Distributed containment control of multi‐agent systems with general linear dynamics in the presence of multiple leaders

This paper considers the containment control problems for both continuous‐time and discrete‐time multi‐agent systems with general linear dynamics under directed communication topologies. Distributed dynamic containment controllers based on the relative outputs of neighboring agents are constructed for both continuous‐time and discrete‐time cases, under which the states of the followers will asymptotically converge to the convex hull formed by those of the leaders if, for each follower, there exists at least one leader that has a directed path to that follower. Sufficient conditions on the existence of these dynamic controllers are given. Static containment controllers relying on the relative states of neighboring agents are also discussed as special cases. Copyright © 2011 John Wiley & Sons, Ltd.     

高阶有向积分器网络的包含控制

研究高阶固定有向积分器网络的包含控制问题。针对静止和运动的领导者,分别为跟随者提出了不同的控制协议。领导者是静止的情况下,应用拉普拉斯变换终值定理提出线性控制协议,驱使跟随者渐近进入由多个领导者所构成的静态凸包中;领导者是移动的情况下,提出非线性控制协议,以保证跟随者能够渐近进入由多个领导者所构成的动态凸包中,并使跟随者一直保持在该凸包中,同时提出了能够达到这一目标的充分条件。仿真结果验证了该理论分析的正确性。     

Finite-time containment control of perturbed multi-agent systems based on sliding-mode control

Aimed at faster convergence rate, this paper investigates finite-time containment control problem for second-order multi-agent systems with norm-bounded non-linear perturbation. When topology between the followers are strongly connected, the nonsingular fast terminal sliding-mode error is defined, corresponding discontinuous control protocol is designed and the appropriate value range of control parameter is obtained by applying finite-time stability analysis, so that the followers converge to and move along the desired trajectories within the convex hull formed by the leaders in finite time. Furthermore, on the basis of the sliding-mode error defined, the corresponding distributed continuous control protocols are investigated with fast exponential reaching law and double exponential reaching law, so as to make the followers move to the small neighbourhoods of their desired locations and keep within the dynamic convex hull formed by the leaders in finite time to achieve practical finite-time containment control. Meanwhile, we develop the faster control scheme according to comparison of the convergence rate of these two different reaching laws. Simulation examples are given to verify the correctness of theoretical results.     

Distributed Containment Control for Nonlinear Multi‐Agent Systems with Time‐Delayed Protocol

In this paper, the containment control problem is considered for nonlinear multi‐agent systems with directed communication topology. Under the guidance of designed distributed communication protocols with/without previous state information, the followers are expected to converge to a dynamic convex hull spanned by multiple leaders. Two multi‐step algorithms are proposed to construct the corresponding protocols, the state feedback protocol and the delay‐coupled protocol, under which the containment control can be achieved asymptotically. Furthermore, it is found that the delay‐coupled protocol is rather sensitive to time delays. That is, real‐time tracking will become impossible by only using long‐dated previous state information. Finally, a numerical example is given to demonstrate the applicability and efficiency of the proposed schemes.     

Adaptive containment control of second-order multi-agent systems with nonlinear dynamics and multiple input-bounded leaders

This paper considers the adaptive containment control problem of second-order multi-agent systems with inherent nonlinear dynamics. In particular, the leaders’ control inputs are nonzero, bounded, and not available to any follower. Based on the relative states among neighbouring agents, a discontinuous adaptive protocol is first proposed to ensure that the containment errors of each follower converge to zero asymptotically, i.e. the states of the followers asymptotically converge to the convex hull spanned by those of the leaders. To eliminate the chattering effect caused by the discontinuous protocol, a continuous adaptive protocol is further designed based on the boundary layer technique and the σ-modification technique. Numerical examples are provided to demonstrate the effectiveness of our theoretical results.     

Event-triggered distributed containment control of heterogeneous linear multi-agent systems by an output regulation approach

In this paper, the event-triggered distributed containment control of heterogeneous linear multi-agent systems in the output regulation framework is studied. The leaders are treated as exosystems and the containment control problem will be converted into an output regulation problem. An event-triggered protocol is then designed for each follower by the output information of neighbours. It is proved that the followers can asymptotically converge to the dynamic convex hull spanned by multiple leaders under the designed protocol and triggered strategy. Furthermore, it is shown that the proposed protocol and triggered condition can exclude Zeno behaviour, so the feasibility of the control strategy is verified. Finally, a numerical simulation is given to illustrate the effectiveness of the proposed protocol.     

Observer-based event-triggered containment control of multi-agent systems with time delay

This paper studies the containment control of general linear multi-agent systems with or without time delay. The observer-based event-triggered control schemes will be considered. For the conventional distributed containment control protocol, we will not update the relative state continuously, i.e. the relative state will be updated by some events which happen intermittently. A completely decentralised event trigger will be designed for leader–follower systems. Under the proposed protocol, if we design some appropriate feedback gain matrices, all followers will asymptotically converge to the convex hull spanned by the dynamic leaders. Numerical simulations are also provided and the results show highly consistent with the theoretical results.     

二阶有向多智能体网络的可控包含控制

研究随机给定拓扑结构的二阶有向多智能体网络的可控包含控制问题.针对当前包含控制研究成果大多没有考虑多智能体网络领导者和跟随者的可控配置问题,结合复杂网络可控性理论和二分图最大匹配算法给出满足网络可控的领导者和跟随者集合,并为跟随者智能体设计相应的控制协议,驱使跟随者能够渐近收敛到由多个领导者构成的静态凸包中,从而实现网络的可控包含控制.仿真结果验证了理论分析的正确性.     

Formation‐containment analysis and design for high‐order linear time‐invariant swarm systems

Formation‐containment analysis and design problems for high‐order linear time‐invariant swarm systems with directed interaction topologies are dealt with respectively. Firstly, protocols are presented for leaders and followers respectively to drive the states of leaders to realize the predefined time‐varying formation and propel the states of followers to converge to the convex hull formed by the states of leaders. Secondly, formation‐containment problems of swarm systems are transformed into asymptotic stability problems, and an explicit expression of the formation reference function is derived. Sufficient conditions for swarm systems to achieve formation containment are proposed. Furthermore, necessary and sufficient conditions for swarm systems to achieve containment and time‐varying formation are presented respectively as special cases. An approach to determine the gain matrices in the protocols is given. It is shown that containment problems, formation control problems, consensus problems and consensus tracking problems can all be treated as special cases of formation‐containment problems. Finally, numerical simulations are provided to demonstrate theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

Output containment analysis and design for high‐order linear time‐invariant swarm systems

Output containment analysis and design problems for high‐order linear time‐invariant swarm systems are dealt with. Firstly, output containment protocols are presented for leaders and followers, respectively, to partially assign the output motion modes of leaders and propel the outputs of followers to converge to the convex hull formed by the outputs of leaders. Then, output containment problems for swarm systems are transformed into stability problems, and sufficient conditions for swarm systems to achieve output containment are proposed. Moreover, an approach to determine the gain matrix in the protocol for swarm systems to achieve output containment is given, which has less calculation complexity. Finally, numerical simulations are presented to demonstrate theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

A low‐complexity design for distributed containment control of networked pure‐feedback systems and its application to fault‐tolerant control

This paper addresses a low‐complexity distributed containment control problem and its extension to fault‐tolerant control for networked nonlinear pure‐feedback systems under a directed graph. The multiple dynamic leaders are neighbors of only a subset of the followers described by completely non‐affine multi‐input multi‐output pure‐feedback dynamics. It is assumed that all followers' nonlinearities are heterogeneous and unknown. The proposed containment controller is implemented by using only error surfaces integrated by performance bounding functions and does not require any differential equations for compensating uncertainties and faults. Thus, compared with the previous containment control approaches for multi‐agent systems with unknown non‐affine nonlinearities, the distributed containment control structure is simplified. In addition, it is shown that the proposed control scheme can be applied to the fault‐tolerant containment control problem in the presence of unexpected system and actuator faults, without reconstructing any control structure. It is shown from Lyapunov stability theorem that all followers nearly converge to the dynamic convex hull spanned by the dynamic leaders and the containment control errors are preserved within certain given predefined bounds. Copyright © 2016 John Wiley & Sons, Ltd.     

Distributed H∞ containment control of multiagent systems over switching topologies with communication time delay

In this study, we consider the problem of distributed H_∞ containment control for multiagent systems over switching communication topologies. There exists a constant time‐delay and the energy‐bounded communication disturbances in the information transmission process, which are considered. Using the relative output, we develop an observer‐based containment control scheme such that the followers asymptotically converge to the convex hull formed by the leaders with a guaranteed H_∞ performance level. By constructing a Lyapunov functional and using the inequality technique, sufficient conditions for the existence of such dynamic controllers are obtained in terms of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed control protocol.     

Distributed adaptive containment control of networked flexible-joint robots using neural networks

This study presents a distributed adaptive containment control approach for a group of uncertain flexible-joint (FJ) robots with multiple dynamic leaders under a directed communication graph. The leaders are neighbors of only a subset of the followers. The derivatives of the leaders are unknown, namely, the position information of the leaders is only available for implementing the proposed control approach. The local adaptive dynamic surface containment controller for each follower is designed using only neighbors’ information to guarantee that all followers converge to the dynamic convex hull spanned by the dynamic leaders. The function approximation technique using neural networks is employed to estimate the model uncertainties of each follower. It is proved that the containment control errors converge to an adjustable neighborhood of the origin regardless of model uncertainties and the lack of shared communication information. Simulation results for FJ manipulators are provided to illustrate the effectiveness of the proposed adaptive containment control scheme.     

Event-based containment control for multi-agent systems with packet dropouts

This paper is concerned with the event-triggered containment control for stochastic multi-agent systems subject to packet dropouts. The adopted event-triggered protocol is with absolutely triggered conditions catering for the requirement of real-time to an extreme. In light of such a protocol, a novel definition of containment control in mean-square sense, named as χ-containment, is proposed to better describe the tracking dynamics of followers. Based on relative measurement outputs, the purpose of this paper is to design an output-feedback controller such that all followers converge into the convex hull spanned by leaders. First, with the help of the property of the Laplacian matrix, the containment control problem is transformed to an easily setting step by step. Then, sufficient conditions with the form of matrix inequalities are derived to guarantee the desired χ-containment which depends on the initial values and the event-triggered thresholds. By introducing a free matrix combined with an orthogonal basis of the null space of control matrix, the controller gain can be obtained by solving a set of linear matrix inequalities. Finally, a simulation example is given to verify the effectiveness of the designed control protocol.     

Containment control of linear multi‐agent systems with multiple leaders of bounded inputs using distributed continuous controllers

This paper considers the containment control problem for multi‐agent systems with general linear dynamics and multiple leaders whose control inputs are possibly nonzero and time varying. Based on the relative states of neighboring agents, a distributed static continuous controller is designed, under which the containment error is uniformly ultimately bounded and the upper bound of the containment error can be made arbitrarily small, if the subgraph associated with the followers is undirected and, for each follower, there exists at least one leader that has a directed path to that follower. It is noted that the design of the static controller requires the knowledge of the eigenvalues of the Laplacian matrix and the upper bounds of the leaders’ control inputs. In order to remove these requirements, a distributed adaptive continuous controller is further proposed, which can be designed and implemented by each follower in a fully distributed fashion. Extensions to the case where only local output information is available and to the case of multi‐agent systems with matching uncertainties are also discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Event-based broadcasting containment control for multi-agent systems under directed topology

The event-based broadcasting containment control problem for both first-order and second-order multi-agent systems under directed topology is investigated. Based on certain event, each agent decides when to transmit its current states to its neighbours and the agents’ distributed control algorithms are based on these sampled state measurements, which can significantly decrease the number of the controllers’ updates. All the agents are divided into two groups, namely, the leaders and the followers. The formation control is introduced. The leaders exchange their information to converge to a formation. The followers utilise the information from both their leader neighbours and their follower neighbours and are driven to the convex hull of the leaders using the proposed control algorithms. Numerical simulations are provided to illustrate the effectiveness of the obtained theoretical results.     

带强连通分支的多智能体系统可控包含控制

研究有强连通子图拓扑结构的有向多智能体系统领导者选择及可控包含控制问题.根据网络拓扑结构,智能体被分为两类:单元智能体和一般智能体.首先设计强连通子图中个体组成的单元智能体的一致性协议实现各个单元的一致;后由单元智能体和一般智能体构成新的拓扑结构,结合复杂网络可控性理论与二分图最大匹配算法确定满足网络可控的最少领导者集合,并为所有智能个体设计相应的控制协议,驱使跟随者渐近收敛到多个领导者组成的动态凸包中,从而实现网络的可控包含控制.仿真结果验证了理论分析的正确性.     

二阶有向网络的鲁棒有限时间包容控制

在考虑非线性干扰和跟随者速度不可测的情况下,研究二阶有向网络的有限时间估计和鲁棒包容控制问题.提出含有估计层、控制层和跟随者层的新型分布式控制结构.应用有限时间稳定性理论和等效输出注入的思想,设计幂次估计器和滑模估计器,并给出网络误差的有限时间估计.进而采用齐次理论和滑模方法设计非光滑控制协议,驱使跟随者在有限时间内收敛并保持在领航者所构成的动态凸包中,且有效抑制干扰.仿真结果验证了理论分析的正确性.     

Containment Control of General Linear Multi-agent Systems with Multiple Dynamic Leaders: a Fast Sliding Mode Based Approach

In this paper, distributed containment control problems of general linear multi-agent systems are investigated. The objective is to make the followers in a multi-agent network converge to the convex hull spanned by some leaders whose control inputs are nonzero and not available to any followers. Sliding mode surfaces are defined for the cases of reduced order and non-reduced order, respectively. For each case, fast sliding mode controllers are designed. It is shown that all the error trajectories exponentially reach the sliding mode surfaces in a finite time if for each follower, there exists at least one of the leaders who has a directed path to the follower, and the leaders' control inputs are bounded. The control Lyapunov function for exponential finite time stability, motivated by the fast terminal sliding mode control, is used to prove reachability of the sliding mode surfaces. Simulation examples are given to illustrate the theoretical results.      

Robust containment tracking of uncertain linear multi-agent systems: a non-smooth control approach

This paper addresses the distributed robust containment tracking problem of networked systems with uncertain linear dynamics and multiple controlled leaders. The uncertainty class considered in this paper satisfies some matched conditions. To achieve containment tracking in such a multi-agent system, some distributed controllers consisting of using relative state-based continuous feedback and a non-smooth manoeuvre are developed. By transforming the containment tracking problem into the global robust stabilisation problem of interconnected systems, it is shown that the states of followers will asymptotically converge to a convex hull formed by those of the leaders if the control parameters in the proposed controllers are appropriately selected. It is clearly pointed out that the involved control parameters can be successfully found for solving the containment tracking problem if each follower can be directly or indirectly influenced by at least one leader, and the nominal dynamics of followers are stabilisable. The important issue of how fast containment can be achieved is also addressed. Finally, some numerical simulations are given for illustration.