A parameter-unknown chaotic system adaptive synchronization control method based on DNA strand displacement

Abstract

The application discloses a parameter-unknown chaotic system adaptive synchronization control method based on DNA strand displacement, and comprises the following steps: a chemical reaction module of a nonlinear differential equation with square terms and product terms is obtained by describing emergent reactions, trigger reactions and degradation reaction processes through double-molecule chemical reactions; a Yang chaotic system is constructed by using a DNA strand displacement reaction; an adaptive controller capable of realizing synchronization and an adaptive law expression for unknown parameter estimation are obtained; and the effectiveness of the adaptive controller is verified according to the Lyapunov stability principle. Through the DNA strand displacement mechanism, an internal synchronization mode without the participation of an external chaotic system, i.e. the synchronization between subsystems in a chaotic system, is realized; and under random reactions or other disturbances, the synchronization between the subsystems and the estimation of the unknown parameters by the adaptive law are verified. The method has robustness to parameter deviation, and through triggering a series of DNA reactions, different one-to-one and one-to-many combinations between the subsystems are synchronized.

Description

Technical Field

[0001] This invention relates to the field of chaotic synchronization technology based on DNA strand substitution in biological systems, and specifically to an adaptive synchronization control method for chaotic systems with unknown parameters based on DNA strand substitution. Background Technology

[0002] DNA molecules are widely considered ideal engineering materials for constructing molecular devices based on chemical reaction networks (CRNs), especially DNA strand displacement reactions (DSDs), which have become a formal means of programming and analyzing DNA devices. Furthermore, CRNs are an effective programming language for designing complex network behaviors, meaning that any chemical reaction can be implemented and approximated using DSDs. Therefore, constructing CRNs to represent the dynamics of systems has become a primary goal in designing applications such as chaotic systems and biochemical controllers. Combined with DNA strand displacement mechanisms, this can further enable digital circuits, signal processing computations, and simulations.

[0003] Chaos is an unpredictable, random phenomenon exhibited by nonlinear systems. Different nonlinear functions of chaotic systems result in different dynamic characteristics. Chaotic synchronization is a prerequisite for achieving secure chaotic communication; the result of synchronization is that the trajectories of the response system and the driving system gradually become consistent. With the development of chaotic synchronization theory, scholars have proposed different types of synchronization methods. For example, Shahzad M.'s article "Internal synchronization using adaptive sliding mode," published in the International Journal of Robust and Nonlinear Control, 33(3), 2320-2335 in 2023, introduces the concept of internal synchronization, achieving synchronization of a specific subsystem or the entire system, eliminating dependence on external systems. These are supplementary improvements to complete synchronization and have been widely applied in fields such as secure communication, physics, and biology. Therefore, exploring new synchronization methods for chaotic systems and studying control methods for these new synchronization methods has significant practical value.

[0004] DSD technology can not only realize the compilation of chaotic systems, but also enhance the security of chaotic system synchronization. Zou C, Zhang Q, and Wei X published an article entitled "Synchronization of hyper-Lorenz system based on DNA strand Displacement" in IEEE / ACM Transactions on Computational Biology and Bioinformatics, 19(3), 1897-1908 in 2021. They constructed a hyper-Lorenz system with an ideal chemical network, designed coupling terms through auxiliary DNA strands and substitution reactions, and realized system synchronization. In addition, Sun J, Shan Z, Liu P, and Wang Y published an article entitled "Backstepping Synchronization Control for Three-dimensional Chaotic Oscillatory System via DNA Strand Displacement" in IEEE Transactions on NanoBioscience, doi:10.1109 / TNB.2022.3213946 in 2022. Based on inversion control theory and DNA reaction modules, they designed three synchronization controllers to ensure the synchronization of two three-dimensional DNA chaotic systems. This is the first time that inverse control of a chaotic system has been achieved in the field of DSD, and the control speed is superior to PI control and coupled combination synchronization.

[0005] Existing research on synchronization control of chaotic systems based on Distributed Scaling Deterministics (DSD) rarely addresses the case of chaotic systems with unknown parameters and varying structures. However, in practical applications, neglecting these uncertainties can negatively impact and weaken the system's synchronization performance. Adaptive control methods can achieve robust synchronization of chaotic systems with uncertain parameters; therefore, research on adaptive synchronization of chaotic systems based on DSD is of practical significance. Summary of the Invention

[0006] The purpose of this invention is to propose an adaptive synchronization control method for a chaotic system with unknown parameters based on DNA strand substitution. This method is robust to parameter deviations and can achieve synchronization of internal subsystems without the participation of an external chaotic system.

[0007] To achieve the above objectives, this application proposes an adaptive synchronization control method for a chaotic system with unknown parameters based on DNA strand substitution, comprising:

[0008] By describing emergent reactions, triggering reactions, and degradation reactions through bimolecular chemical reactions, a chemical reaction module with nonlinear differential equations containing square and product terms is obtained.

[0009] Constructing the Yang chaotic system using DNA strand displacement reaction;

[0010] Obtain the adaptive controller that enables synchronization and the adaptive law expression for estimating unknown parameters.

[0011] Furthermore, the effectiveness of the adaptive controller is verified based on the Lyapunov stability principle.

[0012] Furthermore, through the DNA strand substitution mechanism, an internal synchronization method that does not require the participation of an external chaotic system can be achieved, that is, synchronization between subsystems within a chaotic system can be realized.

[0013] Under random reactions or other perturbations, verify the estimation of unknown parameters by the synchronization and adaptive laws between subsystems.

[0014] Furthermore, the emergent reaction, triggering reaction, and degradation reaction processes are described through bimolecular chemical reactions, specifically as follows:

[0015] Emergent response:

[0016]

[0017]

[0018]

[0019] Where, q i and q m It represents the DNA reaction rate, X and Y represent the substrate, Ka and Ga represent the enzyme, Ea and F represent the enzyme-substrate complex, Ha and sp2 represent the output substance;

[0020] Triggering reaction:

[0021]

[0022]

[0023]

[0024] Where Ba and Da represent enzymes, and Ca and B represent enzyme-substrate complexes;

[0025] Degradation reaction:

[0026]

[0027]

[0028]

[0029] Where Pa and Ta represent enzymes, Sa and H represent enzyme-substrate complexes, and Ra represents the output substance;

[0030] Based on mass action kinetics (MAK), the concentration changes of Y in emergent reactions, triggering reactions, and degradation reactions can be abstracted as follows: and These three types of nonlinear differential equations are used to construct chemical reaction modules with square terms and product terms.

[0031] Furthermore, the Yang chaotic system is constructed using DNA strand substitution reaction, specifically: let x1, x2 and x3 be the state variables, and a, b and c be the chaotic system parameters;

[0032] The differential dynamic equations of the Yang chaotic system are as follows:

[0033]

[0034]

[0035]

[0036] The corresponding CRNs are:

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043]

[0044]

[0045] Among them, the first type of reaction The corresponding DNA implementation is represented as:

[0046]

[0047] Second type of reaction The corresponding DNA implementation is represented as:

[0048]

[0049] Third type of reaction The corresponding DNA is implemented as a degradation reaction module. Here, B1, E, and F1 are intermediate products; A, C, D, and G are auxiliary substances participating in the reaction; waste represents inert waste that does not interact with other substances; and q... m k represents the reaction rate of maximum chain displacement. 1- k3 represents the reaction rate.

[0050] Furthermore, to obtain the adaptive controller expression that enables synchronization, specifically: to make the response system... With drive system If synchronized, the error e1 = x2 - x1 is zero, and the error dynamic system... Asymptotic stability is achieved, and the results are as follows:

[0051]

[0052] Where u1 is the adaptive controller and k is an adaptive parameter greater than 0; This represents the estimated values ​​of the unknown parameters c and a.

[0053] Furthermore, we obtain the adaptive law expression that can estimate the unknown parameters, specifically:

[0054]

[0055] Among them, a l It is an adaptive law, e a and e c It is the difference between the unknown parameter and the estimated parameter.

[0056] Furthermore, the effectiveness of the adaptive controller is verified based on the Lyapunov stability principle. Specifically, a positive definite Lyapunov function PDLF is selected. in, v1 along the error dynamic system The time derivative of the trajectory is as follows:

[0057]

[0058] Because v1 is positive definite, It is semi-negative definite, therefore the error dynamic system Global asymptotic stability is achieved, and the driving and response systems are synchronized.

[0059] Furthermore, an internal synchronization mechanism that does not require the participation of an external chaotic system is achieved through DNA strand substitution, i.e., synchronization between subsystems within a chaotic system. Specifically, DNA-based internal synchronization is achieved by providing an adaptive controller and the corresponding CRNs for the adaptive law expression. The CRNs of the adaptive controller u1 are constructed as follows:

[0060]

[0061]

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[0068] Adaptive law a l The CRNs are constructed as follows:

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[0075] For the fourth type of reaction The DNA is implemented as a trigger reaction module, and internal synchronization remains effective in one-to-many scenarios.

[0076] As a further step, under random reactions or other perturbations, the estimation of unknown parameters by the synchronization and adaptive laws between subsystems is verified. Specifically, in actual chemical reactions, the unknown parameter is assumed to be δ = δ + w. i Where δ is an unknown parameter, w i For disturbance;

[0077]

[0078]

[0079] The estimated value of the unknown parameter is continuously modified as the product of the system error, the adaptive parameter and the parameter error is used to adjust the control effect of the controller u1, thereby completing the synchronization of the controlled chaotic system and the estimation of the unknown parameters of the disturbance.

[0080] Compared with existing technologies, the technical solutions adopted in this invention have the following advantages: For the construction of the adaptive controller and the Yang chaotic system, emergence, triggering, and degradation chemical reaction modules are designed, which have the ability to express differential equations containing product terms and quadratic terms. Furthermore, the derivation of the adaptive controller and the unknown parameter identification rules is completed, giving their respective expressions in the chemical reaction network. This controller does not require the parameters of the controlled chaotic system. The adaptive synchronization control method proposed in this invention is robust to parameter deviations and also provides an internal synchronization strategy that does not require the participation of an external chaotic system. By triggering a series of DNA reactions, different one-to-one and one-to-many combinations between subsystems are synchronized. Attached Figure Description

[0081] Figure 1 DSD reaction diagram for emergent chemical reaction module;

[0082] Figure 2 The DSD reaction diagram for triggering the chemical reaction module;

[0083] Figure 3 DSD reaction diagram of the degradation chemical reaction module;

[0084] Figure 4 This is a molecular schematic diagram of an adaptive controller.

[0085] Figure 5 This is a diagram illustrating the working process of the molecular adaptive law.

[0086] Figure 6 For the subsystem synchronization process based on ideal CRNs and DSD, (a) is the evolution diagram of x1 and x2 over time, (b) is the evolution diagram of e1 over time, and (c) is the parameter estimate change diagram.

[0087] Figure 7 To select the evolution plots of error variable concentration over time for different error values, the parameters [a,b,c] are [35, 3, 35], [26, 1, 28], and [40, 6, 50], respectively;

[0088] Figure 8 The evolution plots of the estimated values ​​of unknown parameters over time are selected for different error values. The parameters [a,b,c] are [35, 3, 35], [26, 1, 28] and [40, 6, 50], respectively.

[0089] Figure 9 This is a schematic diagram of a DNA-based internal synchronization molecule. Specific implementation methods

[0090] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of this application and are not intended to limit the application; that is, the described embodiments are only a part of the embodiments of this application, and not all of them.

[0091] Therefore, the following detailed description of the embodiments of this application provided in the accompanying drawings is not intended to limit the scope of the claimed application, but merely to illustrate selected embodiments of the application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without inventive effort are within the scope of protection of this application.

[0092] This embodiment provides an adaptive synchronization control method for a chaotic system with unknown parameters based on DNA strand substitution, specifically including:

[0093] S1: A chemical reaction module describing emergent, triggered, and degradation processes through bimolecular chemical reactions, resulting in nonlinear differential equations with square and product terms.

[0094] Specifically, an emergent reaction refers to a reaction in which substrate X reacts with substrate Y at a rate k1, while the concentration of substrate X remains constant and the concentration of substrate Y increases exponentially. An ideal emergent reaction is... Its DNA implementation is specifically as follows:

[0095]

[0096]

[0097]

[0098] Where, q i and q m This refers to the DNA reaction rate, where X and Y represent the substrate, Ka and Ga represent the enzyme, Ea and F represent the enzyme-substrate complex, and Ha and sp2 represent the output substance, such as... Figure 1 As shown.

[0099] A triggered reaction occurs when substrate X reacts with different auxiliary substances, resulting in a constant concentration of substrate X while substrate Y is gradually formed. An ideal triggered reaction is... Its DNA implementation is specifically as follows:

[0100]

[0101]

[0102]

[0103] Where Ba and Da represent enzymes, and Ca and B represent enzyme-substrate complexes, such as... Figure 2 As shown.

[0104] The degradation reaction involves the gradual consumption of substrate Y by substrate X, while the concentration of substrate X remains constant throughout the reaction. The ideal degradation reaction is... Its DNA implementation is specifically as follows:

[0105]

[0106]

[0107]

[0108] Where Pa and Ta represent enzymes, Sa and H represent enzyme-substrate complexes, and Ra represents the output substance, such as... Figure 3 As shown.

[0109] Based on mass action kinetics (MAK), the concentration changes of Y in emergent reactions, triggering reactions, and degradation reactions can be abstracted as follows: and These three types of nonlinear differential equations are used to construct chemical reaction modules with square terms and product terms.

[0110] S2: Constructing the Yang chaotic system using DNA strand displacement reaction;

[0111] Specifically, let x1, x2, and x3 be the state variables, and a, b, and c be the system parameters; the differential dynamic equations of the Yang system are as follows:

[0112]

[0113]

[0114]

[0115] The corresponding CRNs are:

[0116]

[0117]

[0118]

[0119]

[0120]

[0121]

[0122]

[0123]

[0124] Among them, the first type of reaction The corresponding DNA implementation is represented as:

[0125]

[0126] Second type of reaction The corresponding DNA implementation is represented as:

[0127]

[0128] Third type of reaction The corresponding DNA is implemented as a degradation reaction module; where B1, E, and F1 are intermediate products, A, C, D, and G are auxiliary substances participating in the reaction, waste represents inert waste that does not interact with other substances, and q m k1-k3 represents the reaction rate of maximum chain displacement.

[0129] S3: Obtain the adaptive controller that enables synchronization and the adaptive law expression for estimating unknown parameters;

[0130] Specifically, to make the response system With drive system Synchronization results in zero error e1 = x2 - x1, indicating a dynamic error system. Asymptotic stability is achieved, and the results are as follows:

[0131]

[0132]

[0133] Where u1 is the adaptive controller, a l It is an adaptive law, e a and e c It is the difference between the unknown parameter and the estimated parameter, and k is an adaptive parameter that is greater than 0.

[0134] S4: Verify the effectiveness of the adaptive controller based on the Lyapunov stability principle;

[0135] Specifically, we choose a positive definite Lyapunov function PDLF. in, and These are estimated parameters. v1 follows the error dynamic system. The time derivative of the trajectory is as follows:

[0136]

[0137] Because v1 is positive definite, It is semi-negative definite, therefore the error dynamic system Global asymptotic stability is achieved, and the driving and response systems are synchronized.

[0138] S5: Through DNA strand substitution mechanism, an internal synchronization method is achieved that does not require the participation of an external chaotic system, that is, synchronization between subsystems within a chaotic system is achieved;

[0139] Specifically, within a chaotic system, each subsystem possesses some of the dynamic characteristics of the entire system. The synchronization achieved by different subsystems acting as driving and response systems respectively is called internal synchronization. DNA-based internal synchronization is achieved by providing an adaptive controller and the corresponding CRNs with the adaptive law expression. The adaptive law a... l The CRNs are constructed as follows:

[0140]

[0141]

[0142]

[0143]

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[0146] The CRNs of controller u1 are constructed as follows:

[0147]

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[0155] Among them, the fourth type of reaction The DNA is implemented as a trigger response module. Internal synchronization remains effective in one-to-many scenarios, and the controller and adaptive law work as follows: Figure 4-5 As shown.

[0156] Adaptive internal synchronization process of DNA-based chaotic systems with unknown parameters, such as... Figure 6 As shown, the concentration changes of the subsystem state variables in the ideal CRNs and DSD are basically consistent, and the error dynamic system e1 approaches 0, indicating that the subsystems have achieved synchronization. The curves showing the changes of the estimated values ​​of parameters a and c in the ideal CRNs and DSD are as follows. Figure 6 As shown in (c), they all converge to a certain constant.

[0157] S6: Under random reactions or other perturbations, verify the estimation of unknown parameters by the synchronization and adaptive laws between subsystems;

[0158] Specifically, in the process of constructing analog circuits using DNA, the different cascaded chemical reaction modules can influence each other, causing changes in DNA strand concentrations, such as the signal strand and the auxiliary strand. Therefore, the impact of parameter errors on synchronization should be studied. In actual chemical reactions, the unknown parameter is assumed to be δ = δ + w. i Where δ is an unknown parameter, w i This represents parameter error. The relevant implementation results are as follows: Figure 7 As shown, the concentration change curves of the error variable match well and all stabilize to 0 within a certain time, indicating that synchronization has been achieved between subsystems under the influence of different parameter errors. Figure 8 As shown, although there are errors in the concentration curves of DNA strands and ideal CRNs, the estimated values ​​of the parameters all converge to a certain constant, thus completing the estimation of the unknown parameters.

[0159] The working principle of adaptive synchronization is shown in Figure 9 In this process, the estimated values ​​of the unknown parameters are continuously modified as the system error and the product of the adaptive parameter and the parameter error are increased, adjusting the control effect of controller u1, thereby achieving synchronization of the controlled system and estimation of the unknown parameters of the disturbance. Furthermore, the adaptive parameter k... i The adaptive parameter can take any positive value, which determines the convergence speed of the corresponding error system. Choosing a larger adaptive parameter can achieve fast synchronization but will increase control costs. Designers can choose an appropriate adaptive parameter value according to their needs.

[0160] In this invention, the adaptive control method is applied to the field of biomolecules, which can achieve internal synchronization of chaotic systems with unknown DNA strand substitution parameters.

[0161] In this application, the terms “comprising,” “including,” or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0162] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. An adaptive synchronization control method for a chaotic system with unknown parameters based on DNA strand substitution, characterized in that, include: Emergent, triggered, and degradation reaction processes are described through bimolecular chemical reactions, resulting in a chemical reaction module with nonlinear differential equations containing square and product terms. Constructing the Yang chaotic system using DNA strand displacement reaction; Obtain the adaptive controller that enables synchronization and the adaptive law expression for estimating unknown parameters; By using the DNA strand substitution mechanism, an internal synchronization method that does not require the participation of an external chaotic system is achieved, that is, synchronization between subsystems within a chaotic system is realized. Under random reactions or other perturbations, verify the estimation of unknown parameters by the synchronization and adaptive laws between subsystems; Emergent reactions, triggered reactions, and degradation reactions are described using bimolecular chemical reactions, specifically: Emergent response: in, and It is the DNA reaction rate. , Indicates the substrate. and Indicates enzyme, , Indicates an enzyme-substrate complex. and Indicates the output substance; Triggering reaction: in, and Indicates enzyme, , This represents an enzyme-substrate complex; Degradation reaction: in, and Indicates enzyme, , Indicates an enzyme-substrate complex. Indicates the output substance; Combining mass action kinetics (MAK), emergent reactions, triggered reactions, and degradation reactions The concentration changes are abstracted as follows: , and These three types of nonlinear differential equations are used to construct chemical reaction modules with square terms and product terms. Constructing a Yang chaotic system using a DNA strand displacement reaction, specifically: Let... , and These are state variables, , and These are parameters of a chaotic system; The differential dynamic equations of the Yang chaotic system are as follows: The corresponding CRNs are: Among them, the first type of reaction The corresponding DNA implementation is represented as: Second type of reaction The corresponding DNA implementation is represented as: Third type of reaction The corresponding DNA is implemented as a degradation reaction module; in, , and It is an intermediate product. , , and It is an auxiliary substance that participates in the reaction. Indicates inert waste that does not interact with other substances. This indicates the reaction rate of maximum chain displacement. Indicates the reaction rate.

2. The adaptive synchronization control method for a chaotic system with unknown parameters based on DNA strand substitution according to claim 1, characterized in that, It also includes verifying the effectiveness of the adaptive controller based on the Lyapunov stability principle.

3. The adaptive synchronization control method for a chaotic system with unknown parameters based on DNA strand substitution according to claim 1, characterized in that, To obtain the adaptive controller expression that enables synchronization, specifically: to make the response system... With drive system Synchronization, then error Zero error dynamic system Asymptotic stability is achieved, and the results are as follows: in, It is an adaptive controller. It is an adaptive parameter greater than 0; , Representing unknown parameters and The estimated value.

4. The adaptive synchronization control method for a chaotic system with unknown parameters based on DNA strand substitution according to claim 3, characterized in that, The expression for the adaptive law that can estimate unknown parameters is obtained as follows: in, It is an adaptive law. and It is the difference between the unknown parameter and the estimated parameter.

5. The adaptive synchronization control method for a chaotic system with unknown parameters based on DNA strand substitution according to claim 1, characterized in that, The effectiveness of the adaptive controller is verified based on the Lyapunov stability principle, specifically by selecting a positive definite Lyapunov function PDLF. ;in, , , Along the error dynamic system The time derivative of the trajectory is as follows: because It is positive definite. It is semi-negative definite, therefore the error dynamic system Global asymptotic stability is achieved, and the driving and response systems are synchronized.

6. The adaptive synchronization control method for a chaotic system with unknown parameters based on DNA strand substitution according to claim 1, characterized in that, A DNA strand substitution mechanism is used to achieve an internal synchronization method that does not require the participation of an external chaotic system, i.e., synchronization between subsystems within a chaotic system. Specifically, DNA-based internal synchronization is achieved by providing an adaptive controller and CRNs corresponding to the adaptive law expression. The CRNs are constructed as follows: Adaptive law The CRNs are constructed as follows: Fourth type of reaction The DNA is implemented as a trigger reaction module, and internal synchronization remains effective in one-to-many scenarios.

7. The adaptive synchronization control method for a chaotic system with unknown parameters based on DNA strand substitution according to claim 1, characterized in that, Under random reactions or other perturbations, the estimation of unknown parameters by the synchronization and adaptive laws among subsystems is verified. Specifically, in actual chemical reactions, the unknown parameters are assumed to be... ,in For unknown parameters, For disturbance; The estimated values ​​of the unknown parameters are continuously modified as the product of the system error, the adaptive parameters, and the parameter error is applied, thus adjusting the adaptive controller. The control effect is achieved, thereby synchronizing the controlled chaotic system and estimating the unknown parameters of the disturbance.

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