2. Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, New York NY 11201, USA
Distributed control for group coordination of multiagent 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 multiagent 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 multiagent 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 multiagent 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 strictfeedback 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 cyclicsmallgain approach to robust distributed controller design for nonlinear multiagent systems. More precisely,we will use the notions of inputtostate stability (ISS) ^{[23, 24]} and inputtooutput stability (IOS) ^{[25, 26]} to describe the dynamic interaction between the controlled agents,and use the recently developed cyclicsmallgain 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 cyclicsmallgain theorems for networks of nonlinear systems,and ^{[25, 26, 27, 28, 29, 30]} for the original smallgain 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 closedloop multiagent 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 timedelays and disturbances in information exchange. Section VII contains some concluding remarks.
To make the paper selfcontained,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 strictfeedback 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) 
For distributed control of the multiagent 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 $y_iy_k$ 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 multiagent 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) 
${\bf Remark 1.}$ 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 INGREDIENTThe main result in this paper is based on a design ingredient for measurement feedback control of a class of firstorder nonlinear uncertain systems.
Consider a nonlinear system
\begin{align} \dot{\xi}=\nu+\phi(\xi,\omega), \end{align}  (5) 
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) 
${\bf Remark 2.}$ 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 inputtostate
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) 
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) 
\[\dot{\zeta }=\theta (\xi +\delta ,\zeta ),\]  (12) 
\[\mu =\kappa (\zeta ),\]  (13) 
\[Z(t)\le {{\alpha }^{\text{UO}}}({{Z}_{0}}+\\delta {{\}_{[0,t]}}+\\omega {{\}_{[0,t]}})+D_{0}^{\text{UO}},\]  (14) 
\[\xi (t)\le \max \left\{ \beta ({{Z}_{0}},t),\gamma (\\delta {{\}_{[0,t]}}),\chi (\\omega {{\}_{[0,t]}}) \right\}\]  (15) 
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 DESIGNIn 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 cyclicsmallgain 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) 
\begin{align} d_i=y_0y_i^m, \end{align}  (17) 
The control law for each agent $i$ is in the form of
\begin{align} u_i=x_{i(n_i+1)}^*, \end{align}  (18) 
\begin{align} x_{ij}^*=\varphi_{i(j1)}(\eta_{i(j1)}),j=2,\cdots,n_i+1, \end{align}  (19) 
The variables $\eta_{i(j1)}$ 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(j1)}=\rho_{i(j1)}(\eta_{i(j1)},e_{i(j1)}),j=3,\cdots,n_i+1, \end{align}  (21) 
\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) 
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) 
\begin{align} \dot{e}_{i1}=x_{i2}^*+\bar{\Delta}_{i1}(e_{i1},z_{i1}). \end{align}  (25) 
Suppose that the $e_{i(j1)}$system is in the form of
\begin{align} \dot{e}_{i(j1)}=x_{ij}^*+\bar{\Delta}_{i(j1)}(e_{i(j1)},z_{i(j1)}), \end{align}  (26) 
\begin{align} \dot{e}_{ij}=&\ \dot{x}_{ij}\frac{\partial\varphi_{i(j1)}(\eta_{i(j1)})} {\partial\eta_{i(j1)}}\dot{\eta}_{i(j1)}=\nonumber\\ &\ \dot{x}_{ij}\frac{\partial\varphi_{i(j1)}(\eta_{i(j1)})}{\partial\eta_{i(j1)}} \rho_{i(j1)}(\eta_{i(j1)},e_{i(j1)})=\nonumber\\ &\ x_{i(j+1)}+\Delta_{ij}(\bar{x}_{ij},w_i)\nonumber\\ &\ \frac{\partial\varphi_{i(j1)}(\eta_{i(j1)})}{\partial\eta_{i(j1)}}\rho_{i(j1)}(\eta_{i(j1)},e_{i(j1)}), \end{align}  (27) 
\begin{align} \dot{e}_{ij}=x_{i(j+1)}^*+\bar{\Delta}_{ij}(e_{ij},z_{ij}), \end{align}  (28) 
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) 
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) 
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) 
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) 
\begin{align} d_i(t)\leq\delta_i\max_{k\in\mathcal{N}_i}\{a_{ik}e_{k1}(t)\}, \end{align}  (34) 
\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 closedloop multiagent system is transformed into a network of IOS subsystems. We study the condition under which the closedloop multiagent system satisfies the cyclicsmallgain 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 cyclicsmallgain 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 cyclicsmallgain 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 cyclicsmallgain
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 smallgain 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 closedloop distributed system satisfies the cyclicsmallgain condition. The main result of the paper is given by Theorem 1.
${\bf Theorem 1.}$ Consider the multiagent 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 closedloop multiagent 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 cyclicsmallgain theorem in ^{[27]},the closedloop 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 closedloop 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 $e_{i1}=y_iy_0$. 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 closedloop multiagent 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 firstorder dynamics. Moreover,as shown in the following section,the proposed distributed control strategy is also robust with respect to timedelays and disturbances in information exchange. In this paper,we focus on the ultimate achievement of output agreement,while the converging rate of the closedloop distributed system is also of interest. Based on IOS cyclicsmallgain methods,it is possible to employ the $\mathcal{KL}$ functions to represent the converging rates of the subsystems and the closedloop distributed system.
V. ROBUSTNESS TO TIMEDELAYS AND DISTURBANCES IN INFORMATION EXCHANGEIn this section,we discuss the influence of timedelays 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 timedelays 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) 
In this case,$y_i^m(t)$ can still be written in the form of
$y_i^m(t)$ $=$ $y_0d_i(t)$ 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) 
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
timedelay,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) 
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 smallgain 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 timevarying 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) 
In this case,for the smallgain synthesis,we still rewrite
$y_i^m(t)$ in the form of $y_i^m(t)=y_0d_i(t)$ 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) 
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) 
\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) 
\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) 
${\bf Remark 4.}$ An advantage of the smallgain 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 EXAMPLEIn this section,we employ an example to show the effectiveness of the cyclicsmallgain approach to distributed control.
Consider a multiagent system composed of three firstorder
agents:
\begin{align} \dot{x}_i=u_i+\Delta_i(x_i), \end{align}  (50) 
\begin{align} \Delta_i(x_i)\Delta_i(a_i)\leq\psi_{\Delta_i}(x_ia_i) \end{align}  (51) 
For control design,we first define $e_i=x_iy_0$. 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) 
\begin{align} \dot{u}_i=v_i, \end{align}  (53) 
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_id_i), \end{align}  (54) 
\begin{align} &\dot{\xi}_{i2}=v_iL(\xi_{i2}+L\xi_{i1}), \end{align}  (55) 
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_iL(\xi_{i2}+L\xi_{i1}). \end{align}  (59) 
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) 
\begin{align} e_i(t)\leq\max\left\{\beta_i(Z_i(0),t),\gamma_i(\d_i\_{ [0,t]})\right\}, \end{align}  (61) 
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) 
\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 cyclicsmallgain 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) 
\begin{align} b_1b_2b_3<2. \end{align}  (65) 
\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) 
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 $x_3(0)=3$.
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Fig. 1.The state trajectories of the agents. 
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Fig. 2.The control signals of the agents. 
This paper has presented a cyclicsmallgain approach to robust distributed control of nonlinear multiagent systems. With the novel distributed control law,the closedloop multiagent system is rendered to be IOS with the external disturbances as the inputs. Asymptotic output agreement can be achieved if the system is disturbancefree. The robustness with respect to bounded timedelays and disturbances in information exchange has also been studied. Future research directions may include distributed control with timevarying/switching information exchange topology and the applications to multivehicle systems. Another line of future research is to refine the current design when the agreement value is timevarying.
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