A method for determining optimal control input trajectory under multi-layer complex network

Abstract

This invention discloses a method for determining the optimal control input trajectory in a multi-layered complex network. For a cyber-physical system characterized as a multi-layered complex network model, under the condition of minimizing control energy, a method for calculating the control input trajectory by inputting control signals into the shallow layer to control layers that cannot be directly accessed is proposed. First, for a cyber-physical system, it is modeled as a mathematical model of a multi-layered complex network. Based on this, a linear system theory framework is applied. Given the initial state, final state, control time, and sampling duration of the system, the control input trajectory is calculated under the constraint of minimizing control energy. In the calculation of the control trajectory, there is no need to calculate the complex coupling relationships between the eigenvalues ​​of the system's adjacency matrix. In the system, when the condition that there is a connection between any two layers is met, the surface layer can be controlled to achieve a given target state in inaccessible deeper layers.

Description

Technical Field

[0001] This invention relates to the field of cyber-physical system control technology, specifically to a method for determining the optimal control input trajectory under a multi-layered complex network. Background Technology

[0002] In recent years, with the rapid development of computing power and network technology, cyber-physical systems (CPSs) have been widely used in various industries, such as smart home appliances, self-driving cars, and drones. Many complex cyber-physical systems can be analyzed efficiently and universally within the framework provided by complex network science. In past research, for the control problem of the system, most scholars have established the mathematical model of the actual complex cyber-physical system as a single-layer complex network model, and determined its control characteristics based on the system's response within the framework of linear system theory by establishing the system's dynamic equations. Then, control schemes are designed, and the optimal control input is solved.

[0003] In most real-world cyber-physical systems, system components do not exist in isolation; different types of interactions between them can coexist and influence each other. While establishing a single-layer complex network model can meet some control requirements, it still deviates significantly from actual systems. Specifically, as model complexity increases and the model size grows, control schemes designed using single-layer network models do not perform well in practical applications. Individual system components do not exist in isolation but influence and are affected by surrounding components through various forces, influences, and causal interactions. For example, in intelligent manufacturing production systems, different system components can be selectively used to achieve the same goal. Therefore, mathematically establishing multi-layer network models to study the control problems of complex cyber-physical systems is a reliable direction for development.

[0004] In recent years, the control problem of multilayer networks has gradually attracted the attention of most scholars. In 2016, Giulia Menichetti et al. proposed a mathematical model for studying the control problem of two-layer networks in the paper (Control of Multilayer Networks. Sci Rep 6, 20706 (2016).). However, in this model, they considered it as a macroscopic background and did not consider the influence of the two layers on each other. They only placed the two layers in a hyperadjacency matrix for consideration, which still required the calculation and design of control schemes for each layer separately. Subsequently, Pragya Srivastava et al. established a fully connected two-layer network in the paper (Structural underpinnings of control in multiplexnetworks), which is closer to the real system. In real systems, the control of the system is such that controlling only one component can satisfy the overall system controllability. Based on this, they restricted the input of control signals to the input layer network in this model to make the deep network reach the target state. Under this condition, they studied the optimal control input problem. This fully connected two-layer network model considers the connections and influences between different systems in a complex system. However, solving the control input trajectory requires extensive calculations based on diagonalization and Euclidean angles. In practical applications, this model demands high accuracy and is time-consuming and costly. Furthermore, real-world systems are closer to network models with more layers and greater complexity. A system model based solely on a two-layer network also differs from real-world systems. Therefore, control problems under multi-layered, universally applicable network models urgently need further investigation.

[0005] The scale of current industrial control systems continues to grow, and computational methods facing high computational demands, complexity, and accuracy requirements are encountering significant challenges. Furthermore, basic two-layer network models cannot adequately replace control systems with ever-increasing complexity. To address these difficulties, considering that in practical systems, the surface layer can be controlled to achieve the target state of deeper, inaccessible layers, research on optimal control input trajectory calculation methods based on a multi-layer complex network holistic model with minimum control energy is essential. This involves constructing a mathematical model of a multi-layer complex network that satisfies practical deep-layer controllability, and then considering the calculation of the optimal control input trajectory from a holistic perspective, under the constraints of minimum control energy, control input limitations, and standard state-space equations. Summary of the Invention

[0006] In view of this, the present invention provides a method for determining the optimal control input trajectory under a multi-layer complex network. For cyber-physical systems that can be characterized as multi-layer complex network models, this method solves the problem of calculating the control input trajectory by inputting control signals into shallow layers to control layers that cannot be directly accessed, while minimizing control energy.

[0007] To achieve the above objectives, the technical solution of the present invention includes the following steps:

[0008] For a cyber-physical system, it is first modeled as a mathematical model of a multi-layer complex network. The parameters of the complex network model are given in advance, and the hyperadjacency matrix of the multi-layer complex network model is established.

[0009] Construct an augmented matrix consisting of a superadjacency matrix and an input matrix, and calculate and divide the matrix exponential expression of the augmented matrix under a given control target time.

[0010] Initialize the initial values ​​of the common-state vector and the state vector of the augmented matrix, and initialize the state trajectory matrix of the augmented matrix.

[0011] The state trajectory matrix of the augmented matrix is ​​calculated using the augmented matrix. The state trajectory matrix of the augmented matrix is ​​then calculated iteratively using the state trajectory matrix of the augmented matrix. After the iteration terminates, the trajectory matrix of the augmented matrix is ​​completed, and the obtained trajectory matrix is ​​a block trajectory matrix.

[0012] By using the segmented trajectory matrix, the system state trajectory is obtained, and finally the control input trajectory for the cyber-physical system is obtained.

[0013] Furthermore, the parameters of the complex network model are given in advance, including:

[0014] The complex network model is given in advance with the following parameters: number of layers M; number of nodes N in each layer; and N-dimensional identity matrix I. N Zero matrix 0 N ;

[0015] The interlayer connection weight κ from layer i to layer j ij , κ ij κ should be satisfied ij >0;

[0016] i∈(1,2,…,M-1),j∈(2,3,…,M);

[0017] The in-layer real symmetric matrix A of each network layer k ;k∈(1,2,3,…,M);

[0018] The interlayer connectivity matrix K from layer i to layer j ij =κ ij I N ;

[0019] The target control time is T, and the control input matrix is ​​B = (I N 0 N K 0 N ) T ∈R MN×N The time step t of the discrete numerical solution F Given an initial state x0∈R MN×1 and target state x F ∈R MN×1 ;

[0020] Based on the pre-given parameters of the complex network model, construct the hyperadjacency matrix O of the multi-layer complex network model.

[0021]

[0022] Furthermore, an augmented matrix is ​​constructed by combining the superadjacency matrix and the input matrix, where the augmented matrix is: B' is the transpose of B; O' is the transpose of O;

[0023] The specific steps for calculating the matrix exponential expression of the augmented matrix under a given control target time and dividing it into blocks are as follows:

[0024]

[0025] Among them, E 11 E 12 E 21 E 22 ∈R MN×MN These are the first to fourth block matrices, respectively.

[0026] Furthermore, initialize the initial values ​​of the common-state vector and the augmented matrix state vector, and initialize the state trajectory matrix of the augmented matrix, specifically as follows:

[0027] Initial value of common state vector For E 12 The generalized inverse matrix;

[0028] Establish a time series t = (0t) F 2t F …Tt F T), totaling T / t F +1 sampling point;

[0029] Initialize augmented matrix and initial state vector

[0030] Initialize the augmented state trajectory matrix and assign initial values:

[0031] Where the i-th column is denoted as v i-1 .

[0032] Furthermore, the index of the discrete interval matrix is ​​calculated using the augmented matrix, and the state trajectory matrix of the augmented matrix is ​​iteratively calculated using the index of the discrete interval matrix. After the iteration terminates, the trajectory matrix of the augmented matrix is ​​calculated, and the obtained trajectory matrix is ​​a block trajectory matrix, specifically:

[0033] Calculate the discrete interval matrix exponent: Et = exp(AT·t) F )

[0034] Iterative calculation, from i=2 to i=T / t F +1, calculate v in each iteration i =Et·v i-1 The value is then assigned to the trajectory matrix v, and the trajectory matrix calculation is completed after the iteration terminates.

[0035] The trajectory matrix after block calculation is:

[0036] Using the segmented trajectory matrix, the system state trajectory is obtained, and the final calculated control input trajectory is given as follows:

[0037]

[0038] The final calculated control input trajectory is

[0039]

[0040] The control input trajectory is represented as follows: In the form of, In other words, throughout the entire control process, the goal of enabling the deeper network to reach the target state is achieved solely through the control input layer.

[0041] Beneficial effects:

[0042] 1. This invention provides a method for determining the optimal control input trajectory in a multi-layered complex network. By constructing a mathematical model of a multi-layered complex network with realistic deep controllability, and then considering the requirements of minimum control energy, the constraints of control input, and the standard state-space equations, the optimal control input trajectory is determined. Under the condition of minimizing control energy, this method solves the problem of calculating the control input trajectory for shallow layers to control layers that cannot be directly accessed.

[0043] 2. This invention provides a method for determining the optimal control input trajectory under a multi-layer complex network. It can build a suitable multi-layer complex network model based on known information. Under the premise that there is a full-rank full connection between any two layers of the network, it can give the optimal control input trajectory with a small amount of computation and the minimum control energy from the overall perspective.

[0044] 3. This invention provides a method for determining the optimal control input trajectory in a multi-layer complex network. The input layer and control target layer states obtained by applying the control input trajectory calculation method proposed in this invention meet the requirements under a given state. Simulation results show the effectiveness and feasibility of this invention in calculating the control input trajectory for a specific network system. Attached Figure Description

[0045] Figure 1 This is a schematic diagram illustrating the specific calculation process of the control input trajectory method provided by the present invention;

[0046] Figure 2 This is a graph showing the calculation results of an embodiment of the control input trajectory provided by the present invention. Detailed Implementation

[0047] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0048] In practical engineering problems, the solution cost for diagonalizing a system network model increases significantly with the size of the model, and the deviation from the actual physical situation also increases. Therefore, solving the optimal control input trajectory problem under state-space equation constraints based on network model diagonalization in component form is a computationally intensive problem. Furthermore, for more complex network models with multiple layers, the coupling relationships become more intricate, making computation even more difficult. Therefore, for optimal control problems in multi-layered networks, calculating the control input trajectory based on the overall model offers significant advantages and greater universality. This invention provides a method for calculating the optimal control input trajectory in a multi-layered complex network model that minimizes control energy. This method can construct a suitable multi-layered complex network model based on known information. Assuming full-rank full connectivity between any two layers, it provides a computationally efficient method for calculating the optimal control input that minimizes control energy. This invention provides a method for constructing a suitable multi-layered complex network model that minimizes control energy while enabling the calculation of the optimal control input by controlling surface-level inputs to reach inaccessible deeper layers and achieve the target state.

[0049] like Figure 1 As shown, this embodiment of the invention provides a method for calculating the control trajectory by minimizing control energy in a multi-layer complex network model, including the following steps:

[0050] Step 1: For a cyber-physical system, first model it as a mathematical model of a multi-layer complex network, pre-define the parameters of the complex network model, and establish the hyperadjacency matrix of the multi-layer complex network model.

[0051] The complex network model is given in advance with the following parameters: number of layers M; number of nodes N in each layer; N-dimensional identity matrix and zero matrix I. N 0 N The interlayer connection weight κ from layer i to layer j ij , κ ij κ should be satisfied ij >0; i∈(1,2,…,M-1),j∈(2,3,…,M); the in-layer real symmetric matrix A of each network layer. k ;k∈(1,2,3,…,M); K is the interlayer connectivity matrix from layer i to layer j. ij =κ ij I N ; Control target time T, control input matrix B = (I N 0 N K 0 N ) T ∈R MN×N The time step t of the discrete numerical solution F Given an initial state x0∈R MN×1 and target state x F ∈R MN×1 .

[0052] Establish the hyperadjacency matrix of a multi-layer complex network model based on a pre-given condition:

[0053]

[0054] Step 2: Construct an augmented matrix consisting of the superadjacency matrix and the input matrix, and calculate and divide the matrix exponent expression of the augmented matrix under a given control target time.

[0055] In this embodiment of the invention, the augmented matrix is ​​calculated as follows:

[0056] Calculate the matrix exponential expression of the augmented matrix at a given control target time and divide it into blocks:

[0057]

[0058] Among them, E 11 E 12 E 21 E 22 ∈R MN×MN These are the first to fourth block matrices, respectively.

[0059] Step 3: Initialize the initial values ​​of the common-state vector and the state vector of the augmented matrix, and initialize the state trajectory matrix of the augmented matrix.

[0060] Calculate the initial value of the common state vector

[0061] Establish a time series t = (0t) F 2t F …Tt F T), totaling T / t F +1 sampling point

[0062] Calculate the augmented matrix and initial state vector

[0063] Initialize the augmented state trajectory matrix and assign initial values:

[0064] Where the i-th column is denoted as v i-1 .

[0065] Step 4: Calculate the discrete interval matrix exponent using the augmented matrix, and iteratively calculate the state trajectory matrix of the augmented matrix using the discrete interval matrix exponent. After the iteration terminates, the trajectory matrix of the augmented matrix is ​​completed, and the obtained trajectory matrix is ​​the block trajectory matrix.

[0066] Calculate the exponent of the discrete interval matrix:

[0067] Et=exp(AT·t F )

[0068] Iterative calculation, from i=2 to i=T / t F +1, calculate v in each iteration i =Et·v i-1 The value is then assigned to the trajectory matrix v, and the trajectory matrix calculation is completed after the iteration terminates.

[0069] The trajectory matrix after block calculation is:

[0070]

[0071] The final system state trajectory is then given as follows:

[0072]

[0073] The final calculated control input trajectory for the cyber-physical system is

[0074]

[0075] The control input trajectory is represented as follows: In the form of, In other words, throughout the entire control process, the goal of enabling the deeper network to reach the target state is achieved solely through the control input layer.

[0076] Figure 2 This is an example of the optimal control input trajectory calculation method for a multi-layer complex network model provided by the present invention. First, given the complex network model with M=2 layers, N=4 nodes per layer, and K as the inter-layer connection matrix from the first to the second layer... 12 =I N , i.e. κ 12 =1; control target time T=5, control input matrix B=(I N 0 N ) T The time step t of the discrete numerical solution F =0.1, the in-layer real symmetric matrix A of the first layer network 1 The in-layer real symmetric matrix A of the second layer network 2 Superadjacency matrix O; Given initial state x0 and target state x F They are as follows:

[0077]

[0078]

[0079] For the above two-layer network system, given the initial state and final target state required for control, with a final control time of 5s and a sampling interval of 0.1s, the input layer and control target layer states obtained by applying the control input trajectory calculation method proposed in this invention meet the requirements at the given state (see...). Figure 2 (a) in the middle, Figure 2 In (b) of the overall control flow, the control inputs are as follows: Figure 2 As can be seen in (c) above. Simulation results demonstrate the effectiveness and feasibility of this invention in calculating the control input trajectory for a specific network system example.

[0080] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for determining the optimal control input trajectory under a multi-layer complex network, characterized in that, For cyber-physical systems that can be characterized as multi-layered complex network models, a suitable multi-layered complex network model is constructed based on known information. Assuming full-rank full connectivity between any two layers, a computationally efficient method for calculating the optimal control input with minimal control energy is presented, taking a holistic approach. This method includes the following steps: For a cyber-physical system entity, it is first modeled as a mathematical model of a multi-layer complex network. The parameters of the complex network model are given in advance, and the hyperadjacency matrix of the multi-layer complex network model is established. The parameters of the pre-given complex network model include: The number of layers in the complex network model is given in advance. Number of nodes per layer of network ; 3D unit array Zero matrix ; No. layer to the first Inter-layer connection weights , Should meet ; ； In-layer real symmetric matrix of each network layer ; ; No. layer to the first Interlayer connection matrix ; Control target time Control input matrix Time step of discrete numerical solution Given an initial state and target state ; Based on the pre-given parameters of the complex network model, construct the hyperadjacency matrix of the multi-layer complex network model. : ； Construct an augmented matrix consisting of a hyperadjacency matrix and an input matrix, and calculate and partition the matrix exponent expression of the augmented matrix under a given control target time; wherein the augmented matrix is ​​constructed as follows: ; for B The transpose of the matrix; for O The transpose of the matrix; The calculation of the augmented matrix's matrix exponential expression and block division under a given control target time is specifically as follows: in, These are the first to fourth block matrices, respectively; Initialize the initial values ​​of the common-state vector and the augmented matrix state vector, and initialize the state trajectory matrix of the augmented matrix as follows: Initial value of common state vector ; For E 12 The generalized inverse matrix; Establishing time series ,total One sampling point; Initialize augmented matrix and initial state vector Initialize the augmented state trajectory matrix and assign initial values: , among which, the List as ; The state trajectory matrix of the augmented matrix is ​​calculated using the augmented matrix, and then iteratively calculated using the discrete interval matrix exponent. After the iteration terminates, the trajectory matrix of the augmented matrix is ​​completed, and the resulting trajectory matrix is ​​a block trajectory matrix, specifically: Calculate the exponent of the discrete interval matrix: Iterative computation, from Iterate to Each iteration calculates And assign values ​​to the trajectory matrix. The trajectory matrix calculation is completed after the iteration terminates; The trajectory matrix after block calculation is: ; Using the segmented trajectory matrix, the system state trajectory is obtained, and the final calculated control input trajectory is given as follows: The final calculated control input trajectory is ； The control input trajectory is represented as follows: In the form of, In other words, throughout the entire control process, the goal of enabling the deep network to reach the target state is achieved solely through the control input layer.

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