Method and system for detecting nonlinear system attack based on constraint polynomial

Abstract

The invention relates to a nonlinear system attack detection method and system based on a constraint polynomial, and the method comprises the steps: converting a polynomial holosymmetric polytope into a constraint polynomial holosymmetric polytope through the order reduction of the constraint polynomial holosymmetric polytope according to the definition of the constraint polynomial holosymmetric polytope; constructing a state observer based on the constrained polynomial holosymmetric polytope; constructing a discrete time system model based on the state variable and the system matrix of the target system; and obtaining an observation result of the state observer in real time, and judging whether the discrete time system model is attacked or not according to the observation result. According to the method, the nonlinear state estimation problem is realized through the state observer constructed by the constraint polynomial holosymmetric polytope, and the nonlinear state estimation problem is used for solving the attack detection problem of a nonlinear system.

Description

Technical Field

[0001] The present invention belongs to the technical field of wireless communication security, and particularly relates to a detection method and system for attacking a nonlinear system based on a constrained polynomial. Background Art

[0002] In recent years, state estimation has become increasingly important in many fields, especially nonlinear state estimation. Among the existing achievements, state estimation is usually divided into two types. One is Bayesian state estimation, which is used to process the state estimation of a system with interference / noise described by random variables; the other is set-valued state estimation, which mainly deals with the state estimation of a system with unknown but bounded noise. The set-valued estimation of linear systems has been relatively mature, but there are still quite a few challenges for the set-valued estimation of nonlinear systems, and it is very important to conduct relevant research on it. Summary of the Invention

[0003] To solve the problem of nonlinear state estimation in a communication system, in a first aspect of the present invention, a detection method for attacking a nonlinear system based on a constrained polynomial is provided, including: According to the definition of the constrained polynomial full symmetric polytope, reducing the order of the constrained polynomial full symmetric polytope, and converting the polynomial full symmetric polytope into the constrained polynomial full symmetric polytope; Based on the constrained polynomial full symmetric polytope, constructing a state observer; Based on the state variables and system matrix of the target system, constructing a discrete-time system model; obtaining the observation result of the state observer in real time, and judging whether there is an attack on the discrete-time system model according to the observation result.

[0004] In some embodiments of the present invention, the reducing the order of the constrained polynomial full symmetric polytope includes: optimizing the constrained polynomial full symmetric polytope through matrix simplification; converting the optimized constrained polynomial full symmetric polytope into a polynomial full symmetric polytope; sequentially performing order reduction processing on the polynomial full symmetric polytope and converting it into a constrained polynomial full symmetric polytope.

[0005] Furthermore, the optimizing the constrained polynomial full symmetric polytope includes: removing the columns with all elements being zero in the constrained polynomial full symmetric polytope; reducing the number of columns and constraints of the generating matrix.

[0006] In some embodiments of the present invention, the constructing a state observer based on the constrained polynomial full symmetric polytope includes: determining the prediction step of the state observer based on a second-order continuously differentiable function and the constrained polynomial full symmetric polytope; determining the update step of the state observer based on a second-order continuously differentiable function and the constrained polynomial full symmetric polytope.

[0007] Further, the step of determining the update of the state observer based on the second-order continuously differentiable function and the constrained polynomial fully symmetric polytope includes: determining the update step of the state observer based on the second-order continuously differentiable function and two constrained polynomial fully symmetric polytopes.

[0008] In the above embodiment, the method for determining whether there is an attack on the discrete-time system model according to the observation result includes: calculating the system state and sensor output after the discrete-time system model is attacked based on the state observer and the discrete-time system model; determining whether there is an attack on the discrete-time system model by calculating whether there is an intersection between the system state and sensor output after the attack.

[0009] In a second aspect of the present invention, there is provided a detection system for attacks on a non-linear system based on constrained polynomials, including: A conversion module, configured to reduce the order of the constrained polynomial fully symmetric polytope according to the definition of the constrained polynomial fully symmetric polytope, and convert the polynomial fully symmetric polytope into a constrained polynomial fully symmetric polytope; A first construction module, configured to construct a state observer based on the constrained polynomial fully symmetric polytope; A second construction module, configured to construct a discrete-time system model based on the state variables and system matrix of the target system; A judgment module, configured to obtain the observation result of the state observer in real time, and determine whether there is an attack on the discrete-time system model according to the observation result.

[0010] Further, the conversion module includes: an optimization unit, configured to optimize the constrained polynomial fully symmetric polytope through matrix simplification; a conversion unit, configured to convert the optimized constrained polynomial fully symmetric polytope into a polynomial fully symmetric polytope; and a second conversion unit, configured to perform order reduction processing on the polynomial fully symmetric polytope in sequence to convert it into a constrained polynomial fully symmetric polytope.

[0011] In a third aspect of the present invention, there is provided an electronic device, including: one or more processors; a storage device, configured to store one or more programs, and when the one or more programs are executed by the one or more processors, enable the one or more processors to implement the detection method for attacks on a non-linear system based on constrained polynomials provided by the present invention in the first aspect.

[0012] In a fourth aspect of the present invention, there is provided a computer-readable medium, on which a computer program is stored, wherein the computer program, when executed by a processor, implements the detection method for attacks on a non-linear system based on constrained polynomials provided by the present invention in the first aspect.

[0013] The beneficial effects of the present invention are: The present invention proposes an estimation method based on a constrained polynomial fully symmetric polytope, which solves the problem of attack detection in nonlinear systems with unknown but bounded noise; through the reduction order technology of the constrained polynomial fully symmetric polytope, the curse of dimensionality problem in the iterative estimation process is solved. The proposed generalized intersection theory can not only handle convex noise sets but also non-convex noise sets. The present invention is applicable to application scenarios of state estimation and attack detection in nonlinear systems with unknown but bounded noise, such as unmanned aerial vehicle systems, robot systems, etc., and has high application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 is a schematic diagram of the basic process of the method for detecting attacks on nonlinear systems based on constrained polynomials in some embodiments of the present invention; Figure 2 is a schematic diagram of the specific process of the method for detecting attacks on nonlinear systems based on constrained polynomials in some embodiments of the present invention; Figure 3 is a schematic diagram of the structure of the secure estimation device for attack separation and reconstruction based on unknown inputs in some embodiments of the present invention; Figure 4 is a schematic diagram of the structure of an electronic device in some embodiments of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0015] The principles and features of the present invention will be 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.

[0016] Referring to Figure 1 and Figure 2 , in the first aspect of the present invention, a method for detecting attacks on a nonlinear system based on constrained polynomials is provided, including: S100. According to the definition of the constrained polynomial fully symmetric polytope, by reducing the order of the constrained polynomial fully symmetric polytope, the polynomial fully symmetric polytope is converted into a constrained polynomial fully symmetric polytope; S200. Based on the constrained polynomial fully symmetric polytope, construct a state observer; S300. Based on the state variables and system matrix of the target system, construct a discrete-time system model; S400. Obtain the observation results of the state observer in real time, and determine whether there is an attack in the discrete-time system model according to the observation results.

[0017] Without loss of generality, the constrained polynomial fully symmetric polytope is defined as follows: where c is the center point, Gis a generating matrix, A is a real matrix, E and R is a non - negative integer matrix, h 、 q and p are non - negative integers.

[0018] Property 1: Assume that , , , , , ; where, represents dimensional real matrix, represents dimensional non - negative integer matrix. The following properties can be obtained: , where, .

[0019] Property 2: Assume that , , , where, , , , , , , , . Then, the following can be obtained: where, .

[0020] Property 3: For each , , , , , , , , , , , the generalized intersection can be obtained according to the following algorithm: , where, , , , , .

[0021] Property 4: Denote , for each All have the following relationships: , Among them, , , , .

[0022] In step S100 of some embodiments of the present invention, the order reduction of the polynomial totally symmetric polytope includes: S101. Optimize the constrained polynomial totally symmetric polytope through matrix simplification; Furthermore, optimizing the constrained polynomial totally symmetric polytope includes: removing the columns in which all elements of the constrained polynomial totally symmetric polytope are zero; Specifically, reduce the columns of the generating matrix as follows: Among them, .

[0023] Denote , then there is , , , , , .

[0024] Then reduce the columns of the generating matrix and the number of constraints: , Among them, . Denote , then there is , , , , , .

[0025] S102. Convert the optimized constrained polynomial totally symmetric polytope into a polynomial totally symmetric polytope; perform order reduction processing on the polynomial totally symmetric polytope in sequence and convert it into a constrained polynomial totally symmetric polytope.

[0026] Specifically, converting the constrained polynomial totally symmetric polytope into a polynomial totally symmetric polytope is represented by the following function: (1) Among them, , , .

[0027] Then, the polynomial fully symmetric polytope is reduced in order by using existing methods, and then it is converted into a constrained polynomial fully symmetric polytope.

[0028] It should be noted that the existing methods for reducing the order of the polynomial fully symmetric polytope include: the reduction method based on the dynamic event-triggering mechanism, the reduction method by optimizing the parameter matrix of the fully symmetric polytope, the intersection polynomial reduction method, the Pade approximation method, the time moment method, the continued fraction method, the reduction method based on matrix decomposition, etc.

[0029] In step S200 of some embodiments of the present invention, constructing the state observer based on the constrained polynomial fully symmetric polytope includes: S201. Determine the prediction step of the state observer based on the second-order continuously differentiable function and the constrained polynomial fully symmetric polytope. Specifically, denote as a class of second-order continuously differentiable functions, with its parameters, . Select any , then there is: (2) where, , , , , , , , , , , , , , , , .

[0030] S202. Determine the update step of the state observer based on the second-order continuously differentiable function and the constrained polynomial fully symmetric polytope.

[0031] Furthermore, determining the update step of the state observer based on the second-order continuously differentiable function and the constrained polynomial fully symmetric polytope includes: determining the update step of the state observer based on the second-order continuously differentiable function and two constrained polynomial fully symmetric polytopes.

[0032] Specifically, denote as a class of second-order continuously differentiable functions. Assume , as two constrained polynomial fully symmetric polytopes, which respectively have generating vectors and constraints. Define , Select any , then there is: (3) Wherein, , , , , , , , , , , , , , , , , .

[0033] In step S300 of some embodiments of the present invention, a discrete-time system model is constructed based on the state variables and system matrix of the target system.

[0034] Specifically, the following discrete-time system model is established: (4) (5) Wherein, is the system state, is the sensor measurement output when under attack, is the system disturbance, is the noise of the sensor; is the false data injected by the attacker; the function f and are continuously differentiable functions.

[0035] In step S400 of the above embodiment, the judging whether there is an attack on the discrete-time system model according to the observation result includes: S401. Based on the state observer and the discrete-time system model, calculate the system state and sensor output after the discrete-time system model is attacked; Assume , , .

[0036] Define , assume is known at k -1 moment. Then, it can be calculated that: (6) Among them, , , , , , , , , , , , , .

[0037] After calculation, define , and it can be calculated that: (7) Among them, , , , , , , , , , , .

[0038] S402. Determine whether there is an attack on the discrete-time system model by calculating whether there is an intersection between the system state after the attack and the sensor output.

[0039] Specifically, after calculating and , if the condition holds, then it can be determined that an attack exists, where represents the empty set, .

[0040] It can be understood that the present disclosure aims at a nonlinear discrete system with unknown but bounded noise. By designing a corresponding set-valued observer, the nonlinear state estimation problem is solved and used to solve the attack detection problem of the nonlinear system. The present invention proposes an observer design method based on a constrained polynomial fully symmetric polytope and a corresponding order reduction method to solve the curse of dimensionality problem.

[0041] Embodiment 2 Referring to Figure 3 , the second aspect of the present invention provides a detection system 1 for attacks on a nonlinear system based on a constrained polynomial, including: A conversion module 11, configured to reduce the order of a constrained polynomial totally symmetric polytope according to the definition of the constrained polynomial totally symmetric polytope, and convert the polynomial totally symmetric polytope into a constrained polynomial totally symmetric polytope; A first construction module 12, configured to construct a state observer based on the constrained polynomial totally symmetric polytope; A second construction module 13, configured to construct a discrete-time system model based on the state variables and system matrix of the target system; A judgment module 14, configured to obtain the observation result of the state observer in real time, and judge whether there is an attack on the discrete-time system model according to the observation result.

[0042] Further, the conversion module 11 includes: an optimization unit, configured to optimize the constrained polynomial totally symmetric polytope through matrix simplification; a conversion unit, configured to convert the optimized constrained polynomial totally symmetric polytope into a polynomial totally symmetric polytope; and a second conversion unit, configured to perform order reduction processing on the polynomial totally symmetric polytope in sequence to convert it into a constrained polynomial totally symmetric polytope.

[0043] Embodiment 3 Reference Figure 4 In a third aspect of the present invention, an electronic device is provided, including: one or more processors; a storage device, configured to store one or more programs, and when the one or more programs are executed by the one or more processors, enable the one or more processors to implement the detection method for attacks on a non-linear system based on constrained polynomials in the first aspect of the present invention.

[0044] The electronic device 500 may include a processing device (such as a central processing unit, a graphics processing unit, etc.) 501, which may perform various appropriate actions and processes according to the program stored in the read-only memory (ROM) 502 or the program loaded from the storage device 508 into the random access memory (RAM) 503. In the RAM 503, various programs and data required for the operation of the electronic device 500 are also stored. The processing device 501, the ROM 502, and the RAM 503 are connected to each other through a bus 504. The input / output (I / O) interface 505 is also connected to the bus 504.

[0045] Generally, the following devices may be connected to the I / O interface 505: an input device 506 including, for example, a touch screen, a touchpad, a keyboard, a mouse, a camera, a microphone, an accelerometer, a gyroscope, etc.; an output device 507 including, for example, a liquid crystal display (LCD), a speaker, a vibrator, etc.; a storage device 508 including, for example, a hard disk, etc.; and a communication device 509. The communication device 509 may allow the electronic device 500 to communicate with other devices wirelessly or wiredly to exchange data. Although Figure 4An electronic device 500 with various devices is shown, but it should be understood that it is not required to implement or have all the shown devices. Instead, more or fewer devices may be implemented or had. Figure 4 Each block shown in Figure 4 may represent a device or, as required, multiple devices.

[0046] In particular, according to an embodiment of the present disclosure, the processes described above with reference to the flowcharts can be implemented as computer software programs. For example, an embodiment of the present disclosure includes a computer program product that includes a computer program carried on a computer-readable medium, and the computer program contains program code for performing the methods shown in the flowcharts. In such an embodiment, 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, the above-mentioned functions defined in the methods of the embodiments of the present disclosure are performed. It should be noted that the computer-readable medium described in the embodiments of the present disclosure can be a computer-readable signal medium, a computer-readable storage medium, or any combination of the two. 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 of the above. More specific examples of a computer-readable storage medium can include, but are not limited to: an electrical connection with one or more wires, a portable computer disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical fiber, a portable compact disk read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the above. In the embodiments of the present disclosure, a computer-readable storage medium can be any tangible medium that contains or stores a program that can be used by or in conjunction with an instruction execution system, apparatus, or device. In the embodiments of the present disclosure, a computer-readable signal medium can include a data signal propagated in a baseband or as part of a carrier wave, in which computer-readable program code is carried. Such a propagated data signal can take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination of the above. A computer-readable signal medium can also be any computer-readable medium other than a computer-readable storage medium, and the computer-readable signal medium can send, propagate, or transmit a program for use by or in conjunction with an instruction execution system, apparatus, or device. The program code contained on the computer-readable medium can be transmitted using any appropriate medium, including but not limited to: wires, optical cables, RF (radio frequency), etc., or any suitable combination of the above.

[0047] The above computer-readable medium may be included in the above electronic device; or it may exist separately without being assembled into the electronic device. The above computer-readable medium carries one or more computer programs, and when the above one or more programs are executed by the electronic device, the electronic device is caused to: Computer program code for performing the operations of the embodiments of the present disclosure may be written in one or more programming languages or combinations thereof. The programming languages include object-oriented programming languages such as Java, Smalltalk, C++, Python, and also include conventional procedural programming languages such as the "C" language or similar programming languages. The program code may be executed entirely on the user's computer, partially on the user's computer, executed as a stand-alone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In the case of a remote computer, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or it may be connected to an external computer (for example, by using an Internet service provider to connect through the Internet).

[0048] The flowcharts and block diagrams in the accompanying drawings illustrate the possible architectures, functions, and operations of systems, methods, and computer program products according to various embodiments of the present disclosure. In this regard, each block in the flowchart or block diagram may represent a module, a program segment, or a part of code that contains one or more executable instructions for implementing the specified logical function. It should also be noted that in some alternative implementations, the functions marked in the blocks may occur in a different order than that marked in the accompanying drawings. For example, two consecutive blocks shown may actually be executed substantially in parallel, and they may sometimes be executed in the reverse order, depending on the functions involved. It should be noted that each block in the block diagram and / or flowchart, as well as the combination of blocks in the block diagram and / or flowchart, may be implemented by a dedicated hardware-based system for performing the specified functions or operations, or may be implemented by a combination of dedicated hardware and computer instructions.

[0049] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A detection method for attacks on nonlinear systems based on constraint polynomials, characterized in that, including: According to the definition of the constrained polynomial totally symmetric polytope, reducing the order of the constrained polynomial totally symmetric polytope to convert the polynomial totally symmetric polytope into the constrained polynomial totally symmetric polytope; Based on the constrained polynomial totally symmetric polytope, constructing a state observer; Based on the state variables and system matrix of the target system, constructing a discrete-time system model; Obtaining the observation results of the state observer in real time and judging whether there is an attack on the discrete-time system model according to the observation results.

2. The detection method for attacks on nonlinear systems based on constraint polynomials according to claim 1, wherein The reducing the order of the constrained polynomial totally symmetric polytope includes: Optimizing the constrained polynomial totally symmetric polytope through matrix simplification; Converting the optimized constrained polynomial totally symmetric polytope into a polynomial totally symmetric polytope; Performing successive order reduction processing on the polynomial totally symmetric polytope to convert it into a constrained polynomial totally symmetric polytope.

3. The detection method for attacking a non-linear system based on a constraint polynomial according to claim 2, characterized in that The optimizing the constrained polynomial totally symmetric polytope includes: Removing the columns with all elements being zero in the constrained polynomial totally symmetric polytope; Reducing the number of columns of the generating matrix and the number of constraints.

4. The detection method for attacking a non-linear system based on a constraint polynomial according to claim 1, characterized in that, The constructing a state observer based on the constrained polynomial totally symmetric polytope includes: Based on the second-order continuously differentiable function and the constrained polynomial totally symmetric polytope, determining the prediction step of the state observer; Based on the second-order continuously differentiable function and the constrained polynomial totally symmetric polytope, determining the update step of the state observer.

5. The detection method for attacks on nonlinear systems based on constraint polynomials according to claim 4, wherein, The determining the update step of the state observer based on the second-order continuously differentiable function and the constrained polynomial totally symmetric polytope includes: Based on the second-order continuously differentiable function and two constrained polynomial totally symmetric polytopes, determining the update step of the state observer.

6. The detection method for attacking a nonlinear system based on a constraint polynomial according to claim 1, characterized in that The judging whether there is an attack on the discrete-time system model according to the observation results includes: Based on the state observer and the discrete-time system model, calculating the system state and sensor output after the discrete-time system model is attacked; Judging whether there is an attack on the discrete-time system model by calculating whether there is an intersection between the system state and sensor output after the attack.

7. A detection system for attacks on nonlinear systems based on constraint polynomials, characterized in that, including: A conversion module for reducing the order of the constrained polynomial totally symmetric polytope according to the definition of the constrained polynomial totally symmetric polytope to convert the polynomial totally symmetric polytope into the constrained polynomial totally symmetric polytope; A first construction module for constructing a state observer based on the constrained polynomial totally symmetric polytope; A second construction module for constructing a discrete-time system model based on the state variables and system matrix of the target system; A judging module for obtaining the observation results of the state observer in real time and judging whether there is an attack on the discrete-time system model according to the observation results.

8. The detection system for attacking a nonlinear system based on a constraint polynomial according to claim 7, characterized in that, The conversion module includes: An optimization unit for optimizing the constrained polynomial totally symmetric polytope through matrix simplification; A conversion unit for converting the optimized constrained polynomial totally symmetric polytope into a polynomial totally symmetric polytope; A second conversion unit for performing successive order reduction processing on the polynomial totally symmetric polytope to convert it into a constrained polynomial totally symmetric polytope.

9. An electronic device, comprising: One or more processors; A storage device for storing one or more programs, which when executed by the one or more processors, cause the one or more processors to implement the method for detecting a non-linear system attack based on a constraint polynomial according to any one of claims 1 to 6.

10. A computer-readable medium having a computer program stored thereon, wherein, When the computer program is executed by a processor, it implements the method for detecting a non-linear system attack based on a constraint polynomial according to any one of claims 1 to 6.

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