Over the last decade,cooperative control of multi-agent systems has attracted much attention from various disciplines. This is partially due to that it has many potential applications in many engineering systems,such as unmanned air vehicle,robotics,transportation and sensor networks. One fundamental problem in this area is the consensus problem,which contains the leader-less and leader-following consensus (or consensus tracking) problems. It aims to design distributed control laws using the relative information between neighboring agents such that the states or outputs of all agents reach an agreement^{[1, 2, 3, 4]}.

Compared with the leader-less consensus problem,the leader-following consensus problem is more challenging and has more applications. With an active second-order leader agent,^{[5, 6]} studied the leader-following consensus problem of first-order and second-order multi-agent systems,respectively. Reference ^[7] investigated the second-order leader-following consensus problem with multiple time-varying delays,where some necessary and (or) sufficient conditions were obtained for both fixed and switching topologies. In ^[8],the authors proposed discontinuous consensus tracking and swarm tracking algorithms for first-order or second-order multi-agent systems without velocity or acceleration measurements. Reference ^[9] proposed topology-independent consensus protocols for second-order multi-agent systems. References ^{[10, 11, 12]} investigated the leader-following consensus problem for multi-agent systems with general linear dynamics.

Note that most of the existing works focus on the consensus problem of linear multi-agent systems under the assumption that the exact model knowledge of the agent dynamics is known. However,in reality, all physical systems are inherently nonlinear and there may exist disturbances and unmodeled uncertainties. In the existing literature,there are few results on the robust consensus problem of uncertain nonlinear multi-agent systems. An attempt on this problem can be found in ^{[13, 14]},where continuous robust consensus tracking control laws were constructed for a class of first-order and second-order integrator-type multi-agents systems,respectively. The proposed construction was based on the recently developed continuous nonlinear robust control technique called the robust integral of the sign of the error^[15]. The main advantage of this technique is that a continuous control law rather than a usually discontinuous control law can be constructed for a class of systems with bounded disturbances. Therefore,many results have recently been developed^{[16, 17, 18]}.

In this paper,we further study the leader-following consensus problem for a class of multi-agent systems with disturbances and unmodeled uncertainties. Contrary to ^{[13, 14]},where only first-order and second-order integrator-type multi-agents systems were investigated,we consider a class of more general multi-agent systems. We firstly consider a linear single input multi-agent system in the controllable canonical form with disturbances and unmodeled uncertainties satisfying matching condition,which can be see as an extension of ^[15] to multi-agent systems. By employing the robust integral of the sign of the error technique,a continuous distributed control law is constructed which achieves semiglobal leader-following consensus. We then extend the above result to a class of multi-input multi-agent systems with disturbances and unmodeled uncertainties satisfying matching condition. Moreover,we show that if the unmodeled uncertainties vanish or satisfy global Lipschitz condition,then global leader-following consensus can be achieved. It is worth pointing out that different from some previous results such as ^{[7, 10, 11]},where the leader$'$s control input was assumed to be available to all follower agents,we consider a more general case,that is,the leader$'$s control input is nonzero and not available to any follower agent. Note that ^{[5, 6, 8, 12]} have also studied this problem. However,they only considered the linear multi-agent systems cases. Moreover,the resulted closed-loop system solutions of ^{[5, 6]} were only bounded with any positive constant while the proposed control laws in ^{[8, 12]} were discontinuous.

Let $\mathcal {G}( \mathcal {V},\mathcal {E},\mathcal {A})$ be an undirected graph of order $N$,where $ \mathcal {V}=\{1,2,\cdots, N\}$ is the set of nodes,$\mathcal {E}\subset \mathcal {V}\times \mathcal {V}$ is the set of undirected edges and $\mathcal {A}=[a_{ij}]_{N\times N}$ is the adjacency matrix with $a_{ii}=0$. An edge denoted by pair $(j,i)$ represents a communication channel from $j$ to $i$. For undirected graph $\mathcal {G}$,if $(j,i)\in\mathcal {E}$,then $(i,j)\in\mathcal {E}$. The neighborhood of the $i$-th agent is denoted by $\mathcal {N}_i=\{j\in \mathcal {V}|(j,i)\in\mathcal {E}\}$. For any $i, j\in\mathcal {V}$,$a_{ij}\geq0$ and $a_{ij}>0$ if and only if $j$ $\in$ $\mathcal {N}_i$. The Laplacian matrix of graph $\mathcal {G}$ is defined as $\mathcal {L}=\mathcal {D}-\mathcal {A}$,where $\mathcal {D}=\mathrm{diag}\{d_1,d_2,\cdots,d_N\}$ is called the degree matrix of $\mathcal {G}$ with $d_i=\sum_{j\in \mathcal {N}_i}a_{ij}$,$i=1,\cdots,N$. A path from node $i_1$ to node $i_{k}$ is a sequence of edges of the form $(i_1,i_2),(i_2,i_3)$, $\cdots$,$(i_{k-1},i_{k})$,where $i_j\in \mathcal {V}$, $j=1,\cdots,k$. If there exists a path between any two vertices of $\mathcal {G}$,then $ \mathcal {G}$ is said to be connected, otherwise disconnected. A subgraph $\mathcal {G}_1$ of $\mathcal {G}$ is an induced subgraph if two vertices of $\mathcal {G}_1$ are adjacent in $\mathcal {G}_1$ only if they are adjacent in $\mathcal {G}$.

Consider a graph $\mathcal {\bar{G}}$ consisting of $N$ agents and a leader agent. When regarding the $N$ agents as the vertices in $\mathcal {V}$,the relationships between agents can be described by a simple and undirected graph $\mathcal {{G}}$. $(i,j)$ is an edge of $\mathcal {{G}}$ if and only if agents $i$ and $j$ are neighbors. $\mathcal {\bar{G}}$ contains $\mathcal {{G}}$ and a leader with edges between some agents and the leader. The connection weight matrix is denoted by $\mathcal {B}=$ $\mathrm{diag}\{b_1$,$\cdots$, $b_N\}$,where $b_i\geq0$,$i=1,\cdots,N$ and $b_i>0$ if and only if agent $i$ is connected to the leader. $\mathcal {\bar{G}}$ is connected if at least one agent in each component is connected with the leader. Denote $H=\mathcal {L}+\mathcal {B}$. A useful lemma about $H$ is given as follows.

$\mathbf{Lemma}$ 1^[5]. If graph $\mathcal {\bar{G}}$ is connected,then the symmetric matrix $H$ associated with $\mathcal {\bar{G}}$ is positive definite.

Consider a class of uncertain multi-agent systems with $N$ $+$ $1$ agents as follows

where ${\pmb x}_i\in\mathbf{R}^n$ represents the state and ${\pmb u}_i\in\mathbf{R}^m$ represents the control input of agent $i$. $A\in\mathbf{R}^{n\times n}$ and $B\in\mathbf{R}^{n\times m}$ are constant matrices. ${\pmb f}_{i1}(t)\in\mathbf{R}^m$ represents the disturbance,and $\mathcal {C}^2$ function ${\pmb f}_{i2}: \mathbf{R}^n\rightarrow\mathbf{R}^m$ represents the unmodelled dynamics. The agent indexed by $0$ is called the leader,and the rest agents indexed by $1,\cdots,N$ are referred to as the followers. The communication topology of $N$ follower agents and one leader agent is denoted by $\mathcal {\bar{G}}$. Assume that the undirected graph $\mathcal {\bar{G}}$ is connected.

The objective of this paper is to solve the semiglobal leader-following consensus problem of system (1).

$\mathbf{Definition}$ 1 (Semiglobal leader-following consensus problem). For any (arbitrarily large) compact set $\Omega$ $\subseteq$ $\mathbf{R}^n$,design continuous distributed control laws ${\pmb u}_i$,$i=1$,$\cdots$,$N$ based on the local relative state information between neighboring agents such that for any initial conditions ${\pmb x}_i(0)$ $\in$ $\Omega$,the following holds

To facilitate the stability analysis,the following mild assumptions are given for system (1).

$\mathbf{Assumption}$ 1. ($A$,$B$) is stabilizable.

$\mathbf{Assumption}$ 2. The disturbance terms ${\pmb f}_{i1}(t)$,$i=0$,$\cdots$,$N$ of system (1) and their first-order and second-order time derivatives are bounded.

$\mathbf{Assumption}$ 3. ${\pmb x}_0$,$\dot{{\pmb x}}_{0}$ and $\ddot{{{\pmb x}}}_{0}$ are bounded.

$\mathbf{Assumption}$ 4. The unknown control input of the leader agent ${\pmb u}_0$ of system (1) and its first-order and second-order time derivatives are bounded.

In this section,we will design continuous distributed control laws to solve the semiglobal leader-following consensus problem for system (1) by employing the robust integral of the sign of the error technique.

Firstly,we consider a class of single input uncertain multi-agent systems that can be seen as a special case of system (1). The dynamics of $i$-th agent is given by

where ${\pmb x}_i=[x_{i1},x_{i2},\cdots,x_{in}]^{\rm T}\in\mathbf{R}^n$ represents the state and $u_i\in\mathbf{R}$ represents the control input of agent $i$.

The relative state between $i$-th agent and its neighboring agents is defined as follows:

Its dynamics is described by

Let $\bar{z}_{in}=z_{in}+\sum\nolimits_{j=1}^{n-1}g_jz_{ij}$,where $g_j$,$j=1,\cdots,n-1$ are constants such that the polynomial $s^{n-1}+g_{n-1}s^{n-2}+\cdots+g_{1}$ is Hurwitz. Then,we have

Define the filtered tracking error $r_i\in\mathbf{R}$ as follows:

where $\alpha$ is a positive control gain. Denote ${\pmb r}=[r_1,\cdots,r_N]^{\rm T}$ and $\bar{{\pmb z}}_n=[\bar{z}_{1n},\cdots,\bar{z}_{Nn}]^{\rm T}$. Then,we get

where

Let

According to the mean value theorem,there exists a nondecreasing function $\chi$ such that the following holds

where $\|\cdot\|$ denotes the Euclidean norm,${\pmb z}=[\bar{{\pmb e}}_{1,n-1}^{\rm T},\cdots$,$\bar{{\pmb e}}_{N,n-1}^{\rm T}$, $\bar{{\pmb z}}^{\rm T}_n$,${\pmb r}^{\rm T}]^{\rm T}$,$\bar{{\pmb e}}_{i,n-1}=[z_{i1},\cdots,z_{i,n-1}]^{\rm T}$.

Based on the above analysis,we can now design a continuous distributed control law as follows:

where $k$,$\beta$ are positive constants,and $\mathrm{sgn}(\cdot)$ denotes the standard signum function. Then,we have

$\mathbf{Lemma}$ 2^{[14, 15]}. Let the auxiliary function $L(t)\in\mathbf{R}$ be defined as follows:

If positive constant $\beta$ is selected to satisfy the following condition

where $\|\cdot\|_{\mathcal {L}_{\infty}}$ denotes the ${\mathcal {L}_{\infty}}$ norm,then

$\mathbf{Theorem}$ 1. Consider system (2),and assume Assumptions 2 $\sim$ 4 hold. The control law (5) solves the semiglobal leader-following consensus problem of system (2), provided that the control gain $\alpha>\alpha_c>0$,$\beta$ is selected according to inequality (6),and $k$ is chosen sufficiently large,where positive constant $\alpha_c$ will be given in the following analysis.

$\mathbf{Proof.}$ Define $Q(t)=\varpi-\int_{0}^tL(\tau){\rm d}\tau\in\mathbf{R}$. By Lemma 2, $Q(t)\geq0$ if $\beta$ is selected according to inequality (6). Consider the following Lyapunov candidate function

where ${\pmb y}=[{\pmb z}^{\rm T},~\sqrt{Q}]^{\rm T}$ and $P$ is a positive definite matrix satisfying $PA+A^{\rm T}P=-2I$ with

By (3),we have $\dot{\bar{{\pmb z}}}_n=H{\pmb r}-\alpha\bar{{\pmb z}}_{n}$. Then,the time derivative of $V$ along the solutions of system (3) is given by

where $\bar{B}=[0,\cdots,0,1]^{\rm T}$, $\alpha_c=\frac{1}{2}\|P\bar{B}\|^2$,$\lambda_0=\min\{\frac{1}{2}$, $\alpha$-$\alpha_c$,$\lambda_m\}$,and $\lambda_m$ denotes the minimum eigenvalue of positive definite matrix $H^{2}$. Therefore, if

then there exists a positive constant $\gamma$ such that

By Theorem 8.4 of ^[19],$V({\pmb y},t)\in\mathcal {L}_{\infty}$ in the region of $\mathcal {D}$

So,$\bar{{\pmb e}}_{i,n-1}(t)$,${\bar{{\pmb z}}}_n(t)$,${\pmb r}(t)\in\mathcal {L}_{\infty}$,$i=1,\cdots,N$. From (3), $\dot{\bar{{\pmb z}}}_n(t)$ $\in$ $\mathcal {L}_{\infty}$. Therefore,$x_{i1}^{(j)}(t)\in\mathcal {L}_{\infty}$, $j=0,1,\cdots,n$,and hence,${\pmb u}(t)\in\mathcal {L}_{\infty}$.

To show that the leader-following consensus problem is solved locally,consider the following region

where $\lambda_1$ and $\lambda_2$ are two positive constants such that

By Theorem 8.4 of ^[19],we know that ${\pmb z}(t)\rightarrow0$ as $t\rightarrow\infty$,$\forall {\pmb y}(0)\in\mathcal {S}$. That is,$\bar{{\pmb e}}_{i,n-1}(t),{\bar{{\pmb z}}}_n(t),{\pmb r}(t)\rightarrow0$, $t\rightarrow\infty$,$i=1$,$\cdots$,$N$ and $\forall {\pmb y}(0)\in\mathcal {S}$. Therefore,$\|{\pmb x}_i(t)-{\pmb x}_0(t)\|\rightarrow0$ as $t$ $\rightarrow$ $\infty$, $\forall {\pmb y}(0)\in\mathcal {S}$. So,the leader-following consensus problem is solved locally by the distributed control law (5). Note that semiglobal leader-following consensus can be achieved by properly choosing the control gain $k$ that depends on the initial conditions.

$\mathbf{Remark}$ 1. Note that,if ${f}_{i2}(\cdot)\equiv0$ or ${f}_{i2}(\cdot)$,$i=1,\cdots$,$N$ satisfy the global Lipschitz growth condition,then we can choose the nondecreasing function $\chi(\|{\pmb z}\|)$ to be positive constant $c$ that depends on $H$ and $g_j$,$j=1,\cdots,n-1$. Let $k$ $\geq$ $\frac{c^2}{2\lambda_0\lambda_m}$. By inequality (8),we have

Therefore,${\pmb z}(t)\rightarrow0$ as $t\rightarrow\infty$, $\forall {\pmb y}(0)$. That is,the leader-following consensus problem is solved globally.

In this subsection,we study the semiglobal leader-following consensus problem of system (1) based on the last subsection.

Since $(A,~B)$ is stabilizable,by ^[20],there exists an invertible matrix $\Gamma\in\mathbf{R}^n$,such that

where $\mathbf{0}'{\rm s}$ represent zero matrices with compatible dimensions,$B_i=[0,\cdots,0,1]^{\rm T}$,$i=1,\cdots,p$,

and $A_{\bar{c}}\in\mathbf{R}^{n_{\bar{c}}}$ is a Hurwitz matrix. By linear transformation theory,there exists an invertible matrix $\Lambda\in\mathbf{R}^{m-p}$ such that $\left[ \bar{B}_{1\bar{c}}^{\rm T},\cdots,\bar{B}_{p\bar{c}}^{\rm T} \right]^{\rm T}\Lambda=D_{\bar{c}}$,

where $\bar{B}_{j\bar{c}}$ represents the matrix composed of the first $n_j-1$ rows of $B_{j\bar{c}}$. Consider the following state transformation

Then,we have

where

$D_{j\bar{c}}$,$j=1,\cdots,p$ are some corresponding submatrices of $D_{\bar{c}}$,and $B_{j\bar{c}n_j}$ denotes the last row of matrix $B_{j\bar{c}}$.

To solve the semiglobal leader-following consensus problem of system (9),we next consider a class of multi-agent systems as follows:

where $u_{ij},f_{i1j},f_{i2j}\in\mathbf{R}$, $j=0,1,\cdots,n_4-n_3+n_2-n_1$ represent the control input, disturbances and unmodelled dynamics as in system (2),${\pmb y}_{in_3}=[y_{i1},\cdots,y_{in_3}]^{\rm T}\in\mathbf{R}^{n_3}$, $\bar{{\pmb u}}$ $\in$ $\mathbf{R}^{n_{\bar{u}}}$,$y_{i,n_4+1}$ is a component of ${\pmb w}_i$,${\pmb w}_i\in\mathbf{R}^{n_{{\pmb w}_i}}$ can be seen as some disturbances,$\rho_{jk}\in\mathbf{R}$ and $\bar{R}\in\mathbf{R}^{n_{\bar{u}}}$, $K_{ij}\in\mathbf{R}^{n_{{\pmb w}_i}}$,${K}_{in_3}$ $\in$ $\mathbf{R}^{n_3}$ are some constant matrices. Consider the following coordinate transformation

Then,we get

where $\bar{{\pmb y}}_i=[y_{i1},\cdots,y_{in_4}]^{\rm T}$, $\bar{\rho}_{jk}$ are some constants and $\bar{K}_{ij}$, ${\bar{K}}_{in_3}$ are some matrices that are dependent on $\rho_{jk}$ and $K_{ij}$,${K}_{in_3}$.

As in the last subsection,design distributed control laws similar to (5). Then,there exists a Lyapunov function $\bar{V}$ in the form of (7) such that

where $\sigma_1,\sigma_2$ are two positive constants,$\bar{{\pmb z}}$ is defined similar to ${\pmb z}$,

Now,we are ready to study the leader-following consensus problem of system (9). By the special form of $A_j,B_j$,$D_{\bar{c}j}$, $j=1,\cdots,p$,we know that subsystem $\tilde{{\pmb x}}_{ij}$ is similar to system (10) (which may have several subsystems that are in the form of $[y_{i1},\cdots,y_{n_3-1}]^{\rm T}$) with ${\pmb w}_i=[\tilde{{\pmb x}}_{i,j+1}^{\rm T},\cdots$,$\tilde{{\pmb x}}_{ip}^{\rm T}$,$\tilde{{\pmb x}}_{i\bar{c}}^{\rm T}]^{\rm T}$. By the previous analysis for system (10),we can similarly prove that there exists a continuous distributed control law,such that under the control law,there exists a positive definite and radially unbounded function $V_j$ satisfying

where $\check{{\pmb z}}_j,\check{{\pmb w}}_j,{\pmb \xi}$ and $\chi_j$ have the same meanings as for system (10), $\sigma_{1j},\sigma_{2j},\kappa$ are some positive constants. Moreover,$\kappa$ can be chosen to be sufficiently large and independent of $\sigma_{1j}$,$\sigma_{2j}$ and $\chi_j$.

Consider the following Lyapunov candidate function for system (1), or equivalently,system (10)

where

$\check{\alpha}_j$,$j=1,\cdots,p+1$ are positive constants and $P_{\bar{c}}$ is a positive definite matrix satisfying the following equation

Then,by choosing $\check{\alpha}_j$ recursively (that is,choosing $\check{\alpha}_j$ based on $\check{\alpha}_k$,$k=1,\cdots,j-1$ with $\check{\alpha}_1=1$),we can obtain

for a nondecreasing function $\hat{\rho}$ and a positive constant $\hat{\lambda}$.

By the proof of Theorem 1,we have the following result.

$\mathbf{Theorem}$ 2. Under Assumptions 1 $\sim$ 4,there exists a continuous distributed control law that solves the semiglobal leader-following consensus problem of system (1).

$\mathbf{Remark}$ 2. As in Remark 1,we know that if ${\pmb f}_{i2}(\cdot)\equiv0$ or ${\pmb f}_{i2}(\cdot)$,$i=1,\cdots,N$ satisfy the global Lipschitz growth condition,then we can choose control gains properly such that the corresponding continuous distributed control law solves the leader-following consensus problem globally.

$\mathbf{Remark}$ 3. It is worthwhile to mention two related works^{[8, 12]},which have studied the leader-following consensus problem with unknown but bounded input. Compared with those two papers,the contribution of our paper is at least twofold. First,we consider a class of more general systems with unknown nonlinear dynamics and unknown but disturbances. Second,the distributed control law developed in ^{[8, 12]} are discontinuous, while in our paper,continuous distributed control laws are designed by employing the robust integral of the sign of the error technique.

Consider a multi-agent system with four followers labeled 1,2, 3,4 and a leader labeled 0. The dynamics of the $i$-th agent is described by (1) with

The Laplacian matrix of the corresponding connected information exchange graph $\mathcal {{G}}$ is given by

and the leader adjacency matrix is given by $\mathcal {B}= \mathrm{diag}\{1,0$,0,$0\}$.

Consider the following state transformation

then,we have

Note that system (11) are in the form of (10). Denote

According to our design procedure,we construct the control laws $u_{ij}$ as follows:

where we choose $\alpha=2$,$\beta_1=\beta_2=5$,and $\kappa=20$. The robust tracking results are depicted in Fig. 1. It can be seen that the leader-following consensus is achieved.

Fig. 1

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Fig. 1 Relative states $z_{ij}$, $i=1,2,3,4$, $j=1,2,3$.

In this paper,we have considered the leader-following consensus problem for a class of uncertain multi-agent systems. Based on the relative state between neighboring agents,continuous distributed control laws have been designed by employing the robust integral of the sign of the error technique,which solve the semiglobal leader-following consensus problem. For future research, it is interesting to consider the leader-following consensus problem with only relative output between neighboring agents that are measurable.