I. INTRODUCTION

Distributed control for group coordination of multi-agent systems has recently attracted significant attention from the control community; see,for example,^{[1, 2, 3, 4]} based on Lyapunov methods,^[5] using a passivity approach,^{[6, 7, 8, 9, 10, 11]} based on linear algebra and graph theory,and ^{[12, 13, 14]} using output regulation theory. The main objective of distributed control is to achieve some desired group behavior for multi-agent systems by taking advantage of local system information and information exchanges among neighboring systems. Distributed control may find applications in sensor networks ^[15],vehicle coordination and formation ^{[16, 17, 18, 19]} and smart power grids ^[20],to name only a few. One group behavior of wide interest is the agreement property,for which the interested variables of multi-agent systems are steered to a common value. It should be noted that most of the previously published papers focus on linear models.

In this paper,we study robust distributed control of nonlinear multi-agent systems. The objective is to steer the outputs of the agents to a desired agreement value. In our problem setting,each agent can use its own output and the outputs of its neighbors for the local control law design,while only the informed agents can access the desired agreement value. In addition,the agents studied in this paper are in the disturbed strict-feedback form ^[21] and do not take the identical dynamical model. This makes the distributed control problem in this paper significantly different from the decentralized control problem,in which each decentralized controller often assumes the accurate knowledge of the reference signal and does not take advantage of the available information of neighboring agents; see e.g.,^[22].

The main contribution of this paper is to present a cyclic-small-gain approach to robust distributed controller design for nonlinear multi-agent systems. More precisely,we will use the notions of input-to-state stability (ISS) ^{[23, 24]} and input-to-output stability (IOS) ^{[25, 26]} to describe the dynamic interaction between the controlled agents,and use the recently developed cyclic-small-gain theorem to guarantee the convergence of the agents' outputs to the agreement value. The reader is referred to ^{[27, 28, 29]} for more details on cyclic-small-gain theorems for networks of nonlinear systems,and ^{[25, 26, 27, 28, 29, 30]} for the original small-gain theorems for interconnections of two nonlinear systems.

The rest of the paper is organized as follows. Section II gives the problem formulation. In Section III,we present a design ingredient based on which the closed-loop multi-agent system can be transformed into a network of IOS subsystems. The main result of the paper is given in Section IV. In Section V,we show the robustness of the proposed distributed control strategy with respect to time-delays and disturbances in information exchange. Section VII contains some concluding remarks.

To make the paper self-contained,we give some notations and definitions that are commonly used in the paper here. ${\bf R}^n$ and ${\bf R}_+$ represent the $n$-dimensional Euclidean space and the set of nonnegative real numbers,respectively. $|x|$ represents the Euclidean norm of $x\in{\bf R}^n$. For $u:{\bf R}_+\rightarrow {\bf R}^n$ and $\Delta\subseteq{\bf R}_+$, $\|u\|_{\Delta}$ represents ${\rm esssup}_{t\in \Delta}|u(t)|$. To simplify the notations,we denote $\|u\|_{\infty}=\|u\|_{ [0,\infty)}$. A function $\alpha$: ${\bf R}_+\rightarrow{\bf R}_+$ is said to be positive definite if $\alpha(0)=0$ and $\alpha(s)$ $>$ $0$ for $s>0$. A continuous function $\alpha:{\bf R}_+\rightarrow{\bf R}_+$ is said to be a class $\mathcal{K}$ function,denoted by $\alpha\in\mathcal{K}$, if it is strictly increasing and $\alpha(0)=0$; it is said to be a class $\mathcal{K}_{\infty}$ function,denoted by $\alpha\in\mathcal{K}_{\infty}$,if it is a class $\mathcal{K}$ function and satisfies $\alpha(s)\rightarrow\infty$ as $s\rightarrow\infty$. A continuous function $\beta$: ${\bf R}_+$ $\times$ ${\bf R}_+$ $\rightarrow$ ${\bf R}_+$ is said to be a class $\mathcal{KL}$ function,denoted by $\beta\in\mathcal{KL}$, if,for each fixed $t\in{\bf R}_+$,function $\beta(\cdot,t)$ is a class $\mathcal{K}$ function and,for each fixed $s\in{\bf R}_+$, function $\beta(s,\cdot)$ is decreasing and $\lim_{t\rightarrow\infty}\beta(s,t)=0$.

II. PROBLEM FORMULATION

In this paper,we study the distributed control problem of a group of $N$ nonlinear agents,of which agent $i$ ($1\leq i\leq N$) is described in the strict-feedback form (see,e.g.,^[21]):

\[{{{\dot{x}}}_{ij}}={{x}_{i(j+1)}}+{{\Delta }_{ij}}({{{\bar{x}}}_{ij}},{{w}_{i}}),1\le j\le {{n}_{i}}\]

(1)

\[{{x}_{i({{n}_{i}}+1)}}={{u}_{i}}\]

(2)

\[{{y}_{i}}={{x}_{i1}}\]

(3)

where $ [x_{i1},\cdots,x_{in_i}]^{\rm T}=x_i\in{\bf R}^{n_i}$,with $x_{ij}\in{\bf R}$ ($1\leq j$ $\leq$ $n_i$) is the state, $u_i\in{\bf R}$ is the control input,$y_i\in{\bf R}$ is the output, $\bar{x}_{ij}= [x_{i1},\cdots,x_{ij}]^{\rm T}$,$w_i\in{\bf R}^{n_{w_i}}$ represents external disturbances,and $\Delta_{ij}$'s ($1\leq j\leq n_i$) are unknown locally Lipschitz functions.

For distributed control of the multi-agent nonlinear system (1)$\sim$(3),we use a directed graph (digraph) $\mathcal{G}^c$ to represent the information exchange topology between the agents. Digraph $\mathcal{G}^c$ contains $N$ vertices corresponding to the $N$ agents and $M$ directed edges corresponding to the information exchange links. Specifically,if $y_i-y_k$ is available for local controller design of agent $i$, then there is a directed link from agent $k$ to agent $i$ and agent $k$ is called a neighbor of agent $i$; otherwise,there is no link from agent $k$ to agent $i$. Denote $\mathcal{N}=\{1,\cdots,N\}$. We use $\mathcal{N}_i\subseteq\mathcal{N}$ to represent agent $i'$s neighbor set. In this paper,an agent is not considered as a neighbor of itself and thus $i\notin\mathcal{N}_i$ for $i\in\mathcal{N}$. Agent $i$ is called an informed agent if it has access to the knowledge of the agreement value $y_0$ for its local controller design. We use $\mathcal{L}\subseteq\mathcal{N}$ to represent the set of the informed agents. In some works ^{[10, 12, 14]},$y_0$ is referred to as a virtual leader.

The objective of this paper is to develop a new class of distributed controllers for the multi-agent system based on the available information such that the outputs $y_i$ for $1$ $\leq$ $i$ $\leq$ $N$ converge to the same desired agreement value $y_0$.

The following assumption is made on the agreement value and system (1)$\sim$(3).

${\bf Assumption 1.}$ There exists a nonempty set $\Omega\subseteq{\bf R}$ such that:

1) $y_0\in\Omega$;

2) for each $1\leq i\leq N$,$1\leq j\leq n_i$,

\begin{align} |\Delta_{ij}(\bar{x}_{ij},w_i)-\Delta_{ij}(a_{ij},0)|\leq\psi_{\Delta_{ij}}(| [\bar{x}_{ij}-a_{ij},w_i^{\rm T}]^{\rm T}|) \end{align}

(4)

with $a_{ij}= [a^0,0,\cdots,0]^{\rm T}$ for all $\bar{x}_{ij}\in{\bf R}^j$,$w_i\in{\bf R}^{n_{w_i}}$ and all $a^0\in\Omega$,where $\psi_{\Delta_{ij}}\in\mathcal{K}_{\infty}$ is locally Lipschitz and known.

${\bf Remark 1.}$ It should be noted that a priori information on the bounds of $y_0$ (and thus $\Omega$) is usually known in practice. In this case,property 2) in Assumption 1 can be guaranteed by application of the mean value theorem.

We also assume the boundedness of the external disturbances.

${\bf Assumption 2.}$ For each $i\in\mathcal{N}$,there exists a constant $\bar{w}_i$ $\geq$ $0$ such that $|w_i(t)|\leq\bar{w}_i$ for all $t\geq 0$.

It should be noted that constant $\bar{w}_i$ is not required to be known.

III. A DESIGN INGREDIENT

The main result in this paper is based on a design ingredient for measurement feedback control of a class of first-order nonlinear uncertain systems.

Consider a nonlinear system

\begin{align} \dot{\xi}=\nu+\phi(\xi,\omega), \end{align}

(5)

where $\xi\in{\bf R}$ is the state,$\nu\in{\bf R}$ is the control input,$\omega\in{\bf R}^m$ represents the external disturbance, and $\phi:{\bf R}\times{\bf R}^m\rightarrow{\bf R}$ is an unknown, locally Lipschitz function.

It is assumed that there exists a known,locally Lipschitz $\psi_{\phi}$ $\in$ $\mathcal{K}_{\infty}$ such that for all $\xi,\omega$,

\begin{align} |\phi(\xi,\omega)-\phi(0,0)|\leq\psi_{\phi}(| [\xi,\omega^{\rm T}]^{\rm T}|). \end{align}

(6)

Here,we do not assume the a priori knowledge of $\phi(0,0)$.

${\bf Remark 2.}$ Condition (6) can always be satisfied by a locally Lipschitz $\phi$. Specifically,one may choose $\psi_{\phi}(s)=\max_{| [\xi,\omega^{\rm T}]^{\rm T}|\leq s}|\phi(\xi,\omega)-\phi(0,0)|+\epsilon s$ with $\epsilon$ being a positive constant.

The objective of this section is to present a measurement feedback control law by using $\xi+\delta$ to realize input-to-state stabilization of system (5) with $\delta$ and $\omega$ as the inputs. Here,$\delta$ can be considered as a measurement error. As shown in the following section,the measurement error in the distributed control of agent $i$ is caused by the unavailability of the accurate $y_0$. Specifically,the desired control law for system (5) is in the form of

\[\dot{\eta }=\rho (\eta ,\xi +\delta ),\]

(7)

\[\nu =\varphi (\eta ),\]

(8)

where $\rho$ and $\varphi$ are appropriately designed functions and $\eta$ is the internal state of the control law.

We introduce a dynamic compensator to handle the unknown $\phi(0,0)$:

\begin{align} \dot{\nu}=\mu. \end{align}

(9)

Define $\nu'=\nu+\phi(0,0)$ and $\bar{\phi}(\xi,\omega)=\phi(\xi,\omega)-\phi(0,0)$. The control problem is solvable if we can design a feedback control law to stabilize the following system with $\mu$ as the control input:

\[\dot{\xi }={\nu }'+\bar{\phi }(\xi ,\omega ),\]

(10)

\[{\dot{\nu }}'=\mu .\]

(11)

Since $\phi(0,0)$ is unknown,$\nu'$ is not available for feedback. Note that system (10) and (11) is in the well-known output-feedback form ^[21]. In our recent paper ^[31],we proposed an ISS small-gain approach to measurement output-feedback control of nonlinear uncertain systems in the general output-feedback form by using decentralized observers,and it can be readily applied to the control of system (10) and (11). In ^[32],this technique has been applied to distributed output-feedback control. It should be noted that the observer designs in ^{[31, 32]} are originally motivated by ^[33]. Due to space limitation,we directly give the result here without detailed proofs. For system (10) and (11),we can design a measurement output-feedback control law in the form of

\[\dot{\zeta }=\theta (\xi +\delta ,\zeta ),\]

(12)

\[\mu =\kappa (\zeta ),\]

(13)

with $\theta$ and $\kappa$ being continuously differentiable functions such that the closed-loop system composed of (10)$\sim$(13) is ISS with $Z= [\xi,\nu',\zeta^{\rm T}]^{\rm T}$ as the state and $\delta$ and $\omega$ as the inputs. Moreover,the closed-loop system is unboundedness observable (UO) and IOS with $\xi$ as the output. Specifically,there exist $\alpha^{\text{UO}},\gamma,\chi\in\mathcal{K}$, $\beta\in\mathcal{KL}$ and constant $D_0^{\text{UO}}\geq 0$,such that for any initial state $Z(0)=Z_0$ and any $\delta,\omega$,

\[|Z(t)|\le {{\alpha }^{\text{UO}}}(|{{Z}_{0}}|+\|\delta {{\|}_{[0,t]}}+\|\omega {{\|}_{[0,t]}})+D_{0}^{\text{UO}},\]

(14)

\[|\xi (t)|\le \max \left\{ \beta (|{{Z}_{0}}|,t),\gamma (\|\delta {{\|}_{[0,t]}}),\chi (\|\omega {{\|}_{[0,t]}}) \right\}\]

(15)

hold for all $t\geq 0$. See ^[25] for the original definitions of UO and IOS. Moreover,the IOS gain $\gamma$ corresponding to $\delta$ can be designed to be arbitrarily close to the identity function $\mathrm{Id}$ and the IOS gain $\chi$ corresponding to $\omega$ can be designed to be arbitrarily small. Here,(12) is designed as the observer to deal with the unavailability of $\nu'$.

Clearly,the control law composed of (9),(12) and (13) is in the form of (1)$\sim$(3) with $\eta= [\nu,\zeta^{\rm T}]^{\rm T}$.

IV. DISTRIBUTED CONTROL DESIGN

In this section,we show that each agent $i$ defined by (1) $\sim$(3) can be recursively designed by using the technique proposed in Section III for system (5). As $y_0$ may not be available to each agent,coordination between the agents is necessary. Thus,the control error of one agent may lead to measurement errors of other agents. The cyclic-small-gain theorem is employed to handle such interconnection between the agents.

In this paper,the local controller for each agent $i$ will be designed by directly using $y_i^m$,defined as follows:

\begin{align} y_i^m(t)=\begin{cases}\dfrac{1}{N_i+1}\sum\limits_{k\in\mathcal{N}_i}(y_k(t)+y_0),&i\in\mathcal{L},\\ [4mm] \dfrac{1}{N_i}\sum\limits_{k\in\mathcal{N}_i}y_k(t),&i\in\mathcal{N}\backslash\mathcal{L}, \end{cases} \end{align}

(16)

for $t\geq 0$,where $N_i$ is the size of $\mathcal{N}_i$. For convenience of discussions,we define

\begin{align} d_i=y_0-y_i^m, \end{align}

(17)

as the difference between the desired agreement signal and the actually available signal. Then,$y_i^m=y_0-d_i$.

The control law for each agent $i$ is in the form of

\begin{align} u_i=x_{i(n_i+1)}^*, \end{align}

(18)

with $x_{i(n_i+1)}^*$ recursively defined as

\begin{align} x_{ij}^*=\varphi_{i(j-1)}(\eta_{i(j-1)}),j=2,\cdots,n_i+1, \end{align}

(19)

with each $\varphi_{i(j-1)}$ being a continuously differentiable function.

The variables $\eta_{i(j-1)}$ for $j=2,\cdots,n_i+1$ are generated by

\begin{align} &\dot{\eta}_{i1}=\rho_{i1}(\eta_{i1},e_{i1}+d_i), \end{align}

(20)

\begin{align} &\dot{\eta}_{i(j-1)}=\rho_{i(j-1)}(\eta_{i(j-1)},e_{i(j-1)}),j=3,\cdots,n_i+1, \end{align}

(21)

with $\rho_{i1}$ and $\rho_{i(j-1)}$ for $j=3,\cdots,n_i+1$ being continuously differentiable functions and

\begin{align} &e_{i1}=x_{i1}-y_0, \end{align}

(22)

\begin{align} &e_{ij}=x_{ij}-x_{ij}^*,~j=2,\cdots,n_i. \end{align}

(23)

Note that $e_{i1}+d_i=x_{i1}-y_0+d_i=x_{i1}-y_i^m$. It can be observed that the control law uses the measurements of the known variables $y_i^m$,$x_{i1},\cdots,x_{in_i}$.

To prove the effectiveness of the control law above,we consider the dynamics of $e_{i1},\cdots,e_{in_i}$.

By taking the derivative of $e_{i1}$,we have

\begin{align} \dot{e}_{i1}&=x_{i2}+\Delta_{i1}(\bar{x}_{i1},w_i)\nonumber \end{align} \begin{align} &=x_{i2}^*+\Delta_{i1}(e_{i1}+y_0,w_i)+e_{i2}. \end{align}

(24)

Note that $y_0$ is a constant. By defining $\bar{\Delta}_{i1}(e_{i1},z_{i1})=$ $\Delta_{i1}(e_{i1}+y_0, w_i)+e_{i2}$ with $z_{i1}= [e_{i2},w_i^{\rm T}]^{\rm T}$,the $e_{i1}$-subsystem can be rewritten in the form of

\begin{align} \dot{e}_{i1}=x_{i2}^*+\bar{\Delta}_{i1}(e_{i1},z_{i1}). \end{align}

(25)

With $e_{i1}$ as the state and $x_{i2}^*$ as the control input, system (25) is in the form of (5). Note that $e_{i1}$ is not available for feedback. Instead, $e_{i1}+d_i=x_{i1}-y_0+d_i=x_{i1}-y_i^m$ is available.

Suppose that the $e_{i(j-1)}$-system is in the form of

\begin{align} \dot{e}_{i(j-1)}=x_{ij}^*+\bar{\Delta}_{i(j-1)}(e_{i(j-1)},z_{i(j-1)}), \end{align}

(26)

with $z_{i(j-1)}= [e_{i1},\cdots,e_{i(j-2)},e_{ij},\eta_{i1}^{\rm T},\cdots,\eta_{i(j-2)}^{\rm T},w_i^{\rm T},d_i]^{\rm T}$. Then, with $e_{ij}=x_{ij}-x_{ij}^*$ defined in (23),we have

\begin{align} \dot{e}_{ij}=&\ \dot{x}_{ij}-\frac{\partial\varphi_{i(j-1)}(\eta_{i(j-1)})} {\partial\eta_{i(j-1)}}\dot{\eta}_{i(j-1)}=\nonumber\\ &\ \dot{x}_{ij}-\frac{\partial\varphi_{i(j-1)}(\eta_{i(j-1)})}{\partial\eta_{i(j-1)}} \rho_{i(j-1)}(\eta_{i(j-1)},e_{i(j-1)})=\nonumber\\ &\ x_{i(j+1)}+\Delta_{ij}(\bar{x}_{ij},w_i)-\nonumber\\ &\ \frac{\partial\varphi_{i(j-1)}(\eta_{i(j-1)})}{\partial\eta_{i(j-1)}}\rho_{i(j-1)}(\eta_{i(j-1)},e_{i(j-1)}), \end{align}

(27)

which can be represented by

\begin{align} \dot{e}_{ij}=x_{i(j+1)}^*+\bar{\Delta}_{ij}(e_{ij},z_{ij}), \end{align}

(28)

with $z_{ij}= [e_{i1},\cdots,e_{i(j-1)},e_{i(j+1)},\eta_{i1}^{\rm T},\cdots,\eta_{i(j-1)}^{\rm T},w_i^{\rm T},d_i]^{\rm T}$. Thus,for $j=1,\cdots,n_i$,each $e_{ij}$-subsystem can be rewritten in the form of (28). Note that in the case of $j=n_i$,$e_{i(j+1)}$ $=$ $0$.

Under Assumption 1,by using the definitions above,we can directly prove that there exists a known,locally Lipschitz $\psi_{\bar{\Delta}_{ij}}\in\mathcal{K}_{\infty}$ such that

\begin{align} |\bar{\Delta}_{ij}(e_{ij},z_{ij})-\bar{\Delta}_{ij}(0,0)|\leq\psi_{\bar{\Delta}_{ij}}(| [e_{ij},z_{ij}^{\rm T}]^{\rm T}|). \end{align}

(29)

Thus,each $e_{ij}$-subsystem defined by (28) is in the form of (5) and property (29) is in the form of (6).

Now,we study the transformed $e_{ij}$-subsystems and show that the control laws defined by (18)$\sim$(23) are the desired ones solving our distributed control problem. With the technique proposed in Section IV,the feedback control law

\begin{align} x_{i(j+1)}^*=\varphi_{i1}(\eta_{ij}), \end{align}

(30)

where $\eta_{ij}$ generated by (20) or (21) guarantees that the closed-loop system corresponding to the $e_{ij}$-subsystem is ISS. Moreover,it is UO and IOS with $z_{ij}$ as the input and $e_{ij}$ as the output.

We give the IOS properties of the $e_{i1}$-subsystem and the $e_{ij}$-subsystems for $j=2,\cdots,n_i$ separately:

\begin{align} |e_{i1}(t)|&\leq\notag \\ &\max\left\{\beta_{i1}(|Z_{i1}(0)|,t), \gamma_{i1}(\|d_i\|_{ [0,t]}),\chi_{i1}(\|z_{i1}\|_{ [0,t]})\right\}, \end{align}

(31)

\begin{align} |e_{ij}(t)|&\leq\max\left\{\beta_{ij}(|Z_{ij}(0)|,t),\chi_{ij}(\|z_{ij}\|_{ [0,t]})\right\}, \end{align}

(32)

where $Z_{ij}= [e_{ij},\eta_{ij}^{\rm T}]^{\rm T}$, $\beta_{ij}\in\mathcal{KL}$ and $\chi_{ij}\in\mathcal{K}$ for $j=$ 1,$\cdots$,$n_i$,and $\gamma_{i1}\in\mathcal{K}$. According to the discussions in Section IV,$\gamma_{i1}$ can be designed to be arbitrarily close to the identity function $\mathrm{Id}$ and $\chi_{ij}$ for $j=1,\cdots,n_i$ can be designed to be arbitrarily small.

Recall the definitions of $d_i$ in (17) and $e_{i1}$ in (22). We have

\begin{align} d_i(t)=\begin{cases}-\dfrac{1}{N_i+1}\sum\limits_{k\in\mathcal{N}_i}e_{k1}(t),&i\in\mathcal{L}, \\ [4mm] -\dfrac{1}{N_i}\sum\limits_{k\in\mathcal{N}_i}e_{k1}(t),&i\in\mathcal{N}\backslash\mathcal{L},\end{cases} \end{align}

(33)

for all $t\geq 0$. Note that for any constants $a_1,\cdots,a_n>0$ satisfying $\sum_{i=1}^{n}\frac{1}{a_i}\leq n$,it holds that $\sum_{i=1}^{n}d_i=$ $\sum_{i=1}^{n}\frac{1}{a_i}a_id_i$ $\leq$ $n\max_{1\leq i\leq n}\{a_id_i\}$ for all $d_1,\cdots,d_n\geq 0$. Thus,we have

\begin{align} |d_i(t)|\leq\delta_i\max_{k\in\mathcal{N}_i}\{a_{ik}|e_{k1}(t)|\}, \end{align}

(34)

for all $t\geq 0$,where $\delta_i={N_i}/{(N_i+1)}$ if $i\in\mathcal{L}$,$\delta_i=1$ if $i$ $\notin$ $\mathcal{L}$,and $a_{ik}$ are positive constants satisfying

\begin{align} \sum_{k\in\mathcal{N}_i}\frac{1}{a_{ik}}\leq N_i. \end{align}

(35)

Thus,through the distributed control design,the closed-loop multi-agent system is transformed into a network of IOS subsystems. We study the condition under which the closed-loop multi-agent system satisfies the cyclic-small-gain condition. We consider two classes of simple cycles in the network:

1) the simple cycles only containing the $e_{i1}$-subsystems;

2) other simple cycles.

Along a simple cycle belonging to the second class,there is at least one $e_{ij}$-subsystem with $j\neq 1$,whose IOS gains can be designed to be arbitrarily small. Thus,the cyclic-small-gain condition can be easily satisfied for the second class of simple cycles. In the following procedure,we propose a condition on the information exchange graph $\mathcal{G}^c$ to guarantee the satisfaction of the cyclic-small-gain condition for the simple cycles of the first class.

To explicitly study the interconnections between the $e_{i1}$-subsystems,we substitute (34) into property (31). Also,we choose $\gamma_{i1}$ to be in the form of $\gamma_{i1}(s)=b_is$, where constant $b_i$ $>$ $1$ can be designed to be arbitrarily close to one. Direct calculation yields:

\begin{align} |e_{i1}(t)|\leq\max\Bigl\{&\beta_{i1}(|Z_{i1}(0)|,t), b_i\delta_i\max_{k\in\mathcal{N}_i}\left\{a_{ik}\|e_{k1}\|_{ [0,t]}\right\},\nonumber\\ &\chi_{i1}(\|z_{i1}\|_{ [0,t]})\Bigr\}. \end{align}

(36)

It can be observed that the interconnection topology of the $e_{i1}$-subsystems is in accordance with the information exchange topology,represented by digraph $\mathcal{G}^c$. For $i\in\mathcal{N}$,$k$ $\in$ $\mathcal{N}_i$,we assign the positive value $a_{ik}$ to edge $(k,i)$ in $\mathcal{G}^c$. Denote $\mathcal{C}$ as the set of all simple loops in $\mathcal{G}^c$ and $\mathcal{C}_{\mathcal{L}}$ as the set of all simple loops through the vertices belonging to $\mathcal{L}$. Denote $A_{\mathcal{O}}$ as the product of the positive values assigned to the edges of the loop $\mathcal{O}\in\mathcal{C}$.

Note that $b_i$ can be designed to be arbitrarily close to $1$. According to ^{[27, 29]},we have the following cyclic-small-gain condition for the interconnected $e_{i1}$-subsystems:

\begin{align} &A_{\mathcal{O}}\frac{N}{N+1}<1,\mathcal{O}\in\mathcal{C}_{\mathcal{L}}, \end{align}

(37)

\begin{align} &A_{\mathcal{O}}<1, \mathcal{O}\in\mathcal{C}\backslash\mathcal{C}_{\mathcal{L}}. \end{align}

(38)

Lemma 1 presents a small-gain result in digraphs which leads to a condition on the structure of the information exchange digraph $\mathcal{G}^c$ for the existence of the $a_{ik}$'s to satisfy (37) and (38). The proof of Lemma 1 can be found in ^[34].

${\bf Lemma 1.}$ For digraph $\mathcal{G}^c$,each edge $(j,i)$ is assigned a positive variable $a_{ij}$. Denote $A_{\mathcal{O}}$ as the product of the positive values assigned to the edges of a simple loop $\mathcal{O}$. For $i$ $\in$ $\mathcal{N}$,denote $\mathcal{C}(i)$ as the set of simple loops of $\mathcal{G}^c$ through vertex $i$. If $\mathcal{G}^c$ has a spanning tree $\mathcal{T}$ with vertices $i_1^*$,$\cdots$,$i_q^*$ as the roots, then for any $\epsilon>0$,there exist $a_{ij}>0$ for $i$ $\in$ $\mathcal{N}$,$j\in\mathcal{N}_i$,such that

\begin{align} &\sum_{j\in\mathcal{N}_i}\frac{1}{a_{ij}}< N_i,i\in\mathcal{N}, \end{align}

(39)

\begin{align} &A_{\mathcal{O}}<1+\epsilon,\mathcal{O}\in\mathcal{C}({i_1}\ast)\cup\cdots\cup\mathcal{C}({i_q}\ast), \end{align}

(40)

\begin{align} &A_{\mathcal{O}}<1,\mathcal{O}\in\left(\bigcup_{i\in\mathcal{N}}\mathcal{C}(i)\right)\backslash \left(\mathcal{C}({i_1}\ast)\cup\cdots\cup\mathcal{C}({i_q}\ast)\right). \end{align}

(41)

Based on Lemma 1,if the information exchange digraph $\mathcal{G}^c$ has a spanning tree with (some of) the informed agents as the roots,then the closed-loop distributed system satisfies the cyclic-small-gain condition. The main result of the paper is given by Theorem 1.

${\bf Theorem 1.}$ Consider the multi-agent system in the form of (1)$\sim$(3) satisfying Assumptions 1 and 3. If there is at least one informed agent,i.e.,$\mathcal{L}\neq\emptyset$,and the communication digraph $\mathcal{G}^c$ has a spanning tree with the informed agents as the roots,then we can design distributed control laws defined by (18)$\sim$(23) such that all the signals in the closed-loop multi-agent system are bounded,and the output $y_i$ of each agent $i$ can be steered to within an arbitrarily small neighborhood of the desired agreement value $y_0$. Moreover,if $w_i=0$ for $i\in\mathcal{N}$,then each output $y_i$ asymptotically converges to $y_0$.

${\bf Proof.}$ If $\mathcal{G}^c$ has a spanning tree with vertices belonging to $\mathcal{L}$ as the roots,then according to Lemma 1, there exist positive constants $a_{ik}$ satisfying (35),(37) and (38). Then,with the cyclic-small-gain theorem in ^[27],the closed-loop distributed system is UO and IOS with $w_i$ as the inputs and $e_{i1}$'s as the outputs. With Assumption 2,the external disturbances $w_i$ are bounded. The boundedness of the signals of the closed-loop distributed system can be directly verified under Assumption 2.

By designing the IOS gains $\chi_i$ arbitrarily small (this can be done by using the technique proposed in Section III),the influence of the external disturbances $w_i$ can be made arbitrarily small, and under Assumption 2,the $e_{i1}$'s can be driven to within arbitrarily small neighborhoods of the origin. Recall $e_{i1}=y_i-y_0$. As a result,each $y_i$ can be driven to within an arbitrarily small neighborhood of $y_0$. In the case of $w_i=0$ for $i\in\mathcal{N}$,it can be proved that the closed-loop multi-agent system is globally asymptotically stable at the origin ^[35] and each output $y_i$ asymptotically converges to $y_0$.

${\bf Remark 3.}$ The proposed design is capable of dealing with both the uncertainties and external disturbances in the system dynamics. No global Lipschitz condition on the system dynamics is assumed. The main result seems to be new even if the agents are with first-order dynamics. Moreover,as shown in the following section,the proposed distributed control strategy is also robust with respect to time-delays and disturbances in information exchange. In this paper,we focus on the ultimate achievement of output agreement,while the converging rate of the closed-loop distributed system is also of interest. Based on IOS cyclic-small-gain methods,it is possible to employ the $\mathcal{KL}$ functions to represent the converging rates of the subsystems and the closed-loop distributed system.

V. ROBUSTNESS TO TIME-DELAYS AND DISTURBANCES IN INFORMATION EXCHANGE

In this section,we discuss the influence of time-delays and disturbances in information exchange separately. It should be noted that the design is still valid when the discussions are combined for the complex case with the coexistence of time-delays and disturbances in information exchange.

A. Communication Delays

If there are communication delays,$y_i^m$ defined in (16) should be modified as

\begin{align} y_i^m(t)=\begin{cases}\dfrac{1}{N_i+1}\sum\limits_{k\in\mathcal{N}_i} (y_k(t-\tau_{ik}(t))+y_0),&i\in\mathcal{L},\\ [4mm] \dfrac{1}{N_i}\sum\limits_{k\in\mathcal{N}_i}y_k (t-\tau_{ik}(t)),&i\in\mathcal{N}\backslash\mathcal{L},\end{cases} \end{align}

(42)

for $t\geq 0$,where $\tau_{ik}:{\bf R}_+\rightarrow{\bf R}_+$ represents non-constant time-delays of exchanged information.

In this case,$y_i^m(t)$ can still be written in the form of $y_i^m(t)$ $=$ $y_0-d_i(t)$ with

\begin{align} d_i(t)=\begin{cases}-\dfrac{1}{N_i+1}\sum\limits_{k\in\mathcal{N}_i}e_{k1} (t-\tau_{ik}(t)),&i\in\mathcal{L},\\ [4mm] -\dfrac{1}{N_i}\sum\limits_{k\in\mathcal{N}_i}e_{k1} (t-\tau_{ik}(t)),&i\in\mathcal{N}\backslash\mathcal{L},\end{cases} \end{align}

(43)

for $t\geq 0$,which correspond to (33).

We assume that there exists a $\bar{\tau}\geq 0$ such that,for $i\in\mathcal{N}$,$k$ $\in$ $\mathcal{N}_i$, $\tau_{ik}(t)\leq\bar{\tau}$ holds for all $t\geq 0$. Due to the time-delay,the critical IOS property (36) for the $e_{i1}$-subsystem should be modified as

\begin{align} |e_{i1}(t)|\leq\max\Bigl\{&\beta_{i1}(|Z_{i1}(0)|,t),b_i\delta_i\max_{k\in\mathcal{N}_i}\left\{a_{ik}\|e_{k1}\|_{ [-\bar{\tau},\infty)}\right\},\nonumber\\ &\chi_{i1}(\|z_{i1}\|_{ [0,\infty)})\Bigr\}. \end{align}

(44)

Such modification does not affect the validity of the cyclic-small-gain conditions (37) and (38). By using the time-delay version of the cyclic-small-gain theorem in ^{[29, 36]},we can still guarantee the IOS of the closed-loop multi-agent system with the $e_{i1}$'s as the outputs and the $w_i$'s as the inputs. The main result presented in Theorem 1 can still be proved to be valid by following a similar analysis as for the proof of Theorem 1.

B. Disturbances in Information Exchange

In the distributed control system,the information exchanged between the distributed controllers is used for feedback control. If the exchanged information is disturbed,then measurement feedback control issues should be well handled. Thanks to the small-gain design,in this subsection,we show that our distributed control design is also robust with respect to the disturbances in information exchange.

For each $i\in\mathcal{N}$,$k\in\mathcal{N}_i$,we use $\lambda_{ik}$ to represent the bounded time-varying disturbance acting on the signal $y_k$ which is transmitted to agent $i$. Then, the $y_i^m$ defined in (16) should be modified as

\begin{align} y_i^m(t)=\begin{cases}\dfrac{1}{N_i+1}\sum\limits_{k\in\mathcal{N}_i}(y_k(t)+ \lambda_{ik}(t)+y_0),&i\in\mathcal{L},\\ [4mm] \dfrac{1}{N_i}\sum\limits_{k\in\mathcal{N}_i}(y_k(t)+ \lambda_{ik}(t)),&i\in\mathcal{N}\backslash\mathcal{L},\end{cases} \end{align}

(45)

for $t\geq 0$.

In this case,for the small-gain synthesis,we still rewrite $y_i^m(t)$ in the form of $y_i^m(t)=y_0-d_i(t)$ with

\begin{align} d_i(t)=\begin{cases}-\dfrac{1}{N_i+1}\sum\limits_{k\in\mathcal{N}_i}(e_{k1}(t)+ \lambda_{ik}(t)),&i\in\mathcal{L},\\ [4mm] -\dfrac{1}{N_i}\sum\limits_{k\in\mathcal{N}_i}(e_{k1}(t)+ \lambda_{ik}(t)),&i\in\mathcal{N}\backslash\mathcal{L},\end{cases} \end{align}

(46)

for $t\geq 0$,which correspond to (33).

Note that for any constant $c_{ik}>0$,

\begin{align} |e_{k1}+\lambda_{ik}|\leq&\ |e_{k1}|+|\lambda_{ik}|\leq \nonumber\\ &\ \max\left\{(1+c_{ik})|e_{k1}|,\frac{1+c_{ik}}{c_{ik}}|\lambda_{ik}|\right\}. \end{align}

(47)

Thus,in the presence of disturbances in information exchange, property (34) can be modified as

\begin{align} |d_i(t)|\leq\delta_i\max_{k\in\mathcal{N}_i}\left \{a_{ik}(1+c_{ik})|e_{k1}(t)|,a_{ik}\frac{1+c_{ik}}{c_{ik}}|\lambda_{ik}(t)|\right\}, \end{align}

(48)

where constants $\delta_i$ and $a_{ik}$ are defined as for (34),and constant $c_{ik}$ should be positive. Thus,the critical IOS property (36) for the $e_{i1}$-subsystem can be modified as

\begin{align} |e_{i1}(t)|\leq\max\biggl\{&\beta_{i1}(|Z_{i1}(0)|,t),\nonumber\\ &b_i\delta_i\max_{k\in\mathcal{N}_i}\left\{a_{ik}(1+c_{ik})\|e_{k1}\|_{ [0,t]}\right\},\nonumber\\ &b_i\delta_i\max_{k\in\mathcal{N}_i}\left\{a_{ik}\frac{1+c_{ik}}{c_{ik}}\|\lambda_{ik}\|_{ [0,t]}\right\},\nonumber\\ &\chi_{i1}(\|z_{i1}\|_{ [0,t]})\biggr\}. \end{align}

(49)

The cyclic-small-gain conditions (37) and (38) is still valid if we choose the constants $c_{ik}$ which are small enough. In this case, the closed-loop multi-agent system is IOS with the $w_i$'s and $\lambda_{ik}$'s as the inputs and the $e_{i1}$'s as the outputs.

${\bf Remark 4.}$ An advantage of the small-gain design is that although the communication delay and the disturbance in information exchange are assumed to be bounded,no a priori knowledge on the bounds is needed for the control design and the bounds can be arbitrarily large.

VI. AN EXAMPLE

In this section,we employ an example to show the effectiveness of the cyclic-small-gain approach to distributed control.

Consider a multi-agent system composed of three first-order agents:

\begin{align} \dot{x}_i=u_i+\Delta_i(x_i), \end{align}

(50)

for $i=1,2,3$,where $x_i\in{\bf R}$ is the state,$u_i\in{\bf R}$ is the control input,and $\Delta_i:{\bf R}\rightarrow{\bf R}$ is an unknown,locally Lipschitz function. In this simple case,state $x_i$ is also the output of agent $i$. The objective is to design distributed control laws for the agents to steer the outputs to the agreement value $y_0$. Suppose that there exists a nonempty set $\Omega\subseteq{\bf R}$ such that $y_0\in\Omega$ and for each $i=1,2,3$,

\begin{align} |\Delta_i(x_i)-\Delta_i(a_i)|\leq\psi_{\Delta_i}(|x_i-a_i|) \end{align}

(51)

holds for all $x_i\in{\bf R}$ and all $a_i\in\Omega$. This assumption corresponds to Assumption 1 for systems in the general form.

For control design,we first define $e_i=x_i-y_0$. Then,

\begin{align} \dot{e}_i=&\ u_i+\Delta(e_i+y_0)=\nonumber\\ &\ u_i+(\Delta(e_i+y_0)-\Delta(y_0))+\Delta(y_0), \end{align}

(52)

where the term $(\Delta(e_i+y_0)-\Delta(y_0))$ satisfies $|\Delta(e_i+y_0)-\Delta(y_0)|\leq\psi_{\Delta_i}(|e_i|)$. To deal with the uncertain term $\Delta(y_0)$,we employ a dynamic compensator

\begin{align} \dot{u}_i=v_i, \end{align}

(53)

and consider $v_i$ as the new control input of agent $i$. We now design a control law by using $e_i+d_i$ such that each controlled agent is IOS with $d_i$ as the input and $e_i$ as the output. Then, following the discussions in Section IV,we can realize distributed control. The design procedure below is based on the method proposed in ^[32] based on the gain assignment technique. See,e.g., ^{[25, 37, 38]} for the development of the gain assignment technique.

For the system composed of (52) and (53),we first design a nonlinear observer:

\begin{align} &\dot{\xi}_{i1}=\xi_{i2}+L\xi_{i1}+\pi_{i1}(\xi_{i1}-e_i-d_i), \end{align}

(54)

\begin{align} &\dot{\xi}_{i2}=v_i-L(\xi_{i2}+L\xi_{i1}), \end{align}

(55)

where $\rho_{i1}:{\bf R}\rightarrow{\bf R}$ is an odd,strictly decreasing function,$L$ is a positive constant,variables $\xi_{i1}$ and $\xi_{i2}$ are used to estimate $e_i$ and $u_i+\Delta_i(y_0)-Le_i$,respectively.

Define $\tilde{\xi}_{i1}=e_i-\xi_{i1}$ and $\tilde{\xi}_{i2}=u_i+\Delta_i(y_0)-Le_i-\xi_{i2}$ as the estimation errors. Then,direct calculation yields:

\begin{align} &\dot{\tilde{\xi}}_{i1}=\pi_{i1}(\tilde{\xi}_{i1}+d_i)+ L\tilde{\xi}_{i1}+\tilde{\xi}_{i2}+(\Delta_i(e_i+y_0)-\Delta_i(y_0)), \end{align}

(56)

\begin{align} &\dot{\tilde{\xi}}_{i2}=-L(\tilde{\xi}_{i2}+ \Delta_i(e_i+y_0)-\Delta_i(y_0)+L\tilde{\xi}_{i1}), \end{align}

(57)

\begin{align} &\dot{\xi}_{i1}=\xi_{i2}+L\xi_{i1}-\pi_{i1}(\tilde{\xi}_{i1}+d_i), \end{align}

(58)

\begin{align} &\dot{\xi}_{i2}=v_i-L(\xi_{i2}+L\xi_{i1}). \end{align}

(59)

In system (56)$\sim$(59),the $(\xi_{i1},\xi_{i2})$-subsystem is in the well known strict-feedback form ^[21] involving uncertainties and the $(\tilde{\xi}_{i1},\tilde{\xi}_{i2})$-subsystem can be considered as dynamic uncertainty. The stabilization of such system can be solved by using the small-gain design technique proposed in ^[37]. Moreover, we can design a control law such that the closed-loop system is IOS with $d_i$ as the input and $e_i=\xi_{i1}+\tilde{\xi}_{i1}$ as the output. Specific discussions in the context of distributed control can be found in ^[32].

The control law for system (56)$\sim$(59) is designed as

\begin{align} v_i=L(\xi_{i2}+L\xi_{i1})+\pi_{i3}(\xi_{i2}-\pi_{i2}(\xi_{i1})), \end{align}

(60)

where $\pi_{i2},\pi_{i3}:{\bf R}\rightarrow{\bf R}$ are odd, strictly decreasing functions. Basically,the term $L(\xi_{i2}+L\xi_{i1})$ is used to cancel the same term in (59) and the term $\pi_{i3}(\xi_{i2}-\pi_{i2}(\xi_{i1}))$ is based on the small-gain design. The closed-loop system composed of (56)$\sim$(60) has the following IOS property:

\begin{align} |e_i(t)|\leq\max\left\{\beta_i(|Z_i(0)|,t),\gamma_i(\|d_i\|_{ [0,t]})\right\}, \end{align}

(61)

where $Z_i= [\tilde{\xi}_{i1},\tilde{\xi}_{i2},\xi_{i1},\xi_{i2}]^{\rm T}$, and $\gamma_i\in\mathcal{K}$ can be designed to be arbitrarily close to $\mathrm{Id}$. Property (61) corresponds to property (31) for systems in the general form.

In this example,we consider the case with $\Omega$= ^{[-1, 1]} and the system dynamics satisfying (51) with $\psi_{\Delta_i}(s)=$ $0.5(s+s^2)$ for $s\in{\bf R}_+$. Also,the information exchange digraph is assumed as: $\mathcal{N}_1=\{3\}$,$\mathcal{N}_2=\{1\}$, and $\mathcal{N}_3=\{2\}$. Agent $1$ is the only informed agent, i.e.,$\mathcal{L}=\{1\}$. Then,according to (34),$d_i$ satisfies

\begin{align} |d_i(t)|\leq\delta_i\max_{k\in\mathcal{N}_i}\{a_{ik}|e_k(t)|\}, \end{align}

(62)

where $\delta_1=1/2$,$\delta_2=\delta_3=1$,and the positive constants $a_{13}$,$a_{21}$,$a_{32}$ $\geq$ $1$. By designing $\gamma_i$ to be in the form of $\gamma_i(s)$ $=$ $b_is$ with $b_i>1$,we have

\begin{align} |e_i(t)|\leq\max\left\{\beta_i(|Z_i(0)|,t),b_i\delta_i\max_{k\in\mathcal{N}_i} \left\{a_{ik}\|e_k\|_{ [0,t]}\right\}\right\}. \end{align}

(63)

The cyclic-small-gain condition for this system is

\begin{align} b_1\delta_1a_{13}b_3\delta_3a_{32}b_2\delta_2a_{21}<1. \end{align}

(64)

By using $\delta_1=1/2$,$\delta_2=\delta_3=1$ and choosing $a_{13},a_{21},a_{32}=1$,the condition is reduced to

\begin{align} b_1b_2b_3<2. \end{align}

(65)

To satisfy condition (65),we choose $L=0.8$ and

\begin{align} &\pi_{i1}(r)=-5(1+|r|)r, \end{align}

(66)

\begin{align} &\pi_{i2}(r)=-7(1+|r|)r, \end{align}

(67)

\begin{align} &\pi_{i3}(r)=-10(1+|r|+|r|^2)r, \end{align}

(68)

for $r\in{\bf R}$.

We employ a simulation to demonstrate the validity of the theoretical design. In the simulation,the dynamics of the agents are chosen as: $\Delta_1(x_1)=0.25x_1^2$,$\Delta_2(x_2)=$ $0.15x_2^2$ $+$ $0.1\sin(x_2)$ and $\Delta_3(x_3)=0.15x_3^2+0.1x_3$. The desired agreement value is chosen as $y=1$. Figs.1 and 2 show the state trajectories and the control signals of the three agents with initial states $x_1(0)=-1$,$x_2(0)=2$ and $x_3(0)=3$.

VII. CONCLUSIONS

This paper has presented a cyclic-small-gain approach to robust distributed control of nonlinear multi-agent systems. With the novel distributed control law,the closed-loop multi-agent system is rendered to be IOS with the external disturbances as the inputs. Asymptotic output agreement can be achieved if the system is disturbance-free. The robustness with respect to bounded time-delays and disturbances in information exchange has also been studied. Future research directions may include distributed control with time-varying/switching information exchange topology and the applications to multi-vehicle systems. Another line of future research is to refine the current design when the agreement value is time-varying.