An observer-based adaptive output feedback tracking control method for spacecraft orbit safety

Abstract

The present invention belongs to the technical field of spacecraft approaching orbit safety control, and specifically relates to an observer-based spacecraft orbit safety adaptive output feedback tracking control method. The present invention uses an observer to estimate the unknown state of the system, uses a virtual controller and an actual control law and a parameter adaptive law to establish a spacecraft orbit safety controller, and uses the safety controller to control the spacecraft approaching orbit tracking system, effectively solving the output feedback control problem of the system being attacked by false data injection from the actuator, and ensuring the stability of the system.

Description

Technical Field

[0001] The present invention belongs to the technical field of spacecraft approaching orbit safety control, and in particular relates to an observer-based spacecraft orbit safety adaptive output feedback tracking control method. Background Art

[0002] With the continuous advancement of aerospace technology, research in related fields has deepened. Spacecraft approach orbit control systems enable controlled spacecraft to track their targets, meeting a variety of mission requirements. However, in practical applications, control systems often face unpredictable system states due to complex environments and actuator performance constraints. While a spacecraft's orbit control system is operating, its communication network may be vulnerable to false data injection attacks, causing actuator failure, disrupting the normal operation of the spacecraft, and even causing serious damage. Therefore, research on safe output feedback control methods for spacecraft approach orbits is particularly important.

[0003] Chinese patent publication number CN117262252B discloses a method for controlling autonomous rendezvous and docking of spacecraft that can achieve fuel optimization. Based on the CW equation, a relative motion dynamics model of the tracking spacecraft in a near-circular orbit is established. Considering the slow full transmission of the target spacecraft, a target port motion model is established. According to the mission requirements of autonomous rendezvous and docking of spacecraft, multiple constraints including thrust constraints, soft docking constraints and line of sight cone constraints are established. An optimal rendezvous and docking controller based on variable range model predictive control is designed, and a strategy for solving the optimal control problem is obtained. However, this invention only applies to situations where the system state is known. For spacecraft orbit systems with unknown states, this method is not applicable. At the same time, this method does not consider the security control issues of the system being attacked by cyber attacks.

[0004] Chinese patent publication number CN114995150B discloses an adaptive output feedback control method for spacecraft rendezvous under thrust limitation. This method uses a reference trajectory generator to give a system dynamic tracking model, combines anti-saturation technology to design a passivity-based adaptive state feedback controller, and uses a filter to generate a pseudo-velocity signal to propose an adaptive output feedback control law. This method can solve the output feedback tracking control problem of the control system, but does not consider the response strategy of the spacecraft orbit tracking system when it is attacked by a network, and cannot solve the security control problem of the system.

[0005] In summary, while existing technologies have made some progress in spacecraft approaching orbit control, security and stability challenges remain when faced with unknown system states and false data injection attacks. Therefore, developing an output feedback control method that can effectively resist false data injection attacks and ensure the safe and stable operation of spacecraft approaching orbit systems is an unresolved issue in the current field of aerospace technology. Summary of the Invention

[0006] The purpose of the present invention is to provide a spacecraft orbit safety adaptive output feedback tracking control method based on an observer, which effectively solves the output feedback control problem of the spacecraft approaching orbit system facing false data injection attacks and ensures the safety and stability of the system.

[0007] The technical solutions adopted by the present invention are as follows:

[0008] An observer-based spacecraft orbit safety adaptive output feedback tracking control method, characterized by comprising the following steps:

[0009] Step 1: Use the line-of-sight coordinate system to establish a spacecraft approaching orbit dynamics model, and consider the actuator false data injection attack to establish a trajectory tracking error mathematical model;

[0010] Step 2: Use the state observer to estimate the unknown system state through the system output;

[0011] Step 3: Use the tracking error and estimated state to construct a virtual controller and actual control law, and design a parameter adaptive law to estimate the upper bound of the attack signal to complete the adaptive safety control of the spacecraft orbit under the attack of unknown state and unknown false data injection.

[0012] Preferably, in step 1, a line-of-sight coordinate system is used to establish a spacecraft approaching orbit dynamics model, and a trajectory tracking error mathematical model is established by considering the actuator false data injection attack; for the line-of-sight coordinate system, its x-axis points from the center of mass of the controlled spacecraft to the target, which can be obtained by rotating the non-rotating system fixed to the controlled spacecraft with a rotation angle β and a rotation angle ε in the order of 2-3 rotations; let the distance between the controlled spacecraft and the target be ρ, and only consider ρ>0 and In the case of spacecraft approaching orbit control, the target has no power, and the difference in disturbance and gravity acceleration between the target and the controlled spacecraft is ignored. In the line of sight coordinate system, there is:

[0013]

[0014] where ρ s =[ρ00] T is the relative position vector in the line of sight coordinate system, a s is the expression of the controlled spacecraft control acceleration in the line of sight coordinate system, ω∈R 3×1 is the angular velocity vector in the line of sight, is the angular acceleration vector in the line of sight system.

[0015] In step 1, for the angular velocity vector in the line of sight system and the angular acceleration vector in the line of sight system:

[0016]

[0017] where ω × ∈R 3×1 and are the cross product matrices of the angular velocity vector in the line of sight system and the angular acceleration vector in the line of sight system, respectively, and are defined as:

[0018]

[0019] in Then the dynamic equation in the line of sight system can be obtained:

[0020]

[0021] The control law is a s =u=[u1 u2 u3] T ∈R 3×1 Suffering from bounded false data injection attack η=[η1η2η3] T ∈R 3×1 , whose upper bound is η m =[η m1 η m2 η m3 ] T ∈R 3×1 .

[0022] The dynamic equation of the spacecraft approaching orbit under the line of sight attack with actuator false data injection is:

[0023]

[0024] Let the spacecraft be controlled to track the expected trajectory and define ρ r , ε r and β r are the expected values ​​of relative distance ρ, sight angle ε and sight deflection angle β respectively, let y r =[ρ r ε r β r ] T and μ = [ρεβ] T , then the orbit tracking error is:

[0025]

[0026] From the above formula, the orbit tracking error equation can be obtained as:

[0027]

[0028] Let the system state x1=μ, Then we can get:

[0029]

[0030] Where y is the system output, the nonlinear term Satisfies the Lipschitz condition Where δ>0 is the Lipschitz constant; the time-varying matrix The second norm satisfies in It is its upper bound.

[0031] Preferably, in step 2: using a state observer to estimate the unknown system state through the system output;

[0032] First, the state space form of the system is obtained as follows:

[0033]

[0034] where x = [x1 T x2 T ] T ∈R 6×1 ,

[0035] Among them, k1>0, k2>0, k3>0, k4>0, k5>0, k6>0 are design parameters;

[0036] Design the state observer as follows:

[0037]

[0038] in is an estimate of x, and the gain matrix K is selected so that the matrix A is Hurwitz, that is, for a given positive definite matrix Q∈R 6×6 There exists a positive definite matrix P∈R 6×6 Satisfies the following equation:

[0039] A T P+PA=-Q; let To estimate the error, choose the Lyapunov function: The derivative of the Lyapunov function is obtained as:

[0040]

[0041] where λ min (Q) is the smallest eigenvalue of the matrix Q, ||PB|| is the binorm of PB, ||η m || is η m The mold length.

[0042] Preferably, the step three includes:

[0043] The tracking error and estimated state are used to construct a virtual controller and actual control law, and a parameter adaptive law is designed to estimate the upper bound of the attack signal.

[0044] According to the backstepping method, the design coordinate changes are as follows:

[0045]

[0046] Among them, α∈R 3×1 It is a virtual controller;

[0047] The first step of the backstepping method: by selecting the Lyapunov function:

[0048]

[0049] Design the virtual control law α as:

[0050] Where c1>0 is the design parameter; the derivative of the Lyapunov function is:

[0051]

[0052] The second step of the backstepping method: take the derivative and get:

[0053]

[0054] in The Lyapunov function is selected as follows:

[0055]

[0056] Where γ>0 is a design parameter, is the estimated error of the attack upper bound, is the upper bound of attack η m The actual control law and parameter adaptive law are designed as follows:

[0057]

[0058] in, Where c2>0 is the design parameter, (g(x1)) -1 is the inverse matrix of g(x1), The derivative of the Lyapunov function of the closed-loop system is obtained as:

[0059]

[0060] in

[0061] and satisfy Then the closed-loop system is eventually uniformly bounded; and by selecting appropriate values ​​for parameters k1, k2, k3, k4, k5, k6, c1, c2, and γ, adaptive safety control of the spacecraft orbit of the observer under unknown states and unknown false data injection attacks is achieved.

[0062] The technical effects achieved by the present invention are:

[0063] The present invention provides an observer-based spacecraft orbit safety adaptive output feedback tracking control method, which uses the observer to estimate the unknown state of the system, uses a virtual controller and an actual control law and a parameter adaptive law to establish a spacecraft orbit safety controller, and uses the safety controller to control the spacecraft approaching orbit tracking system, effectively solving the output feedback control problem of the system being attacked by false data injection from the actuator, and ensuring the stability of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 This is a flow chart of the observer-based spacecraft orbit safety adaptive output feedback tracking control method disclosed in an embodiment of the present invention.

[0065] Figure 2 This is a schematic diagram of the overall structure of the controller designed for an embodiment of the present invention.

[0066] Figure 3 is a tracking error curve diagram of an embodiment of the present invention;

[0067] Figure 4 4 is a parameter adaptive law curve diagram of an embodiment of the present invention.

[0068] Figure 5 This is a graph showing the actual control law of an embodiment of the present invention. DETAILED DESCRIPTION

[0069] In order to make the purpose and advantages of the present invention more clearly understood, the present invention is described in detail below with reference to the following examples. It should be understood that the following text is only used to describe one or more specific embodiments of the present invention and does not strictly limit the scope of protection of the present invention.

[0070] like Figure 1-Figure 5 As shown in the figure, an observer-based adaptive output feedback tracking control method for spacecraft orbit safety is proposed. In the line-of-sight coordinate system, a spacecraft approaching orbit dynamics model with actuator false data injection attack is established using relative distance, line-of-sight deflection angle and line-of-sight inclination angle. The unknown system state is estimated using the observer, the upper bound of the attack signal is estimated using the parameter adaptive law, and the system is controlled using a virtual controller and the actual control law, thus realizing adaptive safety control in the face of unknown state and unknown false data injection attack.

[0071] like Figure 1 As shown in FIG, the observer-based spacecraft orbit safety adaptive output feedback tracking control method includes the following steps:

[0072] Step 1: Use the line of sight coordinate system Ox s y s z s Establish a dynamic model of spacecraft approaching orbit, and consider the attack of actuator false data injection to establish a mathematical model of trajectory tracking error;

[0073] The spacecraft is brought close to the orbital system to meet the following conditions:

[0074] Condition 1: The controlled spacecraft approaches the target at a close distance, the difference in gravitational acceleration between the two is negligible, and the target has no power;

[0075] Condition 2: The approaching orbit system meets the following boundary conditions:

[0076]

[0077] Where ρ is the relative distance between the controlled spacecraft and the target, ε is the line of sight inclination, and β is the line of sight deflection.

[0078] Condition 3: Expected trajectory y r ∈R 3×1 , its derivative and the second-order derivative Known, bounded and continuous;

[0079] Condition 4: The false data injection attack energy suffered by the spacecraft is limited, and the false data injection attack η=[η1 η2η3] T ∈R 3×1 The absolute value of each component of has an upper bound, which is η m =[η m1 η m2 η m3 ] T ∈R 3×1 .

[0080] Condition 5: The nonlinear function f(x1,x2) satisfies the Lipschitz condition:

[0081]

[0082] where δ>0 is the Lipschitz constant.

[0083] Condition 6: The time-varying matrix g(x1) satisfies in It is its upper bound.

[0084] First, define the geocentric inertial coordinate system Ox i y iz i , whose origin is fixed to the center of mass of the Earth, Ox i The axis points to the vernal equinox, Oz i The axis points to the North Pole, Oy i The axis is determined by the right-hand rule. i y i z i In the example, let the position vectors of the controlled spacecraft and the target be r c ∈R 3×1 and r t ∈R 3×1 , and their orbital dynamics are modeled as:

[0085]

[0086] where || r c || and ||r t || are r c and r t The modulus of the earth, κ is the gravitational constant, a c ∈R 3×1 is the control acceleration of the controlled spacecraft. In the body coordinate system Ox fixed to the center of mass of the controlled spacecraft b y b z b In the example, let its coordinate axis be parallel to Ox i y i z i , let the relative position vector be ρ b =r t -r c , the relative orbital dynamics model is obtained as:

[0087]

[0088] in It is the difference in gravitational acceleration between the controlled spacecraft and the target.

[0089] For the line of sight coordinate system Ox s y s z s , whose x-axis points from the center of mass of the controlled spacecraft to the target, can be obtained by rotating the system in the order of 2-3 with the line of sight deflection angle β and the line of sight inclination angle ε. b The axis is rotated by an angle β to obtain the intermediate coordinate system Ox′y′z′:

[0090]

[0091] in Then rotate the angle ε around the Oz′ axis to obtain the sight coordinate system Ox s y s zs :

[0092]

[0093] The transformation equation can be obtained from the rotation process:

[0094]

[0095] in is the transformation matrix, and the line of sight Ox s y s z s The following dynamic equation:

[0096]

[0097] where a s =R bs a c is the acceleration of the controlled spacecraft in the line of sight. According to the vector derivative law, we know that:

[0098]

[0099] where ρ s ∈R 3×1 is the relative position vector in the line of sight, ω∈R 3×1 is the angular velocity vector in the line of sight, is the angular acceleration vector in the line of sight system, which are defined as:

[0100]

[0101] where ω × ∈R 3×1 and are their cross product matrices, defined as:

[0102]

[0103] in Then the dynamic equation in the line of sight system can be obtained:

[0104]

[0105] Assume a g →0; acceleration is a s , that is, control law a s =u=[u1 u2 u3] T ∈R 3×1 If the spacecraft is attacked by false data injection η, the dynamic equation of the spacecraft approaching the orbit in the line of sight system is:

[0106]

[0107] Let the spacecraft be controlled to track the expected trajectory and define ρ r , ε r and β r are the expected values ​​of relative distance ρ, sight angle ε and sight deflection angle β respectively, let y r =[ρ r ε r β r ] T and μ = [ρεβ] T , then the orbit tracking error is:

[0108]

[0109] From the above formula, the orbit tracking error equation can be obtained as:

[0110]

[0111] Let the system state x1=μ, Then we can get:

[0112]

[0113] Where y is the system output, the nonlinear term Time-varying matrix

[0114]

[0115] Step 2: Use the state observer to estimate the unknown system state through the system output.

[0116] Combining (18), the state space form of the system is as follows:

[0117]

[0118] where x = [x1 T x2 T ] T ∈R 6×1 ,

[0119]

[0120] Among them, k1>0, k2>0, k3>0, k4>0, k5>0, k6>0 are design parameters.

[0121] Rewrite the system formula (19) as follows:

[0122]

[0123] in is an estimate of x,

[0124] Design the state observer as follows:

[0125]

[0126] Combining (8) and (9), the observation error equation can be obtained as follows:

[0127]

[0128] in is the observation error.

[0129] The gain matrix K is selected so that the matrix A is Hurwitz, that is, for a given positive definite matrix Q∈R 6×6 There exists a positive definite matrix P∈R 6×6 Satisfies the following equation:

[0130] A T P+PA=-Q (24)

[0131] Select the Lyapunov function as follows:

[0132]

[0133] Taking the derivative of V0, we get:

[0134]

[0135] According to Young's inequality:

[0136]

[0137] Where ||PB|| is the bi-norm of PB. Substituting (14)-(15) into (13), we get:

[0138]

[0139] where λ min (Q) is the smallest eigenvalue of the matrix Q.

[0140] Step 3: Use the tracking error and estimated state to construct a virtual controller and actual control law, and design a parameter adaptive law to estimate the upper bound of the attack signal.

[0141] The design coordinates change as follows:

[0142]

[0143] Among them, α∈R 3×1 It is a virtual controller.

[0144] Step 1: Select the Lyapunov function as follows:

[0145]

[0146] Where c1>0 is a design parameter. Combining (17) and (18), we can get:

[0147]

[0148] in, is the estimation error.

[0149] From Young's inequality we can get:

[0150]

[0151] Substituting (20) into (19) yields:

[0152]

[0153] Design the virtual control law α as:

[0154]

[0155] Among them, c1>0 is a design parameter.

[0156] Substituting (22) into (21) we get:

[0157]

[0158] Step 2: From (17)(22), we can get:

[0159]

[0160] in The Lyapunov function is selected as follows:

[0161]

[0162] Where γ>0 is a design parameter, is the estimated error of the attack upper bound, is the upper bound of attack η m estimated value.

[0163] Combining (23)-(25) we can get:

[0164]

[0165] Combining (16), (23), (24), and (26), we can obtain:

[0166]

[0167] According to Young's inequality:

[0168]

[0169] Combining (27) with (28) and (29) we can get:

[0170]

[0171] The actual control law and parameter adaptation law are designed as follows:

[0172]

[0173] Where c2>0 is the design parameter, (g(x1)) -1 is the inverse matrix of g(x1),

[0174] Combining (30)-(32) we can get:

[0175]

[0176] Combined with the attack signal, we can get an upper bound:

[0177]

[0178] Among them, Λ and g(x1) are both diagonal matrices, and From the assumption, we know that |η i |≤η mi ,i=1,2,3

[0179] According to Young's inequality:

[0180]

[0181] Combining (33)-(35) we get

[0182]

[0183] in

[0184] and satisfy The closed-loop system is ultimately uniformly bounded. By selecting appropriate values ​​for the parameters k1, k2, k3, k4, k5, k6, c1, c2, and γ, the system tracking error can be kept within a small neighborhood of the origin. This solves the problem of safe adaptive output feedback control of spacecraft approaching orbits in the presence of actuator false data injection attacks.

[0185] In this case, if Figure 2The overall schematic diagram shown is composed of a state observer, a parameter adaptive law, a virtual controller, an actual control law, and a spacecraft approaching orbit system with an actuator false data injection attack. The control process of the method of the present invention is as follows: the system output is sent to the state observer to obtain an estimated state, the estimated state is used to generate a state adaptive law, the tracking error signal is used to generate a virtual control law, the virtual control signal and the adaptive parameter are sent to the actual controller to generate an actual control signal, the spacecraft approaching orbit system with an actuator false data injection attack is controlled, and the system output is compared with a reference signal (expected trajectory) to generate a tracking error signal.

[0186] In order to verify the effectiveness of the solution proposed in this case, the system attack signal is given as

[0187] η=[0.2+0.1sin0.3t 0.1cos0.2t 0.1-0.1sin0.1t] T (50)

[0188] The expected trajectory is set as:

[0189] y r =[ρ r ε r β r ] T =[4+cos0.1t 0.6sin0.2t 0.6cos0.2t] T (51)

[0190] The initial state information of the system is

[0191] x1(0)=μ(0)=[8-0.9-1.8] T (52)

[0192]

[0193] Select parameters k1=k2=k3=20, k4=k5=k6=600, c1=0.2, c2=0.8, γ=20, Q=5I 6×6 , verified Right now If the parameter requirements are met, the spacecraft control effect diagram is as follows: Figure 3-5 shown. Figure 3 is the spacecraft tracking error trajectory diagram; z1=ρ-ρ r , z2=β-β r , z3=ε-ε r are the three components of tracking error. Figure 3 It can be seen that the adaptive safety output feedback tracking control method of the present invention can enable the spacecraft to track the desired trajectory, and the tracking error converges within 20s, with good tracking effect. Figure 4 The curve diagram of the parameter adaptive law, They are the three components of the parameter adaptation law; Figure 5 is the true control law curve, where u1, u2, and u3 are the three components of the true control law. It can be seen that the adaptive safety output feedback tracking control scheme proposed in this invention can enable a spacecraft with unknown system state to track the target trajectory under the attack of actuator false data injection.

[0194] Through the above technical solution, the present invention provides an observer-based spacecraft orbit safety adaptive output feedback tracking control method, which uses the observer to estimate the unknown state of the system, uses the virtual controller and the actual control law and the parameter adaptive law to establish the spacecraft orbit safety controller, and uses the safety controller to control the spacecraft approaching orbit tracking system, effectively solving the output feedback control problem of the system being attacked by false data injection from the actuator, and ensuring the stability of the system.

[0195] The foregoing is merely a preferred embodiment of the present invention. It should be noted that those skilled in the art may make various improvements and modifications without departing from the principles of the present invention, and such improvements and modifications are also within the scope of protection of the present invention. Structures, devices, and operating methods not specifically described or explained herein shall, unless otherwise specified or limited, be implemented in accordance with conventional means in the art.

Claims

1. An observer-based spacecraft orbit safety adaptive output feedback tracking control method, characterized by: The following steps are involved: Step 1: Use the line-of-sight coordinate system to establish a spacecraft approaching orbit dynamics model, and consider the actuator false data injection attack to establish a trajectory tracking error mathematical model; In step 1, a line-of-sight coordinate system is used to establish a spacecraft approaching orbit dynamics model, and a trajectory tracking error mathematical model is established considering the actuator false data injection attack; for the line-of-sight coordinate system, its x-axis points from the center of mass of the controlled spacecraft to the target, which can be obtained by rotating the non-rotating system fixed to the controlled spacecraft with a rotation angle β and a rotation angle ε in the order of 2-3 rotations; let the distance between the controlled spacecraft and the target be ρ, and only consider ρ>0 and In the case of spacecraft approaching orbit control, the target has no power, and the difference in disturbance and gravity acceleration between the target and the controlled spacecraft is ignored. In the line of sight coordinate system, there is: where ρ s =[ρ00] T is the relative position vector in the line of sight coordinate system, a s is the expression of the controlled spacecraft control acceleration in the line of sight coordinate system, ω∈R 3×1 is the angular velocity vector in the line of sight, is the angular acceleration vector in the line of sight; In step 1, for the angular velocity vector in the line of sight system and the angular acceleration vector in the line of sight system: where ω × ∈R 3×1 and are the cross product matrices of the angular velocity vector in the line of sight system and the angular acceleration vector in the line of sight system, respectively, and are defined as: in Then the dynamic equation in the line of sight system can be obtained: The control law is a s =u=[u1 u2 u3] T ∈R 3×1 Suffering from bounded false data injection attack η=[η1η2η3] T ∈R 3×1 , whose upper bound is η m =[η m1 η m2 η m3 ] T ∈R 3×1 ; The dynamic equation of the spacecraft approaching orbit under the line of sight attack with actuator false data injection is: Let the spacecraft be controlled to track the expected trajectory and define ρ r , ε r and β r are the expected values ​​of relative distance ρ, sight angle ε and sight deflection angle β respectively, let y r =[ρ r ε r β r ] T and μ = [ρεβ] T , then the orbit tracking error is: From the above formula, the orbit tracking error equation can be obtained as: Let the system state x1=μ, Then we can get: Where y is the system output, the nonlinear term Satisfies the Lipschitz condition Where δ>0 is the Lipschitz constant; the time-varying matrix The second norm satisfies in It is its upper bound; Step 2: Use the state observer to estimate the unknown system state through the system output; Step 3: Use the tracking error and estimated state to construct a virtual controller and actual control law, and design a parameter adaptive law to estimate the upper bound of the attack signal to complete the adaptive safety control of the spacecraft orbit under the attack of unknown state and unknown false data injection.

2. The observer-based adaptive output feedback tracking control method for spacecraft orbit safety according to claim 1, wherein in step 2: using a state observer to estimate the unknown system state through the system output; First, the state space form of the system is obtained as follows: where x = x1 T x2 TT ∈R 6×1 , Among them, k1>0, k2>0, k3>0, k4>0, k5>0, k6>0 are design parameters; Design the state observer as follows: in is an estimate of x, and the gain matrix K is selected so that the matrix A is Hurwitz, that is, for a given positive definite matrix Q∈R 6×6 There exists a positive definite matrix P∈R 6×6 Satisfies the following equation: A T P+PA=-Q; let To estimate the error, choose the Lyapunov function: The derivative of the Lyapunov function is obtained as: where λ min (Q) is the smallest eigenvalue of the matrix Q, ||PB|| is the binorm of PB, ||η m || is η m The mold length.

3. The observer-based spacecraft orbit safety adaptive output feedback tracking control method according to claim 1, characterized in that: The step 3 comprises: The tracking error and estimated state are used to construct a virtual controller and actual control law, and a parameter adaptive law is designed to estimate the upper bound of the attack signal. According to the backstepping method, the design coordinate changes are as follows: Among them, α∈R 3×1 It is a virtual controller; The first step of the backstepping method: by selecting the Lyapunov function: Design the virtual control law α as: Where c1>0 is the design parameter; the derivative of the Lyapunov function is: The second step of the backstepping method: take the derivative and get: in The Lyapunov function is selected as follows: Where γ>0 is a design parameter, is the estimated error of the attack upper bound, is the upper bound of attack η m The actual control law and parameter adaptive law are designed as follows: in, Where c2>0 is the design parameter, (g(x1)) -1 is the inverse matrix of g(x1), The derivative of the Lyapunov function of the closed-loop system is obtained as: in and satisfy Then the closed-loop system is eventually uniformly bounded; and by selecting appropriate values ​​for parameters k1, k2, k3, k4, k5, k6, c1, c2, and γ, adaptive safety control of the spacecraft orbit of the observer under unknown states and unknown false data injection attacks is achieved.

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