Chaotic control method for tether satellite based on ogy control and delayed feedback control

Abstract

The application discloses a tethered satellite chaos control method based on OGY control and delay feedback control and belongs to the technical field of spacecraft control. The application aims at solving the problem that the existing tethered satellite chaos control cannot consider both fast capture and high-precision locking. The application comprises the following steps: calculating a tethered satellite state point, tether length and tether length change rate based on a tethered satellite system dynamics model; calculating the Euclidean distance between the current tethered satellite state point and a target fixed point on a Poincare section; if the Euclidean distance is greater than a distance threshold value, performing delay feedback control in step three; otherwise, performing OGY control in step four; the delay feedback control is based on the current tethered satellite state point and a gain matrix to obtain a control amount, and the tether winding and unwinding speed is adjusted according to the control amount; the OGY control is based on the Jacobian matrix of the system dynamics equation to calculate a small speed disturbance amount for offsetting the unstable direction deviation; and the tether winding and unwinding speed is calculated according to the small speed disturbance amount. The application realizes high-precision steady control under low energy consumption.

Description

Technical Field

[0001] This invention relates to a chaotic control method for tethered satellites based on OGY control and delay feedback control, belonging to the field of spacecraft control technology. Background Technology

[0002] A tethered satellite system is a special spacecraft system consisting of two or more spacecraft connected by tethers. Due to its enormous potential in space debris removal, orbit transfer, space physics exploration, and deep space missions, it has become a research hotspot in the current aerospace field.

[0003] In practical missions, satellites often operate in elliptical orbits. As the orbital eccentricity increases, the tethered satellite's orbital angular dynamics exhibit strong nonlinear characteristics due to the periodic changes in the Earth's gravitational gradient torque. Particularly when the eccentricity exceeds a certain critical value, the system displays significant chaotic motion. Chaotic motion is unpredictable and extremely sensitive to initial conditions, which can lead to excessively large tether angles or even tether slack and entanglement, seriously threatening the satellite's safe and stable operation.

[0004] Currently, control methods for chaotic motion include the Ott-Grebogi-Yorke (OGY) method and Delayed Feedback Control (DFC) method. The principle of the OGY method is to stabilize the system on an unstable periodic orbit by applying a small parameter perturbation on the Poincaré section. However, since it belongs to local control logic, the control can only take effect when the system state evolves randomly into a very small linear neighborhood of the target fixed point. This results in a very long waiting time for the system to enter this neighborhood in the global chaotic environment of high eccentricity orbit, and poor real-time performance. Although the DFC method uses the difference between the current state and the delayed state for feedback, it has advantages in guiding a wide range. However, in the complex time-varying environment of elliptical orbits, single delayed feedback control often has the problem of limited accuracy in locking the final stable state, making it difficult to ensure that the system can achieve high-precision long-term residence under perturbation.

[0005] In summary, existing chaotic control methods struggle to simultaneously achieve both rapid acquisition and high-precision locking when dealing with tethered satellites in highly eccentric elliptical orbits. Therefore, designing a control algorithm that can both quickly guide the system out of the chaotic region and stabilize it on a predetermined, regular orbit is a significant technical challenge currently facing the on-orbit operation of tethered satellites. Summary of the Invention

[0006] To address the problem that existing chaotic control methods for tethered satellites cannot simultaneously achieve rapid acquisition and high-precision locking, this invention provides a chaotic control method for tethered satellites based on OGY control and delay feedback control.

[0007] This invention provides a chaotic control method for tethered satellites based on OGY control and delay feedback control, comprising:

[0008] Step 1: Calculate the state points, tether length, and rate of change of tether length of the tethered satellite based on the dynamic model of the tethered satellite system; the state points of the tethered satellite include the in-plane angles of the tethered satellite and the rate of change of the in-plane angles of the tethered satellite.

[0009] Step 2: Calculate the Euclidean distance between the current tethered satellite state point and the target stationary point on the Poincaré section. If the Euclidean distance is greater than the distance threshold, execute the delay feedback control in Step 3; otherwise, execute the OGY control in Step 4.

[0010] Step 3, Delay Feedback Control: Calculate the control quantity based on the current tether satellite state point and gain matrix. The control quantity is the ratio of the tether length change rate to the tether length. Adjust the tether release and take-up speed according to the control quantity. After each control cycle is completed, return to Step 2.

[0011] Step 4, OGY Control: Calculate the small velocity disturbance to compensate for the unstable direction offset based on the Jacobian matrix of the system dynamics equations; calculate the tether deployment and retrieval speeds based on the small velocity disturbances; after each control cycle, calculate the Euclidean distance between the current tethered satellite state point on the Poincaré section and the target stationary point. If the Euclidean distance is greater than the stability tolerance, continue OGY control until the Euclidean distance is no greater than the stability tolerance, maintaining the OGY control law to ensure the tethered satellite maintains a stable operating state.

[0012] According to the chaotic control method for tethered satellites based on OGY control and delay feedback control of the present invention, the dynamic model of the tethered satellite system in step one is as follows:

[0013] ,

[0014] ,

[0015] In the formula For the interior angle of the tethered satellite surface, For orbital eccentricity, For true near point angle, The rate of change of the in-plane angle of the tethered satellite. The rate of change of the rope length. This refers to the length of the rope.

[0016] ,

[0017] In the formula To control the quantity;

[0018] Tethered Satellite Status Points for:

[0019] .

[0020] According to the chaotic control method for tethered satellites based on OGY control and delay feedback control of the present invention, in step two, it is assumed that the state point of the tethered satellite in the nth control cycle is... :

[0021] The Euclidean distance between the tethered satellite state point and the target stationary point on the Poincaré section during the nth control period is:

[0022] ,

[0023] In the formula For Euclidean distance, The target is a fixed point;

[0024] Set distance threshold as ,like Execute delayed feedback control, if Execute OGY control.

[0025] According to the chaotic control method for tethered satellites based on OGY control and delay feedback control of the present invention, the target fixed point satisfy:

[0026] ,

[0027] In the formula This is a mapping function.

[0028] According to the chaotic control method for tethered satellites based on OGY control and delay feedback control of the present invention, in step three, the control quantity is calculated. The method is as follows:

[0029] ,

[0030] ,

[0031] In the formula Here is the gain matrix. This is the time delay value.

[0032] According to the present invention, a chaotic control method for tethered satellites based on OGY control and delay feedback control, the gain matrix... The calculation method is as follows:

[0033] Define nonlinear functions :

[0034] ;

[0035] The dynamic equations of the tethered satellite system are obtained as follows:

[0036] ,

[0037] At the target fixed point Linearization yields:

[0038] ,

[0039] In the formula Let Jacobian matrix be the current state. The Jacobian matrix represents the time-delay state;

[0040] ,

[0041] ,

[0042] Assumption For the steady-state solution, the characteristic equation is derived as follows:

[0043] ,

[0044] In the formula The characteristic matrix, For eigenvalues, It is the identity matrix; if the eigenvalues maximum value real part , The steady-state solution is asymptotically stable if , The steady-state solution is unstable; the eigenvalues ​​are solved using numerical methods. And judge The stability of the steady-state solution is obtained, and thus the gain matrix that stabilizes the tethered satellite system is obtained. .

[0045] According to the chaotic control method for tethered satellites based on OGY control and delay feedback control of the present invention, the method for calculating the small velocity disturbance amount to compensate for the unstable direction offset in step four is as follows:

[0046] The discrete form of the dynamic equations of the tethered satellite system is:

[0047] ,

[0048] In the formula For Poincaré mapping;

[0049] Target stationary point on an unstable periodic orbit Linearize the position:

[0050] ,

[0051] In the formula This is the control quantity for the nth control cycle. For the target control quantity, Let Jacobian be the state matrix. The Jacobian matrix is ​​the control quantity;

[0052] ,

[0053] ,

[0054] The state Jacobian matrix is ​​obtained through singular value decomposition. stable eigenvalues and unstable eigenvalues and stable eigenvectors and unstable eigenvectors ;in , ;

[0055] Define the equation:

[0056] ,

[0057] In the formula For stable basis vectors, These are unstable basis vectors. and They are mutually orthogonal; and satisfy:

[0058] and ;

[0059] Then we get:

[0060] ,

[0061] ,

[0062] In the formula This is the maximum control quantity.

[0063] According to the tethered satellite chaotic control method based on OGY control and delay feedback control of the present invention, the calculation method for the tether release and take-up speed in step four is as follows:

[0064] ,

[0065] In the formula This refers to the speed at which the rope is released and retrieved.

[0066] According to the chaotic control method for tethered satellites based on OGY control and delay feedback control of the present invention, in step four, the stability tolerance is set to... If the current If so, it is determined that the tethered satellite has successfully locked onto the target's unstable periodic orbit position.

[0067] According to the chaotic control method for tethered satellites based on OGY control and delay feedback control of the present invention, in step four, if Return to step two.

[0068] The beneficial effects of this invention are as follows: The method of this invention overcomes the limitations of single control methods in global chaotic environments, such as small capture area, long convergence time, and difficulty in achieving steady-state accuracy. By coupling the delayed feedback control in velocity mode with the OGY method in stages, the method directly uses the real-time mapping state of the system on the Poincaré section for logical switching, eliminating the complex global trajectory planning process. It can also ensure that the system can be quickly guided into a stable neighborhood and accurately locked under large chaotic fluctuations, thereby obtaining a regularized operating state that meets the task requirements.

[0069] This invention enables rapid capture and suppression of large-scale chaotic motion. Addressing the global chaotic characteristics exhibited by tethered satellites in high eccentricity orbits, this invention introduces delayed feedback control (DFC) as a guiding phase. This overcomes the limitation of traditional control methods (such as the single OGY method) which require waiting for the system state to randomly enter the minimum neighborhood of the target fixed point, significantly shortening the waiting time for control to take effect and improving mission execution efficiency.

[0070] The method of this invention balances robustness and steady-state accuracy. A hybrid control strategy is employed: when the system is far from the target, coarse adjustment is performed using the DFC method, which has low dependence on model parameters, enhancing the system's robustness to complex spatial disturbances; after entering the neighborhood, the method switches to the OGY method for precise locking, ensuring that the final orbital pendulum angle can be accurately stabilized on the unstable periodic orbit, achieving high-precision steady-state control with low energy consumption.

[0071] This invention effectively solves the problem of system instability under high eccentricity environments. By dynamically adjusting the tether release and take-up speed, the invention can effectively suppress nonlinear oscillations caused by drastic orbital changes without altering the basic satellite configuration, preventing the risk of tether slack or entanglement, and broadening the applicability of tethered satellites in high eccentricity scientific exploration orbits.

[0072] It involves minimal computation and is easy to implement in engineering. Based on Poincaré mapping theory, this invention features concise and clear control expressions. The DFC stage only requires delayed state feedback, and the OGY stage involves only simple linear projection calculations, placing low demands on the computing power of the satellite's onboard computer and demonstrating strong feasibility for engineering applications. Attached Figure Description

[0073] Figure 1This is a flowchart of the chaotic control method for tethered satellites based on OGY control and delay feedback control as described in this invention;

[0074] Figure 2 This is a schematic diagram of the chaotic control method for tethered satellites;

[0075] Figure 3 In a specific embodiment, the inner angle of the tethered satellite surface Following the true nearest point angle A graph showing the changes;

[0076] Figure 4 The rate of change of the in-plane angle of the tethered satellite in a specific embodiment Following the true nearest point angle A graph showing the changes;

[0077] Figure 5 The control quantity in a specific embodiment A graph showing how the time t changes the time.

[0078] Figure 6 yes Figure 3 Scatter plot of the Poincaré map;

[0079] Figure 7 yes Figure 4 Scatter plot of the Poincaré map. Detailed Implementation

[0080] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0081] Specific Implementation Method 1: Combination Figure 1 and Figure 2 As shown, this invention provides a chaotic control method for tethered satellites based on OGY control and delay feedback control, including:

[0082] Step 1: Calculate the state points, tether length, and rate of change of tether length of the tethered satellite based on the dynamic model of the tethered satellite system; the state points of the tethered satellite include the in-plane angles of the tethered satellite and the rate of change of the in-plane angles of the tethered satellite.

[0083] Step 2: Calculate the Euclidean distance between the current tethered satellite state point and the target stationary point on the Poincaré section. If the Euclidean distance is greater than the distance threshold, execute the delay feedback control in Step 3; otherwise, execute the OGY control in Step 4.

[0084] Step 3, Delay Feedback Control: Calculate the control quantity based on the current tether satellite state point and gain matrix. The control quantity is the ratio of the tether length change rate to the tether length. Adjust the tether release and take-up speed according to the control quantity. After each control cycle is completed, return to Step 2.

[0085] Step 4, OGY Control: Calculate the small velocity disturbance to compensate for the unstable direction offset based on the Jacobian matrix of the system dynamics equations; calculate the tether deployment and retrieval speeds based on the small velocity disturbances; after each control cycle, calculate the Euclidean distance between the current tethered satellite state point on the Poincaré section and the target stationary point. If the Euclidean distance is greater than the stability tolerance, continue OGY control until the Euclidean distance is no greater than the stability tolerance, maintaining the OGY control law to ensure the tethered satellite maintains a stable operating state.

[0086] Step one begins with system initialization and state parameter acquisition. This includes obtaining the initial in-plane angles of the tethered satellite. Target fixed point That is, the point on the Poincaré section where the unstable periodic orbit is expected to stabilize, and the initial true anomaly angle. =0. Preset gain matrix for Delay Feedback Control (DFC). Time delay value OGY control gain and distance threshold for mode switching. During satellite flight, the in-plane angle of the tethered satellite is calculated or acquired in real time. and its rate of change And map it onto the Poincaré section according to the orbital period.

[0087] Step two involves determining the motion area and selecting the control mode. If the current state point is outside the linear neighborhood of the target stationary point, i.e., the distance is greater than the set switching distance threshold, step three, DFC control, is executed; if the state point enters the threshold range of the target stationary point, it is determined that the system has the conditions for acquisition, and step four, OGY control, is executed.

[0088] In step three, the system is driven to rapidly approach the target point by adjusting the tether release and retraction speed. After each control cycle is completed, step two is executed again to continuously evaluate whether the mode switching conditions are met.

[0089] In step four, based on the Jacobian matrix of the system's dynamic equations, the small velocity disturbance required to keep the system's eigenvalues ​​within the unit circle is calculated. By applying a small velocity control signal, the offset in the unstable direction is compensated, and the system trajectory is locked onto the originally unstable periodic orbit.

[0090] During step four, stability monitoring is performed. It verifies whether the distance between the system state point and the target fixed point remains consistently less than the preset stability tolerance. If the tolerance requirement is met, the current control law is maintained to suppress chaos; if a severe external disturbance causes the deviation to exceed the switching threshold again, step two is returned for capture.

[0091] Furthermore, in step one, the initial state parameters of the tethered satellite system are given in the J2000 geocentric inertial coordinate system, and the dynamic model of the tethered satellite system used is as follows:

[0092] ,

[0093] ,

[0094] In the formula For the interior angle of the tethered satellite surface, For orbital eccentricity, For true near point angle, The rate of change of the in-plane angle of the tethered satellite. The rate of change of the rope length. This refers to the length of the rope.

[0095] ,

[0096] In the formula To control the quantity, the size can be adjusted as needed;

[0097] Tethered Satellite Status Points for:

[0098] .

[0099] During initialization, it is necessary to pre-calculate the true anomaly angle using long-term orbital evolution data. Searching for unstable periodic orbits on the Poincaré section, where The number of times the nearest point was reached;

[0100] In step two, it is assumed that the tethered satellite state point in the nth control cycle is... :

[0101] The Euclidean distance between the tethered satellite state point and the target stationary point on the Poincaré section during the nth control period is:

[0102] ,

[0103] In the formula For Euclidean distance, The target is a fixed point;

[0104] Set distance threshold as ,like Execute delayed feedback control, if Execute OGY control.

[0105] Distance threshold The setting of [the parameter] directly determines the convergence of the control system, and can be determined based on the effective region of linearization in the OGY control stage. If [the parameter]... If the setting is too large, forcing OGY control in the highly nonlinear region will lead to iterative divergence; if If the setting is too small, the system may linger in the chaotic region for a long time, making it difficult to trigger the lock mode. Typically, a setting of [missing value] is used. It is the maximum radius within the neighborhood of the fixed point where the linear approximation error of the Jacobian matrix is ​​less than 5%.

[0106] The target fixed point satisfy:

[0107] ,

[0108] In the formula Let be the mapping function. At this point, the system's state remains unchanged after one orbital period.

[0109] In step three, the control quantity is calculated. The method is as follows:

[0110] ,

[0111] ,

[0112] In the formula Here is the gain matrix. This is the time delay value.

[0113] Gain matrix The calculation method is as follows:

[0114] Define nonlinear functions :

[0115] ;

[0116] The dynamic equations of the tethered satellite system are obtained as follows:

[0117] ,

[0118] At the target fixed point Linearization yields:

[0119] ,

[0120] In the formula Let Jacobian matrix be the current state. The Jacobian matrix represents the time-delay state;

[0121] ,

[0122] ,

[0123] Assumption For the steady-state solution, the characteristic equation is derived as follows:

[0124] ,

[0125] In the formula The characteristic matrix, For eigenvalues, It is the identity matrix;

[0126] Solving the characteristic equation yields eigenvalues, which determine the stability of the system's steady-state solution. It should be noted that time-delay differential equations have infinitely many eigenvalues. Analyzing the largest eigenvalue... The real part, if the eigenvalue maximum value real part , The steady-state solution is asymptotically stable if , The steady-state solution is unstable; although there are infinitely many eigenvalues, the eigenvalues ​​located in the right half-plane of the complex plane are finite, and numerical methods are used to solve for the eigenvalues. And judge The stability of the steady-state solution is obtained, and thus the gain matrix that stabilizes the tethered satellite system is obtained. .

[0127] Furthermore, the method for calculating the small velocity disturbance in step four to compensate for the unstable direction offset is as follows:

[0128] The discrete form of the dynamic equations of the tethered satellite system is:

[0129] ,

[0130] In the formula For Poincaré mapping;

[0131] Target stationary point on an unstable periodic orbit Linearize the position:

[0132] ,

[0133] In the formula This is the control quantity for the nth control cycle. For the target control quantity, Let Jacobian be the state matrix. The Jacobian matrix is ​​the control quantity;

[0134] ,

[0135] ,

[0136] The state Jacobian matrix is ​​obtained through singular value decomposition. stable eigenvalues and unstable eigenvalues and stable eigenvectors and unstable eigenvectors ;in , ;

[0137] Define the equation:

[0138] ,

[0139] In the formula For stable basis vectors, These are unstable basis vectors. and They are mutually orthogonal; and satisfy:

[0140] and ;

[0141] By continuously applying small perturbations to the system, the state variables are made to be completely aligned with the stable direction of the fixed point, i.e., the component in the unstable direction is 0. The result is:

[0142] ,

[0143] The left side of the above equation represents the disturbance applied to the system. Since the premise of control is that the system state variables must fall near the fixed point, it is necessary to define the size of this range, or directly give the range of control inputs that meet this condition.

[0144] ,

[0145] In the formula To maximize control, the system should be rationally designed based on its dynamic characteristics. When the control input does not meet the above conditions, no control should be applied to the rope system.

[0146] In step four, the final physical velocity command acting on the tethering and releasing mechanism consists of the nominal velocity and the disturbance amount:

[0147] The method for calculating the rope release and retrieval speed is as follows:

[0148] ,

[0149] In the formula This refers to the speed at which the rope is released and retrieved.

[0150] Among them, the length of the rope This is the measured length of the current tether. Because... The take-up and take-down process is a variable, and it is necessary to ensure that within the sampling period... The value of and state vector The sampling times are strictly aligned. If length is ignored... The real-time product feedback will cause a proportional gain deviation in the actual applied control torque, resulting in limit cycle oscillations in the system near the target fixed point.

[0151] In step four, the stability tolerance is set to... If the current If so, it is determined that the tethered satellite has successfully locked onto the target's unstable periodic orbit position.

[0152] In step four, if Return to step two.

[0153] Specific Implementation Example: Assume a tethered satellite system is located at an eccentricity... The system is in an elliptical orbit at an altitude of 2000 km. The system consists of two satellites: a mother satellite with a mass of 2 kg and a daughter satellite with a mass of 20 kg, and an initial tether length of 10 m. These system parameters are not specific; the controller is adaptable to tethered systems with other parameters. Using the PNF method, the coordinates of the fixed point are calculated to be (0, 0.2527), and the Floquet multiplier for this periodic orbit is calculated to be (-0.5245, -1.9067), indicating that the orbit is unstable. The eigenvalues ​​of the delay differential equation are calculated, and a time delay value is chosen to ensure system stability. Control gain matrix .

[0154] Using the method of the present invention, the following is obtained: Figures 3 to 7 The result. Figure 3 and Figure 4 As shown, the inner angle of the tethered satellite surface Rate of change of interior angles of satellite system The change was from chaos to order. Figure 5 control quantity Ultimately, it also amounts to almost zero. From Figure 6 and Figure 7 The scatter plot of the Poincaré map clearly shows that the chaotic motion of the system is effectively suppressed, the interior angles eventually converge to values ​​close to 0, and the rate of change of the interior angles converges to values ​​close to 0.2527. This is the value of the fixed point of an unstable periodic orbit of the system initially obtained.

[0155] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A chaotic control method for tethered satellites based on OGY control and delay feedback control, characterized in that... include, Step 1: Calculate the state points, tether length, and rate of change of tether length of the tethered satellite based on the dynamic model of the tethered satellite system; the state points of the tethered satellite include the in-plane angles of the tethered satellite and the rate of change of the in-plane angles of the tethered satellite. Step 2: Calculate the Euclidean distance between the current tethered satellite state point and the target stationary point on the Poincaré section. If the Euclidean distance is greater than the distance threshold, execute the delay feedback control in Step 3; otherwise, execute the OGY control in Step 4. Step 3, Delay Feedback Control: Calculate the control quantity based on the current tether satellite state point and gain matrix. The control quantity is the ratio of the tether length change rate to the tether length. Adjust the tether release and take-up speed according to the control quantity. After each control cycle is completed, return to Step 2. Step 4, OGY Control: Calculate the small velocity disturbance to compensate for the unstable direction shift based on the Jacobian matrix of the system dynamics equations; calculate the tether deployment and retrieval speeds based on the small velocity disturbances; after each control cycle, calculate the Euclidean distance between the current tethered satellite state point on the Poincaré section and the target stationary point. If the Euclidean distance is greater than the stability tolerance, continue OGY control until the Euclidean distance is no greater than the stability tolerance, maintaining the OGY control law to ensure the tethered satellite maintains a stable operating state.

2. The chaotic control method for tethered satellites based on OGY control and delay feedback control according to claim 1, characterized in that, The dynamic model of the tethered satellite system in step one is as follows: ， ， In the formula For the interior angle of the tethered satellite surface, For orbital eccentricity, For true near point angle, The rate of change of the in-plane angle of the tethered satellite. The rate of change of the rope length. This refers to the length of the rope. ， In the formula To control the quantity; Tethered Satellite Status Points for: 。 3. The chaotic control method for tethered satellites based on OGY control and delay feedback control according to claim 2, characterized in that, In step two, it is assumed that the tethered satellite state point in the nth control cycle is... : The Euclidean distance between the tethered satellite state point and the target stationary point on the Poincaré section during the nth control period is: ， In the formula For Euclidean distance, The target is a fixed point; Set the distance threshold to ,like Execute delayed feedback control, if Execute OGY control.

4. The tethered satellite chaotic control method based on OGY control and delay feedback control according to claim 3, characterized in that, The target fixed point satisfy: ， In the formula This is a mapping function.

5. The tethered satellite chaotic control method based on OGY control and delay feedback control according to claim 4, characterized in that, In step three, the control quantity is calculated. The method is as follows: ， ， In the formula Here is the gain matrix. This is the time delay value.

6. The tethered satellite chaotic control method based on OGY control and delay feedback control according to claim 5, characterized in that, Gain matrix The calculation method is as follows: Define nonlinear functions : ； The dynamic equations of the tethered satellite system are obtained as follows: ， At the target fixed point Linearization yields: ， In the formula Let Jacobian matrix be the current state. The Jacobian matrix represents the time-delay state; ， ， Assumption For the steady-state solution, the characteristic equation is derived as follows: ， In the formula The characteristic matrix, For eigenvalues, It is the identity matrix; If eigenvalue maximum value real part , The steady-state solution is asymptotically stable if , The steady-state solution is unstable; the eigenvalues ​​are solved using numerical methods. And judge The stability of the steady-state solution is obtained, and thus the gain matrix that stabilizes the tethered satellite system is obtained. .

7. The tethered satellite chaotic control method based on OGY control and delay feedback control according to claim 6, characterized in that, The method for calculating the small velocity disturbance to compensate for the unstable direction offset in step four is as follows: The discrete form of the dynamic equations of the tethered satellite system is: ， In the formula For Poincaré mapping; Target stationary point on an unstable periodic orbit Linearize the position: ， In the formula This is the control quantity for the nth control cycle. For the target control quantity, Let be the state Jacobian matrix. The Jacobian matrix is ​​the control quantity; ， ， The state Jacobian matrix is ​​obtained through singular value decomposition. stable eigenvalues and unstable eigenvalues and stable eigenvectors and unstable eigenvectors ;in , ; Define the equation: ， In the formula For stable basis vectors, These are unstable basis vectors. and They are mutually orthogonal; and satisfy: and ; Then we get: ， ， In the formula This is the maximum control quantity.

8. The chaotic control method for tethered satellites based on OGY control and delay feedback control according to claim 7, characterized in that, In step four, the calculation method for the rope release and reel speed is as follows: ， In the formula This refers to the speed at which the rope is released and retrieved.

9. The chaotic control method for tethered satellites based on OGY control and delay feedback control according to claim 8, characterized in that, In step four, the stability tolerance is set to... If the current If so, it is determined that the tethered satellite has successfully locked onto the target's unstable periodic orbit position.

10. The chaotic control method for tethered satellites based on OGY control and delay feedback control according to claim 9, characterized in that, In step four, if Return to step two.

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