KANG Jianling()， YU Lingling()

College of Science， Donghua University， Shanghai 201620， China

Abstract： The distributed leader-following consensus for nonlinear multi-agent systems in strict-feedback forms is investigated under directed topology. Firstly， each follower node is modeled by an integrator incorporating with nonlinear dynamics. The leader node is modeled as an autonomous nonlinear system which sends its information to one or more followers. Then， a simple and novel distributed protocol is proposed based only on the state feedback， under which the states of the followers ultimately synchronize to the leader. By using Lyapunov stability theorem and matrix theory， it is proved that the distributed leader-following consensus of nonlinear multi-agent systems with strict-feedback form is guaranteed by Lipschitz continuous control laws. Finally， some numerical simulations are provided to show the effectiveness of the developed method.

Key words： leader-following consensus； nonlinear multi-agent system； strict-feedback form； distributed control

Introduction

The multi-agent system has been an important branch of distributed artificial intelligence from the late 20th century to the early 21st century. The goal is to solve large and complex practical problems that are beyond the capacity of a single agent. Its research involves knowledge， objectives， skills， planning and so on.The researchers mainly focus on the interactive communication， coordination and cooperation， conflict resolution among the agents， emphasize the close group cooperation among multiple agents， rather than the autonomy and exertion of individual abilities. The multi-agent system has autonomy， distribution， coordination， self-organizing ability， learning ability and reasoning ability. Using the multi-agent system to solve practical problems has strong robustness， reliability and high efficiency.

The consensus problem， as the basis of the coordinated control among multi-agents， has become a research hotspot in the field of control. Degroot firstly proposed the consensus theory[1]. The consensus[2]of multi-agent systems refers to that some states of agent will reach the same value or synchronization after a period of convergence time. Most of the existing research on the consensus control of multi-agent systems are limited to the analysis of individual dynamics of first-order or second-order integrators. However， the low-order linearization model ignores the high-order and nonlinear characteristics of the actual physical system， so that the dynamic characteristics of the system can not be accurately described.

Existing consensus problems can be roughly categorized into three classes， namely， leaderless consensus[3-6]， leader-follower consensus with one leader(also called cooperative tracking)[7-13]and containment control with multiple leaders[14-19]. In general， the first-order nonlinear multi-agent systems are mostly chained systems， and the second-order nonlinear multi-agent systems can be extended to high-order systems. Distributed containment control problems of high-order multi-agent systems with nonlinear dynamics are investigated， and the distributed adaptive nonlinear protocol is proposed based only on the relative state information， under which the states of the followers converge to the dynamic convex hull spanned by those of the leaders[14]. By using recursive method， the finite-time consensus control is developed， and the finite-time leader-following consensus problem is addressed for a class of high-order multi-agent systems with uncertain nonlinear dynamics[20]. Based on the sliding-mode auxiliary systems， an adaptive near-optimal protocol is finally presented to control high-order nonlinear multi-agent systems with fully unknown parameters[21].

In reality， all physical systems are inherently nonlinear， such as the robot system， air-craft system and induction motor system[22]. In the existing literatures， there are only few results about the consensus of nonlinear multi-agent systems. Hence， the nonlinearities in dynamics have been taken into consideration for many researchers recently， and they concern neuro-adaptive cooperative tracking control of unknown high-order affine nonlinear systems[23]. Tracking consensus is studied for a class of high-order nonlinear multi-agent systems with directed switching networks[24]. Compared with the results in Refs.[23-24]， the nonlinear term with the strict-feedback form is further introduced for the nonlinear systems in this paper. There is no doubt that this type of system can simulate many actuals in many fields， including wheeled mobile robots， dynamics of mainpulatorsetc.[25-26]Based on the discussion above， leader-following consensuses for a type of nonlinear systems with the strict-feedback form are proposed in this work. Motivated by our previous controllers designed for nonlinear systems[27-28]， we construct a novel distributed control algorithm for each follower agent， under which sufficient conditions are obtained for reaching a consensus. Under the directed topology， the leader sends its message to one or more followers. The main contributions of this paper are composed of three aspects. Firstly， compared with results in Refs.[23, 29]， we consider a class of more general nonlinear multi-agent systems in the strict-feedback form. Secondly， different from the results in Refs.[20, 25, 30]， neither the virtual controller nor the observer-based protocol is imposed on the systems. Therefore， a novel and simple control law is proposed in this paper. Thirdly， contrary to the methods in Refs.[25-26, 28, 31]， our consensus protocol is in a distributed fashion without using any global information， such as the eigenvalues of the corresponding Laplacian matrix or the agent’s own state information. Moreover， the directed topology is considered rather than undirected topology in Refs.[20-21].

This paper is organized as follows. The problem statements and preliminaries including some definitions， assumptions and lemmas are illustrated in section 1. In section 2， we design the controller for the system， and stability analyses are provided to ensure the consensus of the system. Simulation results are presented in section 3. The paper ends with a conclusion in section 4.

1 Problem Formulation and Preliminary

In this section， we give a brief account of some graph theory basis and present the leader-following consensus problem.

1.1 Notations

We use the following properties of Kronecker product.

(A?B)(C?D)=(AC)?(BD)，
(A+B)?C=(A?C)+(B?C)，

A?(B+C)=(A?B)+(A?C)，
‖IN?A‖=‖A‖.

1.2 Graph theory

A brief introduction on graph theory will be shown next.Assuming that a node represents a agent， directed graphG(A)={V，E，A} represents the exchange of information between agents.V={vi，i=1， 2， …，N} is the set of nodes，E?V×Vis the set of edges，A=[aij]∈RN×Nis the weighted adjacency matrix for the graph andaij≥0. If there exists an edge from nodejto nodei，i.e.， (vj，vi)∈E， thenaij>0. Assuming thataii=0 for alli∈Γ.Ni={j∈V(j，i)∈E} represents the set of neighbors of nodei， if the edge connecting the two nodes is directed， then graphGis a directed topology. Conversely， it is an undirected topology.

1.3 Problem statement

Consider a multi-agent system composed of one leader agent andN(N≥2) followers. The dynamics of theith (i=1， 2， …，N) agent can be described by the following differential equations in strict-feedback form

(1)

Dynamics of the leader node， labelled 0， is described in strict-feedback form as

(2)

Definition1[32]For any initial conditionxi(0)，i=1， 2， …，N， the leader-following consensus of the systems(1)-(2) is achieved if there exists a control inputuisuch that

The communication topology connecting the agent is considered to satisfy the assumption as below.

Assumption 1 implies that all the follower nodes have access to the leader node either directly or indirectly through directed paths. Without such a assumption， there would be at least one follower node that is isolated or acted as a leader， making the synchronization among all of the nodes impossible.

Themth order tracking error for nodeiis defined as

di， m=xi， m-x0， m，i∈Γ，m=1， 2， …，n.

Letdm=(d1， m，d2， m， …，dN， m)，

then

di=[di， 1，di， 2， …，di， n]T=xi-x0∈Rn×1，

D=[d1，d2， …，dN]T∈RNn×1，

where

xi=[xi， 1，xi， 2， …，xi， n]T∈Rn×1，

x0=[x0， 1，x0， 2， …，x0， n]T∈Rn×1，

xm=[x1， m，x2， m， …，xN， m]T∈RN×1，

In this paper， we consider the scenario in which the individual nodes are connected through a digraph. A particular node can access the state information of its neighbor nodes only. The neighborhood synchronization error is defined as

Algebraic derivation manipulations of systems (1)-(2) lead to

Let

ei=[ei， 1，ei， 2， …，ei， n]T∈Rn×1，m=1， 2， …，n.

where

U=u1，u2， …，uNT∈RN，

Let

where

A0=[-r1， -r2， …， -rn-1]T，

K=[r1，r2， …，rn]T，C=(1， 0， …， 0)，

G=[0， …， 0， 1]T，

fi=[fi， 1，fi， 2， …，fi， n]T∈Rn×1，

f0=[f0， 1，f0， 2， …，f0， n]T∈Rn×1，i=1， 2， …，N，His a Hurwitz matrix.

Thenei， mcan be collectively described as

Assumption2There exists a nonnegative constantl， and the nonlinear dynamicsf(x) satisfies the Lipschitz condition as follows.

‖f(a)-f(b)‖≤l‖a-b‖， ?a，b∈Rm.

2 Controller Design

In this section， based on the local state information of neighborhood agents， we propose the following distributed protocol to each follower.

ui=

(3)

whereK=[r1，r2， …，rn]T，r1，r2， …，rnare designed constants and chosen such thatsn+r1sn-1+…+rn-1s+rnis a Hurwitz polynomial， and it can be collectively described as

Theorem1Consider the followers and the leader whose dynamics are respectively described by systems(1) and(2). Considering the Assumptions 1 and 2， the consensus of the equations(1)-(2) is achieved under the protocol if there exist a positive definite matrixPthat makes the following condition hold.

HTP+PH=-I，

and

ProofChoose the Lyapunov function as follows.

whereP>0. DifferentiatingV(t) yields that

ET(IN?P)(IN?H)E-ET(IN?P)(IN?G)(L+B)U+

ET(IN?(PH))E-ET(IN?(PG))(L+B)U+

According to Assumption 2 and Lemma 1， we have

whereVz(Z) is a positive definite function.

3 Simulation and Results Analyses

Consider a three-node digraphGand a leader node described in Fig. 1. Obviously, Assumption 1 is satisfied.

The dynamics of the leader node is described as follows.

The follower nodes are described by the second-order nonlinear systems.

The initial states are chosen as follows.

x0=[1， -1]T，x1=[0.5， 0]T，x2=[-1.5， 0.5]T，x3=[-0.5， 1]T.

From Fig. 1， we write down the adjacency matrices and the corresponding Laplacian matrix as

Then， we can obtainHandPas

And

K=[2， 1]T，C=(1， 0).

Furthermore， we have the singular values ofPand 2-norm ofKandC. Hence，

In this simulation demonstration， we choose the design parameters as follows：r1=2，r2=1 such thatHis a Hurwitz matrix， and Assumption 2 and Theorem 1 hold. The simulation results are presented in Figs.2-5.

Fig. 2 State trajectories of xi, 1

As shown in Fig. 4 and Fig. 5， it is proven that， under the distributed control laws defined by Eq.(3)， the leader is followed by the three agents asymptotically and a desired tracking control performance is obtained.

The simulation results show the effectiveness of the distributed consensus control scheme proposed in this paper.

Fig. 3 State trajectories of xi, 2

Fig. 4 Evolution of position error di, 1

Fig. 5 Evolution of position error di, 2

4 Conclusions

In this paper， the leader-following consensus with the strict-feedback form is introduced and investigated for a type of multi-agent systems with some nonlinear dynamics under the directed topology. The distributed controller is proposed and ensures system’s stability， and the control signal is well defined. Future work will consider consensus with the communication delay and switching topology.