A test method and device for tracking control system finite time secrecy attack
By constructing a state observer and attack detector using a fully symmetric multicellular method and optimization algorithm, the problem of covert attacks in wireless communication networks is solved, enabling the system to recover to a safe state within a finite time. This method is applicable to autonomous vehicles and robot control.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- WUHAN INST OF TECH
- Filing Date
- 2025-05-19
- Publication Date
- 2026-07-03
AI Technical Summary
Existing tracking and control systems lack effective detection and countermeasures against covert attacks in wireless communication networks, especially when system noise is unknown, making it difficult to guarantee that the system can be restored to a safe state within a limited time.
Using a fully symmetric multi-cell method and optimization algorithm, a state observer and an attack detector are constructed to determine the conditions for covert attacks. The optimal attack method is designed through a multi-stage objective function to ensure that the system is driven to the desired state within a finite time and the impact of the attack is eliminated.
It enables the system state to be driven to the range given by the attack within a finite time and the impact of the attack to be eliminated within a finite time. It is suitable for tracking and control systems with wireless network connections, and has particular application value in autonomous vehicles and robot control.
Smart Images
Figure CN120993875B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of automation control and wireless communication technology, specifically relating to a test method and apparatus for finite-time covert attacks on tracking control systems. Background Technology
[0002] Tracking control is an important research topic in the field of control, with a wide range of applications. Examples include trajectory tracking control for autonomous vehicles and drones, and optimal trajectory tracking control for robotic arms. However, with the development and advancement of wireless networks, network technology is increasingly being used for remote data transmission. But the existence of wireless communication networks makes transmitted signals vulnerable to malicious attacks by competitors. Therefore, researching corresponding finite-time stealth attacks on tracking control systems can provide a reference for the design of tracking controllers under attack conditions, and has significant practical implications. This invention mainly focuses on the design against stealth injection attacks.
[0003] Traditional tracking control problems typically focus on designing the optimal tracking controller to achieve the best tracking performance, or on designing a corresponding tracking controller to mitigate the impact of attacks. However, a fundamental assumption is that the attack is known and that attack detectors are ineffective against covert attacks, resulting in a lack of research specifically targeting covert attacks. Furthermore, existing tracking control systems often assume that the system noise is Gaussian white noise, making them difficult to apply to situations where statistical characteristics are unknown. Summary of the Invention
[0004] To improve the effectiveness of covert attacks, a first aspect of the present invention provides a system model constructed based on the state variables and system matrix of the target system, comprising: determining a controller gain matrix based on the system model, a reference tracking model, and a preset positive definite symmetric matrix; constructing a state observer and an attack detector of the target system using a fully symmetric polytope method and the controller gain matrix; determining covert attack conditions based on the state observer, the attack detector, and the controller gain matrix; constructing multiple objective functions for a multi-stage covert attack within a preset time period according to a preset expectation set and the covert attack conditions; and determining the optimal attack method for security testing of the target system based on the optimal solution of each objective function.
[0005] In some embodiments of the present invention, the construction of the state observer and attack detector of the target system using the fully symmetric multicellular method and the controller gain matrix includes: constructing a state observer based on the fully symmetric multicellular method and the system model; iterating the state observer through the centroid and the generator matrix using the set membership estimation method to obtain a multicellular set of observers; optimizing the centroid and the generator matrix using the minimum robust positive variable set; calculating the estimated residual using the optimized centroid and the generator matrix; and constructing the attack detector using the estimated residual.
[0006] Furthermore, the determination of covert attack conditions based on state observers and attack detectors includes: determining the polytopic set of control signal observers and estimating residuals after the attack based on the attack reference trajectory and / or control signals of the virtual attacker; and determining the covert attack conditions through the polytopic set of state observers and estimating residuals after the attack.
[0007] In some embodiments of the present invention, the step of constructing multiple objective functions for a multi-stage covert attack within a preset time period based on a preset expectation set and covert attack conditions includes: constructing a first objective function to drive the system to a preset expectation set state within a first preset time period based on the covert attack conditions; and constructing a second objective function to restore the system to the state before the attack within a second preset time period.
[0008] Furthermore, the step of constructing a first objective function that drives the system to a preset desired set state within a first preset time period based on the stealth attack conditions includes: determining the first objective function based on the center point in the preset desired set; and determining multiple constraints of the first objective function based on the generator matrix in the preset desired set and the stealth attack conditions.
[0009] Furthermore, the second objective function for restoring the system to its pre-attack state within a second preset time period includes: determining the second objective function based on the minimum trajectory prediction; and determining multiple constraints of the second objective function based on the system model and the gain control matrix.
[0010] A second aspect of the present invention provides a testing apparatus for a finite-time covert attack on a tracking control system, comprising: a first construction module for constructing a system model based on the state variables and system matrix of a target system; and determining a controller gain matrix based on the system model, a reference tracking model, and a preset positive definite symmetric matrix; a second construction module for constructing a state observer and an attack detector of the target system using a fully symmetric polytope method and the controller gain matrix; a first determination module for determining covert attack conditions based on the state observer, the attack detector, and the controller gain matrix; a third construction module for constructing multiple objective functions for a multi-stage covert attack within a preset time period according to a preset expectation set and the covert attack conditions; and a second determination module for determining the optimal attack method for security testing of the target system based on the optimal solution of each objective function.
[0011] A third aspect of the present invention provides an electronic device comprising: one or more processors; and a storage device for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors implement the test method for finite-time stealth attacks on the tracking and control system provided in the first aspect of the present invention.
[0012] In a fourth aspect, the present invention provides a computer-readable medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements a test method for finite-time stealth attacks on a tracking and control system provided in the first aspect of the present invention.
[0013] The beneficial effects of this invention are:
[0014] This invention relates to a finite-time stealth attack design. The designed attack guarantees that the system state can be driven to a given range within a finite time frame, and the effects of the attack can be eliminated within the same time frame, ensuring stealth. This stealth attack design employs an algorithm based on set member estimation and optimization, and is applicable to situations where the system contains unknown but bounded interference and noise. This invention is suitable for situations where the system and tracking controller are connected via a wireless network, or where preset trajectory data is transmitted via a wireless communication network, and has significant application value in autonomous vehicle control and robot control. Attached Figure Description
[0015] Figure 1 This is a basic flowchart illustrating the testing method for a time-limited stealth attack on a tracking control system in some embodiments of the present invention.
[0016] Figure 2 This is a schematic flowchart illustrating the testing method for a time-limited stealth attack on a tracking control system in some embodiments of the present invention.
[0017] Figure 3 This is a schematic diagram of the structure of a test device for a time-limited stealth attack on a tracking control system according to some embodiments of the present invention.
[0018] Figure 4 This is a schematic diagram of the structure of an electronic device in some embodiments of the present invention. Detailed Implementation
[0019] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.
[0020] refer to Figure 1 and Figure 2In a first aspect of the present invention, a system model is constructed based on the state variables and system matrix of a target system, comprising: S100. Determining a controller gain matrix based on the system model, a reference tracking model, and a preset positive definite symmetric matrix; S200. Constructing a state observer and an attack detector for the target system using a fully symmetric polytope method and the controller gain matrix; S300. Determining stealth attack conditions based on the state observer, the attack detector, and the controller gain matrix; S400. Constructing multiple objective functions for a multi-stage stealth attack within a preset time period according to a preset expectation set and stealth attack conditions; S500. Determining the optimal attack method for security testing of the target system based on the optimal solution of each objective function.
[0021] It should be noted that the tracking and control system involved in this application includes trajectory tracking control for autonomous vehicles and drones, optimal trajectory tracking control for robotic arms, or other automated control devices or equipment. This disclosure utilizes fully symmetric multicell theory and optimization methods to address the design problem of finite-time stealth attacks on linear discrete-time systems with unknown meaning but bounded noise.
[0022] Therefore, in step S100 of some embodiments of the present invention, the controller gain matrix is determined based on the system model, the reference tracking model, and the preset positive definite symmetric matrix; specifically, the following system model is considered:
[0023] (1)
[0024] (2)
[0025] in, A , B , C and F For the system matrix; xk For system state variables, UK For control input; ωk and vk For system interference and measurement noise, and satisfying and , and The matrix is known.
[0026] The reference tracking model is:
[0027] (3)
[0028] in, For reference trajectory state, The reference trajectory control input is used; the tracking controller employs a state feedback controller, i.e.
[0029] (4)
[0030] in, For observation status, K The controller gain matrix is calculated as follows:
[0031] (5)
[0032] in, Given a positive definite symmetric matrix, a positive definite matrix P For the equation The solution, Given a positive semidefinite symmetric matrix.
[0033] It should be noted that positive definite matrices P In addition to the Riccati equation, the above methods also provide solutions using Cholesky decomposition, LDL' decomposition, singular value decomposition, and other methods.
[0034] In step S200 of some embodiments of the present invention, constructing the state observer and attack detector of the target system using the fully symmetric polytope method and the controller gain matrix includes:
[0035] S201. Construct a state observer based on the fully symmetric multicellular method and system model;
[0036] Specifically, using the theory of fully symmetric multicells, a state observer Z of the following form is designed;
[0037] (6)
[0038] in, G The observer gain matrix to be designed, This refers to the observer state.
[0039] S202. Based on the set member estimation method, the state observer is iterated through the center point and the generating matrix to obtain the multi-cell set of the observer;
[0040] Specifically, since the observer Z contains interference and noise, it cannot be directly executed. Based on the set membership estimation theory, the centroid and generating matrix are represented in iterative form as follows:
[0041] (7)
[0042] (8)
[0043] in, for k The estimated multi-cell set of time points. Therefore, k The state estimate at time +1 satisfies .
[0044] S203. Optimize the center point and generating matrix using the minimum robust positive transformation set;
[0045] Specifically, the optimal observer gain matrix is obtained by minimizing the Frobenius norm. G The design is as follows:
[0046] (9)
[0047] Among them, positive definite symmetric matrix The solution to the following equation:
[0048] (10)
[0049] S204. Calculate the estimated residuals using the optimized center points and generator matrix; construct an attack detector using the estimated residuals.
[0050] Specifically, refer to Figure 2 The detector is equipped with an observer W of the following form:
[0051] (11)
[0052] in, L The observer gain matrix to be designed, This is the observer state. This invention does not limit the gain. L The design method only needs to ensure A-LC Stability is sufficient. Therefore, the minimum robust positive invariant set can be... ,in
[0053] , (12)
[0054] (13)
[0055] Calculate the estimated residuals based on the observer W. rk satisfy ,in Therefore, the attack detector can be designed as follows:
[0056] (14)
[0057] It should be noted that the state observer Z and the observer W are for observing or measuring the system state and control input, respectively.
[0058] Based on the above step S200, in some embodiments of the present invention, step S300, determining the stealth attack conditions based on the state observer and attack detector, includes:
[0059] S301. Based on the attack reference trajectory and / or control signals of the virtual attacker, determine the polytopic set of control signal observers after the attack and estimate the residuals;
[0060] Specifically, assuming an attacker can compromise the reference trajectory and control signals, i.e.
[0061] (15)
[0062] (16)
[0063] in, , The output of the state observer Z when attacked. and These are reference trajectory points generated by the attacker. and This is fake data that has been injected. and This describes the system's output and state when it is attacked. Assume the attack occurs within a finite time period, i.e. Inside, among which ks and ke These represent the start and end times of the attack, respectively.
[0064] When attacked, the observer Z changes from (7)-(8) to the following form:
[0065] (17)
[0066] (18)
[0067] S302. Determine the secret attack conditions by using the polytopic set of the state observer after the attack and the estimated residuals.
[0068] Specifically, when attacked, the observer W changes from (11) to the following form:
[0069] (19)
[0070] Therefore, the estimated residuals satisfy ,in
[0071] , ,
[0072] ,
[0073] .
[0074] According to the attack detector (14), the conditions for covert attacks are: ,in
[0075] , (20)
[0076] It is understandable that the design of a time-limited stealth attack mainly consists of two phases. The first phase is within a certain time period. Within, the system state is driven to a given desired set. ,in pd Given a vector, Hd Given a matrix, The second stage is within a certain time period. The internal attack effect is removed, among which This marks the end of the first phase of the attack. The two-stage attack design can transform the following two optimization problems.
[0077] In view of this, in step S400 of some embodiments of the present invention, the step of constructing multiple objective functions for a multi-stage covert attack within a preset time period based on a preset expectation set and covert attack conditions includes:
[0078] S401. Based on the conditions of covert attack, construct a first objective function that drives the system to a preset desired set of states within a first preset time period;
[0079] Furthermore, the step of constructing a first objective function that drives the system to a preset desired set state within a first preset time period based on the stealth attack conditions includes: determining the first objective function based on the center point in the preset desired set; and determining multiple constraints of the first objective function based on the generator matrix in the preset desired set and the stealth attack conditions.
[0080] Specifically, solving the attack within the first phase.
[0081] (twenty one)
[0082] Constraints: , ,
[0083] in, ,
[0084] .
[0085] S402. Construct a second objective function that restores the system to its state before the attack within a second preset time.
[0086] Furthermore, the second objective function for restoring the system to its pre-attack state within a second preset time period includes: determining the second objective function based on the minimum trajectory prediction; and determining multiple constraints of the second objective function based on the system model and the gain control matrix.
[0087] Specifically, solving for the attack in the second phase.
[0088] (twenty three)
[0089] Constraints: ,
[0090] .
[0091] In step S500 of some embodiments of the present invention, the optimal attack method for security testing of the target system is determined based on the optimal solution of each objective function.
[0092] Specifically, the first phase of the attack can be designed as follows:
[0093] , (twenty two)
[0094] The second phase of the attack can be designed as follows:
[0095] , (twenty four)
[0096] Example 2
[0097] refer to Figure 3 In a second aspect, the present invention provides a testing device 1 for tracking and controlling a system against a finite-time covert attack, comprising: a first construction module 11, used to construct a system model based on the state variables and system matrix of the target system; and to determine a controller gain matrix based on the system model, a reference tracking model, and a preset positive definite symmetric matrix; a second construction module 12, used to construct a state observer and an attack detector of the target system using a fully symmetric polytope method and the controller gain matrix; a first determination module, used to determine covert attack conditions based on the state observer, the attack detector, and the controller gain matrix; a third construction module 13, used to construct multiple objective functions for a multi-stage covert attack within a preset time according to a preset expectation set and the covert attack conditions; and a second determination module 14, used to determine the optimal attack method for security testing of the target system based on the optimal solution of each objective function.
[0098] Furthermore, the second construction module 12 includes: a first construction unit for constructing a state observer based on a fully symmetric multicellular method and a system model; an iteration unit for iterating the state observer through the centroid and generator matrix based on a set membership estimation method to obtain a multicellular set of observers; an optimization unit for optimizing the centroid and generator matrix through a minimum robust positive variable set; a second construction unit for calculating the estimated residuals through the optimized centroids and generator matrix; and constructing an attack detector through the estimated residuals.
[0099] Example 3
[0100] refer to Figure 4 In a third aspect, the present invention provides an electronic device comprising: one or more processors; and a storage device for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors implement the test method for finite-time covert attacks on the tracking and control system of the first aspect of the present invention.
[0101] Electronic device 500 may include a processing unit (e.g., a central processing unit, a graphics processing unit, etc.) 501, which can perform various appropriate actions and processes according to a program stored in read-only memory (ROM) 502 or a program loaded from storage device 508 into random access memory (RAM) 503. The RAM 503 also stores various programs and data required for the operation of electronic device 500. The processing unit 501, ROM 502, and RAM 503 are interconnected via bus 504. An input / output (I / O) interface 505 is also connected to bus 504.
[0102] Typically, the following devices can be connected to I / O interface 505: input devices 506 including, for example, touchscreens, touchpads, keyboards, mice, cameras, microphones, accelerometers, gyroscopes, etc.; output devices 507 including, for example, liquid crystal displays (LCDs), speakers, vibrators, etc.; storage devices 508 including, for example, hard disks; and communication devices 509. Communication device 509 allows electronic device 500 to communicate wirelessly or wiredly with other devices to exchange data. Although Figure 4 An electronic device 500 with various devices is shown; however, it should be understood that it is not required to implement or possess all of the devices shown. More or fewer devices may be implemented or possessed alternatively. Figure 4 Each box shown can represent a device or multiple devices as needed.
[0103] Specifically, according to embodiments of this disclosure, the processes described above with reference to the flowcharts can be implemented as computer software programs. For example, embodiments of this disclosure include a computer program product comprising a computer program carried on a computer-readable medium, the computer program containing program code for performing the methods shown in the flowcharts. In such embodiments, the computer program can be downloaded and installed from a network via a communication device 509, or installed from a storage device 508, or installed from a ROM 502. When the computer program is executed by a processing device 501, it performs the functions defined in the methods of embodiments of this disclosure. It should be noted that the computer-readable medium described in embodiments of this disclosure can be a computer-readable signal medium or a computer-readable storage medium, or any combination thereof. A computer-readable storage medium can be, for example,—but not limited to—an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof. More specific examples of computer-readable storage media may include, but are not limited to: electrical connections having one or more wires, portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof. In embodiments of this disclosure, a computer-readable storage medium may be any tangible medium containing or storing a program that can be used by or in connection with an instruction execution system, apparatus, or device. In embodiments of this disclosure, a computer-readable signal medium may include a data signal propagated in baseband or as part of a carrier wave, carrying computer-readable program code. Such propagated data signals may take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination thereof. A computer-readable signal medium may also be any computer-readable medium other than a computer-readable storage medium, which can send, propagate, or transmit a program for use by or in connection with an instruction execution system, apparatus, or device. Program code contained on a computer-readable medium may be transmitted using any suitable medium, including but not limited to: wires, optical fibers, RF (radio frequency), etc., or any suitable combination thereof.
[0104] The aforementioned computer-readable medium may be included in the aforementioned electronic device; or it may exist independently and not assembled into the electronic device. The aforementioned computer-readable medium carries one or more computer programs, which, when executed by the electronic device, cause the electronic device to:
[0105] Computer program code for performing the operations of embodiments of this disclosure can be written in one or more programming languages or a combination thereof. Programming languages include object-oriented programming languages—such as Java, Smalltalk, C++, and Python—and conventional procedural programming languages—such as the "C" language or similar programming languages. The program code can be executed entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In cases involving remote computers, the remote computer can be connected to the user's computer via any type of network—including a local area network (LAN) or a wide area network (WAN)—or can be connected to an external computer (e.g., via the Internet using an Internet service provider).
[0106] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of this disclosure. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of code containing one or more executable instructions for implementing a specified logical function. It should also be noted that in some alternative implementations, the functions indicated in the blocks may occur in a different order than those indicated in the drawings. For example, two consecutively indicated blocks may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. It should be noted that each block in the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, can be implemented using a dedicated hardware-based system that performs the specified function or operation, or using a combination of dedicated hardware and computer instructions.
[0107] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.
Claims
1. A test method for finite-time stealth attacks on tracking and control systems, characterized in that, include: Construct a system model based on the state variables and system matrix of the target system; Based on the system model, the reference tracking model, and the preset positive definite symmetric matrix, the controller gain matrix is determined; A state observer and attack detector for the target system are constructed using a fully symmetric multicellular method and the controller gain matrix. Based on the state observer, attack detector, and controller gain matrix, the conditions for covert attacks are determined. Based on the preset expectation set and the conditions for covert attacks, construct multiple objective functions for multi-stage covert attacks within a preset time period; Based on the optimal solution of each objective function, determine the best attack method for security testing of the target system; Based on the system model, the reference tracking model, and the preset positive definite symmetric matrix, the controller gain matrix is determined as follows: in, A , B , C and F For the system matrix; For system state variables, For control input; and For system interference and measurement noise, and satisfying and , and The matrix is known. The reference tracking model is: in, For reference trajectory state, The reference trajectory control input is used; the tracking controller employs a state feedback controller, i.e.: in, For observation status, K The controller gain matrix is calculated as follows: in, Given a positive definite symmetric matrix, a positive definite matrix P For the equation The solution, Given a positive semidefinite symmetric matrix; The construction of the state observer and attack detector of the target system using the fully symmetric polytope method and the controller gain matrix includes: S201. Construct a state observer based on the fully symmetric multicellular method and system model; Specifically, the state observer Z is designed using the theory of fully symmetric multicells; in, G Let be the gain matrix of the observer to be designed. The observer state; S202. Based on the set member estimation method, the state observer is iterated through the center point and the generating matrix to obtain the multi-cell set of the observer; Since the observer Z contains interference and noise, it cannot be directly executed. Based on the set membership estimation theory, the centroid and generating matrix are represented in an iterative form as follows: in, for k The estimated multicellular set of time, k The state estimate at time +1 satisfies ; S203. Optimize the center point and generating matrix using the minimum robust positive invariant set; Specifically, the optimal observer gain matrix is obtained by minimizing the Frobenius norm. G The design is as follows: Among them, positive definite symmetric matrix The solution to the following equation: S204. Calculate the estimated residuals using the optimized center points and generator matrix; construct an attack detector using the estimated residuals; The detector is equipped with an observer W of the following form: in, L Let be the gain matrix of the observer to be designed. Given the observer state, the minimum robust positive invariant set is: ,in ， Calculate the estimated residuals based on the observer W. satisfy , Therefore, the attack detector is designed as follows: State observer Z and observer W are used to observe or measure the system state and control input, respectively; The conditions for determining covert attacks based on state observers and attack detectors include: S301. Based on the attack reference trajectory and / or control signals of the virtual attacker, determine the polytopic set of control signal observers after the attack and estimate the residuals; Specifically, assume that an attacker can compromise the reference trajectory and control signals, i.e.: in, , The output of the state observer Z when attacked. and These are reference trajectory points generated by the attacker. and This is fake data that has been injected. and This represents the system's output and state when it is attacked; assuming the attack occurs within a finite time period, i.e. Inside, among which and These are the start and end times of the attack, respectively. When attacked, observer Z becomes as follows: S302. Determine the conditions for covert attacks by using the polytopic set of the state observers after the attack and the estimated residuals; Specifically, when attacked, the observer W becomes as follows: Therefore, the estimated residuals satisfy ,in, ， ， ， ； According to the attack detector, the conditions for a stealth attack are: ,in ， 。 2. The test method for finite-time stealth attacks on the tracking control system according to claim 1, characterized in that, The construction of multiple objective functions for a multi-stage stealth attack within a preset time period, based on a preset expectation set and stealth attack conditions, includes: Based on the conditions of covert attacks, a first objective function is constructed to drive the system to a preset set of desired states within a first preset time period; A second objective function is constructed to restore the system to its pre-attack state within a second preset time period.
3. The test method for finite-time stealth attacks on the tracking control system according to claim 2, characterized in that, The first objective function, which constructs the system to a preset desired set of states within a first preset time period based on the stealth attack conditions, includes: The first objective function is determined based on the center point of the preset expected set; Based on the generator matrix and stealth attack conditions in the preset expectation set, multiple constraints of the first objective function are determined.
4. A test device for testing the limited-time stealth attack on a tracking and control system, characterized in that, The testing apparatus, using the testing method as described in claim 1, comprises: The first construction module is used to construct a system model based on the state variables and system matrix of the target system; and to determine the controller gain matrix based on the system model, the reference tracking model, and the preset positive definite symmetric matrix. The second construction module is used to construct the state observer and attack detector of the target system using the fully symmetric multicellular method and the controller gain matrix; The first determination module is used to determine the conditions for covert attacks based on the state observer, attack detector, and controller gain matrix. The third construction module is used to construct multiple objective functions for a multi-stage covert attack within a preset time period based on the preset expectation set and covert attack conditions. The second determination module is used to determine the best attack method for security testing of the target system based on the optimal solution of each objective function.
5. The testing device for finite-time stealth attacks on the tracking control system according to claim 4, wherein the second building module comprises: The first building unit is used to construct a state observer based on the fully symmetric multi-cell method and system model; The iterative unit is used to iterate the state observer through the center point and the generation matrix based on the set membership estimation method to obtain a multi-cell set of observers; The optimization unit is used to optimize the center point and the generating matrix using the minimum robust positive invariant set; The second building unit is used to calculate the estimated residuals using the optimized center point and the generator matrix; and to build an attack detector using the estimated residuals.
6. An electronic device, comprising: One or more processors; A storage device for storing one or more programs, which, when executed by one or more processors, cause the one or more processors to implement a test method for a time-limited covert attack on a tracking control system as described in any one of claims 1 to 3.
7. A computer-readable medium having a computer program stored thereon, wherein, When the computer program is executed by the processor, it implements the test method for finite-time stealth attacks on the tracking and control system as described in any one of claims 1 to 3.