A multi-agent system time-varying formation tracking method and system based on internal model principle
By designing a distributed preset time observer and an internal model controller, the time-varying formation tracking problem of unknown disturbances and non-autonomous leaders in a multi-agent system was solved, achieving high-precision and fast formation tracking results and enhancing the system's anti-interference ability and adaptability.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- BEIHANG UNIV
- Filing Date
- 2026-03-12
- Publication Date
- 2026-06-09
Smart Images
Figure CN122172850A_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of disturbance suppression technology, and in particular to a time-varying formation tracking method and system for multi-agent systems based on the internal model principle. Background Technology
[0002] With the rapid development of artificial intelligence and automation technologies, distributed collaborative control of multi-agent systems has demonstrated irreplaceable value in numerous key fields. From regional collaborative perception in environmental monitoring and multi-device linkage in unmanned operation scenarios, to fleet collaborative scheduling in intelligent transportation, and to UAV formations performing complex tasks such as reconnaissance and mapping, multi-agent systems, with their flexibility, fault tolerance, and efficiency, have become core technological supports for promoting industrial intelligence and national defense modernization. Among these, formation tracking control, as a core application direction of multi-agent collaboration, directly determines whether the system can accurately execute preset tasks, and breakthroughs in its technology are of great significance for improving operational efficiency and mission success rates in related fields.
[0003] In practical applications of multi-agent formation tracking, time-varying formation tracking has become a research focus due to its better alignment with dynamic task requirements. However, this field faces several core challenges. On the one hand, the leader is often a non-cooperative agent, and its motion state cannot be directly obtained by all followers. Furthermore, unknown inputs (such as sudden maneuvers or task switching) make it difficult for followers to accurately capture the leader's trajectory. On the other hand, unknown disturbances in the external environment (such as airflow interference and electromagnetic noise) can directly disrupt formation stability and reduce tracking accuracy. In addition, many practical tasks require formation convergence and tracking to be completed within a preset time, while traditional methods struggle to achieve accurate estimation and control within this timeframe. These challenges make time-varying formation tracking a critical technical bottleneck that urgently needs to be overcome in engineering applications.
[0004] In recent years, scholars both domestically and internationally have conducted extensive research on multi-agent formation control, achieving a series of preliminary results. Regarding state estimation, some studies have proposed design strategies based on distributed observers, using information exchange between neighboring nodes to estimate the leader's state. However, most of these methods assume the leader is an autonomous system, neglecting its unknown inputs and making it difficult to adapt to the complex maneuvering behavior of leaders in real-world scenarios. Other scholars have explored finite-time or fixed-time observers, attempting to meet the need for rapid estimation. However, the convergence time of finite-time observers heavily depends on initial estimation errors and design parameters, making it impossible to pre-set precise convergence times. Fixed-time observers, on the other hand, require the introduction of multiple nonlinear gains, leading to complex parameter tuning and hindering their widespread application in large-scale multi-agent systems. In the field of disturbance suppression, existing methods mainly include H∞ control, sliding mode control, and observer-based disturbance compensation. While H∞ control can attenuate the effects of disturbances, it is prone to non-zero steady-state errors. Sliding mode control suffers from chattering problems, and observer-based disturbance compensation methods often require known upper bounds on the disturbances, which are difficult to obtain accurately in practical applications. Disturbance suppression methods based on the internal model principle have attracted attention due to their excellent steady-state performance, but existing internal model-based controllers are mostly designed for autonomous leader scenarios and cannot effectively handle the coupling problem between unknown inputs of non-autonomous leaders and time-varying formations. In terms of control protocol design, most existing schemes rely on global topology information, which reduces the flexibility and scalability of the system. Moreover, most studies only verify the theoretical effectiveness through numerical simulations, lacking physical experimental verification, making it difficult to fully demonstrate their engineering practicality. Overall, existing research has failed to fully cover coupled scenarios with core requirements such as unknown leader inputs, unknown external disturbances, preset time estimation, and fully distributed implementation. Effective time-varying formation tracking methods under such complex conditions remain undeveloped. Summary of the Invention
[0005] The purpose of this application is to provide a time-varying formation tracking method and system for multi-agent systems based on the internal model principle. By designing a distributed preset time observer and an internal model-based controller, time-varying formation tracking of multi-agent systems with unknown disturbances and non-autonomous leaders can be realized.
[0006] To achieve the above objectives, this application provides the following solution.
[0007] In a first aspect, this application provides a time-varying formation tracking method for multi-agent systems based on the internal model principle, which includes the following steps.
[0008] Initialize all relevant parameters; the relevant parameters include: time-varying formation parameters, multi-agent system parameters, communication topology weight matrix, preset convergence time, observer parameters, controller parameters, and initial state; the time-varying formation parameters refer to the desired formation.
[0009] Based on the relevant parameters, two observers are designed to estimate the state and input of the unknown leader, respectively, ensuring that the observation is completed within the preset convergence time.
[0010] Design an internal mold dynamic compensator to suppress unknown external disturbances.
[0011] Based on the state estimation results of the unknown leader, the time-varying formation tracking error is determined.
[0012] Based on the parameters of the multi-agent system, the formation tracking compensation term is determined.
[0013] Based on the time-varying formation tracking error, an adaptive estimation update law is determined.
[0014] The final control law is determined based on the input estimation results of the unknown leader, the internal model dynamic compensator, the time-varying formation tracking error, the formation tracking compensation term, and the adaptive estimation update law.
[0015] Optionally, the multi-agent system includes: a follower dynamics model and a leader dynamics model.
[0016] The expression for the follower dynamics model is shown below.
[0017] .
[0018] .
[0019] .
[0020] The expression for the leader dynamics model is shown below.
[0021] .
[0022] .
[0023] in, For the first i One follower t The state vector at any given time; For control input; n The dimension of the state; m The dimension of the input; For the system matrix; The input matrix; This is an unknown external disturbance; For the first i An unknown disturbance; For constant bias and initial phase; The amplitude and frequency of the disturbance are both unknown. The dimension of the unknown perturbation; The leader's state vector; Unknown input; The matrix is an unknown constant matrix; p The dimension representing the unknown state; The basis function vectors are known.
[0024] Optionally, the expression for the state estimation equation of the unknown leader is as follows.
[0025] .
[0026] .
[0027] .
[0028] .
[0029] in, The state vector of the unknown leader; They are respectively The estimated value; The leader's state vector; The matrix is an unknown constant matrix; For the system matrix; The input matrix; n The dimension of the state; m The dimension of the input; p The dimension representing the unknown state; It is a symmetric positive definite matrix; This is due to local collaborative observation errors; For adaptive gain; Given a known basis function vector; It is the first positive design constant.
[0030] Optionally, the expression for the input estimation equation of the unknown leader is as follows.
[0031] .
[0032] .
[0033] .
[0034] in, The input vector is for an unknown leader; for The estimated value; The matrix is an unknown constant matrix; Given a known basis function vector; It is the second positive design constant; The input matrix; n The dimension of the state; m The dimension of the input; p The dimension representing the unknown state; It is a symmetric positive definite matrix; This is due to local collaborative observation errors; N The number of followers; For the first A follower can receive information from the leader; otherwise... ; For the first j The number of followers can accept the first i One follower information, otherwise .
[0035] Optionally, the expression for the internal mold dynamic compensator is as follows.
[0036] .
[0037] in, Indicates the state of the internal mold. express dimensionality; A controllable matrix pair selected by the user; This is the initial control law.
[0038] Optionally, the expression for the time-varying formation tracking error is as follows.
[0039] .
[0040] in, For time-varying formation tracking error based on observer results; The state vector of the unknown leader; Let the desired formation vector be . For the first i One follower t The state vector at time t.
[0041] Optionally, the expression for the formation tracking compensation term is as follows.
[0042] .
[0043] in, For formation tracking compensation items; For the system matrix; The input matrix; Let be the desired formation vector.
[0044] Optionally, the expression for the adaptive estimation update law is as follows.
[0045] .
[0046] .
[0047] .
[0048] .
[0049] in, To The estimate, These are the unknown parameters contained in the interference model; It is a positive definite diagonal matrix; The input matrix; For time-varying formation tracking error based on observer results; Let be the first function matrix. It is the independent variable of the first function; It is a linearized constant matrix; The state of a compensation system; Indicates the state of the internal mold. express dimensionality; Represents the Tracy-Singh product; A controllable matrix pair selected by the user; This is the second function matrix. It is the independent variable of the second function; For the followers to gather.
[0050] Secondly, this application provides a time-varying formation tracking system for multi-agent systems based on the internal model principle. The time-varying formation tracking system for multi-agent systems based on the internal model principle is used to implement the aforementioned time-varying formation tracking method for multi-agent systems based on the internal model principle. The time-varying formation tracking system for multi-agent systems based on the internal model principle includes the following modules.
[0051] An initialization module is used to initialize all relevant parameters, including: time-varying formation parameters, multi-agent system parameters, communication topology weight matrix, preset convergence time, observer parameters, controller parameters, and initial state; the time-varying formation parameters refer to the desired formation.
[0052] The observer design module is used to design two observers based on the relevant parameters, which are used to estimate the state and input of the unknown leader respectively, and ensure that the observation is completed within the preset convergence time.
[0053] The internal mold dynamic compensator design module is used to design an internal mold dynamic compensator, which is used to suppress unknown external disturbances.
[0054] The time-varying formation tracking error determination module is used to determine the time-varying formation tracking error based on the state estimation results of the unknown leader.
[0055] The formation tracking compensation term determination module is used to determine the formation tracking compensation term based on the parameters of the multi-agent system.
[0056] The adaptive estimation update law determination module is used to determine the adaptive estimation update law based on the time-varying formation tracking error.
[0057] The final control law determination module is used to determine the final control law based on the input estimation result of the unknown leader, the internal model dynamic compensator, the time-varying formation tracking error, the formation tracking compensation term, and the adaptive estimation update law.
[0058] Based on the specific embodiments provided in this application, the following technical effects are disclosed.
[0059] This application provides a time-varying formation tracking method and system for multi-agent systems based on the internal model principle. The method includes: initializing all relevant parameters, including time-varying formation parameters, multi-agent system parameters, communication topology weight matrix, preset convergence time, observer parameters, controller parameters, and initial state. The time-varying formation parameters refer to the desired formation. This provides a stable and standardized initial operating foundation for subsequent observer design, controller construction, and formation tracking implementation, ensuring the orderly start of system modeling and control processes. Based on these parameters, two observers are designed to estimate the state and input of the unknown leader, ensuring observation is completed within the preset convergence time. This allows for rapid and accurate estimation of the required values even when leader information is not directly available, effectively eliminating control obstacles caused by the unmeasurability of state and input. An internal model dynamic compensator is designed to suppress external unknown disturbances. This dynamic structural compensator can offset the adverse effects of unknown disturbances on the system's formation accuracy and stability, significantly improving the system's anti-interference capability. Based on the state estimation results of the unknown leader, the time-varying formation tracking error is determined; this enables accurate quantification of the deviation between the actual and desired formation, providing a reliable basis for subsequent closed-loop control and error correction. Based on the parameters of the multi-agent system, a formation tracking compensation term is determined; this can specifically compensate for the tracking deviation caused by the system's own dynamic characteristics, improving the adaptability and response accuracy of formation control. Based on the time-varying formation tracking error, an adaptive estimation update law is determined; this enables online dynamic adjustment and optimization of the estimation parameters, enhancing the system's adaptability to model uncertainties and communication changes. Based on the input estimation results of the unknown leader, the internal model dynamic compensator, the time-varying formation tracking error, the formation tracking compensation term, and the adaptive estimation update law, the final control law is determined; this organically integrates estimation, compensation, adaptive adjustment, and formation tracking, achieving closed-loop stable control and ensuring the reliable achievement of the time-varying formation tracking target. This method can achieve high-precision, highly stable, and highly disturbance-resistant time-varying formation tracking control of multi-agent systems within a preset time, even in the presence of unknown leaders, unknown external disturbances, and model uncertainties. It has the comprehensive advantages of accurate observation, strong adaptability, outstanding anti-interference ability, and fast control response. Attached Figure Description
[0060] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0061] Figure 1 This is an application environment diagram of a time-varying formation tracking method for a multi-agent system based on the internal model principle in one embodiment of this application.
[0062] Figure 2 This is a flowchart illustrating a time-varying formation tracking method for a multi-agent system based on the internal model principle, provided as an embodiment of this application.
[0063] Figure 3 This is a schematic diagram of the communication topology used in a simulation provided for an embodiment of this application.
[0064] Figure 4 This is a module architecture diagram of a time-varying formation tracking method for a multi-agent system based on the internal model principle, provided as an embodiment of this application.
[0065] Figure 5 The image shows the tracking trajectory of five intelligent agents within a time frame of 0s-60s, as provided in one embodiment of this application.
[0066] Figure 6 A simulation diagram of the change curve of follower formation tracking error within 0s-60s provided in an embodiment of this application.
[0067] Figure 7 A simulation diagram of the leader state estimation error change curve within the time period of 0s-60s provided in an embodiment of this application.
[0068] Figure 8 This is a schematic diagram of the functional modules of a time-varying formation tracking system for a multi-agent system based on the internal model principle, provided as an embodiment of this application. Detailed Implementation
[0069] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.
[0070] The key challenges of time-varying formation tracking can be summarized into three points: First, how to accurately estimate the preset time of non-autonomous leader states and unknown inputs without global topology information; second, how to effectively suppress complex unknown disturbances and ensure steady-state tracking accuracy without relying on upper bound information of disturbances; and third, how to design a fully distributed control protocol that balances system flexibility, robustness, and engineering feasibility to ensure stable achievement of time-varying formation goals in dynamic scenarios.
[0071] To make the above-mentioned objectives, features and advantages of this application more apparent and understandable, the application will be further described in detail below with reference to the accompanying drawings and specific embodiments.
[0072] The time-varying formation tracking method for multi-agent systems based on the internal model principle provided in this application can be applied to, for example... Figure 1 The application environment shown is illustrated. Terminal 102 communicates with server 104 via a network. A data storage system can store the data that server 104 needs to process. The data storage system can be set up independently, integrated into server 104, or placed in the cloud or on another server. Terminal 102 can send all relevant parameters to server 104. These parameters include: time-varying formation parameters, multi-agent system parameters, communication topology weight matrix, preset convergence time, observer parameters, controller parameters, and initial state. The time-varying formation parameters refer to the desired formation. After receiving all relevant parameters, server 104 initializes them. Based on these parameters, two observers are designed to estimate the state and input of the unknown leader, ensuring observation is completed within the preset convergence time. An internal model dynamic compensator is designed to suppress external unknown disturbances. Based on the state estimation result of the unknown leader, a time-varying formation tracking error is determined. Based on the multi-agent system parameters, a formation tracking compensation term is determined. Based on the time-varying formation tracking error, an adaptive estimation update law is determined. Based on the input estimation result of the unknown leader, the internal model dynamic compensator, the time-varying formation tracking error, the formation tracking compensation term, and the adaptive estimation update law, the final control law is determined. Server 104 can feed back the obtained final control law to terminal 102. Furthermore, in some embodiments, the multi-agent system time-varying formation tracking method based on the internal model principle can also be implemented separately by the server 104 or the terminal 102. For example, the terminal 102 can directly perform multi-agent system time-varying formation tracking based on the internal model principle for all relevant parameters, or the server 104 can obtain all relevant parameters from the data storage system and perform multi-agent system time-varying formation tracking based on the internal model principle for all relevant parameters.
[0073] The terminal 102 can be, but is not limited to, various desktop computers, laptops, smartphones, and tablets. Portable wearable devices can be smartwatches, smart bracelets, head-mounted devices, etc. The server 104 can be implemented using a standalone server or a server cluster composed of multiple servers, or it can be a cloud server.
[0074] In one exemplary embodiment, such as Figure 2 As shown, a time-varying formation tracking method for multi-agent systems based on the internal model principle is provided. This method is executed by a computer device, specifically by a terminal or server alone, or by both a terminal and a server. In this embodiment, the method is applied to... Figure 1Taking server 104 as an example, the following steps are included.
[0075] S1: Initialize all relevant parameters; the relevant parameters include: time-varying formation parameters, multi-agent system parameters, communication topology weight matrix, preset convergence time, observer parameters, controller parameters, and initial state; the time-varying formation parameters refer to the desired formation.
[0076] S2: Based on the relevant parameters, design two observers to estimate the state and input of the unknown leader respectively, ensuring that the observation is completed within the preset convergence time.
[0077] S3: Design an internal mold dynamic compensator, which is used to suppress unknown external disturbances.
[0078] S4: Determine the time-varying formation tracking error based on the state estimation results of the unknown leader.
[0079] S5: Determine the formation tracking compensation term based on the parameters of the multi-agent system.
[0080] S6: Based on the time-varying formation tracking error, determine the adaptive estimation update law.
[0081] S7: Determine the final control law based on the input estimation result of the unknown leader, the internal model dynamic compensator, the time-varying formation tracking error, the formation tracking compensation term, and the adaptive estimation update law.
[0082] By implementing steps S1 to S7 above, and by designing a distributed preset time observer and an internal model-based controller, time-varying formation tracking of a multi-agent system with unknown disturbances and a non-autonomous leader can be achieved.
[0083] The specific plan is as follows.
[0084] (a) System model definition.
[0085] Consider a multi-agent system consisting of one leader and N followers, denoted as . The dynamics models of each intelligent agent are defined as follows.
[0086] The expression for the follower dynamics model is shown below.
[0087] (1).
[0088] in, For the first i One follower t The state vector at any given time; For control input; n The dimension of the state; mThe dimension of the input; For the system matrix; Given an input matrix, (satisfying) ); This is due to an unknown external disturbance. The format is as follows.
[0089] (2).
[0090] in, For the first i An unknown disturbance; For constant bias and initial phase; The amplitude and frequency of the disturbance are both unknown. The dimension of the unknown perturbation.
[0091] The expression for the leader dynamics model is shown below.
[0092] (3).
[0093] in, The leader's state vector; Given unknown input, and It satisfies the following parameterization form.
[0094] (4).
[0095] in, This is an unknown constant matrix (which needs to be estimated later). p The dimension representing the unknown state; Given a known basis function vector, and there exists and Make .
[0096] in, It is a constant that is greater than zero and can take any value; I It is an identity matrix.
[0097] Define the communication topology.
[0098] An agent with no neighbors is called a leader agent; a follower agent has at least one neighbor. A graph containing N follower agents is... It is represented that the vertex set is , v i For the first i There are vertices; the edge set is . , e ij As vertices i and jThe edges; the weighted adjacency matrix is ,like From To obtain information, ,otherwise .
[0099] When a multi-agent system (MAS) includes a leader, the leader is labeled as agent 0. Accordingly, the communication graph is defined as follows: If there exists a leader agent 0 that points to a follower agent... i If there is a directed edge, then ,otherwise ,set up For the image The Laplace matrix.
[0100] The communication topology adopts a graph. This represents a spanning tree with the leader as the root node, and the topology among the followers is an undirected graph, whose Laplace matrix is shown below.
[0101] (5).
[0102] in, It is a symmetric positive definite matrix. , Indicates the first One follower can receive information from the leader; otherwise, the number is 0.
[0103] (ii) Design of distributed preset time observer.
[0104] Design two types of distributed adaptive observers: one for the leader's state and one for unknown inputs to the leader, to realize the leader state. and the preset time for the unknown input matrix. The estimates are as follows.
[0105] 1. Observer parameter definition: Set the preset convergence time. ( (For the initial excitation duration), define the time-varying gain scheduling parameters. Its definition and related parameter definitions are shown below.
[0106] (1) When hour, , , , , , .
[0107] (2) When hour, ,in ,in It is a constant; Represents the smallest eigenvalue of a matrix; arrive The parameters are known. express n Dimensional Array traces, .
[0108] In addition, a preset cutoff time is selected for specific engineering implementation. To avoid the problem of the gain tending to infinity in the actual solution, although this method introduces a final convergence estimation error, it can be adjusted... The value of is used to keep the estimation error within an acceptable range, and thus the new gain scheduling parameters can be obtained as shown below.
[0109] (6).
[0110] 2. Solve for the time-varying parameter matrix based on the time-varying parameter Lyapunov equation.
[0111] (7).
[0112] Solving for symmetric positive definite matrices This means that it can be solved get .
[0113] in, express n An identity matrix of dimension 1 This represents the solution obtained from the above equation.
[0114] 3. Observer design equations: Let They are respectively The estimated value, which means (Input estimation equation for unknown leader).
[0115] The observer equation (the state estimation equation for the unknown leader) is shown below.
[0116] (8).
[0117] (9).
[0118] (10).
[0119] (11).
[0120] in, The input vector is for an unknown leader; Given a basis function vector The state vector of the unknown leader; They are respectively The estimated value; The leader's state vector; The matrix is an unknown constant matrix; For the system matrix; The input matrix; n The dimension of the state; m The dimension of the input; p The dimension representing the unknown state; It is a symmetric positive definite matrix; This is due to local collaborative observation errors; For adaptive gain, its initial value is ; Given a known basis function vector; It is the first positive design constant.
[0121] (iii) The result obtained in step (ii) above , The input is fed into the time-varying formation tracking controller based on the internal model.
[0122] Based on the observer estimation results and the internal model principle, a distributed tracking controller is designed to achieve disturbance suppression and formation tracking. The steps are as follows.
[0123] 1. Design of internal mold dynamic compensator.
[0124] In response to external disturbances In the model Given ,and For any i and s There exists a positive constant. and a series of real numbers This makes the following formula true.
[0125] (12).
[0126] definition We can further obtain: .
[0127] in, .
[0128] It should be noted that col() and row() are shorthand forms for matrix permutations. Assuming there are k matrices... ,So , ; These are values defined in formula (2), and all of them exist in formula (2) using the summation symbol. These are arranged in a col pattern. ), the same m indivual It consists of .
[0129] Further definition and ,in For Hurwitz matrices, For any one of the controllable matrix pairs selected by the user. ,exist The Sylvester equation is satisfied as shown below.
[0130] (13).
[0131] Define coordinate transformation: And give the definition of relevant compact sets: , , , , Therefore, we can obtain the transformation: .
[0132] The internal mold compensator is defined as follows.
[0133] (14).
[0134] in, Indicates the state of the internal mold. express dimensionality ; A controllable matrix pair selected by the user; This is the initial control law.
[0135] 2. Error definition and coordinate transformation.
[0136] The tracking error is defined as follows.
[0137] (15).
[0138] in, For the first i One follower t The state vector at any given time; The leader's state vector; Represent the desired time-varying formation vector and design a compensation input. , yes The nominal value (that is, when) When it is known, there is ).
[0139] 3. Adaptive update law and control law.
[0140] The formation tracking compensation items are given below.
[0141] (16).
[0142] in, For formation tracking compensation items; For the system matrix; The input matrix; Let be the desired formation vector. It satisfies the formation feasibility condition. ,in , I m for m An identity matrix of dimension 1.
[0143] Furthermore, the adaptive estimation update law is given.
[0144] (17).
[0145] (18).
[0146] (19).
[0147] (20).
[0148] in, To The estimate, These are the unknown parameters contained in the interference model; It is a positive definite diagonal matrix; The input matrix; For time-varying formation tracking error based on observer results; Let be the first function matrix. It is the independent variable of the first function; It is a linearized constant matrix; The state of a compensation system; Indicates the state of the internal mold. express dimensionality; Represents the Tracy-Singh product; A controllable matrix pair selected by the user; This is the second function matrix. It is the independent variable of the second function; For the followers to gather.
[0149] Finally, the final control law is given below.
[0150] (twenty one).
[0151] (twenty two).
[0152] in, u i This is the final control law; K i It is a positive definite matrix; For time-varying formation tracking error based on observer results; Let be the first function matrix. It is the independent variable of the first function; To The estimate, These are the unknown parameters contained in the interference model; The input vector is for an unknown leader; For formation tracking compensation items; As an intermediate variable; For a given value; These are values obtained based on the Sylvester equation; A controllable matrix selected by the user; The input matrix; The state of a compensation system; Indicates the state of the internal mold. express The dimension of.
[0153] In summary, the solution proposed in this application can be summarized as follows:
[0154] 1. Initialization: Set the multi-agent system parameters (A, B), communication topology weight matrix G, and limit the convergence time. Observer parameters ( ), controller parameters ( ) and initial state ( ).
[0155] 2. Two observers are designed to estimate the state (Equation (8)) and input (Equation (10)) of the unknown leader, respectively, ensuring that the observers are within a preset time. T p The observation is completed within the time frame. An adaptive parameter needs to be introduced in formula (8). Therefore, there is an additional adaptive gain (Equation (9)) to be used in conjunction with Equation (8) for observation.
[0156] Furthermore, since the model of the leader has assumptions (see Equations (3)-(4) and the preceding and following text), the unknown inputs of the leader can be linearized, thereby allowing the observation of the unknown inputs of the leader to be linearized. The problem is equivalently transferred to observing the unknown parameters after linearization. Therefore, we have the observer formula (10).
[0157] 3. Before designing the control law, it is necessary to design an internal model dynamic compensator (i.e., formula (14)) to suppress external unknown disturbances. This compensator is equivalent to defining a variable. This variable will be used in the control law design later. Meanwhile, formula (15) represents the actual tracking error. However, due to the real Since it is unknown, it is necessary to borrow the unknown leader state observed in the observer. replace Construct a usable local tracking error (time-varying formation tracking error). This is so that it can be used in subsequent control laws.
[0158] 4. Design the control law as shown in formula (21). The control law uses... K i The gain matrix (positive definite matrix) that needs to be set. By using formula (15) Replaced with observer get( ); It is obtained from formula (17); Let be the first function matrix. It is the independent variable of the first function; The output of the observer (Equation (10)) combined with get, It is obtained through formula (16), , among them It is given. It is calculated using formula (13); Given by formula (19); Given by formula (14).
[0159] Specifically, the simulation parameter settings for this application are as follows.
[0160] 1. Consider a system consisting of five agents, with the communication topology as shown in the diagram. Figure 3 As shown, blue represents the leader and red represents the followers, corresponding to a second-order integrator model. The system matrix is: .
[0161] 2. The disturbance modeling and corresponding parameters are as follows: ,in, , , , , , , , , .
[0162] 3. The preset time observer parameters are selected as follows: The basis functions are chosen as follows: The preset time is selected as .
[0163] 4. The controller parameters are selected as follows: , .
[0164] And thus obtain , , , .
[0165] 5. Desired formation: , .
[0166] like Figure 3 As shown, the corresponding Laplace matrix can be obtained from this. Figure 4 This diagram illustrates the modular architecture of the method employed in this application. Each agent interacts with others through a communication network. Each agent observes the unknown inputs and states of the unknown leader within a preset time period. Based on the observation results, a controller based on the internal model principle is designed to mitigate disturbances. Figure 5 This represents the trajectory diagram of five intelligent agents in a formation tracking scenario from 0s to 60s. The leader follows a predetermined trajectory around a circle, while the other four followers are designed according to the method of this application, starting from any initial position, and eventually completing the formation tracking of the leader. Figure 6 This represents the formation tracking error of the follower to the leader during the formation tracking process. It can be seen that the formation tracking error converges to zero at about 34 seconds, indicating that the formation tracking objective has been achieved. Figure 7 This represents the estimation error of each follower agent regarding the leader's state within 0s-60s. It can be seen that before 30s, the state estimation error converges to 0, which matches the preset time, thus completing the state observer that converges within the preset time.
[0167] It should be noted that in the diagram, "leader" refers to the leader and "follower" refers to the follower.
[0168] This application achieves the following technical effects through the collaborative design of a distributed preset time observer and an inner membrane-based time-varying formation tracking controller.
[0169] 1. Solved the problem of pre-set time estimation for non-autonomous leaders; the observer can estimate the pre-set time. T p It accurately estimates the leader's state and unknown inputs, and the convergence time does not depend on the initial estimation error and design parameters, meeting the time constraints of actual tasks.
[0170] 2. Controllers based on the internal model principle do not require known upper bound information of external disturbances, and can effectively suppress unknown disturbances containing constant bias and harmonic components, thus achieving zero steady-state error tracking.
[0171] 3. The control protocol adopts a fully distributed design, which does not require global topology information and can complete control decisions only through local and neighbor node information, thus enhancing the system's flexibility and scalability.
[0172] This application effectively overcomes the shortcomings of existing multi-agent time-varying formation tracking methods in handling unknown inputs from non-autonomous leaders, external disturbances, and preset time estimations, significantly improving the system's tracking accuracy, robustness, and engineering practicality. It can be widely applied in fields such as UAV formation and autonomous robot collaboration.
[0173] Based on the same inventive concept, this application also provides a time-varying formation tracking system for multi-agent systems based on the internal model principle for implementing the aforementioned time-varying formation tracking method based on the internal model principle. The solution provided by this system is similar to the implementation scheme described in the above method. Therefore, the specific limitations of one or more embodiments of the time-varying formation tracking system for multi-agent systems based on the internal model principle provided below can be found in the limitations of the time-varying formation tracking method for multi-agent systems based on the internal model principle described above, and will not be repeated here.
[0174] In one exemplary embodiment, such as Figure 8 As shown, a time-varying formation tracking system for a multi-agent system based on the internal model principle is provided. The time-varying formation tracking system for a multi-agent system based on the internal model principle includes the following modules.
[0175] An initialization module is used to initialize all relevant parameters, including: time-varying formation parameters, multi-agent system parameters, communication topology weight matrix, preset convergence time, observer parameters, controller parameters, and initial state; the time-varying formation parameters refer to the desired formation.
[0176] The observer design module is used to design two observers based on the relevant parameters, which are used to estimate the state and input of the unknown leader respectively, and ensure that the observation is completed within the preset convergence time.
[0177] The internal mold dynamic compensator design module is used to design an internal mold dynamic compensator, which is used to suppress unknown external disturbances.
[0178] The time-varying formation tracking error determination module is used to determine the time-varying formation tracking error based on the state estimation results of the unknown leader.
[0179] The formation tracking compensation term determination module is used to determine the formation tracking compensation term based on the parameters of the multi-agent system.
[0180] The adaptive estimation update law determination module is used to determine the adaptive estimation update law based on the time-varying formation tracking error.
[0181] The final control law determination module is used to determine the final control law based on the input estimation result of the unknown leader, the internal model dynamic compensator, the time-varying formation tracking error, the formation tracking compensation term, and the adaptive estimation update law.
[0182] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.
[0183] This document uses specific examples to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the methods and core ideas of this application. Furthermore, those skilled in the art will recognize that, based on the ideas of this application, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of this application.
Claims
1. A time-varying formation tracking method for multi-agent systems based on the internal model principle, characterized in that, The time-varying formation tracking method for multi-agent systems based on the internal model principle includes: Initialize all relevant parameters; the relevant parameters include: time-varying formation parameters, multi-agent system parameters, communication topology weight matrix, preset convergence time, observer parameters, controller parameters, and initial state; the time-varying formation parameters refer to the desired formation. Based on the relevant parameters, two observers are designed to estimate the state and input of the unknown leader respectively, ensuring that the observation is completed within the preset convergence time. Design an internal mold dynamic compensator to suppress unknown external disturbances; Based on the state estimation results of the unknown leader, the time-varying formation tracking error is determined; Based on the parameters of the multi-agent system, the formation tracking compensation term is determined; Based on the time-varying formation tracking error, an adaptive estimation update law is determined; The final control law is determined based on the input estimation results of the unknown leader, the internal model dynamic compensator, the time-varying formation tracking error, the formation tracking compensation term, and the adaptive estimation update law.
2. The time-varying formation tracking method for multi-agent systems based on the internal model principle according to claim 1, characterized in that, The multi-agent system includes: a follower dynamics model and a leader dynamics model; The expression for the follower dynamics model is: ； ； ； The expression for the leader dynamics model is: ； ； in, For the first i One follower t The state vector at any given time; For control input; n The dimension of the state; m The dimension of the input; For the system matrix; The input matrix; This is an unknown external disturbance; For the first i An unknown disturbance; For constant bias and initial phase; The amplitude and frequency of the disturbance are both unknown. The dimension of the unknown perturbation; The leader's state vector; Unknown input; The matrix is an unknown constant matrix; p The dimension representing the unknown state; The basis function vectors are known.
3. The time-varying formation tracking method for multi-agent systems based on the internal model principle according to claim 1, characterized in that, The expression for the state estimation equation of the unknown leader is: ； ； ； ； in, The state vector of the unknown leader; They are respectively The estimated value; The leader's state vector; The matrix is an unknown constant matrix; For the system matrix; The input matrix; n The dimension of the state; m The dimension of the input; p The dimension representing the unknown state; It is a symmetric positive definite matrix; This is due to local collaborative observation errors; For adaptive gain; Given a known basis function vector; It is the first positive design constant.
4. The time-varying formation tracking method for multi-agent systems based on the internal model principle according to claim 1, characterized in that, The expression for the input estimation equation of the unknown leader is: ； ； ； in, The input vector is for an unknown leader; for The estimated value; The matrix is an unknown constant matrix; Given a known basis function vector; It is the second positive design constant; The input matrix; n The dimension of the state; m The dimension of the input; p The dimension representing the unknown state; It is a symmetric positive definite matrix; This is due to local collaborative observation errors; N The number of followers; For the first A follower can receive information from the leader; otherwise... ; For the first j The number of followers can accept the first i One follower information, otherwise .
5. The time-varying formation tracking method for multi-agent systems based on the internal model principle according to claim 1, characterized in that, The expression for the internal mold dynamic compensator is: ； in, Indicates the state of the internal mold. express dimensionality; A controllable matrix pair selected by the user; This is the initial control law.
6. The time-varying formation tracking method for multi-agent systems based on the internal model principle according to claim 1, characterized in that, The expression for the time-varying formation tracking error is: ； in, For time-varying formation tracking error based on observer results; The state vector of the unknown leader; Let the desired formation vector be . For the first i One follower t The state vector at time t.
7. The time-varying formation tracking method for multi-agent systems based on the internal model principle according to claim 1, characterized in that, The expression for the formation tracking compensation term is: ； in, For formation tracking compensation items; For the system matrix; The input matrix; Let be the desired formation vector.
8. The time-varying formation tracking method for multi-agent systems based on the internal model principle according to claim 1, characterized in that, The expression for the adaptive estimation update law is: ； ； ； ； in, To The estimate, These are the unknown parameters contained in the interference model; It is a positive definite diagonal matrix; The input matrix; For time-varying formation tracking error based on observer results; Let be the first function matrix. It is the independent variable of the first function; It is a linearized constant matrix; The state of a compensation system; Indicates the state of the internal mold. express dimensionality; Represents the Tracy-Singh product; A controllable matrix pair selected by the user; This is the second function matrix. It is the independent variable of the second function; For the followers to gather.
9. The time-varying formation tracking method for multi-agent systems based on the internal model principle according to claim 1, characterized in that, The expression for the final control law is: ； ； in, u i This is the final control law; K i It is a positive definite matrix; For time-varying formation tracking error based on observer results; Let be the first function matrix. It is the independent variable of the first function; To The estimate, These are the unknown parameters contained in the interference model; The input vector is for an unknown leader; For formation tracking compensation items; As an intermediate variable; For a given value; These are values obtained based on the Sylvester equation; A controllable matrix selected by the user; The input matrix; The state of a compensation system; Indicates the state of the internal mold. express The dimension of.
10. A time-varying formation tracking system for multi-agent systems based on the internal model principle, characterized in that, The time-varying formation tracking system for multi-agent systems based on the intrinsic model principle is used to implement the time-varying formation tracking method for multi-agent systems based on the intrinsic model principle as described in any one of claims 1-9. The time-varying formation tracking system for multi-agent systems based on the intrinsic model principle includes: An initialization module is used to initialize all relevant parameters, including: time-varying formation parameters, multi-agent system parameters, communication topology weight matrix, preset convergence time, observer parameters, controller parameters, and initial state; the time-varying formation parameters refer to the desired formation. The observer design module is used to design two observers based on the relevant parameters, which are used to estimate the state and input of the unknown leader respectively, and to ensure that the observation is completed within the preset convergence time. The internal mold dynamic compensator design module is used to design an internal mold dynamic compensator, which is used to suppress unknown external disturbances. The time-varying formation tracking error determination module is used to determine the time-varying formation tracking error based on the state estimation results of the unknown leader. The formation tracking compensation term determination module is used to determine the formation tracking compensation term based on the parameters of the multi-agent system. The adaptive estimation update law determination module is used to determine the adaptive estimation update law based on the time-varying formation tracking error. The final control law determination module is used to determine the final control law based on the input estimation result of the unknown leader, the internal model dynamic compensator, the time-varying formation tracking error, the formation tracking compensation term, and the adaptive estimation update law.