Survey of containment control in multi-agent systems: concepts, communication, dynamics, and controller design

Cooperative robotics has gained considerable traction in the industrial sector with the expeditious use of multi-agent systems. Containment control is an emergent topic in cooperative multi-agent systems, with applications in areas such as warehouse robotics, inventory management, and transportation. Containment control is an amalgamation of formation and consensus control, which predominantly uses groups of multiple agents assigned as leaders and followers. This survey paper provides a comprehensive overview of key topics in containment control of multi-agent systems ranging from containment concepts to controllers. The paper discusses the role of communication and network topologies, the importance of accurate sensor measurements and agent dynamics, and various controller design and stability analysis techniques. The paper concludes by highlighting the promising research directions and challenges in the field along with real-world applications. Overall, this survey paper provides a valuable resource for researchers, practitioners, and students interested in the development of advanced multi-agent systems with containment control.


Introduction
Multi-agent systems play a major role in various industries leading to numerous robotic innovations (K.-C.Chen et al., 2021;Logothetis et al., 2021;X. Xu, Lu, et al., 2021).The development of cooperative controllers has made it possible for these multi-agent systems to work efficiently.Cooperative control systems are a form of networked systems where agents exchange information such as sensor data, actuator commands, and controllers over a communication network to design an all-inclusive control system (Lu & Guo, 2023).These agents can range from collaborative robotic platforms to complex human-in-the-loop robot systems, few such examples in diverse industries are manufacturing (Monostori et al., 2015), construction (Krizmancic et al., 2020), warehouse management (Cardarelli et al., 2017), package delivery (Mathew et al., 2015).The use of networked control systems in a cooperative fashion instead of deploying single agent systems has several advantages (Z.Li & Duan, 2017;Parker, 2008;Ren & Cao, 2011): • Efficiency: Teams of agents can cooperate to achieve greater efficiency than a single-agent system CONTACT Yen-Chen Liu yliu@mail.ncku.edu.twDepartment of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan performing solo missions.By working together, agents can divide tasks among themselves and perform them simultaneously, resulting in faster and more efficient task completion.• Cost-effectiveness: Building a few robots with limited functions is cheaper than building a powerful multipurpose single robot.Instead of designing and building one complex robot, multiple simpler robots can be developed for different tasks, resulting in a more cost-effective solution.
• Redundancy: Multiple robots provide redundancy and robustness while solving problems faster.If one robot fails or is compromised, other robots can continue to work and complete the task, increasing the overall reliability of the system.• Distribution: Distributed systems do not require overall states/control of all agents, reducing computational complexity.Instead of relying on a centralised controller to monitor and manage all agents, distributed systems can operate independently and perform tasks autonomously.• Scalability: A distributed system that relies primarily on local information among agents is easily scalable.Adding or removing agents to the system does not require a significant reconfiguration of the overall system, making it easier to adapt to changing requirements.• Resilience: Distributed systems are less prone to failure when some agents are compromised, and the system can be restructured to accomplish the task.
If one or more agents are compromised, the system can be restructured to use other agents to complete the task.• Heterogeneity: It is easier to equip a complementary sensor suite among multiple robots than to attach overall components for computation, payload, and sensing on each robot.Different robots can be designed for different tasks and equipped with sensors appropriate for the task at hand, resulting in a more efficient and effective system.• Collaboration: Heterogeneous systems can have different capabilities to achieve a cooperative task with low costs, such as aerial-ground robots.Different robots can work together to complete a task, such as an aerial robot providing a birds-eye view while a ground robot navigates and collects data on the ground.This collaboration results in a more comprehensive and efficient solution.
Cooperative control in multi-agent systems can be broadly categorised into four branches: 1. Consensus control (Z.Li & Duan, 2017), focuses on achieving agreement among multiple agents regarding a specific state or value, and finds applications in coordination and synchronisation; 2. Formation control (Oh et al., 2015), deals with the algorithmic study of multi-agent formation into specific geometric patterns and finds applications in reconnaissance, transportation, and surveillance; 3. Coverage control (Galceran & Carreras, 2013), addresses the efficient deployment of multiple agents in a given area of interest, and finds applications in environmental monitoring, surveillance, and mapping; and 4. Containment control (Dong, 2016), which aims to keep a group of agents within a designated region while avoiding collisions, and finds applications in warehouse robotics, wildlife monitoring, crowd control, formation flying, and transportation.
These four branches of cooperative control, namely consensus control, formation control, coverage control, and containment control, draw inspiration from biological systems (Ren & Cao, 2011;Slowik & Kwasnicka, 2018).The synchronisation and coordination observed in the lighting pattern of fireflies exemplify consensus control in nature.Similarly, the coordinated flight of migrating bird flocks showcases formation control, where individual birds align themselves to achieve a collective behaviour.Aquatic life, such as schools of fish, also exhibits formation control to evade predators through synchronised movements.
Coverage control relates to exploration tasks, akin to how ant colonies efficiently forage by following pheromone trails without centralised communication.During exploration, multiple subgroups of ants work together, forming formations to bridge gaps and optimise their search.Finally, containment control has parallels with the protective behaviour observed during mammal migrations, where large herds form a barrier around the vulnerable young to ensure their safety.
It is evident that these branches of cooperative control are closely interconnected and have evolved as both standalone algorithms and combined control approaches.They share common principles of coordination, synchronisation, and cooperation, demonstrating how nature has inspired and provided effective solutions for multi-agent systems.This survey paper provides an overview and the latest developments in containment control for multi-agent systems, which is an emergent area in the field of cooperative control (Duan et al., 2023;W. Li et al., 2023;R. Yang et al., 2023;Zhou et al., 2023b).Containment control is a fundamental problem that extends the consensus algorithm by incorporating formation control, either implicitly or explicitly (L.-M.Chen et al., 2017;Dimarogonas et al., 2006).The formation imposes a geometric constraint by leader agents over consensus.The primary goal of containment control is to utilise multi-agent systems to coordinate agent movements while avoiding collisions and ensuring energy-efficient traversal.
Containment control is a useful approach that can be applied in various applications, including warehouses (Zhou et al., 2023a).For instance, as shown in Figure 1(a), a group of autonomous mobile robots can be deployed in a warehouse environment to transport goods from one location to another.The robots need to navigate through the warehouse while avoiding collisions with other robots and obstacles.Containment control can be used to ensure efficient and safe navigation of these robots in the warehouse (Hu et al., 2021).By organising the robots into groups led by leader robots, follower robots can track the motion of the  leaders while maintaining a safe distance to prevent collisions (Duan et al., 2023).The leader robots can also communicate with other leader robots to avoid congestion in high-traffic areas.The geometric constraint imposed by the leader robots ensures that the follower robots maintain a specified distance from each other while following the leader robot's motion.
Another example of the use of containment control is in the case of drones used in firefighting or delivery (Dong et al., 2019).These drones cannot efficiently inspect an area for safe passage without compromising flight time or reducing endurance by equipping numerous sensors (Y.Zhang et al., 2020).In such cases, the use of cooperative containment control with multiple agents assigned designated tasks is a costeffective and efficient solution.Multi-agent systems with specialised functions provide redundancy, inventory cost-effectiveness, and complementary in achieving the assigned mission, resulting in efficient task completion.
Although containment control appears to be a simple combination of consensus and formation control, it covers a vast range of subtopics.This survey paper aims to identify and summarise these topics as shown in Figures 2 and 3, starting with an overview of the conceptual framework of containment control in Section 2. In Section 3, multi-agent communications with network topologies, communication delays, and event-triggered connections are discussed.Section 4 covers multi-agent sensing and dynamics under measurements, uncertainties, disturbances, and agent modelling.Various controllers under adaptive, event-triggered and fault-tolerant techniques along with containment stability and convergence are provided in Section 5. Finally, Sections 6 and 7 concludes the survey paper with future research directions, applications of containment and conclusion.

Concept of containment control
The term containment control was first proposed in Ferrari-Trecate et al. (2006), where the graph-based consensus algorithm was extended to include multiple leaders.The problem was inspired by the herding strategy of sheep as shown in Figure 1(b), and soon evolved as a separate discipline in cooperative control.The first major work on containment control was published in Ji et al. (2008), where partial differential equations  in conjunction with the hybrid stop-go policy were used.On the basis of reference input to the system, containment can be conceptualy classified as stationary containment and dynamic containment, Figure 4.However, the benefit of containment control lies in its ability to provide a cost-effective solution to control and coordinate multiple agents in various fields.Containment control enables the agents to converge to a common objective while maintaining a specified formation and relative distance among them Figure 1(a).This can be applied to various applications, including warehouse robotics, surveillance, inventory management, satellite control, and cooperative transportation.
Containment control is a likely candidate for several cooperative systems as it improves the efficiency and effectiveness of multi-agent systems by reducing the need for costly sensors and centralised coordination.The use of containment control in largescale systems with multiple agents can improve task allocation and distribution, reducing the communication overhead between agents.This can further enable the use of smaller, less expensive sensors, making it a practical solution for many real-world applications.Furthermore, recent advancements in communication and computation technology have made it possible to design and implement more complex containment control systems, opening up new opportunities for research and development.
Preliminaries and notations: Let G = (V, E, A) denote a graph that models the interaction among n robots or agents in a multi-agent system.Here, V = {v 1 , . . ., v n } is the set of robots, E ⊂ V × V is the set of edges, and A = [a ij ] is the non-negative adjacency matrix that represents the weight of the edges.An edge originating at robot v j and ending at robot v i is denoted as (v j , v i ) ∈ E, which implies that robot v i receives information from robot v j , and the weight of the edge is a ij > 0. If a ij = 0, then there is no interaction between robot v i and robot v j .The neighbour set of robot v i is defined as which is the set of robots that have a direct interaction with robot i.
In the multi-agent system V = v 1 , . . ., v n , let there be m leader agents and n−m follower agents.The set of leader agents is denoted as V E = v 1 , . . ., v m , and the set of follower agents is denoted as The convex hull spanned by leader agents is a geometric constraint which is the smallest convex polygon, that contains a given set of followers.The containment control objective for the i th agent is denoted by u i .The position and velocity states of the i th agent at time t are denoted by x i (t) and v i (t) respectively.

Static leader containment
Static leader containment or static containment refers to the control of multi-agent systems that achieve the desired shape or spatial configuration without agents moving after the desired states are reached.Basically, it considers multiple stationary leaders at or moving towards a specific goal to span a stationary convex hull.The control policy for these leaders is generally independent and broadcasts their states to followers.The control goal is to guarantee asymptotic or finite time convergence of followers from any arbitrary initial states to the convex hull spanned by leaders.Applications of static containment are mainly seen in spacecraft formations, swarm robotics and distributed sensor networks.
In the early stages of research on distributed containment control, Cao et al. (2011) proposed a protocol for stationary leaders under directed topology and its extensions for directed switching topology.The designed control protocol is given in (1) for reference, under directed graph topology with positive constant β > 0 in the controller.A detailed analysis of asymptotic convergence with convexity was given in H. Liu et al. (2012) with switching topology.
The containment of position vectors was studied in Meng et al. (2010) and Weng et al. (2015) under finite-time attitude convergence to asymptotic convergence of followers.Static containment is practical when scalable systems of followers need to be driven to a prescribed position without imposing direct control.The leader agents maintain zero input, u i (t) = 0, i ∈ V E for stationary containment.The protocols can be used without velocity measurements, as presented in J. Li et al. (2012), and convergence among the followers is improved with velocity inputs.
( 1 ) The control law in (1) is designed to ensure the containment of followers (agents) around stationary leaders and is mainly dependent on the states of leaders.
The position difference between agents, scaled by the positive constant β, and the velocity difference are combined with the adjacency matrix elements a ij (t) to determine the control input u i (t) for agent i.By including the velocity term, the control law improves convergence among followers and enhances coordination and synchronisation.This enables efficient containment around the stationary leaders in multi-agent systems.
However, this information can be communicated to followers indirectly among them if there exists a directed spanning tree.Containment algorithms in most of the research can be simplified to static containment protocols with advantages such as improved performance, robustness and less computational effort.

Dynamic leader containment
Dynamic containment refers to the control of multiagent systems to achieve a desired spatial configuration or shape over time, with the agents moving dynamically.This means that the positions and velocities of the agents are controlled to form a specific geometric pattern or shape while they move.Control protocols in dynamic containment differ depending on the nature of the leaders' movement.The majority of the literature considered identical velocity or minimal velocity among leaders.Especially in finite-time containment protocols, leaders are to generally have identical velocities to allow observer design which is not always practical.
In X. Wang, Li, et al. (2014), dynamic containment was proposed for leaders with constant but non-identical velocities.In contrast, the same timevarying speed for all leaders was considered in J. Qin et al. (2017).On the other hand, distributed tracking algorithms without the need for velocity measurements were given in J. Li et al. (2012) and Cao et al. (2012) for bounded velocity dynamic leaders.An early set of protocols from Cao et al. (2011) to drive followers V F into a moving convex hull are given in (2) and (3), where leaders have a constant and varying velocity with non-negative constants γ , β in the controller.

Varying velocity
The control laws proposed in Equations ( 2) and ( 3) are designed to achieve dynamic containment of followers around leaders with different velocity characteristics.In Equation ( 2), where identical velocity among leaders is considered, the term γ acts as a positive constant that scales the control influence of position and velocity differences between agents.The term a ij (t) represents the entry in the adjacency matrix, indicating the communication or interaction link between agents i and j.By considering the position and velocity differences between agents, as well as the strength of their interactions, the control law guides the followers towards dynamic containment around the leaders.
In the case of (3), where varying velocity among leaders is considered, the control law becomes more complex.The positive constants α and γ scale the control influence of position and velocity differences between agents.Similar to before, the term a ij (t) denotes the entry in the adjacency matrix, reflecting the communication or interaction link between agents i and j.The control law consists of two distinct parts.The first part focuses on the interaction with leaders (V E ), while the second part focuses on the interaction with followers (V F ).By considering the position and velocity differences with neighbouring agents, the control law adjusts the positions and velocities of the followers.This adjustment is crucial for achieving dynamic containment around the time-varying leaders.
Dynamic control has many advantages since it provides a structured approach to controlling multiple agents over time, allowing for improved performance, scalability and robustness to disturbances without requiring significant computational effort.Similar to static containment control, dynamic containment has a few limitations and drawbacks that must be taken into consideration when designing and implementing control algorithms.These setbacks of dynamic containment control include the complexity of the control algorithms, communication constraints, sensing limitations, nonlinear dynamics, and limited control authority.In addition, dynamic containment can be more challenging to design and implement than static containment because it involves the control of both position and velocity over time.

Challenges and future work
One potential future direction in containment control research is to consider the incorporation of secondary tasks for followers once containment control and safety are ensured.Currently, most containment control approaches focus on achieving the desired spatial configuration and maintaining relative distances among agents.However, in many real-world applications, agents may have additional tasks or objectives beyond containment.By introducing a hierarchical framework, where followers can perform secondary tasks while still adhering to the containment constraints, the overall system's flexibility and functionality can be enhanced.Future work can explore how to allocate resources, prioritise tasks, and design control strategies that allow followers to switch between containment and secondary tasks seamlessly, while ensuring the overall system's stability and performance.This research direction will enable the utilisation of multiagent systems in a broader range of applications where agents need to fulfil multiple objectives concurrently.

Multi-agent communication
Multi-agent communication in containment control refers to the exchange of information between agents to coordinate their movement and achieve a desired containment configuration.This communication is key to maintaining a specific geometric pattern or shape while they move.The communication is often wireless and can be modelled as a graph topology as shown in Figure 5, which in turn helps in understanding the network of agents and assessing stability.The communication network topology can be used to exchange information about the agents' states, such as their positions and velocities, or to exchange control signals to coordinate their movements.In practice, communication channels can be limited by range, bandwidth, latency, and other factors that affect the quality and reliability of communication.Therefore, it is important to consider the communication aspects of containment control such as network topology, communication delay, information loss and effective event-triggered connections when designing and implementing the containment control algorithms.

Fixed network topology
In the early days of containment control research, fixed undirected connected graphs were commonly used in accordance with the hybrid stop-go policy or with the Laplacian controller for the follower subset.Researchers have proven the equilibrium in containment with solutions on partial differential equations in Dimarogonas et al. (2006) and Ji et al. (2008).In H. Liu et al. (2012), the necessary and sufficient conditions were outlined for the convergence as a stronger case of directed graphs among followers.This study emphasised that convergence is dependent only on topology in the case of stationary leaders where followers enter the convex hull.Moreover, for any high-dimensional space, followers shall converge to a minimal hyperrectangle with stationary leaders whose hyper-planes are normal to the frame of chosen inertial coordinate reference.
In the case of dynamic leader containment, convergence depends on the topology and gain matrices or parameters.Hence, followers can enter the spanned convex hull and maintain the desired relative velocity in reference to leader agents.For discrete containment, the sampling size also becomes essential for convergence.The Kronecker-based decoupling technique is used for an extension to higher dimensions, as described in Cao et al. (2011).However, in J. Qin et al. (2017), the graphs were required to be undirected for certain protocols in heterogeneous networks.Further, separate protocol extensions for varying velocity leaders with directed graphs were induced over position and velocity interaction topologies.Output feedback containment was studied in Y. Li, Hua, et al. (2017) for directed and undirected fixed graphs with high-order nonlinear agents.
In Dong et al. (2019) the formation-containment problem was investigated with directed topologies and sufficient conditions were provided for convergence.The approach to finding gain matrices was given by solving the algebraic Riccati equation.A game theoretic approach was used in Hu et al. (2021) to develop a unified two-layer cluster formation containment, assuming fixed and connected topologies for every new cluster.However, in real-world scenarios, the interaction between agents can be fixed or switched due to factors such as time delays, limited sensing, and unreliable communication, leading to topology changes.To achieve containment under a switching topology similar to the fixed topology, a directed spanning tree must be guaranteed at each time interval.However, conventional analysis tools such as Laplacian partition or Kronecker decoupling may not be directly applicable.

Switching network topology
Switching topology has been extensively studied in the context of containment control for multi-agent systems.In Cao et al. (2012), the coordinate transformation technique was used to derive convergence results for both dynamic and stationary leaders under directed switching topology, as well as leaders with time-varying velocity.Later, (Lou & Hong, 2012) considered random switching interconnections described as a continuous-time irreducible Markov chain, as opposed to sequential directed switching.One advantage of switching topology is its flexibility and robustness in the face of uncertain network conditions, making it a popular research direction.However, it can also introduce additional complexity in the controller design and may require more sophisticated analysis techniques for stability verification.
In Su and Chen (2015), the assumption of a jointly connected switching topology was made, and semiglobal state feedback containment and output feedback containment were attained.The tracking problem in containment was considered in Y. Wang, Zhou, et al. (2017), where a time-varying weight-unbalanced digraph with an explicit convergence rate was studied.In Liang et al. (2021), a semi-Markovian system with semi-Markovian switching topology was investigated.Various controller schemes were developed based on linear matrix inequality, which can handle partly unknown transition rates that mirror practical applications of containment with semi-Markovian random switching.A sequential switching sequence was used for multiple stationary leaders in Xiong et al. (2019) under discrete single integrator agents.Directed switching topologies under inconsistent bounded delays were studied in Y. Wang et al. (2022).
Other variations of switching topology include as follows, W. Zhang et al. (2017) considered switching topology with an event-triggered scheme, and Z. Yang et al. (2017) studied containment tracking over timevarying weight-unbalanced digraphs where the typical Lyapunov method could not be used.In F. Wang, Yang, Liu, et al. (2017), switching topology was considered for agents with single integrator and double integrator dynamics in the presence of time-varying delays.A set of necessary conditions on system dynamics and network topology were imposed in J. Qin et al. (2019) for switching networks to achieve exponential output containment with dynamic controllers.A switching signed network was addressed in R. Yang et al. (2022) as bipartite containment with the controller given below: The control law presented in Equation ( 4) addresses the problem of bipartite containment in a switching signed network.Each term in the control law plays a specific role in achieving the desired containment behaviour.The term a η(t) ij represents the entry in the adjacency matrix of the switching graph at time t.This matrix captures the communication or interaction links between agents i and j within the network.By considering the switching nature of the graph, the control law can adapt to dynamic changes in the interaction topology.The function sgn(a η(t) ij ) determines the sign of the interaction between agents i and j at time t.It takes into account whether the interaction is positive or negative, which influences the direction of the containment forces.By considering the sign of the interaction, the control law can effectively steer the agents towards containment.
The terms x j (t − τ ) and x i (t − τ ) represent the positions of agents j and i at a delayed time instance t − τ .These delayed positions are essential for calculating the difference between the positions of the agents.By comparing the positions, the control law determines the containment forces needed to align the agents appropriately.The term |a η(t) ij | ensures that the interaction strength is positive, regardless of the sign of a η(t) ij .This guarantees that consistent containment forces are exerted on the followers, irrespective of the specific signs of the interactions.By taking the absolute value, the control law considers the magnitude of the interaction strength for effective containment.The parameter k scales the overall influence of the containment forces.It determines the strength with which the followers are steered towards containment around the leaders.By adjusting the value of k, the control law can regulate the containment behaviour according to the specific requirements of the system.
Overall, the control law described in Equation ( 4) combines the adjacency matrix, interaction signs, delayed positions, and a scaling parameter to achieve bipartite containment in a switching signed network.Each term contributes to the overall behaviour of the control law, enabling effective containment and tracking of the leaders by the followers, even in the presence of dynamic changes in the network topology.The research on switching topology in containment control offers various research opportunities and challenges, and its application in real-world systems, such as satellite formations, power networks, and cooperative transportation has the potential to bring significant benefits in terms of efficiency, robustness, and scalability.

Communication delays
Communication delays are inherent in many realworld multi-agent systems and can have a significant impact on the system's performance.In cooperative control, time delays can arise due to communication latencies, processing times, or even physical delays such as transportation delays.These delays can result in instability, poor convergence, and decreased robustness, making it crucial to account for them in the system design.In the case of networked systems, where agents communicate over a shared communication network, time delays can be modelled as a bounded, time-varying function τ (t) < τ max , constants τ and inter-agent delays τ ij .
Further, research articles usually consider time delays among agent states x i (t − τ ) or control inputs u i (t − τ ), which are comprehensively provided in Table 1.In the table, x ij (t − τ ), u ij (t − τ ) are delays in neighbouring agent states and control inputs respectively.Additionally, the term (k − τ ) denotes delay in discrete-time systems, and f (x i (t − τ )), x (α)  i (t − τ ) are delay in nonlinear and fractional order system dynamics respectively.The time-varying nature of the delays can make the system's dynamics time-varying as well, which further complicates the system analysis.
In S. Liu, Xie, and Zhang (2014), protocols were developed for continuous and discrete systems with delays in the control input of neighbours that are robust to constant time delays.In K. Liu, Xie, and Wang (2014), second-order multi-agent systems were examined with time-varying delays under both stationary and dynamic containment.The paper employed the Lyapunov-Razumikhin function and Lyapunov-Krasovskii function methods to derive sufficient conditions for allowable upper bound delay for stationary and dynamic leader containment, respectively.In Miao et al. (2017), constant time delay under event trigger conditions was investigated for first and second-order systems using the sum of squares method, and sufficient containment conditions were obtained for both single and multiple time delays.
Second-order systems with inherent nonlinear dynamics and time delays were studied in B. Li, Chen, In B. Li and Zhang (2016), fixed time delay with fixed directed topology was investigated, and algorithms for continuous and discrete systems under single integrator dynamics were proposed along with sufficient conditions.Time-varying delays over communication in first and second-order systems were studied in F. Wang, Yang, Liu, et al. (2017) under switching topology, providing sufficient conditions using the Lyapunov-Krasovskii function for stability analysis.Time-varying delays under sensor faults and unmodelled dynamics were studied in Z. Li et al. (2022), for which an adaptive resilient controller was developed with a sliding-mode estimator.
In Shi et al. (2017), time-varying delays in secondorder discrete-time systems were investigated with asynchronicity in independent agent state updates that are not equispaced.Consequently, necessary and sufficient conditions for asymptotic stability were obtained using non-negative matrix theory and graph theory.Communication delays were focused in Xiong et al. (2019) for the containment of a single integrator discrete dynamics system under switching topologies.Input time delays in discrete-time linear fractional order systems were considered in Shahamatkhah and Tabatabaei (2020).Delayed control protocols were provided in R. Yang et al. (2022) for nonlinear agents to solve bipartite containment.Input delays for agents with collective dynamics described as parabolic partial differential equations were studied in H. Zhang et al. (2023) with a dynamic event-trigger scheme.
In H.-Y. Yang et al. (2019), frequency domain analysis was applied to obtain the upper bound on communication delays for fractional order systems, while in D. Wang, Wang, et al. (2019) Routh array criterion was used to derive necessary and sufficient conditions for the first-order system.In H. Liu et al. (2019), necessary and sufficient conditions for containment were established for fractional order single integrator systems with constant transmission delays.Sufficient conditions for constrained containment were established in Y. Wang et al. (2022) with a projection-based controller.Here inconsistently bounded communication time delays were studied assuming followers have at least one stationary leader interaction.
Various algorithms and methods for addressing time delays in different types of multi-agent systems are surveyed as shown in Table 1.The use of Lyapunov functions, non-negative matrix theory, and graph theory have been popular in deriving sufficient conditions for system stability and containment.The analysis of communication delays with unmodelled dynamics and sensor faults has also been considered.These studies have made significant contributions towards enhancing the robustness and performance of cooperative control systems in the presence of time delays.Time delays pose considerable challenges in cooperative control systems, and their impact on the system's performance is imperative in real-world applications.
Apart from communication delays, other networkinduced problems include packet dropouts, bitrate constraint and encoding-decoding schemes for secure data transmission, upon which there has not been a direct investigation in containment control.These factors can significantly impact the performance and stability of the containment control system and pose a challenging problem.
Preliminary results on polytope containment problem with linear encoding schemes for decision variables were studied in Sadraddini and Tedrake (2019) and encoding-decoding schemes based on quantisation communication topology were presented in L. Li, Shi, et al. (2019).Usually, these schemes are employed to avoid potential inconsistencies in agent states and control actions induced by time lags.The schemes could include strategies such as delay compensation techniques, predictive control methods, or the use of robust control algorithms.These approaches aim to mitigate the adverse effects of communicationinduced errors to ensure the convergence and coordination of the agents in the containment task.
The bit rate constraint on the other hand introduces limitations on the available bandwidth for communication.This constraint affects the amount of data that can be transmitted between agents, potentially impacting the accuracy and timeliness of information exchange.Dealing with bit rate constraints requires careful resource allocation and optimisation strategies.A network with constraint data rate was studied in Mu and Liu (2016), where quantised communication was followed by agents.Techniques such as data compression, prioritisation of critical information, and efficient coding schemes can be employed to maximise the utilisation of the limited bandwidth while maintaining the necessary communication for containment control.

Event-triggered communication
Event-triggered communication has been widely studied as an alternative to the traditional time-triggered communication approach in cooperative control systems.Event-triggered communication can reduce the number of transmissions between agents, which can lead to a reduction in communication overhead and resource consumption.
In Miao et al. (2017), the authors investigated the problem of containment control for multi-agent systems with constant time delay under event-triggered communication.Event-based broadcasting was implemented in K. Liu et al. (2016), where each agent transmits the state information to neighbours based on a certain event.In Zou and Xiang (2017), the framework of output regulation was studied for containment control of linear heterogeneous systems.Based on output information, the event-triggered protocol was designed for each follower.A decentralised event-trigger was designed in W. Liu et al. (2017) with constant communication time delays.Package dropouts of probabilistic nature were studied in W. Chen et al. (2018) for prescribed containment under an event-based communication scheme to update the output-feedback controller.
An event-triggered scheduling protocol was proposed in W. Chen et al. (2020) for communication among agents where the estimated state was transmitted.Here an issue of finite-horizon H ∞ control to achieve containment was discussed for discrete timevarying linear systems.In Y. Xu, Fang, et al. (2021), agent communication was scheduled with an event trigger under undirected network topology among followers.Sufficient conditions were developed to solve containment with extensions to localised adaptive event triggers.A dynamic event-triggered communication scheme with an error-dependent term was given in Lv et al. (2022) for hypersonic flight swarms.A nonrecursive output feedback design was established to keep the geometric configuration of formation containment without requiring full-state information.
Event-triggered communication has been utilised for containment under various network topologies and time delays, as described in this section of multi-agent communication.It offers an alternative approach to continuous or periodic communication by exchanging information based on specific events or triggers.In fixed network topologies, agents selectively communicate when certain conditions or events occur, reducing overall communication load and improving efficiency.Similarly, in switching network topologies, event-triggered communication allows for selective and efficient information transmission when the network topology changes dynamically or agents switch between communication channels.By incorporating event-triggered communication in multi-agent systems, both fixed and switching network topologies can benefit from more efficient and responsive communication strategies that adapt to system requirements and minimise unnecessary message exchange.
Furthermore, event-triggered communication can help mitigate communication delays by sending messages only when necessary, thereby reducing the impact of network congestion or other communication challenges.However, it is worth noting that eventtriggered communication is susceptible to transmission delays, which may arise due to various factors such as network congestion or limited bandwidth.An adaptive event-trigger mechanism as discussed in Wei et al. (2021) proposed dynamic triggering law by relating to triggering errors.This effectively reduced data transmissions in turn alleviating the induced transmission delays.Other recent work in Bi et al. (2022Bi et al. ( , 2023) ) employed distributed observers to account for unbounded transmission delays in standalone containment and formation-containment systems respectively.
Event-triggered communication can provide robustness against network failures and delays to an extent, as the communication is only triggered when certain conditions are met, rather than being time-based.However, designing an event-triggering mechanism that balances the trade-off between communication frequency and control performance is a challenging problem.Several approaches proposed for eventtriggered communication were outlined in this section of the survey paper.Overall, event-triggered communication has shown promising results in improving the performance and robustness of multi-agent systems, and it is a topic that merits further research in the field of cooperative control.

Challenges and future work
In multi-agent communication, exploring the incorporation of dynamic or time-varying weightage for network topologies could open new avenues for optimisation.Researchers have worked on weighted graphs in the context of containment control, where the weights assigned to communication channels determine their influence on the overall system.This concept can be extended to switching topologies, where the weights may not exist or may vary over time.By dynamically adjusting the weightage assigned to each communication channel, agents can adapt their information exchange strategies based on the prevailing system requirements and dynamics.However, stability analysis for this could be increasingly challenging.
Introducing dynamic weightage in multi-agent communication offers several advantages.Firstly, it enables efficient allocation of communication resources by prioritising channels that are most relevant or critical at a given moment.This adaptive approach prevents unnecessary communication overhead and reduces the overall system complexity.Additionally, by considering the temporal dynamics of the weights, agents can adapt their communication patterns to capture time-sensitive information or respond to changes in the environment more effectively.This could be seen as a similar strategy to event-based communication with a special focus on individual agents.Nevertheless, transmission delays could persist in these approaches making it a difficult problem.
In practical applications, this dynamic weightage allocation can be implemented using various strategies such as game-theoretic approaches or auction-based mechanisms.These methods allow agents to strategically determine the importance and value of different communication channels, facilitating efficient resource allocation and minimising communication costs.By embracing these concepts and exploring their potential in multi-agent systems, we can enhance the robustness, scalability, and efficiency of communication protocols, leading to improved overall system performance.

Measurements and agent dynamics
Measurements and agent dynamics play crucial roles in the containment control of multi-agent systems.Measurements acquired by sensors are used to obtain information about the state of the agents and the environment, which are used to control the movement of agents to maintain the desired containment configuration.This movement of agents in response to the control input is dependent on the agent dynamics, thus dynamics and measurements greatly affect the stability of the overall containment.
The commonly used sensory measurements include position, velocity and orientation values.However, the acquisition of these values could involve disturbances and uncertainties in various capacities.As a result, the design of estimators and observers has become another essential topic in containment.Among agent dynamics, integrator models, unicycle dynamics and higher-order dynamics such as fractional order systems are commonly used to replicate real-world systems.In short, measurements and agent dynamics are essential components of containment control in multi-agent systems towards ensuring the stability, performance, and feasibility of the control algorithms.

Sensory measurements
In the area of multi-agent containment systems, different studies have proposed various methods for achieving containment without using velocity and acceleration measurements.A controller without velocity measurements was developed in Cao et al. (2012) for containment under both stationary and dynamic leaders.In J. Li et al. (2012), two control protocols were designed based on only position measurements, assuming that the velocity and acceleration of both followers and leaders are unavailable.These protocols were proven to guarantee finite-time convergence even with non-identical velocities of leaders and without any knowledge of the bound of acceleration.A formation-containment protocol with only sampled position information was used in B. Zheng and Mu (2016) with sufficient conditions of convergence.
Similarly, Y. Zhao and Duan (2015) proposed protocols that do not rely on velocity and acceleration measurements but do require knowledge of the bound on acceleration.The study also theoretically estimated the finite settling time for containment.Other studies have exploited the control inputs of neighbours and proposed adaptive control algorithms without using velocity information.For instance, S. Liu, Xie, and Zhang (2014) designed containment protocols by exploiting the control inputs of neighbours, while Mei et al. (2015) proposed an adaptive control algorithm without using the velocity information of neighbours.
Furthermore, heterogeneous networks were considered in J. Qin et al. (2017), where protocols were developed for leaders that remained stationary, moved at a constant velocity, and varied in speed.Local information was used for solving the formation-containment problem in L.-M.Chen et al. (2017), while measurement size reduction and aperiodic sampling intervals were proposed in D. Zhang et al. (2018) to reduce communication among agents.Containment control in nonidentical networks was addressed in Liang et al. (2019) for tracking, while velocity and position constraints were introduced in Y. Wang et al. (2022) for followers where velocities were of the non-convex set.
The design of effective containment control protocols in multi-agent systems is a challenging task, and the choice of sensory measurements plays a pivotal role.Position measurements are the most commonly used for containment control, while velocity and acceleration measurements can provide additional information to improve the performance of the control algorithms.However, in practical scenarios, it may not be feasible to obtain all these measurements due to sensor limitations or communication constraints.Therefore, the development of novel containment control strategies that rely only on position measurements or exploit the information provided by limited sensory measurements with estimators remains an active area of research.

Uncertainties and estimators
The effectiveness of the containment control approach for multi-agent systems largely depends on the accuracy of the information exchanged among agents.However, uncertainties, such as model uncertainties, external disturbances, and sensor faults, can affect the performance of the system.Therefore, the development of robust control strategies and efficient estimators is crucial for effective containment control.
In Meng et al. (2010), a distributed sliding mode estimator and non-singular sliding surface were proposed for dynamic leaders to achieve convergence of attitudes and angular velocities.Similarly, Mei et al. (2012) presented an adaptive controller with sliding mode estimators for the containment of Lagrangian systems under parametric uncertainties.In X. Wang, Li, et al. (2014), distributed finite-time observers were developed for followers to estimate the weighted average velocity of leaders, while external disturbances were considered.
In the presence of communication noises, Y. Wang, Cheng, et al. (2014) proposed a time-varying gain for attenuation, and further gave three necessary conditions for ensuring mean square containment.Sliding mode control and back-stepping control were combined with neural networks for control design in C. Wang, Wen, et al. (2018).A sliding mode observer was developed in Yu et al. (2020) to estimate a reference for each follower UAV, considering sensor faults.In H. Liu et al. (2015), the small gain theorem was used to obtain sufficient conditions for the stability of uncertain discrete-time systems.Containment in the presence of unknown nonlinearities and external disturbances was studied in Mei et al. (2015) based on neural network approximations using adaptive gain.
Aperiodic sampled-data-based protocols with uncertainty in time-varying sampling intervals were developed in H. Liu et al. (2015), and sufficient conditions were obtained using the small-gain theorem.A predefined performance design was proposed in Yoo and Park (2017) for a heterogeneous under-actuated and uncertain system, where the non-linearities and external forces of each agent were unknown.Unknown heterogeneous nonlinearities among followers were considered in Yoo (2017) and the controller was implemented using only error surfaces by integrating performance bounding functions.External disturbances with uncertain discrete linear systems were investigated in Liang et al. (2017), where a discrete compensator was used to estimate the convex hull, and uncertain parts of dynamics were tackled by an internal model compensator.
In Cong et al. (2018), a distributed observer was used to estimate the convex hull for each follower under output regulation and state feedback control frameworks.A containment error and adaptive distributed observer were presented in Liang et al. (2019).In order to estimate system matrices and convex hulls of leaders, with external disturbances restrained by a global regulation equation.A finite time sliding mode estimator was designed in D. Li, Zhang, et al. (2019) for position and velocity estimations, using a high-gain observer and adaptive neural networks to account for model uncertainties.Adaptive updating laws for parametric uncertainties were given in L. Chen et al. (2019) under event trigger schemes for formation containment.
In N. Li et al. (2020), projection-type estimation algorithms were used to estimate unknown parameters and control gains.The Lyapunov approach was used to ensure that bounded estimation errors converge.In H. Qin et al. (2020), neural networks were used for estimation in the presence of external disturbances, model uncertainties, and thruster faults for underwater vehicles.Sensor faults were taken into account in Z. Li et al. (2022) using a sliding mode estimator to improve transient performance and reduce the chattering phenomenon for an unmodelled dynamical multi-agent system.Distributive velocity estimators were designed for followers in L. Chen et al. (2022) achieving asymptotic convergence to the convex hull.Additionally, adaptive updating laws against external disturbances and uncertainties were designed for all agents.
Various uncertainties and estimators have been proposed in the literature to address the challenges of containment control in multi-agent systems.These methods have considered various types of uncertainties, such as parametric uncertainties, external disturbances, model uncertainties, and sensor faults, and have proposed various estimators, such as sliding mode estimators, finite-time observers, and distributed observers, to estimate the states of the agents and the convex hull.These proposed methods have shown significant promise in achieving containment control in multi-agent systems and can serve as a valuable resource for future research in the field.

Linear agent dynamics
One of the simplest models for multi-agents is the single and double integrators, which are frequently used in the design of control protocols for their simplicity.To apply general tools of analysis and convergence, these models can be modified to fit a system of linear equations.Researchers have investigated the containment of continuous linear multi-agent systems with input saturation (Su & Chen, 2015), aperiodic sampled-based containment protocol for continuous linear multi-agent systems (H.Liu et al., 2015), containment of agents with general linear dynamics under bounded control inputs with time-varying leaders (Z.Li et al., 2015), and the containment of linear multiagent systems in the presence of external disturbances and uncertainties (Liang et al., 2017).Other studies have investigated discrete time-fractional order linear systems with input time delay (Shahamatkhah & Tabatabaei, 2020) and asynchronous containment with arbitrary topologies (Shi et al., 2020).

Nonlinear agent dynamics
Studies have focused on unknown second-order nonlinear systems (Mei et al., 2015) and networked Lagrangian dynamic systems in the presence of nonlinearities and external disturbances (Q.Wang, Fu, et al., 2017).Nonlinear adaptive control protocols have been proposed based on relative state information for dynamic containment.Containment of second-order systems under nonlinear dynamics was studied in Y. Wang, Zhou, et al. (2017).The containment of strict feedback uncertain nonlinear systems was discussed in W. Wang, Wang, and Peng (2017), while Yoo (2017) addressed nonlinear pure feedback systems of low complexity.Researchers have also investigated secondorder systems with inherent nonlinear dynamics and time delays (B.Li, Chen, et al., 2017).
In L.-M.Chen et al. (2017), the focus was on networked Euler-Lagrange systems, where the agents' dynamics could differ significantly.The paper proposed a cooperative formation-containment problem, which involved designing nonlinear control protocols using the backstepping method and a reduced-order dynamic gain observer to estimate the unmeasured states.In Y. Li, Hua, et al. (2017), the focus was on high-order nonlinear systems, where only output signals could be measured.The paper proposed an output feedback containment approach using the backstepping method and a reduced-order dynamic gain observer to estimate the unmeasured states.
In D. Li, Zhang, et al. (2019), the authors considered the problem of formation containment under model uncertainties using output feedback for Euler-Lagrange systems.In L. Chen et al. (2019), the focus was on the same Euler-Lagrange system, but an event trigger scheme was used to address the problem of formation containment.In H. Liu et al. (2019), fractional order containment was considered by establishing some necessary and sufficient conditions.Finally, in Zou and Xiang (2019) and N. Li et al. (2020), the authors considered the problem of containment under an event trigger scheme for second-order nonlinear systems and a nonlinearly coupled system, respectively.

Heterogeneous agent dynamics
Linear and nonlinear control protocols were presented in Y. Zheng and Wang (2014) for a heterogeneous multi-agent system where leaders had firstorder and second-order integrator dynamics, respectively.Haghshenas et al. (2015) considered a heterogeneous linear multi-agent system with an output regulation framework.This work was based on containment error, where a dynamic compensator was introduced that guaranteed convergence.Containment of heterogeneous multi-agent systems with linear dynamics was studied in Zuo et al. (2017), where two control protocols were designed based on internal model principles with state feedback and static output feedback.
The dynamics of followers in Yoo (2017) were described by multi-input multi-output systems that were pure-feedback and completely affine.Linear heterogeneous systems were considered in Zou and Xiang (2017) with event trigger protocols.Heterogeneous systems with unknown leaders were studied in Zuo et al. (2018), where the dynamics of leaders were available only to neighbour followers.Further, output regulator equations were solved with an adaptive tuning law without the knowledge of leaders' dynamics.In Cong et al. (2018) singular heterogeneous systems were considered, where two distributed frameworks were presented for containment with and without access to full state information of followers.
To account for diverse individual features of different agents, H.-Y. Yang et al. (2019) considered fractional-order dynamics.Containment was solved for a compounded fractional-order heterogeneous system with communication delays.The study in Jiang et al. (2019) addressed time-varying formation containment of linear heterogeneous systems under directed topology.In J. Qin et al. (2019), output containment for a linear heterogeneous system was studied under fixed and switching topologies.Necessary conditions were imposed on the network and system dynamics, incorporating the internal model principle.Optimal control protocols were proposed using the algebraic Riccati equation and reinforcement learning.
Overall, the research on containment control for heterogeneous multi-agent systems has yielded various control protocols based on different principles, including the internal model principle, adaptive tuning law, and event-triggered schemes.These protocols have been successfully applied to various types of multi-agent systems with different dynamics, including fractional-order dynamics and unknown leaders.

Challenges and future work
Sensor fusion is a critical aspect of multi-agent systems, particularly when dealing with heterogeneous sensing platforms and agents.It involves combining information from various sensors and agents to enhance the overall system performance.Cooperative sensing in distributed protocols allows agents to augment their individual sensing capabilities by leveraging the collective information.This approach overcomes the limitations of individual sensors and enables a more accurate perception of the environment.
The use of non-homogeneous sensing platforms and heterogeneous agents presents research opportunities in sensor fusion.Different sensors provide complementary data from various facets of the environment, such as visual, acoustic, thermal, or chemical information.Integrating these diverse sensor modalities enhances decision-making, object tracking, localisation, mapping, and target recognition.Furthermore, fusing sensor data from heterogeneous agents allows for leveraging their strengths and compensating for individual limitations, enabling efficient operation in complex and dynamic environments.
In summary, sensor fusion research in multiagent systems aims to combine information from diverse sensors and agents to improve system performance.By leveraging cooperative sensing and incorporating heterogeneous capabilities, sensor fusion enhances perception, decision-making, and overall system effectiveness, even in the presence of unmodelled dynamics.

Controller design and analysis techniques
This section elaborates on various control techniques, including adaptive control, event-triggered control, and fault-tolerant control.A Venn diagram is provided in Figure 6, to show an overview of the relationship and intersections between three control techniques.The overlapping regions highlight the topics that are shared between control techniques, emphasising the areas of intersection and mutual relevance.By analysing the overlapped parts, one can identify the specific topics and concepts that bridge the different control techniques and provide a comprehensive understanding of their combined benefits.The section also presents a convergence analysis of a generic containment controller that is based on consensus formation.

Adaptive control
Adaptive control is a widely researched topic as it can address challenges posed by uncertainties, nonlinearities, and time-varying dynamics that are common in multi-agent systems.One approach proposed in Mei et al. (2012) presented a distributed adaptive control algorithm for the Lagrangian system with sliding mode estimators.Another work in Z. Li et al. (2015) introduced an adaptive continuous controller that does not require knowledge of upper bounds on control input for leaders and avoids calculating eigenvalues of the matrix.In Mei et al. (2015), an adaptive control algorithm was developed for the containment of nonlinear multi-agent systems using neural networks.This work proposed an adaptive gain design for the controller.Inherent nonlinear dynamics was investigated in P. Wang and Jia (2015) with adaptive control guaranteeing containment error convergence to an adjustable residual set.Another approach was presented in Q. Wang, Fu, et al. (2017), which proposed a distributed adaptive nonlinear protocol based on relative state information.The control gains were updated locally based on the relative state information.
In L.-M.Chen et al. ( 2017), a formation-containment algorithm was proposed without using relative velocity information.The control gains were updated adaptively using only local information.Fault-tolerant adaptive control was studied in Ye et al. (2017), which proposed an observer-based method to estimate bias faults and states.In Zuo et al. (2018), adaptive output containment was investigated for heterogeneous systems with unknown leaders.This work proposed a distributed adaptive observer-based controller that estimated the dynamics of leaders using local information.
Adaptive containment with input quantisation for nonlinear multi-agent systems was studied in C. Wang, Wen, et al. (2018).This work proposed a distributed adaptive observer-based controller that does not require information on quantisation parameters.In Jiang et al. (2019), a distributed adaptive observerbased controller was proposed for time-varying formation containment, where agents had knowledge of their own outputs.An adaptive control strategy and non-smooth control were used for the containment of second-order nonlinear systems in Sun et al. (2020) with distributed protocols.In N. Li et al. (2020), a distributed adaptive control protocol was proposed for agents with nonlinear coupled dynamics using only local information.Finally, in H. Qin et al. (2020), an adaptive law was designed to compensate for the upper bounds of estimation errors caused by uncertainties, disturbances, and faults.
In general, adaptive control techniques can be used to adjust the control parameters of the system in real-time, based on feedback from the system and knowledge of the uncertainties and disturbances.This can help to ensure the robust performance of the system despite changes in the system's dynamics or uncertainties.Additionally, adaptive control can help to avoid the need for explicit knowledge of the system's parameters.Consequently, the need to compute the eigenvalues of the system matrix, which can be computationally expensive or difficult to obtain in some cases.Overall, adaptive control is a useful tool in the design of containment control strategies for multi-agent systems, as it can help to address the challenges posed by uncertainties, nonlinearities, and time-varying dynamics.

Event-triggered control
Event-triggered control is a method that can reduce communication and computation costs in multi-agent systems.In W. Zhang et al. (2017), event-triggered containment was investigated for fixed and switching topologies with two cases of opinionated leaders.A pull-based event trigger was used where the control of the agent was not needed to update when the state of its neighbours was updated.In Miao et al. (2017), sufficient conditions were obtained for containment in the presence of single, multiple time delays and event trigger conditions.Both centralised and decentralised strategies without Zeno behaviours were proposed for event-triggered containment in discrete systems by Tang and Li (2018).
An event-triggered control scheme was used for formation containment in L. Chen et al. (2019) where the controller gains were adaptively tuned with only local information.In Zou and Xiang (2019), containment control of second-order nonlinear systems under an event-trigger scheme was investigated, and both centralised and decentralised protocols were proposed with the nonsmooth analysis used to show convergence.Self/event-trigger containment control protocols over directed topology that relieve the chattering effect were discussed in T. Xu et al. (2020).
Event-trigger can reduce communication and computation costs and increase the system's stability and robustness.The key problems in event triggers usually lie in the choice of triggering conditions with suitable parameters and control conditions for robustness.

Fault-tolerant control
Fault tolerance and resilience have emerged as important subfields in the area of containment control.In Yoo (2017), a fault-tolerant control approach was presented for nonlinear pure-feedback systems, where the followers have heterogeneous and unknown nonlinearities.The controller was designed using error surfaces that were integrated over performance bounding functions, without the need for compensating for uncertainties and faults through differential equations.In W. Wang, Wang, and Peng (2017), a fault-tolerant containment control strategy was proposed in the presence of actuator faults with a prescribed performance bound.This approach ensured that the errors were kept within predefined limits, independent of any actuator faults that may occur.
In Ye et al. (2017), an adaptive fault-tolerant controller was developed to handle actuator bias faults in multi-agent systems with unknown and bounded input from dynamic leaders.An observer was used to estimate states and bias faults, while Yu et al. (2020) investigated finite-time fault-tolerant containment control in the presence of actuator faults and input saturation.
In H. Qin et al. (2020), the authors proposed a non-singular fast terminal sliding surface for distributed finite-time control in the presence of thruster faults, model uncertainties, and external disturbances, for the containment of small autonomous underwater vehicles.An adversarial environment with faulty agents was considered in Yan and Wen (2020), where resilient control protocols were proposed for first and second-order systems, with the assumption that malicious agents were locally upper bounded.An adaptive resilient containment was addressed in Z. Li et al. (2022) to mitigate unknown sensor fault impacts, along with time-varying delays under unmodelled dynamics.The design guaranteed the boundedness of all signals in the closed-loop system under sensor faults.
Fault-tolerant control is important in containment control since in several real-world scenarios, faults and uncertainties are inevitable.These faults can be due to actuator failures, sensor faults, communication losses, or other external disturbances.If such faults are not addressed properly, they can lead to the loss of the whole system due to instability.Therefore, incorporating fault tolerant control in containment control helps to ensure that the system remains stable and the containment objective is achieved, even in the presence of faults and uncertainties.This is achieved by developing controllers that are able to adapt and compensate for these faults, using various techniques such as fault detection and diagnosis, observer-based fault estimation, and fault-tolerant control strategies.

Containment stability analysis
The containment stability and convergence analysis are essential as they provide insight to predict the performance of the system, verify the containment algorithms, and identify optimal parameters such as network topology, control gains, and bounds.The stability analysis of the multi-agent system typically involves containment stability and convergence analysis.Containment stability aims to ensure that agents remain within predefined boundaries, while convergence analysis ensures that the agents' states converge to a desired state.

Containment analysis
Consider a group of n linear agents among which m are followers.The leader subset of n−m agents are denoted by E and m followers by F. Each agent has double integrator dynamics with position, velocity, and control input vectors denoted by x i (t), v i (t), and u i (t) respectively.Here, B 1 = [1, 0] , B 2 = [0, 1] , and φ i (t) = [x i (t), v i (t)] .The system can be written as: (5) Remark 5.1: A leader agent E has only leaders as its neighbours for information exchange which is usually for explicit formation.
Preliminary containment protocols under directed topology Figure 7, based on formation and consensus (Dong et al., 2019) are defined as, These protocols are designed to enable containment control in a multi-agent system.In (6a), the control law for followers (i ∈ F) is expressed.The term K 1 φ i (t) represents a proportional feedback control, where K 1 is a gain matrix and φ i (t) is the state variable associated with agent i.This term allows individual followers to adjust their behaviour based on their own state.The second term, K 2 j∈N i w ij (φ i (t) − φ j (t)), accounts for the interaction between neighbouring agents.Here, w ij represents the weight associated with the interaction between agent i and its neighbour j.By summing up the weighted differences between the states of the agent and its neighbours, the control law incorporates consensus principles, promoting alignment among the followers.
In (6b), the control law for leaders (i ∈ E) is defined.Similar to the followers, a proportional feedback control term K 1 φ i (t) is included.Additionally, the term K 3 j∈N i w ij ((φ i (t) − h i (t)) − (φ j (t) − h j (t))) accounts for the differences between the leaders' states (φ i (t)) and their associated reference formation signals (h i (t)).This term enables the leaders to guide the followers towards containment by adjusting their positions relative to the reference signals.
The gain matrices K l = [k l1 , k l2 ], (l = 1, 2, 3) are calculated using the algebraic Riccati equation.These matrices determine the overall influence of the respective control terms and can be adjusted to achieve desired performance and stability properties in the containment control system.The control laws, parameterised by the gain matrices, allow for individual adjustments and interaction-based alignment, facilitating the achievement of containment objectives in the multi-agent system.Definition 5.1: Containment is said to be achieved when ( 7) is proven for the overall system of leaders E and followers F. Such that there exists non-negative constants α jk , where n k=m+1 α jk = 1 for any j ∈ F and k ∈ E, Definition 5.2: Time-varying formation for the system is said to be achieved when there exists a vector function r(t) ∈ R 2 and formation vector h i (t) for agents satisfying, The Laplacian matrix L for the overall system with interactions among leaders and a spanning tree towards followers can be given as, where L 1 ∈ R m×m is associated with follower interactions, L 2 ∈ R m× (n−m) shows the directed path from leaders to followers.Finally, m) is the exclusive interaction among the leaders.Here, L 1 and L 3 can be individual Laplacian matrices of leader and follower agents as decoupled sub-graphs.

Lemma 5.3 (Meng et al., 2010):
For directed topology with spanning tree from leaders, all eigenvalues of L 1 have positive real parts with each entry L −1 1 L 2 as non-negative, and each row sum of −L −1 1 L 2 equal to one.
The overall compact system for all the agents is defined in (10) with a Kronecker product '⊗'.Such that, φ F (t) = [φ 1 (t), φ 2 (t), . . ., φ m (t)] are followers and φ E (t) = [φ m+1 (t), φ m+2 (t), . . ., φ n (t)] are leaders.A detailed stability analysis from Dong et al. (2019) shows the system Hurwitz, The stability condition and final states for the containment of followers is, Thus the final asymptotic convergence of containment control for the compact system in ( 10) is shown to be achieved by (11) based on convexity (7), that achieves containment stability and convergence of followers under the geometric convex hull (8) while avoiding collisions.

Finite-time convergence
Common approaches for containment convergence analysis are asymptotic convergence and finite-time convergence.Asymptotic convergence ensures that the agents' states converge to a desired state in the limit as time goes to infinity.On the other hand, finite-time convergence guarantees that the agents' states converge to the desired state within a finite time.The latter is particularly useful in applications where fast convergence is desirable, such as in robotics and autonomous vehicles.Literature based on finite-time convergence is briefly listed.
In Meng et al. (2010), the authors proposed finitetime model-independent containment protocols for stationary leaders and distributed sliding mode estimators for dynamic leaders.The power integrator approach was used to develop distributed finite containment in X. Wang, Li, et al. (2014).In Y. Zhao and Duan (2015), finite-time containment of agents with double integrator dynamics was studied, and the Lyapunov function was computed to estimate the finite settling time.In H. Wang, Wang, et al. (2017), finitetime containment was established for static or dynamic leaders with any preset time using nonlinear feedback control protocols with the generic assumption that a spanning tree is required.
An adaptive finite-time control solution to achieve formation containment with uncertain nonlinear systems was presented in Y. Wang, Song, et al. (2018).A prescribed time-distributed control method for containment and consensus based on the scaling function was proposed in Y. Wang, Song, et al. (2019).In L. Zhao et al. (2022), a finite-time control framework was proposed using a distributive adaptive backstepping strategy for uncertain manipulators.The authors used finite-time command filters to avoid the complexity of conventional backstepping control, and filtering errors were removed using an error compensation mechanism that used only local information.Fixed-time containment protocols were designed in T. Xu et al. (2020) for nonlinear systems.

Challenges and future work
Adaptive control is a promising approach to address uncertainties, nonlinearities, and time-varying dynamics in multi-agent systems.Various adaptive control algorithms have been proposed, such as distributed adaptive control with sliding mode estimators, adaptive gain design using neural networks, and adaptive observer-based controllers.These techniques enable real-time adjustment of control parameters based on system feedback and knowledge of uncertainties, ensuring robust performance and eliminating the need for explicit knowledge of system parameters.Future research in adaptive control can focus on developing more efficient and computationally less demanding algorithms, as well as investigating their applicability to different types of multi-agent systems.
Event-triggered control offers a solution to reduce communication and computation costs in multi-agent systems.It has been applied to containment control, enabling agents to update their control actions only when triggered by specific events.This approach has shown potential in achieving stability and robustness while minimising resource usage.Further research in event-triggered control can explore the development of optimised triggering conditions and control strategies to enhance system performance and resilience.
Fault-tolerant control is crucial for ensuring system stability and containment objective achievement in the presence of faults and uncertainties.Existing research has proposed fault-tolerant control approaches using fault detection, diagnosis, and compensation techniques.Future research in this area can focus on improving fault-tolerant control strategies, addressing more complex fault scenarios, and developing adaptive fault compensation methods to enhance the robustness and resilience of containment control systems.

Research directions and applications
The field of containment control for multi-agent systems has significant practical applications and is a rapidly growing area of research.The research opportunities in containment control have the potential to revolutionise the way we approach complex tasks requiring coordination among multiple agents.To further improve the efficacy of containment control, the research could focus on enhancing agent interactions in heterogeneity, resolving goal conflicts, mitigating uncertainties and external disturbances, developing effective geometric constraints for an enclosure, and creating distributed systems with optimal control policies.Among these, the following research directions are elaborated below that aim to address the practical limitations of containment control and improve its applicability in real-world scenarios.
Heterogenous systems: The research on containment control for heterogeneous multi-agent systems is motivated by various applications, such as formation control of autonomous vehicles, flocking of mobile robots, and cooperative control of unmanned aerial vehicles (UAVs) with better vantage and unmanned ground vehicles (UGVs) with better payload.In these applications, different agents may have different capabilities, such as sensing, communication, and motion dynamics, and they need to cooperate to achieve a common objective.The study so far on containment control for heterogeneous multi-agent systems led to the development of various control protocols based on different principles, such as the internal model principle, adaptive tuning law, and event-triggered schemes, which have been successfully applied to different types of multi-agent systems.These can be complementary in function with the sensor suite and mobility.The research on containment control for heterogeneous multi-agent systems is a direction of interest because it addresses challenging problems and has the potential to provide solutions for real-world applications.
Robustness and fault tolerance: In real-world scenarios, faults and uncertainties are inevitable, and they can negatively impact the performance and stability of the system.Therefore, developing more robust and fault-tolerant containment control systems can help to ensure that the containment objective is achieved, even in the presence of these disturbances.This could involve the development of algorithms that are better able to handle communication failures or other types of disruptions, as well as those that can adapt to changes in the environment or system parameters.Moreover, robustness and fault tolerance are essential for safety-critical applications such as autonomous vehicles, where a single failure could lead to catastrophic consequences.In such scenarios, fault-tolerant control can help to ensure that the system remains safe and stable, even in the presence of faults and uncertainties.In general, as the number of agents in a multi-agent system increases, the likelihood of faults and uncertainties also increases.Thus, researchers could focus on developing more advanced techniques for fault detection and diagnosis, observer-based fault estimation, and fault-tolerant control strategies.Additionally, there is a need for the development of more comprehensive and realistic simulation models to test the effectiveness of these techniques in different scenarios.
Game theoretic approaches: It provides a powerful tool for analysing the behaviour of agents and predicting their actions in response to different scenarios.By modelling the interactions between agents as a game, it is possible to develop optimal strategies for achieving the containment objective.This approach can help address the challenges of achieving consensus part of containment among multiple agents with conflicting objectives, as is often the case in realworld multi-agent systems.The design of cooperative and distributed algorithms that allow a group of agents to accomplish complex tasks while maintaining spatial constraints can be formulated as a gametheoretic problem.Game theory provides a mathematical framework for analysing strategic interactions among agents and can be used to design optimal control strategies that are resilient to adversarial attacks and disturbances.
Artificial Intelligence: Integration of artificial intelligence techniques such as machine learning, reinforcement learning, and neural networks can be used to develop efficient control strategies for multi-agent systems.These techniques can enable agents to learn from their environment and adapt their behaviour to achieve the containment objective.For example, machine learning algorithms can be used to learn the dynamics of the agents and the environment, while reinforcement learning algorithms can be used to learn optimal control policies that maximise the containment objective based on reward.Neural networks, as a powerful tool for function approximation, can also be used in developing efficient and adaptive control strategies for multi-agent systems.By integrating these artificial intelligence techniques, we can potentially achieve more efficient, adaptive and robust containment control in multi-agent systems.
Here are a few real-world applications of containment control ranging from warehouse robotics to public transportation: • Inventory Management: In a warehouse setting, multiple robots or autonomous vehicles can be used to transport goods between different locations.Containment control can ensure that the robots move in a coordinated manner, avoiding collisions and efficiently completing the task.This can result in more efficient use of space and resources, reducing costs and improving overall productivity.• Manufacturing: Containment control can help coordinate robotic arms in a manufacturing setting to optimise production flow and reduce downtime.For instance, in automobile assembly lines, robotic arms can be used to pick up and move parts from one location to another.Containment control can ensure that the robotic arms move efficiently and safely, avoiding collisions and reducing the risk of damage to the parts.• Warehouse Robotics: In a warehouse setting, multiple robots can be used to pick up and transport items from one location to another.Containment control can ensure that the robots move efficiently, avoiding collisions and ensuring that each item is picked up and transported to the correct location with a subset of agents to guide and follow.
• Cooperative Transportation: Containment control can be used to coordinate multiple vehicles or robots in a transportation setting, such as public transportation or emergency response.For instance, in transportation, multiple vehicles need to work together to transport people to different locations.Containment control can ensure that the vehicles move in a coordinated manner, avoiding collisions and efficiently completing the task.Similarly, in emergency response situations, multiple vehicles may need to transport people or supplies to different locations, and containment control can help ensure that this is done safely and efficiently.

Conclusion
This survey paper has conducted a comprehensive investigation of containment control in multi-agent systems, focusing on key research areas and highlighting significant findings.The survey has provided an extensive overview of the containment concept, multi-agent communication, measurement and agent dynamics, and controller design and analysis.Throughout the paper, various aspects of containment control have been examined, showcasing the use of generic controllers with dynamic and static leaders in different network topologies under fixed and dynamic connections.The survey has emphasised the guaranteed convergence to the convex hull from arbitrary initial conditions, considering the dependency of containment on network topology and control gains.Various modelling techniques for switching topologies such as sequential switching, Markov chains and semi-Markovian switching chains with unknown transition rates and weights have been studied.Furthermore, the impact of communication delays, including constant and time-varying delays, as well as the presence of sensor faults, unmodeled dynamics, and nonlinearities were elaborated.The use of standard tools for delay analysis, such as linear matrix inequalities, Lyapunov-Razumikhin, and Lyapunov-Krasovskii, has been demonstrated to provide sufficient conditions.
The survey has also highlighted the importance of estimators in mitigating the need for full-state information on agents, specifically in the context of measurement and agent dynamics.Additionally, it has outlined research on systems with heterogeneous dynamics, fractional-order systems, and Euler-Lagrange dynamics, which present significant research directions for real-world applications.Moreover, the paper has discussed control designs utilising adaptive, event-triggered, and fault-tolerant control techniques, inspired by consensus formation, and provided an analysis of generic containment controllers.The review has also identified promising areas for future research, including the development of control strategies for heterogeneous systems, accounting for uncertainties and disturbances, integrating gametheoretic approaches, and leveraging artificial intelligence techniques such as machine learning, reinforcement learning, and neural networks.
By consolidating the existing literature and highlighting fruitful avenues for further investigation, this survey paper serves as a valuable resource for researchers and practitioners interested in the field of containment control in multi-agent systems.The presented findings and research directions pave the way for advancements in the design and implementation of robust and efficient containment control strategies in various applications.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by the National Science and Technology Council (NSTC), Taiwan, under Grant NSTC 112-2636-E-006-001.

Figure 1 .
Figure 1.Application of containment control for cooperative transport, inspired by shepherd dogs.(a) Safe cooperative transportation of simple package delivery followers using leader agents and (b) Shepherd dogs herding sheep, where dogs are trained about the environment.

Figure 2 .
Figure 2. Structure and outline of key topics in cooperative containment control in multi-agent systems.

Figure 3 .
Figure 3.A block diagram of overall containment control with key elements of the survey paper.

Figure 4 .
Figure 4. Containment control based on reference input given to leader agents.

Figure 5 .
Figure 5. Classification of communication graph topologies in containment control.

Figure 6 .
Figure 6.Venn diagram of Adaptive Control, Event-triggered Control, and Fault-tolerant Control that ensure robust performance when used individually or in conjunction.

Figure 7 .
Figure 7. Directed interaction topology with leaders as hexagons and followers as circular nodes.
Thummalapeta is a Ph.D. candidate at National Cheng Kung University, Taiwan, since September 2019.He received his B.S. in Mechatronics Engineering from SRM University, Chennai, India, in 2015, and his M.S. in Aerospace Engineering from the University of Petroleum and Energy Studies, Dehradun, India, in 2017.He worked as a Research Associate at the Systems and Control Engineering Department of IIT Bombay, India, from 2018 to 2019.His main research topic is Containment and Coverage Control for Multi-Agent Systems.He also works on other research topics including Networked Robotic Systems, Aerial Robotics, and Multi-Robot Systems.Yen-Chen Liu received the B.S. and M.S. degrees in Mechanical Engineering from the National Chiao Tung University, Hsinchu, Taiwan in 2003 and 2005, respectively, and the Ph.D. degree in Mechanical Engineering from the University of Maryland, College Park, MD, USA in 2012.He is currently a Professor in the Department of Mechanical Engineering, at National Cheng Kung University, Tainan, Taiwan.His research interests include vehicle dynamic control, networked robotic system, aerial robotics, multi-robot system, mobile robot network, and human-robot interaction.Dr. Liu was the recipient of the Outstanding Young Faculty, Chinese Society of Mechanical Engineering (CSME), Taiwan in 2015, the Ta-You Wu Memorial Award, Ministry of Science and Technology (MOST), Taiwan in 2016, Outstanding Young Robotics Award, Robotics Society of Taiwan (RST), Taiwan in 2017, Kwoh-Ting Li Researcher Award, National Cheng Kung University, Taiwan in 2018, and Young Scholar Fellowship-Columbus Programme, MOST, Taiwan in 2019.

Table 1 .
Multi-agent communication in containment in various  research.
Chen et al. (2022))t al. (2022)et al.(2017), and sufficient conditions in terms of linear matrix inequalities for asymptotic containment were established.Heterogeneous unbounded communication delays were considered inShen and Lam (2016), with an emphasis on convergence rate analysis, and special cases of communication delays were provided for logarithmic, linear, and sublinear growth rates.Algorithms of formation containment were developed in L.Chen et al. (2022)with communication delay under Euler-Lagrange dynamics.The variable gain technique was used for leaders to eliminate delay and velocity estimators for followers.