Optimal control method for robust set stabilization of Boolean control networks based on robust dynamic programming

Abstract

The present invention relates to the field of biomedical engineering technology, and in particular to an optimal control method for robust set stabilization of a Boolean control network based on robust dynamic programming, comprising the following steps: S1, solving the maximum robust control invariant subset: for a given disturbed Boolean control network, designing an iterative elimination algorithm until the maximum robust control invariant subset is found; S2, determining the optimal control problem: setting the cost function g(x,u) according to actual needs, and constructing an infinite time domain optimal control problem P1; S3, optimal control problem conversion: converting the infinite time domain optimal control problem into a finite time domain optimal control problem P2; S4, solving the finite time domain optimal control problem: solving the finite time domain optimal control problem to obtain an optimal solution. The present invention has lower time complexity, significantly improves computational efficiency, and can process a relatively large-scale Boolean control network.

Description

Technical Field

[0001] The invention relates to the technical field of biomedical engineering, and in particular to an optimal control method for robust set stabilization of a Boolean control network based on robust dynamic programming. Background Art

[0002] Boolean networks were originally used to model gene regulatory networks. When a Boolean network contains external binary inputs, it is called a Boolean control network.

[0003] Boolean networks are used to describe intracellular biochemical networks such as gene regulatory networks and protein networks, and are important for analyzing the dynamic evolution of these networks. External stimuli applied to cells through drugs, radiation, stress, and genetic engineering can be modeled as control inputs of Boolean networks. Therefore, studying the control theory of Boolean control networks is of great significance for the diagnosis and treatment of genetic diseases, targeted drug design, cell directed differentiation and tissue engineering. The matrix semi-tensor product mathematical tool provides a convenient algebraic representation and promotes the development of control theory of Boolean control networks.

[0004] The stabilization problem of Boolean control networks refers to designing a control scheme to make the system reach the target state (stable state) from any initial state, while the set stabilization problem is to reach the target state set (such as an attractor). In the biomedical context, the target state or state set can represent the healthy cell state. Designing a control scheme is equivalent to formulating a treatment plan to transform the cell from a diseased state to a healthy state. Real systems such as gene regulatory networks will always encounter various interferences, such as gene expression noise. It is necessary to design a robust control method so that the Boolean control network can still achieve stabilization or set stabilization under arbitrary interference, thus constituting a robust set stabilization problem. Early studies explored the robust control invariance of Boolean control networks, but did not study their stabilization problem in depth. Recent studies have mainly focused on time-invariant state feedback control schemes, only considering time optimality and ignoring the time complexity of the algorithm. Therefore, existing methods have defects in optimality criteria, computational complexity and running time, and are difficult to handle large-scale Boolean control networks and cannot meet diverse optimization needs. Therefore, it is necessary to design a new general control method to overcome the above shortcomings and achieve efficient, accurate and optimal solutions. Summary of the invention

[0005] Based on the above objectives, the present invention provides an optimal control method for robust set stabilization of Boolean control networks based on robust dynamic programming.

[0006] The optimal control method of robust set stabilization of Boolean control network based on robust dynamic programming includes the following steps:

[0007] S1, solve the maximum robust control invariant subset I C(Z): Design an iterative elimination algorithm for a given disturbed Boolean control network until the maximum robust control invariant subset is obtained;

[0008] S2. Determine the optimal control problem: Set the cost function \(g(x, u)\) according to actual requirements, and construct the infinite-horizon optimal control problem \(P1\): \(\min\) π∈Π G π (x 0 , Z);

[0009] S3. Transformation of the optimal control problem: Transform the infinite-horizon optimal control problem into a finite-horizon optimal control problem \(P2\):

[0010] S4. Solve the finite-horizon optimal control problem: Combine the obtained maximum robust control invariant subset, design a robust dynamic programming algorithm that stops within a finite number of steps, and solve the finite-horizon optimal control problem to obtain the optimal solution.

[0011] Optionally, the maximum robust control invariant subset \(I\) C (Z) is the union of all robust control invariant subsets, and the robust control invariant subset is a subset of the target set \(Z\). For any state belonging to the robust control invariant subset, regardless of the perturbation received, there always exists a control input \(u\) such that the state at the next moment is still in the target set \(Z\).

[0012] Optionally, the iterative elimination algorithm includes:

[0013] Input the Boolean control network and the target set: Input the Boolean control network \(BCN\) and the target set where \(\Delta\) N represents the set of all logical vectors in the \(N\)-dimensional identity matrix \(I\) n arranged in order from left to right, and each column in \(I\) n is a logical vector;

[0014] Initialization: Initialize \(Z_0 = Z, i = 0\);

[0015] Solve the controllable set: Find the states \(x\) in \(z\) i (i \geq 0) that do not belong to \(Z\) i and store the solved states \(x\) in the set \(\Delta Z\) CS (Z i ), where the controllable set \(C\) i (Z CS ) is for the Boolean control network through a specific control input; i )

[0016] Eliminate state variables: Remove the states in \(Z\) i that belong to the set \(\Delta Z\) iEliminate the states in it, and take the remaining states as a new set and store it in Z i+1 ;

[0017] Variable increment: Increment the variable i by 1;

[0018] Determine whether the set is empty: Determine whether ΔZ i-1 is an empty set If the set ΔZ i-1 is empty, it means that Z i is the desired I C (Z), that is, I C (Z) = Z i , otherwise, jump to solve the controllable set and continue to execute.

[0019] Optionally, the cost function g(x, u) in S2 is expressed as:

[0020]

[0021] Optionally, G in the infinite-horizon optimal control problem P1 π (x 0 , Z) is expressed as:

[0022] G π (x 0 , Z) = max ξ∈Ξ G π (x 0 , Z, ξ).

[0023] Optionally, in the finite-horizon optimal control problem P2 is expressed as:

[0024]

[0025] Optionally, the robust dynamic programming algorithm includes:

[0026] Input P2: Input the finite-horizon optimal control problem P2;

[0027] Calculate the maximum robust control invariant subset: Run the iterative elimination algorithm to calculate I C (Z);

[0028] Initialization Initialization

[0029] Initialize d: Initialize d = 0;

[0030] Iterative calculation Calculate

[0031] Variable d increment: Increment the variable d by 1;

[0032] Check convergence conditions: Decision Is it true? If it is true, continue to execute. Otherwise, jump to iterative calculation implement;

[0033] Collect minima: For each x∈Δ N ,pass Collect all minima;

[0034] Return result: return and U * (x),

[0035] Optionally, the The calculation formula is:

[0036]

[0037] Optionally, the The calculation formula is:

[0038]

[0039] Beneficial effects of the present invention:

[0040] The present invention provides a new solution to the robust set stabilization problem of Boolean control networks by introducing an optimal control framework, and establishes the equivalence between the robust set stabilization problem and the optimal control problem.

[0041] The present invention solves the optimal control problem through a robust dynamic programming algorithm. The algorithm is simple and easy to implement, and can obtain an accurate solution within a finite number of iterations. Compared with existing methods, it has lower time complexity, significantly improves computational efficiency, and can handle larger-scale Boolean control networks.

[0042] The present invention, by being applicable to generalized cost functions, is not limited to time optimization and meets diverse practical optimization needs of engineering. The method has good versatility and scalability and can effectively handle other more complex robust set stabilization problems, including robust set stabilization problems of logic systems with state and control quantity constraints, time delays, random switching and other properties. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] In order to more clearly illustrate the technical solutions in the present invention or the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings in the following description are only for the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0044] Figure 1Schematic diagram of the control method process according to an embodiment of the present invention;

[0045] Figure 2 Schematic diagram of the Boolean network model according to an embodiment of the present invention;

[0046] Figure 3 Schematic diagram of the robust steady state of the Boolean network according to an embodiment of the present invention. Detailed implementation manners

[0047] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. At the same time, it should be noted here that in order to make the embodiments more detailed, the following embodiments are the best and preferred embodiments. For some well-known technologies, those skilled in the art can also adopt other alternative methods for implementation; moreover, the accompanying drawings are only for more specifically describing the embodiments and are not intended to specifically limit the present invention.

[0048] It should be noted that in the specification, references to "an embodiment", "embodiments", "exemplary embodiments", "some embodiments", etc. indicate that the described embodiments may include specific features, structures, or characteristics, but not every embodiment necessarily includes the specific feature, structure, or characteristic. Additionally, when combining embodiments to describe a specific feature, structure, or characteristic, implementing such feature, structure, or characteristic in combination with other embodiments (whether explicitly described or not) should be within the knowledge of those skilled in the relevant art.

[0049] Generally, terms can be understood, at least in part, from their use in context. For example, at least in part depending on the context, the term "one or more" as used herein can be used to describe any feature, structure, or characteristic in a singular sense, or can be used to describe a combination of features, structures, or characteristics in a plural sense. Additionally, the term "based on" can be understood as not necessarily intended to convey a set of exclusive factors, but rather, at least in part depending on the context, can allow for the existence of other factors that may not be explicitly described.

[0050] As Figures 1-3 shown, the optimal control method for robust steady-state stabilization of a Boolean control network based on robust dynamic programming includes the following steps:

[0051] S1, solve for the maximum robust control invariant subset I C (Z): For a given perturbed Boolean control network, design an iterative elimination algorithm until the maximum robust control invariant subset is obtained.

[0052] In this embodiment, the ara-operon network in Escherichia coli is used as the research object, and this network plays a crucial role in regulating the arabinose operon in Escherichia coli.

[0053] First, model the ara-operon network, which includes 9 nodes corresponding to 9 state variables x i (1 ≤ i ≤ 9). There are 4 input variables, namely x i {A e , A em , A ra_ , G e}. Its Boolean expression is as shown in the appendix Figure 2 . Among them, variables D and T are affected by random perturbations, thus establishing the Boolean network stabilization problem.

[0054] According to the characteristics of the Boolean network, each state variable has two states. After combining 9 state variables, there are a total of N = 2 9 = 512 different states. Similarly, 4 control inputs have a total of Q = 2 2 = 4 different combinations.

[0055] Execute the iterative elimination algorithm according to the established Boolean control network model. After running, obtain the maximum robust control invariant subset I C .

[0056] I C (Z) = δ 512 {25, 27, 29, 130, 138, 139, 142, 145, 154, 158, 265, 266, 267, 268, 269, 281};

[0057] Among them, δ i {j} represents the j-th logical vector in Δ i .

[0058] The iterative elimination algorithm is an algorithm for solving I C (Z), specifically including:

[0059] The first step: Input the BCN and the target set

[0060] The second step: Initialize Z0 = Z, i = 0;

[0061] The third step: Find the states x in Z i (i ≥ 0) that do not belong to the controllable set C i (Z CS ) and store these state variables in the set ΔZ i i ;

[0062] The fourth step: Remove the state variables in Z i that belong to the set ΔZ i , and take the remaining state variables as a new set and store them in Z i+1 ;​

[0063] Step 5: Variable i increases by 1;

[0064] Step 6: Determine ΔZ i-1 Is it an empty set? If the set ΔZ i-1 If it is empty, it means that Z i For what I want C (Z), that is, I C (Z)=Z i , otherwise, jump to the third step to continue executing the algorithm steps;

[0065] In the above iterative elimination algorithm, Δ N Represents the N-dimensional identity matrix I n The set of all logical vectors in I arranged from left to right. n Each column in is a logical vector, and the controllable set C CS (Z i ) is a Boolean control network that drives the state to the set Z in 1 step through a specific control input. i without being affected by interference.

[0066] I C (Z) is the union of all robust control invariant subsets, and the robust control invariant subset is a subset of the target set Z. For any state belonging to the robust control invariant subset, no matter what kind of disturbance it is subjected to, there is always a control input u so that the state at the next moment is still in the target set Z.

[0067] S2, determine the optimal control problem: set the cost function g(x,u) according to actual needs, and construct the infinite time domain optimal control problem P1: min π∈Π G π (x 0 ,Z).

[0068] This embodiment studies the optimal robust set stabilization of time, and sets the cost function as follows:

[0069]

[0070] G in infinite horizon optimal control problem P1 π (x 0 ,Z) is calculated as follows:

[0071] G π (x 0 ,Z)=max ξ∈Ξ G π (x 0 ,Z,ξ).

[0072] S3, Optimal control problem transformation: Transform the infinite-horizon optimal control problem into a finite-horizon optimal control problem P2:

[0073] In the optimal control problem P2 The calculation formula is as follows:

[0074]

[0075] S4, Solve the finite-horizon optimal control problem P2: Combine the maximum robust control invariant subset I C (Z) obtained in S1, design a robust dynamic programming algorithm that can stop within a finite number of steps, and solve the finite-horizon optimal control problem P2 to obtain the optimal solution.

[0076] The robust dynamic programming algorithm specifically includes:

[0077] The first step: Input the finite-horizon optimal control problem P2;

[0078] The second step: Run the iterative elimination algorithm to calculate I C (Z);

[0079] The third step: Initialize

[0080] The fourth step: Initialize d = 0;

[0081] The fifth step: Calculate

[0082] The sixth step: Increment the variable d by 1;

[0083] The seventh step: Judge Whether it holds. If it holds, continue to execute the eighth step; otherwise, jump back to the fifth step to continue executing the algorithm steps;

[0084] The eighth step: For each x ∈ Δ N , collect all the minimum values through the formula ;

[0085] The ninth step: Return and U * (x),

[0086] The calculation formula of

[0087]

[0088] The calculation formula of

[0089]

[0090] The cost function is brought into the robust dynamic programming algorithm and executed. Eventually, the robust dynamic programming algorithm is completed in 5 iterations. The specific process is as Figure 3 shown. The return value of the robust dynamic programming algorithm is the obtained solution.

[0091] From the analysis results, since the This means that P2 is feasible for each initial state, and thus is also feasible for each initial state in P1. That is, the ara-operon network can be robustly stabilized to a specific target set Z.

[0092] The above example has the following technical effects:

[0093] From the perspective of optimal control, a new method for defining the cost function is designed. In the embodiment, time is used as the optimal index, but other indexes can also be selected according to the actual situation. On this basis, the connection between the robust stabilization and optimal control problems of the Boolean control network is established, and thus a robust dynamic programming algorithm specifically for this optimization challenge is developed.

[0094] Different from the traditional iterative algorithm, the method proposed in this invention patent iterates at most N times and has the same solution as the traditional method, ensuring the finite termination of the algorithm while guaranteeing the accuracy of the calculation result.

[0095] In addition, compared with more professional technologies, the time complexity of the method proposed in this invention patent is significantly reduced, and the calculation efficiency is greatly improved compared with the traditional algebraic method. This improvement in efficiency has been verified in dealing with medium-sized Boolean control networks, and also reflects the potential of this method in dealing with large-scale Boolean control networks in the future.

[0096] This invention covers any alternatives, modifications, equivalent methods, and solutions made within the essence and scope of this invention. To enable the public to have a thorough understanding of this invention, specific details are described in detail in the following preferred embodiments of this invention. However, those skilled in the art can fully understand this invention without these detailed descriptions. In addition, well-known methods, processes, procedures, components, and circuits are not described in detail to avoid unnecessary confusion to the essence of this invention.

[0097] The above description is only a preferred embodiment of this invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of this invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of this invention.

Claims

1. An optimal control method for robust set stabilization of Boolean control networks based on robust dynamic programming, characterized in that: It includes the following steps: S1, Solve the maximum robust control invariant subset I C (Z): For a given disturbed Boolean control network, design an iterative elimination algorithm until the maximum robust control invariant subset is obtained; S2. Determine the optimal control problem: Set the cost function \(g(x, u)\) according to the actual requirements, and construct the infinite-horizon optimal control problem \(P1\): \(\min\) π∈Π G π (x 0 , Z); S3, Optimal control problem transformation: Transform the infinite-horizon optimal control problem into a finite-horizon optimal control problem P2: S4. Solve the finite-horizon optimal control problem: Combine the obtained maximum robust control invariant subset, design a robust dynamic programming algorithm that stops within a finite number of steps, and solve the finite-horizon optimal control problem to obtain the optimal solution; The iterative elimination algorithm includes: Input Boolean control network and target set: Input Boolean control network BCN and target set where, Δ N represents the set of all logical vectors in the N - dimensional identity matrix I n arranged in order from left to right, and each column in I n is a logical vector; Initialization: Initialize \(Z_0 = Z\), \(i = 0\); Solve the controllable set: find Z i that does not belong to Z i of the controllable set C CS (Z i ) of the state x, and store the solved state x in the set ΔZ i where the controllable set C CS (Z i ) is for the Boolean control network through specific control inputs; Excluding state variables: Remove the states in Z i that belong to the set ΔZ i , and take the remaining states as a new set, which is stored in Z i+1 ; Variable increment: Increment the variable \(i\) by 1; Determine whether the set is empty: Determine ΔZ i-1 is an empty set If the set ΔZ i-1 is empty, it indicates that Z i is the desired I C (Z), that is, I C (Z) = Z i , otherwise, jump to solve the controllable set and continue execution.

2. The optimal control method for robust set stabilization of Boolean control networks based on robust dynamic programming according to claim 1, characterized in that: The maximum robust control invariant subset I C (Z) is the union of all robust control invariant subsets, and the robust control invariant subset is a subset of the target set Z. For any state belonging to the robust control invariant subset, regardless of the perturbation received, there always exists a control input u such that the state at the next moment remains in the target set Z.

3. The optimal control method of robust set stabilization of Boolean control network based on robust dynamic programming according to claim 2, characterized in that: The cost function \(g(x, u)\) in S2 is expressed as:

4. The optimal control method for robust set stabilization of Boolean control networks based on robust dynamic programming according to claim 3, characterized in that: The \(G\) in the infinite-horizon optimal control problem \(P1\) π (x 0 , Z) is expressed as: G π (x 0 , Z) = max ξ∈Ξ G π (x 0 , Z, ξ).

5. The optimal control method for robust set stabilization of Boolean control networks based on robust dynamic programming according to claim 4, characterized in that: in the finite-time optimal control problem P2 is expressed as:

6. The optimal control method for robust set stabilization of Boolean control networks based on robust dynamic programming according to claim 5, characterized in that: The robust dynamic programming algorithm includes: Input \(P_2\): Input the finite-horizon optimal control problem \(P_2\); Calculate the maximum robust control invariant subset: Run the iterative elimination algorithm to calculate I C (Z); Initialization Initialization Initialization of \(d\): Initialize \(d = 0\); Iterative calculation Calculate Variable \(d\) increment: Increment the variable \(d\) by 1; Check the convergence condition: Determine whether holds. If it holds, continue to execute; otherwise, jump to the iterative calculation and execute; Collect minima: For each x ∈ Δ N , collect all minima via ; Return result: Return and 7. The optimal control method of robust set stabilization of Boolean control network based on robust dynamic programming according to claim 6, characterized in that: The said The calculation formula is as follows:

8. The optimal control method for robust set stabilization of Boolean control networks based on robust dynamic programming according to claim 7, characterized in that: The said The calculation formula is as follows:

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