Compatibility equation-based model dimension reduction method and motion control method

Abstract

The invention provides a model dimension reduction method based on a compatibility equation and a motion control method. According to the dimension reduction method, a directed graph is constructed for a multi-agent cooperative motion control system; classifying nodes in the directed graph into n node groups with similar characteristics by adopting a clustering algorithm; utilizing the reduction vector to construct a compatibility equation and decoupling; solving a reduction vector and other unknown quantities in the decoupling compatibility equation; obtaining a reduced adjacent matrix by using the solved reduced vector and the adjacent matrix of the original digraph; constructing an observable value by using the solved reduction vector and a node state quantity in an original digraph, and further determining a reduction kinetic equation; the reduction adjacency matrix and the reduction kinetic equation form a dimension reduction model. The compatible equation reduction method can map a high-dimensional state to low-dimensional representation on the basis of keeping the core motion characteristics of the system, so that the control algorithm can design a control strategy in a subspace after dimension reduction, thereby reducing the control complexity and improving the response speed of the system.

Description

Technical Field

[0001] The present invention relates to the fields of control theory and network science, and in particular to a model dimension reduction method and a motion control method based on compatible equations. Background Art

[0002] Model dimensionality reduction algorithms aim to find a low-dimensional representation of a high-dimensional system. This representation should retain key properties of the original system while being easier to analyze, providing insights into the system's inner workings and long-term behavior. A suitably reduced dimensionality also reveals the effective dimensionality of the original system—the degree to which it can be compressed into a simpler form.

[0003] Dimensionality reduction is particularly challenging when the original system is highly heterogeneous, meaning that the rules defining the unit dynamics and their interactions vary significantly across units. Therefore, the importance and interactions of different nodes should be optimized so that the reduced system retains the core dynamical characteristics of the original system while minimizing information loss during the dimensionality reduction process.

[0004] In the motion control process, directly controlling a high-dimensional motion system using the global state is not only computationally complex but also causes the control signal to diffuse in the high-dimensional space, reducing control accuracy. To this end, while maintaining the core motion characteristics of the system, the high-dimensional state should be mapped to a low-dimensional representation, thereby reducing control complexity and improving system response speed. Summary of the Invention

[0005] In light of this, the present invention provides a model dimensionality reduction method and motion control method based on compatible equations. This method achieves a low-dimensional representation of a high-dimensional system, preserving key properties of the original system while making it easier to analyze. It also provides a motion control method that enables control algorithms to run efficiently in a low-dimensional space, ensuring real-time, efficient, and accurate system control.

[0006] In order to solve the above technical problems, the present invention is implemented as follows.

[0007] A model dimensionality reduction method based on a compatibility equation, comprising:

[0008] Step 1: Construct a directed graph for the multi-agent cooperative motion control system;

[0009] Step 2: Use clustering algorithm to classify N nodes in the directed graph into n groups of nodes G1,…,G with similar characteristics. n ;

[0010] Step 3: According to the node group G1,…,G n, determine the inter-group adjacency matrix and the inter-group in-degree matrix; construct a reduction vector to simplify the representation of the directed graph node state; use the inter-group adjacency matrix, the inter-group in-degree matrix and the reduction vector to construct a compatibility equation;

[0011] Step 4: Decouple the compatibility equation so that both sides of the decoupled compatibility equation contain the same reduction vector;

[0012] Step 5: Solve the decoupled compatibility equation to obtain all unknown quantities in the compatibility equation including the reduced vector;

[0013] Step 6: Using the solved reduced vector and the adjacency matrix of the original directed graph, a reduced adjacency matrix is ​​obtained; using the solved reduced vector and the node state quantity in the original directed graph, observable values ​​are constructed, and the reduced dynamic equation is determined using the observable values; the reduced adjacency matrix and the reduced dynamic equation constitute a dimensionality reduction model.

[0014] Preferably, the step 2 uses a spectral clustering algorithm to classify the N nodes in the directed graph.

[0015] Preferably, the spectral clustering algorithm specifically includes:

[0016] Step 21: Calculate the normalized Laplace matrix L of the directed graph norm ;

[0017] Step 22: Normalize the Laplace matrix L norm Perform eigendecomposition and select the eigenvectors corresponding to the smallest n eigenvalues ​​to form an N×n dimensional matrix V; n is a positive integer less than N;

[0018] Step 23: Take each row of the matrix V as the coordinate of the directed graph node in the n-dimensional space, perform k-means clustering on the N node coordinates in the matrix V, cluster them into n groups, and obtain n groups of node groups G1,…,G n .

[0019] Preferably, in step 3, the node groups G1, ..., G n , determine the inter-group adjacency matrix and inter-group in-degree matrix as:

[0020] By rearranging the adjacency matrix elements of the directed graph according to the node groups, the inter-group adjacency matrix is ​​obtained as follows:

[0021] Among them, W vρ is m in size v ×m ρ The submatrix of G ρ Node to node group G in vAll interaction weights of the nodes in; v represents the vth group of nodes, ρ represents the ρth group of nodes; m v is the number of elements in the vth node group, m ρ is the number of elements in the ρth node group;

[0022] By rearranging the in-degree matrix elements of the directed graph according to the node groups, the inter-group in-degree matrix is ​​obtained as follows: Among them, K vρ It is a diagonal matrix formed by a series of weighted in-degrees in a directed graph.

[0023] Preferably, in step 3, the compatibility equation constructed using the inter-group adjacency matrix, the inter-group in-degree matrix and the reduction vector is:

[0024]

[0025] in, is the reduced vector, μ vρ and λ vρ are the two matrices to be solved to make the compatibility equations valid.

[0026] Preferably, in step 4, the decoupling compatibility equation is:

[0027]

[0028] where W′ vρ is the decoupled inter-group adjacency matrix, λ′ vρ W′ vρ The main eigenvalue of:

[0029]

[0030] Preferably, in step 5, in the process of solving the decoupling compatibility equation, solving the reduced vectors for the n node groups in sequence includes:

[0031] Selecting a standard basis; the reduced vector is constructed by weighting the standard basis;

[0032] Construct the scalar product matrix C using the standard basis;

[0033] The scalar product matrix C is used to solve the weighting coefficients, and the reduced vector is obtained by weighting with the standard basis.

[0034] Preferably, the scalar product matrix C constructed using the standard basis is:

[0035] The standard basis constructed is u1,…,u r , r=m v ;

[0036] Calculate the element c in the r×r matrix C st for:

[0037]

[0038] Where <·,·> represents the scalar product; u s and u t are standard bases u1,…,u r The sth and tth elements in .

[0039] Preferably, the nodes in the directed graph are intelligent agents, and the edges are communication relationships between nodes; the communication strength between nodes serves as the weight on the edge; and the direction of the directed graph represents the leadership relationship from the leading object to the follower object.

[0040] Preferably, the method comprises:

[0041] Using the model dimensionality reduction method described in any one of claims 1 to 9 to reduce the dimensionality of the multi-agent collaborative motion control system to obtain a dimensionality reduction model;

[0042] designing a controller based on the dimensionality reduction model;

[0043] The controller generates control signals based on the low-dimensional state feedback information of the agent.

[0044] Beneficial effects:

[0045] (1) The compatible equation reduction method of the present invention can map high-dimensional states to low-dimensional representations while maintaining the core motion characteristics of the system. This allows the control algorithm to design control strategies in the reduced-dimensional subspace, thereby reducing control complexity and improving system response speed. In addition, the method can preserve the key dynamic modes of the motion system, allowing the control performed in the low-dimensional space to still effectively act on the original system, achieving efficient motion control.

[0046] (2) A preferred embodiment of the present invention provides a solution for screening groups of nodes with similar characteristics through a spectral clustering algorithm. Spectral clustering first calculates the normalized Laplace matrix, which contains the in-degree matrix and the adjacency matrix, and is a reflection of the network topology. The Laplace matrix is ​​then subjected to eigendecomposition to extract the network node features. Therefore, the clustering solution is equivalent to integrating the characteristics of the two dimensions of network topology and network nodes, making the clustering effect more reasonable. Therefore, spectral clustering reveals the global topological characteristics of the network, maps the original network to a low-dimensional space, and realizes the node division of the network topology.

[0047] (3) The present invention can be used for motion control, while ensuring system stability and control accuracy, while reducing computational requirements and enhancing the system's adaptability to uncertainties. This avoids the problem of directly using high-dimensional global states for control, which is not only computationally complex but also causes the control signal to diffuse in high-dimensional space, reducing control accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 This is a schematic diagram of the present invention using the compatible equation simplification method to achieve system dimensionality reduction.

[0049] Figure 2 This is a flow chart of the present invention using the compatible equation simplification method to achieve system dimensionality reduction.

[0050] Figure 3 Figure 3. Absolute error of uniform reduction and consistent equation reduction methods versus the number of target groups n for different dynamics and network datasets.

[0051] Figure 4 The graph shows how the absolute error of four network topology reduction algorithms varies with the number of target groups n on different network datasets. DETAILED DESCRIPTION

[0052] Model dimensionality reduction algorithms aim to find a low-dimensional representation of a high-dimensional system. This representation should retain key properties of the original system while being easier to analyze, providing insights into the system's inner workings and long-term behavior. A suitably reduced dimensionality also reveals the effective dimensionality of the original system—the degree to which it can be compressed into a simpler form.

[0053] Dimensionality reduction is particularly challenging when the original system is highly heterogeneous, meaning that the rules defining the unit dynamics and their interactions vary significantly across units. Therefore, the importance and interactions of different nodes should be optimized so that the reduced system retains the core dynamical characteristics of the original system while minimizing information loss during the dimensionality reduction process.

[0054] Reasonable network topology node partitioning can better analyze the internal structure of the system, explore the connections between different groups, and optimize control strategies. Existing model dimensionality reduction methods often lack effective dimensionality reduction strategies when faced with highly heterogeneous original systems, that is, when the rules defining unit dynamics and their interactions vary greatly between different units.

[0055] To this end, the present invention implements node division for model dimensionality reduction based on clustering algorithm, which serves as the basis for dimensionality reduction of complex network topology models. It also utilizes compatibility conditions to merge network nodes into groups that keep their dynamic behaviors as consistent as possible, and implements dimensionality reduction at the model dynamics system and network topology levels based on the compatible equation simplification method, which can be effectively applied to improve computational efficiency and control accuracy in the field of motion control.

[0056] In a preferred embodiment, the present invention reveals the global topological characteristics of the network through a spectral clustering algorithm, maps the original network into a low-dimensional space, and realizes node partitioning of the network topology.

[0057] The model dimensionality reduction algorithm of the present invention is summarized into the following three steps:

[0058] (1) A spectral clustering algorithm is used to classify N nodes into n groups of nodes with similar connection attributes.

[0059] (2) According to the node group G1,…,G n , determine the inter-group adjacency matrix and the inter-group in-degree matrix, and use the inter-group adjacency matrix, the inter-group in-degree matrix and the reduced vector to construct the compatibility equation.

[0060] (3) Decouple the compatibility equations and solve them to achieve model dimensionality reduction of dynamic systems and network topologies.

[0061] The following examples describe in detail the design process and usage of the model dimensionality reduction method proposed in the present invention.

[0062] Example 1

[0063] Assumption 1 (complete dynamic system): For a multi-agent cooperative motion control system, it includes N agents. There are N active functions x1,…,x N belong Class, is a real-valued function of time, representing the activity of node i. These functions satisfy the ODE system

[0064]

[0065] where i∈{1,…,N}, yes Function f defines the autodynamics of a node, while g considers the dynamic coupling between pairs of nodes. Encodes the strength of the interaction from node j to node i.

[0066] Definition 1 (adjacency matrix and in-degree matrix): Given a system that meets Assumption 1, we call As the adjacency matrix, a weighted directed network consisting of N nodes and M edges is defined, where M is the number of non-zero elements in W. The in-degree matrix K is defined as an N×N diagonal matrix consisting of the weighted in-degrees of the nodes in the network, that is,

[0067]

[0068] The model assumes that node heterogeneity comes from the adjacency matrix itself, that is, the functions f and g are the same for all nodes. Therefore, it is reasonable to assume that nodes with similar connectivity properties will have similar activities.

[0069] For multi-agent cooperative motion control systems, the nodes in the directed graph are agents, and the edges are the communication relationships between nodes; the communication strength between nodes is the weight w on the edge. ij ; The direction of the directed graph represents the leadership relationship from the leader object to the follower object.

[0070] Assumption 2 (Reduced System): Given a system defined by Assumption 1, there exists And divide N nodes into n non-empty groups G1,…,G n That is, for all v, For v≠ρ,

[0071] Non-negative, normalized vectors a1,…,a n The weighted average of node activities is called the reduction vector. For all v∈{1,…,n}, With Group G v Relevant, satisfying And when Time a vi =0.

[0072] There are n Class observable values ​​χ1,…,χ n , defined by the reduction vector and the activity function:

[0073]

[0074] The present invention aims to construct a simplified version of the system by defining a set of linear observables, each of which represents a weighted average of the activities of nodes within each connection-based community. As a result, the reduced system has the same dimensionality as the number of modules in the original network.

[0075] In order to reduce the dimensionality of the system, it is necessary to specify how to map the original dynamics into the reduced dynamics of n observables. This means that the reduced vectors a1,…,a n , and provides a system of ODEs for the time evolution of the observables.

[0076] Assumption 3 (ordered partition): Let {G1,…,G n} is a partition of {1,…,N}, and the indices in each partition set are consecutive integers:

[0077]

[0078] where m v =|G v |It is G v size.

[0079] Without loss of generality, we can assume that the nodes have been reordered, which enables us to express the adjacency matrix W and indegree matrix K as follows:

[0080] Definition 2 (inter-group adjacency matrix and inter-group in-degree matrix): According to Definition 1 and Assumption 3, the adjacency matrix can be represented in block form, which is called the inter-group adjacency matrix W:

[0081]

[0082] Where W vρ is m in size v ×m ρ The submatrix of G is called the submatrix of the inter-group adjacency matrix, including ρ Nodes in group G v All interaction weights of the nodes in are defined as:

[0083]

[0084] in v represents the vth node group, ρ represents the ρth node group; m v is the number of elements in the vth node group, m ρ is the number of elements in the ρth node group.

[0085] In a similar way, we can define the submatrix K of the between-group in-degree matrix vρ :

[0086]

[0087] in is the weighted in-degree of node i, considering only nodes from group G ρ Therefore, the inter-group in-degree matrix K can be expressed as follows:

[0088]

[0089] Definition 3 (Partially Reduced Vector): According to Assumptions 2 and 3, for each v∈{1,…,n}, define the vth partially reduced vector satisfy

[0090]

[0091] Note that the normalization condition holds for the partially reduced vector Still holds true, that is

[0092] According to Assumption 3, N nodes {1,…,N} are divided into n groups of node sets {G1,…,G n}, which is an important prerequisite for introducing model dimensionality reduction algorithm.

[0093] To this end, a spectral clustering algorithm is proposed, and the implementation steps are as follows:

[0094] Step a1: Calculate the normalized Laplace matrix L of the directed graph of the multi-agent cooperative motion control system norm , defined as

[0095]

[0096] Where I is the identity matrix, W is the adjacency matrix of the directed graph, and K is the in-degree matrix of the directed graph.

[0097] Normalized Laplace matrix L norm It not only reflects the global topological characteristics of the graph, but also normalizes the degree so that high-value nodes and low-value nodes have a more balanced influence in the calculation, avoiding the deviation caused by uneven degree distribution.

[0098] Step a2: Normalize the Laplace matrix L norm Perform eigendecomposition and select the eigenvectors corresponding to the smallest n eigenvalues ​​to form an N×n dimensional matrix V.

[0099] This process is equivalent to reducing the dimensionality of the original network, making the complex structure that was originally difficult to divide more clearly separable in the low-dimensional space.

[0100] Step a3: Consider each row of the matrix V as the coordinate of the directed graph node corresponding to the row in the n-dimensional space, perform k-means clustering on the N node coordinates in the matrix V, cluster them into n clusters, and finally obtain n sets of node sets G1,…,G n .

[0101] Next, we implement model dimensionality reduction and propose the following lemma to describe the dynamic characteristics after reduction:

[0102] Lemma 1 (Approximate reduction dynamics): Let Assumptions 1 and 2 hold, then

[0103]

[0104] in

[0105]

[0106] say To simplify the adjacency matrix, it describes the topological characteristics of the simplified network.

[0107] The present invention proposes a compatible equation simplification method to achieve model dimensionality reduction:

[0108] Proposition 1 (Compatible Equation Simplification Method): Let Assumptions 1, 2, and 3 hold, and let {W vρ} v,ρ ,{K vρ} v,ρ and It is a set of matrices and vectors defined in 2 and 3. If there are two n×n matrices μ=(μ vρ ) v,ρ and λ=(λ vρ ) v,ρ , so that the partially reduced vector satisfies the equation

[0109]

[0110] in, is the reduced vector, μ vρ and λ vρ are the two matrices to be solved to make the compatibility equations valid.

[0111] but

[0112]

[0113] Formula (13,14) is called a compatibility equation, and the solution is as follows:

[0114] First, the original compatibility equation is transformed into a decoupled compatibility equation

[0115]

[0116] where W′ vρ is the decoupled inter-group adjacency matrix, λ′ vρ W′ vρ The main eigenvalue of:

[0117]

[0118] Calculate W′ according to formula (18) vρ , and calculate all W′ vρ The main eigenvalue λ' vρ .

[0119] Then, the steps to solve the above decoupling compatibility equations and achieve model dimensionality reduction of the dynamic system and network topology include:

[0120] For v = 1 to n, execute the following steps:

[0121] Step b1: Select the standard basis r takes m v ;

[0122] Step b2: Calculate the r×r matrix C=(c st ) s,t , whose element c st for:

[0123]

[0124] where <·,·> represents the scalar product, u s and u t are standard bases u1,…,u r The sth and tth elements in ;

[0125] Step b3: Set 1=(1,…,1) T ,y=(x1,…,x r ,K) T , where K is the error in the approximate solution of the compatibility equation. Solve the equation Get the parameters x1,…,x r ;

[0126] Step b4: Calculate the partially reduced vector:

[0127] Step b5: According to the definition Get the reduced vector

[0128] Step b6: Use the solved reduced vector and the adjacency matrix of the original directed graph to calculate ρ∈{1,…,n}; According to the definition Get the reduced adjacency matrix W r ;

[0129] Step b7: Use the solved reduced vector and the node state in the original directed graph to construct the observable value:

[0130] Step b8: Calculation

[0131] Step b9: Calculate the reduced dynamic equation using observable values:

[0132]

[0133] v∈{1,…,n} and W r It is a direct description of the dynamic system and network topology after model dimensionality reduction. Figure 1 A schematic diagram of the steps for achieving system dimensionality reduction using the compatible equation reduction method is shown.

[0134] Based on the above analysis and explanation, Figure 2 The dimensionality reduction process of the present invention is summarized. Figure 2 As shown, the process includes:

[0135] Step 1: Construct a directed graph for the multi-agent cooperative motion control system.

[0136] Step 2: Use clustering algorithm to classify N nodes in the directed graph into n groups of nodes G1,…,G with similar characteristics. n The clustering algorithm is preferably a spectral clustering algorithm, and for details, see steps a1-a3 above.

[0137] Step 3: According to the node group G1,…,G n , determine the inter-group adjacency matrix and the inter-group in-degree matrix, see formula (5) and formula (8); construct a reduced vector to simplify the representation of the node state of the directed graph; use the inter-group adjacency matrix, the inter-group in-degree matrix and the reduced vector to construct the compatibility equation, see formula (13,14).

[0138] Step 4: Decouple the compatibility equation so that both sides of the decoupled compatibility equation contain the same reduction vector, see formula (16,17).

[0139] Step 5: Solve the decoupled compatibility equation to obtain all unknowns in the compatibility equation including the reduced vector.

[0140] Step 6: Using the solved reduced vector and the adjacency matrix of the original directed graph, a reduced adjacency matrix is ​​obtained. Using the solved reduced vector and the node state quantities in the original directed graph, observable values ​​are constructed, and the observable values ​​are used to determine the reduced dynamic equation; the reduced adjacency matrix and the reduced dynamic equation constitute a dimensionality reduction model.

[0141] For the specific implementation process of steps 5 and 6, please refer to steps b1-b9 above.

[0142] Furthermore, the present invention designs relevant experiments to quantitatively analyze the performance of the model after dimensionality reduction in terms of dynamic characteristics retention, network topology consistency, etc.

[0143] 1. Evaluation of dynamic simplification effects

[0144] The uniform reduction method is used as a comparison algorithm, and its reduction vector is defined as

[0145]

[0146] To systematically compare the effectiveness of uniform reduction and consistent equation reduction in dynamical reduction, our main approach is to calculate and compare the error between the average equilibrium observables of the reduced system and the average equilibrium state of the original system. We define the following two evaluation metrics:

[0147] (1) Calculate the average equilibrium observable value of the system after dimensionality reduction under different target group numbers n

[0148]

[0149] The average equilibrium state of the original system

[0150]

[0151] The absolute error between in is the balanced observable value of group v under the target number of groups n, x oi is the equilibrium state of the original system's ith node. The absolute error under the target number n The calculation method is

[0152]

[0153] This metric measures the deviation from the overall average state of the system after dimensionality reduction. A smaller absolute error means that the dimensionality reduction algorithm is better able to maintain the steady-state characteristics of the original system.

[0154] (2) Randomly select l different target groups n k ,k∈{1,…,l}, calculate the root mean square error χ RMSE , used to evaluate the overall error level of the dimensionality reduction algorithm under multiple target groups. RMSE The calculation formula is

[0155]

[0156] This metric reflects the error fluctuations under different target groups, n, and provides a more comprehensive assessment of the stability and accuracy of the dimensionality reduction algorithm under different experimental conditions. A smaller root mean square error indicates that the dimensionality reduction algorithm can better preserve the dynamic characteristics of the original system at different scales.

[0157] This paper designs relevant experiments to quantify the performance of the two methods under different target groups n. The core goal of the experiment is to measure the deviation of the system after dimensionality reduction from the original system at the dynamic level to ensure that the dimensionality reduction does not significantly change the overall dynamic characteristics of the system. The average equilibrium observable value of the two reduction algorithms under different target groups n is Average equilibrium state with the original system The absolute error like Figure 3 As shown, the horizontal axis represents the number of target groups n, and the vertical axis represents the absolute error Method 1 and 2 represent the uniform reduction method and the compatible equation reduction method, respectively; Dynamics 1 and 2 represent the neuronal dynamics and infectious disease dynamics, respectively; Network 1 to 3 represent the Face-to-Face, DM-LC, and email-Eu-core network datasets, respectively.

[0158] Assume that the number of randomly selected groups in formula (24) is l = 50, and calculate the root mean square error χ of the two algorithms under different dynamics and network combinations. RMSE As shown in Table 1.

[0159] Table 1 Root mean square error χ using the uniform reduction method and the consistent equation reduction method on different dynamics and network datasets RMSE Bold values ​​indicate significant values.

[0160]

[0161]

[0162] according to Figure 3 The experimental results show that as the number of target groups n increases and approaches the number of original network nodes N, the average equilibrium observable values ​​of the two reduction algorithms Average equilibrium state with the original system The absolute error The absolute error of the consistent equation reduction method is lower than that of the uniform reduction method in most cases. The experimental results in Table 1 also verify this at the overall error level. At the dynamic level, the consistent equation reduction method achieves better dimensionality reduction than the uniform reduction method.

[0163] 2. Evaluation of network topology simplification effect

[0164] In network dimensionality reduction research, the ability to preserve topological structure is an important criterion for measuring the quality of dimensionality reduction algorithms. To systematically compare the performance of different network topology reduction methods, this paper mainly calculates and compares the differences in spectral characteristics between the reduced network and the original network, especially the changes in eigenvalues. The following two evaluation metrics are defined:

[0165] (1) Calculate the average absolute eigenvalue of the network after dimensionality reduction under different target group numbers n

[0166]

[0167] The mean absolute eigenvalue of the original network

[0168]

[0169] The absolute error between in The reduced adjacency matrix W under the target number n is r The vth eigenvalue of is the i-th eigenvalue of the original network adjacency matrix W. The absolute error under the target number n The calculation method is

[0170]

[0171] This metric measures the degree of change in the overall spectral characteristics of the network after dimensionality reduction. A smaller absolute error means that the dimensionality reduction algorithm is better able to maintain the eigenvalue distribution of the original network, ensuring the stability of the topological structure.

[0172] (2) Randomly select l different target groups n k ,k∈{1,…,l}, calculate the root mean square error λ RMSE , used to evaluate the overall error level of the dimensionality reduction algorithm under multiple target groups. RMSE The calculation formula is

[0173]

[0174] This metric reflects the error fluctuations under different target groups, n, and can more comprehensively evaluate the stability and accuracy of the dimensionality reduction algorithm under different experimental conditions. A smaller root mean square error indicates that the dimensionality reduction algorithm can better preserve the topological characteristics of the original network at different scales.

[0175] This paper designs an experiment to quantify the degree of deviation between the topological characteristics of the network after dimensionality reduction and the original network. The core goal is to ensure that the network after dimensionality reduction can still well maintain the overall topological characteristics of the original network without causing drastic changes in the network structure due to dimensionality reduction. Average absolute eigenvalue with the original network The absolute error like Figure 4 As shown, the horizontal axis represents the number of target groups n, and the vertical axis represents the absolute error Methods 1 to 4 represent the MGC algorithm, GraphZoom algorithm, uniform reduction method, and compatible equation reduction method, respectively. Networks 1 to 3 represent the Face-to-Face, DM-LC, and email-Eu-core network datasets, respectively.

[0176] Assume that the number of randomly selected groups in formula (28) is l = 50, and calculate the root mean square error λ of the four algorithms under different network combinations RMSE As shown in Table 2.

[0177] Table 2 Root mean square error λ of four network topology reduction algorithms on different network datasetsRMSE .

[0178] Bold values ​​indicate significant values.

[0179]

[0180] according to Figure 4 The experimental results show that as the number of target groups n increases and approaches the number of original network nodes N, the average absolute eigenvalues ​​of the four network topology reduction algorithms Average absolute eigenvalue with the original network The absolute error The absolute error of the consistent equation reduction method is lower than that of the other three algorithms in most cases. The experimental results in Table 2 also confirm this at the overall error level. At the network topology level, the consistent equation reduction method also outperforms the uniform reduction method in dimensionality reduction. Experimental results at both the dynamics and network topology levels demonstrate the superiority of the consistent equation reduction method in dimensionality reduction.

[0181] Example 2

[0182] The present invention not only proposes a model dimensionality reduction method based on compatible equations, but also further expands the application of this method in motion control systems and proposes a motion control method based on compatible equations.

[0183] Complex motion systems typically involve a large number of state variables and coupling relationships. Directly performing control calculations on high-dimensional systems is costly and susceptible to noise and system parameter perturbations. Therefore, using the consistent equation reduction method to reduce the dimensionality of motion systems can effectively extract the core dynamic characteristics of the motion system, reduce computational complexity, and enhance the system's robustness to external perturbations.

[0184] In the process of motion control, traditional methods usually rely on the complete system state to construct the control law, such as optimal control or robust control methods based on full-state feedback. However, for high-dimensional motion systems, directly using the global state for control is not only computationally complex, but also causes the control signal to diffuse in the high-dimensional space, reducing the control accuracy. The compatible equation reduction method can map the high-dimensional state to a low-dimensional representation while maintaining the core motion characteristics of the system. This allows the control algorithm to design the control strategy in the reduced-dimensional subspace, thereby reducing the control complexity and improving the system response speed. In addition, this method can retain the key dynamic modes of the motion system, so that the control performed in the low-dimensional space can still effectively act on the original system, achieving efficient motion control.

[0185] In practice, a dynamic model of the high-dimensional motion system is first established. Using the compatible equation reduction method described above, the system is reduced in dimensionality to obtain low-dimensional state variables and their evolution equations. Based on this reduced dimensionality, an adaptive control strategy is designed, enabling the controller to generate control signals solely based on low-dimensional state feedback information.

[0186] Compared to traditional methods, this approach not only reduces computational effort but also improves the system's real-time control performance. For example, in multi-agent collaborative motion control tasks, using compatible equations to reduce dimensionality can map the complex high-dimensional individual state space into low-dimensional feature variables. Coordinated control strategies can then be designed in this low-dimensional space, enabling control signals to efficiently guide the multi-agents toward collaborative goals.

[0187] The advantage of this method lies in its ability to maintain system stability and control accuracy while reducing computational requirements and enhancing the system's adaptability to uncertainties. Therefore, this method is not only applicable to the motion control of rigid systems such as manipulators and mobile robots, but can also be applied to complex motion control scenarios such as flexible structures and autonomous unmanned systems, providing a new solution for efficient and low-cost motion control.

[0188] The above specific embodiments merely illustrate the design principles of the present invention. The shapes and names of the components described herein may vary and are not limiting. Therefore, those skilled in the art may modify or substitute equivalents for the technical solutions described in the above embodiments. Such modifications and substitutions, without departing from the inventive spirit and technical solutions of the present invention, shall fall within the scope of protection of the present invention.

Claims

1. A model dimensionality reduction method based on compatibility equations, characterized in that: include: Step 1: Construct a directed graph for the multi-agent cooperative motion control system; Step 2: Use clustering algorithm to classify N nodes in the directed graph into n groups of nodes G1,…,G with similar characteristics. n ; Step 3: According to the node group G1,…,G n , determine the inter-group adjacency matrix and the inter-group in-degree matrix; construct a reduction vector to simplify the representation of the directed graph node state; use the inter-group adjacency matrix, the inter-group in-degree matrix and the reduction vector to construct a compatibility equation; Step 4: Decouple the compatibility equation so that both sides of the decoupled compatibility equation contain the same reduction vector; Step 5: Solve the decoupled compatibility equation to obtain all unknown quantities in the compatibility equation including the reduced vector; Step 6: Use the solved reduced vector and the adjacency matrix of the original directed graph to obtain the reduced adjacency matrix; Using the solved reduced vector and the node state quantities in the original directed graph, the observable values ​​are constructed, and the reduced dynamic equations are determined using the observable values; The simplified adjacency matrix and the simplified dynamics equation constitute a dimensionality reduction model.

2. The method according to claim 1, wherein The step 2 uses a spectral clustering algorithm to classify the N nodes in the directed graph.

3. The method according to claim 2, wherein The spectral clustering algorithm specifically includes: Step 21: Calculate the normalized Laplace matrix L of the directed graph norm ; Step 22: Normalize the Laplace matrix L norm Perform eigendecomposition and select the eigenvectors corresponding to the smallest n eigenvalues ​​to form an N×n dimensional matrix V; n is a positive integer less than N; Step 23: Take each row of the matrix V as the coordinate of the directed graph node in the n-dimensional space, perform k-means clustering on the N node coordinates in the matrix V, cluster them into n groups, and obtain n groups of node groups G1,…,G n .

4. The method according to claim 1, wherein In step 3, the node groups G1, ..., G n , determine the inter-group adjacency matrix and inter-group in-degree matrix as: By rearranging the adjacency matrix elements of the directed graph according to the node groups, the inter-group adjacency matrix is ​​obtained as follows: Among them, W vρ is m in size v ×m ρ The submatrix of G ρ Node to node group G in v All interaction weights of the nodes in; v represents the vth group of nodes, ρ represents the ρth group of nodes; m v is the number of elements in the vth node group, m ρ is the number of elements in the ρth node group; By rearranging the in-degree matrix elements of the directed graph according to the node groups, the inter-group in-degree matrix is ​​obtained as follows: Among them, K vρ It is a diagonal matrix formed by a series of weighted in-degrees in a directed graph.

5. The method according to claim 4, wherein In step 3, the compatibility equation constructed using the inter-group adjacency matrix, the inter-group in-degree matrix and the reduction vector is: in, is the reduced vector, μ vρ and λ vρ are the two matrices to be solved to make the compatibility equations valid.

6. The method according to claim 5, wherein In step 4, the decoupling compatibility equation is: where W′ vρ is the decoupled inter-group adjacency matrix, λ′ vρ W′ vρ The main eigenvalue of:

7. The method according to claim 6, wherein In step 5, in the process of solving the decoupling compatibility equation, the reduced vectors are solved in sequence for the n node groups, including: Selecting a standard basis; the reduced vector is constructed by weighting the standard basis; Construct the scalar product matrix C using the standard basis; The scalar product matrix C is used to solve the weighting coefficients, and the reduced vector is obtained by weighting with the standard basis.

8. The method according to claim 7, wherein The scalar product matrix C constructed using the standard basis is: The standard basis constructed is u1,…,u r , r=m v ; Calculate the element c in the r×r matrix C st for: Where <·,·> represents the scalar product; u s and u t are standard bases u1,…,u r The sth and tth elements in .

9. The method according to claim 1, wherein The nodes in the directed graph are intelligent agents, and the edges are the communication relationships between nodes; the communication strength between nodes serves as the weight on the edge; and the direction of the directed graph represents the leadership relationship from the leading object to the follower object.

10. A motion control method for obtaining a dimensionality reduction model based on a compatible equation, characterized in that: The method includes: Using the model dimensionality reduction method described in any one of claims 1 to 9 to reduce the dimensionality of the multi-agent collaborative motion control system to obtain a dimensionality reduction model; designing a controller based on the dimensionality reduction model; The controller generates control signals based on the low-dimensional state feedback information of the agent.

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