A nonlinear system false data injection attack input reconstruction resilience model predictive control method

Abstract

The application discloses a kind of nonlinear system false data injection attack input reconstruction elasticity model predictive control methods, including steps: S10: establishing information physical industrial control system model, according to launch time and duration, establish false data attack model;S20: disturbance observer is constructed, according to control system input and output data, estimate system state, and the estimation error upper bound of observer is quantified;S30: according to estimated state, the constraint of optimization problem is constructed and cost function, to construct optimal control problem;S40: according to the optimal control sequence of trigger time, construct input reconstruction strategy, select the key signal of it and apply protection, with the control sequence of protective signal is sent to actuator;S50: according to the attack detection result of system, select to use optimal control sequence or reconstruction control sequence act on controlled object.The application effectively defends false data injection attack, solves the problems, such as the reconstruction error of existing method is larger, and there is limitation in application range.

Description

Technical Field

[0001] This invention belongs to the field of security technology of industrial cyber-physical systems, and in particular relates to a method for reconstructing elastic model predictive control of nonlinear systems based on false data injection attack inputs. Background Technology

[0002] Cyber-physical systems (CPS), as a new generation of systems that deeply integrate computing, sensing, communication, and control, have become a core component of modern critical infrastructure, widely used in power grids, transportation, and industrial control. However, due to their functional criticality, structural complexity, and network openness, CPS face increasingly prominent cybersecurity threats. Cyberattacks against control systems are generally categorized into data leakage attacks, deception attacks, and destructive attacks. Data injection attacks, a typical type of deception attack, are highly covert and dangerous, posing a serious threat to the security of CPS. Such attacks can occur in sensor-to-controller and controller-to-actuator channels. In the controller-to-actuator channel, data injection attacks directly tamper with control signals, leading to more direct and severe system performance degradation or functional failure.

[0003] Model predictive control (MMDC) is an advanced control strategy capable of explicitly handling constraints. Due to its inherent predictive power and robustness, it is increasingly valued in the field of resilient predictive control. Currently, resilient MMDC methods against spoofing attacks on the controller-actuator channel mainly fall into three categories: buffer-based methods, robust control-based methods, and input reconstruction-based methods. The first two methods rely on assumptions about the duration of the attack or the attacker's energy limitations, which may no longer be effective against more capable attackers. Input reconstruction-based methods, typically within a self-triggered MMDC framework, protect key signals in the control sequence and reconstruct feasible inputs upon attack, thereby stabilizing the system during an attack. This approach reduces computational burden and relaxes restrictions on attacks.

[0004] However, existing elastic model predictive control methods based on input reconstruction are designed for deterministic systems where the system state can be directly obtained, failing to adequately consider the widespread disturbances and situations where the state cannot be directly measured in real-world systems. Secondly, when the system has not yet reached steady state, the control sequence undergoes drastic dynamic changes, leading to significant reconstruction errors in existing reconstruction methods, thus impacting control performance. Summary of the Invention

[0005] To address the aforementioned issues, this invention proposes a reconstructive elastic model predictive control method for nonlinear systems subjected to disturbances with practical control constraints and limited resources. This method effectively defends against spoofing data injection attacks and solves the problems of large reconstruction errors and limited application scope of existing methods.

[0006] To achieve the above objectives, the technical solution adopted by this invention is: a method for reconstructing an elastic model predictive control of a nonlinear system based on spurious data injection attack input, comprising the following steps:

[0007] S10: Establish a cyber-physical industrial control system model, and based on the initiation time and duration, establish a fake data attack model;

[0008] S20: Construct a disturbance observer, estimate the system state based on the input and output data of the control system, and quantify the upper bound of the observer's estimation error;

[0009] S30: Based on the obtained estimated state, construct the constraints and cost function of the optimization problem, thereby constructing the optimal control problem;

[0010] S40: Based on the optimal control sequence at the triggering time, construct an input reconstruction strategy, select key signals to apply protection, and send the control sequence with protection signals to the actuator;

[0011] S50: Based on the system's attack detection results, select to use the optimal control sequence or reconstruct the control sequence to act on the controlled object.

[0012] Furthermore, considering a discrete-time linear time-invariant system as the controlled object, a flexible model predictive control framework is established, including sensors, observers, and actuators installed locally, and controllers and observers installed at a remote control center;

[0013] At each sampling time, remote and local observers obtain state estimates using the control input signals acting on the controlled system and the measurement outputs collected by the sensors. At the trigger time, the controller uses the state estimates to generate a control sequence, selects key signals in the sequence to apply additional protection, and sends this sequence, key signal indices, and the next trigger time to the local actuator.

[0014] The system's detector checks whether an attack has occurred at the current moment. If no attack is detected, the actuator directly applies the received control sequence to the controlled system until the next trigger moment. If an attack is detected, the actuator uses the key signal of no attack to reconstruct a feasible reconstructed input sequence and applies it to the controlled system until the next trigger moment. When the system state enters the subsequently defined terminal domain, state feedback is used for control.

[0015] Furthermore, in step S10, a discrete nonlinear cyber-physical industrial control system model with additive perturbations is established:

[0016] ;

[0017] Where x(k) represents the system state, u(k) represents the system input, and y(k) represents the measurement output; d(k) and v(k) represent the process disturbance and measurement disturbance, respectively, where f is a function vector, and B, C, D, and G are known matrices;

[0018] system The control input constraints are:

[0019]

[0020] Among them, set It is a compact set and a convex set, and its interior contains the origin.

[0021] Furthermore, the fake data attack model includes:

[0022] make The time when the fake data injection attack (k) is initiated represents the temporal characteristics of the attack; for each attack instance, the attack duration is... And satisfy Based on the above parameters, the activation time intervals of all fake injection attacks are defined as a set:

[0023]

[0024] To further describe the attack state, a binary indicator variable v(k) is introduced;

[0025]

[0026] Where v(k)=1 indicates that an attack exists at time k; v(k)=0 indicates that there is no attack at time k;

[0027] Under an FDI attack, the actual control input u received by the system actuator f (k):

[0028] ;

[0029] in, It is a normal control signal. It is a fake control signal injected by the attacker.

[0030] Furthermore, in step S20, a disturbance observer is introduced to simultaneously estimate the system state and measure the disturbance based on the input and output data of the control system, and to quantify the upper bound of the observer's estimation error.

[0031] Based on the original system dynamics model, an augmented system model is constructed:

[0032] ;

[0033] in, Let be the augmented system state variables; where the augmented system matrix is ​​defined as:

[0034]

[0035] Where I is an identity matrix of appropriate dimensions;

[0036] definition Then the standard state-space expression for h(k) is:

[0037]

[0038] in, , , , , , yes The false reversal;

[0039] Next, intermediate variables are introduced. ,get:

[0040] ;

[0041] in, ;

[0042] Based on the above system, the disturbance observer is constructed as follows:

[0043] ;

[0044] in, The observer gain matrix is ​​to be determined;

[0045] Estimated system state The measurement disturbance is expressed as follows: and ,in and This is the corresponding extraction matrix;

[0046] definition It is the augmented system state estimation error. It is the original system state estimation error. It is the estimation error of the process disturbance; the gain matrix of the above observer The selection method is as follows ,in It is a positive definite symmetric matrix. and Let be a positive real number, satisfying the following linear matrix inequality:

[0047]

[0048] The observer is a uniform eventually bounded observer, satisfying , .

[0049] Furthermore, in step S30, the cost function for the optimization problem is constructed, including:

[0050] Nominal system:

[0051] Let N denote the prediction time domain, and the cost function for online optimization is:

[0052]

[0053] in, The stage cost function is, and the terminal cost function is. ,in It is a weight matrix.

[0054] Furthermore, in step S30, the optimal control problem of model predictive control is formulated as follows:

[0055]

[0056] in, For the trigger time, For the terminal constraint set;

[0057] Constraint 1: This indicates that the initial predicted state is the estimated state of the observer at the current time.

[0058] Constraint 2: Represents the constraints on the predicted state evolution, satisfying the nominal system model;

[0059] Constraint 3: Represents a constraint on the predictive control input;

[0060] Constraint 4: Represents a constraint on the predicted state of the terminal;

[0061] The solutions to the optimal control problem, namely the optimal control sequence and the corresponding optimal predicted state sequence, are denoted as follows:

[0062]

[0063] An attacker can tamper with the entire control sequence. Combined with the aforementioned fake data injection attack model, the actual control input sequence received by the executor is:

[0064]

[0065] To ensure the feasibility of model predictive control and system stability, the weight matrix... Assumption required: There exists a terminal region. A matrix and a local state feedback controller This makes it possible for all The following equation holds true:

[0066] ;

[0067] Where F and G are the stage cost and terminal cost functions in the aforementioned cost function, respectively.

[0068] This assumption means It is an invariant set of terminals in the nominal system;

[0069] Model predictive control employs a dual-mode strategy to conserve resources: once the system state is stable... Entering the terminal area Control input Will be directly controlled by the local controller The calculation is generated; the controller is located on the actuator side and does not need to be transmitted over a network, so it is not affected by spoofed data injection attacks in the communication channel.

[0070] Furthermore, in step S40, an input reconstruction strategy is constructed, employing a three-point interpolation input reconstruction mechanism, including:

[0071] Based on optimal control sequence The three key signals in the process are the initial control quantity. Intermediate control quantity and terminal control quantity The reconstructed control signal is given by the following piecewise linear interpolation formula:

[0072]

[0073] in, This is the point with the largest error during two-point reconstruction. The selection rule is:

[0074] make Input for the two-point reconstruction strategy:

[0075]

[0076] but .

[0077] Furthermore, based on the estimation error quantification results, the deviation between the reconstructed state and the optimal predicted state is given, including:

[0078] If the reconstruction control sequence generated by the three-point interpolation input reconstruction mechanism is taken from the trigger time... When applied to the system, the state is reconfigured. With the optimal predicted state Deviation between The upper bound is denoted as It is given by the following formula:

[0079]

[0080] in, express Input reconstruction error at time step; Let P be the largest eigenvalue of the matrix, and let P be the terminal weight matrix in the aforementioned cost function. and functions respectively and Lipschitz constant, This is the upper bound of the state estimation error. To measure the upper bound of the disturbance;

[0081] This indicates that the deviation between the reconstructed state and the optimal predicted state mainly comes from three parts: state estimation error, input reconstruction error, and process disturbance;

[0082] Based on the recursive feasibility and stability of model predictive control, the selection criteria for key protection signals are given:

[0083]

[0084] in:

[0085]

[0086]

[0087] in, , This refers to the trigger interval when the system is under attack, and the trigger interval when the system is not under attack. It can be obtained using a self-triggered model predictive control method, where The prediction length for model predictive control. From At any given moment, the upper bound of the state error between the reconstructed input and the optimal input is applied. , and It is the weight matrix in the cost function. and These are the terminal domain parameters in the optimal control problem. It is an adjustable parameter.

[0088] Furthermore, step 5 includes: based on the system's attack detection results, selecting to apply the optimal control sequence or reconstruct the control sequence to the controlled object, including:

[0089] If the system detects that no attack has occurred, the actuator directly applies the received control sequence to the controlled system until the next trigger time; if an attack is detected, the actuator uses the key signal that no attack has occurred to reconstruct a feasible reconstructed input sequence and applies it to the controlled system until the next trigger time; when the system state enters the terminal domain, state feedback is used for control.

[0090] The beneficial effects of adopting this technical solution are:

[0091] This invention targets cyber-physical control systems, considering process noise and measurement noise, and establishes a nonlinear discrete state-space model of the control system. An observer is designed to estimate the system state. Considering attack characteristics, a spoofing data injection attack model is established. An optimal control problem for self-triggered model predictive control is established. For the optimal input sequence obtained from solving the optimal control problem, an input reconstruction strategy is given. Based on the recursive feasibility and stability of model predictive control, conditions for selecting key protection signals are further designed. The proposed observer-based input reconstruction elastic model predictive control method can guarantee the control performance of the system under spoofing data injection attacks, while also having advantages such as relaxing attack assumptions and saving computational resources. Attached Figure Description

[0092] Figure 1 This is a schematic diagram of the process for reconstructing an elastic model predictive control method for nonlinear systems based on input from spurious data injection attacks, as per the present invention.

[0093] Figure 2 This is an example of an elastic model predictive control architecture based on input reconstruction in this invention.

[0094] Figure 3 This is a schematic diagram of the observer error in an embodiment of the present invention;

[0095] Figure 4 This is a schematic diagram of control input in an embodiment of the present invention;

[0096] Figure 5 This is a schematic diagram of state evolution in an embodiment of the present invention;

[0097] Figure 6 This is a schematic diagram of the triggering time and calculation time in an embodiment of the present invention. Detailed Implementation

[0098] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described below with reference to the accompanying drawings.

[0099] In this embodiment, see Figure 1 As shown, this invention proposes a method for reconstructing a resilient model predictive control of a nonlinear system based on input from a spoofed data injection attack, comprising the following steps:

[0100] S10: Establish a cyber-physical industrial control system model, and based on the initiation time and duration, establish a fake data attack model;

[0101] S20: Construct a disturbance observer, estimate the system state based on the input and output data of the control system, and quantify the upper bound of the observer's estimation error;

[0102] S30: Based on the obtained estimated state, construct the constraints and cost function of the optimization problem, thereby constructing the optimal control problem;

[0103] S40: Based on the optimal control sequence at the triggering time, construct an input reconstruction strategy, select key signals to apply protection, and send the control sequence with protection signals to the actuator;

[0104] S50: Based on the system's attack detection results, select to use the optimal control sequence or reconstruct the control sequence to act on the controlled object.

[0105] As an optimization of the above embodiments, such as Figure 2 As shown, considering a discrete-time linear time-invariant system as the controlled object, an elastic model predictive control framework is established, including sensors, observers, and actuators installed locally, and controllers and observers installed at a remote control center;

[0106] At each sampling time, remote and local observers obtain state estimates using the control input signals acting on the controlled system and the measurement outputs collected by the sensors. At the trigger time, the controller uses the state estimates to generate a control sequence, selects key signals in the sequence to apply additional protection, and sends this sequence, key signal indices, and the next trigger time to the local actuator.

[0107] The system's detector checks whether an attack has occurred at the current moment. If no attack is detected, the actuator directly applies the received control sequence to the controlled system until the next trigger moment. If an attack is detected, the actuator uses the key signal of no attack to reconstruct a feasible reconstructed input sequence and applies it to the controlled system until the next trigger moment. When the system state enters the subsequently defined terminal domain, state feedback is used for control.

[0108] As an optimization of the above embodiment, in step S10, a discrete nonlinear cyber-physical industrial control system model with additive perturbation is established:

[0109] ;

[0110] Where x(k) represents the system state, u(k) represents the system input, and y(k) represents the measurement output; d(k) and v(k) represent the process disturbance and measurement disturbance, respectively, where f is a function vector, and B, C, D, and G are known matrices;

[0111] system The control input constraints are:

[0112]

[0113] Among them, set It is a compact set and a convex set, and its interior contains the origin.

[0114] The fake data attack model includes:

[0115] make The time when the fake data injection attack (k) is initiated represents the temporal characteristics of the attack; for each attack instance, the attack duration is... And satisfy Based on the above parameters, the activation time intervals of all fake injection attacks are defined as a set:

[0116]

[0117] To further describe the attack state, a binary indicator variable v(k) is introduced;

[0118]

[0119] Where v(k)=1 indicates that an attack exists at time k; v(k)=0 indicates that there is no attack at time k;

[0120] Under an FDI attack, the actual control input u received by the system actuator f (k):

[0121] ;

[0122] in, It is a normal control signal. It is a fake control signal injected by the attacker.

[0123] As an optimization of the above embodiment, a disturbance observer is introduced in step S20 to simultaneously estimate the system state and the measured disturbance based on the input and output data of the control system, and to quantify the upper bound of the observer's estimation error;

[0124] Based on the original system dynamics model, an augmented system model is constructed:

[0125] ;

[0126] in, Let be the augmented system state variables; where the augmented system matrix is ​​defined as:

[0127]

[0128] Where I is an identity matrix of appropriate dimensions;

[0129] definition Then the standard state-space expression for h(k) is:

[0130]

[0131] in, , , , , , yes The false reversal;

[0132] Next, intermediate variables are introduced. ,get:

[0133] ;

[0134] in, ;

[0135] Based on the above system, the disturbance observer is constructed as follows:

[0136] ;

[0137] in, The observer gain matrix is ​​to be determined;

[0138] Estimated system state The measurement disturbance is expressed as follows: and ,in and This is the corresponding extraction matrix;

[0139] definition It is the augmented system state estimation error. It is the original system state estimation error. It is the estimation error of the process disturbance; the gain matrix of the above observer The selection method is as follows ,in It is a positive definite symmetric matrix. and Let be a positive real number, satisfying the following linear matrix inequality:

[0140]

[0141] The observer is a uniform eventually bounded observer, satisfying , .

[0142] As an optimization scheme of the above embodiment, the cost function of the optimization problem is constructed in step S30, including:

[0143] Nominal system:

[0144] Let N denote the prediction time domain, and the cost function for online optimization is:

[0145]

[0146] in, The stage cost function is, and the terminal cost function is. ,in It is a weight matrix.

[0147] The optimal control problem of model predictive control is formulated as follows:

[0148]

[0149] in, For the trigger time, For the terminal constraint set;

[0150] Constraint 1: This indicates that the initial predicted state is the estimated state of the observer at the current time.

[0151] Constraint 2: Represents the constraints on the predicted state evolution, satisfying the nominal system model;

[0152] Constraint 3: Represents a constraint on the predictive control input;

[0153] Constraint 4: Represents the constraint on the predicted state of the terminal.

[0154] The solutions to the optimal control problem, namely the optimal control sequence and the corresponding optimal predicted state sequence, are denoted as follows:

[0155]

[0156] An attacker can tamper with the entire control sequence. Combined with the aforementioned fake data injection attack model, the actual control input sequence received by the executor is:

[0157]

[0158] To ensure the feasibility of model predictive control and system stability, the weight matrix... Assumption required: There exists a terminal region. A matrix and a local state feedback controller This makes it possible for all The following equation holds true:

[0159] ;

[0160] Where F and G are the stage cost and terminal cost functions in the aforementioned cost function.

[0161] This assumption means It is an invariant set of terminals in the nominal system;

[0162] Model predictive control employs a dual-mode strategy to conserve resources: once the system state is stable... Entering the terminal area Control input Will be directly controlled by the local controller The calculation is generated; the controller is located on the actuator side and does not need to be transmitted over a network, so it is not affected by spoofed data injection attacks in the communication channel.

[0163] As an optimization of the above embodiment, in step S40, an input reconstruction strategy is constructed, employing a three-point interpolation input reconstruction mechanism, including:

[0164] Based on optimal control sequence The three key signals in the process are the initial control quantity. Intermediate control quantity and terminal control quantity The reconstructed control signal is given by the following piecewise linear interpolation formula:

[0165]

[0166] in, The point with the largest error during two-point reconstruction is selected according to the following rules:

[0167] make Input for the two-point reconstruction strategy:

[0168]

[0169] but .

[0170] When the system has not yet reached a steady state, this input reconstruction mechanism can effectively reduce reconstruction error, thereby reducing the state deviation between the reconstructed state and the optimal predicted state.

[0171] Based on the estimation error quantification results, the deviation between the reconstructed state and the optimal predicted state is given, including:

[0172] If the reconstruction control sequence generated by the three-point interpolation input reconstruction mechanism is taken from the trigger time... When applied to the system, the state is reconfigured. With the optimal predicted state Deviation between The upper bound is denoted as It is given by the following formula:

[0173]

[0174] in, express Input reconstruction error at time step; Let P be the largest eigenvalue of the matrix, and let P be the terminal weight matrix in the aforementioned cost function. and functions respectively and Lipschitz constant, This is the upper bound of the state estimation error. To measure the upper bound of the disturbance;

[0175] This indicates that the deviation between the reconstructed state and the optimal predicted state mainly comes from three parts: state estimation error, input reconstruction error, and process disturbance;

[0176] Based on the recursive feasibility and stability of model predictive control, the selection criteria for key protection signals are given:

[0177]

[0178] in:

[0179]

[0180]

[0181] in, , This refers to the trigger interval when the system is under attack, and the trigger interval when the system is not under attack. It can be obtained using the self-triggered model predictive control method, where The prediction length for model predictive control. From At any given moment, the upper bound of the state error between the reconstructed input and the optimal input is applied. , and It is the weight matrix in the cost function. and These are the terminal domain parameters in the optimal control problem. It is an adjustable parameter.

[0182] As an optimization of the above embodiment, step 5 includes: selecting the optimal control sequence or reconstructing the control sequence to act on the controlled object based on the system's attack detection results, including:

[0183] If the system detects that no attack has occurred, the actuator directly applies the received control sequence to the controlled system until the next trigger time; if an attack is detected, the actuator uses the key signal that no attack has occurred to reconstruct a feasible reconstructed input sequence and applies it to the controlled system until the next trigger time; when the system state enters the terminal domain, state feedback is used for control.

[0184] Example:

[0185] Consider a nonlinear cyber-physical system with the following parameters to verify the observer-based nonlinear system spoof data injection attack input reconstruction elastic model predictive control method proposed in this invention:

[0186] ;

[0187] ;

[0188] .

[0189] in, It is the sampling interval. For the mass of the aircraft, It is the acceleration due to gravity. and These represent the aerodynamic drag coefficients of the drone in the horizontal and vertical directions, respectively. The input constraints are .

[0190] The time domain of model predictive control is The weight matrix is , , also, , .

[0191] To illustrate the effectiveness of this invention in relaxing attack restrictions, it is assumed that spoofing attacks occur at all times. Simulation verification is performed under the above settings, and the observer error is as follows: Figure 3 Control input such as Figure 4 The state evolution results are as follows Figure 5 The trigger time and calculation time are as follows: Figure 6 Simulation results show that the proposed control algorithm can effectively resist spoofed data injection attacks while ensuring the system's control constraints and performance, thus verifying the effectiveness of the proposed algorithm.

[0192] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A method of nonlinear system false data injection attack input reconstruction resilience model predictive control, characterized in that, Including the following steps: S10: Establish a cyber-physical industrial control system model, and based on the initiation time and duration, establish a fake data attack model; S20: Construct a disturbance observer, estimate the system state based on the input and output data of the control system, and quantify the upper bound of the observer's estimation error; S30: Based on the obtained estimated state, construct the constraints and cost function of the optimization problem, thereby constructing the optimal control problem; S40: Based on the optimal control sequence at the triggering time, construct an input reconstruction strategy, select key signals to apply protection, and send the control sequence with protection signals to the actuator; S50: Based on the system's attack detection results, select to use the optimal control sequence or reconstruct the control sequence to act on the controlled object.

2. The nonlinear system false data injection attack input reconstruction resilience model predictive control method of claim 1, wherein, Considering a discrete-time linear time-invariant system as the controlled object, an elastic model predictive control framework is established, including sensors, observers, and actuators installed locally, and controllers and observers installed at a remote control center; At each sampling time, remote and local observers obtain state estimates using the control input signals acting on the controlled system and the measurement outputs collected by the sensors. At the trigger time, the controller uses the state estimates to generate a control sequence, selects key signals in the sequence to apply additional protection, and sends this sequence, key signal indices, and the next trigger time to the local actuator. The system's detector checks whether an attack has occurred at the current moment. If no attack is detected, the actuator directly applies the received control sequence to the controlled system until the next trigger moment. If an attack is detected, the actuator uses the key signal of no attack to reconstruct a feasible reconstructed input sequence and applies it to the controlled system until the next trigger moment. When the system state enters the subsequently defined terminal domain, state feedback is used for control.

3. The method for reconstructing an elastic model predictive control of a nonlinear system based on a spurious data injection attack input as described in claim 2, characterized in that, In step S10, a discrete nonlinear cyber-physical industrial control system model with additive perturbation is established: ； Where x(k) represents the system state, u(k) represents the system input, and y(k) represents the measurement output; d(k) and v(k) represent the process disturbance and measurement disturbance, respectively, where f is a function vector, and B, C, D, and G are known matrices; The control input constraints on system (1) are: Among them, set It is a compact set and a convex set, and its interior contains the origin.

4. The method for reconstructing an elastic model predictive control of a nonlinear system based on a spurious data injection attack input as described in claim 3, characterized in that, The fake data attack model includes: make The time when the fake data injection attack (k) is initiated represents the temporal characteristics of the attack; for each attack instance, the attack duration is... And satisfy Based on the above parameters, the activation time intervals of all fake injection attacks are defined as a set: To further describe the attack state, a binary indicator variable v(k) is introduced; Where v(k)=1 indicates that an attack exists at time k; v(k)=0 indicates that there is no attack at time k; In case of an FDI attack, the actual received control input u by the system actuator f (k): ; in, It is a normal control signal. It is a fake control signal injected by the attacker.

5. The method for reconstructing an elastic model predictive control of a nonlinear system based on a spurious data injection attack input as described in claim 4, characterized in that, In step S20, a disturbance observer is introduced to simultaneously estimate the system state and measure the disturbance based on the input and output data of the control system, and quantifies the upper bound of the observer's estimation error. Based on the original system dynamics model, an augmented system model is constructed: ； in, Let be the augmented system state variables; where the augmented system matrix is ​​defined as: Where I is an identity matrix of appropriate dimensions. definition Then the standard state-space expression for h(k) is: in, , , , , , yes The false reversal; Next, intermediate variables are introduced. ,get: ； in, ; Based on the above system, the disturbance observer is constructed as follows: ； in, The observer gain matrix is ​​to be determined; Estimated system state The measurement disturbance is expressed as follows: and ,in and This is the corresponding extraction matrix; definition It is the augmented system state estimation error. It is the original system state estimation error. It is the estimation error of the process disturbance; the gain matrix of the above observer The selection method is as follows ,in It is a positive definite symmetric matrix. and Let be a positive real number, satisfying the following linear matrix inequality: The observer is a uniform eventually bounded observer, satisfying , .

6. The method for reconstructing an elastic model predictive control of a nonlinear system based on a spurious data injection attack input as described in claim 5, characterized in that, In step S30, constructing the cost function for the optimization problem includes: Nominal system: Let N denote the prediction time domain, and the cost function for online optimization is: in, The stage cost is [missing information], and the terminal cost is [missing information]. ,in It is a weight matrix.

7. The method for reconstructing a resilient model predictive control of a nonlinear system based on a spurious data injection attack input as described in claim 6, characterized in that, In step S30, the model predictive control optimal control problem is formulated as follows: in, For the trigger time, For the terminal constraint set; Constraint 1: This indicates that the initial predicted state is the estimated state of the observer at the current time. Constraint 2: Represents the constraints on the predicted state evolution, satisfying the nominal system model; Constraint 3: Represents a constraint on the predictive control input; Constraint 4: Represents a constraint on the predicted state of the terminal; The solutions to the optimal control problem, namely the optimal control sequence and the corresponding optimal predicted state sequence, are denoted as follows: An attacker can tamper with the entire control sequence. Combined with the aforementioned fake data injection attack model, the actual control input sequence received by the executor is: To ensure the feasibility of model predictive control and system stability, the weight matrix... Assumption required: There exists a terminal region. A matrix and a local state feedback controller This makes it possible for all The following equation holds true: ； Where F and G are the stage cost and terminal cost functions in the aforementioned cost function, respectively. This assumption means It is an invariant set of terminals in the nominal system; Model predictive control employs a dual-mode strategy to conserve resources: once the system state is stable... Entering the terminal area Control input Will be directly controlled by the local controller The calculation is generated; the controller is located on the actuator side and does not need to be transmitted over a network, so it is not affected by spoofed data injection attacks in the communication channel.

8. The method for reconstructing a resilient model predictive control of a nonlinear system based on a spurious data injection attack input as described in claim 7, characterized in that, In step S40, an input reconstruction strategy is constructed, employing a three-point interpolation input reconstruction mechanism, including: Based on optimal control sequence The three key signals in the process are the initial control quantity. Intermediate control quantity and terminal control quantity The reconstructed control signal is given by the following piecewise linear interpolation formula: in, The point with the largest error during two-point reconstruction is selected according to the following rules: make Input for the two-point reconstruction strategy: but .

9. The method for reconstructing an elastic model predictive control of a nonlinear system based on a spurious data injection attack input as described in claim 1, characterized in that, Based on the estimation error quantification results, the deviation between the reconstructed state and the optimal predicted state is given, including: If the reconstruction control sequence generated by the three-point interpolation input reconstruction mechanism is taken from the trigger time... When applied to the system, the state is reconfigured. With the optimal predicted state Deviation between The upper bound is denoted as It is given by the following formula: in, express Input reconstruction error at time step; Let P be the largest eigenvalue of the matrix, and let P be the terminal weight matrix in the aforementioned cost function. and They are respectively and Lipschitz constant, This is the upper bound of the state estimation error. To measure the upper bound of the disturbance; This indicates that the deviation between the reconstructed state and the optimal predicted state mainly comes from three parts: state estimation error, input reconstruction error, and process disturbance; Based on the recursive feasibility and stability of model predictive control, the selection criteria for key protection signals are given: in: in, , This refers to the trigger interval when the system is under attack, and the trigger interval when the system is not under attack. This can be obtained using a self-triggered model predictive control method. The prediction length for model predictive control. From At any given moment, the upper bound of the state error between the reconstructed input and the optimal input is applied. , and It is the weight matrix in the cost function. and These are the terminal domain parameters in the optimal control problem. It is an adjustable parameter.

10. The method for reconstructing an elastic model predictive control of a nonlinear system based on a spurious data injection attack input according to claim 9, characterized in that, Step 5 includes: based on the system's attack detection results, selecting to apply the optimal control sequence or reconstruct the control sequence to the controlled object, including: If the system detects that no attack has occurred, the actuator directly applies the received control sequence to the controlled system until the next trigger time; if an attack is detected, the actuator uses the key signal that no attack has occurred to reconstruct a feasible reconstructed input sequence and applies it to the controlled system until the next trigger time; when the system state enters the terminal domain, state feedback is used for control.

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