A formation control method and system based on ODE-PDE

Abstract

The present invention discloses a formation control method and system based on ODE-PDE, which belongs to the field of multi-agent cluster control, including: establishing a heterogeneous multi-agent system model of a control system to be controlled, and calculating the ODE tracking error system model of each agent, and obtaining the ODE-PDE error system model shared by sparse-dense distributed agents through partial continuous processing. At any time, the nonlinear function of the sparsely distributed agent is fitted by a first RBF neural network and the control signal is calculated, which is substituted into the corresponding ODE tracking error system model, and the nonlinear function of the densely distributed agent is fitted by a second RBF neural network and the control signal is calculated, which is substituted into the corresponding PDE tracking error system model. If the ODE-PDE tracking error system converges, the formation control ends; otherwise, the control signal continues to be updated and the entity object performs the corresponding action. The present invention can effectively utilize the collaborative control of the multi-agent system to realize the formation control of systems of any scale and any distribution.

Description

Technical Field

[0001] The present invention belongs to the field of multi-agent cluster control, and more specifically, relates to a formation control method and system based on ODE-PDE. Background Art

[0002] In recent years, with the rapid development of communication technology and artificial intelligence technology, the cooperative control of multi-agent systems has attracted widespread attention due to its application prospects in the fields of drone swarms, intelligent transportation, and microgrid groups. Formation control refers to the movement of each entity object in the control system from the initial position to the specified target position or to form a specified formation. It is a typical cooperative control problem of multi-agent systems. The increasing complexity of actual needs has led to the inevitable growth of the scale of multi-agent systems. In order to apply the agent system to solve the control problem of the actual cluster system, it is necessary to first establish a multi-agent system model corresponding to the cluster system. In the multi-agent system model, each agent corresponds to an entity object in the cluster system. For example, in the multi-agent system corresponding to the drone swarm, each agent corresponds to a drone; in the multi-agent system corresponding to the intelligent transportation system, each agent corresponds to an unmanned vehicle or a traffic light; in the multi-agent system corresponding to the microgrid group, each agent corresponds to a microgrid. After establishing the corresponding multi-agent system model, the subsequent control can be carried out with the help of the relevant theoretical methods of the multi-agent system.

[0003] At present, most of the research on multi-agent collaborative control adopts the ODE (Ordinary Differential Equation) method. This method requires the establishment of an ODE model for each agent to describe the dynamic behavior of each agent. When this method is used for large-scale multi-agent systems, the established ODE model is high-dimensional and complex, which brings huge challenges to the related stability analysis and controller design.

[0004] To solve this problem, some scholars have recently proposed a collaborative control method for large-scale multi-agent systems based on PDE (Partial Differential Equation). Compared with the ODE-based method, the main advantage of the PDE method is that it can effectively represent the collective dynamics of large-scale multi-agent systems using a single PDE, and can always maintain consistent analytical complexity regardless of how the number of agents changes.

[0005] At present, the PDE method is only applicable to scenarios where there are enough agents in a limited space, that is, the agents in the space must be densely distributed. However, the distribution forms of actual multi-agent systems are diverse, that is, some agents are densely distributed, while others are likely to be sparsely distributed, so the scope of application of the existing PDE method is very limited.

[0006] In addition, actual multi-agent systems often face the following three problems: (1) Actual systems often require that the time to achieve control goals is limited, and collaborative control tasks need to be completed quickly; (2) Due to the limitations of software and hardware facilities, the communication topology cannot be guaranteed to remain constant; (3) Due to the complexity of the agents themselves, the nonlinearity of actual agents is often complex and unknown. However, these three factors have not been considered in existing research on multi-agent systems based on PDE, resulting in theoretical results that are too idealized and not very practical.

[0007] In general, how to effectively utilize the collaborative control of multi-agent systems to achieve formation control of systems of any scale and distribution is an urgent problem to be solved. Summary of the invention

[0008] In view of the defects of the prior art and the need for improvement, the present invention provides a formation control method and system based on ODE-PDE, the purpose of which is to effectively utilize the collaborative control of multi-agent systems to achieve formation control of systems of any scale and any distribution.

[0009] To achieve the above object, according to one aspect of the present invention, a formation control method based on ODE-PDE is provided, comprising:

[0010] Establish a heterogeneous multi-agent system model corresponding to the control system to be controlled; in the heterogeneous multi-agent system model, each agent corresponds to an entity object in the control system to be controlled;

[0011] Based on the ODE model, the dynamic equations of each agent in the heterogeneous multi-agent system model are established to describe the relationship between the speed of the agent, the communication protocol between agents, and the nonlinear function value of the agent's position; the communication protocol is related to the control signal used to control the speed change; in the heterogeneous multi-agent system model, the nonlinear functions corresponding to sparsely distributed agents and densely distributed agents are different;

[0012] Based on the dynamic equations of each agent, the ODE tracking error system model of the actual position of each agent relative to the target position is obtained, and the ODE tracking error system model corresponding to all densely distributed agents is processed continuously to obtain the ODE error system model corresponding to each sparsely distributed agent and the PDE error system model shared by the densely distributed agents;

[0013] In any t At this moment, follow the steps below to control the formation:

[0014] Step S1: For each sparsely distributed agent, assign its position m i ( t) is input into the first RBF neural network to obtain a univariate nonlinear function value f i ( m i ( t )), and calculate the control signal used to control its speed change as: ;

[0015] For each densely distributed agent, its position s ( z , t ) is input into the second RBF neural network to obtain the binary nonlinear function value g ( s ( z , t )), and calculate the control signal used to control its speed change as: ;

[0016] Step S2: For each sparsely distributed agent, the corresponding univariate nonlinear function value f i ( m i ( t )) and control signals Substitute the corresponding ODE error system model to calculate the corresponding tracking error; for each densely distributed agent, the corresponding binary nonlinear function value g ( s ( z , t )) and control signals Substitute the PDE error system model shared by densely distributed agents to calculate the corresponding tracking error;

[0017] Step S3: Determine whether the tracking errors corresponding to all agents have converged. If so, the formation control ends; if not, continue to update the control signal to each agent so that the tracking errors corresponding to all agents converge and the physical object performs the corresponding action;

[0018] in, represents the relative positions of densely distributed agents, l represents the distribution range of densely distributed agents; is the weight vector of the first RBF neural network; the second RBF neural network is a binary RBF neural network, and is the binary weight vector of the second RBF neural network; and are all known continuous RBF vectors.

[0019] Furthermore, in the heterogeneous multi-agent system model, the change in signal strength when sparsely distributed agents communicate with other agents is characterized by the first semi-Markov process, and the change in signal strength when densely distributed agents communicate with other agents is characterized by the second semi-Markov process; the first semi-Markov process and the second semi-Markov process are independent of each other, and in the first semi-Markov process and the second semi-Markov process, each mode corresponds to a signal strength.

[0020] Furthermore, the communication protocol between sparsely distributed agents and other agents u i ( t )for:

[0021]

[0022] Communication protocols between densely distributed agents and other agents c i ( t )for:

[0023] ;

[0024] in, i represents the agent number, represents the number of sparsely distributed agents, represents the number of densely distributed agents, n represents the number of sparsely distributed agents in the heterogeneous multi-agent system model, v Represents the heterogeneous number of densely distributed agents in a heterogeneous multi-agent system model; m k ( t ) indicates that the sparsely distributed agents are t The location at the moment; represents densely distributed agents in t The location at the moment; represents the distribution interval of densely distributed agents; represents the target position of the sparsely distributed agent, Represents the target location of sparsely distributed agents; represents densely distributed agents in t The control signal at the moment, represents densely distributed agents in t Control signal at every moment; represents the number of sparsely distributed agents, Indicates the number of densely distributed agents; , , and are all semi-Markov switching topology weights, and ; Indicates communication delay, ,and The rate of change satisfies , represents the maximum communication delay, Indicates the maximum rate of change of communication delay.

[0025] Furthermore, the ODE error system model corresponding to the sparsely distributed agent is:

[0026] ;

[0027] The ODE error system model corresponding to densely distributed agents is:

[0028] ;

[0029] in, express t The tracking error of the sparsely distributed agent at all times, , express The rate of change of express t Tracking error of densely distributed agents at all times, , express The rate of change of

[0030] After the ODE error system models corresponding to all densely distributed agents are processed continuously,

[0031] The ODE error system model corresponding to each sparsely distributed agent and the PDE error system model shared by the densely distributed agents are:

[0032] ;

[0033] in, express t Time and location are z Tracking error of densely distributed agents; , , ; By setting two anchor points at both ends of the densely distributed agents, the boundary conditions of the PDE error system model shared by densely distributed agents can be described as: .

[0034] Further, and All are updated according to the adaptive update rate based on the projection rule;

[0035] Adaptive update rate and Adaptive update rate They are:

[0036] ;

[0037] ;

[0038] in,

[0039] ,

[0040] ,

[0041] ,

[0042] ,

[0043] in, , , , , , , 、 , , All represent neural network parameters; , Represents the ideal weight vector of the first RBF neural network; , Represents the ideal binary weight vector of the second RBF neural network.

[0044] Further, , , , , and As the controller gain, the controller gain is determined by:

[0045] Construct the following linear matrix inequality that satisfies finite-time stability:

[0046] ;

[0047] If there is a positive scalar , , and there exists a positive definite constant , , , , , , negative definite constant , and any scalar and , , so that for any , and , the above linear matrix inequalities all hold true, then , , , , and The value of is taken as the corresponding controller gain;

[0048] is an upper triangular matrix, and the expressions of the non-zero elements are: , , , , , , , , , , , , , , , , , , , , ; , represents the semi-Markov switching topology weight; , represents the transition rate between modes in the first semi-Markov process, express The mathematical expectation of , represents the transition rate between modes in the second semi-Markov process, express The mathematical expectation of and denote the mode sets of the first and second semi-Markov processes, respectively. M 1 and M 2 represents the total number of modes of the first and second semi-Markov processes, respectively.

[0049] According to another aspect of the present invention, there is provided a formation control system based on ODE-PDE, comprising: an initialization module, an ODE model building module, an ODE-PDE error model building module, a univariate nonlinear function approximator, a binary nonlinear function approximator, a first neural adaptive controller, a second neural adaptive controller, a first error tracking module, a second error tracking module and a control module;

[0050] An initialization module is used to establish a heterogeneous multi-agent system model corresponding to the control system to be controlled; in the heterogeneous multi-agent system model, each agent corresponds to an entity object in the control system to be controlled;

[0051] The ODE model building module is used to build the dynamic equations of each agent in the heterogeneous multi-agent system model based on the ODE model, which is used to describe the relationship between the speed of the agent, the communication protocol between agents, and the nonlinear function value of the agent's position; the communication protocol is related to the control signal used to control the speed change; in the heterogeneous multi-agent system model, the nonlinear functions corresponding to sparsely distributed agents and densely distributed agents are different;

[0052] The ODE-PDE error model building module is used to obtain the ODE tracking error system model of the actual position of each intelligent agent relative to the target position based on the dynamic equation of each intelligent agent, and to perform continuous processing on the ODE tracking error system model corresponding to all densely distributed intelligent agents to obtain the ODE error system model corresponding to each sparsely distributed intelligent agent and the PDE error system model shared by the densely distributed intelligent agents;

[0053] A univariate nonlinear function approximator for t At this moment, for each sparsely distributed agent, its position m i ( t ) is input into the first RBF neural network to obtain a univariate nonlinear function value f i ( m i ( t ));

[0054] A binary nonlinear function approximator for t At this moment, for each densely distributed agent, its position s ( z , t ) is input into the second RBF neural network to obtain the binary nonlinear function value g ( s ( z , t ));

[0055] The first neural adaptive controller for t At this moment, for each sparsely distributed agent, the control signal used to control its speed change is calculated as: ;

[0056] The second neural adaptive controller is used to t At this moment, for each densely distributed agent, the control signal used to control its speed change is calculated as: ;

[0057] The first error tracking module is used to track the corresponding univariate nonlinear function value of each sparsely distributed agent. f i ( m i ( t )) and control signals Substitute it into the corresponding ODE error system model to calculate the corresponding tracking error;

[0058] The second error tracking module is used to track the corresponding binary nonlinear function value of each densely distributed agent. g ( s ( z , t )) and control signals Substitute the PDE error system model shared by densely distributed agents to calculate the corresponding tracking error;

[0059] Control module for t Always judge whether the tracking errors corresponding to all agents have converged. If so, end the formation control; if not, continue to update the control signal for each agent so that the tracking errors corresponding to all agents converge and the physical objects perform the corresponding actions;

[0060] in, represents the relative positions of densely distributed agents, l represents the distribution range of densely distributed agents; the weight vector of the first RBF neural network; the second RBF neural network is a binary RBF neural network, and is the binary weight vector of the second RBF neural network; and are all known continuous RBF vectors.

[0061] According to another aspect of the present invention, a computer program product is provided, including a computer program; when the computer program is executed by a processor, the above-mentioned ODE-PDE-based formation control method provided by the present invention is implemented.

[0062] According to another aspect of the present invention, a computer-readable storage medium is provided, comprising a stored computer program; when the computer program is executed by a processor, the device where the computer-readable storage medium is located is controlled to execute the above-mentioned ODE-PDE-based formation control method provided by the present invention.

[0063] According to another aspect of the present invention, there is provided an electronic device, comprising:

[0064] A computer-readable storage medium for storing a computer program;

[0065] and a processor, configured to read a computer program in a computer-readable storage medium and execute the above-mentioned ODE-PDE-based formation control method provided by the present invention.

[0066] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects:

[0067] (1) The ODE-PDE-based formation control method provided by the present invention, after establishing the heterogeneous multi-agent system model corresponding to the control system, first establishes the ODE model corresponding to each agent and then establishes the ODE error system model for reflecting the tracking error change rate of each agent based on the model. Then, by continuously processing the ODE error system models corresponding to all densely distributed agents, a PDE error system model shared by the densely distributed agents is obtained. Subsequent formation control can be achieved based on the corresponding error system model. In the present invention, each sparsely distributed agent corresponds to an ODE error system model, and all densely distributed agents share a PDE error system model. Thus, a new ODE-PDE analysis method is proposed. On the one hand, compared with the commonly used ODE method, since the number of sparsely distributed agents in the multi-agent system is often small, the present invention can accurately characterize the collective dynamics of a large-scale multi-agent system with one PDE error system model and a few ODE error system models, which has obvious advantages in terms of simplicity of analysis and can also perform more effective control on large-scale systems. On the other hand, compared with the existing PDE method, the present invention is applicable to the situation where the agents are sparsely / densely distributed, so the present invention has a wider applicability. In general, the present invention can effectively utilize the cooperative control of multi-agent systems to realize the formation control of systems of any scale and any distribution.

[0068] (2) In the heterogeneous multi-agent system model established by the present invention, the nonlinear functions corresponding to sparsely distributed agents and densely distributed agents are different, which fully considers the heterogeneity of the system. For sparsely distributed agents and densely distributed agents, a univariate RBF neural network and a binary RBF neural network are used to approximate the corresponding nonlinear functions, respectively. The RBF neural network approximates the target curve through multiple RBFs. When the nonlinearity of the agent is complex and unknown, the corresponding nonlinear function value can be accurately obtained. In addition, the present invention directly approximates the binary nonlinear function by constructing a binary basis function, which effectively reduces the accuracy loss in the process of the existing RBF neural network approximating the binary nonlinear function.

[0069] (3) In the preferred embodiment of the present invention, two independent semi-Markov processes are used to respectively characterize the changes in signal strength between sparsely distributed agents and densely distributed agents in communication with other agents. The semi-Markov switching rules will vary according to different agent topological structures, thereby being able to describe the inconsistent topological switching phenomenon in heterogeneous multi-agent systems and further improve the practicality of formation control.

[0070] (4) In the preferred embodiment of the present invention, an adaptive update rate based on the projection rule is designed for the weight vector of the RBF neural network on which the control signal calculation depends, thereby realizing a new neural adaptive control scheme based on the projection rule, and constructing a linear matrix inequality that can realize the finite-time stability of the error system. Based on the inequality, the relevant controller gain is determined, thereby realizing the actual finite-time stability of the error system, which can better meet the timeliness requirements of the actual system and further improve the practicality of formation control. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 A block diagram of a formation control method based on ODE-PDE provided in an embodiment of the present invention.

[0072] Figure 2 The semi-Markov process provided by the embodiment of the present invention and Controlled mode switching process.

[0073] Figure 3 The invention provides an approximate error trajectory of an unknown nonlinear function approximated by a RBF neural network.

[0074] Figure 4 The running trajectory of an intelligent agent in a three-dimensional space provided by an embodiment of the present invention.

[0075] Figure 5 An error trajectory of the intelligent agent position tracking in a single spatial dimension is provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0076] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0077] In the present invention, the terms "first", "second", etc. (if any) in the present invention and the drawings are used to distinguish similar objects but not necessarily to describe a specific order or sequence.

[0078] In order to solve the problem that the existing ODE-based control method has high dimensions and complexity when facing large-scale multi-agent systems, and the stability analysis and controller design are difficult, and the existing PDE-based control method is only applicable to densely distributed multi-agent systems and has limited applicability, the present invention provides a formation control method and system based on ODE-PDE. The overall idea is to fully consider the actual system corresponding to the multi-agent system. The agents often conform to the sparse / dense mixed distribution situation, and the number of sparsely distributed agents is small. For each sparsely distributed agent, an ODE error system model is established, and for all densely distributed agents, a shared PDE error system model is established. Thus, a PDE error system model and a few ODE error system models can be used to accurately characterize the collective dynamics of large-scale multi-agent systems, reducing the difficulty of analysis while improving applicability. On this basis, the finite-time formation control problem of multi-agent systems is further studied, the asynchronous semi-Markov switching phenomenon of topological weights is considered, and a neural adaptive control strategy is designed to solve the problems faced by multi-agent systems in practical applications and improve the practicality of formation control.

[0079] The ODE-PDE-based formation control method and system provided by the present invention can be applied to the formation control of any entity system such as drone swarms, intelligent transportation, microgrid groups, etc. Without loss of generality, in the following embodiments, the formation control of drone swarms is taken as an example for explanation. It is easy to understand that in order to realize the formation control of drone swarms, it is necessary to establish a heterogeneous multi-agent system model corresponding to the actual drone swarm to be controlled, in which each agent corresponds to a drone entity object, and the number of sparsely distributed agents and the number of densely distributed agents are respectively recorded as n and v , and define and are the sequence sets of sparsely distributed agents and densely distributed agents respectively.

[0080] gather The initial positions of the sparsely distributed agents in are expressed as , and the target position is represented as .gather The densely distributed agents in the space are initially distributed along a On the curve: or , l represents the distribution range of densely distributed agents; the initial position of each agent is represented by ,in, represents the distribution interval of densely distributed agents; the target positions of densely distributed agents are distributed in another space On the curve: or , the target position of each densely distributed agent is expressed as .

[0081] like Figure 1 As shown, after establishing the above heterogeneous multi-agent system model, the following operations are further included:

[0082] The initial ODE model of the heterogeneous multi-agent system model is established, that is, the dynamic equations of each agent on each coordinate axis are established based on the ODE model as follows:

[0083] ;

[0084] Among them, the superscript Indicates the serial number of the spatial coordinate axis. When considering two-dimensional space, , when considering three-dimensional space, ; Subscript i Indicates the corresponding agent number; and Represents sparsely distributed agents and densely distributed agents in t The absolute position at the moment; and are the communication protocols that need to be designed, representing the communication protocols between sparsely distributed agents and other agents, and the communication protocols between densely distributed agents and other agents; represents the nonlinear function corresponding to the sparsely distributed agent, Represents the nonlinear function corresponding to densely distributed agents.

[0085] Since the dynamics in each dimension are expressed in the same way, we can only study the dynamics in one dimension, and then obtain the dynamic equation of the entire intelligent agent as follows:

[0086] ;

[0087] The above dynamic equations describe the relationship between the speed of the agent, the communication protocol between the agents, and the nonlinear function value of the agent's position.

[0088] When considering the communication topology between agents, existing methods often only consider constant topology or consistent switching topology, which is inconsistent with the actual situation. In response to this problem, as a preferred implementation, the present invention uses two independent semi-Markov processes to respectively characterize the changes in signal strength between sparsely distributed agents and densely distributed agents and other agents. On the one hand, semi-Markov has a time-varying residence time, which is more flexible and more applicable; on the other hand, the semi-Markov switching rules will change according to different agent topological structures, thereby effectively describing the inconsistent topological switching phenomenon in the heterogeneous multi-agent system model, and effectively improving the practicality of formation control.

[0089] Specifically, in the present invention, the first semi-Markov process is used to characterize the signal strength variation of sparsely distributed intelligent agents communicating with other intelligent agents, and the second semi-Markov process is used to characterize the signal strength variation of densely distributed intelligent agents communicating with other intelligent agents.

[0090] by and Denote the first half Markov process and the second half Markov process respectively, then the first half Markov process and the second half Markov process can be defined as:

[0091] ;

[0092] ;

[0093] Among them, ":=" means the definition is, M 1 and M 2 represents the number of modes in the first and second half Markov processes, respectively. and They represent the mode sets of the first semi-Markov process and the second semi-Markov process respectively. Each mode corresponds to a signal strength. The residence time distribution of each mode can be set according to the actual communication situation, and the transfer rate is determined accordingly. The transfer rates of the first semi-Markov process and the second semi-Markov process are respectively expressed as and .

[0094] On this basis, the present invention designs a communication protocol for the agent under the asynchronous switching of semi-Markov topology, and a communication protocol between sparsely distributed agents and other agents. u i ( t) and communication protocols between densely distributed agents and other agents c i ( t ) are:

[0095] ;

[0096] ;

[0097] in, , , and are all semi-Markov switching topology weights, and ; Indicates communication delay, ,and The rate of change satisfies , represents the maximum communication delay, Indicates the maximum rate of change of communication delay. and They respectively represent the control signal for controlling the speed change of sparsely distributed agents and the control signal for controlling the speed change of densely distributed agents.

[0098] In the above communication protocol, for sparsely distributed agents, not only the location information of the sparsely distributed agents themselves is considered, but also the influence of the location information of each densely distributed agent on the location information of the current sparsely distributed agent is considered, and the communication delay between the sparsely distributed agents and the densely distributed agents is considered. For densely distributed agents, the location information of the densely distributed agent itself is considered, as well as the influence of the location information of adjacent densely distributed agents and sparsely distributed agents on the location information of the current densely distributed agent, and the communication delay between sparsely distributed agents and densely distributed agents is considered. ; The corresponding communication protocol is consistent with the actual communication situation between entity objects in the system.

[0099] Based on the established ODE model, the tracking error of the actual position of each agent at each time relative to the target position can be determined. For sparsely distributed agents, the tracking error is: ; For densely distributed agents, the tracking error is: ; After substituting the designed communication protocol into the corresponding ODE model, the change rate of the tracking error is calculated as the ODE error system model corresponding to each agent. The ODE error system model corresponding to the sparsely distributed agent and the ODE error system model corresponding to the densely distributed agent are respectively:

[0100] ;

[0101] After the ODE error system models corresponding to all densely distributed agents are processed continuously, the ODE error system models corresponding to each sparsely distributed agent and the PDE error system model shared by the densely distributed agents are obtained as follows:

[0102] ;

[0103] The ODE error system model corresponding to the sparsely distributed agents and the PDE error system model shared by the densely distributed agents are both related to nonlinear functions and control signals, where: represents the relative positions of densely distributed agents, l represents the distribution range of densely distributed agents; express t Time and location are z Tracking error of densely distributed agents; , , ; By setting two anchor points at both ends of the densely distributed agents, the boundary conditions of the PDE error system model shared by densely distributed agents can be described as: By setting this boundary condition, the calculation can be simplified and the smoothness of the endpoints of the curve where the densely distributed agents are located can be guaranteed, which is consistent with the actual situation that there are other agents before the first densely distributed agent and there are other agents after the last densely distributed agent.

[0104] In order to use the above error system model to determine the tracking error of each intelligent agent, it is also necessary to determine the corresponding nonlinear function value. Considering that the nonlinearity of the actual intelligent agent is often complex and unknown due to the complexity of the intelligent agent itself, the present invention uses the RBF neural network as a nonlinear function approximator to complete the approximation of the unknown nonlinear function;

[0105] for ,definition The weight vector of its RBF neural network is, and its ideal value is , which can be gradually adjusted and optimized through the training of the neural network, represents the number of neurons, is a known continuous RBF vector. Then It can be expressed as;

[0106] ;

[0107] in, represents the approximation error and satisfies , It represents the upper limit of the square of the approximation error. Definition ,but It can be further expressed as,

[0108] ;

[0109] RBF neural networks are mostly used for approximation of univariate nonlinear functions. In the present invention, in order to adapt to the continuous processing of the error system model, the position information of the densely distributed intelligent agent will also be converted to , accordingly, the nonlinear function is a binary nonlinear function; for a binary nonlinear function, the traditional method often converts it into a univariate nonlinear function first, and then uses the RBF neural network to approximate it, but this will cause a certain loss of accuracy. To address this problem, in the present invention, for the nonlinear function ,definition is the binary weight vector of the RBF neural network, and its ideal value is , can be gradually adjusted and optimized through neural network training, represents the number of neurons, is a known continuous RBF vector. Then It can be expressed as;

[0110] ;

[0111] in, represents the approximation error and satisfies , Represents a known constant. Definition ,but It can be further expressed as;

[0112] ;

[0113] The above-mentioned one-element RBF neural network and two-element RBF neural network are respectively recorded as the first RBF neural network and the second RBF neural network. t At this moment, for each sparsely distributed agent, its position m i ( t ) is input into the first RBF neural network to obtain a univariate nonlinear function value f i ( m i ( t )); For each densely distributed agent, its position s ( z , t ) is input into the second RBF neural network to obtain the binary nonlinear function value g ( s ( z ,t )).

[0114] In order to determine the control signal for controlling the speed change in the communication protocol, the present invention further considers the communication delay and designs an adaptive neural network controller based on the projection rule. Specifically, based on the designed adaptive neural network controller, the control signal for controlling the speed change of the sparsely distributed agent is: ; The control signal used to control the speed change of densely distributed agents is: ;

[0115] in, and All are updated according to the adaptive update rate based on the projection rule; Adaptive update rate and Adaptive update rate They are:

[0116]

[0117] in, and are two projection operators designed by the present invention. Specifically,

[0118] ;

[0119] ;

[0120] in,

[0121] ,

[0122] ,

[0123] ,

[0124] ;

[0125] in, , , , , , , 、 , , All represent neural network parameters; , Represents the ideal weight vector of the first RBF neural network; , Represents the ideal binary weight vector of the second RBF neural network.

[0126] by , , , , and As the controller gain, by properly designing the controller gain, the actual finite time stability of the error system can be achieved. Specifically, the present invention determines the controller gain in the following manner:

[0127] The following linear matrix inequality is constructed to ensure the finite-time stability of the error system:

[0128] ;

[0129] If there is a positive scalar , , and there exists a positive definite constant , , , , , , negative definite constant , and any scalar and , , so that for any , and , the above linear matrix inequalities all hold true, then , , , , and The value of is taken as the corresponding controller gain;

[0130] is an upper triangular matrix, and the expressions of the non-zero elements are:

[0131] , , , , , , , , , , , , , , , , , , , , ; , represents the semi-Markov switching topology weight; , represents the transition rate between modes in the first semi-Markov process, express The mathematical expectation of , represents the transition rate between modes in the second semi-Markov process, express The mathematical expectation of and denote the mode sets of the first semi-Markov process and the second semi-Markov process respectively.

[0132] In the present invention, the linear matrix inequality is derived based on the consideration of finite-time formation control. The present invention can realize finite-time formation control while completing the controller design by solving the controller gain that satisfies the linear matrix inequality.

[0133] The following are some examples.

[0134] Embodiment 1:

[0135] A formation control method based on ODE-PDE, comprising:

[0136] Establish a heterogeneous multi-agent system model corresponding to the control system to be controlled; in the heterogeneous multi-agent system model, each agent corresponds to an entity object in the control system to be controlled;

[0137] Based on the ODE model, the dynamic equations of each agent in the heterogeneous multi-agent system model are established to describe the relationship between the speed of the agent, the communication protocol between agents, and the nonlinear function value of the agent's position; the communication protocol is related to the control signal used to control the speed change; in the heterogeneous multi-agent system model, the nonlinear functions corresponding to sparsely distributed agents and densely distributed agents are different;

[0138] Based on the dynamic equations of each agent, the ODE tracking error system model of the actual position of each agent relative to the target position is obtained, and the ODE tracking error system model corresponding to all densely distributed agents is processed continuously to obtain the ODE error system model corresponding to each sparsely distributed agent and the PDE error system model shared by the densely distributed agents;

[0139] In any t At this moment, follow the steps below to control the formation:

[0140] Step S1: For each sparsely distributed agent, assign its position m i ( t ) is input into the first RBF neural network to obtain a univariate nonlinear function value f i ( m i ( t )), and calculate the control signal used to control its speed change as: ;

[0141] For each densely distributed agent, its position s ( z , t ) is input into the second RBF neural network to obtain the binary nonlinear function value g ( s ( z , t )), and calculate the control signal used to control its speed change as: ;

[0142] Step S2: For each sparsely distributed agent, the corresponding univariate nonlinear function value f i ( m i ( t )) and control signals Substitute the corresponding ODE error system model to calculate the corresponding tracking error; for each densely distributed agent, the corresponding binary nonlinear function value g ( s ( z , t )) and control signals Substitute the PDE error system model shared by densely distributed agents to calculate the corresponding tracking error;

[0143] Step S3: Determine whether the tracking errors corresponding to all agents have converged. If so, the formation control ends; if not, continue to update the control signal to each agent so that the tracking errors corresponding to all agents converge and the physical object performs the corresponding action;

[0144] in, represents the relative positions of densely distributed agents, l represents the distribution range of densely distributed agents; is the weight vector of the first RBF neural network; the second RBF neural network is a binary RBF neural network, and is the binary weight vector of the second RBF neural network; and are all known continuous RBF vectors.

[0145] In this embodiment, the specific implementation of each step can refer to the above analysis and will not be repeated here.

[0146] Embodiment 2:

[0147] A formation control system based on ODE-PDE, this embodiment is used to implement the formation control method based on ODE-PDE provided in the above embodiment 1, including: an initialization module, an ODE model establishment module, an ODE-PDE error model establishment module, a univariate nonlinear function approximator, a binary nonlinear function approximator, a first neural adaptive controller, a second neural adaptive controller, a first error tracking module, a second error tracking module and a control module;

[0148] A control module is used to establish a heterogeneous multi-agent system model corresponding to the control system to be controlled; in the heterogeneous multi-agent system model, each agent corresponds to an entity object in the control system to be controlled;

[0149] The ODE model building module is used to build the dynamic equations of each agent in the heterogeneous multi-agent system model based on the ODE model, which is used to describe the relationship between the speed of the agent, the communication protocol between agents, and the nonlinear function value of the agent's position; the communication protocol is related to the control signal used to control the speed change; in the heterogeneous multi-agent system model, the nonlinear functions corresponding to sparsely distributed agents and densely distributed agents are different;

[0150] The ODE-PDE error model building module is used to obtain the ODE tracking error system model of the actual position of each intelligent agent relative to the target position based on the dynamic equation of each intelligent agent, and to perform continuous processing on the ODE tracking error system model corresponding to all densely distributed intelligent agents to obtain the ODE error system model corresponding to each sparsely distributed intelligent agent and the PDE error system model shared by the densely distributed intelligent agents;

[0151] A univariate nonlinear function approximator for t At this moment, for each sparsely distributed agent, its position m i ( t ) is input into the first RBF neural network to obtain a univariate nonlinear function value f i ( m i ( t ));

[0152] A binary nonlinear function approximator for t At this moment, for each densely distributed agent, its positions ( z , t ) is input into the second RBF neural network to obtain the binary nonlinear function value g ( s ( z , t ));

[0153] The first neural adaptive controller for t At this moment, for each sparsely distributed agent, the control signal used to control its speed change is calculated as: ;

[0154] The second neural adaptive controller is used to t At this moment, for each densely distributed agent, the control signal used to control its speed change is calculated as: ;

[0155] The first error tracking module is used to track the corresponding univariate nonlinear function value of each sparsely distributed agent. f i ( m i ( t )) and control signals Substitute it into the corresponding ODE error system model to calculate the corresponding tracking error;

[0156] The second error tracking module is used to track the corresponding binary nonlinear function value of each densely distributed agent. g ( s ( z , t )) and control signals Substitute the PDE error system model shared by densely distributed agents to calculate the corresponding tracking error;

[0157] Control module for t Always judge whether the tracking errors corresponding to all agents have converged. If so, end the formation control; if not, continue to update the control signal for each agent so that the tracking errors corresponding to all agents converge and the physical objects perform the corresponding actions;

[0158] in, represents the relative positions of densely distributed agents, l represents the distribution range of densely distributed agents; The weight vector of the first RBF neural network; the second RBF neural network is a binary RBF neural network, and is the binary weight vector of the second RBF neural network; and are all known continuous RBF vectors.

[0159] In this embodiment, the specific implementation of each module can refer to the above analysis and will not be repeated here.

[0160] Embodiment 4:

[0161] A computer-readable storage medium includes a stored computer program. When the computer program is executed by a processor, the device where the computer-readable storage medium is located is controlled to execute the ODE-PDE-based formation control method provided in the above-mentioned embodiment 1.

[0162] Embodiment 5:

[0163] An electronic device, comprising:

[0164] A computer-readable storage medium for storing a computer program;

[0165] and a processor, configured to read a computer program in a computer-readable storage medium and execute the ODE-PDE-based formation control method provided in the above-mentioned embodiment 1.

[0166] The following takes the formation control of a drone cluster in three-dimensional space as an example to further explain the beneficial effects that can be achieved by the present invention. The drone cluster includes 44 drones, and a global XYZ rectangular coordinate system is established in the space where the 44 drones are active. Each drone is regarded as a mass point, and its kinematic model is established according to the system model considered by the present invention. The five drones are sparsely distributed, and the other 39 agents are densely distributed.

[0167] First, assume a semi-Markov process and There are three modes, namely For a semi-Markov process , assuming that the residence time of modes 1 and 2 follows a Weibull distribution, and the residence time of mode 3 follows an exponential distribution, that is, The probability density function of , , On the other hand, for a semi-Markov process , assuming that the dwell time of mode 1 follows a Gaussian distribution, and the dwell times of modes 2 and 3 follow an exponential distribution, that is, The probability density function of , , , then we can get The transfer rate matrix and The transfer rate matrix They are as follows:

[0168] ;

[0169] ;

[0170] Based on the above parameter settings, we can get the semi-Markov process and The control mode switching process is as follows: Figure 2 As shown by Figure 2 It can be seen that the semi-Markov process and The modal switching rules are inconsistent.

[0171] Secondly, for sparsely distributed agents, the nonlinear functions of each dimension are set as , , , for densely distributed agents, the nonlinear functions of each dimension are set to , , The weights of the semi-Markov switching topology in the designed communication protocol are set as , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , k 1=-7.5, k 2=-6.1, k 3=-9.2.

[0172] The first RBF neural network and the second RBF neural network are used to approximate the one-dimensional nonlinear function corresponding to the sparse distribution agent and the two-dimensional nonlinear function corresponding to the dense distribution agent respectively. The approximation error of the first RBF neural network is , and the approximate error of the second RBF neural network in three dimensions , and Respectively Figure 3 shown. Figure 3 The results shown show that the approximation errors are close to 0, which means that both the first RBF neural network and the second RBF neural network approximate the nonlinear function more accurately.

[0173] In addition, for the linear matrix inequality, set , , then, using the LMI toolbox in MATLAB, a set of feasible controller gains can be obtained as , 、 , , , , , , , .

[0174] In addition, the target position of the agent is set to:

[0175]

[0176] The initial position of the agent is set to:

[0177]

[0178] Under the above parameters and initial conditions, the running trajectory of the multi-agent system in 3D space can be obtained through MATLAB simulation, such as Figure 4 As shown in the figure, the connecting line between the initial position and the target position of each agent represents the running trajectory of the agent. In addition, in order to better demonstrate the control effect, Figure 5 The state trajectory of the tracking error in a single spatial dimension is given. Figure 4 and Figure 5 It can be seen from the figure that the communication protocol and control strategy designed by the present invention are effective and can realize the actual finite-time formation task of large-scale heterogeneous multi-agent systems under time-delay conditions.

[0179] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A formation control method based on ODE-PDE, characterized in that: include: Establishing a heterogeneous multi-agent system model corresponding to the control system to be controlled; in the heterogeneous multi-agent system model, each agent corresponds to an entity object in the control system to be controlled; Based on the ODE model, the dynamic equations of each agent in the heterogeneous multi-agent system model are established to describe the relationship between the speed of the agent, the communication protocol between the agents and the nonlinear function value of the agent position; the communication protocol is related to the control signal used to control the speed change; in the heterogeneous multi-agent system model, the nonlinear functions corresponding to the sparsely distributed agents and the densely distributed agents are different; Based on the dynamic equations of each agent, the ODE tracking error system model of the actual position of each agent relative to the target position is obtained, and the ODE tracking error system model corresponding to all densely distributed agents is processed continuously to obtain the ODE error system model corresponding to each sparsely distributed agent and the PDE error system model shared by the densely distributed agents; In any t At this moment, follow the steps below to control the formation: Step S1: For each sparsely distributed agent, assign its position m i ( t ) is input into the first RBF neural network to obtain a univariate nonlinear function value f i ( m i ( t )), and calculate the control signal used to control its speed change as: ; For each densely distributed agent, its position s ( z , t ) is input into the second RBF neural network to obtain the binary nonlinear function value g ( s ( z , t )), and calculate the control signal used to control its speed change as: ; Step S2: For each sparsely distributed agent, the corresponding univariate nonlinear function value f i ( m i ( t )) and control signals Substitute the corresponding ODE error system model to calculate the corresponding tracking error; for each densely distributed agent, the corresponding binary nonlinear function value g ( s ( z , t )) and control signals Substitute the PDE error system model shared by densely distributed agents to calculate the corresponding tracking error; Step S3: Determine whether the tracking errors corresponding to all agents have converged. If so, the formation control ends; if not, continue to update the control signal to each agent so that the tracking errors corresponding to all agents converge and the physical object performs the corresponding action; in, represents the relative positions of densely distributed agents, l represents the distribution range of densely distributed agents; represents the weight vector of the first RBF neural network; the second RBF neural network is a binary RBF neural network, and represents a binary weight vector of the second RBF neural network; and are all known continuous RBF vectors.

2. The ODE-PDE-based formation control method according to claim 1, characterized in that: In the heterogeneous multi-agent system model, the change in signal strength when sparsely distributed agents communicate with other agents is characterized by the first semi-Markov process, and the change in signal strength when densely distributed agents communicate with other agents is characterized by the second semi-Markov process; the first semi-Markov process and the second semi-Markov process are independent of each other, and in the first semi-Markov process and the second semi-Markov process, each mode corresponds to a signal strength.

3. The ODE-PDE-based formation control method according to claim 2, characterized in that: Communication protocols between sparsely distributed agents and other agents u i ( t )for: ； Communication protocols between densely distributed agents and other agents c i ( t )for: ； in, i represents the agent number, represents the number of sparsely distributed agents, represents the number of densely distributed agents, n represents the number of sparsely distributed agents in the heterogeneous multi-agent system model, v represents the number of densely distributed agents in the heterogeneous multi-agent system model; m k ( t ) indicates that the sparsely distributed agents are t The location at the moment; represents densely distributed agents in t The location at the moment; represents the distribution interval of densely distributed agents; represents the target position of the sparsely distributed agent, Represents the target location of sparsely distributed agents; represents densely distributed agents in t The control signal at the moment, represents densely distributed agents in t Control signal at every moment; represents the number of sparsely distributed agents, Indicates the number of densely distributed agents; , , and are all semi-Markov switching topology weights, and ; Indicates communication delay, ,and The rate of change satisfies , represents the maximum communication delay, Indicates the maximum rate of change of communication delay.

4. The ODE-PDE-based formation control method according to claim 3, characterized in that: The ODE error system model corresponding to the sparsely distributed agent is: ； The ODE error system model corresponding to densely distributed agents is: ； in, express t The tracking error of the sparsely distributed agent at all times, , express The rate of change of express t Tracking error of densely distributed agents at all times, , express The rate of change of After the ODE error system models corresponding to all densely distributed agents are processed continuously, the ODE error system models corresponding to each sparsely distributed agent and the PDE error system model shared by the densely distributed agents are obtained as follows: ； in, express t Time and location are z Tracking error of densely distributed agents; , , ; The boundary conditions of the PDE error system model shared by densely distributed agents are: .

5. The ODE-PDE-based formation control method according to claim 4, characterized in that: and All are updated according to the adaptive update rate based on the projection rule; Adaptive update rate and Adaptive update rate They are: ； ； in, ， ， ， ， in, , , , , , , r、 , , All represent neural network parameters; , represents the ideal weight vector of the first RBF neural network; , Represents the ideal binary weight vector of the second RBF neural network.

6. The ODE-PDE-based formation control method according to claim 5, characterized in that: by , , , , and is the controller gain, then the controller gain is determined by: Construct the following linear matrix inequality that satisfies finite-time stability: ； If there is a positive scalar , , and there exists a positive definite constant , , , , , , negative definite constant , and any scalar and , , so that for any , and , the above linear matrix inequalities all hold true, then , , , , and The value of is taken as the corresponding controller gain; is an upper triangular matrix, and the expressions of the non-zero elements are: , , , , , , , , , , , , , , , , , , , , ; , represents the semi-Markov switching topology weight; , represents the transition rate between modes in the first semi-Markov process, express The mathematical expectation of , represents the transition rate between modes in the second semi-Markov process, express The mathematical expectation of and denote the mode sets of the first and second semi-Markov processes, respectively. M 1 and M 2 denotes the total number of modes of the first and second semi-Markov processes, respectively.

7. A formation control system based on ODE-PDE, characterized in that: include: An initialization module, an ODE model building module, an ODE-PDE error model building module, a univariate nonlinear function approximator, a binary nonlinear function approximator, a first neural adaptive controller, a second neural adaptive controller, a first error tracking module, a second error tracking module, and a control module; The initialization module is used to establish a heterogeneous multi-agent system model corresponding to the control system to be controlled; in the heterogeneous multi-agent system model, each agent corresponds to an entity object in the control system to be controlled; The ODE model building module is used to establish the dynamic equations of each agent in the heterogeneous multi-agent system model based on the ODE model, which is used to describe the relationship between the speed of the agent, the communication protocol between the agents and the nonlinear function value of the agent position; the communication protocol is related to the control signal used to control the speed change; in the heterogeneous multi-agent system model, the nonlinear functions corresponding to the sparsely distributed agents and the densely distributed agents are different; The ODE-PDE error model building module is used to obtain the ODE tracking error system model of the actual position of each intelligent agent relative to the target position based on the dynamic equation of each intelligent agent, and to perform continuous processing on the ODE tracking error system model corresponding to all densely distributed intelligent agents to obtain the ODE error system model corresponding to each sparsely distributed intelligent agent and the PDE error system model shared by the densely distributed intelligent agents; The one-dimensional nonlinear function approximator is used to t At this moment, for each sparsely distributed agent, its position m i ( t ) is input into the first RBF neural network to obtain a univariate nonlinear function value f i ( m i ( t )); The binary nonlinear function approximator is used to t At this moment, for each densely distributed agent, its position s ( z , t ) is input into the second RBF neural network to obtain the binary nonlinear function value g ( s ( z , t )); The first neural adaptive controller is used to t At this moment, for each sparsely distributed agent, the control signal used to control its speed change is calculated as: ; The second neural adaptive controller is used to t At this moment, for each densely distributed agent, the control signal used to control its speed change is calculated as: ; The first error tracking module is used to track the corresponding univariate nonlinear function value of each sparsely distributed agent. f i ( m i ( t )) and control signals Substitute it into the corresponding ODE error system model to calculate the corresponding tracking error; The second error tracking module is used to track the binary nonlinear function value corresponding to each densely distributed intelligent agent. g ( s ( z , t )) and control signals Substitute the PDE error system model shared by densely distributed agents to calculate the corresponding tracking error; The control module is used to t Always judge whether the tracking errors corresponding to all agents have converged. If so, end the formation control; if not, continue to update the control signal for each agent so that the tracking errors corresponding to all agents converge and the physical objects perform the corresponding actions; in, represents the relative positions of densely distributed agents, l represents the distribution range of densely distributed agents; represents the weight vector of the first RBF neural network; the second RBF neural network is a binary RBF neural network, and represents a binary weight vector of the second RBF neural network; and are all known continuous RBF vectors.

8. A computer program product, characterized in that It includes a computer program; when the computer program is executed by a processor, it implements the ODE-PDE-based formation control method described in any one of claims 1 to 6.

9. A computer-readable storage medium, characterized in that: Including a stored computer program; when the computer program is executed by a processor, the device where the computer-readable storage medium is located is controlled to execute the ODE-PDE-based formation control method according to any one of claims 1 to 6.

10. An electronic device, characterized in that: include: A computer-readable storage medium for storing a computer program; and a processor, configured to read a computer program in a computer-readable storage medium and execute the ODE-PDE-based formation control method according to any one of claims 1 to 6.

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