A hierarchical control system for a leader-follower platoon

Abstract

A hierarchical control system for a leader-follower platoon, comprising: an actual communication network between a leader and each follower in the leader-follower platoon, the actual communication network being switchable among a plurality of alternative communication networks during travel of the leader-follower platoon; an adaptive fixed-time state observer corresponding to each follower, the adaptive fixed-time state observer being configured to observe and estimate a state of the leader based on the switchable actual communication network, each follower employing the adaptive fixed-time state observer to observe the leader satisfying an actual fixed-time stability when the actual communication network is any of the alternative communication networks; and a tracking controller corresponding to each follower, the tracking controller being configured to drive the corresponding follower to track the leader based on an observation result of the corresponding adaptive state observer. The hierarchical control system provided by the present application can ensure reliable observation and tracking of the leader by each follower under switchable conditions of the communication network.

Description

Technical Field

[0001] This application belongs to the field of collaborative control technology for multi-agent motion systems, and relates to fixed-time observation and tracking control technology for multi-agent systems under real-time switching of communication networks. Specifically, it provides a hierarchical control system for a leader-follower formation. Background Technology

[0002] Navigation-following formation control is a widely used control scheme for multi-agent cooperative motion, such as UAV formations, robot swarm collaboration, and traffic flow control. In this system, a navigator plans the trajectory, while the followers sense the navigator's state and track it, achieving coordinated movement of the multi-agent formation. To meet the real-time requirements of formation motion control, it is generally necessary to ensure that the observation and control processes converge within a fixed time bound. That is, the convergence time bound is independent of the system's initial state and is determined solely by the design parameters, thus providing a high degree of time determinism for formation tasks.

[0003] However, during actual formation movement, the observations of the navigator by each follower are affected by various factors, such as the need to switch formations according to environmental changes or mission updates, dynamic interruptions and restorations of communication links due to obstacles, wireless channel fading, and the limited direct communication capabilities of some followers in large formations, requiring indirect communication with neighboring followers. Under these factors, the communication network topology of the navigator-follower formation needs to be continuously and dynamically switched, posing two core challenges to the fixed-time control system: First, most followers cannot directly obtain the navigator's state and must rely on distributed observers for indirect estimation. When the communication network topology switches, the set of neighbor information obtained by the followers changes accordingly, and the fixed-parameter observer cannot adapt to the changes in information exchange efficiency in real time, causing observation errors to accumulate near the switching moment. Second, the coupling between external disturbances and topology switching further exacerbates system uncertainty. The fixed observer gain cannot adapt to the changes in information structure brought about by topology switching, nor can it actively suppress the impact of disturbances. The superposition of these two factors severely reduces observation accuracy, leading to inaccurate subsequent control laws.

[0004] Existing methods have significant limitations in addressing the aforementioned challenges: First, most existing fixed-time observers employ constant-coefficient, fixed-gain structures, which can operate under fixed topology and weak disturbance conditions, but the gain cannot be adjusted synchronously when facing topology switching, lacking proactive adaptability. Second, fixed-time controllers often use constant exponential coefficients, failing to automatically switch control strength when communication network topology switching causes increased observation errors, leading to phase lag or steady-state oscillations in the tracking trajectory. Therefore, there is an urgent need to propose a hierarchical, cascaded, and collaborative scheme that integrates adaptive fixed-time state observation with variable exponential coefficient fixed-time control for high-order multi-agent systems subjected to external disturbances in switched network environments. Summary of the Invention

[0005] This application provides a hierarchical control system for navigating and following formations through embodiments, the hierarchical control system comprising: The communication layer includes the actual communication network between the navigator and each follower in the navigator-follower formation, wherein the actual communication network can switch between multiple alternative communication networks during the navigator-follower formation's movement; The observation layer includes an adaptive fixed-time state observer corresponding to each follower, which observes and estimates the state of the leader based on a switchable actual communication network. Under the condition that the actual communication network used by the communication layer is any alternative communication network, when each follower uses its corresponding adaptive fixed-time state observer to observe, the observation error of the leader satisfies the actual fixed-time stability. The control layer includes a tracking controller for each follower, which drives its corresponding follower to track the navigator based on the observation results of its corresponding adaptive state observer.

[0006] The hierarchical control system for navigating and following formations provided in the embodiments of this application, by constructing an adaptive fixed-time state observer, can not only effectively suppress the interference of external disturbances on the observation accuracy, but also maintain the observation performance during network topology switching, thereby achieving accurate estimation of the navigator's state. Attached Figure Description

[0007] Figure 1 This is a schematic diagram of the architecture of a hierarchical control system for navigating and following formations provided according to an embodiment of this application; Figure 2 This is a schematic diagram illustrating the switching between lead-follow formations across multiple communication networks according to an embodiment of this application; Figure 3 This is a schematic diagram of the time-varying observation curve of the navigator information provided according to the embodiments of this application; Figure 4 This is a schematic diagram illustrating the time evolution of observation errors according to embodiments of this application; Figure 5 This is a schematic diagram illustrating the adaptive value evolution process of an adaptive fixed-time state observer provided according to an embodiment of this application; Figure 6 This is a schematic diagram of the tracking trajectory of the follower to the navigator according to an embodiment of this application; Figure 7 This is a schematic diagram illustrating the error convergence between the navigator state estimated by the observer according to an embodiment of this application and the actual states of each follower. Figure 8This is a schematic diagram of the control inputs for each follower according to embodiments of this application; Figure 9 This is a schematic diagram of the tracking trajectory of each follower to the navigator under a constant exponential coefficient controller. Figure 10 This is a schematic diagram illustrating the error convergence under a constant exponential coefficient control strategy. Figure 11 This is a schematic diagram of the control input under a constant exponential coefficient control strategy. Figure 12 This is a schematic diagram comparing the error convergence speed of the control system provided in this application with that of the constant exponential coefficient control strategy. Detailed Implementation

[0008] The present application will now be further described based on preferred embodiments and with reference to the accompanying drawings.

[0009] Furthermore, for ease of understanding, various components on the drawings have been enlarged or reduced, but this is not intended to limit the scope of protection of this application.

[0010] In the description of the embodiments of this application, it should be noted that if terms such as "upper," "lower," "inner," or "outer" are used to indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings, or the orientation or positional relationship commonly used when the product of this application is in use, they are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on this application. In addition, in the description of this application, in order to distinguish different units, the terms "first," "second," etc. are used in this specification, but these are not limited by the manufacturing order, nor should they be construed as indicating or implying relative importance. Their names may differ in the detailed description and claims of this application.

[0011] The vocabulary used in this specification is for illustrative purposes and is not intended to limit the scope of this application. It should also be noted that, unless otherwise expressly specified and limited, the terms "set," "connected," and "linked" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection, a direct connection, or an indirect connection via an intermediate medium; or they can refer to the internal communication between two components. Those skilled in the art will understand the specific meaning of these terms in this application.

[0012] This application provides a hierarchical control system for a navigator-follower formation. This control system is used to adaptively adjust the observation accuracy of each follower on the navigator's state based on the switching of the communication network topology between the navigator and the follower formation during the navigator-follower formation's movement, thereby achieving adaptive adjustment of the control law for each follower.

[0013] Figure 1 A schematic diagram of the architecture of a hierarchical control system for a navigator-follower formation according to some embodiments of this application is shown. As shown in the figure, the control system includes: The communication layer includes the actual communication network between the navigator and each follower in the navigator-follower formation, wherein the actual communication network can switch between multiple alternative communication networks during the navigator-follower formation's movement; The observation layer includes an adaptive fixed-time state observer corresponding to each follower, which observes and estimates the state of the leader based on a switchable actual communication network. Under the condition that the actual communication network used by the communication layer is any alternative communication network, when each follower uses its corresponding adaptive fixed-time state observer to observe, the observation error of the leader satisfies the actual fixed-time stability. The control layer includes a tracking controller for each follower, which drives its corresponding follower to track the navigator based on the observation results of its corresponding adaptive state observer.

[0014] The specific implementation of this control system will be described in detail below with reference to the accompanying drawings.

[0015] <Navigation - The overall framework for following the formation and coordinating movement> like Figure 1 As shown, the control system is used to control the movement of a navigator-follower formation consisting of a navigator and several followers. In the navigator-follower formation, the navigator is responsible for planning the movement path or determining the movement target, and moves according to the planned path or target. Each follower obtains the navigator's status information in real time by observing the navigator, and tracks the navigator under the drive of the control law based on the deviation between the navigator's status and its own status, so as to form a formation that follows the navigator in a coordinated manner.

[0016] It is understandable that both the navigator and the follower can be intelligent agents in various forms, such as unmanned vehicles, unmanned ships, and drones, and both possess the following core functional units: (1) Wireless communication device Each intelligent agent (including the navigator and all followers) is equipped with a device capable of wireless communication, such as a 4G / 5G module, a WiFi module, or a private network communication module. Each follower establishes a navigator-follower formation communication network (i.e., forms a navigator-follower formation) through communication with its neighboring followers within its neighborhood and direct communication with the navigator. Figure 1 (In the communication layer), each follower interacts with the navigator directly or indirectly through the communication network to support the observation function of the navigator.

[0017] (2) Body state sensing device Each intelligent agent is equipped with one or more body state perception devices such as GPS / RTK positioning system, IMU inertial navigation unit, and speed sensor to obtain its own real-time status such as position, speed, and heading. Accordingly, the navigator uses its real-time status as the reference benchmark for the control law of moving along the preset path or moving toward the target, while each follower uses its real-time status as the reference benchmark for the control law of tracking the navigator.

[0018] (3) State observer Each follower is equipped with a state observer. Based on the accessed communication network topology, it can directly obtain the navigator's state information when it can communicate directly with the navigator, or obtain the navigator's state information from neighboring followers that can communicate with it via relay communication when it cannot communicate directly with the navigator. The obtained information is then used to reconstruct the navigator's various states (position, velocity, etc.).

[0019] The state observers equipped by each follower constitute Figure 1 In the observation layer of this application, each follower's corresponding state observer is an adaptive fixed-time state observer. Its gain parameter can be automatically adjusted according to the information interaction efficiency changes under the real-time switching communication network topology, and it can ensure that the upper bound of the convergence time of the distributed observation error between each observer is independent of the initial state, that is, it can converge to zero within a fixed time.

[0020] (4) Controller The navigator's controller can output control commands according to a preset control law (such as ensuring that the real-time trajectory deviates from the preset path or approaches the target with the shortest path while avoiding obstacles) based on the preset motion path or target; each follower's tracking controller outputs corresponding control commands according to the preset tracking control law based on its own state obtained by the body state sensing device and the navigator's state obtained by the state observer.

[0021] The tracking controllers equipped on each follower constitute Figure 1In the control layer of this application, each follower's corresponding tracking controller is a variable exponential fixed-time tracking controller, which adopts a bounded monotonically variable exponential coefficient. When the error amplitude is large, the exponent automatically increases to the upper bound to strengthen the control strength and achieve rapid convergence. When the error amplitude approaches zero, the exponent automatically decreases, making the control behavior gentle and smooth, effectively suppressing steady-state oscillations, and ensuring that the control error is independent of the initial state, that is, it can converge to zero within a fixed time.

[0022] (5) Instruction execution unit Each intelligent agent is equipped with an instruction execution unit to drive the intelligent agent to execute the control commands output by its controller. Generally, the instruction execution unit includes, but is not limited to, a power unit, a steering unit, an attitude adjustment unit, etc. For example, for an unmanned vehicle, the instruction execution unit may include a motor, an electric steering mechanism and its drive system, which can drive the motor and electric steering mechanism to move according to the received torque and steering commands; as another example, for an unmanned ship, the instruction execution unit may include a propeller, a rudder and its drive system, which can adjust the propeller speed and rudder direction according to the control commands.

[0023] It should be noted that, in the embodiments of this application, the control method of the navigator is not limited. Those skilled in the art can drive the navigator to travel along a preset path or toward a preset target based on various control algorithms they have mastered, such as trajectory tracking control, path planning and routing control, target point navigation, etc.

[0024] <Navigation-Following Formation Multi-Agent Dynamics Model and Its Constraints> For a multi-agent system consisting of a navigator and multiple followers, the design of the observer and controller needs to be based on a model that can accurately describe the dynamic characteristics of each agent: the observer needs to reconstruct the navigator's state based on the dynamic model, and the control law needs to drive the followers to follow the navigator based on the dynamic model.

[0025] To improve control accuracy, in the embodiments of this application, the dynamic characteristics of the navigator and each follower are characterized by a high-order dynamic model of order 3 or higher.

[0026] Specifically, The navigator's dynamic characteristics can be expressed by the following formula: The class-linked dynamics equations describe: (1), in, For time variables, Let this be the navigator's state vector. This refers to the first-order state (i.e., position) of the navigator. This represents the navigator's second-order state (i.e., speed), and so on. For the Navigator As described above, in the embodiments of this application, the next state is... ,Right now This includes at least the navigator's position and speed, and as... The increase can also include navigator acceleration, jerk, etc.

[0027] The control inputs for the navigator can be generated based on the controllers equipped in the navigator; These are the internal dynamic uncertainties and external disturbances acting on the navigator, including model uncertainties, parameter changes, and environmental disturbances.

[0028] In embodiments of this application, the number of followers is greater than or equal to 2, wherein, for the th A follower, whose dynamic characteristics can be expressed by the following formula The class-linked dynamics equations describe: (2), in, For the first The state vector of each follower For the first The first-order state (position) of a follower. For the first The second-order state (velocity) of each follower, and so on. For the first a follower first state, For the first A follower's control input, For the action in the Internal dynamic uncertainties and external disturbances on a follower.

[0029] In the process of controlling the navigator-follower formation, the dynamic characteristics of each agent are constrained by the laws of physics. Therefore, the internal dynamics and external disturbances of both the follower and the navigator are bounded quantities, meaning there exists a known upper bound on their positive constants. , , so that: (3).

[0030] Similarly, the navigator's control input It is also subject to the laws of physics, therefore, It is also a bounded quantity, meaning there exists an upper bound for a positive constant. , so that satisfaction .

[0031] <Communication Layer and Communication Network Topology Switching Mechanism> In the embodiments of this application, the communication network topology between the navigator and each follower is located at the communication layer. This communication network topology describes the communication state between the navigator and each follower. During formation movement, the navigator continuously sends its own status through the communication network. When the navigator communicates with a follower... When direct communication is possible between the followers, then the followers... It can directly obtain the status information sent by the navigator, thus enabling direct observation of the navigator's status; when a follower... When direct communication with the navigator is impossible, the navigator's state needs to be estimated by exchanging information with its neighbors (i.e., other followers with whom it can communicate), which involves indirect observation of the navigator. However, if the navigator's state can be obtained through at least one follower capable of direct communication with it, then that follower... The navigator can only be observed indirectly.

[0032] During navigating and following the formation, the communication quality between agents is not constant due to factors such as terrain, obstacles, and electromagnetic interference. Therefore, using a fixed topology communication network may lead to communication interruptions between agents, affecting observation and control. To address this, in the embodiments of this application, the communication layer adopts a switchable topology mechanism: several alternative communication network topologies are pre-established to cope with different obstacles and communication interference issues. During navigating and following the formation, each agent can switch between the alternative communication network topologies as needed, either by the navigator or by a majority vote. Figure 2 This diagram illustrates, in a specific embodiment, the switching between three communication network topologies (a), (b), and (c) between the navigator (numbered 0) and each follower (numbered 1-5). Without loss of generality, the embodiments of this application can be used... Indicates the communication layer at any time The communication network topology is numbered such that, obviously, in order to ensure that any follower in a leader-follower formation can directly or indirectly observe the leader, the communication network topology in the communication layer should meet the following conditions: Condition 1: The navigator is able to communicate directly with at least one follower; Condition 2: Any follower communicates directly with the navigator either through a direct communication path or indirectly with the navigator through at least one relay communication path that includes other followers.

[0033] During the switching of communication networks, the above conditions should be met at any topology switching time. That is, at any time, there is a directed path from the leader to any follower in the communication network topology to ensure the continuous reachability of the communication network. In addition, there should be a sufficient time interval between two adjacent communication network topology switching to avoid communication quality degradation or even connection loss caused by frequent switching.

[0034] The communication network topology between the navigator and its followers can be described using a graphical structure. Considering that communication between the navigator and its followers is unidirectional, while communication between the navigators is bidirectional, the communication network within the formation can be first described using a graphical structure. The communication state among followers is defined as an undirected graph. ,in For the set of follower nodes, Let be the set of edges between each follower. This is a weighted adjacency matrix, representing the communication states between each follower. The elements of the weighted adjacency matrix are... for: (4), Furthermore, define the first The neighborhood of a follower Let the set of all other followers that can communicate with it be defined as an undirected graph. degree matrix for Its diagonal elements are ; and define an undirected graph Laplace matrix for .

[0035] Next, consider the navigator (set as the navigator node). The one-way communication relationship with each follower will Expanded into Navigator In-degree Map Define the adjacency matrix of the leader and followers. for Among them, when followers When able to receive information from the navigator, ,otherwise .

[0036] Finally, you can The topology is a weighted matrix containing navigator in-degree information. It indicates that it is used to describe the communication network topology between the leader and followers, as well as among the followers. The maximum and minimum eigenvalues ​​are denoted as follows: and .

[0037] Obviously, in the application of this application, since the network communication topology can be switched, the matrix in the above formula... , , , , The values ​​of all change with the switching of the communication network. Therefore, in this scenario where the communication network is switchable, each matrix can be represented as follows: (5).

[0038] In addition, the A follower at any given time The neighbor set can be represented as .

[0039] <Observation Layer> As analyzed above, with the real-time switching of the communication network topology in the communication layer, each follower in the leader-follower formation may need to obtain the leader's status information through relay communication links, that is, they can only indirectly observe the leader. Therefore, when designing the observers for the followers, it should be ensured that regardless of the communication network topology used in the communication layer (i.e., whether the follower can directly observe the leader or can only observe the leader indirectly), the observers have fixed-time convergence: the observation error can converge to zero in a finite and constant time, and this observation error convergence time is independent of the initial conditions.

[0040] In some preferred embodiments of this application, for any follower As mentioned above, let's assume , The total number of observers and At any time In its first The observer under the given network communication topology is: (6).

[0041] Table 1 below lists the definitions and explanations of the parameters and functions in equation (4): Table 1 Observer Parameter Description Without loss of generality, the error terms within the parentheses in equation (6) can be integrated and expressed as follows: and : (7), (8), and Each is a real communication network Conditions, No. The first follower to the leader The first state and the second The total observation error of the observations at the first state (including the direct observation error, i.e., the error of directly observing the leader, and the cumulative result of the error of indirect observations through all neighbors) is such that equation (6) can be simplified as: (6').

[0042] It is precisely because of the followers The degree of observational error of the navigator is affected by different communication network topologies. The influence of conditions, in order to ensure that in all Under the condition that the observers are all actual fixed-time state observers, that is, the observation errors of each observer can converge to the neighborhood of the origin within a fixed time, it is necessary to... , The design is intended to adaptively update based on the error level, and the update method can be proven to satisfy the fixed-time convergence condition.

[0043] In the embodiments of this application, and The update law is designed as follows: (9).

[0044] Using equations (7) and (8), equation (9) can be simplified as: (9').

[0045] In equations (9) and (9'), It is a positive number.

[0046] Observing equations (9) and (9'), we can see that and The update speed is affected by two different types of adjustment terms, specifically, for : and For followers In communication network topology Under the conditions of the navigator's first The amplitude adjustment term related to the total observation error of the first-order state, the first... The larger the magnitude of the total observation error of the first-order state, the more effective the magnitude adjustment term becomes. The greater the degree of update, the greater the impact; The damping adjustment term is always opposite to the amplitude adjustment term, and its function is to... The update applies reverse adjustment to avoid The cumulative increase in observation error leads to integral saturation. It should also be noted that the expression for this damping adjustment term is: The polynomial, where The degree and coefficient (the coefficient of the 1st degree term is 1, the coefficient of the 5th degree term is 1). The value is determined based on the actual fixed-time convergence requirement of the observation error.

[0047] for : and To follower In communication network topology Under the conditions of the navigator's first The amplitude adjustment term related to the total observation error of the first-order state, the first... The larger the magnitude of the total observation error of the first-order state, the more effective the magnitude adjustment term becomes. The greater the degree of update, the greater the impact; The damping adjustment term is always opposite to the amplitude adjustment term, and its function is to... The update applies reverse adjustment to avoid As the cumulative increase in observation error leads to integral saturation, the expression for this damping adjustment term is similarly: The polynomial, where The degree and coefficient (the coefficient of the 1st degree term is 1, the coefficient of the 5th degree term is 1). This is also determined based on the requirement of fixed-time convergence of observation errors.

[0048] Therefore, in the embodiments of this application, the update will be performed according to equation (9). and called followers The first for the navigator Rank and first The adaptive gain for the first-order state observation error will use and An observer that performs adaptive gain is called an adaptive fixed-time state observer.

[0049] The following proves, by constructing a Lyapunov function, that the observer form is (6), and that the Lyapunov function used in the function is... and When the update law is (9), any communication network can be used at the communication layer. Under these conditions, all observers are fixed-time observers, and their observation errors satisfy the actual fixed-time stability.

[0050] It should be noted that "fixed-time stability" is a concept already known to those skilled in the art of control systems. Specifically, for a nonlinear dynamic system: (10) in Represents the system state vector. It is a continuous function used to characterize the state changes of a system. If the equilibrium point of the system satisfies two conditions, it is said to be fixed-time stable: First, the system is finite-time stable, that is, there exists a convergence time. Makes all have Secondly, there exists a positive constant. Make ,and Independent of the initial state .

[0051] Generally, the nonlinear dynamic system shown in equation (10) can be proven to be a practically fixed-time stable system by proving the existence of a positive definite Lyapunov function that satisfies specific conditions. Specifically: If for a nonlinear dynamic system, there exists a positive definite Lyapunov function. This makes its time derivative satisfy the following inequality: (11), in It is a positive constant and satisfies the following globally bounded convergence time: (12).

[0052] Furthermore, if the Lyapunov function The time derivative satisfies: (13) in ,and If the dynamic system is stable in real fixed time, then it satisfies real fixed-time stability. Specifically, the system state will be restricted to the following bounded set within a fixed time. ,in, For a small number of parameters that are stable over a fixed time and Enter the set fixed time Satisfy the following upper bound constraints .

[0053] Based on the above proof of practical fixed-time stability, it is first necessary to analyze the dynamics of observation error, for followers. Define the observation errors of each order: (14) The dynamic expression of the observation error is then: (15).

[0054] make: (16); (17) To prove that the observer is actually fixed-time stable, we can prove in sequence... Order observation and various All observations were stable at a fixed time.

[0055] First, construct the following... Lyapunov function of order: (18) Taking the derivative of equation (18), and substituting the observer's expression (6) and the adaptive gain update law expression (9) into the derivative result, we can obtain: (19) in, For a moment In communication network topology Under the given conditions, a weighted topology matrix including navigator in-degree information, for The smallest eigenvalue.

[0056] B. Ning et al. (B. Ning, QL Han, Z. Zuo, L. Ding, Accelerated secondaryfrequency regulation and active power sharing for islanded microgrids with external disturbances: A fully distributed approach, Automatica 174 (2025).doi: 10.1016 / j.automatica.2025.112146.) have proven that for all The following inequalities hold: (20) in It is a constant. It is an exponent and satisfies .

[0057] Applying the inequality relation of equation (20) to equation (19), we get: (twenty one), in, Furthermore, it can be deduced that: (twenty two), Based on equation (21), in equation (22) ,therefore, Ultimately, it can be simplified to the standard form: (twenty three).

[0058] in, By comparing the final standard expressions of equations (13) and (23) one by one, the result can be determined. m The settling time of the observation error satisfies .

[0059] It should be noted that the following proven conclusions were used in the derivation of equation (23): If the constant satisfies Then the following inequality holds: .

[0060] Next, prove ( m -1) Dimensional observation errors exhibit a fixed-time convergence to a neighborhood of the origin, i.e. The Lyapunov function, its derivative, and Substituting the update law expressions for the order observer and adaptive gain, we get: (twenty four), when hour, Next, we can observe that: (25) in Based on this, we can further conclude that: (26).

[0061] Based on equation (25), in equation (26) Finally, equation (24) can be simplified to its standard form: (27) By comparing equations (13) and (27) item by item, we can determine... The settling time of the observation error satisfies .

[0062] for , ,…,until The same proof method can be used to ultimately prove the upper bound of the convergence time of the observation error. satisfy: (28).

[0063] In some embodiments, it can be ,but .

[0064] Equation (28) shows that, under the condition that the actual communication network used in the communication layer is any alternative communication network, when each follower uses its corresponding adaptive fixed-time state observer to make observations, the observation error of the leader satisfies the actual fixed-time stability.

[0065] The above proof process shows that... , The damping adjustment term in the update law must also include a linear term. , And 5th term , Both correspond to the derivatives required in Lyapunov's derivatives. and The attenuation structure, therefore, can only be achieved by using... , This special form of damping adjustment term can precisely offset the positive increase in the observation error amplitude adjustment term after differentiating the Lyapunov function and substituting it into the adaptive law, so that the derivative of the Lyapunov function satisfies the fixed-time stability criterion and ensures the existence of an upper bound on the convergence time.

[0066] It should be noted that the coefficient of the 5th term The existence of this term is to ensure that the higher-order terms are large enough to offset the positive growth portion. See equations (21) and (25). After differentiating the Lyapunov function, the form of the damping adjustment term necessitates the handling of cross terms, resulting in the appearance of multiple intermediate constants. ,in, , Contains , as well as , , The essence is the communication network topology parameters (number of followers) The combined representation of the smallest eigenvalue of the Laplace matrix reflects the influence of the communication network topology on the convergence process of observation errors of each order, as well as the contribution from the fifth-order damping term. Contains , which is the remainder after the 5th-degree term and the linear term are jointly canceled out. Obviously, in order to satisfy convergence, It should be greater than 0, that is .

[0067] Furthermore, these constants are ultimately combined into a comprehensive remainder term. , ,in , It already contains all of the above. , , , The relevant bounded constants, in the subsequent fixed-time criterion, precisely serve as the bounded perturbation term. The role, due to the comprehensive remaining items , Since it is a finite constant, it does not affect the magnitude of the upper bound of the fixed time, but only the size of the final convergence neighborhood. Therefore, by adjusting... This allows for the adjustment of the radius of the convergence neighborhood.

[0068] <Control Layer> The control layer includes a tracking controller corresponding to each follower. Each tracking controller is deployed on its corresponding follower, receives the estimated output of the corresponding adaptive state observer as a reference signal, combines it with the follower's own motion state feedback to obtain the tracking error, and then generates control commands through a control law to drive the corresponding follower to track the leader.

[0069] like Figure 1 As shown in the embodiments of this application, each tracking controller is a variable exponential fixed-time tracking controller. The specific form and construction process of the tracking controller are described in detail below.

[0070] For followers Its tracking error is , Differentiating the tracking error yields the dynamic expression of the tracking error: (29).

[0071] Referring to the previous proof of the stability of the observation layer at a fixed time, when At that time, observation error , Established, and accordingly, introduced and The error dynamics can be restated as follows: (30) in, , .

[0072] For the pilot-follower formation, a sliding mode control architecture is used to control each follower. Therefore, it is necessary to first determine the form of the sliding mode surface and then obtain the control law based on the sliding mode surface.

[0073] Specifically, error dynamics can be expressed as The composite tracking error term, consisting of the first-order tracking error plus the weighted power integrals of the tracking errors of each order, is for the follower. Its composite tracking error term It can be represented as: (31), in, , For the first a follower The exponent of the tracking error. , It provides high gain near the origin, eliminates convergent ridges, and ensures precise state locking near the equilibrium point. A larger control input is generated at the far origin, compressing the time it takes for the system to move from the far end to the origin. , These are proportionality coefficients that are greater than zero.

[0074] Based on the above expression, design a follower. sliding surface : (32), in, , , , All are constants greater than zero.

[0075] Observing equation (32), it can be seen that the sliding surface is composed of a composite tracking error term. And about The weighted integral terms of the sig function are constituted, and The exponent of the sig function is based on The state changes in real time. Therefore, in the embodiments of this application, the sliding surface of equation (32) is called the sliding surface with variable exponential coefficient.

[0076] To determine the control law, for Differentiating the expression, we get: (33), Equation (33) clearly shows all the factors affecting the convergence of the sliding surface: The first and second terms are nonlinear feedback terms of the tracking error in the double power structure; The third item is based on composite tracking error. The variable exponential coefficient term of the tracking error whose state changes in real time; The fourth term is the tracking error dynamics term, which is determined by the follower's own control and uncertainties. , It consists of the observation error component.

[0077] Based on the need to suppress each term in equation (33), and Moving the terms to the left, the final construct yields the following pair of followers. Control laws for implementing control: (34), in, , , , All are constants greater than zero, and Greater than followers The upper bound of internal dynamics and external disturbances ,Right now ,also, .

[0078] Observing equation (34), we can see that the control law also contains four terms, among which: The first and second terms are nonlinear error suppression terms, used to offset errors within the integral sliding surface. The third term is a tracking error suppression term, used to suppress the variable exponent coefficient term in the sliding surface derivative. The fourth term is a robust term with variable exponent coefficients; this term is for the sliding surface... The sig function, and the sig function also has the following characteristics: The variable exponential coefficient changes in real time to adaptively offset the uncertainties caused by internal and external disturbances.

[0079] For any i A follower, whose actual communication network at the communication layer is any alternative communication network, and under the condition of using the aforementioned adaptive fixed-time state observer for observation, is subject to a variable exponential coefficient fixed-time control law, resulting in a tracking error. A bounded neighborhood that converges to the equilibrium point, with an upper bound on the convergence time. ,in , , , , All have an upper limit.

[0080] The following is a detailed proof.

[0081] First, define the Lyapunov function. Calculate its time derivative and substitute the control law into the result to obtain: (35), in, To combine the control law gain term with The combined equivalent result after multiplication and merging of the observation error coupling terms is: , These are constants greater than zero, used to adjust the convergence rate.

[0082] Based on the actual fixed-time stability of the observer, for ,have Using this result, equation (35) can be simplified to: (36).

[0083] when At this time, the sliding surface is in the large deviation convergence stage, and at this time there is ,because and Scaling equation (36) yields: (37) The upper bound of the convergence time in this stage is: (38).

[0084] when At this time, the sliding surface is in the small deviation convergence stage. At this point, due to... ,available Using this minimum value, we can deduce that: (39) in The convergence time for this stage is: (40).

[0085] Combining equations (38) and (40), it can be seen that under the action of the adaptive fixed-time state observer and the variable exponential coefficient control law, the sliding surface in It then converges to the equilibrium point.

[0086] As the sliding surface reaches the equilibrium point Then, it enters the auxiliary variable, namely the composite tracking error. The large deviation convergence phase and the small deviation convergence phase are specifically defined by the Lyapunov function: (41), Calculating its derivative, we get: (42).

[0087] when At that time, the system is in the auxiliary variable. During the large deviation convergence phase, there are .because and We can obtain: (43), The convergence time for this stage is: (44).

[0088] when At that time, the system is in the auxiliary variable. During the small deviation convergence phase, due to , can be obtained Using this minimum value, we can derive: (45) The convergence time for this stage is: (46).

[0089] Therefore, the system will Then it enters the neighborhood of the equilibrium point.

[0090] Finally, when Sometimes, At this point, we enter the final tracking stage, and the error dynamics can be simplified to: (47).

[0091] Z. Zuo et al. (Z. Zuo, QL Han, B. Ning, An explicit estimate for the upper bound of the settling time in fixed-time leader-following consensus of high-order multivariable multiagent systems, IEEE Trans. Ind. Electron. 66(8) (2019) 6250-6259. doi: 10.1109 / TIE.2018.2873510.) have demonstrated that the error shown in equation (47) will be within a fixed time bound. It converges inward to the origin's domain.

[0092] Therefore, considering the convergence of observation and tracking errors at each stage, we can conclude that when At that time, tracking errors of each order , Converging to the neighborhood of the origin, that is: given that the actual communication network used in the communication layer is any alternative communication network, and under the condition of using the aforementioned adaptive fixed-time state observer for observation, any... Under the control of a variable exponential coefficient fixed-time controller, the tracking error of a single follower also satisfies the actual fixed-time stability.

[0093] <Example 1> This embodiment simulates the tracking performance of a leader-follower formation consisting of one leader and five followers under changing network communication topology. The dynamic equations of the leader and followers are in second-order form, where the dynamic equation of the leader is: (48) No. i The dynamic equations for the followers are: (49).

[0094] The states and control inputs of the navigator and follower are respectively used as... and This indicates that the navigator's control inputs are set to... The initial state is configured as follows The initial values ​​for each follower are as follows: , , , , External disturbances adopt and .

[0095] Figure 2 The switching process of the communication network between various agents is shown. At that time, the communication network topology corresponds to Figure 2 In the topology (a), when When the communication network topology switches to topology (b), At that time, the communication network topology is switched to topology (c).

[0096] The observer parameters are designed as follows , The controller parameters are set to , , .

[0097] After completing the above settings, the navigation-following formation process was simulated. To verify the superiority of the method provided in this application, a conventional constant exponential coefficient control strategy was also adopted. Simulation comparisons were performed under the same network communication switching conditions. All similar parameter settings of the conventional strategy remained consistent. The constant exponential coefficient controller is as follows: (50).

[0098] Figure 3 The time-varying observation curves show that the observer can accurately estimate the navigator's state. This high-precision observation stems from the inherent advantages of an adaptive fixed-time state observer.

[0099] Figure 4 The temporal evolution of the observation error is shown. The observation error is defined as the difference between the navigator state estimated by the observer and the actual navigator state. It decays rapidly to near zero, which confirms the effectiveness of the adaptive fixed-time state observer in achieving high-precision state observation.

[0100] Figure 5 The adaptive value evolution process of the adaptive fixed-time state observer is shown. With the help of the adaptive tuning mechanism of the observer gain, the observer can accurately adapt to the system environment of dynamic network topology switching.

[0101] Figure 6 The following diagram illustrates the tracking trajectory of the followers towards the leader, demonstrating that all followers achieve effective state tracking. This reliable multi-agent consensus performance stems from two key factors: first, the adaptive fixed-time state observer provides accurate leader state information as a foundation; and second, the variable exponential coefficient fixed-time controller plays an indispensable role in consensus tracking within the multi-agent system.

[0102] like Figure 7 The diagram shows the convergence of the error between the navigator state estimated by the observer and the actual state of each follower. It can be seen that the error between the navigator state estimated by the observer and the actual state of each follower converges quickly to zero, proving that the hierarchical control system provided in this application can effectively promote navigator-follower consistent tracking.

[0103] Figure 8 The control inputs of each follower are shown, which serve as the direct driving force for achieving multi-agent tracking.

[0104] Figure 9 The tracking trajectories of each follower to the navigator under a constant exponential coefficient controller are shown. The curves show that, compared with the control system provided in this application, the position and velocity trajectories of the followers have phase lag. This rigid response is due to the fixed characteristics of the constant exponential coefficient, which makes it impossible to adjust the control strength according to the error amplitude.

[0105] Figure 10 , Figure 11 The error convergence and control input under the constant exponential coefficient control strategy are shown respectively. Figure 12A comparison of the error convergence speed of the control system provided in this application and the constant exponential coefficient control strategy shows that the former has a significantly faster error convergence speed than the latter.

[0106] The specific embodiments of this application have been described in detail above. For those skilled in the art, several improvements and modifications can be made to this application without departing from the principle of this application, and these improvements and modifications also fall within the protection scope of the claims of this application.

Claims

1. A hierarchical control system for navigating and following formations, characterized in that, include: The communication layer includes the actual communication network between the navigator and each follower in the navigator-follower formation, wherein the actual communication network can switch between multiple alternative communication networks during the navigator-follower formation's movement; The observation layer includes an adaptive fixed-time state observer corresponding to each follower, which observes and estimates the state of the leader based on a switchable actual communication network. Under the condition that the actual communication network used by the communication layer is any alternative communication network, when each follower uses its corresponding adaptive fixed-time state observer to observe, the observation error of the leader satisfies the actual fixed-time stability. The control layer includes a tracking controller for each follower, which drives its corresponding follower to track the navigator based on the observation results of its corresponding adaptive state observer.

2. The hierarchical control system for navigating and following formations according to claim 1, characterized in that, The observer gain update rate of the adaptive state observer includes an amplitude adjustment term related to the total observation error, and a damping adjustment term that is always opposite to the amplitude adjustment term.

3. The hierarchical control system for navigating and following formations according to claim 2, characterized in that, The total observation error includes the direct observation error and the cumulative result of the error from indirect observations through all neighbors; The expression for the damping adjustment term is a polynomial of the observer gain, wherein the degree and coefficient of the observer gain are determined based on the fixed-time convergence requirement of the observation error.

4. The hierarchical control system for navigating and following formations according to claim 3, characterized in that, For any i Each follower employs an adaptive fixed-time state observer with an observer gain update rate of: ， in, , Each is a real communication network Conditions, No. The first follower to the leader Rank, First Total observation error of the first state, , The first The first follower to the leader Rank, First Observer gain for observing the first state. The power coefficient is the coefficient of the power. , All are constants greater than zero and , , .

5. The hierarchical control system for navigating and following formations according to claim 4, characterized in that, Under the condition that the actual communication network used in the communication layer is any alternative communication network, when each follower uses the adaptive fixed-time state observer to observe the leader, the upper bound of the observation error convergence time is... satisfy: ， in, , They are respectively , The learning rate It is a small parameter that is stable over a fixed period of time.

6. The hierarchical control system for navigating and following formations according to claim 1, characterized in that, Under the condition that the actual communication network used in the communication layer is any alternative communication network, and the adaptive fixed-time state observer is used for observation, the tracking error of each follower to the navigator under the control of its corresponding variable exponential coefficient fixed-time controller satisfies the actual fixed-time stability.

7. The hierarchical control system for navigating and following formations according to claim 6, characterized in that, The tracking controller is a variable exponential coefficient fixed-time tracking controller, and its control law is determined based on the variable exponential coefficient integral sliding surface.

8. The hierarchical control system for navigating and following formations according to claim 7, characterized in that, For any i For each follower, the corresponding integral sliding surface with variable exponential coefficients is: ， in, For the first The composite tracking error of each follower , , , All are constants greater than zero.

9. The hierarchical control system for navigating and following formations according to claim 7, characterized in that, For any i A follower, whose control law for a fixed-time tracking controller with variable exponential coefficients includes a nonlinear error suppression term, a tracking error suppression term, and a robust control term; The tracking error suppression term and the robust control term are respectively: ， ， in, , , , All are constants greater than zero.

10. The hierarchical control system for navigating and following formations according to claim 9, characterized in that, For each follower, the upper bound of its tracking error convergence time is... All satisfy: ， in, , , , , These represent the error convergence times for the large deviation convergence stage of the sliding surface, the small deviation convergence stage of the sliding surface, the large deviation convergence stage of the composite tracking error, the small deviation convergence stage of the composite tracking error, and the final tracking stage. , , , , All have an upper limit.

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