Event-triggered time-varying gain pulse control method of nonlinear system under DoS attack

Abstract

The invention discloses an event-triggered time-varying gain pulse control method of a nonlinear system under DoS attack. The method comprises the following steps: constructing a state space model of a nonlinear system, constructing a pulse event trigger time model based on a Lyapunov function, and constructing a DoS attack model based on the duration and frequency of a DoS attack; based on the state space model, the pulse event trigger time model and the DoS attack model of the nonlinear system, carrying out stability analysis on the nonlinear system, and solving a time-varying feedback gain matrix through a linear matrix inequality; and designing a time-varying gain pulse controller based on the time-varying feedback gain matrix, and performing event-triggered time-varying gain pulse control of the nonlinear system under the DoS attack based on the time-varying gain pulse controller. According to the method, the DoS attack can be resisted, the convergence rate of the nonlinear system can be improved, and the resources of the nonlinear system are saved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of pulse control technology, in particular to an event-triggered time-varying gain pulse control method for a nonlinear system under a DoS (Denial of Service) attack. BACKGROUND

[0002] Pulse control is a strategy of applying transient control action at discrete time, which has the advantages of high energy utilization rate and simple implementation, and is widely used in engineering systems. However, the traditional pulse control method usually assumes that the network communication channel is perfect and reliable, and does not consider the influence of network attacks. In actual industrial Internet of Things and other open networks, nonlinear systems are vulnerable to DoS attacks, which can lead to resource depletion, performance degradation, or even instability of the nonlinear system.

[0003] Some existing methods consider DoS attacks, but usually assume that the DoS attack satisfies a specific probability distribution, which limits its applicability in actual non-periodic attack scenarios. In addition, the existing methods also have the following defects: the traditional event-triggered mechanism is prone to Zeno phenomenon after the attack ends, causing resource waste; the fixed gain pulse controller lacks flexibility in the face of disturbances or attacks, and has slow convergence speed.

[0004] Therefore, there is an urgent need in the art for a pulse control method that can guarantee the stability of a nonlinear system under non-periodic DoS attacks, reduce the number of triggers, and improve the convergence speed. SUMMARY

[0005] In view of the above deficiencies in the prior art, the present application provides an event-triggered time-varying gain pulse control method for a nonlinear system under a DoS attack.

[0006] To achieve the above-mentioned application purposes, the technical solution adopted by the present application is as follows: The event-triggered time-varying gain pulse control method for a nonlinear system under a DoS attack comprises the following steps: Construct a state space model of the nonlinear system, construct a pulse event-triggered time model based on the Lyapunov function, and construct a DoS attack model based on the duration and frequency of the DoS attack; Based on the state space model of the nonlinear system, the pulse event-triggered time model, and the DoS attack model, analyze the stability of the nonlinear system, and solve the time-varying feedback gain matrix through linear matrix inequality; Design a time-varying gain pulse controller based on the time-varying feedback gain matrix, and perform event-triggered time-varying gain pulse control of the nonlinear system under a DoS attack based on the time-varying gain pulse controller.

[0007] Further, the state space model of the nonlinear system is represented as: wherein: is a first derivative of , is time, is a first parameter matrix of the nonlinear system, is a state vector of the nonlinear system at , is a second parameter matrix of the nonlinear system, is a nonlinear function satisfying a Lipschitz condition, is an output of the controller at .

[0008] Further, the impulse event triggered time model is constructed based on the Lyapunov function, and the specific process is as follows: constructing a Lyapunov function of the nonlinear system; constructing the impulse event triggered time model based on the Lyapunov function of the nonlinear system, and the impulse event triggered time model is represented as: wherein: is a triggered time of the th impulse event, is a lower limit function symbol, is time, is a triggered time of the th impulse event, is a first positive real number, is a Lyapunov function of the nonlinear system at , is a base of a natural logarithm, is a second positive real number, is a third positive real number, is a Lyapunov function of the nonlinear system at , is a right limit of the triggered time of the th impulse event.

[0009] Further, the DoS attack model is constructed based on the duration and frequency of the DoS attack, and the specific process is as follows: setting a first constraint condition model of the DoS attack based on the duration of the DoS attack; setting a second constraint condition model of the DoS attack based on the frequency of the DoS attack; constructing the DoS attack model based on the first constraint condition model and the second constraint condition model.

[0010] Further, the first constraint condition model is represented as: wherein: is the attack duration of the DoS attack in the time interval , is the time interval of the DoS attack, is a first constant, , is the end time of the time interval , is the start time of the time interval , is a second constant, .

[0011] Further, a second constraint condition model is represented as: wherein: is the number of DoS attacks in the time interval , is a third constant, , is the end time of the time interval , is the start time of the time interval , is a fourth constant, .

[0012] Further, based on the state space model of the nonlinear system, the pulse event triggered time model and the DoS attack model, the stability of the nonlinear system is analyzed, and the time-varying feedback gain matrix is solved through the linear matrix inequality, and the specific process is: Based on the state space model of the nonlinear system, the pulse event triggered time model and the DoS attack model, the stability of the nonlinear system is analyzed through Lyapunov stability analysis to obtain a stability condition model of the nonlinear system; The stability condition model of the nonlinear system is converted into a linear matrix inequality to solve the time-varying feedback gain matrix.

[0013] Further, the linear matrix inequality is represented as: wherein: is a first constructed matrix, , is a positive definite symmetric matrix of the Lyapunov function, is a first parameter matrix of the nonlinear system, This is the matrix transpose operator. It is the fifth constant. , Let be the second parameter matrix of the nonlinear system. nonlinear function The condition matrix that satisfies the Lipschitz conditions. For the state vector of a nonlinear system, It is the identity matrix. In order to be in Time-varying parameters, For the first The triggering time of each pulse event This is the second construction matrix.

[0014] Furthermore, the time-varying feedback gain matrix is ​​expressed as: in: In order to be in The time-varying feedback gain matrix at time, For the first The triggering time of each pulse event.

[0015] Furthermore, a time-varying gain pulse controller is designed based on a time-varying feedback gain matrix, expressed as: in: For the controller in Output at time For time, In order to be in The time-varying feedback gain matrix at time, For nonlinear systems The state vector at time, For about The Dirac function, For the first The triggering time of each pulse event This is the DoS hibernation time interval. This refers to the time interval for a DoS attack.

[0016] The beneficial effects of this invention are as follows: (1) This invention considers the situation of communication networks being subjected to DoS attacks, and takes into account that DoS attacks occur randomly and are non-periodic. Based on this, this invention performs stability analysis on nonlinear systems by using the state-space model of nonlinear systems, the pulse event triggering time model and the DoS attack model, and solves the time-varying feedback gain matrix by using linear matrix inequalities. Based on the time-varying feedback gain matrix, a time-varying gain pulse controller is designed. This can not only resist DoS attacks, but also improve the convergence rate of nonlinear systems and save resources of nonlinear systems. (2) The present invention takes into account that when a DoS attack ends, the state error of the nonlinear system will accumulate to a large extent due to communication interruption. Therefore, the present invention introduces a minimum time trigger interval, namely the first positive real number in the present invention, when constructing the pulse event triggering time model. This ensures that even when the state error of the nonlinear system is large, the pulse event triggering must follow the minimum time trigger interval, so as to smoothly transition back to the normal control rhythm and avoid Zeno's phenomenon. Attached Figure Description

[0017] Figure 1 A schematic diagram of an event-triggered time-varying gain pulse control method for a nonlinear system under a DoS attack. Figure 2 A schematic diagram of the state trajectory of Chua's circuit under DoS attack and time-varying gain pulse controller; Figure 3 This is a schematic diagram of the state trajectory of the Chua's circuit under a DoS attack and a fixed-gain pulse controller. Detailed Implementation

[0018] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0019] like Figure 1 As shown, the event-triggered time-varying gain pulse control method for a nonlinear system under a DoS attack includes steps S1-S3, as detailed below: S1. Construct a state-space model of the nonlinear system, build a pulse event triggering time model based on Lyapunov functions, and construct a DoS attack model based on the duration and frequency of DoS attacks.

[0020] In an optional embodiment of the present invention, the state-space model of the nonlinear system is expressed as: in: for The first derivative, For time, Let be the first parameter matrix of the nonlinear system. For nonlinear systems The state vector at time, Let be the second parameter matrix of the nonlinear system. For nonlinear functions that satisfy the Lipschitz conditions, For the controller in Output at that time.

[0021] This invention constructs a pulse event triggering time model based on Lyapunov functions. The specific process is as follows: The Lyapunov function for constructing a nonlinear system is expressed as: in: For nonlinear systems Lyapunov function at time, For nonlinear systems The state vector at time, This is the matrix transpose operator. Let be a positive definite symmetric matrix of the Lyapunov function.

[0022] A pulse event triggering time model is constructed based on the Lyapunov function of a nonlinear system, expressed as follows: in: For the first The triggering time of each pulse event To determine the sign of the lower bound function, For time, For the first The triggering time of each pulse event It is the first positive real number. For nonlinear systems Lyapunov function at time, is the base of the natural logarithm. It is the second positive real number. It is the third positive real number. For nonlinear systems Lyapunov function at time, For the first The right limit of the triggering time of a pulse event.

[0023] This invention constructs a DoS attack model based on the duration and frequency of DoS attacks. The specific process is as follows: The first constraint model for a DoS attack, based on the duration of the DoS attack, is expressed as: in: For DoS attacks within a time range The duration of the attack within, This refers to the time interval of a DoS attack. It is the first constant. , Time interval End time, Time interval The start time, It is the second constant. ; The second constraint model for DoS attacks, based on the frequency of DoS attacks, is expressed as follows: in: Time interval Number of DoS attacks within the region It is the third constant. , Time interval End time, Time interval The start time, It is the fourth constant. ; A DoS attack model is constructed based on the first constraint model and the second constraint model.

[0024] Specifically, the present invention constructs a DoS attack model by simultaneously establishing a first constraint model and a second constraint model.

[0025] S2. Based on the state-space model, pulse event triggering time model and DoS attack model of nonlinear systems, stability analysis of nonlinear systems is performed, and the time-varying feedback gain matrix is ​​solved by linear matrix inequalities.

[0026] In an optional embodiment of the present invention, the present invention performs stability analysis on a nonlinear system based on a state-space model, a pulse event triggering time model, and a DoS attack model, and solves for the time-varying feedback gain matrix using linear matrix inequalities. The specific process is as follows: based on the state-space model, pulse event triggering time model, and DoS attack model of the nonlinear system, Lyapunov stability analysis is used to perform stability analysis on the nonlinear system to obtain the stability condition model of the nonlinear system; the stability condition model of the nonlinear system is converted into linear matrix inequalities to solve for the time-varying feedback gain matrix.

[0027] The stability condition model for a nonlinear system is expressed as: , , , , in: For time Time parameters, For the first The first DoS sleep time interval before the DoS attack Each pulse moment, It is the sixth constant. , It is the seventh constant. , The convergence rate of the nonlinear system. For time-varying parameters, , It is the eighth constant. and The average pulse interval, For the first The end of the DoS attack. For the first The start of the DoS attack For the first The first pulse moment within the DoS sleep time interval prior to the DoS attack. For the first The first DoS sleep time interval before the DoS attack Each pulse moment, For the first The number of pulses during the DoS sleep time interval before the next DoS attack. .

[0028] Specifically, the first two of the five inequalities above are matrix inequalities, and the first matrix inequality is about the positive definite symmetric matrix of the Lyapunov function. The first is a linear matrix inequality, and the second is a matrix inequality concerning the time-varying feedback gain matrix. Positive definite symmetric matrix of Lyapunov functions The matrix inequality, due to the positive definite symmetric matrix of the Lyapunov function in this matrix inequality. The inverse of the matrix is ​​not a linear matrix inequality. The third inequality concerns the first positive real number. The first inequality is the first matrix inequality, the second is the asymptotic stability condition derived from Lyapunov stability analysis, and the third inequality limits the duration of a DoS attack. Because the second of the first two matrix inequalities is not a linear matrix inequality, variable substitution is required, replacing the variable with... , ,and , This is the first construction matrix. This is the second construction matrix. In order to be in Given the time-varying feedback gain matrix, the first matrix inequality becomes: Then perform a contract transformation, that is, multiply both sides of the equation by... ,get: Because it also contains matrices The quadratic term is not yet a linear matrix inequality. Further application of Schul's complement lemma yields: Similarly, the same operation is performed on the second matrix inequality. First, variable substitution is performed, resulting in: Then perform a contract transformation, that is, multiply both sides of the equation by... ,get: Finally, substitute ,get: Therefore, by using variable substitution and Schul's complement lemma, both matrix inequalities were transformed into linear matrix inequalities.

[0029] The linear matrix inequality is expressed as: in: This is the first construction matrix. , Let be a positive definite symmetric matrix of the Lyapunov function. Let be the first parameter matrix of the nonlinear system. This is the matrix transpose operator. It is the fifth constant. , Let be the second parameter matrix of the nonlinear system. nonlinear function The condition matrix that satisfies the Lipschitz conditions. For the state vector of a nonlinear system, It is the identity matrix. In order to be in Time-varying parameters, For the first The triggering time of each pulse event This is the second construction matrix.

[0030] Specifically, this invention uses the Linear Matrix Inequality Toolbox in MATLAB to solve the first inequality in the linear matrix inequality, obtains the first constructing matrix, and substitutes it into the second inequality in the linear matrix inequality to obtain the second constructing matrix.

[0031] The time-varying feedback gain matrix is ​​expressed as: in: In order to be in The time-varying feedback gain matrix at time, For the first The triggering time of each pulse event.

[0032] S3. Design a time-varying gain pulse controller based on a time-varying feedback gain matrix, and perform event-triggered time-varying gain pulse control of a nonlinear system under a DoS attack based on the time-varying gain pulse controller.

[0033] In an optional embodiment of the present invention, the present invention designs a time-varying gain pulse controller based on a time-varying feedback gain matrix, as follows: in: For the controller in Output at time For time, In order to be in The time-varying feedback gain matrix at time, For nonlinear systems The state vector at time, For about The Dirac function, For the first The triggering time of each pulse event This is the DoS hibernation time interval. This refers to the time interval for a DoS attack.

[0034] Simulation experiment: This invention uses a Chua's circuit for simulation experiments and determines the parameters in the Chua's circuit. The state vector of the Chua's circuit after determining the parameters is expressed as follows: , in: for The first derivative, For Chua's circuit in The first component of the state vector at time, For Chua's circuit in The second component of the state vector at time, To satisfy the Lipschitz condition for nonlinear functions, the electrical word response of a nonlinear resistor is described in the Chua's circuit. for The first derivative, For Chua's circuit in The third component of the state vector at time, for The first derivative.

[0035] This invention sets other parameters in the nonlinear system as follows: , , , , , , , , , , , .

[0036] Based on the parameter conditions set above, this invention conducts pulse control experiments on Chua's circuit under DoS attacks using the time-varying gain pulse controller proposed in this invention and the fixed gain pulse controller in the prior art.

[0037] like Figure 2As shown, under DoS attack and time-varying gain pulse controller, the Chua's circuit takes 1.98 seconds to converge and is triggered 3 times.

[0038] like Figure 3 As shown, under DoS attack and fixed gain pulse controller, the Chua's circuit takes 2.52 seconds to converge and is triggered 7 times.

[0039] Based on the above results, it can be seen that using the time-varying gain pulse controller of this invention to conduct pulse control experiments on Chua's circuit under DoS attacks and bring it to a stable state reduces the number of triggers by 57.14% compared to the fixed gain pulse controller in the prior art. Furthermore, the time-varying gain pulse controller of this invention exhibits a faster convergence speed. Therefore, it can be concluded that the event-triggered time-varying gain pulse control of nonlinear systems under DoS attacks based on the time-varying gain pulse controller of this invention is beneficial for saving resources and achieving rapid convergence.

[0040] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. An event-triggered time-varying gain pulse control method for nonlinear systems under DoS attacks, characterized in that, Includes the following steps: A state-space model of a nonlinear system is constructed, an impulse event triggering time model is built based on Lyapunov functions, and a DoS attack model is built based on the duration and frequency of DoS attacks. Based on the state-space model, pulse event triggering time model, and DoS attack model of nonlinear systems, stability analysis of nonlinear systems is performed, and the time-varying feedback gain matrix is ​​solved by linear matrix inequalities. A time-varying gain pulse controller is designed based on a time-varying feedback gain matrix, and event-triggered time-varying gain pulse control of a nonlinear system under a DoS attack is performed based on the time-varying gain pulse controller.

2. The event-triggered time-varying gain pulse control method for nonlinear systems under DoS attacks according to claim 1, characterized in that, The state-space model of a nonlinear system is expressed as: in: for The first derivative, For time, Let be the first parameter matrix of the nonlinear system. For nonlinear systems The state vector at time, Let be the second parameter matrix of the nonlinear system. For nonlinear functions that satisfy the Lipschitz conditions, For the controller in Output at that time.

3. The event-triggered time-varying gain pulse control method for nonlinear systems under DoS attacks according to claim 1, characterized in that, The process of constructing a pulse event triggering time model based on Lyapunov functions is as follows: Constructing Lyapunov functions for nonlinear systems; A pulse event triggering time model is constructed based on the Lyapunov function of a nonlinear system, expressed as follows: in: For the first The triggering time of each pulse event To determine the sign of the lower bound function, For time, For the first The triggering time of each pulse event It is the first positive real number. For nonlinear systems Lyapunov function at time, is the base of the natural logarithm. It is the second positive real number. It is the third positive real number. For nonlinear systems Lyapunov function at time, For the first The right limit of the triggering time of a pulse event.

4. The event-triggered time-varying gain pulse control method for nonlinear systems under DoS attacks according to claim 1, characterized in that, A DoS attack model is constructed based on the duration and frequency of DoS attacks. The specific process is as follows: The first constraint model for DoS attacks is set based on the duration of the DoS attack. The second constraint model for DoS attacks is set based on the frequency of DoS attacks; A DoS attack model is constructed based on the first constraint model and the second constraint model.

5. The event-triggered time-varying gain pulse control method for a nonlinear system under a DoS attack according to claim 4, characterized in that, The first constraint model is expressed as: in: For DoS attacks within a time range The duration of the attack within, This refers to the time interval of a DoS attack. It is the first constant. , Time interval End time, Time interval The start time, It is the second constant. .

6. The event-triggered time-varying gain pulse control method for a nonlinear system under a DoS attack according to claim 4, characterized in that, The second constraint model is expressed as follows: in: Time interval Number of DoS attacks within the region It is the third constant. , Time interval End time, Time interval The start time, It is the fourth constant. .

7. The event-triggered time-varying gain pulse control method for a nonlinear system under a DoS attack according to claim 1, characterized in that, Based on the state-space model, impulse event triggering time model, and DoS attack model of nonlinear systems, stability analysis is performed on the nonlinear systems, and the time-varying feedback gain matrix is ​​solved using linear matrix inequalities. The specific process is as follows: Based on the state-space model, pulse event triggering time model and DoS attack model of nonlinear system, Lyapunov stability analysis is used to perform stability analysis on nonlinear system and obtain the stability condition model of nonlinear system. The stability condition model of the nonlinear system is transformed into a linear matrix inequality to solve for the time-varying feedback gain matrix.

8. The event-triggered time-varying gain pulse control method for a nonlinear system under a DoS attack according to claim 7, characterized in that, The linear matrix inequality is expressed as: in: This is the first construction matrix. , Let be a positive definite symmetric matrix of the Lyapunov function. Let be the first parameter matrix of the nonlinear system. This is the matrix transpose operator. It is the fifth constant. , Let be the second parameter matrix of the nonlinear system. nonlinear function The condition matrix that satisfies the Lipschitz conditions. For the state vector of a nonlinear system, It is the identity matrix. In order to be in Time-varying parameters, For the first The triggering time of each pulse event This is the second construction matrix.

9. The event-triggered time-varying gain pulse control method for a nonlinear system under a DoS attack according to claim 8, characterized in that, The time-varying feedback gain matrix is ​​expressed as: in: In order to be in The time-varying feedback gain matrix at time, For the first The triggering time of each pulse event.

10. The event-triggered time-varying gain pulse control method for a nonlinear system under a DoS attack according to claim 1, characterized in that, A time-varying gain pulse controller based on a time-varying feedback gain matrix is ​​designed as follows: in: For the controller in Output at time For time, In order to be in The time-varying feedback gain matrix at time, For nonlinear systems The state vector at time, For about The Dirac function, For the first The triggering time of each pulse event This is the DoS hibernation time interval. This refers to the time interval for a DoS attack.

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