A method to improve the average controllability of the network by optimizing the control input

Abstract

The present invention provides a method for optimizing control inputs to improve the average controllability of a network, which relates to the fields of control and complex networks. First, a weighted undirected network system state equation is established by constructing a weighted undirected network topology graph; then, a mathematical model of the optimization problem is established based on the weighted undirected network system state equation; secondly, an input selection index is designed based on the state equation and the objective function; finally, the search space of the optimization problem is determined based on the input selection index and the optimization problem is solved. The present invention provides a control input selection optimization algorithm with high computational efficiency, which can effectively improve the average controllability of the network, reduce control costs, and improve computational efficiency. The above technology can be applied to fields such as information transmission in social networks, regulation of gene networks, and motion coordination of mobile robot networks, and has important economic and social value.

Description

Technical Field

[0001] The present invention relates to the fields of control and complex networks, and in particular to a method for optimizing control input to improve average controllability of a network. Background Art

[0002] Complex network models can explain and simulate the real world. With the development of graph theory and the widespread application of technologies such as the Internet, interest in the study of complex networks has grown. The concept of networks has been applied to various fields, such as power grids, biological networks, economic and social networks. The ultimate goal of understanding complex network systems is to control their functions and behaviors. Controllability is a fundamental concept in cybernetics, described as the ability to guide a dynamic system from any initial state to any target state within a finite time. Analyzing and improving the controllability of complex networks is one of the main topics in network science. To quantify the ease with which a control system achieves a desired state, in the context of network systems, the trace of the controllable Gramian matrix can represent the average controllability of each direction in state space, and is inversely proportional to the average energy required by the control system. Therefore, a larger trace of the controllable Gramian matrix of a network indicates better average controllability of the network.

[0003] In practical applications, controlling energy costs is an important and unavoidable issue. Controllers are usually subject to operational constraints. In order to make better use of resources, each control source can drive multiple nodes according to different control intensities. How to select a set of nodes directly connected to the external control source to drive the network to the desired state at the lowest cost is a typical input selection optimization problem. In addition, for high-dimensional networks, calculations become increasingly difficult as the network dimension increases. Therefore, exploring computationally efficient input selection strategies to solve the above optimization problems has important theoretical and application value. In response to the above problems, the present invention proposes a computationally efficient control input selection algorithm to optimize the average controllability of the network. Summary of the Invention

[0004] In response to the deficiencies of the prior art, the present invention provides a method for optimizing control inputs to improve the average controllability of a network. Compared to previous studies, the algorithm proposed in the present invention can effectively improve the average controllability of a network, reduce control costs, and improve computational efficiency. It is also applicable to large networks and has important practical application value in fields such as controlling mobile robots and regulating protein expression. While maintaining the number of controllers unchanged, the present invention improves the average controllability of the network by increasing the number of connections between external control inputs and nodes or increasing the control weights between external control inputs and controlled nodes. For high-dimensional networks, as the scale of the network increases, the number of control input choices that need to be optimized will increase rapidly, which can make the calculation very difficult. The present invention proposes an input selection index that limits the selection of control inputs that need to be optimized to a smaller search space, thereby improving computational efficiency.

[0005] A method for optimizing control input to improve the average controllability of the network, the specific steps are as follows:

[0006] Step 1: Construct a weighted undirected network G = (V, E), where V = {1, ..., n} represents a set of n nodes. represents a set of edges, Represents an n×n dimensional weighted adjacency matrix. If there is an edge connecting node i and node j, then a ij =a ji >0, otherwise a ij =a ji =0;a ij Represents the element in the i-th row and j-th column of the adjacency matrix, that is, the edge weight between node i and node j; represents a set of n×n dimensional real matrices;

[0007] Step 2: Based on the weighted undirected network constructed in step 1, establish the state equation of the network system. The specific form is:

[0008]

[0009] in, is the n-dimensional state vector, is the m-dimensional input vector, is the weighted adjacency matrix, is the input matrix. If the external control source j is connected to node i, then b ij >0, otherwise b ij =0; represents the set of n×m-dimensional real matrices, and Represent the sets of n-dimensional and m-dimensional real vectors respectively;

[0010] Step 3: Based on the state equation of the network system in step 2, define the mathematical model of the optimization problem, and its objective function is as follows:

[0011]

[0012]

[0013] Among them, the objective function max ΔB f(B+ΔB) represents the input increment matrix Maximize the function f(B+ΔB) under the action; f(B+ΔB)=tr(W B+ΔB ),and is the controllable Gramian matrix after optimizing the input matrix B, t f is the controlled time of the system, tr(W B+ΔB ) represents the matrix W B+ΔB Trace; let the i-th node and the j-th control input form an input selection control pair (i, j); candidate input search space Ω S is the set of input selection control pairs (i, j) selected for optimization; N S is the candidate search space Ω S The number of elements in N S <<mn; Is an n×m-dimensional input selection matrix. If the input selection control pair (i, j) belongs to the candidate input search space Ω S , then Δ k The element corresponding to the i-th row and j-th column of is 1, and the rest are 0; η is N S dimensional binary control source selection vector and for all k=1,…,N S , η k Can be 0 or 1; ω is N S dimensional control input weight vector, ω k is the control weight added by the control input to the controlled node, k=1,…,N S ; The maximum control weight between the added external control input and the controlled node is ω ub ; and the maximum total control weight added is ω max ;The maximum number of modification times of control input is N max ;

[0014] Step 4: Design the input selection indicator based on the state equation in step 2 and the objective function in step 3: γ ij =p i φ T b j ,in, e irepresents an n-dimensional basis vector whose i-th element is 1 and the rest are 0, i = 1,…,n; is the j-th column vector of the input matrix B, j = 1,…,m;

[0015] Step 5: Select the indicator γ based on the input designed in step 4 ij , for all i=1,…,n and j=1,…,m, calculate γ ij , and γ ij Sort in descending order and select the first N S γ ij The corresponding node i and controller j constitute the candidate input search space Ω S ;

[0016] Step 6: Based on the candidate input search space Ω obtained in step 5 S , solve the optimal solution of the optimization problem in step 3, and obtain η and ω; calculate the value defined in step 3 Thus, the maximum value of f(B+ΔB) under the constraints is obtained.

[0017] The beneficial effects of adopting the above technical solution are:

[0018] This paper provides a method for optimizing control inputs to improve the average controllability of a network. The input selection strategy designed in this method can effectively improve the average controllability of the network, reduce control costs, and improve computational efficiency. The method is also applicable to large networks and has important practical application value.

[0019] Compared with existing technologies, the control input selection optimization algorithm proposed in this paper can effectively improve the average controllability of the network, reduce computational complexity, and improve computational efficiency. It is also applicable to large networks and has greater feasibility in practical applications. The research results of this paper can be applied to fields such as information transmission in social networks, regulation of gene networks, and motion coordination in mobile robot networks, and have important economic and social value. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 A flowchart of a method for improving average controllability of a network by optimizing control input provided by the present invention;

[0021] Figure 2 Schematic diagram of the weighted undirected network topology structure established for the implementation of the present invention. DETAILED DESCRIPTION

[0022] The present invention is described in detail below with reference to the accompanying drawings and examples.

[0023] A method to optimize the control input to improve the average controllability of the network, such as Figure 1As shown, the following steps are included:

[0024] Step 1: Construct a weighted undirected network G = (V, E), where V = {1, ..., n} represents a set of n nodes. represents a set of edges, Represents a weighted adjacency matrix. If there is an edge connecting node i and node j, then a ij =a ji >0, otherwise a ij =a ji =0;a ij Represents the element in the i-th row and j-th column of the adjacency matrix, that is, the edge weight between node i and node j; represents a set of n×n dimensional real matrices;

[0025] In this embodiment, a weighted undirected connected topology graph is constructed, such as Figure 2 As shown. Among them, the edge weights of the weighted undirected graph are a 12 =a 21 =0.2, a 13 =a 31 =0.1,a 23 =a 32 =0.2, a 24 =a 42 =0.3.

[0026] Step 2: Based on the weighted undirected network constructed in step 1, establish the state equation of the network system. The specific form is:

[0027]

[0028] in, is the n-dimensional state vector, is the m-dimensional input vector, is the weighted adjacency matrix, is the input matrix. If the external control source j is connected to node i, then b ij >0, otherwise b ij =0; represents the set of n×m-dimensional real matrices, and Represent the sets of n-dimensional and m-dimensional real vectors respectively;

[0029] In this embodiment, to ensure Figure 2 The network system shown is controllable. In the initial state, node 1 and node 2 are controlled by control source u1 and control source u2 with a control intensity of 1, respectively. The state matrix A and input matrix B are

[0030]

[0031] Step 3: Based on the weighted undirected network system state equation in step 2, establish the mathematical model of the optimization problem. Define the mathematical model of the optimization problem as follows:

[0032]

[0033]

[0034] Among them, the objective function max ΔB f(B+ΔB) represents the input increment matrix Maximize the function f(B+ΔB) under the action; f(B+ΔB)=tr(W B+ΔB ),and is the controllable Gramian matrix after optimizing the input matrix B, t f is the controlled time of the system, tr(W B+ΔB ) represents the matrix W B+ΔB Trace; let the i-th node and the j-th control input form an input selection control pair (i, j); candidate input search space Ω S is the set of input selection control pairs (i, j) selected for optimization; N S is the candidate search space Ω S The number of elements in N S <<mn; Is an n×m-dimensional input selection matrix. If the input selection control pair (i, j) belongs to the candidate input search space Ω S , then Δ k The element corresponding to the i-th row and j-th column of is 1, and the rest are 0; η is N S dimensional binary control source selection vector and for all k=1,…,N S , η k Can be 0 or 1; ω is N S dimensional control input weight vector, ω k is the control weight added by the control input to the controlled node, k=1,…,N S ; The maximum control weight between the added external control input and the controlled node is ω ub ; and the maximum total control weight added is ω max ;The maximum number of modification times of control input is N max ;

[0035] In this embodiment, the number of elements in the candidate search space is N S =3, the maximum number of modification times of the control input is N max =2, the maximum control weight ω between the increased external control input and the controlled node ub =0.2, the maximum total control weight ω is increased max=0.4, the system is controlled for time t f =1.

[0036] Step 4: Design the input selection indicator based on the state equation in step 2 and the objective function in step 3: γ ij =p i φ T b j ,in, e i represents an n-dimensional basis vector whose i-th element is 1 and the rest are 0, i = 1,…,n; is the j-th column vector of the input matrix B, j = 1,…,m;

[0037] Step 5: Select the indicator γ based on the input designed in step 4 ij , for all i=1,…,n and j=1,…,m, calculate γ ij , and γ ij Sort in descending order and select the first N S γ ij The corresponding node i and controller j constitute the candidate input search space Ω defined in step 3 S ;

[0038] In this embodiment, the candidate input search space calculated according to step 4 is Ω S ={(2,1),(2,2),(4,2)}, the corresponding input selection matrix is:

[0039]

[0040] Step 6: Based on the candidate input search space Ω obtained in step 5 S Solve the optimal solution of the optimization problem in step 3 and obtain η and ω; calculate the Thus, the maximum value of f(B+ΔB) under the constraints is obtained.

[0041] In this embodiment, the candidate input search space Ω in step 5 is used. S , we can get η=[0,1,1] T ,ω=[0,0.2,0.2] T , input increment When f(B+ΔB)=7.85, it is in the candidate input search space Ω S The optimal value within . The average controllability index of the network before the control input is optimized is f(B) = 2.16, and the average controllability growth rate of the system after optimized control is 364%. Therefore, the input selection strategy designed in this invention can effectively improve the average controllability of the network, reduce control costs, and improve computational efficiency.

[0042] It should be emphasized that the above description is only a preferred embodiment of the present disclosure and an illustration of the technical principles used. Those skilled in the art should understand that the scope of the invention involved in the embodiments of the present disclosure is not limited to the technical solutions formed by the specific combination of the above-mentioned technical features, but should also cover other technical solutions formed by any combination of the above-mentioned technical features or their equivalent features without departing from the above-mentioned inventive concept. For example, the above-mentioned features and the technical features with similar functions disclosed in the embodiments of the present disclosure (but not limited to) are replaced with each other to form a technical solution.

Claims

1. A method for optimizing control input to improve average controllability of a network, characterized in that: The following steps are involved: Step 1: Construct a weighted undirected network; Step 2: Based on the weighted undirected network constructed in step 1, establish the state equation of the network system; Step 3: Based on the state equation of the network system in step 2, define the mathematical model of the optimization problem; Step 4: Design the input selection index γ based on the state equation in step 2 and the objective function in step 3 ij ; Step 5: Select the indicator γ based on the input designed in step 4 ij , for all nodes i=1,…,n and controllers j=1,…,m, calculate γ ij , and γ ij Sort in descending order and select the first N S γ ij The corresponding node i and controller j constitute the candidate input search space Ω S ; Step 6: Based on the candidate input search space Ω obtained in step 5 S , solve the optimal solution of the optimization problem in step 3 and get N S dimensional binary control source selection vector η and N S dimensional control input weight vector ω; Calculate the value defined in step 3 Thus, the maximum value of the objective function f(B+ΔB) under the constraints is obtained; The weighted undirected network described in step 1 is set to G = (V, E), where V = {1, ..., n} represents a set of n nodes, represents a set of edges, Represents an n×n dimensional weighted adjacency matrix. If there is an edge connecting node i and node j′, then a ij′ =a j′i >0, otherwise a ij′ =a j′i =0;a ij′ Represents the element in the i-th row and j′th column of the adjacency matrix, that is, the edge weight between node i and node j′; represents a set of n×n dimensional real matrices; The specific form of the state equation in step 2 is: in, is the n-dimensional state vector, is the m-dimensional input vector, is the weighted adjacency matrix, is the input matrix, if controller j is connected to node i, then b ij >0, otherwise b ij =0; represents the set of n×m-dimensional real matrices, and Represent the sets of n-dimensional and m-dimensional real vectors respectively; The objective function of the mathematical model described in step 3 is as follows: Among them, the objective function Indicates that the input increment matrix Maximize the function f(B+ΔB) under the action; f(B+ΔB)=tr(W B+ΔB ),and is the controllable Gramian matrix after optimizing the input matrix B, t f is the controlled time of the system, tr(W B+ΔB ) represents the matrix W B+ΔB Trace; let the i-th node and controller j form an input selection control pair (i, j); candidate input search space Ω S is the set of input selection control pairs (i, j) selected for optimization; N S is the candidate search space Ω S The number of elements in N S <<mn; Is an n×m-dimensional input selection matrix. If the input selection control pair (i, j) belongs to the candidate input search space Ω S , then Δ k The element corresponding to the i-th row and j-th column of is 1, and the rest are 0; η is N S dimensional binary control source selection vector and for all k=1,…,N S , η k Can be 0 or 1; ω is N S dimensional control input weight vector, ω k is the control weight added by the control input to the controlled node, k=1,…,N S ; The maximum control weight between the added external control input and the controlled node is ω ub ; and the maximum total control weight added is ω max ;The maximum number of modification times of control input is N max .

2. The method for optimizing control input to improve network average controllability according to claim 1, characterized in that: Input selection indicator as described in step 4: γ ij =p i φ T b j ,in, e i represents an n-dimensional basis vector whose i-th element is 1 and the rest are 0, i = 1,…,n; is the j-th column vector of the input matrix B, j = 1,…,m.

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