Neuro-adaptive cooperative control for a class of high-order nonlinear multi-agent systems

Abstract

In this paper, we studied the cooperative control problem for a class of high-order nonlinear multi-agent systems (MASs) with external disturbance and system uncertainty. A neuro-adaptive robust controller with sliding mode variable structure method, with an online-learning RBF-like neural network was proposed to approximate the nonlinear terms. Further, sliding mode variable structure method was used to eliminate the influence of external disturbance and system uncertainty. Lyapunov stability theorem verified the capability of system consensus, and the sufficient conditions for cooperatively uniformly ultimately bounded (CUUB) are also given. At last, two numerical simulations on both homogeneous and heterogeneous MASs demonstrated the effectiveness of our proposed method.

Introduction

Leader-following cooperative control of multi-agent systems (MASs) has been an attractive research topic due to its distributed control feature, which can strengthen the group communication, robustness, flexibility, and deipersion.^1–4

Bechlioulis and Rovithakis, ⁵ Yang and Li ⁶ designed a robust control based on observer for a class of high-order nonlinear multi-agent systems that satisfies the Lipschitz condition to solve the problem of state enclosing control of multi-agent systems. Aryankia and Selmic ⁷ proposed an adaptive neural network-based backstepping controller that uses rigid graph theory to address the distance-based formation control problem and target tracking for nonlinear multi-agent systems with bounded time-delay and disturbance. Radial basis function (RBF) neural network is used to overcome and compensate for the unknown nonlinearity and disturbance in the system dynamics. Ni et al. ⁸ designed a type of sliding mode observer under the condition of input delay, which can send leader information to followers within a limited time. Meng et al. ⁹ proposed an adaptive neural distributed synchronization scheme with guaranteed performance, which is used to deals with the synchronization control problem in the leader-follower format of a class of high-order nonaffine nonlinear multiagent systems under a directed communication protocol. In order to handle the unknown nonlinear problem, the adaptive fuzzy control problem for a class of high-order stochastic nonlinear systems is also designed, and the control performance is guaranteed in a prescribed bound. ¹⁰

For linear systems, Ma and Miao ¹¹ proposed a solution that the leader follows the heterogeneous multi-agent system in the network to achieve consistent output. When the state information is not easy to directly measure, a solution based on the dynamic regulator of the state observer is used to reconstruct the state. For the second-order nonlinear system, Liu and Jia ¹² aims at the second-order nonlinear system, uses the universal approximation function of neural network to estimate and approximate unknown functions online, and obtains an adaptive protocol based on neural network.

However, most of existing works only focus on first- or second-order dynamic systems,^13–20 while many practical engineering systems are high-order dynamics based, such as single-link flexible-joint manipulators,^21,22 robot formation cooperation,^23,24 and synchronous generator coordination.^25–27 On the other hand, due to the model uncertainty, nonlinear dynamics, as well as the system noise, makes it difficult even impossible to obtain the accurate control model in mathematical. The existence of the nonlinear dynamics may transform the system from homogeneous to heterogeneous, which brings difficulties to the cooperative control of nonlinear systems. ²⁸

Consider the above mentioned problems, this paper focuses on the Brunovsky-type high-order nonlinear cooperative control tasks of leader-following MASs. Each follower couples the unknown nonlinearity with external disturbance through a high-order integrator, while the dynamics of them may be completely different. ²⁹ Further, the leader agent is also a high-order nonlinear but non-autonomous system, and its dynamics is also unknown to all the followers.

In this paper, we propose a neuro-adaptive cooperative control method for a class of high-order nonlinear MASs with uncertainty. During the controller design, sliding mode variable structure control approach is used to fasten the systems response speed. Further, an online-learning RBF-like neural network is proposed to deal with the system nonlinearity. Theoretical analysis is given to prove the system stability under our proposed controller, and numerical simulations are also constructed to verified the correctness and effectiveness.

The main contribution can be concluded as:

1. We consider the cooperative problem for a class of high-order nonlinear MASs, which contains external disturbance or system uncertainty. This dynamical models are more suitable for practical engineering.

2. We propose a distributed neuro-adaptive cooperative controller with online-learning RBF-like neural network, where the neural network can update its own weights according to the network output and agent state. For both homogeneous and heterogeneous MASs, the proposed online-learning neural network can always approximate to the target signal with a tiny fitting error.

The rest of paper is organized as follows: Section “Preliminaries and problem statement” introduces some preliminaries about graph theory and problem formulation. Section “Proposed neuro-adaptive controller” gives the proposed neuro-adaptive control method. In Section “Main results,” we proves the stability of our controller by using Lyapunov analysis theorem. In Section “Numeric simulation,” two kinds of MAS simulations are conducted to demonstrate the effectiveness of our method. At last, Section “Conclusion” summaries the paper as an end.

Preliminaries and problem statement

Graph theory and notation

According to graph theory, a directed graph $\mathcal{G}=(V,E,A)$ describes the communication topology of MASs, that is, the information interaction between agents. The vertex $V=\{v_1,v_2,\dots ,v_N\}$ is the set of $N$ agents; $E\in V\times V$ is the weighted edges of information transmission among agents, where $a_{\mathrm{ij}}$ denotes the weight of edge $(v_j,v_i)$ ; $\mathit{A}=[a_{\mathrm{ij}}{]}^{N\times N}$ is the weight matrix associated with the communication topology. For a graph $\mathcal{G}$ , $a_{\mathrm{ij}}>0$ if $(v_j,v_i)\in E$ , otherwise $a_{\mathrm{ij}}=0$ ; the corresponding in-degree matrix and Laplacian matrix can be defined as $\mathit{D}=\mathrm{diag}\{\mathrm{de}{\mathrm{g}}_1,\mathrm{de}{\mathrm{g}}_2,\dots ,\mathrm{de}{\mathrm{g}}_N\}$ and $\mathit{L}=\mathit{D}-\mathit{A}$ , respectively, where $\mathrm{de}{\mathrm{g}}_i=\sum _{j=1}^N{a}_{\mathrm{ij}}$ denotes the in-degree of vertex $v_i$ .

In this paper, the following notations are used:

${\mathbb{R}}^n$ : an $n$ -dimensional vector of real number set;

$1_n$ : an $n$ -dimensional column vector with number 1;

${\mathbf{I}}_n$ : an $n\times n$ -dimensional identity matrix;

$\parallel ·\parallel$ : the Euclidean norm of a vector;

$\mathrm{diag}\{m_1,m_2,\dots ,m_n\}$ : a diagonal matrix with diagonal elements of $m_1,m_2,\dots ,m_n$ ;

$\underset{¯}{\sigma }(P)$ , $\overline{\sigma }(P)$ : the smallest and largest singular values of matrix $P$ , respectively;

$\mathrm{tr}\{·\}$ : the trace of a matrix.

$\parallel ·{\parallel }_1$ , $\parallel ·{\parallel }_F$ : the 1-norm and Frobenius norm. Given a matrix $\mathit{A}=[a_{\mathrm{ij}}{]}_{m\times n}\in {\mathbb{R}}^{m\times n}$ , the 1-norm is

$$\parallel \mathit{A}{\parallel }_1=\underset{1\le j\le n}{max}\sum _{i=1}^m|a_{\mathrm{ij}}|,$$

and the Frobenius norm is defined as the sum of the squares of the absolute values of the elements of a matrix, that is,

$$\parallel \mathit{A}{\parallel }_F=\sqrt{\mathrm{tr}({\mathit{A}}^T\mathit{A})}=\sqrt{\sum _{i=1}^m\sum _{j=1}^n|a_{\mathrm{ij}}{|}^2}$$

Problem statement

Given a MAS system consisted of a leader agent and $N$ follower agents, where the nonlinear dynamic model of $i$ th ( $i=1,2,\dots ,N$ ) follower agent is formulated as:

$$\{\begin{array}{c}{\stackrel{·}{x}}_{i,m}(t)=x_{i,m+1}(t),m=1,2,\cdots ,M-1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_{i,m}(t)=f_i(t,x_i)+d_i(t)+u_i(t),m=M}\end{array}$$

(1)

where $x_{i,m}(t)\in \mathbb{R}$ denotes the $m$ th state of agent $i$ at time $t$ ; $d_i(t)$ $\in \mathbb{R}$ denotes the uncertainty term (such as external disturbance or unmodeled dynamics); $u_i(t)\in \mathbb{R}$ is the control input, and $f_i(t,x_i)$ is a continuous nonlinear function that describes the inherent dynamic behavior of agent $i$ . Let state vector $x_i(t)=[x_{i,1}(t),x_{i,2}(t),\dots ,x_{i,M}(t){]}^{\mathrm{T}}\in {\mathbb{R}}^M$ .

The dynamics of leader agent (labeled by 0) is described by

$$\{\begin{array}{c}{\stackrel{·}{x}}_{0,m}(t)=x_{0,m+1}(t),m=1,2,\cdots ,M-1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_{0,m}(t)=f_0(t,x_0),m=M}\end{array}$$

(2)

where $x_{0,m}(t)\in \mathbb{R}$ is the $m$ th order state at time $t$ . $f_0(t,x_i)$ is a continuous nonlinear function that represents the inherent dynamics of the leader. Let $x_0(t)=[x_{0,1}(t),x_{0,2}(t),\dots ,x_{0,M}(t){]}^{\mathrm{T}}\in {\mathbb{R}}^M$ .

The $m$ th order consensus error of agent $i$ is defined as ${\delta }_{i,m}=x_{i,m}-x_{0,m}$ . Let ${\mathbf{\delta }}_m=[{\delta }_{1,m},{\delta }_{2,m},\dots ,{\delta }_{N,m}{]}^T$ , thus we have ${\mathbf{\delta }}_m={\mathit{x}}_m-1_N{x}_{0,m}$ , where ${\mathit{x}}_m=[x_{1,m},x_{2,m},\dots ,x_{N,m}{]}^{\mathrm{T}}\in {\mathbb{R}}^N,m=1,2,\dots ,M$ . The goal of designing a distributed control protocol for each follower is to approach zero consensus error after a certain tracking process.

Definition 1. ³⁰ (Cooperatively uniformly ultimately bounded, CUUB) For any $m(m=1,2,\dots ,M)$ , if there exists a compact set ${\mathrm{\Omega }}_m\subset {\mathbb{R}}^N$ satisfies the following three conditions, it is said that the high-order tracking error ${\delta }_m$ of the followers is CUUB.

1. $\{0\}\subset {\mathrm{\Omega }}_m$ ;

2. $\forall {\delta }_m(t_0)\subset {\mathrm{\Omega }}_m$ ;

3. There exists an upper bound ${\mathrm{\Delta }}_m$ and time $T_m$ holds $\parallel {\delta }_m\parallel \le {\mathrm{\Delta }}_m$ if $\forall t\ge t_0+T_m$ .

For agent $i$ , if the tracking error is CUUB, $x_{i,m}(t)$ can converge to the neighborhood state $x_{0,m}(t)$ of the leader at $t\ge t_0+T_m$ .

The local neighborhood error of the agent $i$ is defined as

$$\begin{array}{c}{e}_{i,m}=\sum _{j\in N_i}{a}_{\mathrm{ij}}(x_{j,m}-x_{i,m})+b_i(x_{0,m}-x_{i,m})\hfill \\ =\sum _{j=1}^N{L}_{\mathrm{ij}}{x}_{j,m}+b_i(x_{0,m}-x_{i,m})\hfill \end{array}$$

(3)

where $m=1,2,\dots ,M$ . If there is a directed edge $(v_0,v_i)$ between the follower agent $i$ and the leader agent $0$ , the follower can “perceive” the information of the leader agent, then this weight of this edge is $b_i>0$ , otherwise $b_i=0$ . Let the adjacency matrix $\mathit{B}=\mathrm{diag}\{b_1,b_2,\dots ,b_N\}$ , and the $m$ th order global neighborhood error is ${\mathit{e}}_m(t)=[e_{1,m},e_{2,m},\dots ,e_{N,m}{]}^{\mathrm{T}}\in {\mathbb{R}}^N$ . Let $\mathit{f}(t,x)=[f_1(t,x_1),f_2(t,x_2),\dots ,f_N(t,x_N){]}^{\mathrm{T}}\in {\mathbb{R}}^N$ , $\mathit{u}(t)=[u_1(t),u_2(t),\dots ,u_N(t){]}^{\mathrm{T}}\in {\mathbb{R}}^N$ , $\mathit{d}(t)=[d_1(t),d_2(t),\dots ,d_N(t){]}^{\mathrm{T}}\in {\mathbb{R}}^N$ . Thus, the derivative of (3) can be obtained as

$$\{\begin{array}{c}{\stackrel{·}{e}}_{i,m}(t)=e_{i,m+1}(t),m=1,2,\cdots ,M-1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{e}}_{i,m}(t)=-(\mathit{L}+\mathit{B})(\mathit{f}+\mathit{d}+\mathit{u}-f_0{1}_N),m=M}\end{array}$$

(4)

Define the extended directed network graph composed of the leader and the follower agents as $\overline{\mathcal{G}}=(\overline{V},\overline{E},\overline{A})$ , to realize the information interaction between the network $G$ and the leader agent $0$ . The tracking consensus needs to make the following assumptions about the network topology.

Assumption 1. There exists a directed spanning tree with leader $0$ as the root node in the extended network $\overline{\mathcal{G}}$ .

Lemma 1.^28,31 Define

$$\begin{array}{c}\mathit{q}={[q_1,q_2,\dots ,q_N]}^{\mathrm{T}}={(\mathit{L}+\mathit{B})}^{-1}{1}_N\hfill \\ \mathit{P}=\mathrm{diag}\{p_i\}=\mathrm{diag}\{\frac{1}{q_i}\}\hfill \\ \mathit{Q}=\mathit{P}(\mathit{L}+\mathit{B})+{(\mathit{L}+\mathit{B})}^{\mathrm{T}}\mathit{P}\hfill \end{array}$$

thus we have both $\mathit{P}$ and $\mathit{Q}$ are positive-definite matrices. According to the Assumption 1, in the extended network $\overline{\mathcal{G}}$ , $\mathit{L}+\mathit{B}$ is a non-singular matrix.

Lemma 2. ³² $\parallel {\mathbf{\delta }}_m\parallel \le \parallel {\mathit{e}}_m\parallel /\underset{¯}{\mathbf{\sigma }}(\mathit{L}+\mathit{B}),m=1,2,\dots ,M$ . Proof: According to (3), define the global error vector

$$\begin{array}{c}\mathbf{\delta }=[{\mathbf{\delta }}_1,{\mathbf{\delta }}_2,\dots ,{\mathbf{\delta }}_M{]}^{\mathrm{T}}\hfill \\ =[{\mathit{x}}_1-1_N{x}_{0,1},{\mathit{x}}_2-1_N{x}_{0,2},\dots ,{\mathit{x}}_M-1_N{x}_{0,M}{]}^{\mathrm{T}}\hfill \end{array}$$

where ${\mathit{x}}_m={[x_{1,m},x_{2,m},\dots ,x_{N,m}]}^{\mathrm{T}}\in {ℝ}^N,m=1,2,\dots ,M$ .

Then, the global error vector of graph $\overline{\mathcal{G}}$ can be given by

$$\{\begin{array}{c}{\mathit{e}}_1=-(\mathit{L}+\mathit{B})({\mathit{x}}_1-1_N{x}_{0,1})=-(\mathit{L}+\mathit{B}){\mathbf{\delta }}_1\hfill \\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\mathit{e}}_m=-(\mathit{L}+\mathit{B})({\mathit{x}}_m-1_N{\mathit{x}}_{0,m})=-(\mathit{L}+\mathit{B}){\mathbf{\delta }}_m}\end{array}$$

(5)

According to Assumption 1, the matrix $\mathit{L}+\mathit{B}$ is non-singular. Due to ${\mathit{e}}_m=-(\mathit{L}+\mathit{B}){\mathbf{\delta }}_m$ in (5), it can be obtained

$${\mathbf{\delta }}_m=-(\mathit{L}+\mathit{B}{)}^{-1}{\mathit{e}}_m$$

(6)

Thus,

$$\Vert {\mathbf{\delta }}_m\Vert =\Vert {(\mathit{L}+\mathit{B})}^{-1}{\mathit{e}}_m\Vert \le \Vert {\mathit{e}}_m\Vert /{\sigma }_{-}(\mathit{L}+\mathit{B})$$

(7)

Proposed neuro-adaptive controller

For MASs with unknown disturbance or unmodeled dynamics, we designed a distributed adaptive controller by using siding mode variable structure control method and online-learning RBF neural network, where the sliding mode variable structure control can rapidly response while ensuring the system stability, and neural network can adaptively fine-tune the weights in real-time.

Sliding mode surface function design

The sliding mode surface function of agent $i(i\in N)$ is designed by

$$s_i={\alpha }_1{e}_{i,1}+{\alpha }_2{e}_{i,2}+\cdots +{\alpha }_{M-1}{e}_{i,M-1}+e_{i,M}$$

(8)

where the $M-1$ -order polynomial with coefficients ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{M-1}$ is Hurwitz. Then if $s_i$ is bounded, ${\mathit{e}}_i$ is also bounded. ${\mathit{e}}_i\to 0$ as $s_i\to 0$ .

Thus, the global error vector of sliding mode surface function can be formulated as $\mathit{s}=[s_1,s_2,\dots ,s_N{]}^{\mathrm{T}}={\alpha }_1{\mathit{e}}_1+\dots +{\alpha }_{M-1}{\mathit{e}}_{M-1}+{\mathit{e}}_M$ .

Define $\mathbf{\alpha }=[{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{M-1}{]}^T$ , ${\mathit{E}}_1=[{\mathit{e}}_1,{\mathit{e}}_2,\dots ,{\mathit{e}}_{M-1}]\in {\mathbb{R}}^{N\times M-1}$ , ${\mathit{E}}_2={\stackrel{·}{\mathit{E}}}_1=[{\mathit{e}}_2,{\mathit{e}}_3,\dots ,{\mathit{e}}_M]\in {\mathbb{R}}^{N\times M-1}$ , $\mathit{I}=[0,0,\dots ,1{]}^{\mathrm{T}}\in {\mathbb{R}}^{M-1}$ , and

$$\mathbf{\Gamma }=\left(\begin{array}{cc}0\hfill & \mathit{I}\hfill \\ \multicolumn{1}{c}{-{\alpha }_1}& (-{\alpha }_2-\dots -{\alpha }_{M-1})\hfill \end{array}\right)$$

(9)

Thus we have

$${\mathit{E}}_2={\mathit{E}}_1{\mathbf{\Gamma }}^{\mathrm{T}}+\mathit{s}{\mathit{I}}^{\mathrm{T}}$$

(10)

Since $\mathbf{\Gamma }$ is a Hurwitz matrix, given any positive number $\beta$ , there exists a symmetric matrix ${\mathit{P}}_1>0$ makes the Lyapunov function hold, that is,

$${\mathbf{\Gamma }}^{\mathrm{T}}{\mathit{P}}_1+{\mathit{P}}_1\mathbf{\Gamma }=-\beta {\mathit{I}}_N,$$

(11)

The derivative of the dynamic sliding mode error $\mathit{s}$ is

$$\stackrel{·}{\mathit{s}}=\mathbf{\rho }-(\mathit{L}+\mathit{B})(\mathit{f}+\mathit{d}+\mathit{u}-f_0{1}_N)$$

(12)

where $\mathbf{\rho }={\alpha }_1{\mathit{e}}_2+{\alpha }_2{\mathit{e}}_3+\dots +{\alpha }_{M-1}{\mathit{e}}_M={\mathit{E}}_2\mathbf{\alpha }$ .

Lemma 3. For agent $i(i=1,2,\dots ,N)$ , assume

$$\begin{array}{c}|s_i(t)|\le {\psi }_i,\forall t\ge t_0\hfill \\ \multicolumn{1}{c}{|s_i(t)|\le {\theta }_i,\forall t\ge T_{{\theta }_i}}\end{array}$$

(13)

where time $T_{{\xi }_i}>t_0$ , $t_0$ is the initial time, the upper bound ${\psi }_i>0,{\xi }_i>0$ . There exists time $T_{{\theta }_i}>t_0$ , ${\theta }_i>0$ makes

$$\begin{array}{c}\Vert {\mathit{e}}_i(t)\Vert \le {\psi }_i,\forall t\ge t_0\hfill \\ \multicolumn{1}{c}{\Vert {\mathit{e}}_i(t)\Vert \le {\theta }_i,\forall t\ge T_{{\theta }_i}}\end{array}$$

(14)

Proof: Let ${\mathit{e}}_i(t)=[e_{i,1},e_{i,1},\dots ,e_{i,M-1}{]}^{\mathrm{T}}\in {\mathbb{R}}^{M-1}$ , according to (8), we have

$${\stackrel{·}{\mathit{e}}}_i=\mathbf{\Gamma }{\mathit{e}}_i+\mathit{I}{s}_i$$

(15)

By (15),

$${\mathit{e}}_i(t)={\mathrm{e}}^{\mathbf{\Gamma }(t-t_0)}{\mathit{e}}_i(t_0)+{\int }_{t_0}^t{\mathrm{e}}^{\mathbf{\Gamma }(t-\tau )}\mathit{I}{s}_i(\tau )\mathrm{d}\tau$$

(16)

Considering $\mathbf{\Gamma }$ is a Hurwitz matrix, there exists $\varphi >0,\lambda >0$ , such that

$$\Vert {\mathrm{e}}^{\mathbf{\Gamma }(t-t_0)}\Vert \le \varphi {\mathrm{e}}^{-\lambda (t-t_0)}$$

(17)

So we have the inequality of (16),

$$\begin{array}{c}\parallel {\mathit{e}}_i(t)\parallel \le \varphi {\mathrm{e}}^{-\lambda (t-t_0)}\parallel {\mathit{e}}_i(t_0)\parallel \hfill \\ +{\int }_{t_0}^t\varphi {\mathrm{e}}^{-\lambda (t-\tau )}\parallel \mathit{I}\parallel |s_i(\tau )|\mathrm{d}\tau \hfill \\ \le \varphi {\mathrm{e}}^{-\lambda (t-t_0)}\parallel {\mathit{e}}_i(t_0)\parallel +\varphi \frac{\parallel \mathit{B}\parallel }{\lambda }\mathrm{su}{\mathrm{p}}_{t_1<\tau <t}|s_i(\tau )|\hfill \\ =\varphi {\mathrm{e}}^{-\lambda (t-t_0)}\parallel {\mathit{e}}_i(t_0)\parallel +\frac{\varphi }{\lambda }\mathrm{su}{\mathrm{p}}_{t_1<\tau <t}|s_i(\tau )|\hfill \end{array}$$

(18)

It can be seen from (18) that if $s_i(t)$ is bounded, $\parallel {\mathit{e}}_i(t)\parallel <\infty$ , so that for all $m=1,2,\dots ,M$ , $e_{i,m}(t)$ is bounded. And

$$e_{i,M}(t)=s_i-{\alpha }_1{e}_{i,1}(t)-{\alpha }_2{e}_{i,2}(t)-\cdots -{\alpha }_{M-1}{e}_{i,M-1}(t)$$

is also bounded. Therefore, $\parallel {\mathit{e}}_i(t)\parallel <\infty$ if $s_i(t)<\infty$ , that is, $\parallel {\mathit{e}}_i(t)\parallel$ is bounded.

Then it is proved that if $s_i(t)\to 0$ , $\parallel e_i\parallel \to 0$ holds. Since $s_i(\tau )\to 0$ , for any given sufficiently small constant ${\epsilon }_s>0$ , there is a time step $t_1$ such that when $\tau \ge t_1$ , $(\varphi /\lambda )|s_i(\tau )|\le {\epsilon }_s$ , it holds $(\varphi /\lambda )\mathrm{su}{\mathrm{p}}_{t_1\le \tau \le t}\le {\epsilon }_s$ . Similarly, from the exponential stability of $e^{-\lambda (t-t_1)}$ , it can be obtained that, for any given sufficiently small constant ${\epsilon }_e>0$ , there is a time step $t_2$ such that when $t-t_1\ge t_2$ , $\varphi e^{-\lambda (t-t_1)}\parallel e_i(t_1)\parallel \le {\epsilon }_e$ holds. Substituting the variable $t_0$ of the inequality (18) by $t_1$ , it can be obtained that when $t\ge t_1+t_2$ , it has

$$\begin{array}{c}\Vert {\mathit{e}}_i(t)\Vert \le \varphi {\mathrm{e}}^{-\lambda (t-t_1)}\Vert {\mathit{e}}_i(t_1)\Vert +\frac{\varphi }{\lambda }\underset{t_1\le \tau \le t}{\sup }|s_i(\tau )|\hfill \\ \le \varphi {\mathrm{e}}^{-\lambda (t-t_1)}\Vert {\mathit{e}}_i(t_1)\Vert +\frac{\varphi }{\lambda }\underset{\tau \ge t_1}{\sup }|s_i(\tau )|\hfill \\ \le {\epsilon }_e+{\epsilon }_s.\hfill \end{array}$$

(19)

According to the arbitrariness of ${\epsilon }_e$ and ${\epsilon }_s$ , when $t\to \infty$ , $\parallel e_i(t)\parallel \to 0$ holds. Thus we get $e_{i,m}(t)\to 0$ for all $m=1,2,...,M-1$ . Therefore, $e_{i,M}(t)=s_i-{\alpha }_1{e}_{i,1}(t)-{\alpha }_2{e}_{i,2}(t)-\dots -{\alpha }_{M-1}{e}_{i,M-1}(t)\to 0$ can be obtained by (8). In summary, when $s_i(t)\to 0$ , ${\mathit{e}}_i\to 0$ holds, the Lemma 3 is proved.

Distributed control protocol design

Before designing the neuro-adaptive control protocol for each follower agent, the following assumptions should be made.

Assumption 2. There is a positive value $\overline{X}>0$ , the leader state satisfies $\parallel x_0(t)\parallel \le \overline{X}$ .

Assumption 3. There is a continuous function $g(·):{\mathbb{R}}^m\to \mathbb{R}$ holds $|f_0(t,x_0(t))|\le |g(x_0(t))|$ .

Assumption 4. The external disturbance $d_i(t)$ of each follower is unknown but bounded, that is, $\sup \{\parallel d_1(t)\parallel ,\parallel d_2(t)\parallel ,\dots ,\parallel d_N(t)\parallel ,\parallel d_0(t)\parallel \}\le d$ , where $d$ is an unknown normal number.

By Assumption 3, there exists an upper limit $F,\forall t\ge t_0$ for the leader’s nonlinearity $f_0(t,x_0(t))$ . It also should be noted that the precise values of supremums $\overline{X},\overline{F}$ , and $\overline{d}$ do not need to be known for controller design, they are only used for the Lyapunov method to analyze system stability.

The kernel function of our proposed online-adaptive RBF-like neural network is Gaussian function,

$${\mathit{\phi }}_j=\exp \left(\frac{{\Vert {\mathit{x}}_i-{\mathit{c}}_j\Vert }^2}{2{{\hat{b}}_j}^2}\right),j=1,2,\dots ,z$$

(20)

where $z$ is the number of nodes in hidden layer, ${\mathit{c}}_j$ and ${\hat{b}}_j$ are the center and spread width, respectively.

The network output is calculated by

$$o_i(x_i)={\mathit{W}}_i^{\mathrm{T}}{\mathit{\phi }}_i(x_i)+{\epsilon }_i$$

where the output weights ${\mathit{W}}_{\mathit{i}}={[w_{i,1},w_{i,2},\dots ,w_{i,z}]}^{\mathrm{T}}$ , ${\mathit{\phi }}_i(x_i)=[{\phi }_{i,1}(x_i),{\phi }_{i,2}(x_i),\dots ,{\phi }_{i,z}(x_i){]}^{\mathrm{T}}$ . Thus, the output vector can be written as

$$\mathit{o}(x_i)={\mathit{W}}^{\mathrm{T}}\mathit{\phi }(x)+\mathit{\epsilon }$$

(21)

According to the Stone-Weierstrass theorem, ³³ given a compact set $\mathrm{\Omega }$ , for any positive number ${\epsilon }_h$ , there exists a sufficiently large positive integer $z^{*}$ , so that for any $z>z^{*}$ , an ideal weight vector ${\mathit{W}}^i$ and a suitable radial basis function vector ${\phi }_i$ can be found such that $\parallel {\epsilon }_i\parallel \le {\epsilon }_h$ .

The system state $x_{i,1}(t),x_{i,2}(t),\dots ,x_{i,M}(t)$ is used as the network input, so actual output of network is

$${\hat{o}}_i(x_i)={\hat{\mathit{W}}}_i^{\mathrm{T}}(t){\mathit{\phi }}_i(x_i)$$

(22)

where ${\hat{\mathit{W}}}_i\left(t\right)\in {\mathbb{R}}^z$ is the estimated weights of agent $i$ . Let the global network output vector be

$$\stackrel{·}{\mathit{o}}(x)={\hat{\mathit{W}}}^{\mathrm{T}}\mathit{\phi }(x)$$

(23)

where the approximation error $\mathit{\epsilon }=[{\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_N]$ .

Design the weight adaptive update method for proposed online-learning neural network as

$${\stackrel{·}{\hat{\mathit{W}}}}_i=-{\mathit{F}}_i{\mathit{\phi }}_i{\mathit{s}}_i{p}_i(\mathrm{de}{\mathrm{g}}_i+b_i)-\zeta {\mathit{F}}_i{\hat{\mathit{W}}}_i$$

(24)

Thus the vector form of all the network weights is

$$\stackrel{·}{\hat{\mathit{W}}}=-\mathit{F}\mathit{\phi }\mathit{sP}(\mathit{D}+\mathit{B})-\zeta \mathit{F}\hat{\mathit{W}}$$

(25)

where ${\mathit{F}}_i={\mathit{F}}_i^T\in {\mathbb{R}}^{z\times z}$ is any positive definite matrix, the positive number $\zeta$ is an adjustable scalar, and the matrix $\mathit{P}$ is defined in Lemma 1.

Further, we define the maximal value of Gaussian function output, ideal weights to be ${\overline{\phi }}_i=\underset{x_i\in \mathrm{\Omega }}{max}\parallel {\mathit{\phi }}_i(x_i)\parallel$ and ${\overline{W}}_i=max\parallel {\mathit{W}}_i\parallel$ , respectively. Thus there exist positive numbers $\overline{\phi },\overline{W}$ , and $\overline{\epsilon }$ , satisfy $\parallel \mathit{\phi }\parallel \le \overline{\phi },\parallel \mathit{W}\parallel \le \overline{W}$ and $\parallel \mathit{\epsilon }\parallel \le \overline{\epsilon }$ .

We propose the neuro-adaptive control protocol for each agent $i$ by

$$\begin{array}{c}\multicolumn{1}{c}{u_i=\frac{1}{\mathrm{de}{\mathrm{g}}_i+b_i}({\alpha }_1{e}_{i,2}+{\alpha }_2{e}_{i,3}+\cdots +{\alpha }_{M-1}{e}_{i,M})}\\ \multicolumn{1}{c}{-{\hat{o}}_i(x_i)+k{s}_i}\end{array}$$

(26)

Thus, the vector form for all follower agents can be written as

$$\mathit{u}=-(\mathit{D}+\mathit{B}{)}^{-1}\mathbf{\rho }-\hat{\mathit{o}}(x)+k\mathit{s}$$

(27)

where the control gain satisfies

$$k>\frac{2}{\sigma (\mathit{Q})}(\frac{r^2}{\zeta }+\frac{2}{\beta }{\mathit{g}}^2+h)$$

(28)

where $r=-\frac{1}{2}\overline{\phi }\overline{\sigma }(\mathit{P})\overline{\sigma }(\mathit{A})$ , $g=-\frac{1}{2}\left(\frac{\overline{\sigma }\left(\mathit{P}\right)\overline{\sigma }\left(\mathit{A}\right)}{\underset{\_}{\sigma }\left(\mathit{D}+\mathit{B}\right)}{\Vert \mathbf{\Gamma }\Vert }_F\Vert \overline{\mathit{\alpha }}\Vert +\overline{\sigma }\left({\mathit{P}}_1\right)\right)$ , $h=\frac{\overline{\sigma }\left(\mathit{P}\right)\overline{\sigma }\left(\mathit{A}\right)}{\underset{¯}{\sigma }(\mathit{D}+\mathit{B})}\Vert \overline{\alpha }\Vert$ , ${\mathit{P}}_1$ has been defined in (11), $\beta >0$ , $\mathit{Q}$ has been explained in Lemma 1, and the coefficient $\zeta$ has been explained in (25).

Main Results

Let $\tilde{\mathit{W}}$ be the error between the ideal weight $\mathit{W}$ and estimated weights $\hat{\mathit{W}}$ of the online-learning neural network. Design the Lyapunov candidate as

$$\mathit{V}(t)={\mathit{V}}_1(t)+{\mathit{V}}_2(t)+{\mathit{V}}_3(t)$$

(29)

where $\mathit{P}={\mathit{P}}^{\mathrm{T}}>0;{\mathit{F}}^{-1}={\mathit{F}}^{-T}>0$ , and

$${\mathit{V}}_1(t)=\frac{1}{2}{\mathit{s}}^{\mathrm{T}}\mathit{Ps}$$

(30)

$${\mathit{V}}_2(t)=\frac{1}{2}\mathrm{tr}\left\{{\tilde{\mathit{W}}}^{\mathrm{T}}{\mathit{F}}^{-1}\tilde{\mathit{W}}\right\}$$

(31)

$${\mathit{V}}_3(t)=\frac{1}{2}\mathrm{tr}\left\{{\mathit{E}}_1{\mathit{P}}_1{\mathit{E}}_1^{\mathrm{T}}\right\}$$

(32)

The derivative of ${\mathit{V}}_1(t)$ is

$$\begin{array}{c}{\stackrel{·}{\mathit{V}}}_1(t)={\mathit{s}}^{\mathrm{T}}\mathit{P}\stackrel{·}{\mathit{s}}\hfill \\ ={\mathit{s}}^{\mathrm{T}}\mathit{P}[\mathbf{\rho }-(\mathit{L}+\mathit{B})(\mathit{f}+\mathit{d}+\mathit{u}-f_0{1}_N)]\hfill \end{array}$$

(33)

Considering $\mathit{L}=\mathit{D}-\mathit{A}$ , by substitute (27) to (33), we have

$$\begin{array}{c}{\stackrel{·}{\mathit{V}}}_1(t)={\mathit{s}}^{\mathrm{T}}\mathit{P}[\mathbf{\rho }-(\mathit{L}+\mathit{B})(\mathit{f}+\mathit{d}-(\mathit{D}+\mathit{B}{)}^{-1}\mathbf{\rho }\hfill \\ -\hat{\mathit{o}}(x)+k\mathit{s}-f_0{1}_N)]\hfill \\ ={\mathit{s}}^{\mathrm{T}}\mathit{P}\mathbf{\rho }-{\mathit{s}}^{\mathrm{T}}\mathit{P}(\mathit{L}+\mathit{B})(\mathit{\epsilon }+\mathit{d}-f_0{1}_N)\hfill \\ -k{\mathit{s}}^{\mathrm{T}}\mathit{P}(\mathit{L}+\mathit{B})\mathit{s}\hfill \\ -{\mathit{s}}^{\mathrm{T}}\mathit{P}[(\mathit{D}+\mathit{B})-\mathit{A}][(\mathit{D}+\mathit{B}{)}^{-1}\mathbf{\rho }+{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{\phi }]\hfill \\ ={\mathit{s}}^{\mathrm{T}}\mathit{P}(\mathit{L}+\mathit{B})(\mathit{\epsilon }+\mathit{d}-f_0{1}_N)-k{\mathit{s}}^{\mathrm{T}}\mathit{P}(\mathit{L}+\mathit{B})\mathit{s}\hfill \\ -{\mathit{s}}^{\mathrm{T}}\mathit{P}(\mathit{D}+\mathit{B}){\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{\phi }+{\mathit{s}}^{\mathrm{T}}\mathit{PA}{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{\phi }\hfill \\ +{\mathit{s}}^{\mathrm{T}}\mathit{PA}(\mathit{D}+\mathit{B}{)}^{-1}\mathbf{\rho }\hfill \end{array}$$

(34)

According to Lemma 1 and ${\mathit{x}}^{\mathrm{T}}\mathit{y}=\mathrm{tr}\{\mathit{y}{\mathit{x}}^{\mathrm{T}}\}$ , (34) can be rewritten by

$$\begin{array}{c}{\stackrel{·}{V}}_1(t)=-{\mathit{s}}^{\mathrm{T}}\mathit{P}(\mathit{L}+\mathit{B})(\mathit{\epsilon }+\mathit{d}-f_0{1}_N)\hfill \\ -\frac{1}{2}k{\mathit{s}}^{\mathrm{T}}\mathit{Qs}-\mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{\phi }{\mathit{s}}^{\mathrm{T}}\mathit{P}(\mathit{D}+\mathit{B})\}\hfill \\ +\mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{\phi }{\mathit{s}}^{\mathrm{T}}\mathit{PA}\}+{\mathit{s}}^{\mathrm{T}}\mathit{PA}(\mathit{D}+\mathit{B}{)}^{-1}\mathbf{\rho }\hfill \end{array}$$

(35)

Because $\stackrel{·}{\stackrel{~}{\mathit{W}}}=\stackrel{·}{\mathit{W}}-\stackrel{·}{\hat{\mathit{W}}}=-\stackrel{·}{\hat{\mathit{W}}}$ , by substituting it to (25), the derivative of $V_2(t)$ w.r.t. time is

$$\begin{array}{c}{\stackrel{·}{V}}_2(t)=\frac{1}{2}\mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}{\mathit{F}}^{-1}\tilde{\mathit{W}}\}\hfill \\ =\mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}{\mathit{F}}^{-1}[\mathit{F}\mathbf{\varphi }\mathit{rP}(\mathit{D}+\mathit{B})+\zeta \mathit{F}\hat{\mathit{W}}]\}\hfill \\ =\mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{\phi }\mathit{sP}(\mathit{D}+\mathit{B})\}+\zeta \mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}\hat{\mathit{W}}\}\hfill \\ =\mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{\phi }\mathit{sP}(\mathit{D}+\mathit{B})\}+\zeta \mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{W}\}-\zeta \mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}\tilde{\mathit{W}}\}\hfill \end{array}$$

(36)

Solving the derivative of $V_3(t)$ w.r.t. time, we have

$${\stackrel{·}{V}}_3(t)=\mathrm{tr}\{{\mathit{E}}_2{\mathit{P}}_1{\mathit{E}}_1^{\mathrm{T}}\}$$

(37)

Substituting (10) to (37), and considering (11),

$$\begin{array}{c}{\stackrel{·}{V}}_3(t)=-\frac{\beta }{2}\mathrm{tr}\{{\mathit{E}}_1{\mathit{E}}_1^{\mathrm{T}}\}+\mathrm{tr}\{\mathit{s}{\mathbf{\Gamma }}^{\mathrm{T}}{\mathit{P}}_1{\mathit{E}}_1^{\mathrm{T}}\}\hfill \\ \le -\frac{\beta }{2}\Vert {\mathit{E}}_1{\Vert }_{\mathrm{F}}^2+\overline{\sigma }({\mathit{P}}_1)\Vert \mathit{s}\Vert \Vert {\mathit{E}}_1{\Vert }_{\mathrm{F}}\hfill \end{array}$$

(38)

Thus, the overall derivative of Lyapunov function $V(t)$ can be summarized asC

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(t)={\stackrel{·}{V}}_1(t)+{\stackrel{·}{V}}_2(t)+{\stackrel{·}{V}}_3(t)}\\ =-{\mathit{s}}^{\mathrm{T}}\mathit{P}(\mathit{L}+\mathit{B})(\mathit{\epsilon }+\mathit{d}-f_0{1}_N)-\frac{1}{2}k{\mathit{s}}^{\mathrm{T}}\mathit{Qs}\hfill \\ +\mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}\phi {\mathit{s}}^{\mathrm{T}}\mathit{PA}\}+\zeta \mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{W}\}-\zeta \mathrm{tr}\{{\tilde{\mathit{W}}}^{\mathrm{T}}\tilde{\mathit{W}}\}\hfill \\ +{\mathit{s}}^{\mathrm{T}}\mathit{PA}(\mathit{D}+\mathit{B}{)}^{-1}\mathbf{\rho }-\frac{\beta }{2}\mathrm{tr}\{{\mathit{E}}_1{\mathit{E}}_1^{\mathrm{T}}\}+\mathrm{tr}\{\mathit{s}{\mathbf{\Gamma }}^{\mathrm{T}}{\mathit{P}}_1{\mathit{E}}_1^{\mathrm{T}}\}\hfill \\ \le \overline{\sigma }(\mathit{P})\overline{\sigma }(\mathit{L}+\mathit{B})\overline{T}\parallel \mathit{s}\parallel -\frac{1}{2}k\underset{¯}{\sigma }(\mathit{Q})\parallel \mathit{s}{\parallel }^2\hfill \\ +\overline{\phi }\overline{\sigma }(\mathit{P})\overline{\sigma }(\mathit{A})\parallel \tilde{\mathit{W}}{\parallel }_F\parallel s\parallel +\zeta \overline{\mathit{W}}\parallel \tilde{\mathit{W}}{\parallel }_F\hfill \\ -\zeta \parallel \tilde{\mathit{W}}{\parallel }_F^2+\frac{\overline{\sigma }(\mathit{P})\overline{\sigma }(\mathit{A})}{\underset{¯}{\sigma }(\mathit{D}+\mathit{B})}(\parallel s\parallel \parallel {\mathit{E}}_1{\parallel }_F\parallel \mathbf{\Gamma }{\parallel }_F\parallel \overline{\alpha }\parallel \hfill \\ +\parallel s{\parallel }^2\parallel \mathbf{\Gamma }\parallel \parallel \overline{\alpha }\parallel )\hfill \\ -\frac{\beta }{2}\parallel {\mathit{E}}_1{\parallel }_F^2+\overline{\sigma }(P_1)\parallel \mathit{s}\parallel \parallel {\mathit{E}}_1{\parallel }_F\hfill \end{array}$$

(39)

where $\overline{T}=\overline{\epsilon }+\overline{d}+\overline{F}$ . Remark 2. As stated in Section of Distributed Control Protocol Design, $\overline{\epsilon },\overline{d},\overline{F}$ are the boundaries of neural network approximation error, system uncertainty, and the leader initial state respectively. Even though we can determine the value of these parameters, we can guarantee the stability of the whole system with the property of boundedness.

Note the definition of $r,g$ , and $h$ in (28), (39) can be rewritten by

$$\begin{array}{c}\stackrel{·}{V}(t)\le -(\frac{1}{2}k\underset{¯}{\sigma }(\mathit{Q})-\frac{\overline{\sigma }(\mathit{P})\overline{\sigma }(\mathit{A})}{\underset{¯}{\sigma }(\mathit{D}+\mathit{B})}\Vert \overline{\mathit{a}}\Vert )·\Vert \mathit{s}{\Vert }^2\hfill \\ -\zeta \Vert \tilde{\mathit{W}}{\Vert }_{\mathrm{F}}^2-\frac{\beta }{2}\Vert {\mathit{E}}_1{\Vert }_{\mathrm{F}}^2+\overline{\phi }\overline{\sigma }(\mathit{P})\overline{\sigma }(A)·\Vert \tilde{\mathit{W}}{\Vert }_{\mathrm{F}}\Vert \mathit{s}\Vert \hfill \\ +(\frac{\overline{\sigma }(\mathit{P})\overline{\sigma }(\mathit{A})}{\overline{\sigma }(\mathit{D}+\mathit{B})}{\Vert \mathbf{\Lambda }\Vert }_{\mathrm{F}}\Vert \overline{\alpha }\Vert +\overline{\sigma }({\mathit{P}}_1))\Vert \mathit{s}\Vert \Vert {\mathit{E}}_1{\Vert }_{\mathrm{F}}\hfill \\ +\overline{\sigma }(\mathit{P})\overline{\sigma }(\mathit{L}+\mathit{B})\overline{T}\Vert \mathit{s}\Vert +\zeta \overline{W}\Vert \tilde{\mathit{W}}{\Vert }_{\mathrm{F}}\hfill \\ =-(\frac{1}{2}k{\sigma }_{-}(Q)-h)\Vert \mathit{s}{\Vert }^2-\zeta \Vert \tilde{\mathit{W}}{\Vert }_F^2\hfill \\ -\frac{\beta }{2}\Vert {\mathit{E}}_1{\Vert }_{\mathrm{F}}^2-2r\Vert \tilde{\mathit{W}}{\Vert }_{\mathrm{F}}\Vert \mathit{s}\Vert -2g\Vert \mathit{s}\Vert \Vert {\mathit{E}}_1{\Vert }_{\mathrm{F}}\hfill \\ +\overline{\sigma }(P)\overline{\sigma }(L+B)\overline{T}\Vert \mathit{s}\Vert +\zeta \overline{W}\Vert \tilde{\mathit{W}}{\Vert }_{\mathrm{F}}\hfill \\ =-{\left[\begin{array}{c}{\Vert {\mathit{E}}_1\Vert }_{\mathrm{F}}\hfill \\ \multicolumn{1}{c}{{\Vert \tilde{\mathit{W}}\Vert }_{\mathrm{F}}}\\ \multicolumn{1}{c}{\Vert \mathit{s}\Vert }\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{ccc}\frac{\beta }{2}\hfill & 0\hfill & g\hfill \\ \multicolumn{1}{c}{0}& \zeta \hfill & r\hfill \\ \multicolumn{1}{c}{g}& r\hfill & \frac{1}{2}k\overline{\sigma }(Q)-h\hfill \end{array}\right]\left[\begin{array}{c}{\Vert {\mathit{E}}_1\Vert }_{\mathrm{F}}\hfill \\ \multicolumn{1}{c}{{\Vert \tilde{\mathit{W}}\Vert }_{\mathrm{F}}}\\ \multicolumn{1}{c}{\Vert \mathit{s}\Vert }\end{array}\right]\hfill \\ +\left[\begin{array}{ccc}0\hfill & \zeta \overline{W}\hfill & \overline{\sigma }(\mathit{P})\overline{\sigma }(\mathit{L}+\mathit{B})\overline{T}\hfill \end{array}\right]\left[\begin{array}{c}{\Vert {\mathit{E}}_1\Vert }_{\mathrm{F}}\hfill \\ \multicolumn{1}{c}{{\Vert \mathit{W}\Vert }_{\mathrm{F}}}\\ \multicolumn{1}{c}{\Vert \mathit{s}\Vert }\end{array}\right]\hfill \end{array}$$

(40)

Let

$$\begin{array}{c}\mathit{z}={[{\Vert {\mathit{E}}_1\Vert }_{\mathrm{F}}{\Vert \tilde{\mathit{W}}\Vert }_{\mathrm{F}}\Vert \mathit{s}\Vert ]}^{\mathrm{T}},\hfill \\ =\left[\begin{array}{ccc}\frac{\beta }{2}\hfill & 0\hfill & g\hfill \\ \multicolumn{1}{c}{0}& \zeta \hfill & r\hfill \\ \multicolumn{1}{c}{g}& r\hfill & \frac{1}{2}k\underset{¯}{\sigma }\left(Q\right)-h\hfill \end{array}\right],\hfill \\ \mathit{\omega }={\left[0\zeta \overline{W}\overline{\sigma }\left(\mathit{P}\right)\overline{\sigma }(\mathit{L}+\mathit{B})\overline{T}\right]}^{\mathrm{T}}\hfill \end{array}$$

(41)

Therefore, we have

$$\stackrel{·}{V}(t)\le -{\mathit{z}}^{\mathrm{T}}\mathbf{\Xi }z+{\mathit{\omega }}^{\mathrm{T}}\mathit{z}=-V_z(\mathit{z})$$

(42)

Achieving asymptotic stability of the above mentioned MAS requires $V_z(z)$ being a positive-definite function, thus, the following two conditions should be met:

1. The matrix $\Xi$ is positive-definite;

2. $\Vert \mathit{z}\Vert >\frac{\Vert \mathit{\omega }\Vert }{\overline{\sigma }\left(\mathbf{\Xi }\right)}$ .

Remark 1. If all the main sub-determinants of matrix $\mathbf{\Xi }$ are greater than 0, it can be said that $\mathbf{\Xi }$ is positive-definite.

$$\{\begin{array}{c}\beta >0\hfill \\ \multicolumn{1}{c}{\beta \zeta >0}\\ \multicolumn{1}{c}{\zeta [\beta (\frac{1}{2}k{\sigma }_{-}(Q)-h)-2g^2]-\beta r^2>0}\end{array}$$

(43)

It is easy to know that $\parallel \omega {\parallel }_1>\parallel \omega \parallel$ , if $\parallel \mathbf{z}\parallel \ge \mathrm{Bd}$ , the above condition 2 holds, it can be obtained

$$\mathrm{Bd}=\frac{{\Vert \mathit{\omega }\Vert }_1}{{\sigma }_{-}(\mathbf{\Xi })}=\frac{\overline{\sigma }(\mathit{P})\overline{\sigma }(\mathit{L}+\mathit{B})\overline{T}+\zeta \overline{W}}{{\sigma }_{-}(\mathbf{\Xi })}$$

(44)

Therefore, both conditions 1 and 2 are met, so we have

$$\stackrel{·}{V}(t)\le -V_z(\mathit{z}),\forall \Vert \mathit{z}\Vert \ge \mathrm{Bd},$$

(45)

and $V_z(\mathit{z})$ is a continuous positive-definite function.

Numeric simulation

In this section, numeric results on both homogeneous and heterogeneous multi-agent systems are provided to verify the effectiveness of our proposed control method. Homogeneous system means that the mathematical models of each agent dynamics are the same, while the heterogeneous means that the models are different with each other. Figure 1 shows the directed communication topology of the agents for the two numeric experiments, it includes one leader and six follower agents. All the communication weights among agents are assumed to be 1.

Figure 1.

Directed communication topology with six followers and one leader.

The mathematical models of the dynamics for all the six followers are assumed to be third-order nonlinear systems with uncertainty:

$$\{\begin{array}{c}{\stackrel{·}{x}}_{i,1}(t)=x_{i,2}(t)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_{i,2}(t)=x_{i,3}(t)}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_{i,3}(t)=f_i(t,x_i)+d_i(t)+u_i(t)}\end{array}$$

The parameters, including number of hidden nodes and connection weights of the proposed online-learning neural network are randomly initialized in advance.

Consensus of homogeneous MASs

Let the nonlinear function $f_i,(i=0,1,\dots ,6)$ of the homogeneous MAS be

$$\begin{array}{c}{f}_i(t,x_i(t))=20\sin (2t+0.2)+10\sin (2x_{i,2})\hfill \\ -0.2x_{i,1}+0.5x_{i,2}\hfill \end{array}$$

where $f_0$ is the nonlinear term of the leader agent.

We initialize the state of the leader and follower agents as

$$\begin{array}{c}{\mathit{x}}_0=[x_{0,1},x_{0,2},x_{0,3}{]}^{\mathrm{T}}=[2.6,-24.6,0.1{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_1=[x_{1,1},x_{1,2},x_{1,3}{]}^{\mathrm{T}}=[12.2,-0.2,1.0{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_2=[x_{2,1},x_{2,2},x_{2,3}{]}^{\mathrm{T}}=[-5.1,0,0.2{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_3=[x_{3,1},x_{3,2},x_{3,3}{]}^{\mathrm{T}}=[-9,17.5,0{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_4=[x_{4,1},x_{4,2},x_{4,3}{]}^{\mathrm{T}}=[-8.1,6.4,0.9{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_5=[x_{5,1},x_{5,2},x_{5,3}{]}^{\mathrm{T}}=[1.8,-9.2,0.2{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_6=[x_{6,1},x_{6,2},x_{6,3}{]}^{\mathrm{T}}=[1.5,7.7,0.1{]}^{\mathrm{T}}\hfill \end{array}$$

For the $i$ th agent, it is assumed that it contains uncertainties such as external disturbances and sensor noise, which are uniformly modeled as $0.05\sin (t)$ . The state trajectories of all the agents of the third-order system are shown in Figure 2. The position state of the six homogeneous followers are gradually approaching the position state of leader $0$ , that is, the tracking consensus is achieved.

Figure 2.

Homogeneous systems state trajectories: (a) state trajectories of $x_{i,1},i=0,1,\dots ,5$ , (b) state trajectories of $x_{i,2},i=0,1,\dots ,5$ , and (c) state trajectories of $x_{i,3},i=1,\dots ,5$ .

Consensus of heterogeneous MASs

For the heterogeneous MAS, reconsider the third-order system of with six followers, where the nonlinear function $f_i$ for each agent $i(i=1,\dots ,6)$ is different,

$$\begin{array}{c}{f}_1(t,x_1(t))=0.5\sin x_{1,1}-0.1x_{1,2}+\cos (1.2t)\hfill \\ f_2(t,x_2(t))=-0.2\sin x_{2,1}-0.5x_{2,2}\hfill \\ f_3(t,x_3(t))=-1.5\sin x_{3,1}-x_{32}+0.5\cos (t)\hfill \\ f_4(t,x_4(t))=0.8\sin x_{4,1}-\cos x_{4,3}\hfill \\ f_5(t,x_5(t))=-\sin x_{5,1}-0.1\cos (0.1t)\hfill \\ f_6(t,x_6(t))=-1.8\sin x_{6,1}+0.1\sin x_{6,2}+\cos (2t)\hfill \end{array}$$

We initialize the state of the leader and follower agents as

$$\begin{array}{c}{\mathit{x}}_0=[x_{0,1},x_{0,2},x_{0,3}{]}^{\mathrm{T}}=[-4.9,-0.2,14.3{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_1=[x_{1,1},x_{1,2},x_{1,3}{]}^{\mathrm{T}}=[3.2,-1.0,49.1{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_2=[x_{2,1},x_{2,2},x_{2,3}{]}^{\mathrm{T}}=[-2.3,-0.5,-17.2{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_3=[x_{3,1},x_{3,2},x_{3,3}{]}^{\mathrm{T}}=[-12.4,-1.8,-15.4{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_4=[x_{4,1},x_{4,2},x_{4,3}{]}^{\mathrm{T}}=[-2.2,-1.2,-34.6{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_5=[x_{5,1},x_{5,2},x_{5,3}{]}^{\mathrm{T}}=[-5.0,0,-27.8{]}^{\mathrm{T}}\hfill \\ {\mathit{x}}_6=[x_{6,1},x_{6,2},x_{6,3}{]}^{\mathrm{T}}=[-0.5,-1.4,21.9{]}^{\mathrm{T}}\hfill \end{array}$$

The controller and the parameters of neural network, network topology, the uncertain term $d_i(t)$ , and the dynamics model of the leader are the same as those in Section “Consensus of homogeneous MASs” . The state trajectories of all agents in the third-order system are shown in Figure 3. The position states of the six heterogeneous followers are gradually approaching the position state of leader 0, which means that they achieve state consensus finally, that is,

$$\underset{t\to \infty }{\lim }\Vert {\mathit{x}}_i(t)-{\mathit{x}}_j(t)\Vert =0,\forall i,j=0,1,\cdots ,N.$$

Figure 3.

Heterogeneous systems state trajectories: (a) state trajectories of $x_{i,1},i=0,1,\dots ,5$ , (b) state trajectories of $x_{i,2},i=0,1,\dots ,5$ , and (c) state trajectories of $x_{i,3},i=1,\dots ,5$ .

Through the numeric simulations, it is verified that the proposed adaptive consensus control protocol is not only applicable to homogeneous multi-agent systems, but also to heterogeneous multi-agent systems.

It should be noted in view of the problems of external disturbances and uncertain items in the system, currently there are mainly robust control methods and state observer methods. Robust control has a certain inhibitory effect on external disturbances, but it cannot effectively eliminate the influence of external disturbances on consistency. At present, the more commonly used method is to use a state observer to estimate the uncertain items of the multi-agent system to compensate for the unknown items, to achieve the consistency of the multi-agent, and the design is for the linear systems. In this paper, researches on high-order nonlinear uncertain multi-agent systems are carried out. The nonlinear term of the system is approximated by RBF neural network. A sliding mode controller is designed to compensate for external disturbances, so that the multi-agent can be stabilized, thereby realizing the leader-follower consistency problem.

Conclusion

In this paper, we proposed a neuro-adaptive cooperative controller for high-order nonlinear leader-following MASs with uncertainty, where sliding mode variable structure control is used. The highlights are the method does not need to precisely know the upper bounds of both nonlinear terms and uncertain terms. Further, the proposed online-learning neural network can approximate the unknown nonlinearity of dynamical system and disturbance adaptively. Theoretical analysis based on Lyapunov stability theory shows the CUUB of consensus error. The simulation results verified the performance on both homogeneous and heterogeneous MASs, the proposed controller can not only effectively deal with unknown nonlinear dynamics, but also has good anti-interference ability to ensure the convergence of tracking consensus.