# A Distributed Formation Control Scheme with Obstacle Avoidance for Multiagent Systems

Authors: Xiaowu Yang; Xiaoping Fan
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This study considers the problem of formation control for second-order multiagent systems. We propose a distributed nonlinear formation controller where the control input of each follower can be expressed as a product of a nonlinear term that relies on the distance errors under the leader–follower structure. In the leader–follower structure, a small number of agents are assumed to be the leaders, and they are responsible for steering a group of agents to the specific destination, while the rest of the agents are called followers. The stability of the proposed control laws is demonstrated by utilizing the Lyapunov function candidate. To solve the obstacle avoidance problem, the artificial potential approach is employed, and the agents can avoid each possible obstacle successfully without getting stuck in any local minimum point. The control problem of multiagent systems in the presence of unknown constant disturbances is also considered. To attenuate such disturbances, the integral term is introduced, and the static error is eliminated through the proposed PI controller, which makes the system stable; the adaptive controller is designed to reduce the effect of time-varying disturbances. Finally, numerical simulation results are presented to support the obtained theoretical results.


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## Body

## 1. Introduction

Distributed or decentralized coordination control of multiagent systems has received increasing research attention in the control community; it has been widely used in many practical applications, such as unmanned surface vessels, unmanned aerial vehicles, satellite clusters, and military surveillance [1–7]. The multiagent systems are generally composed of a team of agents, in which all the agents are required to interact with others and then perform various kinds of tasks.Three main structures are commonly utilized in the existing literature for the formation control of multiagent systems, i.e., leader–follower, behavioral, and virtual structures. The leader–follower structure is widely used because of its simplicity and implementation, in which a small percentage of agents (or multiple leaders) are designated as the leaders, and the rest are called followers. The reference signals for a group of agents is assumed to be available to the designed leaders. In the behavioral structure, the specified behaviors, for example, collision and obstacle avoidance, are first specified for each agent. The agents possess a rigid geometric configuration using a virtual leader to present the formations in the virtual structure, where the virtual leader will never fail. The more advantages and disadvantages, as well as the distinctions among these three structures can be found in [8–10]. Two methods are considered for the formation control of multiagent systems, namely, the centralized [11] and decentralized controllers [12, 13], according to the interaction topology of sensing graph. In the centralized controller, each agent is assumed to have access to all the states collected from the central controller. This controller may fail to achieve the control objective in the presence of faults. In the decentralized controller, each agent can achieve its own tasks defined according to the desired formation only based on the local measurements, which might provide reliable control when communication between agents is limited or even unavailable [14]. As discussed above, the decentralized controller devised for each agent under the leader–follower structure is considered in this study. The geometrical structure of the desired formation can be defined by inter-distances, relative-positions, and inter-bearings. The distance-constrained formation control methods can be used to fulfill translational and rotational formation maneuvers [15], which can also be used to solve the obstacle avoidance problem; further, relative position constrained formation control methods are used to achieve the translational formation maneuvers [16], and bearing constrained formation control methods are employed to achieve the translational and scaling formation maneuvers [17], according to the properties of invariant of the constraints. Generally, the distance- and bearing-only-based formation control are nonlinear control problems, whereas the relative position-based formation control and bearing-based formation control are linear control problems [18–21]. The consensus patterns of multiagent systems have appeared in the literature [22, 23]. In [11], the authors proposed a class of nonlinear consensus control laws, which were applied in smart buildings, autonomous space satellites, and frequency of power systems, and PI controllers were designed for eliminating the state errors. In [10], distributed distance-based methods are employed to deal with the formation control for first order dynamics multiagent systems. The consensus problem with unknown disturbance is addressed in [24]. The obstacle avoidance problem is not considered in [10, 11]. Compared with the consensus, the formation control is more challenging as other physical constraints need to be satisfied.As mentioned, one of the important problems in the formation control of multiagent systems is the requirement to consider acquiring the desired control performance, which is referred to as obstacle avoidance problem. Cooperative obstacle avoidance of multiagent systems needs to be solved in many practical formation controls owing to its difficulty and complexity. To solve this problem, the artificial potential field (APF) (or) repulsive force approaches are applied. Geometrically, the obstacles can be viewed as static agents, which steer all the dynamic agents away from the static agents (obstacles). The obstacles can be modeled as a single particle/point [25], elliptical shapes, and rectangular shapes, in general [8, 13]. In [13], the distributed control strategy based on an artificial potential function is proposed to address the formation control and obstacle avoidance with the rectangular agents, which solved the obstacle avoidance problem. In [8], the authors investigated the formation control of a group of robots under virtual and behavioral structures, in which a rotational potential field is employed to drive the autonomous mobile agent away from the obstacle without getting stuck in the local minimum points. APF approaches are also utilized in [25] to deal with the obstacle avoidance problem. This study also considers the relative position-based formation control of multiagent systems in the presence of unknown constant disturbances. As is well known, multiagent systems are generally vulnerable to disturbances, which leads to failure in achieving the control objective. To eliminate the static errors caused by the bounded unknown constant disturbances, PI controllers are usually introduced for obtaining the desired performance when the systems suffer from constant disturbances, because the PI controllers have high-frequency gain and thus filter any noise produced by the input signals. In [11], distributed PI controllers are employed to address the consensus for first- and second- order dynamics of systems, which can attenuate the static errors caused by the unknown disturbances. In [17], using PI controllers, the authors proposed bearing-constrained formation control in the presence of input disturbances. Other practical problems such as acceleration saturation, and collision avoidance are also considered in that study. A similar idea was adopted in this study.As discussed above, formation control with obstacle avoidance problems for second-order dynamics of multiagent systems under the leader–follower control structure is addressed, and the formation control under the constant disturbances are also considered in this study. The main contributions are stated as follows.(a) The distributed control scheme for achieving and maintaining the desired formation is presented, which only depends on the local measurements, where the nonlinear interaction weights are employed to characterize the convergent behavior for agents appropriately.(b) The obstacle avoidance problem can be solved by employing the artificial potential function. The agents cannot get stuck into the local minimum points with the assistance of the navigation function, i.e., the attraction function created by the following agents and the leader agents, where the leader agents are assumed to have access to the reference signal for a team of agents.(c) Multiagent systems are susceptible to various disturbances, such as constant- and time-varying disturbances, while the bounded properties of these disturbances are required in general. We focus on the systems with the bounded disturbances in this study, which will cause a drift effect on the formation characterized by the relative measurements. To address this additional issue, we apply the PI controller to attenuate the bounded unknown constant disturbances, and the static errors can then be eliminated. The control systems with time varying disturbances are also discussed by using the adaptive technique.The remainder of this paper is organized as follows. In Section2, the basic concepts of graph theory and artificial potential function are given. The second-order dynamics of multiagent (or system model) are defined in Section 3. The distributed control algorithms for formation control of a multiagent system and artificial potential function approaches for obstacle avoidance of the agents based on leader–follower structure are proposed, respectively. The practical issue of constant disturbances for multiagent systems is studied by using the integral control action in Section 4. In Section 5, the simulation results are provided to illustrate the effectiveness of the proposed control strategy, and conclusions are provided in Section 6.


## 2. Preliminaries

LetG =(V, E) denote a directed graph consisting ofnvertices (nodes), where V={v1,…,vn} andE ⊆ {(vi,vj)∈V×V} containing the ordered pairs of vertices are called the node set and edge set, respectively; M=mijn×n is the weighted adjacency matrix. Each agent can be abstracted as a node in the graph. The edgeeij = (vi;vj) is created if the position information of agentjis available to agenti, which implies agentj is a neighbor of agenti; thus, the interconnection topology of a multiagent system can be modeled as a graph. The information interaction weights among the agents are characterized by the adjacency matrix M=mijn×n. Note thatmij > 0 ifeij ∈ E, otherwisemij = 0. Furthermore, the possibility of a self-loop is not considered, by convention; i.e.,mii = 0. The property of time-invariance ofGis assumed.In view ofM, the diagonal matrix D={σ1,⋯,σN} can be easily obtained, i.e., each entry σi is calculated as the sum ofi-th row ofM. Laplacian matrix of the directed graph is then given as(1)L=D-MFrom (1), one obtainslii > 0 andlij < 0,lii,lij ∈ L.The properties of the interaction topology of a graph are characterized by the Laplacian matrix. Note that the Laplacian matrix has the following useful properties:(i)L1 = 0, where1n=1,…,1T and0n=0,…,0T.(ii) only one eigenvalue is equal to 0, which is also referred to as the trivial eigenvalue. The real part of the rest of eigenvalues are positive.(iii)L is a positive semidefinite.With the above properties at hand, the following result is easily obtained; namely, the graph is connected if the smallest eigenvalue ofL is equal to zero, which is a straightforward consequence according to the property (ii).To achieve and maintain the desired formation, we further assume that the followers have access to the local information from the leaders under the leader–follower structure. We denote the weight matrix of interaction topology as ΔL = diag{δ1,…,δm}, wherem denotes the cardinality of a set of the leaders with a slight abuse of notion, and the sum of the diagonal entries of ΔL is greater than zero, i.e., δ1 + δ2 +…+ δm > 0. It is observed that the size ofL equals to size of ΔL, i.e., size(L) = size(ΔL). We choosem ≤ 0.2n, for the sake of generality, in this study.According to the properties of LaplacianL and weight matrix ΔL, the following result is easily obtained:(2)L+ΔL=l11+σ1⋯l1n⋮⋱⋮ln1⋯lnn+σnNote that the new matrix (L +ΔL) is a positive definite, which is a stable matrix, because all the eigenvalues of (L +ΔL) are always positive.Agentjis said to be a neighbor of agentiif agentjmoves to the feasible sensing area of agenti under the assumption that each agent has the limited sensing rangeRs. The neighborhood ofxis defined asΩz: = {p∈Rnd∣p–x<ϵ} withΩx and ϵ denoting the non-empty set and sufficiently small positive constant, respectively. Then, the following result is obtained; that is, the neighborhood setNi of agentiis defined as the set of agents (or nodes) that are adjacent to agenti, that isNi = {(i,j)∈E,E∈(V,V)}.Figure1 shows the information topologyG for directed graph of six nodes (viewed as agents). As per the directed graph theory, the following results are directly obtained as follows:V= {l,1,2,3,4,5},E= {(1,l),(2,l),(2,1),(3,1),(3,2),(4,2),(4,3),(5,3),(5,4)}. Assume that the total neighbor set for six agents, i.e., denoted by N(G) =Nl ∪ N1 ∪ N2 ∪ N3 ∪ N4 ∪ N5 withNl = {∅},N1 = {l},N2 = {1},N3 = {1,2},N4 = {2,3}, andN5 = {3,4}, respectively. There is no loop, i.e.,mii = 0 and self-loop in the directed graph.Figure 1
Information interaction of network under the leader follower structure. The blue circle and red circles denote the leaderland followers, labelled by 1,…, 5, respectively.Assumption 1.
To generate the sensing interaction topologyG of the directed graph, e.g., as shown in Figure 1, there exist any three edgeseij,ejk, andeki in the edge setE such that the directed graph satisfies the following conditions: eij+ejk<eki, eij+eki<ejk, and ejk+eki<eij. The connected graph considered in this study is assumed.We further assume that the desired (or target) formation is infinitesimally rigid by applying the Henneberg sequence in this study [10, 26].Assumption 2.
The control gainγij(·) of agenti is continuous and Lipschitz, which only depends on the differenceyij of inter-distance between all pairs of agents, for all (i,j) ∈ E, namely, γij(αij,yij) =sij αij (exp⁡(yij)-1) fory; otherwise, γij(αij,yij) =sijyij, where αij ∈ R+ are hyper-parameters, and γij ∈ R+ are scale factors. Note that αij is also interpreted as interaction function.Remark 3.
To obtain the control law for each follower agenti, we introduce the nonlinear control gain γij(αij,yij). As expected, agenti can follow its neighbors with the same direction (or orientation), which can guarantee the desired convergence performance for the overall system under the Assumption 2. In other words,yij = 0 implies γij(αij,yij) = 0, the desired formation is achieved, as will be shown in Section 4. Note that the distributed control law of agenti for the part of controlling the geometrical formation is nonlinear.To solve the obstacle avoidance problem, we introduce the concept of artificial potential function as follows.The artificial potential function (APF)VAPF(yik) is a nonnegative differentiable and radially unbounded between agentsi and obstaclesk such that(a)VAPF(yik) → 0 if and only ifyik ≥RAPF,(b)VAPF(yik) → ∞ asyik approaches to zero,whereyik denotes the distance errors between the actual distancedik and the minimal repulsive distance threshold δik, i.e.,yik = dik-δik,RAPF denotes the maximal distance that the artificial potential can be generated. It is further assumed thatRAPF is less than the sensing rangeRs of each agent, i.e.,RAPF <Rs.Consider an artificial potential function (APF) for agentsi as(3)Vi_APF=∑k=1mVAPFyikThe time derivative ofVi_ APF(yik) is given as(4)V˙i_APF=∇qi∑k=1mVAPFyikApplying the gradient-based control approach, the repulsive force for each agenti is defined as(5)uobs=-V˙i_APFIf agent is approaching the obstaclek, the repulsive force is generated to steer agenti to move toward gradient direction of the repulsive potential function (3), i.e., the agenti is driven away from the obstacle, which implies that obstacle avoidance problem can be solved by implementing the obstacle control lawuobs, implying that the agents can be steered away from any possible encountered obstacle.


## 3. Problem Statement and Control Objective

### 3.1. System Model

It is assumed that each agent is endowed with second-order integrator dynamics. Therefore, the multiagent system consisting ofn agents is characterized as(6)q˙i=pi,p˙i=fqi,pi+gqi,piuiwhereqi ∈ Rm is the position andpi ∈ Rm is the velocity. The position and control input of the agents can be stacked into q=q1T,…,qiT,…,qnTT with qi=qi1,…,qimT ∈ Rm, and p=p1T,…,piT,…,pnTT with pi=pi1,…,pimT ∈ Rm, respectively. ui=ui1,…,uimT ∈ Rm is the control input, and u=[u1,…,um].f(·) ∈ Rm is the continuously differentiable function withf(·) = 0;g(·) ∈ Rm are the smooth functions withg(·) ≠ 0 for allq andp.To continue the analysis, we define the relative position and their error vectors as(7)zir=qi-qr,zij=qi-qj.and(8a)eqir=zir-zird,epir=pi-pr.(8b)andeqij=zij-zijd,epij=pi-pj.respectively.Taking the first derivative of (8a) and (8b), we obtain the error dynamics(9a)e˙qir=epir,e˙pir=p˙i-p˙r(9b)ande˙qij=epij,e˙pij=p˙i-p˙jAs discussed in the above sections, it is assumed that the leader dynamics is given as(10)q˙r=pr,p˙r=fqr,prwhereqr ∈ Rm andpr ∈ Rm represent the position and velocity parameters of leader agent, respectively. Note that the knowledge of (10) is now available to the follower agents.Remark 4.
Under the leader following control scheme, equation (6) is used to characterize the reference trajectory for a team of agents. Then, the desired formation can be achieved and maintained under the manipulation of the control law proposed in the sequel.Assumption 5 (see [11, 25]).
A functionf(·) with domainsP andQ is Lipschitz if there exist two positive constants ζL=[ζLq,ζLp] ∈ R+: forp∈ P,q∈ Q, andi = 1, …,n.(11)f~·≤ζLqqi-qr+ζLppi-prwith(12)f~·=fqi,pi-fqr,prNote that ζL=[ζLq,ζLp] are viewed as the Lipschitz operators.Assumption 6.
We assume that two system reference variablesqr andpr for the leader agent satisfy the following properties:(13)-ς1≤qr≤ς1,-ς2≤pr≤ς2.for some positive constants ς1 ∈ R+ and ς2 ∈ R+.Assumption 7 (see [18]).
For a given desired formation control, a Lyapunov function candidateV(z) andz(p) satisfy the following constraints.
(a) Θ(γ) is compact for any γ ≥ 0.
(b) There existγe=[γep,γeq] > 0 such thatV(z) = 0 if and only if z = γe in Θ(γ). Note that the converse is also true, i.e., z = γe impliesV(z) = 0.
(c) For any given boundedz, ∂z(p)/∂p and u(z,p) are bounded.
Assumption7 means that z = γe are the asymptotically stable equilibrium values for the system. Θ(γ) is viewed as the attraction region owing to the invariance set principle. The formation control system is locally stable if Θ(γ) is the local space; the system is globally stable, otherwise. In practice, it is difficult to obtain the attraction region of the whole space for nonlinear control systems. It is noted that the constraint (b) in Assumption 7 implies Assumption 6. For example, in distance-based formation control, as addressed in [8, 10, 27], the desired formation can be achieved if and only if the actual interdistance converges to the desired interdistance as time tends to infinity. Thus, Assumption 7 is reasonable and verified.With the above results at hand, we introduce the following useful definition.Definition 8 (achieving the desired formation).
The desired formation for a multiagent is said to be achieved if the following constraints are satisfied under the Assumptions5 – 7.(14)limt→∞⁡qi-qr=qid-qrdlimt→∞⁡qi-qj=qid-qjd,j∈Niand (15)limt→∞⁡pi-pr=0,limt→∞⁡pi-pj=0,j∈Ni.respectively. Here, qid–qrd is the desired local displacement between agentiand the leader, and qid–qjd is the desired local displacement (or position) between agentiand follower agentj. The velocitypi of agenti converges to the reference velocitypr as timet evolves. Note that equation (9a) and (9b) not only determines the orientation of the desired formation but also specifies the desired interdistances among agents by calculating the magnitudes of qid – qrd and qid – qjd, respectively.We are now prepared to state the control objective for multiagent system as follows.


### 3.2. Control Objective

The control objective in this study is to design the distributed control law such that the system satisfies the physical constraints given as follows:(a) achieve the desired formation shape;(b) track the reference trajectory with the same reference velocity for each follower agent;(c) solve the obstacle avoidance problem.To fulfill the control objective, we first present the error dynamics as follows.Substituting (6), (10), and (12) into (9a) and (9b), yields(16)e˙qir=epir,e˙pir=f~·+gqi,piuir.which is expressed in a condensed form as(17)e˙qire˙pir=0m×mIm0m×m0m×meqirepir+0n×mf~·+0n×mgqi,piuirand(18)e˙pij=eqij,e˙qij=f~·+gqi,piuij.which is also rewritten in a condensed form as(19)e˙qije˙pij=0m×mIm0m×m0m×meqijepij+0n×mf~·+0n×mgqi,piuijrespectively, where e=eqT,epTT, eqT=eq1T,…,eqnTT, epT=ep1T,…,epnTT, f(·)=f1T·,…,fnT·T, and u(·)=u1T,…,unTT.


## 3.1. System Model

It is assumed that each agent is endowed with second-order integrator dynamics. Therefore, the multiagent system consisting ofn agents is characterized as(6)q˙i=pi,p˙i=fqi,pi+gqi,piuiwhereqi ∈ Rm is the position andpi ∈ Rm is the velocity. The position and control input of the agents can be stacked into q=q1T,…,qiT,…,qnTT with qi=qi1,…,qimT ∈ Rm, and p=p1T,…,piT,…,pnTT with pi=pi1,…,pimT ∈ Rm, respectively. ui=ui1,…,uimT ∈ Rm is the control input, and u=[u1,…,um].f(·) ∈ Rm is the continuously differentiable function withf(·) = 0;g(·) ∈ Rm are the smooth functions withg(·) ≠ 0 for allq andp.To continue the analysis, we define the relative position and their error vectors as(7)zir=qi-qr,zij=qi-qj.and(8a)eqir=zir-zird,epir=pi-pr.(8b)andeqij=zij-zijd,epij=pi-pj.respectively.Taking the first derivative of (8a) and (8b), we obtain the error dynamics(9a)e˙qir=epir,e˙pir=p˙i-p˙r(9b)ande˙qij=epij,e˙pij=p˙i-p˙jAs discussed in the above sections, it is assumed that the leader dynamics is given as(10)q˙r=pr,p˙r=fqr,prwhereqr ∈ Rm andpr ∈ Rm represent the position and velocity parameters of leader agent, respectively. Note that the knowledge of (10) is now available to the follower agents.Remark 4.
Under the leader following control scheme, equation (6) is used to characterize the reference trajectory for a team of agents. Then, the desired formation can be achieved and maintained under the manipulation of the control law proposed in the sequel.Assumption 5 (see [11, 25]).
A functionf(·) with domainsP andQ is Lipschitz if there exist two positive constants ζL=[ζLq,ζLp] ∈ R+: forp∈ P,q∈ Q, andi = 1, …,n.(11)f~·≤ζLqqi-qr+ζLppi-prwith(12)f~·=fqi,pi-fqr,prNote that ζL=[ζLq,ζLp] are viewed as the Lipschitz operators.Assumption 6.
We assume that two system reference variablesqr andpr for the leader agent satisfy the following properties:(13)-ς1≤qr≤ς1,-ς2≤pr≤ς2.for some positive constants ς1 ∈ R+ and ς2 ∈ R+.Assumption 7 (see [18]).
For a given desired formation control, a Lyapunov function candidateV(z) andz(p) satisfy the following constraints.
(a) Θ(γ) is compact for any γ ≥ 0.
(b) There existγe=[γep,γeq] > 0 such thatV(z) = 0 if and only if z = γe in Θ(γ). Note that the converse is also true, i.e., z = γe impliesV(z) = 0.
(c) For any given boundedz, ∂z(p)/∂p and u(z,p) are bounded.
Assumption7 means that z = γe are the asymptotically stable equilibrium values for the system. Θ(γ) is viewed as the attraction region owing to the invariance set principle. The formation control system is locally stable if Θ(γ) is the local space; the system is globally stable, otherwise. In practice, it is difficult to obtain the attraction region of the whole space for nonlinear control systems. It is noted that the constraint (b) in Assumption 7 implies Assumption 6. For example, in distance-based formation control, as addressed in [8, 10, 27], the desired formation can be achieved if and only if the actual interdistance converges to the desired interdistance as time tends to infinity. Thus, Assumption 7 is reasonable and verified.With the above results at hand, we introduce the following useful definition.Definition 8 (achieving the desired formation).
The desired formation for a multiagent is said to be achieved if the following constraints are satisfied under the Assumptions5 – 7.(14)limt→∞⁡qi-qr=qid-qrdlimt→∞⁡qi-qj=qid-qjd,j∈Niand (15)limt→∞⁡pi-pr=0,limt→∞⁡pi-pj=0,j∈Ni.respectively. Here, qid–qrd is the desired local displacement between agentiand the leader, and qid–qjd is the desired local displacement (or position) between agentiand follower agentj. The velocitypi of agenti converges to the reference velocitypr as timet evolves. Note that equation (9a) and (9b) not only determines the orientation of the desired formation but also specifies the desired interdistances among agents by calculating the magnitudes of qid – qrd and qid – qjd, respectively.We are now prepared to state the control objective for multiagent system as follows.


## 3.2. Control Objective

The control objective in this study is to design the distributed control law such that the system satisfies the physical constraints given as follows:(a) achieve the desired formation shape;(b) track the reference trajectory with the same reference velocity for each follower agent;(c) solve the obstacle avoidance problem.To fulfill the control objective, we first present the error dynamics as follows.Substituting (6), (10), and (12) into (9a) and (9b), yields(16)e˙qir=epir,e˙pir=f~·+gqi,piuir.which is expressed in a condensed form as(17)e˙qire˙pir=0m×mIm0m×m0m×meqirepir+0n×mf~·+0n×mgqi,piuirand(18)e˙pij=eqij,e˙qij=f~·+gqi,piuij.which is also rewritten in a condensed form as(19)e˙qije˙pij=0m×mIm0m×m0m×meqijepij+0n×mf~·+0n×mgqi,piuijrespectively, where e=eqT,epTT, eqT=eq1T,…,eqnTT, epT=ep1T,…,epnTT, f(·)=f1T·,…,fnT·T, and u(·)=u1T,…,unTT.


## 4. Formation Control and Obstacle Avoidance

As discussed in the previous sections, it is supposed that each follower has no more than two neighbors in the information interaction topology of the directed graph. The leader has access to all the information, i.e., position and velocity, of the reference trajectory, while the followers do not know the information from the reference trajectory. To achieve the control target, we propose the control law as follows, the stability analysis of multiagent system is provided, then the distributed control algorithm for obstacle avoidance problem is presented. As a result, we analyze the stability properties of system according to the formation and obstacle control, without loss of generality, followed by the associated proofs.


### 4.1. Design of Formation Control

The general formation control lawui is given as(20)ui=1gqi,piuij+uir︸Formationcontrolterm+ui_obs︸obstacleavoidanceterm-f~·where(21)uij=mijqj-qi+qid-qjd+mijpj-pi,uir=mikqj-qi+qid+mikpj-pi.with qid-qid=dijd∈R+, qid=dird∈R+, and qjd=djrd∈R+ denoting the desired distance of edges which connect agenti toj and the leadersl, respectively, which satisfy Assumptions 1 and 2. The control in (20) is clearly distributed as the control of agenti only relies on the local measurements from its neighboring agents (the leader or other followers). Here,mij ∈ R andmir ∈ R denote some constant numbers, respectively; they are taken from the adjacency matrixM. Physically,mij represents the interaction weights between agentsi andj, andmik represents the interaction weights between agenti andk-thleader agent.ui_obs will be given in Section 4.3. As observed in (21), the control gains are some unknown positive constants; therefore, equation (21) is clearly linear, while in many practical applications, the control gains (or called weights) are varied with the interdistances between the follower agenti and their neighbors. To better characterize these behaviors, mathematically speaking, we will explicitly give the notion of γij(αij,yij) and γir(αir,yir) as follows.The control gainsγij(αij,yij) and γir(αir,yir) [11] for the follower agenti and the leaders and follower agentj are assumed as(22)γirαir,yir=sir11+exp⁡yir,yir<0,dir,otherwise.and(23)γijαij,yij=sij11+exp⁡yij,yij<0,dij,otherwise.respectively.Here,sij and αij remain the same as Assumption 2.yij is defined asyij = zij-zijd=dij-dijd, for alli,j = {m-1,m-2,…,n}.Consider the follower agent and the leaders, for ease of notation, simply calledj. As a result, the geometrical interpretation of (18) is provided and stated as follows: (1) agenti moves away from the agentj if the initial distance zij is less than the desired distance zijd, and then the control gains are chosen assij/(exp(yij) + 1); (2) agenti moves toward from the agentj if the initial distance zij is greater than the desired distance zijd, and then the control gains are selected assijdij. Therefore, the desired formation is achieved if and only ifyij = 0 as timet evolves, i.e., zij=zijd, which implies the actual velocityvi converges to the reference velocityvr in terms of (7), (8a), and (8b).As outlined in the previous sections, in order to achieve the control objective, assume that at least one follower has a neighbor of the leader agent. By replacingmij andmik with γij(αij,yij) and γik(αik,yik), respectively, in terms of (8a) and (8b), we obtain(24)eqi·=mij·γijαij,yijqj-qi+qid-qjd+mij·γijαij,yijqk-qi+qidand(25)epi·=mij·γijαij,yijpj-pi+mij·γijαij,yijpk-pirespectively.Differentiating (24) and (25), we have(26)e˙qi·=mij·γijαij,yijpj-pi+mij·γijαij,yijpk-pi=epiand(27)e˙pi·=ui+f~·respectively.To continue the analysis, define the formation control as(28)ui=1gqi,pieqi+epi+ui_obs-f~·It is noted that (28) represents distributed (or decentralized) nonlinear control laws as (28) only depends on the local information from their neighbors.Substituting (28) into (27) yields(29)e˙pi·=eqi+epi+ui_obsWith the above results in hand, we are now ready to present the formation stability analysis as follows. As a rule, the stability analysis for the overall system is divided into two parts: (1) formation stability analysis and (2) obstacle avoidance analysis. The analyses for the above two parts are presented in Sections 4.2 and 4.3, respectively, as follows.


### 4.2. Formation Stability Analysis

Define the Lyapunov function candidate as(30)ViF=12∑i=1neqi2+epi2which is continuously differentiable.To proceed, we define the formation error dynamics as(31)e˙qie˙pi=-γi·0n×n-InL+ΔLL+ΔL︸AL⊗Imeqiepiwhere γi(·) = diag{γik,…,γij,…,γin}.Taking the first time derivative ofViF, one gets(32)V˙iF=eqiTQ-e˙qi+epiTQ-e˙pi=-eqiTQ-epi+epiTQ-eqi+epiSubstituting (31) into (32) yields(33)V˙iF=-eqiepiTQeqiepi≤0where(34)Q=γi·0n×nInL+ΔLL+ΔL⊗ImTQ-+Q-T0n×nInL+ΔLL+ΔL⊗ImSince the positive semidefinite of (34), it follows that (33) is verified.Theorem 9.
Consider the multiagent system (6) with the reference trajectory (10) under the leader-following structure. For any given bounded initial condition, the control objective is achieved under the action of the proposed control law (20).Proof.
The Lyapunov function candidate in (30) attains the minimal value if and only ifeqi andepi are simultaneously equal to zero, i.e.,zij − zijd=0, zik-qid=0, andpi =pj =pr because of nonnegative control gains γij(αij,yij) and γik(αik,yik). In terms of (33), the first time derivative ofViF is seminegative positive, i.e., ∂ViF/∂t ≤ 0, implying thatViF(t) −ViF(0) ≤ 0. Therefore, the system is asymptotically stable. This concludes the proof.Lemma 10.
The matrixAL in (31) is positive definite, implying that −AL is Hurwitz (or stability), for anymij andsij; namely, the system is asymptotically stable.Proof.
Lettingλ be the eigenvalues of −AL, then(35)detλIn+AL=detλIn-γi·γi·L+ΔLλIn+γi·L+ΔL=detλ2In+λγi·L+ΔL+γi2·L+ΔL
As a consequence, det(λIn +AL) = 0, and we obtain(36)λ=12-γi·L+ΔL+γi·L+ΔL2-4L+ΔLThe eigenvalues λ in (36) have a strictly negative real part according to Definition8, implying that −AL is Hurwitz, which completes the proof.


### 4.3. Obstacle Avoidance in the Leader-Follower Structure

In this subsection, we propose a distributed control strategy for avoiding obstacles of a group of autonomous agents based on the leader–follower structure. In this study, geometrically, it is assumed that each obstacle k is viewed as a circle. Without loss of generality, the boundary of each obstacle can be recognized by each agent equipped with visual sensor.Consider the artificial potential function as(37)Vi_APF=∑k=1mdik-δik-2,δik<dik≤RAPF.0,dik>RAPF.where dik=qi–qk denotes the distance between agenti and obstaclek. δik remains the same as in Definition8.From (37), the artificial potential function is unbounded when agenti gets close to the obstaclek, namely,dik → δik. The artificial potential is produced or vanishes if the distancedik lies in (δik,RARF) ordik >RARF. As a result, the associated repulsive force for each agenti is devised as(38)ui_obs=0,dik>RAPF.-∑k=1m2dik-δik-3pi-pjdik,δik<dik≤RAPF.Using the control law (38), we can solve the obstacle avoidance problem.Agent 1 can avoid two obstacles without getting stuck in the local minimum points by using the attractive force produced by agent 1 and the reference trajectory, which is illustrated in Figure2.Figure 2
Obstacle avoidance of agent 1. The blue circle and black square denote the initial and final positions of agent 1, respectively. Two different obstacles are considered, which are colored with red and blue, respectively. The local minimum points are given.


### 4.4. Bounded Constant and Time-Varying Disturbances

In many practical applications, multiagent systems are vulnerable to exterior constant disturbances. The constant disturbances (which can sometimes be viewed as constant velocities) cannot be rejected by the distributed proportional control laws. Toward this end, we apply the PI controller for double-integrator dynamics to achieve the desired formation under the constant (or static) disturbances.To continue our analysis, we analyze a distributed (or decentralized) PI formation controller with the constant disturbances for a multiagent system, where each agenti only uses the local position and velocity measurements from its neighbors. Hence, this controller not only provides an advantage of eliminating the static errors but also improves the convergence performance. Therefore, the desired formation is achieved under the proposed control laws in the presence of constant disturbances. We first devise the distributed controller for agenti as(39)ξ˙ij=qj-qi+qid-qjd,ui=mikqj-qi+qid+mikpj-pi+ciξij.withci denoting some positive constant number, which is referred to as integral gains. Equation (39) relies only on the local measurements. The diagram of the distributed PI controller is shown in Figure 3. Note that the system for each agenti may be unstable when a minor local measurement error occurs.Figure 3
Block diagram of distributed PI controller. Agenti only has access to the local measurements.Integrating the first equation of (39), we obtain(40)ξij=qid-qjdt-∫0tqiτ-qjτdτFrom (36), each agenti has only access to local relative positions (or measurements) and the time for obtaining the desired estimate of the integral error states.Consider again the double-integrator dynamics for multiagent systems as(41)q˙i=pip˙i=ui+diwithdi denoting the unknown constant disturbances.The main result of this part is given by Theorem11 as follows.Theorem 11.
Consider the multiagent system (41) under the connected interaction graphG. Using the formation control law (39), the static errors are eliminated, and the desired formation is achieved under the Assumption 4.1, i.e., limt→∞⁡qi-qj=qdi –qdj and limt→∞⁡pi-pj=0 fori = 1…,n,j∈ Ni.Remark 12.
To address the problem in the presence of the constant disturbances, we introduce the control term of integral action, which can eliminate the static errors and guarantee the desired performance. In other words, the actual relative position vector converges to the specified relative position vector; thus, the control objective is achieved.Proof.
From (39) and (41), by introducing a new state vector ξi=ξ1j,…,ξnjT, we may write the corresponding augmented matrix in a compact form as(42)ξ˙ie˙qie˙pi=0n×nIn0n×n0n×n0n×nIn-kqiL+ΔL-kpiL+ΔL-ciInξieqiepi+0n×10n×1cn×1qid-qjd+0n×10n×1dn×1fori = {1,…,n}, together with the bounded initial condition. Here,kqi ∈ R+ andkpi ∈ R+ are positive numbers, respectively, qid-qjd is the desired relative position vector given in advance, and dn×1 are constant (static) disturbances. Note that the aim of (42) is to stabilize system with respect to the parameters, i.e., ξi,eqi,epi.
Taking the time derivative of (42), we obtain(43)ξ¨ije¨qie¨pi=0n×nIn0n×n0n×n0n×nIn-kqiL+ΔL-kpiL+ΔL-ciIn︸≔AAugξ˙ije˙qie˙pi
To obtain the eigenvalues ofAAug, the characteristic polynomial ofAAug is calculated as(44)det⁡AAug=λIn-AAug=sIn-In0n×n-0n×nsIn-InkqiL+ΔLkpiL+ΔLsIn+ciIn=s3+cis2+kpiL+ΔLs+kqiL+ΔL=0
To continue the analysis, consider the characteristic polynomial of (L+ ΔL) as(45)s3+cis2+kpiλis+kqiλi=0where λi ∈ R+ are the eigenvalues of (L + ΔL), i.e., ςi > 0.
Comparing (44) with (45), we learn that all the eigenvalues ofAAug have strictly negative real parts because of the positive eigenvalues of λi as mentioned before, which impliesAAug is Hurwitz (or stable), and the system is therefore asymptotically stable. Furthermore, one can also place the poles according to the characteristic polynomial (47) such that the system is asymptotically stable. To this end, the control gain can be easily computed by using the command of place (·) in MATLAB. As discussed,AAug is Hurwitz, and we then obtain(46)s3+cis2+kpiλis+kqiλi=0
From (46), one can get(47)limt→∞qj-qi=qid-qjdwhich implies that the proposed distributed PI controller (39) can eliminate the static errors caused by constant but unknown disturbances and implies that the relative positions between agentsi andj converge to the desired relative positions, the desired formation is achieved and, maintained, meanwhile the velocity errors between agentsi andj converge to zero. The proof is thus completed.Remark 13.
If the errors are equal to zero, i.e., the actual relative position equals the desired relative position, the proposed controller stops to calculate the changes, implying that the desired formation is achieved. It is noted that the convergence rate of the overall system is dominated by the smallest eigenvalue ofAAug.Example 14.
Without loss of generality and for brevity, consider an agent 1 with initial positionq1(0) = 0.420.50T and its second-order dynamics (41) with the bounded constant disturbances given asd1 = 0.780.25T andd2 = 3.264.92T, and the control gains arekq = 1.35 andkp = 1.35 and integral gainci = 0.75, respectively.Figure4 illustrates the trajectories of agent 1 in the presence of disturbances, where the left and right figures show the comparison results concerning the static performance between with and without the aim of integral control terms. As expected, the static error is successfully eliminated by introducing the integral control terms, depicted by blue lines in the left subfigures, whereas the static error is not eliminated without the aim of the integral control, shown in red lines in the right subfigures, in x and y coordinates, respectively. The right two subfigures show the convergence performance of the system in the presence of two different constant disturbances. The convergence rate is bigger when the disturbance is smaller, i.e.,d1 = 0.780.25T, and the resulting trajectories of agent 1 are depicted by blue lines. Therefore, the system is asymptotically stable under the proposed control law.Figure 4
State trajectory of agent 1 in the presence of constant disturbances. Left: using the integral control, Right: without the help of the integral control for the disturbances.With the above results in hand, to attenuate the bounded time-varying disturbances, assume thatdit ≤θiφi(t), whereθi is unknown constant,φi(t) is bounded function of timet, the robust controller is derived by the following procedure:(48)Vtd=∑j∈Ni12aijeqi2+12βijepi2+12θ~iγi-1θ~iwhere θ^i is an estimate of θi, θ~=θ-θ^ is the parameter error; αij, βij, and γi are positive constants. Equation (48) is continuously differentiable. The time derivative ofVta is(49)V˙td=∑j∈Niaijeqie˙qi+βijepie˙pi+θ~iγi-1θ~˙=∑j∈Niaijeqiepi+βijepiui+βijepidi+θ~iγi-1θ~˙=∑j∈Nieqiepi+βijepiui+βijepiθiφt+θ~iγi-1θ~˙Design the adaptive controlui as (50)ui=∑j∈Ni-aijβijeqi-epi-θ^iφtSubstituting (50) into (49), we have (51)V˙td=∑j∈Ni-βijepi2+βijepiθi-θ^iφt+θ~iγi-1θ~˙To guarantee the negative semidefinite of V˙td, the adaptation law is chosen as(52)θ~˙=-γiβijepiφt-γiθi-θ^iSubstituting (52) into (51) yields(53)V˙td=∑j∈Ni-βijepi2-θi-θ^iTθi-θ^iwhich is continuously differentiable function and negative semi-definite, implying that Vtd(t) − Vtd(0) ≤ 0, with the equality is satisfied if and only ifepi = 0 and θ^i= θi; it follows thatepi = 0. Therefore, the system is asymptotically stable, the desired formation is achieved, and the time-varying disturbances are attenuated under the adaptive control law (50).


## 4.1. Design of Formation Control

The general formation control lawui is given as(20)ui=1gqi,piuij+uir︸Formationcontrolterm+ui_obs︸obstacleavoidanceterm-f~·where(21)uij=mijqj-qi+qid-qjd+mijpj-pi,uir=mikqj-qi+qid+mikpj-pi.with qid-qid=dijd∈R+, qid=dird∈R+, and qjd=djrd∈R+ denoting the desired distance of edges which connect agenti toj and the leadersl, respectively, which satisfy Assumptions 1 and 2. The control in (20) is clearly distributed as the control of agenti only relies on the local measurements from its neighboring agents (the leader or other followers). Here,mij ∈ R andmir ∈ R denote some constant numbers, respectively; they are taken from the adjacency matrixM. Physically,mij represents the interaction weights between agentsi andj, andmik represents the interaction weights between agenti andk-thleader agent.ui_obs will be given in Section 4.3. As observed in (21), the control gains are some unknown positive constants; therefore, equation (21) is clearly linear, while in many practical applications, the control gains (or called weights) are varied with the interdistances between the follower agenti and their neighbors. To better characterize these behaviors, mathematically speaking, we will explicitly give the notion of γij(αij,yij) and γir(αir,yir) as follows.The control gainsγij(αij,yij) and γir(αir,yir) [11] for the follower agenti and the leaders and follower agentj are assumed as(22)γirαir,yir=sir11+exp⁡yir,yir<0,dir,otherwise.and(23)γijαij,yij=sij11+exp⁡yij,yij<0,dij,otherwise.respectively.Here,sij and αij remain the same as Assumption 2.yij is defined asyij = zij-zijd=dij-dijd, for alli,j = {m-1,m-2,…,n}.Consider the follower agent and the leaders, for ease of notation, simply calledj. As a result, the geometrical interpretation of (18) is provided and stated as follows: (1) agenti moves away from the agentj if the initial distance zij is less than the desired distance zijd, and then the control gains are chosen assij/(exp(yij) + 1); (2) agenti moves toward from the agentj if the initial distance zij is greater than the desired distance zijd, and then the control gains are selected assijdij. Therefore, the desired formation is achieved if and only ifyij = 0 as timet evolves, i.e., zij=zijd, which implies the actual velocityvi converges to the reference velocityvr in terms of (7), (8a), and (8b).As outlined in the previous sections, in order to achieve the control objective, assume that at least one follower has a neighbor of the leader agent. By replacingmij andmik with γij(αij,yij) and γik(αik,yik), respectively, in terms of (8a) and (8b), we obtain(24)eqi·=mij·γijαij,yijqj-qi+qid-qjd+mij·γijαij,yijqk-qi+qidand(25)epi·=mij·γijαij,yijpj-pi+mij·γijαij,yijpk-pirespectively.Differentiating (24) and (25), we have(26)e˙qi·=mij·γijαij,yijpj-pi+mij·γijαij,yijpk-pi=epiand(27)e˙pi·=ui+f~·respectively.To continue the analysis, define the formation control as(28)ui=1gqi,pieqi+epi+ui_obs-f~·It is noted that (28) represents distributed (or decentralized) nonlinear control laws as (28) only depends on the local information from their neighbors.Substituting (28) into (27) yields(29)e˙pi·=eqi+epi+ui_obsWith the above results in hand, we are now ready to present the formation stability analysis as follows. As a rule, the stability analysis for the overall system is divided into two parts: (1) formation stability analysis and (2) obstacle avoidance analysis. The analyses for the above two parts are presented in Sections 4.2 and 4.3, respectively, as follows.


## 4.2. Formation Stability Analysis

Define the Lyapunov function candidate as(30)ViF=12∑i=1neqi2+epi2which is continuously differentiable.To proceed, we define the formation error dynamics as(31)e˙qie˙pi=-γi·0n×n-InL+ΔLL+ΔL︸AL⊗Imeqiepiwhere γi(·) = diag{γik,…,γij,…,γin}.Taking the first time derivative ofViF, one gets(32)V˙iF=eqiTQ-e˙qi+epiTQ-e˙pi=-eqiTQ-epi+epiTQ-eqi+epiSubstituting (31) into (32) yields(33)V˙iF=-eqiepiTQeqiepi≤0where(34)Q=γi·0n×nInL+ΔLL+ΔL⊗ImTQ-+Q-T0n×nInL+ΔLL+ΔL⊗ImSince the positive semidefinite of (34), it follows that (33) is verified.Theorem 9.
Consider the multiagent system (6) with the reference trajectory (10) under the leader-following structure. For any given bounded initial condition, the control objective is achieved under the action of the proposed control law (20).Proof.
The Lyapunov function candidate in (30) attains the minimal value if and only ifeqi andepi are simultaneously equal to zero, i.e.,zij − zijd=0, zik-qid=0, andpi =pj =pr because of nonnegative control gains γij(αij,yij) and γik(αik,yik). In terms of (33), the first time derivative ofViF is seminegative positive, i.e., ∂ViF/∂t ≤ 0, implying thatViF(t) −ViF(0) ≤ 0. Therefore, the system is asymptotically stable. This concludes the proof.Lemma 10.
The matrixAL in (31) is positive definite, implying that −AL is Hurwitz (or stability), for anymij andsij; namely, the system is asymptotically stable.Proof.
Lettingλ be the eigenvalues of −AL, then(35)detλIn+AL=detλIn-γi·γi·L+ΔLλIn+γi·L+ΔL=detλ2In+λγi·L+ΔL+γi2·L+ΔL
As a consequence, det(λIn +AL) = 0, and we obtain(36)λ=12-γi·L+ΔL+γi·L+ΔL2-4L+ΔLThe eigenvalues λ in (36) have a strictly negative real part according to Definition8, implying that −AL is Hurwitz, which completes the proof.


## 4.3. Obstacle Avoidance in the Leader-Follower Structure

In this subsection, we propose a distributed control strategy for avoiding obstacles of a group of autonomous agents based on the leader–follower structure. In this study, geometrically, it is assumed that each obstacle k is viewed as a circle. Without loss of generality, the boundary of each obstacle can be recognized by each agent equipped with visual sensor.Consider the artificial potential function as(37)Vi_APF=∑k=1mdik-δik-2,δik<dik≤RAPF.0,dik>RAPF.where dik=qi–qk denotes the distance between agenti and obstaclek. δik remains the same as in Definition8.From (37), the artificial potential function is unbounded when agenti gets close to the obstaclek, namely,dik → δik. The artificial potential is produced or vanishes if the distancedik lies in (δik,RARF) ordik >RARF. As a result, the associated repulsive force for each agenti is devised as(38)ui_obs=0,dik>RAPF.-∑k=1m2dik-δik-3pi-pjdik,δik<dik≤RAPF.Using the control law (38), we can solve the obstacle avoidance problem.Agent 1 can avoid two obstacles without getting stuck in the local minimum points by using the attractive force produced by agent 1 and the reference trajectory, which is illustrated in Figure2.Figure 2
Obstacle avoidance of agent 1. The blue circle and black square denote the initial and final positions of agent 1, respectively. Two different obstacles are considered, which are colored with red and blue, respectively. The local minimum points are given.


## 4.4. Bounded Constant and Time-Varying Disturbances

In many practical applications, multiagent systems are vulnerable to exterior constant disturbances. The constant disturbances (which can sometimes be viewed as constant velocities) cannot be rejected by the distributed proportional control laws. Toward this end, we apply the PI controller for double-integrator dynamics to achieve the desired formation under the constant (or static) disturbances.To continue our analysis, we analyze a distributed (or decentralized) PI formation controller with the constant disturbances for a multiagent system, where each agenti only uses the local position and velocity measurements from its neighbors. Hence, this controller not only provides an advantage of eliminating the static errors but also improves the convergence performance. Therefore, the desired formation is achieved under the proposed control laws in the presence of constant disturbances. We first devise the distributed controller for agenti as(39)ξ˙ij=qj-qi+qid-qjd,ui=mikqj-qi+qid+mikpj-pi+ciξij.withci denoting some positive constant number, which is referred to as integral gains. Equation (39) relies only on the local measurements. The diagram of the distributed PI controller is shown in Figure 3. Note that the system for each agenti may be unstable when a minor local measurement error occurs.Figure 3
Block diagram of distributed PI controller. Agenti only has access to the local measurements.Integrating the first equation of (39), we obtain(40)ξij=qid-qjdt-∫0tqiτ-qjτdτFrom (36), each agenti has only access to local relative positions (or measurements) and the time for obtaining the desired estimate of the integral error states.Consider again the double-integrator dynamics for multiagent systems as(41)q˙i=pip˙i=ui+diwithdi denoting the unknown constant disturbances.The main result of this part is given by Theorem11 as follows.Theorem 11.
Consider the multiagent system (41) under the connected interaction graphG. Using the formation control law (39), the static errors are eliminated, and the desired formation is achieved under the Assumption 4.1, i.e., limt→∞⁡qi-qj=qdi –qdj and limt→∞⁡pi-pj=0 fori = 1…,n,j∈ Ni.Remark 12.
To address the problem in the presence of the constant disturbances, we introduce the control term of integral action, which can eliminate the static errors and guarantee the desired performance. In other words, the actual relative position vector converges to the specified relative position vector; thus, the control objective is achieved.Proof.
From (39) and (41), by introducing a new state vector ξi=ξ1j,…,ξnjT, we may write the corresponding augmented matrix in a compact form as(42)ξ˙ie˙qie˙pi=0n×nIn0n×n0n×n0n×nIn-kqiL+ΔL-kpiL+ΔL-ciInξieqiepi+0n×10n×1cn×1qid-qjd+0n×10n×1dn×1fori = {1,…,n}, together with the bounded initial condition. Here,kqi ∈ R+ andkpi ∈ R+ are positive numbers, respectively, qid-qjd is the desired relative position vector given in advance, and dn×1 are constant (static) disturbances. Note that the aim of (42) is to stabilize system with respect to the parameters, i.e., ξi,eqi,epi.
Taking the time derivative of (42), we obtain(43)ξ¨ije¨qie¨pi=0n×nIn0n×n0n×n0n×nIn-kqiL+ΔL-kpiL+ΔL-ciIn︸≔AAugξ˙ije˙qie˙pi
To obtain the eigenvalues ofAAug, the characteristic polynomial ofAAug is calculated as(44)det⁡AAug=λIn-AAug=sIn-In0n×n-0n×nsIn-InkqiL+ΔLkpiL+ΔLsIn+ciIn=s3+cis2+kpiL+ΔLs+kqiL+ΔL=0
To continue the analysis, consider the characteristic polynomial of (L+ ΔL) as(45)s3+cis2+kpiλis+kqiλi=0where λi ∈ R+ are the eigenvalues of (L + ΔL), i.e., ςi > 0.
Comparing (44) with (45), we learn that all the eigenvalues ofAAug have strictly negative real parts because of the positive eigenvalues of λi as mentioned before, which impliesAAug is Hurwitz (or stable), and the system is therefore asymptotically stable. Furthermore, one can also place the poles according to the characteristic polynomial (47) such that the system is asymptotically stable. To this end, the control gain can be easily computed by using the command of place (·) in MATLAB. As discussed,AAug is Hurwitz, and we then obtain(46)s3+cis2+kpiλis+kqiλi=0
From (46), one can get(47)limt→∞qj-qi=qid-qjdwhich implies that the proposed distributed PI controller (39) can eliminate the static errors caused by constant but unknown disturbances and implies that the relative positions between agentsi andj converge to the desired relative positions, the desired formation is achieved and, maintained, meanwhile the velocity errors between agentsi andj converge to zero. The proof is thus completed.Remark 13.
If the errors are equal to zero, i.e., the actual relative position equals the desired relative position, the proposed controller stops to calculate the changes, implying that the desired formation is achieved. It is noted that the convergence rate of the overall system is dominated by the smallest eigenvalue ofAAug.Example 14.
Without loss of generality and for brevity, consider an agent 1 with initial positionq1(0) = 0.420.50T and its second-order dynamics (41) with the bounded constant disturbances given asd1 = 0.780.25T andd2 = 3.264.92T, and the control gains arekq = 1.35 andkp = 1.35 and integral gainci = 0.75, respectively.Figure4 illustrates the trajectories of agent 1 in the presence of disturbances, where the left and right figures show the comparison results concerning the static performance between with and without the aim of integral control terms. As expected, the static error is successfully eliminated by introducing the integral control terms, depicted by blue lines in the left subfigures, whereas the static error is not eliminated without the aim of the integral control, shown in red lines in the right subfigures, in x and y coordinates, respectively. The right two subfigures show the convergence performance of the system in the presence of two different constant disturbances. The convergence rate is bigger when the disturbance is smaller, i.e.,d1 = 0.780.25T, and the resulting trajectories of agent 1 are depicted by blue lines. Therefore, the system is asymptotically stable under the proposed control law.Figure 4
State trajectory of agent 1 in the presence of constant disturbances. Left: using the integral control, Right: without the help of the integral control for the disturbances.With the above results in hand, to attenuate the bounded time-varying disturbances, assume thatdit ≤θiφi(t), whereθi is unknown constant,φi(t) is bounded function of timet, the robust controller is derived by the following procedure:(48)Vtd=∑j∈Ni12aijeqi2+12βijepi2+12θ~iγi-1θ~iwhere θ^i is an estimate of θi, θ~=θ-θ^ is the parameter error; αij, βij, and γi are positive constants. Equation (48) is continuously differentiable. The time derivative ofVta is(49)V˙td=∑j∈Niaijeqie˙qi+βijepie˙pi+θ~iγi-1θ~˙=∑j∈Niaijeqiepi+βijepiui+βijepidi+θ~iγi-1θ~˙=∑j∈Nieqiepi+βijepiui+βijepiθiφt+θ~iγi-1θ~˙Design the adaptive controlui as (50)ui=∑j∈Ni-aijβijeqi-epi-θ^iφtSubstituting (50) into (49), we have (51)V˙td=∑j∈Ni-βijepi2+βijepiθi-θ^iφt+θ~iγi-1θ~˙To guarantee the negative semidefinite of V˙td, the adaptation law is chosen as(52)θ~˙=-γiβijepiφt-γiθi-θ^iSubstituting (52) into (51) yields(53)V˙td=∑j∈Ni-βijepi2-θi-θ^iTθi-θ^iwhich is continuously differentiable function and negative semi-definite, implying that Vtd(t) − Vtd(0) ≤ 0, with the equality is satisfied if and only ifepi = 0 and θ^i= θi; it follows thatepi = 0. Therefore, the system is asymptotically stable, the desired formation is achieved, and the time-varying disturbances are attenuated under the adaptive control law (50).


## 5. Simulation Results

In this section, numerical simulation results are given to support the obtained theoretical results. Consider a five-agent system under the leader–follower structure as shown in Figure1.Consider the second-order dynamics for multiagent system as(54)q˙i=pi,p˙i=-2qi1-3pi1-2qi2-3pi2+0.52pi14+pi2ui+ditTo characterize the desired formation, the desired relative position vector is provided, i.e.,q1d=40T; q2d=22T; q3d=2-2T; q4d=-22T; q5d=-2-2T. The initial positions of the agents are randomly given asq1(0) = 0.420.50T,q2(0) = 0.35-0.67T,q3(0) = 0.04-0.46T,q4(0) = 0.12-0.06T, andq5(0) = 0.180.19T, which is illustrated in Figure 5. The other control parameters are chosen as γ4 = 1.462.73 and β4 = 1.532.04, and the external bounded time-varying disturbanced4(t) = 5.17cos⁡(0.75t)8.68sin⁡(0.38t) is imposed on the 4th agent as time evolves, without loss of generality and for brevity. Note that obstacle avoidance problem is considered in the simulation examples.Figure 5
Initial positions of agents.To proceed, the desired trajectory for the leader is then provided as(55)q˙r=pr,p˙r=0.12cos⁡0.15pr20.26.Without loss of generality and for simplicity, the Laplacian matrix is defined as(56)L=10000-11000-1-12000-1-12000-1-12and the weight matrix ΔL is given as ΔL = diag{1,1,0,0,0}. It is easily verified that all the eigenvalues of (L + ΔL) are positive.The decentralized controller is given as(57)ui=γijαij,yijqj-qi+qid-qjd+γikαik,yikqk-qi+qidwhere γi(·, ·) remains unchanged and depends on the distance errorsyij andyik.It is assumed that three obstacles points are considered and placed atqo1=7.184.25T, qo2=11.5212.25T, and qo3=13.9818.94T, respectively, which is shown in Figure 6. Figure 6 illustrates the formation trajectory for a group of agents, where the squares denote the final positions of the agent at each simulation instant. As seen, the desired formation cannot be achieved at 4 s. As timet evolves, the target formation is achieved and maintained from at approximately 8 s. The corresponding distance errors converge to zero as timet evolves, which is shown in Figure 7.Figure 6
Formation trajectory without using the repulsive force.Figure 7
Errors between the actual distances and the desired distances without considering obstacle avoidance.As the agents sense each obstacle, the corresponding obstacle avoidance control term is then activated, which leads to the desired formation trajectory being changed. After avoiding the obstacle, the desired formation of five agents is rebuilt and achieved under the attractive force of the proposed formation control law at approximately 22 s. Figure8 depicts the formation for multiagent system approaches to three obstacles, respectively.Figure 8
Formation control with obstacle avoidance by using the repulsive force.Figure9 shows the distance errors between the actual distances and the corresponding desired distances among agents. Clearly, the distances errors tend to zero after avoiding each obstacle, implying that the actual distances converge to the corresponding desired distances as timet evolves. The desired formation is therefore achieved. Moreover, we can see that the distance errors increase sharply when agents approach the obstacles, which implies that the repulsive potential field increases and steers the agents away from obstacles, as expected.Figure 9
Distance errors of formation with obstacle avoidance. The period of obstacle avoidance is depicted.Without loss of generality, consider the multiagent system with constant bounded disturbance is given asd=15.160000;19.480000T; namely, agent 1 is assumed to be disturbed in this setup.The comparative results between without and with integral actions are given in Figures10 and 11, respectively. The desired formation is not achieved and maintained in Figure 10, implying that the system is unstable, whereas the desired formation is achieved and maintained under the integral action, which is shown in Figure 11. For agent 1, the static errore21 is not eliminated without using the integral control term, which is illustrated in Figure 12. The static errore21 can be eliminated with the aim of integral action, which is shown in Figure 13.Figure 10
Formation trajectory in the presence of constant disturbance without employing the integral action.Figure 11
Formation trajectory in the presence of constant disturbance with the aim of integral action.Figure 12
Distance errors in the presence of constant disturbance without integral action.Figure 13
Distance errors in the presence of constant disturbance under the assistance of integral action.The comparative results between without and with adaptive control actions are given in Figures14 and 15, respectively. The desired formation is not achieved and maintained in Figure 14, implying that the system is unstable, whereas the desired formation is achieved and maintained under the adaptive control action, which is shown in Figure 15, which is further supported by the distance error shown in Figures 16 and 17, respectively. Figures 18 and 19 show the trajectories of estimates of parametersθ41 andθ42, and the curves of parameter estimation errors are depicted in Figure 20, which implies that the errors converge asymptotically to zero. Figure 21 shows the trajectory of the adaptation laws.Figure 14
Formation trajectory in the presence of time-varying disturbance without employing the adaptive control term.Figure 15
Formation trajectory in the presence of time-varying disturbance with the aim of the adaptive control term.Figure 16
Distance errors in the presence of time-varying disturbance without the adaptive control term.Figure 17
Distance errors in the presence of time-varying disturbance under the assistance of the adaptive control term.Figure 18
The estimate of parameterθ41.Figure 19
The estimate of parameterθ42.Figure 20
The parameter estimation errors.Figure 21
Adaptation laws.


## 6. Conclusion

A distributed nonlinear formation scheme was proposed for second-order dynamical systems, where the control input of each follower can be expressed as a product of a nonlinear term under the leader–follower structure. To deal with the obstacle problem, the artificial potential function approaches were applied to steer the agents away from the obstacles without getting stuck in local minimum positions. The stability analysis for a multiagent system is proved based on the theory of Lyapunov stability. Using the proposed control law, the desired formation is achieved and maintained while avoiding the obstacles. The formation control for multiagent systems in the presence of unknown bounded disturbances is investigated as well. Finally, the numerical simulation results are provided to demonstrate the effectiveness of the proposed distributed strategies.



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*Source: 3252303-2019-06-23.xml*