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

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.

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 discusses the robust predefined output containment (RPOC) control problem for heterogeneous nonlinear multiagent systems having multiple uncertain nonidentical leaders. In order to solve this problem, a new kind of distributed observer-based RPOC control framework is presented. First, for obtaining the information of nonidentical leaders' dynamics, including uncertain parameters in leaders' system matrices, output matrices, states, and outputs, four kinds of adaptive observers are constructed in a fully distributed form without any knowledge of the dynamics of nonidentical leaders, exactly. Second, on the basis of adaptive learning technique, a new RPOC controller is then developed by using the presented observers, where the adaptive observers can make up for the uncertain parameter in followers' dynamics, and the solutions of output regulation equations can be obtained adaptively by the developed adaptive strategy. Furthermore, with the help of the output regulation method and Lyapunov stability theory, the RPOC criteria for the considered system under unknown nonidentical leaders' dynamics are derived from the constructed controller. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed RPOC controller.

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 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.

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.

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.

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 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.

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.

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.