In this article, the fixed-time formation-containment tracking (FXFCT) problem is addressed for heterogeneous multi-agent systems (HMASs) under a directed interaction topology. Agents are divided into a reference leader, providing a trajectory for the entire HMASs; formation-leaders, achieving the desired formation by following this trajectory; and followers. Novel distributed fixed-time observers are developed for the formation-leaders and followers with the directed interaction topology, respectively. Distributed fixed-time control protocols are then proposed for the formation-leaders and followers using the coordinate transformation methods, eliminating the restrictive but commonly adopted full-row rank assumption of agent input matrices. It is shown that under the proposed control protocols, the concerned FXFCT problem can be solved. Simulations verify the effectiveness of the obtained theoretical results. Note to Practitioners - Fixed-time cooperative control of HMASs, e.g., autonomous aerial vehicles (AAVs) and autonomous ground vehicles (AGVs), enables rapid execution of complex collective tasks, efficient resource allocation, and improved operational efficiency. Such capabilities are particularly relevant to applications in logistics, energy management, and smart agriculture, where time-critical coordination delivers significant social and economic benefits. As a unified cooperative control framework, the formation-containment tracking (FCT) problem encompasses several classical problems, such as consensus tracking, formation tracking, and containment control. Solving the FCT problem therefore provides a comprehensive solution that addresses multiple coordination objectives simultaneously, making it particularly relevant for complex real-world missions that require hierarchical coordination in HMASs. This paper addresses the FXFCT problem for HMASs under a directed interaction topology. We develop novel distributed fixed-time protocols that integrate distributed fixed-time observers and controllers for both formation-leaders and followers. For practitioners, a key practical advantage lies in an explicit, a priori computable upper bound on the convergence time that is independent of initial conditions, enabling predictable, deadline-aware mission planning. In addition, directed communication and the removal of restrictive full-row rank assumptions on agent dynamics alleviate communication overhead and improve practical applicability to heterogeneous platforms. © 2026 IEEE.

This article studies a multiplayer capture-the-flag (CTF) differential game, where multiple pursuers try to intercept evaders whose objectives are to first reach a flag and then reach a return region. The critical point is that the flag and the return region are half-planes. Our goal is to address the problem of determining the game winner and computing the pursuit winning strategies. By decomposing the complex multiplayer game into many manageable subgames involving multiple pursuers and one evader, we present the strategies under which the pursuers guarantee to win against the evader, regardless of the evader's strategy, with the necessary and sufficient conditions to determine the game winner. We then extend the results to the cases of the flag-staying time and the minimum safe flag position. To reduce the computational burdens, we prove that if multiple pursuers can ensure the pursuit winning against an evader, then at most two pursuers in this coalition are required. Finally, we solve the multiplayer game by evaluating pairwise subgame outcomes for pursuer-evader matchings. Numerical and experimental results are presented to illustrate the theoretical conclusions.

This article addresses the problem of distributed output formation optimal tracking for heterogeneous multiagent systems (MASs). Unlike the main approach in most existing formation tracking studies, which relies on prespecified trajectories, this work aims to enable heterogeneous MASs to achieve desired formation while tracking the optimal reference trajectory generated by a distributed optimization algorithm. First, a distributed time-varying optimization algorithm is proposed as the distributed optimal reference trajectory generator, which accounts for inequality constraints and ensures fixed-time convergence in consensus and asymptotic convergence in optimality. Then, a distributed output formation optimal tracking control protocol is developed for heterogeneous MASs, with the integration of the distributed optimal reference trajectory generator. Subsequently, the convergence of the proposed distributed output formation tracking control algorithm-based on time-varying optimization-is rigorously proven using Lyapunov stability analysis. Finally, simulation examples are provided to validate the theoretical results.

In this article, fully distributed prescribed-time control methods for time-varying formation tracking problems of general linear systems with directed interactions are investigated. The considered leaders have nonzero control inputs, whose upper bounds are initially unknown to the followers. For the given multiple-leader scenario, a formation tracking protocol is designed, which incorporates three components: a distributed controller, an adaptive law, and a distributed estimator. Specifically, based on the adaptive technique, the controller can be implemented without requiring global topology knowledge and the leaders’ input bounds. The estimator can also be executed in a distributed fashion, and algebraic loop phenomena are avoided herein. To guarantee prescribed-time convergence of the controller and estimator, two time-varying parameters are designed. Rigorous convergence analyses are conducted using Lyapunov functions and parametric Lyapunov equations. Furthermore, to tackle typical numerical implementation issues associated with prescribed-time control methods, a modified version of the protocol is presented. Finally, simulation examples are provided to verify the proposed theoretical results.

This article focuses on resilient time-varying optimal tracking problems of output formation in open multiagent systems (MASs). Agents can join or exit at any time and may be subject to switching between normal and malicious identities. Normal agents in the open MAS aim to minimize the sum of their local time-varying composite objective functions, each consisting of an output-related term and a state-related nonsmooth term. Simultaneously, agents are required to maintain a given output formation configuration. Based on relative outputs from neighbors, a distributed tracking protocol is proposed, combining the subgradient method with proximal mapping and an adaptive aggregation technique. By analyzing the upper bounds of total dynamic regret and individual dynamic regrets, it is proved that resilient optimal tracking of output formation can be achieved without knowledge of agent identities. Simulations validate these results.

Autonomous flight in unknown environments requires precise spatial and temporal trajectory planning, often involving computationally expensive nonconvex optimization prone to local optima. To overcome these challenges, we present the Neural-Enhanced Trajectory Planner (NEO-Planner), a novel approach that leverages a Neural Network (NN) Planner to provide informed initial values for trajectory optimization. The NN-Planner is trained on a dataset generated by an expert planner using batch sampling, capturing multimodal trajectory solutions. It learns to predict spatial and temporal parameters for trajectories directly from raw sensor observations. NEO-Planner starts optimization from these predictions, accelerating computation speed while maintaining explainability. Furthermore, we introduce a robust online replanning framework that accommodates planning latency for smooth trajectory tracking.Extensive simulations demonstrate that NEO-Planner reduces optimization iterations by 20%, leading to a 26% decrease in computation time compared with pure optimization-based methods. It maintains trajectory quality comparable to baseline approaches and generalizes well to unseen environments. Real-world experiments validate its effectiveness for autonomous drone navigation in cluttered, unknown environments.Code: https://github.com/Amos-Chen98/neo-plannerVideo: https://youtu.be/UoroRe-euDk

Ensuring the safety of various real-world applications based on reinforcement learning (RL), such as quadcopter control, robotic manipulators, and autonomous robots, remains a critical challenge, despite RL’s remarkable success in solving complex decision-making tasks. Existing on-policy Lagrangian optimization methods in safe RL typically use a single policy to balance the trade-off between safety and return without taking the potential benefits of adopting multiple policies into account. In this paper, a new on-policy method is proposed, named Safe-Adjusted Policy Optimization(SAPO), which is a dual-policy framework designed to address safety constraint violations in RL. By incorporating a cost-oriented policy to dynamically adjust a reward-oriented policy, the SAPO effectively resolves the trade-off between safety and return. Moreover, to enhance performance in carrying out high-dimensional tasks, the Kullback-Leibler (KL) divergence and a Gaussian kernel are employed in the distance functions to facilitate the training. In addition, a quadcopter-safe-navigation task is designed to overcome the drawback of previous research on quadcopter-safe-navigation with RL that only pays attention to reward function design without considering policy-level optimization. Finally, experimental results verify the feasibility of the designed task. Meanwhile, indicated by the test on real device, the proposed algorithm is easy to be implemented, offers performance guarantees, and outperforms existing safe RL baselines.

This paper presents a distributed observer-based non-linear control framework to address a prescribed-time output formation tracking problem of heterogeneous nonlinear networks under a nonlinear leader system and a digraph with a spanning tree, wherein dynamics and dimensions of leader-follower systems can be different. To solve this issue, prescribed-time nonlinear distributed observers are firstly developed to estimate the nonlinear leader’s state, and the prescribed-time distributed control law is further presented. Under this framework, the zero-error output formation tracking is obtained within a pre-specified and user-defined time by leveraging the time transformation approach. The proposed algorithm is free of global knowledge, and its convergence time is irrelevant to initial conditions, control parameters or network structures. Simulation and experimental results are given to verify the designs’ effectiveness. The main features of developed designs lie in that: (1) prescribed-time output formation tracking of heterogeneous nonlinear systems can be achieved under directed graphs, wherein both dynamics and dimensions of leader-follower nonlinear systems can be different; (2) the convergence time is pre-specified and thus, irrelevant to any initial conditions, control parameters, or network structures; (3) the proposed algorithm that does not depend on any global information, is distributed under a general communication graph with a directed spanning tree; and (4) the prescribed-time output formation tracking solution can be extended beyond the terminal time.

This article addresses a prescribed-time time-varying output formation tracking (TVOFT) problem for heterogeneous multiagent systems (MASs) under directed topologies. Formation tracking in heterogeneous linear MASs is critical for practical applications, such as cooperative robotics, autonomous transportation, and surveillance. However, many existing related designs often fail to guarantee convergence within a prescribed time. To overcome this limitation, a distributed prescribed-time TVOFT protocol associated with a corresponding design algorithm is presented. In the designed protocol, a distributed output-feedback observer is constructed for each follower to estimate the state of the leader within a prescribed time. Then, a local output-feedback controller is developed by incorporating a local state observer. It is proved that the heterogeneous MASs can achieve the desired TVOFT within the prescribed time. Furthermore, a heterogeneous experimental platform consisting of two autonomous aerial vehicles and three autonomous ground vehicles, is constructed to verify the effectiveness of the proposed prescribed-time TVOFT design. Comparative experiment results highlight the advantages of the proposed design over existing methods from the perspective of accurate prescribed-time convergence and practical feasibility.

This paper investigates the robust consensus scheme for a specific type of leader-follower multi-agent systems (MASs) in the context of a human-cyber-physical interactive system. The system encounters frequent communication challenges characterized by packet loss and data leakage. To address these challenges, the paper proposes a secure decentralized scheme that leverages game theory to determine optimal policies in a zero-sum game. The scheme utilizes the adaptive dynamic programming (ADP) method, which involves a decentralized iterative process. The paper demonstrates that the generated sequence exhibits exponential convergence and provides the maximum iteration number based on the convergence error. To mitigate data leakage among players, the scheme incorporates a secure operation that involves encrypted data. This integration seamlessly combines data conversion and encryption-decryption into decentralized computation. Finally, the paper presents a simulation example to validate the efficacy of the consensus scheme.

Formation constrained optimal tracking problems for heterogeneous multi-agent systems under Byzantine attacks are studied. The objective of the honest agents, unaffected by Byzantine attacks, is to find optimal trajectories that minimize the cumulative local cost functions of all honest agents, while simultaneously satisfying the cumulative local inequality constraints and maintaining the formation configuration. A distributed resilient formation tracking controller utilizing preview control and primal-dual strategy is proposed without requiring each agent to know which agents are affected by Byzantine attacks. The performance of the proposed algorithm is analyzed in terms of the upper bounds of dynamic regret and constraint violations. Numerical simulations and experiments, involving one unmanned aerial vehicle and four unmanned ground vehicles, are performed to verify the effectiveness of the obtained results.

This article investigates the Nash equilibrium (NE) seeking problems for multicoalition games with heterogeneous general linear dynamics over unbalanced digraphs. The coalition can be regarded as a virtual player but the true decision-makers are the actual players themselves which can only access to their own cost functions. To deal with the unbalanced digraphs, the left eigenvector is estimated and its prescribed-time convergence is illuminated with the time transfer approach. Then the NE seeking part and the output regulation part are designed to adapt the dynamics of the players. The initial values of the auxiliary vectors are selected to avoid utilizing the out-degree information. The steady state of the closed-loop system is analyzed and the prescribed-time convergence is proved based on the Lyapunov method. Finally, the simulation results of both the mobile sensor connectivity game and the electricity market game are presented to show the effectiveness of the proposed algorithm.

In practical applications, not all nodes in networked swarm systems are cooperative. The noncooperative nodes are transformed from healthy ones because of cyber-attacks launched by malicious adversaries, hardware faults caused by low reliability individuals, or communication delay. In this article, an approximate fault detection method using residual threshold is shown to judge which agent is cooperative or not, and to reconstruct the communication topology. Then, the event-triggered technique is utilized to design the multileaders formation tracking protocol with resilience to noncooperative nodes. The Zeno behavior is considered to constrain the trigger condition. Finally, the comparison simulation results show the effectiveness and advantage for the proposed secure control method.

This article proposes a finite-time distributed state estimation (DSE) algorithm for discrete-time stochastic nonlinear systems with heterogeneous sensors. Considering the network with heterogeneous sensors, the distributed estimate framework is designed by three phases, namely, priori prediction, measurement update, and consensus fusion. To obtain the accurate priori prediction results, the interactive multiple model (IMM) method is adopted to calculate the priori state value in the priori prediction phase. By introducing the measurement probability matrix, a novel heterogeneous measurement information fusion algorithm is designed. Then the measurement information of each sensor is used to update the priori prediction estimates to calculate the estimate results in the measurement update phase. Based on the consensus method, the estimate results of each sensor are fused with consensus weight to calculate the distributed state estimates of nonlinear systems in the consensus fusion phase. Besides, with finite consensus fusion steps, the bounds of the proposed distributed estimate algorithm are proved to be existed. Finally, distributed state estimate simulation example for nonlinear system is set to validate the performance.

This article investigates the tracking control problem of heterogeneous multiagent systems (MASs) with intrinsic nonlinear dynamics in noisy and time-delayed environments. First, a stability criterion for nonlinear stochastic delay systems with multiplicative noise and time-varying delay is proposed by applying the appropriate Lyapunov-Krasovskii functional. Then, based on the proposed stability criterion, sufficient conditions are derived for mean square (m.s.) and almost sure (a.s.) tracking of heterogeneous MASs with intrinsic nonlinear dynamics. Afterward, the above sufficient conditions further degenerate to integrator heterogeneous MASs. In particular, when the time-delay vanishes, the explicit conditions are obtained for the integrator heterogeneous MASs in the form of scalar inequalities, which can intuitively reflect the relationship between noise intensity and control gains. Finally, simulation results validate the effectiveness of the proposed control protocol.

Time-varying formation tracking (TVFT) control problem for multiple Euler–Lagrange (EL) systems with an uncertain leader is investigated in this article. The objective of this problem is for the followers' states to track the uncertain leader's output and achieve the desired formation. One of the major challenges in solving such a problem lies in estimating the information of the leader. In this article, an observer-based TVFT control framework is presented in this article, which does not require all or part of the follower agents to know the system dynamic knowledge of the leader agent directly in advance. First, two fully distributed adaptive observers are presented for estimating the uncertain leader's state, output, and the unknown system matrices under an IE condition, which relaxes the existing restrictive persistently exciting or cooperative finite-time excitation condition. Then, on the basis of the constructed adaptive observers, state and output variables of the leader, a fully distributed TVFT for EL systems with an uncertain leader is developed, which means the proposed formation tracking controller operates independently of the system dynamic knowledge of the leader agent; with the help of the Lyapunov stability theory, the TVFT criterion for the considered system is derived. Finally, in order to demonstrate the theoretical results derived in this article, a practical air–ground formation tracking platform, which consists of an unmanned aerial vehicle and four unmanned ground vehicles, is introduced, and physical experimental results are obtained.

This article concentrates on the design of an active resilient formation tracking control strategy for heterogeneous swarm systems (HSS) under malicious cyber-attacks. The attack signals, which are injected into both actuator and sensor randomly, can be detected and estimated by using an observer-based attack estimation scheme. The compromised measured outputs of individuals in the swarm are used to design the distributed secure control protocol and the consensus-based formation condition. For each follower, a compensator is designed to address the secure formation tracking problem for HSS using an approximate model following strategy. Moreover, the event-triggered technique is utilized to achieve the time-varying formation and to track the leader, and saves the communication resource in practice. Finally, a numerous simulation for a heterogeneous swarm system with three different dynamics nodes is presented to verify the effectiveness of the proposed secure approach.

This article provides a solution to time-varying formation (TVF) tracking control problems of nonholonomic mobile robots (NMRs) with asymmetric input constraints. Different from existing works, the configuration of the formation for NMRs is time-varying, which requires formation tracking errors to keep convergent even if the configuration of the desired formation changes over time. A novel sliding mode controller is proposed to handle this problem, and the closed-loop stabilities of NMRs under given control protocols are proven by Lyapunov theory. As special cases of TVF, the above control methods can be directly applied to flexible formation and fixed formation studied in existing works. It should be pointed out that if the desired formation is fixed, the angular velocity of the tracking target is not necessary when designing the corresponding control algorithm. In addition, there is no singularity in the methods provided in this article. By collaborative selection of parameters in these control protocols, the adjustment ranges of linear and angular velocities associated with the leader or virtual leader are almost the same as that of followers. Numerical simulations and hardware-in-the-loop (HiL) simulations further verify the theoretical results.

This article is concerned with resilient formation tracking problems for general linear multiagent systems, where the leader's control input is unavailable to all the followers and partial followers' behaviors are malicious due to the node attacks. Despite the presence of the nonautonomous leader and the adversarial followers, the remaining benign followers are still expected to track the leader's trajectory with the prescribed time-varying formation. To this end, a resilient scheme comprising an attack detection and isolation strategy and a formation tracking protocol is proposed. The detection strategy enables every benign follower to identify its adversarial neighbors with two-hop communication information, as long as the underlying topology meets given conditions. Then, the detected adversarial agents are directly removed to avoid the spread of their influence, which induces a new problem called node loss. To accommodate possible node loss events, the designed tracking protocol is independent of certain global knowledge, and its convergence is demonstrated by means of the impulsive Lyapunov functions. Finally, the proposed resilient scheme is verified by two simulation examples.

This article investigates a fully data-driven method to solve the robust output formation tracking control problem for the multiagent system (MAS) under actuator faults. The outputs of the followers are controlled to track those of multiple leaders with respect to a convex point while achieving an expected time-varying formation. To obviate the requirement of various system prior knowledge in typical MAS control, a hierarchical frame is developed with three learning and control stages using the online measured data. First, a distributed adaptive observer is designed to coordinate the state convex of multiple leaders while estimating unknown dynamics. The adaptive mechanism relaxes the demand for global topology. Second, by collecting and reusing the online system data, an off-policy reinforcement learning (RL) method is proposed in a continuous form to acquire nominal feedback gains from partial observations of the followers. Essential system models are learned along with the RL process, while solutions to the output regulation equations are implicitly obtained. Third, a comprehensive robust controller is further presented based on the previous learning results. To address the actuator faults with efficiency loss and bias, the adaptive neural networks and robust compensations are utilized in a model-free manner. The output formation tracking is achieved under a derived feasibility condition while stabilities of the learning and control methods are analyzed. Finally, simulation results demonstrate the validity of this fully data-driven control frame.

Distributed output formation optimal tracking problems for multiagent systems over time-varying topologies with asynchronous and intermittent communications are investigated. Each agent collaboratively computes and tracks the optimal output formation reference that minimizes a global objective function formed by summing local objective functions. Simultaneously, this reference satisfies global constraints composed of local nonlinear inequality constraints and local closed convex set constraints. An asynchronous distributed estimator-based tracking control protocol is designed utilizing the constrained stochastic subgradient random projection method and the Lyapunov stability theory. Sufficient conditions for asymptotic convergence are given. It is revealed that the states of agents with constraints under asynchronous and intermittent communications converge asymptotically to the optimal reference signal using only neighboring information within the predefined formation. Finally, a numerical example is provided to validate the theoretical results.

The finite-time output time-varying formation tracking (TVFT) problem for heterogeneous nonlinear multiagent system (MAS) is investigated in this article, where the dynamics of the agents can be nonidentical, and leader's input is unknown. The target of this article is that the outputs of followers need to track leader's output and realize the desired formation in finite time. First, for removing the assumption that all agents are required to know the information of leader's system matrices and the upper boundary of its unknown control input in previous studies, a kind of finite-time observer is constructed by exploiting the neighboring information, which can estimate not only the leader's state and system matrices but also can compensate for the effects of unknown input. On the basis of the developed finite-time observers and adaptive output regulation method, a novel finite-time distributed output TVFT controller is proposed with the help of the technique of coordinate transformation by introducing an extra variable, which removes the assumption that the generalized inverse matrix of follows' input matrix needs to be found in the existing results. By means of the Lyapunov and finite-time stability theory, it is proven that the expected finite-time output TVFT can be realized by the considered heterogeneous nonlinear MASs within a finite time. Finally, simulation results demonstrate the efficacy of the proposed approach.

This article investigates the time-varying group formation tracking problems for multiagent systems where the agents are divided into multiple groups and each one achieves different goals. Specially, competition is allowed between different subgroups and there exists cooperation within each group. On this premise, distributed Nash equilibrium seeking strategy is utilized to search for the optimal relative evolutionary trend of each group, and formation tracking control protocol is designed to enable followers in each group to track their leader. Moreover, considering that the velocity signals are usually not available in practical situations, Nash equilibrium seeking strategy and formation tracking control are modified without velocity measurements. According to Lyapunov-based theory, it is proven that both the convergence of the Nash equilibrium seeking between subgroups and the convergence of formation tracking error within each group can be fulfilled. A simulation is provided to demonstrate the theoretical results.

This article investigates the practical time-varying output formation tracking (TVOFT) problem for heterogeneous nonlinear multiagent systems (MASs) having multiple leaders, where agents herein could have heterogeneous dynamics and interact with each other under event-triggered communications. It is required that the outputs of followers not only track the predefined convex combination of multiple leaders but also achieve the desired time-varying formation simultaneously. The existing works on formation tracking problems for MASs with multiple leaders depend on the assumption that each follower is a well-informed or uninformed follower, where the well-informed follower is required to have all the leaders as its neighbor. To remove the limitation, a fully distributed observer-based formation tracking control protocol is developed and employed. First, an adaptive state observer with an edge-based event-triggered mechanism for estimating the states of multiple leaders is proposed based on the neighboring interactions, which eliminates the unexpected Zeno behavior. Second, a novel observer is constructed for each follower by exploiting the output information of the follower, in which the adaptive neural network (NN)-based approximation is exploited to compensate for the unknown nonlinearity. A practical TVOFT control protocol is then generated by the proposed observers, where the parameters are determined by an algorithm including five steps. With the help of Lyapunov stability theory and output regulation method, a practical TVOFT criterion for the considered closed-loop system is derived. Finally, the effectiveness of the proposed control scheme is illustrated by a numerical example.

This article addresses finite-time connectivity-preserving rendezvous problems of networked uncertain Euler-Lagrange systems, where two types of time-varying leaders are investigated, and only a subset of followers can have access to the leader’s trajectory. The distributed estimation and control architecture is then established to solve this problem with an emphasis on the settling-time estimation. In particular, in the first layer, the finite-time distributed estimators are developed to estimate and reconstruct the states of both linear and nonlinear leaders, respectively. In the second layer, distributed controllers are designed for consensus tracking in a finite-time using estimated leader information. Further, to account for limited sensing ranges, another distributed algorithm is given via an artificial potential field to guarantee finite-time rendezvous. Numerical simulation results are given to validate the effectiveness of the proposed designs.

In this paper, the trajectory tracking control problem of quadrotor subject to unknown external disturbance and input constraints is addressed. In order to achieve the active disturbance rejection, a composite nonlinear extended state observer (ESO) consisting of both linear and nonlinear functions is developed, and the rigorous convergence analysis is also presented. It is worth mentioning that the proposed composite nonlinear ESO brings together the advantages of both the linear and nonlinear ESOs. Furthermore, the position controller is proposed in the nested saturation control framework, and the disturbance estimate is incorporated in the controller to reject disturbances actively, which provides a composite disturbance rejection control strategy for systems with input constraints and unknown disturbances. The domain of attraction and the rigorous stability analysis of whole system are also presented. Finally, simulation and experiment are provided to show the efficiency of the proposed methods.

The practical output containment problem for heterogeneous nonlinear multiagent systems under external disturbances generated by an exosystem is investigated in this article. It is required that the outputs of followers converge to the predefined convex combination of leaders' outputs. One of the major challenges in solving such a problem lies in dealing with the coupling among different nonlinearities, state dimensions, and system matrices of heterogeneous agents. To overcome the aforementioned challenge, a distributed observer-based control protocol is developed and employed. First, an adaptive state observer for estimating the states of all the leaders is constructed based on the neighboring interactions. Second, two new classes of observers are constructed for each follower exploiting the output information of the follower, in which the adaptive neural networks (NNs)-based approximation is exploited to compensate for the unknown nonlinearity in the followers' dynamics. A practical output containment control protocol is then generated by the proposed observers, where the control parameters are determined by an algorithm including two steps. Furthermore, with the help of the Lyapunov stability theory and the output regulation method, the practical output containment criteria for the considered closed-loop system under the influences of external disturbances are derived on the basis of the presented control protocol. Finally, the derived theoretical results are illustrated by a simulation example.

The distributed formation tracking control problem with multiple leaders under actuator faults and constraints is investigated in this article. All followers in the multiagent system should achieve a desired time-varying formation and track the convex combination of multiple leaders. To accomplish the control task, an active reconfigurable control scheme is proposed using the local information between agents, as well as the fault values of individuals provided by fault estimation observers. Combining with the Lyapunov stability theorem and the property of the Laplacian matrix, the control gains are calculated using the adaptive technique with a formation tracking feasibility condition. The original reconfigurable protocol is modified by utilizing anti-windup compensators to against saturation phenomenons (both magnitude and rate) in actuators. The simulation results validate that the presented scheme can address the faults as well as the actuator saturation.

This paper investigates practical time-varying formation tracking problems for high-order multiagent systems in an adversarial environment, where the leader's control input is unknown and the followers are prone to agent-attacks. To guarantee the remaining benign followers not suffering attacks still can track the leader's trajectory with an expected formation configuration, the resilient practical time-varying formation (RPTVF) tracking is examined in this work. Firstly, to avoid the influence of attacked follower agents, a security strategy is proposed, which is an enhanced version of the weighted mean-subsequence-reduced algorithm. Secondly, using relative information of neighbors, a resilient tracking protocol is designed for each benign follower to track the leader. Then, it is proved that according to the above strategy and protocol, the multiagent systems with a leader of unknown input can reach convergence in a finite time. Finally, simulation examples are given to verify the theoretical results.

This paper studies the multi-agent differential game based problem and its application to cooperative synchronization control. A systematized formulation and analysis method for the multi-agent differential game is proposed and a data-driven methodology based on the reinforcement learning (RL) technique is given. First, it is pointed out that typical distributed controllers may not necessarily lead to global Nash equilibrium of the differential game in general cases because of the coupling of networked interactions. Second, to this end, an alternative local Nash solution is derived by defining the best response concept, while the problem is decomposed into local differential games. An off-policy RL algorithm using neighboring interactive data is constructed to update the controller without requiring a system model, while the stability and robustness properties are proved. Third, to further tackle the dilemma, another differential game configuration is investigated based on modified coupling index functions. The distributed solution can achieve global Nash equilibrium in contrast to the previous case while guaranteeing the stability. An equivalent parallel RL method is constructed corresponding to this Nash solution. Finally, the effectiveness of the learning process and the stability of synchronization control are illustrated in simulation results.

The adaptive practical optimal time-varying formation tracking problems of the disturbed high-order multi-agent systems with a noncooperative leader are considered. Different from the former achievements, the effects of the leader’s unknown control input and followers’ external disturbances are both considered in the optimal time-varying formation tracking issues. Firstly, an adaptive practical optimal time-varying formation tracking protocol is proposed. The extended state observers and adaptive neural networks are introduced to estimate the integrated uncertainty and value function for the adaptive protocol, respectively. Then, an algorithm is presented to determine the control parameters for the adaptive optimal protocol and neural networks weights update laws. Thirdly, the stability and the optimal formation tracking property are analyzed for the closed loop disturbed high-order multi-agent system. Finally, the numerical simulation results are presented for revealing the effectiveness of the obtained theoretical methods.

Time-varying group formation-containment tracking problems for general linear multiagent systems with unknown control input are investigated. Agents are classified into tracking leaders, formation leaders, and followers and assigned in groups. Tracking leaders with unknown control inputs provide unpredictable trajectories as macroscopic moving references. Formation leaders accomplish desired subformations while following the trails of tracking leaders. At the same time, followers converge into different convex hulls spanned by formation leaders. First, formation-containment tracking protocols are designed with neighboring relative information and effects of unknown input of tracking leaders. Then, the design of group division is analyzed by adjusting the properties in Laplacian matrices, which represent interaction relationships. An algorithm to determine the parameters in control protocols is proposed, and the formation tracking feasible constraint is presented. Next, it is proved that the general linear multiagent system can achieve time-varying group formation-containment control effectively with errors uniformly asymptotically converging to zero under designed protocols. Finally, a numerical simulation is given to verify the effectiveness of obtained theoretical results.

In this article, time-varying group formation (TVGF) tracking problems for general linear multiagent systems (GLMASs) with switching interaction topologies are investigated. Different from previous studies, a novel TVGF tracking approach is proposed, where all agents are divided into three types: the virtual leader, the group leader, and the follower. The virtual leader is designed to assign the trajectory of GLMASs. Subgroups can be interacted with each other by cooperation among group leaders such that the relative configuration between different groups can be adjusted simultaneously. The followers in each group can achieve time-varying subformations. Moreover, under the influence of external disturbances and switching topologies, based on the distributed observer, two different distributed adaptive control protocols are constructed without using any global information such as the eigenvalue of the Laplacian matrix related to the communication topologies, the upper boundness of the leader’s input, and so on. The algorithms to determine parameters of control protocols are also presented. Furthermore, the closed-loop stability of GLMASs is proven by the Lyapunov theory. Finally, numerical simulations are given to verify the effectiveness of theoretical results.

Predefined mymargin finite-time output containment control problem for nonlinear multi-agent systems with multiple dynamical leaders under directed topology is investigated, where the outputs of followers can converge to the predefined convex hull formed by the multiple leaders within a finite time, and the leaders can have unknown control inputs. Firstly, for the directed topological structure among the followers, a distributed adaptive observer is designed to estimate the whole states of all the leaders under the influences of the leaders' unknown inputs. By utilizing Hardy's inequality and common Lyapunov theory, the finite-time convergence of the proposed observer is proved. On the basis of this conclusion, a predefined distributed containment control protocol including the desired convex combinations of the leaders is developed for each follower by using the given weights. Then an algorithm is proposed to design the control parameters in the proposed containment control protocol. With the help of the output regulation theory, the finite-time output containment criterion for nonlinear multi-agent systems in the presence of the leaders' unknown inputs is derived. Finally, a numerical simulation example is presented to demonstrate the effectiveness of the theoretical results.

This article is concerned with the co-design of fault detection and consensus control protocol for a class of multi-agent systems (MASs) subject to Denial-of-Service attack, where the attack behavior is hidden to the defender as the adversary may launch the attack with different durations but the defender may not know the real situation at each time instant. Due to the complicated attack behavior, a hidden semi-Markov process is introduced where only the observed attack modes emitted by the inaccessible ones are available. The probability density functions of sojourn time and the semi-Markov kernel are adopted to help analyze the combined fault detection and consensus control system. Sufficient conditions on the σ-error mean square stability of MASs with mixed H∞/H-performance are established by using a set of Lyapunov functions that depend on the hidden and the observed attack modes. Moreover, the gain matrices of controller and detector are obtained by solving an optimization problem with some matrix inequality constraints. Finally, the simulation of autonomous unmanned underwater vehicles is used to show the effectiveness of the proposed results.

Fully adaptive practical time-varying output formation tracking issues of high-order nonlinear stochastic multiagent systems with multiple leaders are researched, where the adaptive fuzzy-logic system (FLS) is introduced for estimating the mismatched integrated uncertain items. Distinctive with former results, stochastic noise is considered in the dynamics, and the followers are required for achieving the time-varying output formation tracking in probability of the convex combination of the leaders' outputs. First, a fully adaptive practical time-varying output formation tracking protocol is put forward, which only utilizes the neighboring relative information, and the global interaction topology information is not used. Besides, the designed protocol employs the adaptive FLSs to estimate the mismatched uncertainties of the followers and the leaders, and the uncertain boundary functions of the stochastic noise. Then, the design process of control protocol and parameter adaptive update law is summarized within four steps in an algorithm. Third, the stability and the properties of the proposed protocol and algorithm are analyzed by employing the Lyapunov theories and stochastic stability theories. Finally, numerical simulation results illustrate the effectiveness of achieved protocol and algorithm.

This article addresses the distributed secure platoon control of connected vehicles with denial-of-service (DoS) attack phenomena, which may occur at some sampling time instant. First, a switched time-delay system model is introduced, which captures the time-varying sampling and the DoS attack phenomena simultaneously. Then, sufficient conditions are obtained based on the Lyapunov stability theory, the Jensen’s Inequality method, and the topology matrix decoupling technique, such that the vehicle platoon system under consideration achieves an exponential tracking performance. In this article, the presented system design conditions establish several quantitative relationships between attack parameters and system performance. Moreover, the critical values of attack frequency (AF) and the sampling interval (SI) are also derived, respectively. Finally, both of the simulation and experiment studies on a network of four vehicles are introduced to validate the design.

In this paper, the problem of distributed state estimation for nonlinear networked system with correlated noises is investigated. First, a distributed weighted consensus-based cubature information filtering algorithm is designed, of which the goal is to achieve accurate estimated states with the presence of correlated noises in a fully distributed fashion. Then based on the statistical linear approximation method, it is further proved that, the estimated states of the proposed filtering algorithm are consistent. Finally, the improved performance of the proposed filtering algorithm are confirmed by the numerical simulation.

This paper focuses on the finite-time output containment problem for a kind of nonlinear multi-agent systems with multiple dynamic leaders. Firstly, considering the topological structure among the followers, a kind of adaptive distributed observer is designed to estimate the whole states of all the leaders. By utilizing common Lyapunov theory, the finite-time convergence of proposed distributed observer is proved. On the basis of this conclusion, a containment control protocol including the desired convex combinations of the leaders is developed for each follower by using the given weights. With the help of the output regulation theory, the finite-time output containment criterion for nonlinear multi-agent systems is derived. Finally, a numerical example is presented to demonstrate the effectiveness of the theoretical results.

This paper studies the finite-time time-varying formation tracking problems for high-order multiagent systems (MASs) under the influences of both mismatched disturbances and the leader's unknown input. The outputs of the followers can not only realize the prespecified finite-time formation but also track the desired trajectory generated by the leader in finite time. First, a disturbance observer is designed for every follower to estimate the disturbances in finite time. Then, based on the homogeneous finite-time control, the integral sliding mode control, and the super-twisting algorithm, a distributed formation tracking protocol is presented utilizing the neighboring interaction. The proposed protocol is continuous, so the large chattering of the control inputs can be avoided effectively. Furthermore, it is proved that the desired formation tracking can be realized in finite time by MASs in the presence of mismatched disturbances and the leader's unknown input. Finally, a simulation example is designed to verify the theoretical results.

Robust H∞guaranteed cost time-varying formation tracking problems for high-order multiagent systems with external disturbances and time-varying delays are considered. It is required that the followers' states need to track the leader's state and to actualize the desired time-varying formation simultaneously. First, this paper devises a linear time-varying formation tracking protocol considering the effects of the delays constituted by the neighboring information. Then, an integral linear quadratic cost function constructed by the control input constraint and the formation tracking error constraint is utilized to analyze the suboptimal robust H∞guaranteed cost time-varying formation tracking performance. Third, sufficient conditions for actualizing the guaranteed cost time-varying formation tracking with H∞disturbance attenuation performance are derived, and the conditions for the feasible time-varying formation tracking are raised. To obtain the gain matrix of the protocol, only four linear matrix inequalities are required to be solved. Finally, a numerical simulation example reveals the effectiveness of the acquired results, where five agents achieve the robust H∞guaranteed cost time-varying formation tracking.

Formation-containment control problems for multiple multirotor unmanned aerial vehicle (UAV) systems with directed topologies are studied, where the states of leaders form desired formation and the states of followers converge to the convex hull spanned by those of the leaders. First, formation-containment protocols are constructed based on the neighboring information of UAVs. Then, sufficient conditions for multi-UAV systems to achieve formation-containment are presented. An explicit expression to describe the relationship among the states of followers, the time-varying formation for the leaders and the formation reference is derived. It is shown that the states of followers not only converge to the convex hull formed by those of leaders but also keep certain formation specified by the convex combination of the formation for the leaders. Moreover, an approach to determine the gain matrices of the formation-containment protocol is proposed by solving an algebraic Riccati equation. Finally, a formation-containment platform with five quadrotor UAVs is introduced, and both the simulation and experimental results are presented to demonstrate the effectiveness of the obtained results. Note to Practitioners-This paper addresses the problem of formation-containment control for multi-UAV systems over directed topologies. In practical applications, there may exist multiple leaders and multiple followers in a multi-UAV system. Formation-containment means that the states of leaders form the desired time-varying formation and at the same time the states of the followers converge to the convex hull spanned by those of the leaders. Formation-containment control provides a unified framework for formation control and containment control, and has potential applications in the cooperative source seeking, load transportation, and surveillance. Although formation control and containment control problems have been studied a lot, the formation-containment control problem for multi-UAV system is still open and challenging. This paper proposed a distributed formation-containment protocol for the multi-UAV system using local neighboring information. Sufficient conditions for multi-UAV systems to achieve formation-containment are presented. It is proven that the states of followers not only converge to the convex hull formed by those of leaders but also keep certain formation specified by the convex combination of the formation for the leaders. An approach to design the formation-containment protocol is given. A remarkable point for this paper is that the obtained results are demonstrated by practical experiments with five quadrotor UAVs.

This paper addresses fully distributed formation control problems for high-order linear multiagent systems (MASs) with directed communication topology. In order to overcome the shortcomings of designing time-varying formation (TVF) protocols via the global information of the communication network and the full states of all the agents in existing formation results, a novel adaptive TVF protocol is developed. First, dynamic output feedback information and sequential observers are used to construct the adaptive TVF protocol. Then, a distributed algorithm which includes a TVF feasibility constraint is proposed. Only local outputs of neighboring agents are used in the algorithm. Moreover, applying the proposed approach and the Lyapunov stability theory, it is found that the fully distributed TVF can be achieved. Finally, numerical simulations are given to demonstrate the theoretical results.

Time-varying formation tracking (TVFT) control problems for a team of unmanned aerial vehicles (UAVs) with switching and directed interaction topologies are investigated, where the follower UAVs realize a given time-varying formation while tracking the leader UAV. A TVFT control protocol is firstly constructed utilizing local neighboring information, where the information of the leader UAV is only available to partial followers and the neighborhood can be switching. An algorithm composed of four steps is provided to design the TVFT protocol. It is proved that the UAV swarm system can realize the TVFT using the designed protocol if the dwell time for the switching directed topologies is larger than a fixed threshold and the TVFT feasibility condition is satisfied. Based on the ultrawideband positioning technology, a quadrotor UAV formation control platform with four quadrotor UAVs is given. The obtained theoretical results are applied to solve the target enclosing problems of the UAV swarm systems. A flying experiment for three follower quadrotor UAVs to enclose a leader quadrotor UAV is carried out to verify the effectiveness of the presented results.

Fully distributed time-varying formation (TVF) control problems are addressed in this paper using an adaptive output-feedback approach for general linear swarm systems with fixed and switching topologies. In contrast to the earlier results on formation control problems, the general linear swarm system can achieve the predefined TVF using only local output information and independent of the Laplacian matrix associated with the communication topologies. The implementation cost of calculating the global information and requiring full state information is avoided when achieving the desired TVF. First, a node-based TVF control protocol is constructed via dynamic output feedback for the case with fixed communication topology, where adaptive-based coupling weights are introduced to eliminate the dependence on the global information about the topology. Then an algorithm is presented to determine the gain matrices in the node-based adaptive TVF control protocol, and a feasible formation constraint is provided. The stability of the algorithm is proved based on the Lyapunov theory. Furthermore, under the case of switching communication topologies, an edge-based TVF control protocol is constructed with dynamic output feedback and an adaptive law for adjusting the coupling weights among agents. A sufficient condition is derived using the common Lyapunov theory for general linear swarm systems to achieve the predefined TVF satisfying the feasible formation constraint. Two numerical examples are presented to illustrate the theoretical results.

This paper studies the containment control problems for fully heterogeneous multi-agent systems (HMASs) with switching topologies, where both the leaders and the followers have different dynamics. Firstly, a distributed observer is constructed for each follower to estimate the states of all the non-identical leaders using the neighboring interaction. Then, based on the estimated states of the multiple leaders, an output containment control protocol is proposed for HMASs using the output regulation strategy, where several predefined weights are applied for the followers to specify the desired convex combinations of the leaders. Thus, the given containment format is independent of the interaction topology. In light of the common Lyapunov stability and output regulation theory, it is proved that the desired output containment can be realized by fully HMASs under the influences of switching topologies. Finally, a simulation example is provided to verify the effectiveness of the theoretical results.

This paper investigates event-triggered formation problems of general linear multi-agent systems subject to sampled-data. The time-varying formation this paper studied can be described by a bounded piecewise differentiable function. Firstly, a time-varying formation control protocol is proposed based on event-triggered scheme with the sampled states of the neighboring agents. Each agent broadcasts its state information to neighbor nodes at an sampled instant if the triggering condition is satisfied, and the communication load is decreased significantly. Then an algorithm consisting of three steps is proposed to design the event-triggered formation control protocol. Moreover, it is proven that the multi-agent systems can achieve the desired time-varying formation which belongs to the feasible formation set with the bounded formation error under the designed event-triggered formation protocol. Finally, the effectiveness of the theoretical analysis is verified by a simulation.

Practical time-varying formation tracking problems for second-order nonlinear multiagent systems with multiple leaders are investigated using adaptive neural networks (NNs), where the time-varying formation tracking error caused by time-varying external disturbances can be arbitrarily small. Different from the previous work, there exists a predefined time-varying formation formed by the states of the followers and the formation tracks the convex combination of the states of the leaders with unknown control inputs. Besides, the dynamics of each agent has both matched/mismatched heterogeneous nonlinearities and disturbances simultaneously. First, a practical time-varying formation tracking protocol using adaptive NNs is proposed, which is constructed using only local neighboring information. The proposed control protocol can process not only the matched/mismatched heterogeneous nonlinearities and disturbances, but also the unknown control inputs of the leaders. Second, an algorithm with three steps is introduced to design the practical formation tracking protocol, where the parameters of the protocol are determined, and the practical time-varying formation tracking feasibility condition is given. Third, the stability of the closed-loop multiagent system is proven by using the Lyapunov theory. Finally, a simulation example is showed to illustrate the effectiveness of the obtained theoretical results.

Formation-containment tracking problems for high-order linear multi-agent system on directed graphs are studied, where the multi-agent system is composed of a virtual leader, several real leaders and followers. The real leaders need to not only accomplish a predefined time-varying formation but also track the expected trajectory generated by the virtual leader, while the followers are required to converge to the convex hull formed by the real leaders. Firstly, a formation-containment tracking protocol is presented by using the neighboring relative information. Then an algorithm to design the control parameters is given, where the formation-containment tracking feasible constraint is proposed. Furthermore, based on the Lyapunov theory, it is proved that the multi-agent system can achieve the given formation-containment tracking under the designed protocol. Finally, the effectiveness of the theoretical results is verified by a simulation example.

Time-varying formation control problems for mobile robot swarm systems are investigated. Firstly, for the formation control, the mobile robot is regarded as a point-mass system, and the dynamics of each mobile robot is modeled by single integrator. Then, consensus-based formation protocols are presented for mobile robots to achieve a predefined time-varying formation. Necessary and sufficient conditions for mobile robot swarm systems to achieve time-varying formations are proposed. Moreover, a formation platform, which consists of four omni-directional mobile robots is introduced, where a Ultra-Wind Band (UWB) based indoor navigation system (INS) is used. Finally, simulation and experimental examples are presented for the mobile robot swarm systems to achieve a time-varying formation respectively.

This paper investigates the time-varying formation robust tracking problems for high-order linear multiagent systems with a leader of unknown control input in the presence of heterogeneous parameter uncertainties and external disturbances. The followers need to accomplish an expected time-varying formation in the state space and track the state trajectory produced by the leader simultaneously. First, a time-varying formation robust tracking protocol with a totally distributed form is proposed utilizing the neighborhood state information. With the adaptive updating mechanism, neither any global knowledge about the communication topology nor the upper bounds of the parameter uncertainties, external disturbances and leader's unknown input are required in the proposed protocol. Then, in order to determine the control parameters, an algorithm with four steps is presented, where feasible conditions for the followers to accomplish the expected time-varying formation tracking are provided. Furthermore, based on the Lyapunov-like analysis theory, it is proved that the formation tracking error can converge to zero asymptotically. Finally, the effectiveness of the theoretical results is verified by simulation examples.

Formation control problems for linear multi-agent systems with switching and directed topologies are investigated. A consensus based protocol is constructed using local neighboring information. An algorithm consisting of three steps is presented to design the formation control protocol. The stability of the proposed algorithm is proven using the piecewise Lyapunov function theory. It is obtained that the predefined time-varying formation can be achieved by the linear multi-agent systems with switching directed topologies if the given time-varying formation belongs to the feasible formation set and the dwell time is larger than a positive threshold. A numerical simulation is provided to demonstrate the effectiveness of the theoretical results.

Output containment control problems for high-order linear time-invariant swarm systems are investigated. Firstly, output containment protocols are presented for leaders and followers respectively to make sure that the output dynamics property of leaders can be improved and the outputs of followers can 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 output containment protocol is given, which has less calculation complexity. Finally, numerical simulations are presented to demonstrate theoretical results.