}}\) and \(\hat{\omega } \) are bounded.

4 Stability Analysis

Theorem 1

Consider multi-agent system (1) and leader node (2) under Assumptions 1–5. Using distributed control law (22) and adaptive laws (30)–(32), the tracking errors \({\delta _i}(i = 1, \ldots ,n)\) are cooperative uniformly ultimately bounded and the tracking error \({\delta _i}\) converges to the small neighborhoods of the origin.

Proof

Define the Lyapunov function

$$\begin{aligned} {V_0} = {s^\mathrm{T}}Ps/2 + tr\{ {{\tilde{\theta }} ^\mathrm{T}}{\tilde{\theta }} \} /2{\eta _1} + {\tilde{{\bar{M}}}}_{\varepsilon hf}^\mathrm{T}{\tilde{{\bar{M}}}_{\varepsilon hf}}/2{\eta _2} + {{\tilde{\omega }} ^\mathrm{T}}{\tilde{\omega }} /2{\eta _3} \end{aligned}$$

Differentiating \({V_{\mathrm{0}}}\) with respect to time t, we have

$$\begin{aligned} \begin{aligned} {{{\dot{V}}}_0}&= {s^\mathrm{T}}P\gamma - {s^\mathrm{T}}P(L + B)u - {s^\mathrm{T}}P(D + B){\theta ^{*\mathrm{T}}}\xi \\&\quad + {s^\mathrm{T}}PA{\theta ^{*\mathrm{T}}}\xi + {s^\mathrm{T}}P(D + B){\xi _f} - {s^\mathrm{T}}PA{\xi _f}\\&\quad - {s^\mathrm{T}}P(D + B)(\varepsilon + h - {\underline{f}_0}) + {s^\mathrm{T}}PA(\varepsilon + h - {\underline{f}_0})\\&\quad - tr\{ {{\tilde{\theta }}^\mathrm{T}}\dot{\hat{\theta }} \} /{\eta _1} - {{\tilde{{\bar{M}}}}}_{\varepsilon hf}^\mathrm{T}{{\dot{\hat{{\bar{M}}}}}_{\varepsilon hf}}/{\eta _2} - {{{\tilde{\omega }} }^\mathrm{T}}\dot{\hat{\omega }} /{\eta _3} \end{aligned} \end{aligned}$$
(33)

Substituting control law (22) and adaptive law (30)–(32) in (33), one has

$$\begin{aligned} \begin{aligned} {{{\dot{V}}}_0} \le&- m{s^\mathrm{T}}P(L + B)s + ((5r + 2rm + {{{\bar{\lambda }} }^2}/4r{{\underline{\lambda }}^2}){\bar{\sigma }} (A) + r){\bar{\sigma }} (P){s^\mathrm{T}}s\\&- \frac{{{\eta _\theta }}}{{{\eta _1}}}tr\{ {{{\tilde{\theta }} }^\mathrm{T}}\hat{\theta } \} - \frac{{{\eta _M}}}{{{\eta _2}}}{{\tilde{{\bar{M}}}}}_{\varepsilon hf}^\mathrm{T}{{\hat{{\bar{M}}}}_{\varepsilon hf}} - \frac{{{\eta _\omega }}}{{{\eta _3}}}{{{\tilde{\omega }} }^\mathrm{T}}\hat{\omega } + {\mu _s} \end{aligned} \end{aligned}$$
(34)

Since

$$\begin{aligned} \begin{aligned} - \frac{{{\eta _\theta }}}{{{\eta _1}}}tr\{ {{\tilde{\theta }} ^\mathrm{T}}\hat{\theta } \}&\le - \frac{{{\eta _\theta }}}{{2{\eta _1}}}tr\{ {{\tilde{\theta }} ^\mathrm{T}}{\tilde{\theta }} \} + \frac{{{\eta _\theta }}}{{2{\eta _1}}}tr\{ {\theta ^{*\mathrm{T}}}{\theta ^*}\}\\ - \frac{{{\eta _M}}}{{{\eta _2}}}{\tilde{{\bar{M}}}}_{\varepsilon hf}^\mathrm{T}{\hat{{\bar{M}}}_{\varepsilon hf}}&\le - \frac{{{\eta _M}}}{{2{\eta _2}}}{\tilde{{\bar{M}}}}_{\varepsilon hf}^\mathrm{T}{\tilde{{\bar{M}}}_{\varepsilon hf}} + \frac{{{\eta _M}}}{{2{\eta _2}}}{{\bar{M}}^\mathrm{T}}_{\varepsilon hf}{{\bar{M}}_{\varepsilon hf}}\\ - \frac{{{\eta _\omega }}}{{{\eta _3}}}{{\tilde{\omega }} ^\mathrm{T}}\hat{\omega }&\le - \frac{{{\eta _\omega }}}{{2{\eta _3}}}{{\tilde{\omega }} ^\mathrm{T}}{\tilde{\omega }} + \frac{{{\eta _\omega }}}{{2{\eta _3}}}{\omega ^\mathrm{T}}\omega \end{aligned} \end{aligned}$$

one has

$$\begin{aligned} \begin{aligned} {{{\dot{V}}}_0} \le&- \frac{1}{2}m{s^\mathrm{T}}Qs + ((5r + 2rm + {{{\bar{\lambda }} }^2}/4r{{\underline{\lambda }}^2}){\bar{\sigma }} (A) + r){\bar{\sigma }} (P){s^\mathrm{T}}s\\&- \frac{{{\eta _\theta }}}{{2{\eta _1}}}tr\{ {{{\tilde{\theta }} }^\mathrm{T}}{\tilde{\theta }} \} - \frac{{{\eta _M}}}{{2{\eta _2}}}{{\tilde{{\bar{M}}}}}_{\varepsilon hf}^\mathrm{T}{{\tilde{{\bar{M}}}}_{\varepsilon hf}} - \frac{{{\eta _\omega }}}{{2{\eta _3}}}{{{\tilde{\omega }} }^\mathrm{T}}{\tilde{\omega }} \\&+ \frac{{{\eta _\theta }}}{{2{\eta _1}}}tr\{ {\theta ^{*\mathrm{T}}}{\theta ^*}\} + \frac{{{\eta _M}}}{{2{\eta _2}}}{{{\bar{M}}}^\mathrm{T}}_{\varepsilon hf}{{{\bar{M}}}_{\varepsilon hf}} + \frac{{{\eta _\omega }}}{{2{\eta _3}}}{\omega ^\mathrm{T}}\omega + {\mu _s} \end{aligned} \end{aligned}$$
(35)

From the previous analysis, we know \({\theta ^ * },{{\bar{M}}_{\varepsilon hf}},\omega \) are bounded. If \({\eta _\theta },{\eta _M},{\eta _\omega }\) and \({\eta _1},{\eta _2},{\eta _3}\) are chosen appropriately, then

$$\begin{aligned} \frac{{{\eta _\theta }}}{{2{\eta _1}}}tr\{ {\theta ^{*\mathrm{T}}}{\theta ^*}\} + \frac{{{\eta _M}}}{{2{\eta _2}}}{{\bar{M}}^\mathrm{T}}_{\varepsilon hf}{{\bar{M}}_{\varepsilon hf}} + \frac{{{\eta _\omega }}}{{2{\eta _3}}}{\omega ^\mathrm{T}}\omega \le {\mu _{\theta M}} \end{aligned}$$

where \({\mu _{\theta M}} > 0 \in R\) is a design parameter.

Let \({\mu _0} = {\mu _s} + {\mu _{\theta M}}\), then

$$\begin{aligned} {{\dot{V}}_0} \le - {\lambda _0}{V_0} + {\mu _0} \end{aligned}$$

Further, one has

$$\begin{aligned} 0 \le {V_0}(t) \le \frac{{{\mu _0}}}{{{\lambda _0}}} + \left( {V_0}({t_0}) - \frac{{{\mu _0}}}{{{\lambda _0}}}\right) {e^{ - {\lambda _0}(t - {t_0})}} \end{aligned}$$

where \({\lambda _0} = \min \left\{ {\frac{{1/2m\underline{\sigma }(Q)}}{{{\bar{\sigma }} (P)}} - (5r + 2rm{\mathrm{+ }}{{{\bar{\lambda }} }^2}/4r{{\underline{\lambda }}^2}){\bar{\sigma }} (A)} \right. - r, \left. {\frac{{{\eta _\theta }}}{{2{\eta _1}}},\frac{{{\eta _M}}}{{2{\eta _2}}},\frac{{{\eta _\omega }}}{{2{\eta _3}}}} \right\} .\)

Let \({V_0} = \frac{1}{2}\underline{\sigma }(P){M^2}, M > 0 \in R\) is a design parameter. Since \(\underline{\sigma }(P)||s(t)|{|^2} \le 2{V_s}(t) \le 2{V_0}(t)\), then \(||s(t)|| \le \sqrt{2{V_0}(t)/\underline{\sigma }(P)}, ||s(t)|| \le M\) . Similarly \(||{\tilde{\theta }} (t)|| \le \sqrt{{\eta _1}} M,||{\tilde{{\bar{M}}}_{\varepsilon hf}}(t)|| \le \sqrt{{\eta _2}} M , ||{\tilde{\omega }} || \le \sqrt{{\eta _3}} M\). According to the previous analysis, we know s and \(\omega \) are bounded, because \({s_f}= s + \omega \), so \({s_f}\) also is bounded. Since \( |{e_{k,i}}| \le {2^{i - 1}}\lambda _k^{i - n}M\), one has \( ||{e_i}|| \le \sqrt{N{{({2^{i - 1}}\lambda _k^{i - n}M)}^2}} \). According to (16), we can know \(||{e_{a,i}}|| \le \sqrt{N{{({2^{i - 1}}\lambda _k^{i - n}M)}^2}} + {\beta _\omega }\;i = 1, \ldots , ~ n\;k = 1, \ldots ,N \). From Lemma 2, we can get the following result: from the previous analysis, \({s_f}(t)\) is cooperative uniform ultimate boundedness. Further, \({s_{f,k}}(t)\) also are cooperative uniform ultimate boundedness. Because \({e_{a,i}}(t)\) is bounded, from Lemma 2, one has \({\delta _{k,i}}\) are bounded. And since the state of the leader is bounded, the state \({x_{f,k}}\) are bounded as well.

5 Simulation Results

Consider a 5-node digraph G and a leader node described in Fig. 4. The dynamics of the leader node is described as follows:

$$\begin{aligned} \left\{ \begin{aligned}&{{\dot{x}}_{0,1}}(t) = {x_{0,2}}(t)\\&{{\dot{x}}_{0,2}}(t) = {x_{0,3}}(t)\\&{{\dot{x}}_{0,3}}(t) = -{x_{0,1}}(t)-1.5{x_{0,2}}(t) - 2{x_{0,3}}(t) + 2\sin (2t) + 4\cos (2t) \end{aligned} \right. \end{aligned}$$

The follower nodes are described by third-order nonlinear systems in the form of (1) with

$$\begin{aligned} \begin{aligned}&{{\dot{x}}_{1,3}}(t) = - {x_{1,1}}{x_{1,2}} + \sin ({x_{1,3}}) + {u_1} + {h_1}\\&{{\dot{x}}_{2,3}}(t) = {x_{2,1}}\cos ({x_{2,2}}) + 2{x_{2,3}} + {u_2} + {h_2}\\&{{\dot{x}}_{3,3}}(t) = - {x_{3,1}} + \sin ({x_{3,2}}) + {u_3} + {h_3}\\&{{\dot{x}}_{4,3}}(t) = {({x_{4,1}} + {x_{4,2}})^2} + 3{x_{4,3}} + \cos (2t) + {u_4} + {h_4}\\&{{\dot{x}}_{5,3}}(t) = - 2{x_{5,1}} + {x_{5,2}} + {u_5} + {h_5} \end{aligned} \end{aligned}$$

In this paper, the disturbance \({h_k}\) is random constant and bounded by \(|{h_k}| \le 1\). Choose the following initial states: \({x_0} = {[0,1,1]^\mathrm{T}},{x_1} = {[1, - 1,0]^\mathrm{T}}, {x_2} = {[0, - 1, - 2]^\mathrm{T}},{x_3} = {[0,1,0]^\mathrm{T}},{x_4} = [1.5,0,0{]^\mathrm{T}},{x_5} = {[0,1, - 1]^\mathrm{T}}.\)

Fig. 4
figure 4

Topology of the communication

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From Fig. 1 and Lemma 1, we can know

$$\begin{aligned} A= & {} \left[ {\begin{array}{*{20}{c}} 0&{}\quad 0&{}\quad 2&{}\quad 0&{}\quad 0\\ 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 4&{}\quad 0&{}\quad 1&{}\quad 0\\ 3&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] B = \left[ {\begin{array}{*{20}{c}} 5&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1 \end{array}} \right] D = \left[ {\begin{array}{*{20}{c}} 2&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 5&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 3&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] \\ L= & {} \left[ {\begin{array}{*{20}{c}} 2&{}\quad 0&{}\quad { - 2}&{}\quad 0&{}\quad 0\\ { - 1}&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad { - 4}&{}\quad 5&{}\quad { - 1}&{}\quad 0\\ { - 3}&{}\quad 0&{}\quad 0&{}\quad 3&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] L + B = \left[ {\begin{array}{*{20}{c}} 7&{}\quad 0&{}\quad { - 2}&{}\quad 0&{}\quad 0\\ { - 1}&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad { - 4}&{}\quad 5&{}\quad { - 1}&{}\quad 0\\ { - 3}&{}\quad 0&{}\quad 0&{}\quad 3&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1 \end{array}} \right] \\ q= & {} \left[ {\begin{array}{*{20}{c}} {{\mathrm{0}}{\mathrm{.62}}}\\ {{\mathrm{1}}{\mathrm{.62}}}\\ {{\mathrm{1}}{\mathrm{.69}}}\\ {{\mathrm{0}}{\mathrm{.96}}}\\ {{\mathrm{1}}{\mathrm{.00}}} \end{array}} \right] P = \left[ {\begin{array}{*{20}{c}} {{\mathrm{1}}{\mathrm{.59}}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad {{\mathrm{0}}{\mathrm{.61}}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad {{\mathrm{0}}{\mathrm{.59}}}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad {{\mathrm{1}}{\mathrm{.04}}}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {{\mathrm{1}}{\mathrm{.00}}} \end{array}} \right] \\ Q= & {} \left[ {\begin{array}{*{20}{c}} {{{11}}{{.16}}}&{}\quad {{{0}}}&{}\quad {{{- 1}}{{.18 }}}&{}\quad 0&{}\quad 0\\ {{{- 1}}{{.59}}}&{}\quad {{{0}}{\mathrm{.61 }}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad {{{- 2}}{\mathrm{.45}}}&{}\quad {{\mathrm{2}}{\mathrm{.95}}}&{}\quad {{\mathrm{- 1}}{\mathrm{.04 }}}&{}\quad 0\\ {{{- 4}}{\mathrm{.78}}}&{}\quad 0&{}\quad 0&{}\quad {{\mathrm{3}}{\mathrm{.12}}}&{}\quad 0\\ {{{0 }}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {1.00} \end{array}} \right] \end{aligned}$$
Fig. 5
figure 5

The state of the leader 0

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Fig. 6
figure 6

Tracking error \({\delta _1}\)

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Fig. 7
figure 7

Tracking error \({\delta _2}\)

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Fig. 8
figure 8

Tracking error \({\delta _3}\)

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Further, we have the singular values of: 11.01, 4.03, 0.42, 2.36, 1.00. \({\bar{\sigma }} (A) = {{2}}{{.41}}, {\bar{\sigma }} (P) = 1.59 ,{\bar{\sigma }} (Q) = 11.01,{\bar{\sigma }} (D + B) = 1.00\). In this simulation, we choose \(m = 1,{\lambda _1} = 2,{\lambda _2} = 2,{\lambda _3} = 1,{\eta _1} = {\eta _2} = {\eta _3} = 100,{\eta _\theta } = {\eta _M} = {\eta _\omega } = 0.01,{\mu _s} = {\mu _{\theta M}} = 0.1,r = 100.\)

The simulation results are presented in Figs. 5, 6, 7 and 8. From Fig. 5, we can find that the state of the leader is bound. From Figs. 6, 7 and 8, at the beginning, under the normal controller, the tracking errors converge to the neighborhood of the origin. When signal transmission faults occurred, we can find the tracking errors deviate from the neighborhood of the origin. However, if the signal transmission faults are compensated for proposed fault tolerant controllers (22), we can obtain better tracking control performance again. The simulation results have illustrated the effectiveness of the proposed scheme.

6 Conclusions

In this paper, the cooperative adaptive fault-tolerant tracking control problem of high-order nonlinear multi-agent systems with signal transmission faults is studied. Based on the approximation capability of neural networks, an adaptive fault-tolerant control scheme is proposed, which guarantees that all followers asymptotically synchronize to a leader with tracking errors converging to a small adjustable neighborhood of the origin. However, the topology among nodes considered in this paper is fixed. In practical applications, the topology may be variable. For example, a new node is added to or removed from communication network. In the case, how to handle the cooperative control problem of nonlinear multi-agent systems is important and challenging, which is studied in our further research.