Distributed adaptive neural containment control for multi-uav systems with nonlinear uncertainties under a directed graph

Transcription

1 第 3 卷第 10 期 015 年 10 月 DOI: /CA 控制理论与应用 Control heory & Applications Vol. 3 No. 10 Oct. 015 有向图下非线性无人机群自适应合围控制 余瑶, 任昊, 张兰, 孙长银 ( 北京科技大学自动化学院, 北京 ) 摘要 : 本文研究了有向图下具有非线性和干扰的无人机群的分布式合围控制问题. 其中仅部分跟随者是领导者的邻居, 对于每一个跟随者, 至少存在一条从领导者到这个跟随者有向路径. 文中假设无人机的空气动力学特性是非线性不确定的, 并且领导者的输出是时变的. 结合反推设计方法提出了仅利用邻居信息的分布式合围控制方法, 使得跟随者的状态收敛于领导者状态所张成的凸包里. 利用神经网络函数逼近技术补偿无人机系统中的非线性不确定项, 通过李雅普诺夫稳定性理论证明了合围误差可以以任意收敛速度收敛到原点任意小的邻域. 最后通过仿真结果验证了控制协议的有效性. 关键词 : 合围 ; 无人机群系统 ; 非线性不确定性 ; 自适应神经网络控制 ; 图论 ; 反推法中图分类号 : P73 文献标识码 : A Distributed adaptive neural containment control or multi-uav systems with nonlinear uncertainties under a directed graph YU Yao, REN Hao, ZHANG Lan, SUN Chang-yin (School o Automation and Electrical Engineering, University o Science and echnology Beijing, Beijing , China) Abstract: We investigate the distributed containment control problem or multiple unmanned aerial vehicles (UAVs) systems with nonlinear uncertainties and bounded disturbances under a directed graph, where the leaders are neighbors o only a subset o the ollowers. For each ollower, there exists at least one leader that has a directed path to the ollower. It is assumed that aerodynamic characteristics o UAVs are nonlinear uncertainties, and the outputs o leaders are timevarying. A distributed containment control protocol combined with backstepping design method is proposed by using neighbors inormation, so that the states o the ollowers will converge to the convex hull spanned by the dynamic leaders. he unction approximation technique using neural networks is employed to compensate unknown nonlinear terms induced rom the controller design procedure. By Lyapunov stability theorem, it is shown that the containment control errors will converge to an expected neighborhood o the origin with an arbitrary convergence rate. Simulation examples are presented to illustrate the eectiveness o the proposed control algorithm. Key words: containment; multi-uav systems; nonlinear uncertainties; adaptive neural control; graph theory; backstepping 1 Introduction An unmanned aerial vehicle (UAV) is an aircrat with no on-board human pilot. UAVs are usually deployed or military and special operation applications, and used in numerous civil tasks alling within the dull, dirty and dangerous category. Potential and existing applications o UAVs include acrobatic aerial ootage in ilmmaking, orest ire detection, search and rescue missions. he research on cooperative control o multi-uav systems [1] has drawn considerable interests in recent years due to several superiorities o multi-uav systems in contrast with individual UAV systems, such as higher eiciency, better robustness, and larger survival probability. he distributed cooperative control [ 3] or multi-uav systems requires that the protocol or each UAV use only limited local inormation to reach an overall goal in complicated environment, which makes the control a great challenge. o achieve the goals above, much progress has been made in study o consensus control [4 8], containment control [9 10], and ormation control [11 13]. Among these control problems, consensus is an important and undamental problem, which means that all agent states reach an agreement with their neigh- Received 3 March 015; accepted 18 July 015. Corresponding author. cys@ustb.edu.cn; el.: Supported by National Outstanding Youth Science Foundation ( ), National Natural Science Foundation ( ), Beijing Higher Education Young Elite eacher Project (YEP0378), Fundamental Research Funds or the Central Universities (FRF P A) and Beijing Natural Science Foundation ( ).

2 No. 10 YU Yao et al.: Distributed adaptive neural containment control or multi-uav systems with nonlinear uncertainties under a directed graph 1385 bors in a certain sense. Consensus is a typical collective behavior [14 15], which can be classiied into leaderless consensus [16 17] and leader-ollowing consensus [18 1]. In practice, it may be saer and more eicient or multi-uav systems i there are multiple leaders and ollowers in battleield environment, e.g., only some o UAVs are equipped with high precision sensors, so that other UAVs may be saer and easier to avoid obstacles i they converge to the convex hull. For multiple leaders case, containment control problems arise, which require that the states o ollowers converge to the convex hull ormed by the leaders. he containment control problem has recently gained much attention. Containment problems or irst-order and second-order multi-agent systems have been studied in [9, 5]. In [9], the distributed containment control or double-integrator dynamics in the presence o both stationary and dynamic leaders was studied. In [], the problem o distributed containment control o a group o mobile autonomous agents was discussed with multiple stationary or dynamic leaders under both ixed and switching directed network topologies. In [3], a stop-and-go strategy was proposed or a group o single-integrator agents to the convex polytope spanned by the leaders under a ixed undirected network topology. he inite-time containment control problem or secondorder multi-agent systems under a ixed communication topology is addressed in [4]. For doubleintegrator dynamics, distributed containment control in the presence o both stationary and dynamic leaders was discussed in [5]. here have been also some results on the problem o distributed containment control or linear high-order multi-agent systems [6 7]. In [6], the behavior o multiple agents with linear dynamics was investigated by the study o interaction topologies. Formation-containment problem was transormed into asymptotic stability problem by using state transormation and state space decomposition approaches in [7]. All these results ocused on containment control o linear multi-agent systems. For multi-uav systems, there have been a ew results o containment control [8]. In [8], ormationcontainment control problems or multi-uav systems with directed interaction topologies were addressed, where each UAV system was modeled as a second order linear system. Nonlinear uncertainties are inevitable in real systems due to imprecise measurements, unmodeled dynamics, external disturbances, etc. hereore, dierent kinds o control strategies, such as backstepping [9 31], sliding mode control [3], neural networks [33 35], adaptive control [36], have been developed to deal with nonlinear systems. We notice that there are many results on tracking or consensus problems or multi-agent systems with nonlinear uncertainties [37 41], but there are only a ew results on containment control problem with nonlinear uncertain agents. Despite these eorts, little progress has been made in distributed adaptive containment control or multi-uav systems with nonlinear uncertainties under a directed graph, which is the main content o our research. In this paper, containment control is investigated or multi-uav systems with nonlinear uncertainty under a directed graph topology. Nonlinear uncertain aerodynamic characteristics o UAVs are considered. Matching condition is not satisied. And the leaders are neighbors o only a subset o the ollowers. he dynamic characteristics o a single UAV are divided into nominal model and unknown nonlinear uncertainties. A distributed adaptive containment controller combined with backstepping method is proposed so that all o the ollowers converge to the dynamic convex hull spanned by the dynamic leaders. he unction approximation technique using neural networks is employed to compensate unknown nonlinear terms. he stability o the closed-loop systems is analyzed via a Lyapunov-based method, which shows that the containment error can be reduced as small as desired. his paper is organized as ollows. Section introduces useul results o the graph theory and the dynamics model o multi-uav systems. In Section 3, an adaptive neural controller with backstepping design method is proposed and stability analysis is presented by Lyapunov unction. In Section 4, simulation examples are presented to illustrate the analytical results. Finally, conclusions are drawn in Section 5. Preliminaries.1 Graph theory For M + N UAVs, a directed graph G (ν, ε) is a pair (ν, ε), where the set o nodes or vertices is ν {1,, M + N} and the set o edges or arcs is ε ν ν. An edge (j, i) ε means that UAV i can obtain inormation rom UAV j, but not vice versa where j and i are the parent node and child node, respectively. N i = {j (j, i) ε} means the set o neighbors o the node i, which is the set o nodes with edges incoming to node i. A directed path rom node i 1 to node i k is a sequence o edges o the orm (i 1, i ), (i, i 3 ),, (i k 1, i k ) in a directed graph. A directed tree is a special directed graph where every

3 1386 Control heory & Applications Vol. 3 node has exactly one parent except or the root and the root has directed paths to every other node. I there exists at least one UAV that has directed paths to all other UAVs, we can believe that the directed graph has a directed spanning tree [4 43]. For a directed graph G, the adjacent matrix A = (a ij ) R (M+N) (M+N) is deined as a ij > 0 i (j, i) ε, otherwise, a ij = 0. Sel-edges are not allowed, i.e., a ii = 0. he (nonsymmetric) Laplacian matrix L = D A R (M+N) (M+N), where D = diag{d 1, d,, d M, d M+1,, d M+N }, d i = M+N a ij is the diagonal element o the degree,j i matrix D. Lemma 1 For a directed graph G, it is easy to see that zero is always an eigenvalue o L with 1 as a corresponding right eigenvector, and all o the nonzero eigenvalues o L have positive real parts. Besides, zero is a simple eigenvalue o L i and only i the directed graph G has a directed spanning tree [44].. Problem statement It is assumed that there exist M ollowers, labeled as UAVs 1 to M, and N leaders, labeled as an UAV M + 1 to M + N under a directed communication graph topology. Each UAV should be equipped with a sensor suite to determine its own position and orientation in the inertial reerence rame and receive the (relative) position inormation rom its neighbors. Assume that all the UAVs have ixed attitudes. he translational dynamics o the ith UAV can be written as [45] { ṗ i = g i (Θ i )v i, M i v i = D i (v i )v i R i (Θ i ) + τ i, (1) where p i = [X i Y i Z i ] R 3, Θ i = [ϕ i θ i ψ i ] means position and attitude (described by Euler angles, i.e., roll ϕ i, pitch θ i, and yaw ψ i ) vectors in the inertial reerence rame, respectively. g i (Θ i ) is the kinematic transormation matrix rom the bodyixed reerence rame to the inertial reerence rame, v i = [µ i, ν i, ω i ] is translational velocity vector in the body-ixed reerence rame, M i is the inertia matrix, D i (v i ) is the damping matrix, R i (Θ i ) is the restoring orce vector, and τ i is the control orce vector. For an angle α R, denote s α = sin α, c α = cos α or brevity. g i (Θ i ) is deined as g i (Θ i ) = c ψi c θi s ψi c ϕi +s ϕi s θi c ψi s ψi s ϕi +s θi c ψi c ϕi s ψi c θi c ψi c ϕi +s ϕi s θi s ψi c ψi s ϕi +s θi s ψi c ϕi, s θi s ϕi c θi c ϕi c θi M i = diag{m i1, m i, m i3 }, D i (v i ) = diag{d Li1 + d Qi1 µ i, d Li + d Qi ν i, d Li3 + d Qi ω i }, m ij, d Lij, d Qij > 0, R i (Θ i ) = [(W i B i )s θi, (W i B i )c θi s ϕi, (W i B i )c θi c ϕi ], where W i and B i represent the gravitational and buoyancy orces, respectively. For the position containment control o M + N UAVs system, the dynamic models o M ollowers can be simpliied as ẋ i,1 = g i (Θ i )x i,, ẋ i, = u i + Φ i (x i,1, x i,, κ i ), () y i = x i,1, where i = 1,,, M, x i,1 = [X i Y i Z i ], Θ i = [ϕ i θ i ψ i ] means position and attitude (described by Euler angles, i.e., roll ϕ i, pitch θ i, and yaw ψ i ) vectors in the inertial reerence rame, respectively, x i, R 3 is translational velocity vector in the bodyixed reerence rame, u i R 3 is the input vector o the ith ollower, y i R 3 is the output vector, which is the position vector o the ith ollower in the inertial reerence rame, Φ i (x i,1, x i,, κ i ) are unknown nonlinear smooth unctions, which include the aerodynamic orces, the pitch and elevation channels couplings, the parameters perturbation, and time-varying disturbances κ i. It s supposed that the motions o N leaders are independent o the motions o M ollowers, the ollowers 1 to M have at least one neighbor, and the leaders M + 1 to M + N have no neighbors. he communication topology or the M+N UAVs is considered as a directed graph G (ν, ε) with ν {0, 1,,, M, M + 1,, M + N}. We denote ν = 1,,, M and ν l = {M + 1,, M + N} as the node set o the ollowers and the leaders, respectively. It is considered that ν ν l = ν and ν ν l =. o represent the communications among ollowers and the communications between the ollowers and the leaders, the Laplacian matrix L is deined as [ ] L Ll L =, (3) 0 N M 0 N N where L R M M is the matrix related to the communication among the M ollowers and L l R N N is the matrix related to the communication rom the N leaders to the M ollowers. he ollowing technical lemmas and assumptions o bounded input and bounded output property or stable dynamic inequalities are considered or the convenience o stability analysis.

4 No. 10 Remark 1 YU Yao et al.: Distributed adaptive neural containment control or multi-uav systems with nonlinear uncertainties under a directed graph 1387 he set C R n is said to be convex i or any x 1, x C and any [0, 1], the point x 1 + (1 )x is in C. he convex hull Co(X) or a set o points X = [x 1 x n ] id the minimal convex set containing all points in X and is deined as Co(X) = n i = 1 [46]. Assumption 1 Both multiple dynamic leaders r j (t) C, j = M + 1,, M + N and its derivative ṙ j (t) R 3 are bounded, and the leaders r j (t) are only available or the ith ollower satisying j N i, i = 1,, M. Assumption he states x i,1 and x i, o the ith ollower are only known and available or the j-th ollowers satisying i N j, i = 1,, M, j = 1,, M and i j. Assumption 3 For any one o the M ollowers, there exists at least one leader that has a directed path to that ollower. Lemma According to Assumption 3, L is a nonsingular M matrix, and thus it is invertible. Additionally, each entry o L 1 L l is nonnegative and all row sums o L 1 L l equal to one [47]. Let r(t) = [rm+1 (t) r M+N (t)], r d (t) = [rd,1 (t) r d M (t)] = ( L 1 L l I p )r(t), where r d,i (t) R p, i = 1,, M, stands or Kronecker product, and I p is an identity matrix o order p. For ith ollower, r d,i (t) can be regard as a reerence point in the convex hull spanned by the leaders. From Lemma, we get that in r d,i(t) h(t) < ϵ, h(t) R(t) with i = 1,, M or all t 0, where R(t) = Co{r M+1 (t),, r M+N (t)}, and ϵ is a positive constant which can be made suiciently small, while all signals in the total class-loop systems are bounded. hereore, the containment control problem can be regarded as the tracking problem o each ollower such that y i (t) r d,i (t) = y(t) + ( L 1 L l I p )r(t) < ϵ, where i = 1,, M and ϵ is a positive constant which can be made suiciently small [48]. When each ollower tracks to its own reerence point, we can consider that all ollowers converge to the convex hull spanned by the leaders. As we know, radial basis unction (RBF) neural networks are usually employed as the unction approximator to model nonlinear unctions, which we believe to have reached a airly good point in terms o stability and lexibility. he ollowing RBF NN is used to approximate the continuous unction h(z): R q R, h nn (Z) = W S(Z), (4) where the input vector Z Ω R q, weight vector W = [w 1 w w l ] R k, the number o neuron k > 1; and S(Z) = [s 1 (Z) s (Z) s k (Z)] with s i (Z) = exp[ (Z µ i) (Z µ i ) φ ], (5) i where i = 1,,, k, µ i = [µ i1 µ i µ iq ] is the center o the receptive ield and φ i is the width o the Gaussian unction [49]. It has been proven that the aorementioned RBFNN can approximate any smooth unction over a compact set to arbitrarily any accuracy as h(z) = W S(Z) + ξ, Z Ω Z, (6) where the compact set Ω Z R q, Z is the input variables o the NNs, W is the ideal constant weights vector, and ξ is the bounded unction approximation error. I Z remain within some preixed compact set Ω Z which have a suiciently large size, there exists controllers with suiciently large number o NN nodes such that all the signals in the closed-loop remain bounded. For convenience o analysis, the ideal constant weights W is deined as the value o that minimizes or all, i.e., W arg min sup h(z) W W Rk{ S(Z) }, (7) Z Ω Z and there exists W such that ξ ξ with constant ξ > 0 or all Z Ω Z. 3 Adaptive containment controller design and stability analysis For the second-order system o the ith ollower, the idea o backstepping is employed to design controllers. he design procedure o the ith ollower contains two steps. Based on the distributed dynamic surace design, we deine the graph based error suraces s i,k, k = 1,, or the ith ollower as s i,1 = M a ij (y i y j )+ M+N (8) a ij (y i r j ), s i, = x i, v i,, where i = 1,, M, v i, are the virtual control. Step 1 he derivative o s i,1 along () and (8) is ṡ i,1 = M a ij (ẏ i ẏ j ) + M+N a ij (ẏ i ṙ j ) = M a ij g j (Θ j )x j, M+N a ij ṙ j + ( M a ij + M+N a ij )g i (Θ i )x i, =

5 1388 Control heory & Applications Vol. 3 M a ij g j (Θ j )x j, M+N a ij ṙ j + d i g i (Θ i )(s i, + v i, ). (9) Choose the Lyapunov candidate unction V i,1 as V i,1 = 1 s i,1. (10) o stabilize (9), the distributed irst virtual control law v i, or the ith ollower is designed as v i, = 1 d i ( ρs i,1 + M a ij g j (Θ j )x j, + M+N where ρ is a positive constant. Substituting (11) into (9), one obtains ṙ j ), (11) ṡ i,1 = ρs i,1 + d i g i (Θ i )s i,. (1) Dierentiating (10) along (1) yields V i,1 = ρs i,1 + d i g i (Θ i )s i,1 s i,. (13) Step Using () and (8), the derivative o s i, can be represented by ṡ i, = u i + Φ i (x i,1, x i,, κ i ) v i,. (14) o stabilize (14), there exists a desired eedback control u i = ρs i, (Φ i (x i,1, x i,, κ i ) v i, ), where v i, can be expressed as v i, = M v i, (x j, + Φ i (x i,1, x i,, κ i )) + M+N x j,1 v i, r j ṙ j. For the desired eedback control u i, rom Assumption 1 we can get that h i (Z i ) = Φ i (x i,1, x i,, κ i ) v i, are smooth unctions, which denote the unknown part o u i, where Z i = [x i,1 x i, κ i ( v i, x i,1 ) ( v i, x M,1 ), v i, r M+1 ṙ M+1 v i, r M+N ṙ M+N ]. By employing an RBF neural network Wi S i(z i ) to approximate h i (Z i ), u i can be expressed u i = ρs i, Wi S i (Z i ) ξ i, where Wi denote the ideal constants weights, and ξ i ξi is the approximation error with constant ξi > 0. Since Wi is unknown, u i cannot be realized in practice. Consider u i = ρs i, Ŵ i S i (Z i ). (15) hen, we have ṡ i, = u i + Φ i (x i,1, x i,, κ i ) v i, = ρs i, W i S i (Z i ) + ξ i. (16) Consider the Lyapunov unction candidate V i, as V i, = 1 s i, + 1 W i Γi W i. (17) he derivative o V i, is V i, = s i, ṡ i, + W i Γi Ŵ i = ρs i, + s i, ξ i W i S i (Z i )s i, + W i Γi Ŵ i. (18) Consider the adaptation law or Ŵi as Ŵ i = Wi = Γ Wi [S i (Z i )s i, σ i Ŵ i ], (19) where σ i > 0 and Γ Wi = ΓW i > 0 are design constants, Wi = Ŵi Wi. Let ρ = ρ 1 +ρ, where ρ 1 and ρ > 0. hen, (18) becomes V i, = ρ 1 s i, ρ s i, + s i, ξ i σ i W i Ŵ i. (0) By completion o squares, one has σ i W i Ŵ i = σ i W i ( W i + Wi ) σ i W i + σ i W i Wi σ i W i + σ i Wi ρ s i, + s i, ξ i ξ i ξ i. (1) 4ρ 4ρ hen one has the ollowing inequality V i, ρ 1 s i, σ i W i + δ i, () where δ i σ i Wi / + ξi /4ρ. Let V i = V i,1 + V i,. I we choose ρ = ρ + d ig i (Θ i ), ρ 1 = ρ 1 + d ig i (Θ i ), where ρ γ, ρ 1 γ, γ is a positive constant, and choose σ i and Γ i such that σ i γλ{γi 1 }, then rom (13) and () we have the ollowing inequality: V i = ρs i,1+d i g i (Θ i )s i,1 s i, ρs i, σ i W i +δ i (ρ d ig i (Θ i ) σ i W i Γi W i + δ i = ρ s i,1 ρ 1s i, σ i W i Γ i )s i,1 (ρ 1 d ig i (Θ i ) )s i, W i + δ i γ( 1 s i,1 + 1 s i, + 1 W i Γi W i ) + δ i =

6 No. 10 YU Yao et al.: Distributed adaptive neural containment control or multi-uav systems γv i + δ i. (3) Let V = M V i. Its derivative is V = M with nonlinear uncertainties under a directed graph 1389 V i M ( γv i + δ i ) γv + δ, (4) where δ = M δ i is positive constant. he inequality (4) implies V < 0 on V > δ/γ. hereore, i V (0) δ/γ, then V (t) δ/γ or all t 0. In addition, one has that V (t) e γt V (0) + δ/γ(1 e γt ). Using s 1 / V (t) with s 1 = [s 1,1 s M,1 ], one gets s 1 e γt V (0) +δ/γ(1 e γt ). hereore, as time increases, all error suraces s 1 exponentially converge to the compact set Ω s = {s 1 s 1 δ/γ}. he compact set Ω s can be kept arbitrarily small by increasing γ. hen, rom s 1 = ( L I p )y + ( L l I p )r where y = [y1 ym ], since s 1 can be reduced suiciently small, we know that the ollower output vector y(t) converge to the convex hull spanned by the dynamic leaders r(t), i.e., y(t) + ( L 1 L l I p )r(t) < ϵ. So the containment control errors in the overall closedloop system can be converged to an adjustable neighborhood o the origin with an arbitrary convergence rate. 4 Simulation results In this section, three cases are preormed to illustrate the eectiveness o the proposed approach. A multi-uav system with 3 leaders and 4 ollowers in a three-dimensional space is considered. Fig.1 shows the topological directed graph o the interaction among the leaders (L5 to L7) and the ollowers (F1 to F4). It is assumed that a ij = 1 i (j, i) ε, otherwise, a ij = 0. hen one can get the adjacent matrix A as A = , and the degree matrix D = diag{,,,, 0, 0, 0}. he dynamics models o M ollowers are given by (). he RBF neural network approximation technique is used to estimate unknown nonlinear unctions Φ i (x i,1, x i,, κ i ), which include the aerodynamic orces, couplings, the parameters perturbation, and time-varying disturbances. It is assumed that the bound o nonlinear smooth unctions Φ i (x i,1, x i,, κ i ) = x i,1 + x i, + 3. he containment control input (15) o the proposed approach is applied. Case 1 We consider the trajectories o UAVs are in a 3-D space. he reerence trajectories o three leaders are set to be static, i.e., the position states o leaders L5, L6 and L7 are r L5 = [ 1 1 1], r L6 = [5 5 5] and r L7 = [3 3 3], respectively. he initial positions o the our ollowers are random. In the eedback controller (15), we set ρ = 50. he states trajectories o the ollowers in simulations are shown in Fig.. Fig. 1 opological directed graph Fig. he states trajectories o UAVs or Case 1 Case Compared with Case 1, the reerence trajectories one o three leaders to is assumed to be sine or cosine unctions, and other two leaders keep static, i.e., the reerence trajectories o leaders L5, L6 and L7 are r L5 = [ 1 1 1], r L6 = [5 5 5] and r L7 (t) = [ + sin(5t) 3 + cos(5t) + sin t ], respectively. he same distributed containment control protocol is applied. he states trajectories o UAVs is shown in Fig. 3. Case 3 In order to urther illustrate the versatility o the proposed approach, the reerence trajectories o three leaders are set to periodic unctions. We

7 1390 Control heory & Applications choose the reerence trajectories o leaders L5, L6 and L7 as rl5 (t) = [5+ sin(5t+1) 10+cos(8t ) 3+ sin 10t + cos(7t + 1)], rl6 (t) = [10 + cos(8t ) 3 + sin 10t + cos(7t + 1) 5 + sin(5t + 1)] and rl7 (t) = [3 + sin 10t + cos(7t + 1) 5 + sin(5t + 1) 10 + cos(8t )], respectively. One practical problem corresponding to this model is the secure landing o multi-uav systems on the deck o a galley. he multi-uav systems have to protect itsel rom wave-induced oscillations which are usually described by a sinusoid signal with variable requency. he same distributed containment control protocol is applied. Figure 4 shows the states trajectories o UAVs in this simulation. Vol. 3 is obvious that all ollowers can converge to the static convex hull spanned by leaders quickly and the containment control protocol designed by the proposed method is robust and eective. 5 Conclusions In this paper, distributed containment control problem or multi-uav systems with nonlinear uncertainty under a directed graph topology is considered. he dynamic characteristics o a single UAV are divided into nominal model and unknown nonlinear unction. A distributed adaptive containment controller based on the dynamic surace design is proposed so that all o the ollowers converge to the dynamic convex hull spanned by the dynamic leaders. Neural networks method is used as the unction approximation technique to compensate unknown nonlinear terms derived rom the controller design procedure or the ollowers. It has been shown that the system has achieved semi-global uniorm ultimate boundedness and the containment control error can converge to any small neighbourhood o the origin with an arbitrary convergence rate. Simulation examples has been given to veriy the algorithms. Reerences: [1] DUAN H, LI P. Bio-inspired Computation in Unmanned Aerial Vehicles [M]. Berlin: Springer, 014: [] GUAN Y Q, JI Z J, ZHANG L, et al. Recent developments on controllability o multi-agent systems [J]. Control heory & Applications, 015, 3(4): Fig. 3 he states trajectories o UAVs or Case [3] HONG Y G, ZHAI C. Dynamic coordination and distributed control design o multi-agent systems [J]. Control heory & Applications, 011, 8(10): [4] SUN Y G, WANG L. Consensus o multi-agent systems in directed networks with nonuniorm time-varying delays [J]. IEEE ransactions on Automatic Control, 009, 54(7): [5] KURIKI Y, NAMERIKAWA. Consensus-based cooperative ormation control with collision avoidance or a multi-uav system [C] //American Control Conerence. Portland, USA: IEEE, 014: [6] ZHOU Y, YU X, SUN C, et al. Robust synchronisation o secondorder multi-agent system via pinning controlp [J]. IE Control heory & Applications, doi: /iet-cta, 014: 095. [7] ZHOU Y, YU X, SUN C, et al. Higher order inite-time consensus protocol or heterogeneous multi-agent systems [J]. International Journal o Control, 015, 88(): [8] XI J X, YU Y, LIU G B, et al. Guaranteed-cost consensus or singular multi-agent systems with switching topologies [J]. IEEE ransactions on Circuits and Systems, 014, 61(5): [9] CAO Y, SUAR D, REN W, et al. Distributed containment control or multiple autonomous vehicles with double-integrator dynamics: algorithms and experiments [J]. IEEE ransactions on Control Systems echnology, 011, 19(4): Fig. 4 he states trajectories o UAVs or Case 3 In Cases 1,, 3, the same containment control protocol is applied to multi-uav systems with dierent types o reerence trajectories o three leaders. It [10] LI Z, DUAN Z, REN W, et al. Containment control o linear multiagent systems with multiple leaders o bounded inputs using distributed continuous controllers [J]. International Journal o Robust and Nonlinear Control, doi: /rnc, 014: [11] CUI R, GE S S, VOON E, et al. Leader ollower ormation control o underactuated autonomous underwater vehicles [J]. Ocean Engineering, 010, 37(17):

8 No. 10 YU Yao et al.: Distributed adaptive neural containment control or multi-uav systems with nonlinear uncertainties under a directed graph 1391 [1] ABDESSAMEUD A, AYEBI A. Formation control o VOL unmanned aerial vehicles with communication delays [J]. Automatica, 011, 47(11): [13] WANG Q, IAN Y P, XU Y J. Globally asymptotically stable ormation control o three agents [J]. Journal o Systems Science and Complexity, 01, 5(6): [14] REN W. Inormation consensus in multivehicle cooperative control [J]. IEEE Control Systems Magazine, 007, 7(): [15] CHEN W S, WEN C Y, HUA S Y, et al. Distributed cooperative adaptive identiication and control or a group o continuous-time systems with a cooperative PE condition via consensus [J]. IEEE ransactions on Automatic Control, 014, 59(1): [16] RAO S, GHOSE D. Sliding mode control-based autopilots or leaderless consensus o unmanned aerial vehicles [J]. IEEE ransactions on Control Systems echnology, 014, (5): [17] ZHENG Z, SPRY S C, GIRARD A R. Leaderless ormation control using dynamic extension and sliding control [C] //he 17th IFAC World Congress. Korea, South: IFAC, 008: [18] CAI N, CAO J, KHAN M J. A controllability synthesis problem or dynamic multi-agent systems with linear high-order protocol [J]. International Journal o Control, Automation and Systems, 014, 1(6): [19] MENG Z, LIN Z, REN W. Robust cooperative tracking or multiple non-identical second-order nonlinear systems [J]. Automatica, 013, 49(8): [0] GALZI D, SHESSEL Y. UAV ormations control using high order sliding modes [C] //American Control Conerence. Minneapolis, USA: IEEE, 006: [1] YOO S J. Distributed consensus tracking or multiple uncertain nonlinear strict-eedback systems under a directed graph [J]. IEEE ransactions on Neural Networks and Learning Systems, 013, 4(4): [] CAO Y, REN W, EGERSED M. Distributed containment control with multiple stationary or dynamic leaders in ixed and switching directed networks [J]. Automatica, 01, 48(8): [3] JI M, FERRARI-RECAE G, EGERSED M, et al. Containment control in mobile networks [J]. IEEE ransactions on Automatic Control, 008, 53(8): [4] ZHAO Y, DUAN Z, WEN G, et al. Distributed inite-time containment control or second-order multi-agent systems [C] //he 31st Chinese Control Conerence. Heei: IEEE, 01: [5] CAO Y, SUAR D, REN W, et al. Distributed containment control or double-integrator dynamics: algorithms and experiments [C] //American Control Conerence. Marriott Waterront, USA: IEEE, 010: [6] LIU H, XIE G, WANG L. Containment o linear multi-agent systems under general interaction topologies [J]. Systems & Control Letters, 01, 61(4): [7] DONG X, SHI Z, LU G, et al. Formation-containment analysis and design or high-order linear time-invariant swarm systems [J]. International Journal o Robust and Nonlinear Control, 014: DOI: /rnc.374. [8] YU B, DONG X, SHI Z, et al. Formation-containment control or unmanned aerial vehicle swarm system [C] //he 33rd Chinese Control Conerence. Nanjing: IEEE, 014: [9] YU Y, SUN C Y, JIAO Z. Robust-decentralized tracking control or a class o uncertain MIMO nonlinear systems with time-varying delays [J]. International Journal o Robust and Nonlinear Control, 014, 4(18): [30] YU Y, LU G, SUN C, et al. Robust backstepping decentralized tracking control or a 3-DOF helicopter [J]. Nonlinear Dynamics, 015, doi: /s [31] SUN C, YU Y. Robust Control or Strict-Feedback or Nonlinear Systems and Its Application [M]. Beijing: Science Press, 014: [3] SUN C, MU C, ZHANG R. Hypersonic Vehicles erminal Sliding Mode Control echnology [M]. Beijing: Science Press, 013: 1 3. [33] WANG D, HUANG J. Neural network-based adaptive dynamic surace control or a class o uncertain nonlinear systems in stricteedback orm [J]. IEEE ransactions on Neural Networks, 005, 16(1): [34] LI, GUO L, SUN C, et al. Further results on delay-dependent stability criteria o neural networks with time-varying delays [J]. IEEE ransactions on Neural Networks, 008, 19(4): [35] HU X, SUN C, ZHANG B. Design o recurrent neural networks or solving constrained least absolute deviation problems [J]. IEEE ransactions on Neural Networks, 010, 1(7): [36] JIANG Z P, REPPERGER D W. New results in decentralized adaptive non-linear stabilization using output eedback [J]. International Journal o Control, 001, 74(7): [37] CAI N, CAO J W, MA H Y, et al. Swarm stability analysis o nonlinear dynamical multi-agent systems via relative Lyapunov unction [J]. Arabian Journal or Science and Engineering, 014, 39(3): [38] WANG X L, HONG Y G, HUANG J, et al. A distributed control approach to a robust output regulation problem or multi-agent linear systems [J]. IEEE ransactions on Automatic Control, 010, 55(1): [39] LI W, ZHANG J F. Distributed practical output tracking o high-order stochastic multi-agent systems with inherent nonlinear drit and diusion terms [J]. Automatica, 014, 50(1): [40] LI Z, DUAN Z, LEWIS F L. Distributed robust consensus control o multi-agent systems with heterogeneous matching uncertainties [J]. Automatica, 014, 50(3): [41] HOU Z G, CHENG L, AN M. Decentralized robust adaptive control or the multiagent system consensus problem using neural networks [J]. IEEE ransactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 009, 39(3): [4] AN F X, GUAN X P, LIU D R. Consensus protocol in networked multi-agent systems with non-balanced topology [J]. Control heory & Applications, 009, 6(10): [43] PAN H, NIAN X H, GUO L. Consensus control o higher-order multi-agent systems with delays [J]. Control heory & Applications, 013, 30(7): [44] LIN Z, FRANCIS B, MAGGIORE M. Necessary and suicient graphical conditions or ormation control o unicycles [J]. IEEE ransactions on Automatic Control, 005, 50(1): [45] LI S, WANG X. Finite-time consensus and collision avoidance control algorithms or multiple AUVs [J]. Automatica, 013, 49(11): [46] BOYD S, VANDENBERGHE L. Convex Optimization [M]. Cambridge: Cambridge University Press, 004: [47] MENG Z, REN W, YOU Z. Distributed inite-time attitude containment control or multiple rigid bodies [J]. Automatica, 010, 46(1): [48] YOO S J. Distributed adaptive containment control o uncertain nonlinear multi-agent systems in strict-eedback orm [J]. Automatica, 013, 49(7): [49] GE S S, WANG C. Adaptive neural control o uncertain MIMO nonlinear systems [J]. IEEE ransactions on Neural Networks, 004, 15(3): 作者简介 : 余瑶 (198 ), 女, 博士, 副教授, 目前研究方向为非线性控 制 多智能体系统 鲁棒控制 时延系统, yuyao@ustb.edu.cn; 任昊 (1991 ), 男, 硕士研究生, 目前研究方向为无人机群系统 分布式协同控制, renh618@16.com; 张兰 (1968 ), 女, 硕士生导师, 副教授, 目前研究方向为自适 应信号处理 多智能体系统, zhanglan013@ustb.edu.cn; 孙长银 (1975 ), 男, 博士生导师, 教授, 目前研究方向为智能控 制 飞行控制 模式识别 最优理论 多智能体系统等, cys@ ustb.edu.cn.