Distributed robust H ∞ composite-rotating consensus of second-order multi-agent systems

Abstract

In this article, the distributed $H\infty$ composite-rotating consensus problem is concerned for a class of second-order multi-agent systems. First, based on local state feedback and communication feedback, a distributed control algorithm is proposed. Then, sufficient conditions are derived in order to make all agents reach a composite-rotating consensus with the desired $H\infty$ performance. Finally, the simulations are given to show the effectiveness of the theoretical results.

Introduction

In the past decade, consensus problems of multi-agent systems have drawn more and more attention from researchers in different fields. Numerous results have been obtained on consensus problems.^1–22 For example, Ren and Beard ¹ proposed some second-order consensus algorithms to guarantee the state consensus of systems subjected to the saturation of the actuator and limited available information. Pavone and Frazzoli ² and Wang et al. ³ proposed different control algorithms for symmetric formations in multi-agent systems. And a survey of formation control of multi-agent systems was presented in Oh et al. ⁴ Sepulchre et al., ⁵ Ren, ⁶ and Lin and Jia ⁷ studied the collective rotating motions, where the circle centers of all agents are assumed to converge to a fixed point. Lin et al. ⁸ studied the general composite-rotating problem. Several problems may emerge in the practical applications, for example, the multi-agent systems are often subjected to various disturbance, which includes actuator bias, measurement errors, calculation errors, and external disturbance caused by communication or measuring. The disturbance effects or even destroys the stability of multi-agent systems. Therefore, it is significant to investigate the effects of disturbance for multi-agent systems. Lin et al. ⁹ studied the robust $H\infty$ consensus problems of multi-agent systems with disturbances and uncertainties. Lin and Ren ¹⁰ extend previous work to a constraint situation. Mo and Jia ¹¹ studied the $H\infty$ consensus problems of a high-order system. However, their methods cannot be directly applied into the rotating consensus problems.

Inspired by the above literatures, the robust $H\infty$ composite-rotating consensus problem of the second-order multi-agent systems with uncertainties is investigated in this article. This study is of great importance for it has various applications, such as in fly-around satellites systems and space exploration tasks. To solve this problem, first, we design a distributed control protocol based on local state feedback and communication feedback. Then, we transform the composite-rotating consensus problem into the exponentially stability problem of the linear system by a linear transformation. Finally, by applying Lyapunov approach and $H\infty$ control theory of linear system, it is proved that all agents can reach the composite-rotating consensus with the desired $H\infty$ performance asymptotically.

The following notations will be used throughout this article. $\mathbb{R}$ denotes the set of all $m$ -dimensional real column vectors, $I_m$ denotes the $m$ -dimensional unit matrix, ⊗ denotes the Kronecker product, $1$ represents $[1,1,...,1{]}^T$ with compatible dimensions (sometimes, $1_n$ is used to denote $1$ with dimension n), 0 denotes zero value or zero matrix with appropriate dimensions, ||.|| refers to the standard Euclidean norm for vectors, and ${\mathcal{L}}_2[0,\infty )$ denotes the space of square integrable vector functions over $[0,\infty )$ .

Graph theory and robust H_∞ control theory

Some basic knowledge of graph theory and $H\infty$ control theory will be introduced in this section, which will be applied to convergence analysis in the following sections.

Graph theory

Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A})$ be an undirected graph of order $n$ , where $\mathcal{V}=\{s_1,s_2,...,s_n\}$ is the set of vertices, $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is the set of edges, and $\mathcal{A}=[a_{\mathrm{ij}}]\in {\mathbb{R}}^{n\times n}$ is the weighted adjacency matrix. The vertex indexes belong to a finite index set $\mathcal{I}=\{1,2,...,n\}$ . An edge of $\mathcal{G}$ is denoted by $e_{\mathrm{ij}}=(s_i,s_j)$ , where $e_{\mathrm{ij}}\in \mathcal{E}$ . Associated with the edges, the adjacency elements are positive which means $e_{\mathrm{ij}}\in \mathcal{E}\iff a_{\mathrm{ij}}>0$ . Thus, the adjacency matrix $\mathcal{A}$ is a symmetric non-negative matrix. The set of neighbors of vertex $s_i$ is denoted by $N_i=\{s_k\in \mathcal{V}:(s_i,s_k)\in \mathcal{E}\}$ . The out-degree of node $s_i$ is defined as $d_o(s_i)=\sum _{k=1}^n{a}_{\mathrm{ik}}$ . Then, the Laplacian corresponding to the undirected graph is defined as $L=[l_{\mathrm{ik}}]$ , where $l_{\mathrm{ii}}=d_o(s_i)$ and $l_{\mathrm{ik}}=-a_{\mathrm{ik}},i\ne k$ . Obviously, the Laplacian of any undirected graph is symmetric. A path is a sequence of ordered edges of the form $(s_{i1},s_{i2}),(s_{i3},s_{i4}),...$ , where $i_k\in \mathcal{I}$ and $s_{\mathrm{ik}}\in \mathcal{V}$ . If there exists a path from every vertex to every other vertex, the graph is said to be connected. ²³

Lemma 1. If the undirected graph $\mathcal{G}$ is connected, then the Laplacian $L$ of $\mathcal{G}$ has the following properties: ²³

1. Zero is a simple eigenvalue of $L$ , and $1$ is the corresponding eigenvector, that is, $L1\mathbf{=}$ 0.

2. The rest $n-1$ eigenvalues are all positive and real.

Robust H_∞ control theory

In this section, some basic knowledge on robust $H\infty$ control theory will be introduced. ²⁴ Consider the following system

$$\begin{array}{l}\stackrel{.}{x}(t)=(A+\Delta A)x(t)+B\delta (t)\\ z(t)=Cx(t),x_0=0\end{array}$$

(1)

where $x(t)\in {\mathbb{R}}^n$ represents the system state, $z(t)=\mathrm{Cx}(t)\in {\mathbb{R}}^m$ represents the objective signal to be attenuated, and $\delta (t)\in {\mathcal{L}}_2[0,\infty )$ represents the external disturbance. The system matrices $A,B,C$ are constant matrices in appropriate dimensions and $\Delta A$ represents the uncertainty with $\Delta A=F_{1}R(t)F_2$ , in which $F_1$ and $F_2$ are prescribed as constant matrices and $R(t)$ is an unknown matrix function satisfying $R^T(t)R(t)\le I$ . For a prescribed scalar $\gamma$ , the performance index is defined as follows

$$\begin{array}{l}J(\delta )=\underset{0}{\overset{\infty }{{\displaystyle \int }}}[z(t{)}^{T}z(t)-{\gamma }^2\delta (t{)}^T\delta (t)]dt\\ (\parallel T_{\delta z}(s){\parallel }_{\infty }<\gamma )\end{array}$$

(2)

Lemma 2. System (1) is exponentially stable with $(\parallel T_{\delta z}(s){\parallel }_{\infty }<\gamma )$ , for all non-zero $\delta (t)\in {\mathcal{L}}_2[0,\infty )$ , if there exists a positive-definite matrix $P\in {\mathbb{R}}^{n\times n}$ and a scalar $\mu >0$ satisfying ⁹

$$\left[\begin{array}{ccc}{A}^{T}P+\mathrm{PA}+C^{T}C+\mu F_2^T{F}_2\hfill & P{F}_1\hfill & \mathrm{PB}\hfill \\ \multicolumn{1}{c}{F_1^{T}P}& -\mu I\hfill & 0\hfill \\ \multicolumn{1}{c}{B^{T}P}& 0\hfill & -{\gamma }^{2}I\hfill \end{array}\right]<0$$

(3)

Lemma 3. (Schur Complement). For given symmetric matrix $S$ with the form $S=[S_{\mathrm{ij}}],S_{11}\in {\mathbb{R}}^{r\times r},S_{12}\in {\mathbb{R}}^{r\times (n-r)},S_{22}\in {\mathbb{R}}^{(n-r)\times (n-r)}$ , then, $S<0$ if and only if $S_{11}<0$ , $S_{22}-S_{21}{S}_{11}^{-1}{S}_{12}<0$ or $S_{22}<0{,}_{11}-S_{12}{S}_{22}^{-1}{S}_{21}<0$ .

The problem description

Consider a multi-agent system is composed of $n$ agents. Each vertex corresponds to an agent in the undirected graph $\mathcal{G}$ . The available information channel between agents $s_i$ and $s_j$ can be regarded as an edge $(s_i,s_j)\in \mathcal{E}$ in the undirected graph $\mathcal{G}$ . Suppose the $i$ th agent $s_i(i\in \mathcal{I})$ has the following dynamics

$$\begin{array}{c}{\stackrel{·}{r}}_i=v_i\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{v}}_i=u_i+{\delta }_i}\end{array}$$

(4)

where $r_i,v_i,u_i\in \mathbb{C}$ denote the position, velocity, and the control input of agent $s_i$ , and ${\delta }_i\in {\mathcal{L}}_2[0,\infty )$ denote the external disturbance of the agent $s_i$ , which might be caused by measurement or calculation errors.

The main aim of this article is to design a distributed control algorithm to enable all agents to move together in a circular orbit under a desired constant angular velocity ${\omega }_1\in (0,\infty )$ ; meanwhile, the trajectories of each agent’s circle center form a circle under a constant angular velocity ${\omega }_2\in (0,\infty )$ and the closed-loop system satisfies the desired $H\infty$ performance.

Definition 1. For all the $i,j\in \mathcal{I}$ , the multi-agent system (4) is said to achieve composite-rotating consensus, if the system satisfies the following limits ⁸

$$\begin{array}{c}\underset{t\to \infty }{lim}[r_i(t)-r_j(t)]=0\hfill \\ \multicolumn{1}{c}{\underset{t\to \infty }{lim}[v_i(t)-v_j(t)]=0}\end{array}$$

for all $i,j\in \mathcal{I}$ , and each agent $i$ surrounds the point $r_i(t)+{\omega }_1^{-1}j{v}_i(t)$ whose trajectories from a circle, that is, there exists a positive number ${\omega }_2$ such that

$$\underset{t\to \infty }{lim}[{\stackrel{·}{r}}_i(t)+{\omega }_1^{-1}j{\stackrel{·}{v}}_{i}t-j{\omega }_2(r_i(t)+{\omega }_1^{-1}j{v}_i(t))]=0$$

for all $i\in \mathcal{I}$ .

Main results

Control algorithm design and network dynamics

The disturbances are usually unavoidable in practical applications for the multi-agent systems. Therefore, it is significant to study the robust $H\infty$ composite-rotating consensus problem of system (4). In this article, we use the following control algorithm

$$u_i=j{\omega }_1{v}_i-{\omega }_1{\omega }_2{c}_i-\sum _{s_j\in N_i}(a_{\mathrm{ij}}+\Delta a_{\mathrm{ij}}(t))[(v_i-v_j)+(c_i-c_j)]$$

(5)

for $i\in \mathcal{I}$ .

The algorithm (5) contains two parts. The first part can ensure that all agents can move in a composite-rotating way by the effects of $j{\omega }_1{v}_i$ and ${\omega }_1{\omega }_2{c}_i$ . Since $j{\omega }_1{v}_i$ can be regarded as the centripetal acceleration to make each agent circle around the point $c_i$ , and ${\omega }_1{\omega }_2{c}_i$ can make circle center $c_i$ move in a circle, thus, a composite-rotating motion is completed. The second part is the distributed state feedback that enables all agents to eliminate the disagreement dynamics and make all agents to reach a consensus. And $\Delta a_{\mathrm{ij}}(t)$ denote the uncertainty of $a_{\mathrm{ij}}$ with

$$|\Delta a_{\mathrm{ij}}(t)|=\{\begin{array}{cc}\le {\psi }_{\mathrm{ij}}\hfill & i\ne j\mathrm{and}{a}_{\mathrm{ij}}\ne 0\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}\hfill \end{array}$$

where ${\psi }_{\mathrm{ij}}$ is a specified constant for $i,j\in \mathcal{I}$ . Moreover, in this article, we assume that $\Delta a_{\mathrm{ij}}(t)$ is a continuous function of time $t$ .

The output functions can be assigned as $z_i(t)=[z_{i1}(t),z_{i2}(t){]}^T\in {\mathbb{R}}^2$ with $i\in \mathcal{I}$ to describe averages of the relative displacements $r_i$ and circle centers $c_i$ of all agents. Combining the relative information, the output can be naturally given as follows

$$\begin{array}{c}{z}_{i1}(t)=r_i(t)-\frac{1}{n}\sum _{j=1}^n{r}_j\hfill \\ z_{i2}(t)=c_i(t)-\frac{1}{n}\sum _{j=1}^n{c}_j\hfill \end{array}$$

Since all agents can be removed to a fixed point by vectors $j{v}_i/{\omega }_1$ and $j{\stackrel{·}{c}}_i/{\omega }_2$ , and speed $v_i$ has a linear relation with relative displacement $r_i$ as well as the circle center $c_i$ of the agent $v_i=-j{\omega }_1(r_i-c_i)$ , it is clear that consensus can be achieved if and only if

$$\underset{t\to \infty }{lim}{z}_i(t)=0,i=1,2,...,n$$

denote

$$\begin{array}{c}\multicolumn{1}{c}{\xi =[r_1,c_1,r_2,c_2,...,r_n,c_n]}\\ \multicolumn{1}{c}{A=\left[\begin{array}{cc}j{\omega }_1\hfill & -j{\omega }_1\hfill \\ \multicolumn{1}{c}{0}& -j{\omega }_2\hfill \end{array}\right]\in {\mathbb{R}}^{2\times 2},}\\ \multicolumn{1}{c}{B_1=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{-j{\omega }_1}& 1-j{\omega }_1\hfill \end{array}\right]\in {\mathbb{R}}^{2\times 2}}\\ \multicolumn{1}{c}{B_2=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& j/{\omega }_1\hfill \end{array}\right]\in {\mathbb{R}}^{2\times 2}}\\ \multicolumn{1}{c}{C=\left[\begin{array}{cccc}\frac{n-1}{n}\hfill & -\frac{1}{n}\hfill & \dots \hfill & -\frac{1}{n}\hfill \\ \multicolumn{1}{c}{-\frac{1}{n}}& \frac{n-1}{n}\hfill & \ddots \hfill & -\frac{1}{n}\hfill \\ \multicolumn{1}{c}{⋮}& \ddots \hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{-\frac{1}{n}}& -\frac{1}{n}\hfill & \dots \hfill & \frac{n-1}{n}\hfill \end{array}\right]\in {\mathbb{R}}^{n\times n}}\end{array}$$

By applying the algorithm (5), the network dynamics can be rewritten as the following equation

$$\begin{array}{c}\stackrel{·}{\xi }(t)=[I_n\otimes A-(L+\Delta L)\otimes B_1]\xi (t)+(I_n\otimes B_2)\delta (t)\hfill \\ z(t)=(C\otimes I_2)\xi (t)\hfill \end{array}$$

(6)

where $L$ depicts the Laplacian of the graph $\mathcal{G}$ and $\Delta L\in {\mathbb{R}}^{n\times n}$ represents the uncertainty of Laplacian. It can be known from the robust $H\infty$ control theory that $\Delta L$ can be decomposed as $\Delta L=E_{1}M(t)E_2$ , in which $E_1\in {\mathbb{R}}^{n\times |\mathcal{E}|}$ , $E_2\in {\mathbb{R}}^{|\mathcal{E}|\times n}$ represent specified constant matrices and $M(t)\in {\mathbb{R}}^{|\mathcal{E}|\times |\mathcal{E}|}$ is a diagonal matrix whose diagonal elements represent the uncertainties of the edges.

Define the $H\infty$ performance index as follows

$$J=\underset{0}{\overset{\infty }{\int }}[z^T(t)z(t)-{\gamma }^2{\delta }^T(t)\delta (t)]\mathrm{dt}$$

(7)

where $\gamma$ is a positive scalar.

In this section, the definition of the composite-rotating consensus, the distributed control algorithm, and the closed-loop network dynamics of the multi-agent system were proposed. However, it is much harder to give the sufficient conditions for all agents achieving composite-rotating consensus with the $H\infty$ performance index $J<0$ , especially when the uncertainties are included.

$H\infty$ composite-rotating consensus control

Lemma 4. The matrix $C$ satisfies the following statements: ⁹

1. The eigenvalues of $C$ are $1$ with multiplicity $n-1$ and 0 with multiplicity 1. The vectors $1_n^T$ and $1_n$ are the left and right eigenvectors of $C$ associated with the zero eigenvalue respectively.

2. There exists an orthogonal matrix $U\in {\mathbb{R}}^{n\times n}$ such that

$$U^T\mathrm{CU}=\left[\begin{array}{cc}{I}_{n-1}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]$$

and the last column is $1_n/\sqrt{n}$ . Let ${\Xi }_1\in {\mathbb{R}}^{n\times n}$ be the Laplacian of any directed graph, then $U^T{\Xi }_{1}U=[v_1,0]$ , $v_1\in {\mathbb{R}}^{n\times (n-1)}$ .

For convenience, denote $U=[U_1{\overline{U}}_1]$ where ${\overline{U}}_1=1_n/\sqrt{n}$ is the last column of $U$ and $U_1\in {\mathbb{R}}^{n\times (n-1)}$ is the rest part.

Lemma 5. Let

$$\begin{array}{c}\multicolumn{1}{c}{W(t)=\frac{1}{n}\sum _{i=1}^n\underset{0}{\overset{t}{\int }}{e}^{A(t-\tau )}{B}_2{\delta }_i(\tau )d\tau }\\ \multicolumn{1}{c}{\hat{\xi }(t)=\xi (t)-1_n\otimes W(t)}\\ \multicolumn{1}{c}{\phi (t)=(U_1\otimes I_2{)}^T\hat{\xi }(t)}\\ \multicolumn{1}{c}{\overline{\phi }(t)=({\overline{U}}_1\otimes I_2{)}^T\hat{\xi }}\end{array}$$

The system (6) is exponentially stable with $||T_{\delta z}(s)|{|}_{\infty }\le \gamma$ if and only if the following system is exponentially stable with $||T_{\delta z}(s)|{|}_{\infty }\le \gamma$

$$\begin{array}{c}\stackrel{·}{\phi }(t)=[I_{n-1}\otimes A-(\overline{L}+\overline{\Delta }L)\otimes B_1]\phi (t)+(U_1^T\otimes B_2)\delta (t)\hfill \\ z(t)=(U_1\otimes I_2)\phi (t)\hfill \end{array}$$

(8)

Proof: It can be known from the equations in Lemma 5 that $W(t)$ represents the average effect caused by external disturbance upon each agent; $\hat{\xi }(t)$ describes the states of all the agents, which takes out the average of external disturbances; $\overline{\phi }(t)$ describes the average states of all the agents; and $\phi (t)$ depicts the disagreement state of all agents.

First, pre-multiplying the matrix $(U\otimes I_2{)}^T$ to $\stackrel{·}{\xi }$ , we have

$$\begin{array}{c}\multicolumn{1}{c}{{(U\otimes I_2)}^T\stackrel{·}{\xi }(t)=\left[\begin{array}{c}\stackrel{·}{\phi }(t)\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\overline{\phi }}(t)}\end{array}\right]+{(U\otimes I_2)}^T(1_n\otimes \stackrel{·}{W}(t))}\\ \multicolumn{1}{c}{=\left[\begin{array}{c}\stackrel{·}{\phi }(t)\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\overline{\phi }}(t)}\end{array}\right]+\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{\sqrt{n}\mathrm{AW}(t)+\frac{1}{\sqrt{n}}\sum _{i=1}^n{B}_2{\delta }_i(t)}\end{array}\right]}\end{array}$$

(9)

It can be known from Lemma 1 that $L{1}_n=0$ and $\Delta L{1}_n=0$ when the graph $\mathcal{G}$ is connected. Then, pre-multiplying both sides of system (6) with the matrix $(U\otimes I_2{)}^T$ , we have

$$\begin{array}{c}\multicolumn{1}{c}{{(U\otimes I_2)}^T\stackrel{·}{\xi }(t)=(U\otimes I_2{)}^T(I_n\otimes A-(L+\Delta L)\otimes B_1)(\hat{\xi }(t)}\\ \multicolumn{1}{c}{+1_n\otimes W(t))+(U^T\otimes B_2)\delta (t)}\\ \multicolumn{1}{c}{=\left[\begin{array}{cc}{I}_{n-1}\otimes A\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& A\hfill \end{array}\right]\left[\begin{array}{c}\phi (t)\hfill \\ \multicolumn{1}{c}{\overline{\phi }(t)}\end{array}\right]}\\ \multicolumn{1}{c}{-\left[\begin{array}{cc}(U_1^{T}L{U}_1+U_1^T\Delta L{U}_1)\otimes B_1\hfill & 0\hfill \\ \multicolumn{1}{c}{({\overline{U}}_1^{T}L{U}_1+{\overline{U}}_1^T\Delta L{U}_1)\otimes B_1}& 0\hfill \end{array}\right]\left[\begin{array}{c}\phi (t)\hfill \\ \multicolumn{1}{c}{\overline{\phi }(t)}\end{array}\right]}\\ \multicolumn{1}{c}{+\left[\begin{array}{c}(U_1^T\otimes B_2)\delta (t)\hfill \\ \multicolumn{1}{c}{\sqrt{n}\mathrm{AW}(t)+B_2\frac{1}{\sqrt{n}}\sum _{i=1}^n\delta (t)}\end{array}\right]}\end{array}$$

(10)

Denote $\overline{L}=U_1^{T}L{U}_1$ and $\overline{\Delta L}=U_1^T\Delta L{U}_1$ . It is easy to derive the following equation by (9) and (10)

$$\begin{array}{c}\stackrel{·}{\phi }(t)=[I_{n-1}\otimes A-(\overline{L}+\overline{\Delta }L)\otimes B_1]\phi (t)+(U_1^T\otimes B_2)\delta (t)\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\overline{\phi }}(t)=[({\overline{U}}_1^{T}L{U}_1+{\overline{U}}_1^T\Delta L{U}_1)\otimes B_1]\phi (t)+A\overline{\phi }(t)}\end{array}$$

(11)

The output function $z(t)$ can be transformed into the following equation using Lemma 4 and equation (11)

$$\begin{array}{c}\multicolumn{1}{c}{z(t)=(C\otimes I_2)\xi (t)=(C\otimes I_2)(U\otimes I_2)(U\otimes I_2{)}^T\xi (t)}\\ \multicolumn{1}{c}{=\left[\begin{array}{cc}{U}_1\otimes I_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]\left[\begin{array}{c}\phi (t)\hfill \\ \multicolumn{1}{c}{\overline{\phi }(t)}\end{array}\right]}\end{array}$$

(12)

From the equations above, it is clear that the output function is dependent on $\phi (t)$ , and $\phi (t)$ is independent on $\overline{\phi }(t)$ . Hence, it follows that $\underset{t\to \infty }{lim}z(t)=0$ when $\underset{t\to \infty }{lim}\phi (t)=0$ . In conclusion, whether the multi-agent system (6) achieves consensus only depends on the component $\phi (t)$ . The network dynamic (6) is equivalent to

$$\begin{array}{c}\stackrel{·}{\phi }(t)=[I_{n-1}\otimes A-(\overline{L}+\overline{\Delta L})\otimes B_1]\phi (t)+(U_1^T\otimes B_2)\delta (t)\hfill \\ \multicolumn{1}{c}{z(t)=(U_1\otimes I_2)\phi (t)}\end{array}$$

Theorem 1. Consider an undirected network of $n$ agents which has fixed topology and uncertainty. If there exists a symmetric positive-definite matrix $P\in {\mathbb{R}}^{(2n-2)\times (2n-2)}$ and a positive scalar $\mu$ such that the following LMI holds

$$\left[\begin{array}{cccc}P(I_{n-1}\otimes A-\overline{L}\otimes B_1)+{(I_{n-1}\otimes A-\overline{L}\otimes B_1)}^{T}P+\mu [(U_1^T{E}_2^T{E}_2{U}_1)\otimes I_2]+I_{2n-2}& P[(U_1^T{E}_1)\otimes B_1]& P(U_1^T\otimes B_2)& \\ & {[(U_1^T{E}_1)\otimes B_1]}^{T}P& -\mu I& 0\\ & {(U_1^T\otimes B_2)}^{T}P& 0& -{\gamma }^{2}I\end{array}\right]<0$$

(13)

where

$$H_1=I_{n-1}\otimes A-\overline{L}\otimes B_1$$

$$H_2=(U_1^T{E}_1)\otimes B_1$$

$$H_3=U_1^T\otimes B_2$$

$$\overline{E}=(U_1^T{E}_2^T{E}_2{U}_1)\otimes I_2$$

Then, the control algorithm (5) can make the multi-agent system (6) reach composite-rotating consensus with the $H\infty$ performance index $J<0$ for a given positive constant $\gamma$ .

Proof: First, from Lemma 5, we have the multi-agent system (6) is equivalent to system (8). Let $P\in {\mathbb{R}}^{2(n-1)\times 2(n-1)}$ be a symmetric matrix. Take the following Lyapunov function

$$V(t)={\phi }^T(t)P\phi (t)$$

Calculating $\stackrel{·}{V}(t)$ along system (8), it follows that

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(t)=2{\phi }^T(t)P\stackrel{·}{\phi }(t)}\\ \multicolumn{1}{c}{=2{\phi }^T(t)P(I_{n-1}\otimes A)\phi (t)-2{\phi }^T(t)P[(\overline{L}+\overline{\Delta L})\otimes B_1]\phi (t)}\\ \multicolumn{1}{c}{+2{\phi }^T(t)P(U_1^T\otimes B_2)\delta (t)}\end{array}$$

Noted that $M^T(t)M(t)\le I$ and for any $x,y\in {\mathcal{R}}^n$ and any symmetric positive-definite matrix $\Gamma \in {\mathcal{R}}^{n\times n}$

$$2x^{T}y\le x^T{\Gamma }^{-1}x+y^T\Gamma y$$

We have

$$\begin{array}{c}\multicolumn{1}{c}{-2{\phi }^T(t)P(\overline{\Delta L}\otimes B_1)\stackrel{·}{\phi }(t)\le {\phi }^T(t)P(U_1^T{E}_1\otimes B_1)]}\\ \multicolumn{1}{c}{(\frac{1}{\mu }{I}_{2n})(U_1^T{E}_1\otimes B_1{)}^{T}P\phi t}\\ \multicolumn{1}{c}{+{\phi }^T(t)[(U_1^T{E}_2^T{M}^T)\otimes I_2](\mu I_{2n})[(M{E}_2{U}_1)\otimes I_2]\phi (t)}\\ \multicolumn{1}{c}{\le \frac{1}{\mu }{\phi }^T(t)P(U_1^T{E}_1\otimes B_1)(U_1^T{E}_1\otimes B_1{)}^{T}P\phi (t)}\\ \multicolumn{1}{c}{+\mu {\phi }^T(t)\overline{E}\phi (t)}\end{array}$$

Consequently

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(t)\le 2{\phi }^T(t)P(I_{n-1}\otimes A)\stackrel{·}{\phi }(t)-2{\phi }^T(t)P(\overline{L}\otimes B_1)\stackrel{·}{\phi }(t)}\\ \multicolumn{1}{c}{+\frac{1}{\mu }{\phi }^T(t)P(U_1^T{E}_1\otimes B_1)(U_1^T{E}_1\otimes B_1{)}^{T}P\phi (t)}\\ \multicolumn{1}{c}{+\mu {\phi }^T(t)\overline{E}\phi (t)}\end{array}$$

According to the theorem of linear superposition, the response of a linear system is the sum of a zero input response and a zero state response. The former is triggered by the non-zero initial condition, meanwhile the latter is triggered by the external input. Therefore, in order to analyze the effects of the external disturbance $\delta (t)$ on system (8), it is a natural way to suppose that all agents start from the consensus state, that is, $\phi (0)=0$ . Clearly, $V(0)=0$ .

For any $T>0$ , we have

$$\begin{array}{c}{J}_T=\underset{0}{\overset{T}{\int }}[z^T(t)z(t)-{\gamma }^2{\delta }^T(t)\delta (t)]\mathrm{dt}\hfill \\ \multicolumn{1}{c}{=\underset{0}{\overset{T}{\int }}[z^T(t)z(t)-{\gamma }^2{\delta }^T(t)\delta (t)+\stackrel{·}{V}(t)]\mathrm{dt}-(V(T)-V(0))}\\ \multicolumn{1}{c}{\le \underset{0}{\overset{T}{\int }}[{\eta }^T(t)N\eta (t)]\mathrm{dt}-V(T)}\end{array}$$

where $\eta (t)=[{\phi }^T(t){\delta }^T(t){]}^T$ and matrix

$$N=\left[\begin{array}{cc}P{H}_1+H_1^{T}P+\mu \overline{E}+I_{2n-2}+\frac{1}{\mu }P{H}_2{H}_2^{T}P\hfill & P{H}_3\hfill \\ \multicolumn{1}{c}{H_3^{T}P}& -{\gamma }^{2}I\hfill \end{array}\right]$$

$N<0$ holds if and only if (13) holds by Lemma 3. Since $V(T)$ is positive, we have naturally that $J_T<0$ and system (8) is exponentially stable if (13) holds. Then, by Lemma 5 it is easy to know that the system (6) is exponentially stable if and only if system (8) is exponentially stable with the $H\infty$ performance index $J<0$ . This completes the proof.

Simulations

Numerical simulations will be given in this section to illustrate the theoretical results obtained in the previous section. The graphs $G1,G2$ and $G3$ in Figure 1 are communication topologies of multi-agent system (6) when $n=4$ and $n=6$ .

Figure 1.

Undirected graphs of the 4-agent systems and 6-agent system: (a) G1, (b) G2, and (c) G3.

Suppose that the weight of each edge is 1, the uncertainty of each edge is

$$\Delta a_{\mathrm{ij}}=\{\begin{array}{cc}0.01\sin [(i+j)t],\hfill & i\ne j\mathrm{and}{a}_{\mathrm{ij}}\ne 0\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}\hfill \end{array}$$

The external disturbance usually exists in the form of pulse or step signal in the practice. So, take the external disturbance $\delta (t)=[1.512-1{]}^T\overline{\delta }(t)$ when $n=4$ , and $\delta (t)=[212-111{]}^T\overline{\delta }(t)$ when $n=6$ , where

$$\overline{\delta }(t)=\{\begin{array}{cc}1\hfill & 0\le t\le 6\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}\hfill \end{array}$$

is a step signal. Suppose that the $H\infty$ performance index $\gamma =1$ and the desired angular velocity ${\omega }_1=0.8$ and ${\omega }_2=0.02$ . Now, simulation results are presented.

Figures 2–4 illustrate that all the agents can reach composite-rotating consensus with external disturbances and uncertainties.

Figure 2.

Position trajectories of the agents for G1.

Figure 3.

Position trajectories of the agents for G2.

Figure 4.

Position trajectories of the agents for G3.

Figures 5–7 show the energy of the controlled output and the external disturbance of the three topologies in Figure 1. Obviously, the composite-rotating consensus can be achieved with the desired $H\infty$ performance ${\int }_0^{\infty }{z}^T(t)z(t)\mathrm{dt}<{\int }_0^{\infty }{\gamma }^2{\delta }^T(t)\delta (t)\mathrm{dt}$ .

Figure 5.

The energy trajectories of $z(t)$ and disturbance for G1.

Figure 6.

The energy trajectories of $z(t)$ and disturbance for G2.

Figure 7.

The energy trajectories of $z(t)$ and disturbance for G3.

Conclusion

In this article, the robust $H\infty$ composite-rotating consensus problems are investigated for second-order multi-agent systems with uncertainties. To solve this problem, we propose a distributed control algorithm based on the local information received from agent’s neighbors. Some sufficient conditions are derived to make all agents reach composite-rotating consensus with the desired $H\infty$ performance. Finally, numerical simulations are provided to demonstrate the effectiveness of our theoretical results.