A Multi-Spacecraft Secure Cooperative Control Method for Resisting Multi-Path Deception Attacks

Abstract

The present invention belongs to the technical field of multi-spacecraft formation safety control, and particularly relates to a multi-spacecraft safety cooperative control method for resisting multi-channel spoofing attacks, including establishing the attitude kinematics and dynamics models of spacecraft using Rodriguez parameters, establishing the mathematical model of multi-channel spoofing attacks and preprocessing the attacked spacecraft model, designing the spacecraft consensus error variables based on the states of the damaged system, constructing the correlation relationship between the actual attitude output of the spacecraft and the damaged consensus error variables, estimating the unknown terms in the spacecraft attitude model caused by spoofing attacks using an adaptive fuzzy update law, and constructing a first virtual controller and a second virtual controller using a first-order filter. The present invention can effectively handle the problem of instability when the system is under cyber attacks, and thus effectively solves the problem of the system being under spoofing attacks, ensuring the stability of the system.

Description

Technical Field

[0001] The present invention belongs to the technical field of multi - spacecraft formation safety control, and particularly relates to a multi - spacecraft safety collaborative control method for resisting multiple spoofing attacks. Background Art

[0002] In the context of the rapid development of space technology, the field of spacecraft control has become a research hotspot. With the increasing complexity of space exploration missions, the traditional single - spacecraft mission execution mode is difficult to meet the requirements, and multi - spacecraft collaborative control technology has emerged. A multi - spacecraft system can improve the efficiency and reliability of mission execution through division of labor and cooperation. However, when a multi - spacecraft system executes a mission, its wireless communication link is vulnerable to spoofing attacks, which may lead to incorrect information transmission, affecting the normal operation of the spacecraft and even causing incalculable losses. Therefore, it is particularly important to study the safety collaborative control method of multi - spacecraft under spoofing attack environments;

[0003] For example, Chinese Patent Publication No. CN111752292A proposes a distributed spacecraft tracking control method. This method observes the leader information through a distributed finite - time state observer, adopts a fast non - singular terminal sliding - mode control algorithm to handle external disturbances, and designs an input saturation function to limit the attitude control torque to achieve fast collaborative tracking of the spacecraft attitude. Although this method has made breakthroughs in control accuracy and response speed, it does not involve coping strategies when the system is under spoofing attacks;

[0004] Furthermore, Chinese Patent Publication No. CN117806164A discloses a multi - spacecraft formation consensus control method based on dynamic event method under denial - of - service attacks. This method constructs a multi - spacecraft formation system model and designs a dynamic event - triggered mechanism to reduce the communication frequency to cope with time - series denial - of - service attacks and ensure the stability of the system under attacks. However, when modeling, this method uses a linear model, which has a large difference from the highly non - linear and strongly coupled mathematical model of actual spacecraft, resulting in large modeling errors. In addition, this method also fails to solve the problem of system instability caused by spoofing attacks;

[0005] In summary, the existing technologies have made certain progress in multi - spacecraft collaborative control, but there are still challenges in terms of security and stability when facing spoofing attacks. Therefore, developing a collaborative control method that can effectively resist spoofing attacks and ensure the safe and stable operation of multi - spacecraft systems is an urgent problem to be solved in the current space technology field. Summary of the Invention

[0006] The object of the present invention is to provide a multi-spacecraft secure cooperative control method for resisting multi-path spoofing attacks, which can effectively handle the problem of instability when the system is under cyber attacks, and thus effectively solve the problem of the system being under spoofing attacks, ensuring the stability of the system.

[0007] The technical solution adopted by the present invention is specifically as follows:

[0008] A multi-spacecraft secure cooperative control method for resisting multi-path spoofing attacks includes the following steps:

[0009] S1: Use Rodriguez parameters to establish the attitude kinematics and dynamics models of spacecraft, establish the mathematical model of multi-path spoofing attacks, and preprocess the attacked spacecraft model;

[0010] S2: Design the spacecraft consistency error variable based on the state of the damaged system, and construct the correlation relationship between the actual attitude output of the spacecraft and the damaged consistency error variable;

[0011] S3: Use the adaptive fuzzy update law to estimate the unknown terms in the spacecraft attitude model caused by spoofing attacks, and use a first-order filter to construct a first virtual controller and a second virtual controller;

[0012] S4: Construct a spacecraft attitude controller according to the first virtual controller and the second virtual controller, and design a Nussbaum function to handle the problem of unknown control direction caused by spoofing attacks.

[0013] The technical effects achieved by the present invention are as follows:

[0014] The present invention conducts a security analysis on the injection of incorrect data into the spacecraft communication link by spoofing attacks, designs a fuzzy adaptive update law to effectively estimate the unknown function for the problem of some functions being unknown caused by spoofing attacks hijacking spacecraft nodes, constructs a Nussbaum function for direction search for the problem of unknown control direction, effectively handles the problem of instability when the system is under cyber attacks, uses Rodriguez parameters to establish the spacecraft attitude kinematics and dynamics models, uses the first virtual controller, the second virtual controller and the adaptive fuzzy update law to establish a spacecraft security controller, and uses the security controller to conduct formation cooperative control on multiple spacecraft, effectively solving the problem of the system being under spoofing attacks and ensuring the stability of the system. Description of the Drawings

[0015] Figure 1 is the flowchart of the present invention;

[0016] Figure 2 is the attacked communication topology diagram in the present invention;

[0017] Figure 3It is the curve graph of the attitude tracking change of the spacecraft in the present invention;

[0018] Figure 4 It is the curve graph of the adaptive parameter change of the spacecraft in the present invention;

[0019] Figure 5 It is the curve graph of the Nussbaum function parameter of the spacecraft in the present invention. Detailed implementation manners

[0020] In order to make the purpose and advantages of the present invention clearer and more understandable, the present invention will be specifically described below in conjunction with embodiments. It should be understood that the following text only describes one or several specific implementation manners of the present invention, and does not strictly limit the scope of protection of the specific claims of the present invention.

[0021] As Figure 1 shown, a multi-spacecraft secure cooperative control method for resisting multi-path spoofing attacks. The present invention uses a spoofing attack model to construct a damaged spacecraft attitude model, designs a spacecraft consensus error variable based on the state of the damaged system, constructs the correlation relationship between the actual attitude output of the spacecraft and the damaged consensus error variable, uses an adaptive fuzzy update law to estimate the unknown terms in the spacecraft attitude model caused by spoofing attacks, constructs a first virtual controller and a second virtual controller according to a first-order low-pass filter, and designs a Nussbaum function to handle the possible problem of unknown control direction caused by spoofing attacks, effectively resisting the problem of system instability caused by unknown spoofing attacks, and having high practical significance.

[0022] Embodiment 1:

[0023] It includes the following steps:

[0024] S1: Use Rodriguez parameters to establish the attitude kinematics and dynamics models of the spacecraft, establish the mathematical model of multi-path spoofing attacks and preprocess the attacked spacecraft model;

[0025] When establishing the attitude kinematics and dynamics models of the spacecraft, first use the modified Rodriguez parameter vector to model the kinematic attitude of the i-th rigid body spacecraft, and its formula is:

[0026]

[0027] where, q i = [q xi , q yi , q zi T and ρ i are respectively the main axis and the main rotation angle of the spacecraft, ω i ∈ R 3 = [ω​i1 , ω i2 , ω i3 T is the angular velocity of the spacecraft in the body coordinate system, ε i ∈ (-2π, 2π);

[0028] Jacobian matrix where the skew-symmetric matrix

[0029] I3 ∈ R 3×3 is the identity matrix;

[0030] The dynamic attitude of the i-th rigid-body spacecraft is:

[0031]

[0032] where, J i ∈ R 3×3 is the inertia matrix, u i ∈ R 3 is the control torque, and the skew-symmetric matrix satisfies

[0033] ]

[0034] If the leader spacecraft node is not under spoofing attack, then the leader spacecraft is modeled as:

[0035]

[0036] where, x l and y0 are the outputs of the leader spacecraft, and f(x l ) is a known non-linear function of the leader spacecraft;

[0037] Let P i (q i ) = T i -1 (q i ), and the Lagrangian form of the equation is obtained as:

[0038]

[0039] where:

[0040]

[0041]

[0042] Let x i,1 = q i , The i-th follower spacecraft is rewritten as: ​

[0043]

[0044] The network topology of the spacecraft and the spoofing attacks on the spacecraft nodes are modeled as:

[0045]

[0046] where x i,m is the m-th order state of the i-th follower spacecraft, is the m-th order state of the i-th follower spacecraft under spoofing attack, u i is the controller of the i-th follower spacecraft, is the controller of the i-th follower spacecraft under spoofing attack, y0 is the output signal of the leader spacecraft, is the output signal of the leader spacecraft under spoofing attack, are all time-varying functions, is an unknown bounded function;

[0047] Assume satisfies and where and are unknown constants, then it can be obtained that and where and are all unknown constants;

[0048] Combining the spacecraft attitude model and the mathematical model of multi-channel spoofing attacks, the mathematical model of the spoofed spacecraft is described as:

[0049]

[0050] S2: Design the spacecraft consensus error variable based on the damaged system state, and construct the correlation between the actual attitude output of the spacecraft and the damaged consensus error variable;

[0051] The specific process is as follows:

[0052] Use the following formula to complete S2, which is:

[0053]

[0054] Establish the output consensus error for the i-th follower spacecraft, where a i,j represents the connection relationship between the i-th follower spacecraft and the j-th follower spacecraft. If the two are connected or there is data transmission between them, then a i,j > 0, otherwise, a i,j = 0;

[0055] b i represents the communication relationship between the \(i\)-th follower spacecraft and the leader spacecraft, and are the damaged Rodriguez parameter vectors of the \(i\)-th and \(j\)-th follower spacecrafts respectively, N is the number of spacecrafts.

[0056] The correlation relationship between the damaged consensus error vector and the actual consensus error vector of the spacecraft is constructed by the following formula:

[0057]

[0058] where \(D = diag\{d_1,...,d N \}\) is the in-degree matrix of the spacecraft topology, where The Laplacian matrix \(L = D - A\).

[0059] S3: Estimate the unknown terms in the spacecraft attitude model caused by deception attacks using an adaptive fuzzy update law, and construct the first virtual controller and the second virtual controller using a first-order filter;

[0060] The first virtual controller is obtained by the following formula:

[0061]

[0062] where \(k i,1 is the first virtual controller gain \(i = 1,2,3\), is the non-linear compensation term;

[0063] A first-order low-pass filter is established by the following formula to avoid differentiating the first virtual control vector:

[0064]

[0065] where the input \(\alpha i,1 and the output of the filter have equal initial values, i.e.,

[0066] The error vector of the first-order low-pass filter is constructed by the following formula:

[0067]

[0068] The attitude dynamics error vector of the \(i\)-th spacecraft is constructed by the following formula:

[0069]

[0070] The spacecraft attitude controller obtains the original state vector of the system by the following formula:

[0071]

[0072] The mathematical model of the spacecraft in Equation (4) is re - modeled for controller design by using the relationship between the damaged system state and the original system state;

[0073] Where:

[0074]

[0075] Due to the unknown function θ i (t) in the deception attack, the functions and in the re - modeled spacecraft attitude model are both unknown;

[0076] By processing the unknown functions, we get:

[0077]

[0078] Where, σ i,1 (ο i )≥1 is an unknown function, which satisfies is an unknown function, is the unknown upper bound of the function σ i,1 (ο i );

[0079] Due to some unknown functions in the spacecraft model caused by the deception attack, a fuzzy logic system is introduced:

[0080]

[0081] Where, is an unknown function, is the fuzzy parameter vector, is the non - linear function vector, which satisfies:

[0082]

[0083] Where is 's membership function, s = 1, 2;

[0084] The adaptive fuzzy update law is:

[0085]

[0086] Where, is δ i,1 = max{||θ i,1 || 2The estimated value of}, γ i and C i are all unknown parameters;

[0087] Construct the second virtual controller using the following formula:

[0088]

[0089] where k i,2 is the gain of the second virtual controller is a vector of nonlinear functions that satisfies:

[0090]

[0091] where is 's membership function, s = 1, 2.

[0092] S4: Design a Nussbaum function to handle the possible unknown problem of the control direction caused by spoofing attacks. Construct a spacecraft attitude controller based on the first virtual controller and the second virtual controller, and design a Nussbaum function to handle the unknown problem of the control direction caused by spoofing attacks.

[0093] Due to spoofing attacks, in the reconstructed spacecraft mathematical model (4) is unknown, and its direction cannot be directly determined. Use the following formula to establish the actual spacecraft controller:

[0094] u i = N(χ i )α i,2

[0095] where:

[0096]

[0097] sin(a2χ i (t))

[0098]

[0099] And a1 and a2 are design parameters.

[0100] Example 2:

[0101] Prove the stability, that is, feasibility, of the scheme designed by the present invention through the Lyapunov function. Design the Lyapunov function as follows:

[0102] Select the Lyapunov function as:

[0103]

[0104] Derive the derivative of V i,1 with respect to time, and the resulting formula is:

[0105]

[0106] where B i = d i + b i . According to Young's inequality, we have:

[0107]

[0108] Substitute the above inequality into Equation (20) to obtain:

[0109]

[0110] where:

[0111]

[0112] Since the function contains the unknown function and its derivative Therefore, introduce the fuzzy logic function as:

[0113]

[0114] We can get:

[0115]

[0116] Using Young's inequality again, we have:

[0117]

[0118] where δ i,1 = max{||θ i,1 || 2}. Further, we can obtain:

[0119]

[0120] Substitute the first virtual controller α i,1 and the adaptive update law into the above formula to get:

[0121]

[0122] Choose the Lyapunov function as:

[0123]

[0124] Derive the derivative of V i,2Taking the derivative with respect to time, the formula is obtained as follows:

[0125]

[0126] Where:

[0127]

[0128] The fuzzy logic function is introduced as:

[0129]

[0130] It can be obtained that:

[0131]

[0132] Where:

[0133] δ i,2 = max{||θ i,2 || 2}

[0134]

[0135] Using the second virtual controller and the adaptive fuzzy update law, it can be obtained that:

[0136]

[0137] For a multi-spacecraft model with one leader spacecraft and three followers, the overall Lyapunov function is constructed:

[0138]

[0139] Taking the derivative of the above Lyapunov function, it can be obtained that:

[0140]

[0141] Taking the derivative of the error vector of the first-order low-pass filter, it can be obtained that:

[0142]

[0143] Therefore, it can be obtained that:

[0144]

[0145] Where:

[0146]

[0147] is a known function composed of state and parameter estimation values. Then there exists a set, which is:

[0148]

[0149] Satisfied in the set Δ i above is a positive constant. Then it can be obtained that:

[0150]

[0151] where:

[0152]

[0153] s = 1, 2

[0154] Therefore, the uniformly ultimately bounded realization of the closed-loop system can be achieved. By using the correlation relationship between the damaged consensus error vector and the actual consensus error vector of the spacecraft in Step 2 and by selecting appropriate values for the parameters, the attitude of the spacecraft follower can well track the attitude of the leader, and the safe and stable tracking of the follower spacecraft can still be ensured under the condition of multi-path spoofing attacks;

[0155] Figure 2 is the topological link graph of the multi-spacecraft attitude network topology of the present invention under spoofing attacks. The spoofing attacks in this embodiment are modeled as:

[0156]

[0157] i = 1, 2, 3

[0158] By selecting the parameters as:

[0159] k i,1 = 50, k i,2 = 50, C i,1 = 0.1, C i,2 = 0.1, c i,2 = 1, i = 1, 2, 3

[0160] The effect of the spacecraft safety cooperative controller is as Figures 3 - 5 shown. Figure 3 is the Rodriguez parameter change curve graph of the spacecraft; the r1, r2, r3 curves represent the reference signals; x 111 , x 211 , x 311 curves represent the output curve graphs of the first dimension of spacecraft 1 - 3; x 112 , x 212 , x 312 curves represent the output curve graphs of the second dimension of spacecraft 1 - 3; x 113 , x 213 , x 313 curves represent the output curve graphs of the third dimension of spacecraft 1 - 3.

[0161] According to Figure 3 it can be known that the proposed fuzzy adaptive security control scheme of the present invention can enable the attitude of the follower spacecraft to track that of the leader spacecraft, and the tracking effect is very good; Figure 4 It is a curve graph of the adaptive fuzzy parameters of multiple spacecrafts, Figure 5 and it is a parameter curve graph of the Nussbaum function. It can be seen that the Nussbaum function can well search for the direction of the spacecraft. Thus, it can be known that the proposed fuzzy adaptive security control scheme of the present invention can enable the attitude of the follower spacecraft to track that of the leader spacecraft under spoofing attacks.

[0162] Through the above technical solutions, a multi-spacecraft adaptive fuzzy security cooperative control method for resisting multi-channel spoofing attacks provided by the present invention uses Rodriguez parameters to establish the spacecraft attitude kinematics and dynamics models, uses the first virtual controller, the second virtual controller and the adaptive fuzzy update law to establish the spacecraft security controller, and uses the security controller to perform formation cooperative control on multiple spacecrafts, effectively solving the problem that the system is under spoofing attacks and ensuring the stability of the system.

[0163] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can also be made, and these improvements and refinements should also be regarded as the protection scope of the present invention. The structures, devices and operation methods not specifically described and explained in the present invention, unless otherwise specified and limited, are implemented according to the conventional means in the art.

Claims

1. A multi-spacecraft secure cooperative control method for resisting multi-path spoofing attacks, characterized in that: It includes the following steps: S1: Use Rodriguez parameters to establish the attitude kinematics and dynamics models of the spacecraft, establish the mathematical model of multi-path deception attacks, and preprocess the attacked spacecraft model; In S1, when establishing the attitude kinematics and dynamics models of the spacecraft, first, the modified Rodriguez parameter vector is used to model the kinematic attitude of the i-th rigid-body spacecraft, and its formula is: where q i = [q xi , q yi , q zi T and ρ i are the principal axis and the principal rotation angle of the spacecraft, respectively, ω i ∈ R 3 = [ω i1 , ω i2 , ω i3 T is the angular velocity of the spacecraft in the body coordinate system, and ε i ∈ (-2π, 2π);​​ Jacobian matrix where the skew-symmetric matrix I3 ∈ R 3×3 is the identity matrix; The dynamic attitude of the i-th rigid-body spacecraft is: where, J i ∈R 3×3 is the inertia matrix, u i ∈R 3 is the control torque, and the skew-symmetric matrix satisfies If the leader spacecraft node is not under deception attack, the leader spacecraft is modeled as: y0 = x l , where x l and y0 are the outputs of the leader spacecraft, and f(x l ) is a known nonlinear function of the leader spacecraft; Let P i (q i ) = T i -1 (q i ), to obtain the Lagrangian form of the equation, which is: Where: Let x i,1 = q i , The i-th follower spacecraft is rewritten as: y i = x i,1 The network topology of the spacecraft and the deception attacks suffered by the spacecraft nodes are modeled as: where x i,m is the m-th order state of the i-th follower spacecraft, is the m-th order state of the i-th follower spacecraft under spoofing attack, u i is the controller of the i-th follower spacecraft, is the controller of the i-th follower spacecraft under spoofing attack, y0 is the output signal of the leader spacecraft, is the output signal of the leader spacecraft under spoofing attack, are all time-varying functions, is an unknown bounded function; Hypothesis Satisfy And Where And Are unknown constants, then it can be obtained that And Where θ i , And Are all unknown constants; Integrating the spacecraft attitude model and the mathematical model of multi-path deception attacks, the mathematical model of the spacecraft under deception attack is described as: y i = x i,1 S2: Design the spacecraft consensus error variable based on the damaged system state, and construct the correlation between the actual attitude output of the spacecraft and the damaged consensus error variable; The specific process in S2 is: Use the following formula: Establish the output consistency error for the $i$-th follower spacecraft, where $a$ i,j represents the connection relationship between the $i$-th follower spacecraft and the $j$-th follower spacecraft. If the two are connected or there is data transmission between them, then $a$ i,j > 0, otherwise, $a$ i,j = 0; b i Indicates the communication relationship between the \(i\)-th follower spacecraft and the leader spacecraft, and are the damaged Rodriguez parameter vectors of the \(i\)-th and \(j\)-th follower spacecrafts respectively, N is the number of spacecrafts; Use the following formula to construct the correlation between the damaged consensus error vector and the actual consensus error vector of the spacecraft. The formula is: where \(D = \text{diag}\{d_1,\ldots,d\) N \} is the in-degree matrix of the spacecraft topology, where the Laplacian matrix \(L = D - A\); S3: Use the adaptive fuzzy update law to estimate the unknown terms in the spacecraft attitude model caused by deception attacks, and use a first-order filter to construct the first virtual controller and the second virtual controller; S4: Construct the spacecraft attitude controller according to the first virtual controller and the second virtual controller, and design the Nussbaum function to handle the problem that the control direction is unknown due to deception attacks.

2. The multi-spacecraft safety cooperative control method according to claim 1, characterized in that: In S3, the first virtual controller is obtained using the following formula. The specific formula is: where k i,1 is the first virtual controller gain for i = 1, 2, 3; is the non-linear compensation term; Use the following formula to establish a first-order low-pass filter to avoid taking the derivative of the first virtual control vector. The specific formula is: where the input α of the filter i,1 and the output have equal initial values, i.e., Use the following formula to construct the error vector of the first-order low-pass filter. The specific formula is: Use the following formula to construct the attitude dynamics error vector of the i-th spacecraft. The specific formula is:

3. The multi-spacecraft safety collaborative control method according to claim 2, wherein: Use the following formula to enable the spacecraft attitude controller to obtain the original state vector of the system. The specific formula is: y i = x i,1 (14) Use the relationship between the damaged system state and the original system state to re-model the spacecraft mathematical model in Equation (4) for controller design; Where: Since the function θ i (t) is unknown in the deception attack, the functions and in the spacecraft attitude model after re - modeling are both unknown; By processing the unknown function, we get: where, σ i,1 (ο i ) ≥ 1 is an unknown function that satisfies is an unknown function is the function σ i,1 (ο i )'s unknown upper bound; Due to the fact that some functions in the spacecraft model are unknown caused by deception attacks, a fuzzy logic system is introduced: wherein, is an unknown function, is a fuzzy parameter vector, is a non-linear function vector, which satisfies: Among them is 's membership function, s = 1, 2; The adaptive fuzzy update law is: Among them, is 's estimated value, γ i and C i are both unknown parameters; Use the following formula to construct the second virtual controller. The specific formula is: where k i,2 is the second virtual controller gain is a non - linear function vector, which satisfies: Among them, is 's membership function, where s = 1, 2.

4. The multi-spacecraft safety cooperative control method according to claim 1, wherein: In S4, the actual spacecraft controller is established. Its specific formula is: u i = N(χ i )α i,2 Where: sin(a2χ i (t)) And a1 and a2 are design parameters.

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