A secure data-driven predictive control method for dual-channel DoS attacks

Abstract

This invention discloses a secure data-driven predictive control method for dual-channel DoS attacks. The invention includes the following steps: First, a linear control model is constructed based on Koopman operator theory; next, based on the linear control model, a finite-time domain optimization control problem is constructed and solved using historical data of the target nonlinear system to obtain the optimal control sequence at time k; finally, based on the asynchronous DoS attack state of the dual communication channels, the optimal control input is selected from the optimal control sequence at time k and applied to the target nonlinear system. This invention can effectively achieve linearized modeling of unknown dynamic systems, compensate for the effects caused by dual-communication-channel DoS attacks, and ensure system stability.

Description

Technical Field

[0001] This invention relates to a secure data-driven predictive control method for malicious attacks, specifically a secure data-driven predictive control method for dual-communication-channel DoS attacks. Background Technology

[0002] Most existing control methods rely on an accurate control model of the system. However, obtaining an accurate control model remains a challenge for complex, coupled nonlinear systems. To address this issue, data-driven modeling methods are gaining increasing application, especially when an accurate system model is unavailable or the system dynamics are highly nonlinear. The Koopman operator is a linearization tool that can transform a nonlinear dynamic system into a linear system evolving in a high-dimensional space. By analyzing the system's input and output data, the Koopman operator can extract the system's dynamic characteristics, enabling effective modeling. A key advantage of this approach is that it constructs a linear control model, allowing for the design of control methods using existing, mature linear control theory.

[0003] Model Predictive Control (MPC) excels in handling multi-input multi-output (MIMO) systems and naturally introduces constraints (such as input, output, and state constraints) during the control process. This makes MPC particularly suitable for complex, dynamic, and constrained systems, such as chemical processes, autonomous driving, energy systems, and robotics. The predictive model (i.e., the control model) is the core of MPC methods. Utilizing the Koopman linearized control model as the predictive model can improve the accuracy of the predictive model and effectively reduce the computational complexity of MPC methods.

[0004] With the transformation and upgrading of industrial systems towards informatization and intelligence, cyberattacks (denial-of-service attacks, data injection attacks, etc.) have become a major threat to the security of modern infrastructure and critical systems. Especially in the field of control systems, denial-of-service (DoS) attacks can disrupt data availability, thereby jeopardizing the stability, reliability, and security of the system. Therefore, researching secure data-driven predictive control methods under dual-channel DoS attacks is of great significance. Currently, there are many secure control methods for DoS attacks targeting single-channel communication between sensors and controllers, but research results on secure control methods for simultaneous DoS attacks on dual communication channels (sensor-controller and controller-actuator) are relatively scarce. Summary of the Invention

[0005] This invention addresses the shortcomings and deficiencies in current research on security control methods for simultaneous DoS attacks on both the sensor-controller and controller-actuator communication channels. It provides a secure data-driven predictive control method for dual-channel DoS attacks to enhance the system's resistance to attacks and stability.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] I. A secure data-driven predictive control method for dual-channel DoS attacks

[0008] 1) Construct a linear control model based on Koopman operator theory;

[0009] 2) Based on the linear control model, construct a finite-time optimization control problem and solve it using historical data of the target nonlinear system to obtain the optimal control sequence at time k.

[0010] 3) Based on the asynchronous DoS attack state of the dual communication channels, determine the optimal control sequence at time k. The optimal control input is selected and applied to the target nonlinear system.

[0011] In step 1), the linear control model satisfies the following formula:

[0012] z(k+1)=Az(k)+Bu(k)

[0013]

[0014] The matrix [AB] is an infinite-dimensional Koopman operator. The approximate value is calculated using the following formula:

[0015]

[0016] X lift =[x1,ψ(x1),…,ψ(x)] i ),…,ψ(x K )]

[0017] Y lift =[y1,ψ(y1),…,ψ(x)] i ),…,ψ(y K )]

[0018] Where z(k+1) is the system state in the Koopman space at time k+1, and z(k) is the system state in the Koopman space at time k. , where A and B are the predicted actual system states after dimensionality reduction from the Koopman space, respectively, and A and B are the first and second parameter matrices, respectively. C is the mapping matrix from the Koopman space to the target system space. X represents the pseudo-inverse of a matrix, and · represents matrix multiplication; lift For different sampling states x i The matrix composed of lifting functions, Let X be a matrix lift The pseudo-inverse of Y; lift For different system states y i The matrix composed of lifting functions; ψ(x i ) represents the system state x i The corresponding lifting function; K is the number of system states collected; U is the vector composed of K control inputs; X is the vector composed of K system states.

[0019] In step 2), the finite-time domain optimization control problem satisfies the following formula:

[0020]

[0021] z(l+1)=Az(l)+Bu(l),l=0,…,N-1,

[0022]

[0023] Among them, U * (k) represents the optimal control sequence, containing N optimal control inputs, satisfying U * (k)=[u * (0),u * (1),…,u * [(N-1)], u * (0),u * (1),…,u * (N-1) represent the first control input, the second control input, and the Nth control input, respectively. Represents the cost function, Let u(l) represent the system state stored by the controller at time k, where u(l) represents the given control sequence, N represents the prediction time domain, z(l) and z(l+1) are the system states in the Koopman space at times l and l+1, respectively, and z(N) represents the system state in the Koopman space at time N. A and B are the first and second parameter matrices, respectively, and C is the mapping matrix from the Koopman space to the target system space. s Let Q represent the desired state of the system, R be the state weight matrix, and ∠Q be the control input weight matrix. Q The vector 2-norm operation, ‖·‖, represents the weights of the state weight matrix Q. RThe weights represent the vector L2 norm operation that controls the input weight matrix R; For input constraints, For state constraints, The latest system state stored by the controller; z(0) represents the system state in the Koopman space at time 0; Indicates the system state The lifting function that elevates to the Koopman space.

[0024] Specifically, 3) refers to:

[0025] Detect whether a DoS attack occurs on the sensor-controller channel at time k; if no DoS attack occurs, the controller obtains the latest state of the target system and updates the system state value stored in the controller; if a DoS attack occurs, the controller cannot obtain the actual state of the target system, and the stored system state value is equal to the stored value at the previous time.

[0026] Detect whether a controller-actuator channel DoS attack occurs at time k; if no controller-actuator channel DoS attack occurs, clear the memory of the actuator of the target nonlinear system and set the optimal control sequence U at time k. * (k) Passes the first optimal control input u from the actuator's memory. * (0) Acts on the target nonlinear system;

[0027] If a DoS attack occurs on the controller-actuator channel, the l-th optimal control input u in the actuator's memory will be... * (l) Acting on the target nonlinear system.

[0028] II. A secure data-driven predictive control device for dual-channel DoS attacks

[0029] Linear control model building unit, used to build linear control models based on Koopman operator theory;

[0030] The optimal control sequence generation unit is used to construct a finite-time-domain optimization control problem based on a linear control model and solve it using historical data of the target nonlinear system to obtain the optimal control sequence U at time k. * (k); Optimal control input generation unit, used to generate the optimal control sequence U at time k based on the asynchronous DoS attack state of the dual communication channels. * The optimal control input is selected from (k) and applied to the target nonlinear system.

[0031] The beneficial effects of this invention are:

[0032] This invention addresses discrete-time nonlinear systems with unknown dynamics, constructing a data-driven linear control model using Koopman operator theory. For DoS attacks involving simultaneous sensor-controller and controller-actuator dual communication channels, a secure model predictive control method with attack compensation is designed. The secure Koopman model predictive control method proposed in this invention can effectively compensate for the impact of attacks and ensure system stability. Attached Figure Description

[0033] Figure 1 This is a flowchart of the method of the present invention.

[0034] Figure 2 This is a system control block diagram under a dual-communication channel DoS attack.

[0035] Figure 3 This is a schematic diagram of the system state trajectory of the van der Bohr oscillator under two control methods.

[0036] Figure 4 This is a schematic diagram of the control input trajectory of the van der Bohr oscillator under two control methods.

[0037] Figure 5 This indicates the change in motor speed under the two control methods.

[0038] Figure 6 This indicates the change in input stator current under the two control methods. Detailed Implementation

[0039] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0040] This invention proposes a secure data-driven predictive control method for dual-channel DoS attacks, such as... Figure 1 As shown, the method includes the following steps:

[0041] 1) Construct a linear control model based on Koopman operator theory;

[0042] The target nonlinear system is a discrete-time nonlinear model, and the formula is as follows:

[0043] x + =f(x,u)

[0044] Where x represents the current system state, u represents the control input, and x + Let f represent the system state at the next moment, and let f be an unknown function used to represent the system state x. +The relationship between the system state x and the control input u. Since f is unknown, this invention uses the Koopman operator to construct an approximate linear control model using state data and control input data. The construction process is as follows:

[0045] Randomly generate K control inputs to form a vector U, and generate K system states to form a vector X:

[0046] U = [u1, ..., u] i ,…,u K ],X=[x1,…,x i ,…,x K ]

[0047] Under the influence of U, the system state vector Y is generated:

[0048] Y = [y1, ..., y i ,…,y K ]

[0049] Satisfying condition y i =f(x) i ,u i ), u i For the i-th control input, x i Let y be the i-th system state. i For the target nonlinear system in u i The subsequent system state under the influence of the action.

[0050] The linear control model satisfies the following formula:

[0051] z(k+1)=Az(k)+Bu(k)

[0052]

[0053] Koopman operator theory states that by appropriately choosing the lifting function ψ(x), the system state can be raised to a higher-dimensional Koopman space, such that the higher-dimensional state z is within the infinite-dimensional operator. Linear evolution under the action. Matrix [AB] is an infinite-dimensional Koopman operator. The approximate value is calculated using the following formula:

[0054]

[0055] X lift =[x1,ψ(x1),…,ψ(x)] i ),…,ψ(x K )]

[0056] Y lift =[y1,ψ(y1),…,ψ(x)] i),…,ψ(y K )]

[0057] Where z(k+1) is the system state in the Koopman space at time k+1, and z(k) is the system state in the Koopman space at time k. , where A and B are the predicted actual system states after dimensionality reduction from the Koopman space, respectively, and A and B are the first and second parameter matrices, respectively. C is the mapping matrix from the Koopman space to the target system space. X represents the pseudo-inverse of a matrix, and · represents matrix multiplication; lift For different sampling states x i The matrix composed of lifting functions, Let X be a matrix lift The pseudo-inverse of Y; lift For different system states y i The matrix composed of lifting functions; ψ(x i ) represents the system state x i The corresponding lifting function can be composed of radial basis functions and Gaussian functions; K is the number of system states collected; U is a vector composed of K control inputs; X is a vector composed of K system states.

[0058] 2) Based on the linear control model, construct a finite-time-domain optimization control problem and solve it using historical data of the target nonlinear system to obtain the optimal control sequence U at time k. * (k);

[0059] The finite-time optimization control problem satisfies the following formula:

[0060]

[0061] z(l+1)=Az(l)+Bu(l),l=0,…,N-1,

[0062]

[0063] Among them, U * (k) represents the optimal control sequence, containing N optimal control inputs, satisfying U * (k)=[u * (0),u * (1),…,u * [(N-1)], u * (0),u * (1),…,u * (N-1) represent the first control input, the second control input, and the Nth control input, respectively. Represents the cost function, Let u(l) represent the system state stored by the controller at time k, where u(l) represents the given control sequence, N represents the prediction time domain, z(l) and z(l+1) are the system states in the Koopman space at times l and l+1, respectively, and z(N) represents the system state in the Koopman space at time N. s Let Q represent the desired state of the system, R be the state weight matrix, and ∠Q be the control input weight matrix. Q The vector 2-norm operation, ‖·‖, represents the weights of the state weight matrix Q. R The weights represent the vector L2 norm operation that controls the input weight matrix R; For input constraints, For state constraints, The latest system state stored by the controller; z(0) represents the system state in the Koopman space at time 0; Indicates the system state The lifting function that elevates to the Koopman space.

[0064] 3) Based on the asynchronous DoS attack state of dual communication channels with a finite duration, the optimal control sequence U at time k is determined. * (k) selects the optimal control input and applies it to the actuator of the target nonlinear system.

[0065] 3) Specifically:

[0066] Detect whether a DoS attack has occurred on the sensor-controller channel at time k; if no DoS attack has occurred on the sensor-controller channel, the controller can obtain the latest state of the target system and update the system state value stored in the controller (using symbols). (meaning), that is, to order If a DoS attack occurs on the sensor-controller channel, the controller cannot obtain the actual state of the target system, and the stored system state value is equal to the stored value at the previous moment, that is...

[0067] Detect whether a controller-actuator channel DoS attack occurs at time k; if no controller-actuator channel DoS attack occurs, clear the memory of the actuator of the target nonlinear system and set the optimal control sequence U at time k. * (k) Passes the first optimal control input u from the actuator's memory. * (1) It acts on the target nonlinear system;

[0068] If a DoS attack occurs on the controller-actuator channel, the l-th optimal control input u in the actuator's memory will be... * (l) Acting on the target nonlinear system.

[0069] The effects of the present invention will be further described below with reference to two simulation examples.

[0070] 1) The target nonlinear system is a van der Bohr oscillator, and its control block diagram is as follows: Figure 2 As shown. The dynamic equations of the discrete-time system of the van der Bohr oscillator are as follows:

[0071]

[0072] Where, x 1,k x represents the displacement of the van der Bohr oscillator. 2,k u represents the speed of the van der Bohr oscillator. k This represents the control input. The system state constraint is χ = {x: -0.6 ≤ x}. 1,k The system control input constraint is ≤0.6}. The weight matrix in the cost function is chosen as Q = [1, 0; 0, 1], R = 0.01. The initial system state is chosen as x0 = [0.5, -0.6]. T .

[0073] Simulations were performed using Matlab / IPOPT for both the attack-compensated Koopman model predictive control (K-MPC) method and the non-attack-compensated Koopman model predictive control method. Figure 3 and Figure 4 This represents the changes in the system state trajectory and control input trajectory under the two control methods. From... Figure 3 and Figure 4 It is evident that the K-MPC method with DoS attack compensation stabilizes the system state and control input near the origin, while the K-MPC method without DoS attack compensation cannot drive the system state and control input near the origin. This demonstrates that the secure data-driven predictive control method for dual-channel DoS attacks proposed in this invention can effectively guarantee system stability.

[0074] 2) The target nonlinear system is a DC generator, and its continuous-time system dynamic equations are as follows:

[0075]

[0076] in, This represents the differential of the DC motor current x1. Let χ represent the differential of the DC motor speed x2, and u represent the stator current. The system state constraint is χ = {x: -0.6 ≤ x1 ≤ 0.6}. The system control input constraint is... The weight matrix in the cost function is chosen as Q = [1, 0; 0, 1], R = 0.01. The initial system state is chosen as x0 = [0.5, -0.6]. T The reference trajectory is a square wave signal.

[0077] The above continuous-time system is discretized using the fourth-order Runge-Kutta method with a period of T = 0.01s. Figure 5 and Figure 6 This indicates the changes in motor speed and input stator current under the two control methods. From Figure 5 It can be seen that the K-MPC method with DoS attack compensation enables the motor speed to quickly track the given square wave reference trajectory and satisfies the state constraints; while the K-MPC method without DoS attack compensation cannot enable the motor speed to track the non-fixed square wave reference trajectory and exceeds the state constraints. Figure 6 It is evident that the stator current changes significantly in the K-MPC method without DoS attack compensation, making it impossible for the rotational speed to track the reference value. This demonstrates that the event-triggered data-driven predictive control method under DoS attacks proposed in this invention can effectively guarantee system stability.

[0078] The above embodiments are used to explain and illustrate the present invention, but not to limit the present invention. Any modifications and changes made to the present invention within the spirit and scope of the claims shall fall within the protection scope of the present invention.

Claims

1. A secure data-driven predictive control method for dual-channel DoS attacks, characterized in that, Includes the following steps: 1) Construct a linear control model based on Koopman operator theory; 2) Based on the linear control model, construct a finite-time domain optimization control problem and solve it using historical data of the target nonlinear system to obtain... Optimal control sequence at time 1 ; In step 2), the finite-time optimization control problem satisfies the following formula: , , , in, Represents the optimal control sequence, containing An optimal control input, satisfying , These represent the first control input, the second control input, and the third control input, respectively. One control input, Represents the cost function, For controller The system state is stored in real time. Indicates a given control sequence, Indicates the prediction time domain, and They are time points and time The system state in the Koopman space, Indicates time The system state in the Koopman space, These are the first parameter matrix and the second parameter matrix, respectively. Let be the mapping matrix from the Koopman space to the target system space. Indicates the desired state of the system. It is the state weight matrix. It controls the input weight matrix. The weights represent the state-weight matrix. Vector 2 norm operation, The weights represent the control input weight matrix. Vector 2 norm operation; For input constraints, 𝒳 represents state constraints. The latest system state stored for the controller; Indicates time The system state in the Koopman space; Indicates the system state The lifting function for upgrading to Koopman space; 3) Based on the asynchronous DoS attack status of the dual communication channels, from Optimal control sequence at time 1 The optimal control input is selected and applied to the target nonlinear system.

2. The secure data-driven predictive control method for dual-channel DoS attacks according to claim 1, characterized in that, In step 1), the linear control model satisfies the following formula: matrix It is an infinite-dimensional Koopman operator The approximate value is calculated using the following formula: in, It is a moment The system state in the Koopman space, It is a moment The system state in the Koopman space, These are the predicted values ​​of the actual system state after dimensionality reduction from the Koopman space. These are the first parameter matrix and the second parameter matrix, respectively. Let be the mapping matrix from the Koopman space to the target system space. Represents the pseudo-inverse of a matrix. Represents matrix multiplication; For different sampling states The matrix composed of lifting functions, For matrix The false reversal; For different system states A matrix composed of lifting functions; Indicates system state The corresponding promotion function; The number of system states collected; for A vector composed of control inputs; for The system states are composed of a vector.

3. The secure data-driven predictive control method for dual-channel DoS attacks according to claim 1, characterized in that, Specifically, 3) refers to: Detection Check if a DoS attack occurs on the sensor-controller channel at any given time; if no DoS attack occurs on the sensor-controller channel, the controller obtains the latest state of the target system and updates the system state value stored in the controller. If a DoS attack occurs on the sensor-controller channel, the controller cannot obtain the actual state of the target system, and the stored system state value is equal to the stored value at the previous moment. Detection Check if a controller-actuator channel DoS attack occurs at any given time; if not, clear the memory of the actuator in the target nonlinear system. Optimal control sequence at time 1 The first optimal control input in the actuator's memory is passed to the actuator's memory. Acting on the target nonlinear system; If a controller-actuator channel DoS attack occurs, the memory of the actuator will be... Optimal control input It acts on the target nonlinear system.

4. A secure data-driven predictive control device for dual-channel DoS attacks, characterized in that, include: Linear control model building unit, used to build linear control models based on Koopman operator theory; The optimal control sequence generation unit is used to construct a finite-time-domain optimization control problem based on a linear control model and solve it using historical data of the target nonlinear system to obtain the optimal control sequence. Optimal control sequence at time 1 The optimal control input generation unit is used to generate the optimal control input based on the asynchronous DoS attack status of the dual communication channels. Optimal control sequence at time 1 Select the optimal control input and apply it to the target nonlinear system; The finite-time optimization control problem satisfies the following formula: , , , in, Represents the optimal control sequence, containing An optimal control input, satisfying , These represent the first control input, the second control input, and the third control input, respectively. One control input, Represents the cost function, For controller The system state is stored in real time. Indicates a given control sequence, Indicates the prediction time domain, and They are time points and time The system state in the Koopman space, Indicates time The system state in the Koopman space, These are the first parameter matrix and the second parameter matrix, respectively. Let be the mapping matrix from the Koopman space to the target system space. Indicates the desired state of the system. It is the state weight matrix. It controls the input weight matrix. The weights represent the state-weight matrix. Vector 2 norm operation, The weights represent the control input weight matrix. Vector 2 norm operation; For input constraints, 𝒳 represents state constraints. The latest system state stored for the controller; Indicates time The system state in the Koopman space; Indicates the system state The lifting function that elevates to the Koopman space.

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