A Distributed Adaptive Scheme for Multi-Agent Systems

TL;DR

This paper introduces a novel distributed adaptive control scheme for multi-agent systems that achieves asymptotic consensus without relying on the gradient of Lyapunov functions, addressing limitations of traditional adaptive control in distributed settings.

Contribution

It proposes a new adaptive scheme that works for general multi-agent systems and does not depend on gradient-based Lyapunov functions, enabling distributed implementation.

Findings

Achieves asymptotic consensus in second-order uncertain multi-agent systems.

Works on directed graph networks.

Does not rely on gradient of Lyapunov functions.

Abstract

In traditional adaptive control, the certainty equivalence principle suggests a two-step design scheme. A controller is first designed for the ideal situation assuming the uncertain parameter was known and it renders a Lyapunov function. Then, the uncertain parameter in the controller is replaced by its estimation that is updated by an adaptive law along the gradient of Lyapunov function. This principle does not generally work for a multi-agent system as an adaptive law based on the gradient of (centrally constructed) Lyapunov function cannot be implemented in a distributed fashion, except for limited situations. In this paper, we propose a novel distributed adaptive scheme, not relying on gradient of Lyapunov function, for general multi-agent systems. In this scheme, asymptotic consensus of a second-order uncertain multi-agent system is achieved in a network of directed graph.

Equations

$${\dot{x}}_i=f_i(x),i=1,\cdots ,n,$$

$$\frac{\partial V(x)}{\partial x}f(x)\le -\alpha (\Vert x{\Vert }_R)$$

$$\frac{{\Vert \frac{\partial V(x)}{\partial x}\Vert }^2}{\alpha (\Vert x{\Vert }_R)}\le \sigma$$

$${\dot{x}}_i=f_i(x)+g_i(x_i,w_i,{\mu }_i),i=1,\cdots ,n$$

$$g_i(x_i,w_i,{\mu }_i)=h_i(x_i)(w_i-{\mu }_i).$$

$$\dot{x}=f(x)+H(x)(w-\mu )$$

$$\mu$$

$${\dot{\hat{w}}}^{\text{mbox}\mathsf{T}}$$

$$U(x,\tilde{w})=V(x)+{\tilde{w}}^{\text{mbox}\mathsf{T}}\tilde{w}/(2\lambda )$$

$$\dot{U}(x,\tilde{w})\le -\alpha (\Vert x{\Vert }_R),$$

$$\dot{V}(x)=\frac{\partial V(x)}{\partial x}f(x)-\frac{\partial V(x)}{\partial x}H(x)\tilde{w}$$

$$\le -\alpha (\Vert x{\Vert }_R)-\frac{\partial V(x)}{\partial x}H(x)\tilde{w}.$$

$$\dot{U}(x,\tilde{w})\le -\alpha (\Vert x{\Vert }_R)-\frac{\partial V(x)}{\partial x}H(x)\tilde{w}+{\dot{\tilde{w}}}^{\text{mbox}\mathsf{T}}\tilde{w}/\lambda$$

$$=-\alpha (\Vert x{\Vert }_R)$$

$${\mu }_i$$

$${\dot{\hat{w}}}_i^{\text{mbox}\mathsf{T}}$$

$${\dot{\hat{w}}}_i$$

$$\displaystyle L=\left[\begin{array}[]{cc}0&0_{1\times(n-1)}\\ -b&L_{o}+B\end{array}\right]$$

$$2Q=P(L_o+B)+(L_o+B{)}^{\text{mbox}\mathsf{T}}P>0.$$

$$\displaystyle R=\left[\begin{array}[]{cc}-b&L_{o}+B\end{array}\right],$$

$$RL=(L_o+B)R.$$

$$\dot{V}(x)=-x^{\text{mbox}\mathsf{T}}{R}^{\text{mbox}\mathsf{T}}[P(L_o+B)+(L_o+B{)}^{\text{mbox}\mathsf{T}}P]Rx/2\le -x^{\text{mbox}\mathsf{T}}{R}^{\text{mbox}\mathsf{T}}QRx.$$

$$\dot{U}(x)\le -x^{\text{mbox}\mathsf{T}}{R}^{\text{mbox}\mathsf{T}}QRx+[{\dot{\hat{w}}}^{\text{mbox}\mathsf{T}}/\lambda -x^{\text{mbox}\mathsf{T}}{R}^{\text{mbox}\mathsf{T}}PRH(x)]\tilde{w}.$$

$$\dot{\hat{w}}=\lambda H^{\text{mbox}\mathsf{T}}(x)R^{\text{mbox}\mathsf{T}}PRx$$

$$\displaystyle RH(x)=\left[\begin{array}[]{cc}-b&L_{o}+B\end{array}\right]\mbox{diag}\left[\begin{array}[]{cccc}0&h_{2}(x_{2})&\cdots&h_{n}(x_{n})\end{array}\right]$$

$$=(L_o+B)\bar{H}(x)=(D+B)\bar{H}(x)-E\bar{H}(x)$$

$$\dot{\hat{w}}=\lambda {\bar{H}}^{\text{mbox}\mathsf{T}}(x)(D+B)PRx-\lambda \kappa \hat{w}$$

$$\dot{U}(x)\le -x^{\text{mbox}\mathsf{T}}{R}^{\text{mbox}\mathsf{T}}QRx+[-\kappa {\hat{w}}^{\text{mbox}\mathsf{T}}+x^{\text{mbox}\mathsf{T}}{R}^{\text{mbox}\mathsf{T}}PE\bar{H}(x)]\tilde{w}$$

$$\le \left[-x^{\text{mbox}\mathsf{T}}{R}^{\text{mbox}\mathsf{T}}QRx+x^{\text{mbox}\mathsf{T}}{R}^{\text{mbox}\mathsf{T}}PE\bar{H}(x)\tilde{w}-\kappa \Vert \tilde{w}{\Vert }^2\right]+\kappa \Vert w\Vert \Vert \tilde{w}\Vert .$$

$${\dot{\hat{w}}}_i=\lambda (d_i+b_i)p_i{h}_i^{\text{mbox}\mathsf{T}}(x)L_{i}x-\lambda \kappa {\hat{w}}_i,,i=2,\cdots ,n,$$

$${\dot{\tau }}_i$$

$${\dot{w}}_i$$

$${\mu }_{i,ss}$$

Full text

I Introduction

Control of multi-agent systems (MASs) is motivated by collective phenomena in natural systems and extensive engineering applications, for example, cooperative control of multiple unmanned aerial vehicles (UAVs) and mobile robots, distributed sensor networks, load balancing, etc. Consensus is one of the most active research topics in MASs from the systems and control perspective and it has achieved rapid progress in recent years [1]. The goal is to design collective algorithms for a group of agents such that they achieve agreement in a certain sense of obeying common dynamics. A distributed consensus control protocol can generate effective local control for each agent based on the relative measurement from its neighbors via a network.

The research for MASs of homogeneous linear dynamics is mature with the early works traced back to those on single or double integrators. For example, an observer based output feedback consensus controller was constructed in [2] with both agent outputs and observer states transmitted via network. For a lower cost network with only output transmitted, consensus protocols were studied in [3, 4]. A low-gain approach can also be found in [5] using a stable dynamic filter. More general formation control for linear dynamics can be found in, e.g., [6, 7].

In many practical situations, agent dynamics are usually subject to uncertainties that also induce heterogeneity. To handle system uncertainties, an internal model based approach has been proved to be effective. For example, linear internal model based consensus techniques can found in [8, 9, 10, 11] in different settings. The basis idea is to introduce a reference trajectory for each agent and collectively synchronize these references and hence agent outputs.

While certain nonlinearities of agent dynamics might be handled by feedforward compensation, see, e.g., [12], uncertain nonlinearities likely bring more technical challenges. Most existing results are also based on internal model design. For instance, in [13], the authors designed controllers for MASs of second-order nonlinear dynamics with agreement on a constant. More general nonlinear dynamics were studied in [14, 15] that require that all agents exchange full state information. The most sophisticated result was given in [16] in the output communication setting using a small gain theorem. Some other relevant internal model design can be found for cooperative output regulation in a leader-following setting; see, e.g. [17, 18].

Another research line is to deal with system uncertainties, in particular, unknown parameters, using adaptive control. Like in traditional adaptive control, the certainty equivalence principle suggests a two-step design scheme. A controller is first designed for the ideal situation assuming the uncertain parameter was known and it renders a Lyapunov function. Then, the uncertain parameter in the controller is replaced by its estimation that is updated by an adaptive law along the gradient of Lyapunov function.

In literature, such an adaptive control scheme has been investigated for MASs in some situations. For example, a first-order MAS was studied in [19] for a network of undirected graph. The result was presented in a more general framework in [20]. Similar adaptive technique was used in [21] for both first-order and second-order MASs with a Nussbaum gain added to deal with unknown control direction. Also for networks of undirected graphs, but under the jointly connected condition, an adaptive scheme was studied for first-order MASs in [22] and [23] for leader-following and leaderless settings, respectively. In particular, in [22] each agent requires “not only the information of its neighbors but also the information of its neighbors’ neighbors” and then in [23] the approach was improved to a purely distributed design. It is noted that adaptive control was also used to tune the coupling weights of a network in, e.g., [24].

For a network of directed graph, the associated Laplacian is asymmetric, which significantly complicates the problem. Some relevant work can be found in [25] that gave a result for higher-order MASs, but for the leader-following case. Moreover, it is noted that consensus in [25] cannot be achieved asymptotically but with a residual error. The work in [25] also considers neural network (NN) approximation for the unknown nonlinearities. The residual error is caused not only by NN approximation error, but also by the cost of distributed implementation of the adaptive law. In other words, the residual consensus error still exists even if the NN error is zero. The work in [25] includes the early results in [26, 27] as special cases.

Even though an adaptive law along the gradient of Lyapunov function using the certainty equivalence principle has been proved to be successful in the aforementioned scenarios, it does not work for MASs in general as a Lyapunov function is usually centrally constructed. In other words, distributed implementation of the gradient of Lyapunov function is usually impractical except fort limited cases. For instance, it still remains open to design a distributed adaptive law to achieve asymptotic consensus for a second-order MAS in a directed network. As will be explained in detail later in this paper, an adaptive law along the gradient of Lyapunov function has its inherent drawback to solve this open problem due to the lack of its distributed implementation.

In this paper, we propose a novel distributed adaptive scheme, not relying on gradient of Lyapunov function, for general MASs. In the gradient based scheme, the estimation error is expected to have a steady state zero. To drive the agent states together with the estimation error to their steady states, the adaptive law must follow the gradient of Lyapunov function. The novel idea is to introduce an input compensation such that the steady state of the estimation error is not zero but a manifold in the state space of agent states and estimated parameters. By proper selection of the manifold, it can be made attractive without relying the centrally designed Lyapunov function. At the manifold, the agent states also approach their desired steady state. The idea in characterizing the steady-state manifold originates from the steady-state generator design in the output regulation theory for dealing with asymptotic disturbance rejection and reference tracking [28, 29] and immersion and invariance adaptive control of nonlinearly parameterized systems [30]. Within the novel distributed adaptive scheme, the aforementioned open problem on asymptotic consensus of a second-order nonlinear MAS in a directed network is solved.

II Preliminaries and Motivating Examples

Consider a network of MAS with a properly designed controller, described by

[TABLE]

where $x_{i}\in\mathbb{R}^{l}$ is the state of the $i$-th agent and $f_{i}(x)$ is a general function representing the agent dynamics. Denote $x=[x_{1}^{\mbox{\tiny\sf T}},x_{2}^{\mbox{\tiny\sf T}},\cdots,x_{n}^{\mbox{\tiny\sf T}}]^{\mbox{\tiny\sf T}}$ and $f(x)=[f^{\mbox{\tiny\sf T}}_{1}(x),f^{\mbox{\tiny\sf T}}_{2}(x),\cdots,f_{n}^{\mbox{\tiny\sf T}}(x)]^{\mbox{\tiny\sf T}}$. So, the network has the compact form $\dot{x}=f(x)$. This is the nominal closed-loop MAS free of uncertainties. Suppose the MAS has achieved a certain consensus behavior, specifically, with a property in term of a Lyapunov-like function. Throughout the note, the notation $\|\cdot\|$ means the Euclidean norm and $\|x\|_{R}=\|Rx\|$ for a real matrix $R$.

Assumption 1

There exists a continuously differentiable function $V(x)$ satisfying $\underline{\alpha}(\|x\|_{R})\leq V(x)\leq\bar{\alpha}(\|x\|_{R})$ for a matrix $R\in{\mathbb{R}}^{\bar{n}l\times nl}$ with $\bar{n}\leq n$ and class ${\cal K}_{\infty}$ functions $\underline{\alpha}$ and $\bar{\alpha}$, such that,

[TABLE]

for a class ${\cal K}_{\infty}$ function $\alpha$. Moreover,

[TABLE]

for some positive constant $\sigma$.

•  *

Remark II.1

Two typical scenarios of Assumption 1 are explained as follows.

(i) If $R\in{\mathbb{R}}^{nl\times nl}$, i.e., $\bar{n}=n$, is a nonsingular matrix, then $\|x\|_{R}=0$ implies $\|x\|=0$. In this scenario, the function $V(x)$ is a Lyapunov function for the $x$-system and Assumption 1 implies $\lim_{t\rightarrow\infty}\|x(t)\|=0$, i.e., asymptotic stability about the equilibrium at the origin.

(ii) If $R\in\mathbb{R}^{(n-1)l\times nl}$, i.e., $\bar{n}=n-1$, is a full row rank matrix and the rows are perpendicular to ${\rm span}\{{\bf 1}\otimes I_{l}\}$ where $I_{l}\in{\mathbb{R}}^{l\times l}$ is an identity matrix and {\bf 1}=\left[\begin{array}[]{ccc}1&\ldots&1\end{array}\right]^{\mbox{\tiny\sf T}}\in\mathbb{R}^{n}, then $\|x\|_{R}=0$ implies $x={\bf 1}\otimes x_{o}$ for some $x_{o}\in\mathbb{R}^{l}$. In this scenario, the function $V(x)$ is a Lyapunov function for the $Rx$-subsystem and Assumption 1 implies $\lim_{t\rightarrow\infty}\|x(t)\|_{R}=0$, i.e., $\lim_{t\rightarrow\infty}[x(t)-{\bf 1}\otimes x_{o}(t)]=0$, which is a typical consensus phenomenon.

**

Now, we consider the network subject to uncertainties and the objective is to design an adaptive scheme to deal with the uncertainties such that the behavior of the nominal system is still maintained. The design of an adaptive law is expected to be separated from the consensus controller in the nominal system, which is not explicitly shown in the closed-loop structure (1).

Specifically, the network of MAS subject to uncertainties is represented by

[TABLE]

where $w_{i}\in\mathbb{R}^{m_{i}}$ represents constant unknown parameters and $\mu_{i}\in\mathbb{R}^{m_{i}}$ an additional control input to adaptively account for the uncertainties. Suppose the uncertainties have the linearly parameterized structure, i.e.,

[TABLE]

for some function $h_{i}(x_{i})$. We can rewrite the system in a compact form

[TABLE]

where $w=[w_{1}^{\mbox{\tiny\sf T}},w_{2}^{\mbox{\tiny\sf T}},\cdots,w_{n}^{\mbox{\tiny\sf T}}]^{\mbox{\tiny\sf T}}$, $\mu=[\mu_{1}^{\mbox{\tiny\sf T}},\mu_{2}^{\mbox{\tiny\sf T}},\cdots,\mu_{n}^{\mbox{\tiny\sf T}}]^{\mbox{\tiny\sf T}}$, and H(x)=\mbox{diag}\left[\begin{array}[]{cccc}h_{1}(x_{1})&h_{2}(x_{2})&\cdots&h_{n}(x_{n})\end{array}\right].

If the parameter $w_{i}$ were known, $\mu_{i}=w_{i}$ could trivially cancel the uncertainties $g_{i}(x_{i},w_{i},\mu_{i})$. For the practical case with an unknown $w_{i}$, an adaptive law can be designed along the gradient of the Lyapunov function $V(x)$, as summarized as follows.

Theorem II.1

(Centralized Scheme)* For the system (4) with (5) under Assumption 1, with the controller*

[TABLE]

the derivative of

[TABLE]

with $\tilde{w}=\hat{w}-w$ satisfies

[TABLE]

*along the trajectory of the closed-loop system (4)+(5)+(7).  *

Proof: Direct calculation shows that the derivative of $V(x)$ along the dynamics (4) with (5) satisfies

[TABLE]

Hence,

[TABLE]

for $\dot{\tilde{w}}=\dot{\hat{w}}$ given in (7).

The adaptive law (7) can be rewritten as follows, for $i=1,\cdots,n$,

[TABLE]

that is not always distributed as ${\partial V(x)}/{\partial x_{i}}$ depends on not only the local state of agent $i$, but also the full network state $x$ unless $V(x)$ can be properly designed to have a distributed ${\partial V(x)}/{\partial x_{i}}$ on a case by case basis. However, it can be true only for very limited cases because the function $V(x)$ for the nominal system is constructed in a centralized manner. Two motivating examples are given as follows.

Example II.1

Consider a first-order integrator MAS in a network of an undirected graph associated with a symmetric Laplacian $L$. The nominal network dynamics are $\dot{x}=-Lx$ for $x\in{\mathbb{R}}^{n}$. Let $V(x)=x^{\mbox{\tiny\sf T}}Lx/2$ where $L=R^{\mbox{\tiny\sf T}}R$ for a full row rank matrix $R\in{\mathbb{R}}^{(n-1)\times n}$ when the graph is connected. The derivative along the trajectory of $\dot{x}=-Lx$ is $\dot{V}(x)=-x^{\mbox{\tiny\sf T}}LLx=-(Rx)^{\mbox{\tiny\sf T}}(RR^{\mbox{\tiny\sf T}})(Rx)\leq-r_{\rm min}\|x\|^{2}_{R}$ where $r_{\rm min}>0$ is the minimal eigenvalue of $RR^{\mbox{\tiny\sf T}}$. When the network is subject to uncertainties $H(x)w$, i.e., $\dot{x}=-Lx+H(x)(w-\mu)$, following Theorem II.1, the additional adaptive controller $\mu$ in (7) has the specific form

[TABLE]

with $L_{i}$ the $i$-th row of $L$ and ${\cal N}_{i}$ the set of neighbors of $i$. In this scenario, the adaptive scheme is implemented in a distributed fashion. This development can be found in, e.g., [23].

•  *

Example II.2

Consider a first-order integrator MAS $\dot{x}=-Lx$ in a network of a directed graph associated with a Laplacian of the special form

[TABLE]

that represents a leader-following network with agent $1$ as the leader. The matrix $L_{o}$ is the Laplacian of the sub-network of followers and $B=\mbox{diag}(b),b=[b_{2},\cdots,b_{n}]^{\mbox{\tiny\sf T}}$ with $b_{i}\geq 0$ the weight from the leader to agent $i$. Denote $L_{o}=D-E$ where $D=\mbox{diag}(d_{2},\cdots,d_{n})$ a diagonal matrix and $E$ an off-diagonal one. Assume the network has a spanning tree with the root node being the leader node $1$. Then, there exists a diagonal matrix $P=\mbox{diag}(p_{2},\cdots,p_{n})>0$ such that

[TABLE]

Let

[TABLE]

one has

[TABLE]

Let $V(x)=x^{\mbox{\tiny\sf T}}R^{\mbox{\tiny\sf T}}PRx/2$. The derivative along the trajectory of $\dot{x}=-Lx$ is

[TABLE]

When the network is subject to uncertainties $H(x)w$, i.e., $\dot{x}=-Lx+H(x)(w-\mu)$, along which the derivative of $U(x,\tilde{w})=V(x)+\tilde{w}^{\mbox{\tiny\sf T}}\tilde{w}/(2\lambda)$ is

[TABLE]

Following Theorem II.1, the update law in (7) has the specific form

[TABLE]

that however cannot be implemented in distributed fashion. In fact, a distributed adaptive law for this scenario still remains open.

For the scenario studied in [25], the leader is free of uncertainty, i.e., $h_{1}(x)=0$ and $w\in\mathbb{R}^{0}$ trivially. Then, one has

[TABLE]

for \bar{H}(x)=\left[0,\mbox{diag}\left[\begin{array}[]{cccc}h_{2}(x_{2})&\cdots&h_{n}(x_{n})\end{array}\right]\right]. The following update law was applied

[TABLE]

that gives

[TABLE]

The update law is implemented in a distributed fashion by noting that the matrices $P,D$ and $B$ are diagonal, that is,

[TABLE]

with $L_{i}$ the $i$-th row of $L$. In the expression of $\dot{U}(x)$, the terms in the square brackets can be made negative with a sufficiently large $\kappa$ but the positive term $\kappa\|w\|\|\tilde{w}\|$ causes a residual consensus error. In other words, no asymptotic consensus can be achieved using the approach developed in [25].

•  *

III A Distributed Adaptive Scheme

The main contribution of this paper is to bring a novel adaptive scheme that can be implemented in a distributed fashion. For this purpose, let us have a close inspection on the approach in Theorem II.1. For the system (4) with linearly parameterized uncertainties, we introduce a virtual exosystem

[TABLE]

The agent state $x_{i}$ and input $\mu_{i}$ are expected to have the steady states $x_{i,ss}=\tau_{i}$ and $\mu_{i,ss}=w_{i}$, respectively. In this sense, we call

[TABLE]

the steady-state generator for the input $\mu_{i}$, which motivates the update law

[TABLE]

where $\nabla$ is designed along the gradient of Lyapunov function such that the manifold $\{(x,\mu,\tau,w)\;|\;x_{i}=\tau_{i},\;\mu_{i}=w_{i},\;i=1,\cdots,n\}$ is attractive.

The novel idea is to introduce a function $\beta_{i}(x_{i})$ to the input, i.e., $\mu_{i}=-\beta_{i}(x_{i})+\hat{\mu}_{i}$. Along the virtual exosystem (15), the agent state $x_{i}$ and input $\hat{\mu}_{i}$ are expected to have the steady states $x_{i,ss}=\tau_{i}$ and $\hat{\mu}_{i,ss}=\theta_{i}(\tau_{i},w_{i})=\beta_{i}(\tau_{i})+w_{i}$, respectively. As a result, we have a steady-state generator for the input $\hat{\mu}_{i}$

[TABLE]

that motivates the update law

[TABLE]

In this design, $\beta_{i}$ can be properly selected such that the manifold $\{(x,\hat{\mu},\tau,w)\;|\;x_{i}=\tau_{i},\;\hat{\mu}_{i}=\theta_{i}(\tau_{i},w_{i}),\;i=1,\cdots,n\}$ is attractive. The introduction of $\beta_{i}$ avoids the implementation of $\nabla$ that relies on a centrally designed Lyapunov function.

In this new development, if we treat $\hat{w}_{i}$ as the estimated value of $w_{i}$, the steady state of the estimation error $\hat{w}_{i}-w_{i}$ is not zero but $\theta_{i}(\tau_{i},w_{i})-w_{i}=\beta_{i}(\tau_{i})$ where $\tau_{i}$ is the steady state of $x_{i}$. Therefore, we aim to drive $\hat{w}_{i}-w_{i}$ to the manifold $\{(x_{i},\hat{w}_{i})\;|\;\hat{w}_{i}-w_{i}=\beta_{i}(x_{i}),\;i=1,\cdots,n\}$ in the space of agent states and estimated parameters. By proper selection of the manifold, it can be made attractive and the agent state $x_{i}$ can approach its desired steady state $\tau_{i}$ on the manifold. The rigorous formulation of the approach is given in the following theorem.

Theorem III.1

(Distributed Scheme)* Consider the system (4) with (5) under Assumption 1. Let the distributed controller be*

[TABLE]

where $\beta_{i}(x_{i})$ is any continuously differentiable function satisfying

[TABLE]

for some $\lambda_{i}>0$. Then, the derivative of

[TABLE]

with

[TABLE]

satisfies

[TABLE]

for any $0<k<1$, along the trajectory of the closed-loop system (4)+(5)+(16).

**

Proof: The system composed of (4)+(5)+(16) can be rewritten as

[TABLE]

Direct calculation shows

[TABLE]

For any $0<k<1$, pick $a=(1-k)/{\sigma}$. One has

[TABLE]

Moreover

[TABLE]

Next, one has

[TABLE]

Then, the derivative of $z_{i}^{\mbox{\tiny\sf T}}z_{i}/(2\lambda_{i})$ along the above trajectory is

[TABLE]

As a result, the derivative of

[TABLE]

along the trajectory of the closed-loop system, is

[TABLE]

The proof is thus completed.

Remark III.1

In Theorem III.1 the adaptive controller (16) is implemented at each agent $i$. This scheme is distributed as it only relies on the agent state $x_{i}$ and its nominal dynamics $f_{i}(x)$. The nominal dynamics $f_{i}(x)$ is implemented before hand for the ideal situation free of uncertainties, typically in distributed fashion. The effectiveness of Theorem III.1 will be demonstrated by a network of second-order uncertain dynamics in the next section.

•  *

IV Network of Second-Order Uncertain Dynamics

We consider a group of $n\geq 2$ agents governed by a set of second-order nonlinear differential equations

[TABLE]

where $p_{i},v_{i}\in\mathbb{R}$ are the states and $u_{i}\in\mathbb{R}$ is the input of the agent $i$. The function $\xi_{i}(w_{i},v_{i})=\zeta_{i}(v_{i})w_{i}$ for a bounded function $\zeta_{i}(v_{i})$ represents heterogeneous nonlinearities with $w_{i}$ an unknown constant parameter. The two parameters $\alpha_{1}$ and $\alpha_{2}$ are known. For convenience of presentation, we denote

[TABLE]

and

[TABLE]

In this section, the network topology is given by a directed graph $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$, where $\mathcal{V}=\{1,\cdots,n\}$ denotes a finite non-empty set of nodes (i.e., agents) and $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ presents the set of edges (i.e., communication links). The adjacency matrix ${\cal A}=[a_{ij}]$ of a weighted directed graph is defined as $a_{ii}=0$ (no self-loop) and $a_{ij}>0$ if $(j,i)\in\mathcal{E}$ where $i\neq j$. Let the Laplacian $L$ be defined as $L_{ii}=\sum_{j\neq i}a_{ij}$ and $L_{ij}=-a_{ij}$, where $i\neq j$. For a distributed algorithm, each agent $i$ can achieve the information from the network as follows, with $L_{i}$ the $i$-th row of $L$,

[TABLE]

In this section, we study a general directed leaderless setting that includes the leader-following case (with the Laplacian of the special form (12)) as a special case. Throughout the section, we have the following assumption.

Assumption 2

The network has a directed spanning tree.

•  *

The objective is to design a distributed adaptive consensus protocol (i.e., only $p_{i}$, $v_{i}$, $L_{i}p$ and $L_{i}v$ are available measurements for agent $i$) under Assumption 2, such that the MAS has the following asymptotic property

[TABLE]

for some time functions $p_{o}(t),v_{o}(t):[0,\infty)\mapsto\mathbb{R}$.

Under Assumption 2, the Laplacian $L$ has one zero eigenvalue and the other eigenvalues have positive real parts. Let the vectors $r\in\mathbb{R}^{n}$ and ${\bf 1}$ be the left and right eigenvectors corresponding to the eigenvalue zero of $L$, in particular, $r^{\mbox{\tiny\sf T}}L=0$, $L\mathbf{1}=0$, and $r^{\mbox{\tiny\sf T}}{\bf 1}=1$.

There exist matrices $W\in\mathbb{R}^{\left(n-1\right)\times n}$, $U\in\mathbb{R}^{n\times\left(n-1\right)}$ such that

[TABLE]

One has the following similarity transformation

[TABLE]

where $J=WLU\in\mathbb{R}^{\left(n-1\right)\times\left(n-1\right)}$ is a matrix with all eigenvalues having positive real parts. Define the matrix $R$ as follows

[TABLE]

It is easy to check that $R$ has a full row rank and the rows of $R$ are perpendicular to ${\rm span}\{{\bf 1}\otimes I_{2}\}$.

We have the following technical lemma that has been used in [31] with the proof hidden in system analysis. A direct proof on matrix analysis is given in appendix for readers’ convenience.

Lemma IV.1

*Under Assumption 2, there exist $\gamma_{1},\gamma_{2}>0$ such that the matrix \bar{A}=\left[\begin{array}[]{cc}0&I\\ \alpha_{1}I-\gamma_{1}J&\alpha_{2}I-\gamma_{2}J\end{array}\right] is Hurwitz.  *

The next lemma shows the consensus result for the ideal situation.

Lemma IV.2

Under Assumption 2, consider the system (20) with $\xi_{i}(w_{i},v_{i})=0$ and

[TABLE]

where $\gamma_{1}$ and $\gamma_{2}$ are such that the matrix \bar{A}=\left[\begin{array}[]{cc}0&I\\ \alpha_{1}I-\gamma_{1}J&\alpha_{2}I-\gamma_{2}J\end{array}\right] is Hurwitz. Let $P=P^{\mbox{\tiny\sf T}}>0$ be the solution to the Lyapunov equation

[TABLE]

The function

[TABLE]

satisfies $P_{\rm min}\|x\|^{2}_{R}\leq V(x)\leq P_{\rm max}\|x\|^{2}_{R}$ ($P_{\rm min}$ and $P_{\rm max}$ are the minimum and maximum eigenvalues of $P$) and its derivative along the closed-loop system is

[TABLE]

**

Proof: The closed-loop system composed of (20) and (25) is

[TABLE]

denoted as $\dot{x}_{i}=f_{i}(x).$ It can also be put in a compact form

[TABLE]

From the definition of $T$ and $T^{-1}$, one has

[TABLE]

and

[TABLE]

Using this fact, we have the following calculation

[TABLE]

As a result,

[TABLE]

The proof is completed.

The main result on a distributed adaptive controller is stated in the following theorem that is proved by applying Theorem III.1.

Theorem IV.1

Under Assumption 2, consider the system (20) with the controller

[TABLE]

where $\gamma_{1}$ and $\gamma_{2}$ are given in Lemma IV.2,

[TABLE]

and $\rho_{i}(v_{i})$ is any continuously differentiable function satisfying

[TABLE]

*Then, consensus is achieved in the sense of (21) for some time functions $p_{o}(t),v_{o}(t):[0,\infty)\mapsto\mathbb{R}$.  *

Proof: The closed-loop system composed of (20) and (34) is, for $i=1,\ldots,n$,

[TABLE]

or in a compact form (4), i.e.,

[TABLE]

where $\dot{x}_{i}=f_{i}(x)$ is given in (28) and

[TABLE]

In Lemma IV.2, it has been proved that Assumption 1 is satisfied for $\dot{x}_{i}=f_{i}(x)$. It is noted that

[TABLE]

For (36) and $\beta_{i}(x_{i})=\rho_{i}(v_{i})$, one has (17). Also, (16) takes the special form (35). By Theorem III.1, one has

[TABLE]

for

[TABLE]

and $z_{i}=\rho_{i}(v_{i})-\tilde{w}_{i},\;\tilde{w}=\hat{w}-w$.

It is obvious to see that both $\|x(t)\|_{R}$ and $z(t)$ are bounded. Because of

[TABLE]

$\|\dot{x}(t)\|_{R}$ is bounded and hence $-k\|x(t)\|^{2}_{R}$ uniformly continuous in $t$. By Barbalat’s Lemma, one has $\lim_{t\rightarrow\infty}\|x(t)\|_{R}=0$, that is,

[TABLE]

Let $p_{o}(t)=r^{\mbox{\tiny\sf T}}p(t)$ and $v_{o}(t)=r^{\mbox{\tiny\sf T}}v(t)$. From the following relationship

[TABLE]

[TABLE]

one has

[TABLE]

The proof is thus completed.

Remark IV.1

The controller (34) consists of two components. The first component is designed as (25) for the ideal case with $\xi_{i}(w_{i},v_{i})=0$ to achieve consensus. When the uncertainty $\xi_{i}(w_{i},v_{i})$ is taken into account, an additional adaptive compensator $-\zeta_{i}(v_{i})\mu_{i}$ with the update law (35) is added to the controller. The critical advantage of the approach based on Theorem IV.1 is that the aforementioned two components can be designed separately. *  *

V Numerical Simulation

We consider a network of $n=6$ agents described by (20) with $\alpha_{1}=-1$ and $\alpha_{2}=0$. The nonlinear uncertain terms $\xi_{i}\left(w_{i},v_{i}\right)$’s are given as follows

[TABLE]

Assume all the unknown parameters are arbitrarily selected within the interval $[-1,1]$. The network topology is given in Fig. 1 with communication weights marked associated with the edges, also represented by the Laplacian

[TABLE]

According to Lemma IV.1, we can choose $\gamma_{1}=15$ and $\gamma_{2}=1.7$ such that $\bar{A}$ is Hurwitz and then design the controller (34) and (35) with $\rho_{i}(v_{i})$ specified as follows

[TABLE]

The simulation results of the closed-loop system are illustrated in Fig. 2 with $\lambda_{i}=5$. It is demonstrated that consensus is asymptotically achieved as concluded by Theorem IV.1.

VI Conclusion

In this paper, we have presented a distributed adaptive scheme for an MAS that aims to maintain its nominal collective behavior subject to uncertain nonlinearities. The main idea is to drive the estimation error to a deliberately designed manifold in the space of agent states and estimated parameters, which provides significant advantages in distributed implementation compared with the traditional adaptive law based on gradient of a Lyapunov function. The effectiveness of the new scheme has been demonstrated in solving an open asymptotic consensus problem for a second-order MAS in a leaderless directed network. With appropriate design of the manifold, the scheme is expected to handle nonlinearly parameterized uncertainties in the future work.

VII Appendix

Proof of Lemma IV.1: Under Assumption 2, all eigenvalues of $J$ have positive real parts. Let $P_{J}\in\mathbb{R}^{(n-1)\times(n-1)}$ be the positive definite matrix such that

[TABLE]

Let $c$ be a positive constant such that

[TABLE]

which by Schur complement implies

[TABLE]

By choosing $\gamma_{2}=c\gamma_{1}$ and a sufficiently large $\gamma_{1}>0$, we will show $\bar{A}$ is Hurwitz. Denote

[TABLE]

which is positive definite if $c\gamma_{1}>1$. Note that

[TABLE]

where

[TABLE]

is a constant matrix. For a sufficiently large $\gamma_{1}$, $P\bar{A}+\bar{A}^{\mbox{\tiny\sf T}}P=\gamma_{1}Q+Q_{c}<0$ due to (45). Therefore, $\bar{A}$ is Hurwitz.