Observer-Based Adaptive Consensus Protocol Design for Multi-Agent Systems with One-Sided Lipschitz Nonlinearity

Abstract

This paper considers the observer-based leader–follower consensus problem of multi-agent systems with one-sided Lipschitz conditions and quadratic inner-boundedness nonlinearity. Based on the relative outputs of neighboring agents, an adaptive protocol over a directed graph is designed via assigning a time-varying coupling weight to each node. Compared with the existing protocols, the proposed protocol can not only be utilized in the unidirectional flow of information, but it does not require any global graph information for its design. It has been shown that the established criteria can not only make the observer error tendency zero, but one can also achieve the leader–follower consensus of such nonlinear multi-agent systems. Furthermore, the gains of observer and protocol parameters can be constructed by solving the linear matrix inequalities (LMIs), which are conveniently checked. The results are illustrated by numerical simulations.

1. Introduction

In recent years, the consensus problem has attracted a great deal of attention from multidisciplinary research communities. This is owing to the fact that multiple agents’ collective behaviors are greatly widespread in the natural world, the technology field and human society, and also have potential applications in many areas such as UAV formation flying, wireless sensor networks and smart grids. The essence of consensus is to design appropriate protocols such that all the agents reach an agreement about a certain variable of interest [1]. Due to the spatial dispersion of agents and the constrained sensing ranges of sensors, the consensus controller of multi-agent systems should be shifted from central to distributed paradigms. Particularly, a fully distributed controller is fascinating. It removes the dependence of global information and depends only on the local information, such as the states or outputs of each agent and its neighbors [2].

To this day, an abundance of protocols have been put forward to solve the consensus issue for first-order/second-order agents and even high-order multi-agent systems [3,4,5,6]. Recent research interests have been focused on nonlinear multi-agent systems due to motivation by practical applications, such as assembling a group of mobile robots to collaborate with each other to fulfill certain complex tasks [7]. Consensus problems were studied in [8,9] for multi-agent systems consisting of a general linear part and a Lipchitz nonlinear part. However, some examples were presented in [10] to show that certain dynamical systems are not Lipschitz in the classical sense, but are one-sided Lipschitz. Moreover, one-sided Lipschitz systems have important applications in the real world, for instance in numerical analysis [11], the birth–death issue of a population [12] and stiff issues [13,14]. In fact, compared with traditional Lipschitz systems, one-sided Lipschitz systems are more general, and also have more merits in the control and observation field, and particularly in addressing controller design based on an observer [15,16,17,18]. However, the existing results about one-sided Lipschitz systems are mainly limited in the framework of a single system. Until now, the problem of consensus tracking of multi-agent systems with a one-sided Lipschitz design has not been extensively studied, and this motivates our present research.

In most of the existing literature, when designing consensus protocols, it is indispensable to possess some global information, e.g., the spectrum of the Laplacian matrix of topology graphs [19,20,21,22,23]. One can use direct [19,20,21] and indirect methods [22,23] to calculate the eigenvalues of the Laplacian matrix. The direct method is to use the whole topology graphs to compute the spectrums; the indirect method is an exhaustive search method aiming to find all possible graphs for a given number of agents and then calculate their eigenvalues of the Laplacian matrix of each graph. However, in practice, the entire topology graph or the network size is not always available to each agent. Even though it is possible, the calculation task of these eigenvalues may be too large to implement, especially for large-scale networks. Therefore, protocols designed in this manner cannot be implemented in a fully distributed way, i.e., without utilizing global information on topological graphs. Such an issue was discussed in [24,25,26,27,28]. The main idea to solve this issue is to configurate time-varying coupling weights for each node or edge, and these coupling weights adaptively adjust with the consensus evolution. Specifically, during the consensus progress, the coupling weights continue to increase or decrease and eventually achieve some finite values. This idea was applied to cope with the adaptive consensus of a network of lur’e systems in [24,25] and Lipschitz nonlinear systems in [26]. Unfortunately, most of the existing works focus on state feedback protocols and undirected topology graphs. The pinning synchronization of complex one-sided Lipschitz networks with general directed topologies was investigated in [29], where two synchronization criteria with full-state and partial-state coupling were proposed, respectively. However, these criteria depend on some global graph spectrum information, i.e., they are not fully distributed. On the other hand, in general, output feedback based on a dynamic compensator or an observer is superior in function to static output feedback. Therefore, it is considerably urgent to construct fully distributed tracking protocols, especially observer-based protocols, under directed topologies for reaching consensus tracking for multi-agent systems with one-sided Lipschitz nonlinearity, as pointed out in [30].

Motivated by the above analysis, this paper is devoted to designing consensus protocols based on an observer for multi-agent systems subject to one-sided Lipschitz conditions. The key contributions of the present paper are twofold:

(1)

The one-sided Lipschitz condition in [10,15,16,17,18] is introduced into multi-agent systems, and the traditional observer design for an individual agent is extended to the output feedback protocol design for multi-agent systems subject to one-sided Lipschitz conditions.

(2)

Since the agents’ states are not available in many scenarios, we propose a novel observer-based protocol, and this protocol has two distinctive features. One is that it is independent of any global network information and relies only on the dynamics of agents and their relative outputs, leading it to be fully distributed. Different from the case in [31,32], the interplay of adaptive coupling weights and one-sided Lipschitz dynamics makes convergence analysis very intractable; the other is that it is based on unidirectional information exchange rather than bidirectional exchange [33]. This extension from the undirected graph to the directed case is nontrivial, and the main challenge is that the Laplacian matrices of directed graphs are asymmetric, making the design of adaptive protocols and the construction of the Lyapunov function very tough. Thanks to these two advantages, our protocol seems to be more general and more convenient to apply in engineering. Finally, inspired by the Lyapunov techniques in [2,23], a Lyapunov function with an integral fashion is artfully constructed for achieving the stability of tracking error systems.

The remainder of the paper is given as follows. Section 2 involves preliminaries and problem statements. In Section 3, we design the observer-based adaptive protocol, under which the leader–follower consensus of multi-agent systems subject to one-sided Lipschitz conditions is reached. Simulation examples are presented in Section 4, followed by the Conclusion.

2. Some Preliminaries and Problem Statement

2.1. Preliminaries

${ℝ}^n$ is the $n$-dimensional Euclidean space over real numbers. $(\cdot ,\cdot )$ denotes the inner product in ${ℝ}^n$, i.e., for any $x,y\in {ℝ}^n$, then $(x,y)=x^{T}y$, in which $x^T$ indicates the transpose of $x$. ${\lambda }_{\min }(P)$ and ${\lambda }_{\max }(P)$ are, respectively, the minimum and maximum eigenvalues of the symmetric matrix $P$. An asterisk ‘$*$’ represents a symmetric term of symmetric block matrices.

This paper involves a group of agents that consist of $N$ followers and one leader. The information interaction between $N$ followers is depicted by a weighed digraph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$, in which $\mathcal{V}=\left\{v_1,v_2,\cdots ,v_N\right\}$ and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ are, respectively, the node and edge set, and $\mathcal{A}=\left[a_{ij}\right]\in {ℝ}^{N\times N}$ is the adjacency matrix, with $a_{ij}>0$ if there exists an edge from $v_j$ to $v_i$, and $a_{ij}=0$ otherwise. The Laplacian matrix of $\mathcal{G}$ is denoted by $\mathcal{L}=[l_{ij}]$, in which $l_{ij}=-a_{ij}$, $i\ne j$ and $l_{ii}={\displaystyle {\sum }_{j=1,j\ne i}^N{a}_{ij}}$. $N$ followers and a leader constitute the augmented graph $\tilde{\mathcal{G}}$. The accesses of the followers to the leader are represented by $\mathcal{D}\triangleq diag\{d_1,\cdots ,d_N\}$, where $d_i>0$ if the follower $i$ has access to the information of leader, and $d_i=0$ otherwise.

Unlike the existing adaptive protocols in [20,21,22,24,25,33] that can only be applicable to cases in which the whole graphs are undirected or the subgraphs among the followers are undirected, we relax this assumption to the following case.

Assumption 1. 

The augmented graph  $\tilde{\mathcal{G}}$ is directed, containing a spanning tree whose root node is the leader.

Under the above assumption, define $\tilde{\mathcal{L}}\triangleq \mathcal{L}+\mathcal{D}$. In the following lemma, we give its property.

Lemma 1 

([34]). Under Assumption 1, $\tilde{\mathcal{L}}$ is invertible. Denote ${\left({\theta }_1\cdots ,{\theta }_N\right)}^T={\tilde{\mathcal{L}}}^{-1}{1}_N$ and $\mathsf{\Theta }=diag\left\{\frac{1}{{\theta }_1},\cdots ,\frac{1}{{\theta }_N}\right\}$; then, $\mathsf{\Theta }$ and $\mathsf{\Theta }\tilde{\mathcal{L}}+{\tilde{\mathcal{L}}}^T\mathsf{\Theta }$ are positive definite matrices.

Lemma 2 

(Young’s Inequality [35]). For non-negative real numbers $a$ and $b$ and positive real numbers $p$ and $q$, satisfying $\frac{1}{p}+\frac{1}{q}=1$, we have $a\cdot b\le \frac{a^p}{p}+\frac{b^q}{q}$.

2.2. Problem Formulation

In this paper, the control purpose is to address the leader–follower consensus issue for multi-agent systems subject to one-sided Lipschitz conditions. By virtue of the relative outputs between a couple of neighboring agents, we first construct a state observer, and then, using the observed states, design an adaptive protocol over the directed graph.

Consider a group of $N$ nonlinear agents. The system of the $i$-th agent is given by

$$\left\{\begin{array}{c}{\dot{x}}_i=A{x}_i+D\mathsf{\Phi }(x_i,t)+B{u}_i,\\ y_i=C{x}_i,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}i=1,\cdots ,N.\hspace{0.33em}\end{array}\right.$$

(1)

where the state $x_i\in {ℝ}^n$, input $u_i\in {ℝ}^p$, and output $y_i\in {ℝ}^q$. $A,B,C,D$ are constant matrices. Assume that $(A,B,C)$ is controllable and observable. The function $\mathsf{\Phi }(\cdot ,\cdot ):{ℝ}^n\times ℝ\to {ℝ}^n$ is a continuous function characterizing the intrinsic nonlinearity, satisfying the following assumption.

Assumption 2. 

1. The function $\mathsf{\Phi }$ is said to be a one-sided Lipschitz if there exists a $\rho \in ℝ$ such that $\forall x,y\in D_1\subseteq {ℝ}^n$.

$$\left(\mathsf{\Phi }(x,t)-\mathsf{\Phi }(y,t),x-y\right)\le \rho {‖x-y‖}^2.$$

(2)

where $\rho \in ℝ$ is a one-side Lipschitz constant. If (2) is valid everywhere in ${ℝ}^n$, then this function is called a globally one-side Lipschitz.

2. The function $\mathsf{\Phi }$ is called quadratic inner-bounded if there exist $\mu ,\hspace{0.33em}\nu \in ℝ$ such that $\forall x,y\in D_2\subseteq {ℝ}^n$.

$${‖\mathsf{\Phi }(x,t)-\mathsf{\Phi }(y,t)‖}^2\le \mu {‖x-y‖}^2+\nu \left(\mathsf{\Phi }(x,t)-\mathsf{\Phi }(y,t),x-y\right).$$

(3)

Unlike the well-known Lipschitz constant being positive, the parameters $\rho ,\hspace{0.33em}\mu$ and $\nu$ can be positive, negative or zero. Additionally, compared with traditional Lipschitz systems, one-sided Lipschitz systems and quadratically inner-bounded functions are general, including traditional Lipschitz systems as a special case [16].

In practical engineering, the followers are required to synchronize to the virtue leader, whose dynamics are autonomous, given by

$$\dot{s}=As+D\mathsf{\Phi }(s,t),\hspace{0.33em}s(0)=s_0.$$

(4)

Our control objective is to design an observer-based adaptive protocol such that each agent in system (1) can be synced up with trajectory (4) by virtue of a directed communication network.

3. Main Results

In this section, we will first give the theoretical results of current work, and then prove them analytically.

For nonlinear systems (1) and (4), the state observer is proposed:

$$\left\{\begin{array}{c}{\dot{\widehat{x}}}_i=A{\widehat{x}}_i+B{u}_i+D\mathsf{\Phi }({\widehat{x}}_i,t)+F(y_i-{\widehat{y}}_i),\\ {\widehat{y}}_i=C{\widehat{x}}_i,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}i=1,\cdots ,N.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(5)

where ${\widehat{x}}_i$ is the estimate of $x_i$, and ${\widehat{y}}_i$ is the observed output. $F$ is the observer gain matrix to be determined later. Based on the roughly estimated values of a couple of neighboring agents, we designed the adaptive controller with time-varying coupling weights as follows:

$$\begin{array}{l}{u}_i=-c_i{g}_i(z_i^T{Q}^{-1}{z}_i)K{z}_i,\\ {\dot{c}}_i=z_i^T\mathsf{\Gamma }{z}_i,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}i=1,\cdots ,N.\end{array}$$

(6)

where $z_i\triangleq {\displaystyle {\sum }_{j=1}^N{a}_{ij}({\widehat{x}}_i}-{\widehat{x}}_j)+d_i({\widehat{x}}_i-s)$, and $c_i(t)$ is a time-varying coupling weight with initial value $c_i(0)>1$. The positive definite matrix $Q$ and the feedback gain matrices $K$, $\mathsf{\Gamma }\ge 0$ were designed later. The smooth function $g_i(s)$ is monotonically nondecreasing in $s$, satisfying that $g_i(s)\ge 1$ for any $s>0$. Let $e_{1i}=x_i-{\widehat{x}}_i$ and $e_1={[e_{11}^T,e_{12}^T,\cdots ,e_{1N}^T]}^T$. From (1) and (5), one can see that

$${\dot{e}}_1=\left(I_N\otimes (A-FC)\right)e_1+\left(I_N\otimes D\right){\tilde{\mathsf{\Phi }}}_1$$

(7)

where ${\tilde{\mathsf{\Phi }}}_{1i}\triangleq \mathsf{\Phi }(x_i,t)-\mathsf{\Phi }({\widehat{x}}_i,t)$ and ${\tilde{\mathsf{\Phi }}}_1={\left[{\tilde{\mathsf{\Phi }}}_{11}^T,{\tilde{\mathsf{\Phi }}}_{12}^T,\cdots ,{\tilde{\mathsf{\Phi }}}_{1N}^T\right]}^T$. Let $e_{2i}={\widehat{x}}_i-s$, $e_2={[e_{21}^T,e_{22}^T,\cdots ,e_{2N}^T]}^T$  and $z={[z_1^T,\cdots ,z_N^T]}^T$. Then, $z=(\tilde{\mathcal{L}}\otimes I_n)e_2$. Note that $\tilde{\mathcal{L}}$ is nonsingular, satisfying Assumption 1. It is not difficult to see that the observer-based consensus tracking issue is solvable if and only if both $e_1$ and $z$ converge to zero.

By virtue of (1) and (6), it is easy to obtain that $z$ and $c$ are governed by the following dynamics:

$$\begin{array}{l}\dot{z}=\left(I_N\otimes A-\tilde{\mathcal{L}}cg\otimes BK\right)z+(\tilde{\mathcal{L}}\otimes D){\tilde{\mathsf{\Phi }}}_2+(\tilde{\mathcal{L}}\otimes FC)e_1\\ \hspace{0.33em}\dot{c}=z_{}^T\left(I_N\otimes \mathsf{\Gamma }\right)z.\end{array}$$

(8)

where $c\triangleq diag\left\{c_1,\cdots ,c_N\right\}$, $g\triangleq diag\left\{g_1(z_1^T{Q}^{-1}{z}_1)),\cdots ,g_N(z_N^T{Q}^{-1}{z}_N)\right\}$, ${\tilde{\mathsf{\Phi }}}_{2i}\triangleq \mathsf{\Phi }({\widehat{x}}_i,t)-\mathsf{\Phi }(s,t)$ and ${\tilde{\mathsf{\Phi }}}_2={\left[{\tilde{\mathsf{\Phi }}}_{21}^T,{\tilde{\mathsf{\Phi }}}_{22}^T,\cdots ,{\tilde{\mathsf{\Phi }}}_{2N}^T\right]}^T$.

Next, we present an algorithm (Algorithm 1) which we used to construct the parameters of the adaptive protocol.

Algorithm 1. 

Under Assumptions 1 and 2, the controller parameters in (5) and (6) can be constructed as follows:

• The constants of Assumption 2 need to satisfy

  $$\rho \in ℝ, \mu \in ℝ \mathrm{and} \nu >-2$$

  (9)
• Solve LMI,

  $$\Delta \triangleq \left(\begin{array}{cc}PA+A^{T}P-2C^{T}C+({\epsilon }_1\rho +{\epsilon }_2\mu )I_n& PD+\frac{({\epsilon }_2\nu -{\epsilon }_1)I_n}{2}\\ *& -{\epsilon }_2{I}_n\end{array}\right)<0.$$

  (10)

  to obtain $P>0$ and two positive scalars ${\epsilon }_1,\hspace{0.33em}{\epsilon }_2$. Take the observer gain matrix $F=P^{-1}{C}^T$.
• For an appropriately small positive scalar, $\varphi$, solve LMI,

  $$AQ+Q{A}^T+\frac{1}{\varphi }DQ-\phi B{B}^T<0$$

  (11)

  to obtain $Q>0$ and a positive scalar $\phi$. Choose the feedback gain matrices  $K=B^T{Q}^{-1}$ and $\mathsf{\Gamma }=Q^{-1}B{B}^T{Q}^{-1}$ and the auxiliary function $g_i(z_i^T{Q}^{-1}{z}_i)={\left(1+z_i^T{Q}^{-1}{z}_i\right)}^{\gamma }$, where $\gamma >1$ is any given constant.

Next, the main results are given below.

Theorem 1. 

Let Assumptions 1 and 2 hold. If there exist feasible solutions to (9)–(11), the leader–follower consensus of systems (1) and (4) can be realized under observer (5) and adaptive law (6), whose parameters are designed in Algorithm 1. Moreover, as $t\to \infty$, the adaptive function $g_i(z_i^T{Q}^{-1}{z}_i)\to 1$ and the coupling weight $c_i(t)$ approach a steady-state constant value.

Proof. 

We construct a Lyapunov function candidate as follows:

$$V(t)=V_1(t)+\epsilon V_2(t)$$

(12)

where $V_1(t)=e_1^T(I_N\otimes P)e_1$, $V_2(t)={\displaystyle {\sum }_{i=1}^N\frac{c_i}{{\theta }_i}}{\displaystyle {\int }_0^{z_i^T{Q}^{-1}{z}_i}{g}_i(s)ds}+\frac{(\gamma -1){\lambda }_1}{8\gamma }{\displaystyle {\sum }_{i=1}^N{(c_i-\varsigma )}^2}$ and ${\lambda }_1\triangleq {\lambda }_{\min }(\mathsf{\Theta }\tilde{\mathcal{L}}+{\tilde{\mathcal{L}}}^T\mathsf{\Theta })$; one can get that ${\lambda }_1>0$ and ${\theta }_i>0$ by using Lemma 1. A scalar $\varsigma >0$ will be determined later, and the matrices $P>0$ and $Q>0$ are solutions to (10) and (11). $\gamma >1$ is any given constant. Due to $c_i(0)>1$, $\mathsf{\Gamma }\ge 0$, referring to (6), we have that $c_i(t)\ge 1$ holds for $\hspace{0.33em}t>0$. Additionally, note that $g_i(s)\ge 1$ for $s>0$. Summing up, the function $V(t)$ is positive definite.

By using fact that $F=P^{-1}{C}^T$, along the trajectories of (7), we take the derivative of $V_1(t)$ as follows:

$${\dot{V}}_1(t)=e_1^T\left(I_N\otimes (PA+A^{T}P-2C^{T}C)\right)e_1+2e_1^T\left(I_N\otimes PD\right){\tilde{\mathsf{\Phi }}}_1$$

(13)

From (2) and (3), we get

$${\epsilon }_1{\left(\begin{array}{c}{e}_1\\ {\tilde{\mathsf{\Phi }}}_1\end{array}\right)}^T\left(\begin{array}{cc}\rho I_{Nn}& -\frac{I_{Nn}}{2}\\ *& 0\end{array}\right)\left(\begin{array}{c}{e}_1\\ {\tilde{\mathsf{\Phi }}}_1\end{array}\right)\ge 0 \mathrm{and} {\epsilon }_2{\left(\begin{array}{c}{e}_1\\ {\tilde{\mathsf{\Phi }}}_1\end{array}\right)}^T\left(\begin{array}{cc}\mu I_{Nn}& \frac{\nu I_{Nn}}{2}\\ *& -I_{Nn}\end{array}\right)\left(\begin{array}{c}{e}_1\\ {\tilde{\mathsf{\Phi }}}_1\end{array}\right)\ge 0.$$

(14)

where ${\epsilon }_1$ and ${\epsilon }_2$ are positive scalars. Then, adding the terms on the left-hand sides of (14) and (13) yields

$${\dot{V}}_1(t)\le {\underset{{\xi }_1^T}{\underbrace{\left(\begin{array}{c}{e}_1\\ {\tilde{\mathsf{\Phi }}}_1\end{array}\right)}}}^T\underset{{\mathsf{{\rm T}}}_{11}}{\underbrace{\left(\begin{array}{cc}{I}_N\otimes {\Delta }_{11}& I_N\otimes {\Delta }_{12}\\ *& I_N\otimes {\Delta }_{22}\end{array}\right)}}\underset{{\xi }_1}{\underbrace{\left(\begin{array}{c}{e}_1\\ {\tilde{\mathsf{\Phi }}}_1\end{array}\right)}}\triangleq {\xi }_1^T{\mathsf{{\rm T}}}_{11}{\xi }_1$$

(15)

where $\left(\begin{array}{cc}{\Delta }_{11}& {\Delta }_{12}\\ *& {\Delta }_{22}\end{array}\right)\triangleq \Delta$. Together with (10), $\Delta <0$ means ${\mathsf{{\rm T}}}_{11}<0$.

Along the trajectories of (8), we can take the derivative of $V_2(t)$ as follows:

$$\begin{array}{l}{\dot{V}}_2(t)={\displaystyle {\sum }_{i=1}^N\frac{2c_i}{{\theta }_i}}{g}_i(z_i^T{Q}^{-1}{z}_i)z_i^T{Q}^{-1}{\dot{z}}_i+{\displaystyle {\sum }_{i=1}^N\frac{{\dot{c}}_i}{{\theta }_i}{\displaystyle {\int }_0^{z_i^T{Q}^{-1}{z}_i}{g}_i(s)ds}}\\ +\frac{(\gamma -1){\lambda }_1}{4\gamma }{\displaystyle {\sum }_{i=1}^N(c_i-\varsigma ){\dot{c}}_i}\end{array}$$

(16)

Subsequently, we abbreviate $g_i(z_i^T{Q}^{-1}{z}_i)$ to $g_i$ without causing ambiguity. By using (8), it can be obtained that

$$\begin{array}{l}{\displaystyle {\sum }_{i=1}^N\frac{2c_i}{{\theta }_i}\cdot }{g}_i(z_i^T{Q}^{-1}{z}_i)\cdot z_i^T{Q}^{-1}{\dot{z}}_i\\ =z^T\left(cg\mathsf{\Theta }\otimes (Q^{-1}A+A^T{Q}^{-1})-cg(\mathsf{\Theta }\tilde{\mathcal{L}}+{\tilde{\mathcal{L}}}^T\mathsf{\Theta })cg\otimes Q^{-1}B{B}^T{Q}^{-1}\right)z\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+2z^T(cg\mathsf{\Theta }\tilde{\mathcal{L}}\otimes Q^{-1}D){\tilde{\mathsf{\Phi }}}_2+2z^T(cg\mathsf{\Theta }\tilde{\mathcal{L}}\otimes Q^{-1}FC)e_1\end{array}$$

(17)

On the other hand, due to $z=(\tilde{\mathcal{L}}\otimes I_n)e_2$, $D>0$ and together with Lemma A1 in Appendix A, we have

$$\begin{array}{l}2z^T(cg\mathsf{\Theta }\tilde{\mathcal{L}}\otimes Q^{-1}D){\tilde{\mathsf{\Phi }}}_2\\ \le 2e_2^T({\tilde{\mathcal{L}}}^{T}cg\mathsf{\Theta }\tilde{\mathcal{L}}\otimes Q^{-1}D){\tilde{\mathsf{\Phi }}}_2\\ \le \frac{1}{\varphi }{e}_2^T({\tilde{\mathcal{L}}}^{T}cg\mathsf{\Theta }\tilde{\mathcal{L}}\otimes Q^{-1}D)e_2\\ =\frac{1}{\varphi }{z}^T(cg\mathsf{\Theta }\otimes Q^{-1}D)z\end{array}$$

(18)

By using (17) and (18), and together with ${\lambda }_1\triangleq {\lambda }_{\min }(\mathsf{\Theta }\tilde{\mathcal{L}}+{\tilde{\mathcal{L}}}^T\mathsf{\Theta })$, we have

$$\begin{array}{l}{\displaystyle {\sum }_{i=1}^N\frac{2c_i}{{\theta }_i}\cdot }{g}_i(z_i^T{Q}^{-1}{z}_i)\cdot z_i^T{Q}^{-1}{\dot{z}}_i\\ \le z^T\left(\begin{array}{l}cg\mathsf{\Theta }\otimes (Q^{-1}A+A^T{Q}^{-1}+\frac{1}{\varphi }{Q}^{-1}D)\\ -{\lambda }_1{c}^2{g}^2\otimes Q^{-1}B{B}^T{Q}^{-1}\end{array}\right)z\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+2z^T(cg\mathsf{\Theta }\tilde{\mathcal{L}}\otimes Q^{-1}FC)e_1\end{array}$$

(19)

Because $g_i(s)={\left(1+s\right)}^{\gamma }$ with $\gamma >1$ is monotonically increasing and satisfies $g_i(s)\ge 1$, for s > 0, it follows that

$$\begin{gathered} \begin{array}{l}{\displaystyle {\sum }_{i=1}^N\frac{{\dot{c}}_i}{{\theta }_i}{\displaystyle {\int }_0^{z_i^T{Q}^{-1}{z}_i}{g}_i(s)ds}}\\ \le {\displaystyle {\sum }_{i=1}^N\frac{{\dot{c}}_i}{{\theta }_i}{g}_i\cdot z_i^T{Q}^{-1}{z}_i}\\ \le {\displaystyle {\sum }_{i=1}^N\left[\frac{\gamma -1}{2\gamma }\times \frac{1}{{\theta }_i^{\frac{2\gamma }{\gamma -1}}{\lambda }^{\frac{\gamma +1}{\gamma -1}}}+\frac{\gamma +1}{2\gamma }{g}_i^{\frac{2\gamma }{\gamma +1}}\cdot {\left(z_i^T{Q}^{-1}{z}_i\right)}^{\frac{2\gamma }{\gamma +1}}{\lambda }_1\right]}\hspace{0.33em}{\dot{c}}_i\end{array} \\ \begin{array}{l}\le {\displaystyle {\sum }_{i=1}^N\left[\frac{\gamma -1}{2\gamma \cdot {\theta }_i^{\frac{2\gamma }{\gamma -1}}{\lambda }^{\frac{\gamma +1}{\gamma -1}}}+\frac{\gamma +1}{2\gamma }{g}_i^{\frac{2\gamma }{\gamma +1}}\cdot {\left(1+z_i^T{Q}^{-1}{z}_i\right)}^{\frac{2\gamma }{\gamma +1}}{\lambda }_1\right]}\hspace{0.33em}{\dot{c}}_i\\ ={\displaystyle {\sum }_{i=1}^N\left[\frac{\gamma -1}{2\gamma \cdot {\theta }_i^{\frac{2\gamma }{\gamma -1}}{\lambda }^{\frac{\gamma +1}{\gamma -1}}}+\frac{\gamma +1}{2\gamma }{g}_i^2{\lambda }_1\right]}\hspace{0.33em}{z}_i^T{Q}^{-1}B{B}^T{Q}^{-1}{z}_i\end{array} \end{gathered}$$

(20)

where in the first and second inequality, we, respectively, use the mean value theorem of integrals and Lemma 2. Substituting (16) and (19) into (20) yields

$${\dot{V}}_2(t)\le {\displaystyle {\sum }_{i=1}^N{z}_i^T}\left[\begin{array}{l}\frac{c_i{g}_i}{{\theta }_i}\left(Q^{-1}A+A^T{Q}^{-1}+\frac{1}{\varphi }{Q}^{-1}D\right)\\ \frac{c_i{g}_i}{{\theta }_i}\left(Q^{-1}A+A^T{Q}^{-1}+\frac{1}{\varphi }{Q}^{-1}D\right)\\ -\left(\frac{(\gamma -1){\lambda }_1\cdot \varsigma }{4\gamma }-\frac{\gamma -1}{2\gamma \cdot {\theta }_i^{\frac{2\gamma }{\gamma -1}}{\lambda }^{\frac{\gamma +1}{\gamma -1}}}\right)Q^{-1}B{B}^T{Q}^{-1}\end{array}\right]z_i+2z^T(cg\mathsf{\Theta }\tilde{\mathcal{L}}\otimes Q^{-1}FC)e_1$$

(21)

Due to $g_i(s)\ge 1$ and $c_i(s)\ge 1$, for $s>0$ s > 0, we have

$$\begin{array}{l}{c}_i^2{g}_i^2-\frac{\gamma +1}{2\gamma }{g}_i^2-\frac{(\gamma -1)\cdot c_i}{4\gamma }\\ \ge c_i^2{g}_i^2-\frac{\gamma +1}{2\gamma }{c}_i^2{g}_i^2-\frac{\gamma -1}{4\gamma }{c}_i^2{g}_i^2\\ =\frac{\gamma -1}{4\gamma }{c}_i^2{g}_i^2\end{array}$$

(22)

Select

$$\varsigma >\frac{4\gamma \cdot \overline{\alpha }}{\gamma -1}+\underset{i=1,\cdots ,N}{\max }\frac{2}{{\theta }_i^{\frac{2\gamma }{\gamma -1}}{\lambda }^{\frac{2\gamma }{\gamma -1}}},$$

that is,

$$\frac{(\gamma -1){\lambda }_1\cdot \varsigma }{4\gamma }-\frac{\gamma -1}{2\gamma \cdot {\theta }_i^{\frac{2\gamma }{\gamma -1}}{\lambda }^{\frac{\gamma +1}{\gamma -1}}}>\overline{\alpha }{\lambda }_1$$

(23)

where $\overline{\alpha }>0$ will be determined later.

Due to

$$\frac{\gamma -1}{4\gamma }{c}_i^2{g}_i^2{\lambda }_1+\overline{\alpha }{\lambda }_1\ge c_i{g}_i{\lambda }_1\sqrt{(\gamma -1)\overline{\alpha }/\gamma }$$

(24)

substituting (22)–(24) into (21) yields

$$\begin{array}{l}{\dot{V}}_2(t)\le {\displaystyle {\sum }_{i=1}^N{z}_i^T\left[\frac{c_i{g}_i}{{\theta }_i}\left(Q^{-1}A+A^T{Q}^{-1}+\frac{1}{\varphi }{Q}^{-1}D\right)-c_i{g}_i{\lambda }_1\sqrt{(\gamma -1)\overline{\alpha }/\gamma }{Q}^{-1}B{B}^T{Q}^{-1}\right]}{z}_i\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+2z^T(cg\mathsf{\Theta }\tilde{\mathcal{L}}\otimes Q^{-1}FC)e_1\end{array}$$

(25)

Choose $\overline{\alpha }$ to be sufficiently large such that $\underset{i=1,\cdots ,N}{\min }{\theta }_i\lambda \sqrt{(\gamma -1)\overline{\alpha }/\gamma }>\phi$, and then (25) is turned into

$$\begin{array}{l}{\dot{V}}_2(t)\le z^T\left[cg\mathsf{\Theta }\otimes \left(Q^{-1}A+A^T{Q}^{-1}+\frac{1}{\varphi }{Q}^{-1}D-\phi Q^{-1}B{B}^T{Q}^{-1}\right)\right]z\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+2z^T(cg\mathsf{\Theta }\tilde{\mathcal{L}}\otimes Q^{-1}FC)e_1\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\triangleq \hspace{0.33em}{z}^T{\mathsf{{\rm T}}}_{22}z+2{\xi }_1^T{\mathsf{{\rm T}}}_{12}z\end{array}$$

(26)

where ${\mathsf{{\rm T}}}_{12}=\left(\begin{array}{c}{\tilde{\mathcal{L}}}^{T}cg\mathsf{\Theta }\otimes C^T{F}^T{Q}^{-1}\\ 0\end{array}\right)$. From (11), one can get that ${\mathsf{{\rm T}}}_{22}<0$. By (12), (15) and (26), one can obtain that

$$\dot{V}(t)\le {\left(\begin{array}{c}{\xi }_1\\ z\end{array}\right)}^T\left(\begin{array}{cc}{\mathsf{{\rm T}}}_{11}& \epsilon {\mathsf{{\rm T}}}_{12}\\ *& \epsilon {\mathsf{{\rm T}}}_{22}\end{array}\right)\left(\begin{array}{c}{\xi }_1\\ z\end{array}\right)\triangleq {\vartheta }^T{\rm M}\vartheta$$

Note that ${\mathsf{{\rm T}}}_{11}<0$ and ${\mathsf{{\rm T}}}_{22}<0$. We can choose the appropriate positive constant $\epsilon$ such that ${\mathsf{{\rm T}}}_{11}-\epsilon {\mathsf{{\rm T}}}_{12}{\mathsf{{\rm T}}}_{22}^{-1}{\mathsf{{\rm T}}}_{12}^T<0$ holds, which is equivalent to ${\rm M}<0$. Therefore, one concludes that $e_1$ and $z$ both converge to zero. That is, adaptive protocol (6) solves the observer-based consensus tracking problem for systems (1) and (4). Since $\dot{V}<0$, we can get that $V$ is bounded, implying that $c_i(t)$ is also bounded. It follows from $\mathsf{\Gamma }\ge 0$ that $c_i(t)$ is monotonically increasing. Therefore, $c_i(t)$ is bound to converge to a certain finite constant value. This completes the proof. □

Remark 1. 

Different from the protocols in [19,20,21,22,23], which rely on some global information, adaptive protocol (6) relies only on agent dynamics and estimate states, and therefore it can be implemented by each agent in a fully distributed fashion. On the other hand, adaptive protocol (6) was designed based on the directed graph, which is different from the protocols over undirected graphs in [29,33]. On the other hand, although the distributed adaptive protocols in [2,27] can be applicable to the directed graph, those protocols adopted a state feedback strategy. Since the complete state information is not always available due to economical and/or technological reasons, we inserted observer (5) in the control loop to recover the unavailable states. In short, a distinctive merit of our protocols is the integration of a fully distributed communication fashion, unidirectional communication and dynamic output feedback in a unified framework.

Remark 2. 

When Lemma A1 in Appendix A is used to prove Theorem 1, a sufficiently small positive scalar  $\varphi$ automatically guarantees that the following constraint conditions hold:

$\mu +\rho (\nu +2)-\frac{1}{\varphi }<0$ and $\varphi \left[{\lambda }_{\max }(X)-{\lambda }_{\min }(X)\right]-{\lambda }_{\min }(X)<0$ with $X={\tilde{\mathcal{L}}}^{T}cg\mathsf{\Theta }\tilde{\mathcal{L}}\otimes Q^{-1}D$.

Since a positive scalar  $\varphi$ is appropriately small, the existence of the solutions for LMI (11) can be guaranteed as long as a positive scalar  $\phi$ is appropriately large, such that  $\phi >\inf \left\{\phi \left|\phi B{B}^T>AQ+Q{A}^T+\frac{1}{\varphi }DQ\right.\right\}$ holds.

4. Simulation Example

In this section, a simulation example on supercritical Hopf bifurcation is provided to verify the effectiveness of the proposed protocol.

Consider a group of four followers and a leader, governed by (1) and (4) with the parameters

$$A=\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right), D=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right), B=\left(\begin{array}{c}0\\ 1\end{array}\right), C=\left(\begin{array}{cc}0& 1\end{array}\right)$$

and the one-sided Lipschitz function $\mathsf{\Phi }(x_i,t)=\left(\begin{array}{c}-x_{i1}(x_{i1}^2+x_{i2}^2)\\ -x_{i2}(x_{i1}^2+x_{i2}^2)\end{array}\right)$, $i=1,2,3,4.$ The leader’s system is

$$\left(\begin{array}{l}{\dot{x}}_1\hfill \\ {\dot{x}}_2\hfill \end{array}\right)=\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)\left(\begin{array}{l}{x}_1\hfill \\ x_2\hfill \end{array}\right)+\left(\begin{array}{c}-x_1(x_1^2+x_2^2)\\ -x_2(x_1^2+x_2^2)\end{array}\right).$$

(27)

Using the polar coordinate transformation, $x_1=r\cos \theta$ and $x_2=r\sin \theta$, (27) is rewritten as

$$\dot{r}=r-r^3 \mathrm{and} \dot{\theta }=1$$

(28)

System (28) can represent a moving object’s motion. Additionally, some famous examples, such as a supercritical Hopf bifurcation, can be expressed by (28), whose concrete dynamic behaviors are explained in detail in [36]. From the definition of [16], we can get that $\mathsf{\Phi }(x_i,t)$ is a one-sided Lipschitz system in ${ℝ}^2$, with $\rho =0$. Letting $D_2=\{x\in {ℝ}^2\left|‖x‖<r\right.\}$, some calculations in [16] yield $r=\min \left(\sqrt{-\frac{\nu }{4}},\hspace{0.33em}\hspace{0.33em}\sqrt[4]{\mu +\frac{{\nu }^2}{4}}\right)$, and thus have $\nu <0,\hspace{0.33em}\hspace{0.33em}\mu +\frac{{\nu }^2}{4}>0$. We take $u=-0.01,\hspace{0.33em}\nu =-1.5$. For convenience, the directed topology $\tilde{\mathcal{G}}$ is illustrated in Figure 1, where there are one leader, labeled 0 and four followers, and follower $1$ and $3$ can acquire the information of the leader, and all the weights on the edges are ones. Taking $\varphi =0.08$ and solving LMIs (10)–(11), we obtain

$$\begin{gathered} P=\left(\begin{array}{cc}0.0961& -0.2083\\ -0.2083& 0.5290\end{array}\right), Q=\left(\begin{array}{cc}2.4137& 22.8334\\ 22.8334& 333.4980\end{array}\right), \\ {\epsilon }_1=0.0324, {\epsilon }_2=0.3479,\phi =6.0832\times {10}^3. \end{gathered}$$

Then, the observer and controller gain matrices are given by, respectively,

$$F=\left(\begin{array}{c}27.8726\\ 12.8628\end{array}\right), K=\left(\begin{array}{cc}-0.0805& 0.0085\end{array}\right), \mathsf{\Gamma }=\left(\begin{array}{cc}0.0065& -0.0007\\ -0.0007& 0.0001\end{array}\right).$$

From the square $[-4,4]\times [-4,4]$, we choose the initial conditions of $s(0),\hspace{0.33em}{x}_i(0)$ and ${\widehat{x}}_i(0)$ randomly, and $\left(c_1(0),\hspace{0.33em}{c}_2(0),\hspace{0.33em}{c}_3(0),\hspace{0.33em}{c}_4(0)\right)=\left(1.01,\hspace{0.33em}1.02,\hspace{0.33em}1.04,\hspace{0.33em}1.03\right)$. Set $\gamma =1.001$.

As show in Figure 2 and Figure 3, both the estimate errors $x_i-{\widehat{x}}_i$ and the tracking errors $x_i-s$ all converge to zero. Thereby, the leader–follower consensus is realized when adaptive protocol (6) is imposed onto the four followers. Furthermore, Figure 4 indicates that the adaptive coupling weights $c_i(t)$ converge to constants; Figure 5 shows that the auxiliary functions $g_i(z_i^T{Q}^{-1}{z}_i)$ approach one.

5. Conclusions

This paper solved the consensus tracking issue for multi-agent systems split into general linear dynamics and one-sided Lipschitz nonlinearity. A novel observer-based adaptive protocol was proposed, and, further, it was shown that both the estimate errors and the tracking errors converge to zero as the time tends to infinity. Different from the existing protocols, the distinctive merit of our protocol is the combination of the directed network, fully distributed communication fashion and observer-based feedback in a uniform framework. Furthermore, the observer gains and protocol parameters can be obtained by solving LMIs, which is easily and numerically tractable via the standard software (Matlab software 9.13, MathWorks, Natick, MA, USA). Investigating the cooperative control issues for other types of nonlinear multi-agent systems is our future work.