H∞ Control Method for Elliptical Orbit Rendezvous of Data-Driven Non-Cooperative Spacecraft

Abstract

Data-driven H-infinity control method for elliptical orbit rendezvous of non-cooperative spacecraft, which relates to the field of spacecraft control. The present invention is to solve the problem that the current orbit rendezvous control method for non-cooperative target spacecraft is not applicable to the rendezvous control of elliptical orbits. The present invention includes: establishing a dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system; obtaining a linearized orbit dynamic model according to the dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system; based on the linearized orbit dynamic model, using a data-driven method, designing a non-cooperative target spacecraft elliptical orbit rendezvous H ∞ optimal controller. The present invention is used to obtain the H ∞ optimal controller for the non-cooperative target spacecraft elliptical orbit rendezvous control system.

Description

Technical Field

[0001] The present invention relates to the field of spacecraft control, and particularly to a data-driven H-infinity control method for non-cooperative spacecraft elliptical orbit rendezvous. Background Technique

[0002] The spacecraft rendezvous problem is divided into cooperative rendezvous and non-cooperative rendezvous according to whether the information of the target spacecraft is known. Cooperative rendezvous means that the target spacecraft can provide its orbit and attitude information, such as the rendezvous mission with the space station as the target spacecraft. Non-cooperative rendezvous means that the orbit, attitude and other information of the target spacecraft cannot be obtained, and this type of rendezvous mission also exists in large numbers, such as the cleaning of space debris and the capture of out-of-control satellites. The orbit control of non-cooperative rendezvous spacecraft is a basic link of the non-cooperative rendezvous system, and its purpose is to control the tracking spacecraft to coincide with the target spacecraft in position. Since the orbit parameters of the target spacecraft are unknown, the parameters of the non-cooperative rendezvous spacecraft orbit dynamics model are unknown, which makes the traditional model-based controller design method inapplicable to the non-cooperative rendezvous system, and a new controller design method is needed.

[0003] The existing control methods for non-cooperative rendezvous systems mainly adopt adaptive control methods based on parameter identification, optimal control methods based on adaptive dynamic programming, etc., but these methods are all designed for near-circular orbits. When the spacecraft operates along an elliptical orbit, its orbit parameters change with time, that is, the elliptical orbit spacecraft rendezvous system is a time-varying system. At this time, it is difficult to design an effective system controller using the non-cooperative target rendezvous system control method designed for near-circular orbits, resulting in control failure. Therefore, the current non-cooperative target spacecraft orbit rendezvous control method cannot be applied to the elliptical orbit rendezvous control. Summary of the Invention

[0004] The purpose of the present invention is to solve the problem that the current non-cooperative target spacecraft orbit rendezvous control method is not applicable to the elliptical orbit rendezvous control, and a data-driven H-infinity control method for non-cooperative spacecraft elliptical orbit rendezvous is proposed.

[0005] The specific process of the data-driven H-infinity control method for non-cooperative spacecraft elliptical orbit rendezvous is as follows:

[0006] Step 1: Establish a dynamics model of the non-cooperative target spacecraft elliptical orbit rendezvous system;

[0007] Step 2: Obtain a linearized orbit dynamics model according to the dynamics model of the non-cooperative target spacecraft elliptical orbit rendezvous system established in Step 1;

[0008] Step 3: Based on the linearized orbit dynamics model obtained in Step 2, use a data-driven method to design a non-cooperative target spacecraft elliptical orbit rendezvous H∞ Optimal controller:

[0009] Step 3-1: Take u(t) = u0(t) + u e (t) as the input of the closed-loop system. During the time interval [t0, t s , collect u(t), d(t), χ(t), and obtain the intermediate matrix Θ corresponding to the Fourier series of the system state equation and

[0010] where u0(t) is an arbitrary controller that makes the state of the closed-loop system bounded, and u e (t) is the detection noise, is the input variable of the closed-loop system, is the disturbance of the closed-loop system, is the state variable of the closed-loop system, and n, m, p are the numbers of the state variable, input variable, and disturbance variable of the closed-loop system respectively;

[0011] Step 3-2: Based on the intermediate matrix Θ and obtained in Step 3-1, after integrating both sides of the Fourier series of the system state equation with respect to t and adding the error term caused by the truncation error, obtain the extended equation of the Fourier series of the system state equation, and then obtain the estimated value of the coefficient matrix in the corresponding matrix differential equation of the extended equation

[0012] Step 3-3: Select l0 < l1 < … < l q < < 0, and use the obtained in Step 3-2 to define the intermediate matrix Y, and use to define the intermediate matrix X, and then use the intermediate matrices X and Y to obtain

[0013] where l0, l1, …, l q are all negative numbers, is the intermediate variable, T is the system period, l is a variable, and n2 is a positive integer;

[0014] Step 3-4: Use the obtained in Step 3-3 to obtain H ∞ approximate feedback gain K ap (t) of the optimal controller, and use the approximate feedback gain K ap (t) to obtain H ∞ optimal controller.

[0015] The beneficial effects of the present invention are:

[0016] The present invention relates to a non-cooperative target spacecraft elliptical orbit rendezvous control system in the case where the system dynamics model parameters are unknown. The idea of the Newton iteration algorithm is introduced into the design process of the spacecraft orbit controller. After transformation, the dynamics model of the system is effectively avoided, and the measured input values and state values are directly used for iteration, so as to obtain an H that makes the non-cooperative rendezvous control system of the spacecraft elliptical orbit stable and has disturbance rejection ability ∞ optimal controller.

[0017] The H ∞ optimal controller iteration method proposed in step three of the present invention obtains an approximate H that makes the non-cooperative rendezvous control system of the spacecraft elliptical orbit stable and has disturbance rejection ability ∞ optimal controller, and according to Figures 3 - 8 , each state tends to 0, that is, the position and velocity of the tracking spacecraft relative to the target spacecraft both tend to 0, which means that the designed controller can make the tracking spacecraft reach the position of the target spacecraft; according to Figures 12 - 13 , the disturbance attenuation level of the system reaches the expected standard, that is, the L2 gain from the disturbance to the output of the system can finally be less than the given attenuation parameter γ, which proves that the present invention has disturbance rejection ability. Description of the Drawings

[0018] Figure 1 is a schematic diagram of the geocentric equatorial inertial coordinate system and the target spacecraft orbit coordinate system;

[0019] Figure 2 is a schematic diagram of the elliptical orbit spacecraft rendezvous system;

[0020] Figure 3 is a curve graph of the change of the O′X o axis component x of the position of the tracking spacecraft relative to the target spacecraft;

[0021] Figure 4 is a curve graph of the change of the O′Y o axis component y of the position of the tracking spacecraft relative to the target spacecraft;

[0022] Figure 5 is a curve graph of the change of the O′Z o axis component z of the position of the tracking spacecraft relative to the target spacecraft;

[0023] Figure 6 is the O′X o axis component of the velocity of the tracking spacecraft relative to the target spacecraft change curve graph;

[0024] Figure 7 is the O′Y o axis component of the velocity of the tracking spacecraft relative to the target spacecraft Variation curve graph;

[0025] Figure 8 is the O′Z axis component o for tracking the speed of the spacecraft relative to the target spacecraft Variation curve graph;

[0026] Figure 9 is the O′X axis component a o of the control input x Variation curve graph;

[0027] Figure 10 is the O′Y axis component a o of the control input y Variation curve graph;

[0028] Figure 11 is the O′Z axis component a o of the control input z Variation curve graph;

[0029] Figure 12 is the disturbance attenuation level of the system in the orbital plane;

[0030] Figure 13 is the disturbance attenuation level of the system in the direction perpendicular to the orbital plane. Specific implementation manner

[0031] Specific implementation manner 1: The specific process of the data-driven non-cooperative spacecraft elliptical orbit rendezvous H-infinity control method in this implementation manner is as follows:

[0032] Step 1. Establish the dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system;

[0033] Step 11. Establish the geocentric equatorial inertial coordinate system OXYZ and the target spacecraft orbit coordinate system O′X o Y o Z o as shown Figure 1 in;

[0034] Geocentric equatorial inertial coordinate system OXYZ: The coordinate origin O is located at the center of the earth, the OX axis points from the center of the earth to the vernal equinox direction, the OZ axis points from the center of the earth to the north pole, and the OY axis forms a right-handed coordinate system with the above two axes;

[0035] Target spacecraft orbit coordinate system O′X o Y o Z o : The coordinate origin O′ is located at the center of mass of the target spacecraft, the direction of the O′X o axis is the direction from the center of the earth to the target spacecraft, and the O′Y oThe axis points to the velocity direction of the target spacecraft, O′Z o The axis forms a right-handed coordinate system with the above two axes.

[0036] Steps one and two: Based on the geocentric equatorial inertial coordinate system OXYZ and the target spacecraft orbital coordinate system O′X o Y o Z o Establish a dynamic model for the elliptical orbit rendezvous system of a non-cooperative target spacecraft:

[0037] Since the perturbation effects of other celestial bodies on the spacecraft are very small, only the two-body problem between the Earth and the spacecraft is considered during modeling. As Figure 2 shown, denote the distance vector from the geocenter to the center of mass of the target spacecraft as ρ1, the distance vector from the geocenter to the center of mass of the chasing spacecraft as ρ2, and the distance vector from the center of mass of the target spacecraft to the chasing spacecraft as ρ = ρ2 - ρ1.

[0038] Step one and two one: Obtain the motion equation of the chasing spacecraft relative to the target spacecraft in the geocentric equatorial inertial coordinate system OXYZ:

[0039] In the geocentric equatorial inertial coordinate system, according to the law of universal gravitation, the motion of the target spacecraft can be described by the following equation:

[0040]

[0041] where μ = GM is an intermediate variable, G is the gravitational constant, M is the mass of the Earth, and the value of μ is approximately 3.986×10 14 m 3 / s 2 , t is time;

[0042] Similarly, the motion of the chasing spacecraft can be described as:

[0043]

[0044] where a f is the acceleration generated by the thrust acting on the chasing spacecraft, and d is the perturbation term.

[0045] Subtract Equation (2) from Equation (1) to obtain the motion equation of the chasing spacecraft relative to the target spacecraft in the geocentric equatorial inertial coordinate system:

[0046]

[0047] Step one and two two: Based on the formula (3) obtained in step one and two one, obtain the motion equation of the chasing spacecraft relative to the target spacecraft in the target spacecraft orbital coordinate system:

[0048] Let the vector ω be the orbital angular velocity vector of the target spacecraft at a certain point, then the derivative of ρ with respect to time in the orbital coordinate system of the target spacecraft and the derivative of ρ with respect to time in the geocentric equatorial inertial coordinate system satisfy:

[0049]

[0050] Furthermore, the relationship between the second-order derivatives can be obtained:

[0051]

[0052] Substitute Equation (5) into Equation (3), and denote The motion equation of the chaser spacecraft relative to the target spacecraft in the orbital coordinate system of the target spacecraft can be obtained:

[0053]

[0054] where is the velocity of the chaser spacecraft relative to the target spacecraft in the orbital coordinate system of the target spacecraft, is the acceleration of the chaser spacecraft relative to the target spacecraft in the orbital coordinate system of the target spacecraft;

[0055] Steps One, Two, and Three: Obtain the dynamic model of the elliptical orbit rendezvous system of the non-cooperative target spacecraft according to formula (6):

[0056] According to the definition of the orbital coordinate system of the target spacecraft, ω = [0 0 ω] T , ρ1 = [ρ1 0 0] T , where ω and ρ1 are the magnitudes of the orbital angular velocity of the target spacecraft at a certain point and the magnitude of the distance from the earth's center to the center of mass of the target spacecraft, respectively. ρ can be expressed as ρ = [x y z] T , where x, y, and z are the coordinate components of the chaser spacecraft in the orbital coordinate system of the target spacecraft, that is, the respective components of the position of the chaser spacecraft relative to the target spacecraft, a f can be expressed as a f = [a x a y a z T , a x , a y , a z are the coordinate components of a f in the orbital coordinate system of the target spacecraft, d can be expressed as d = [d x d y d z T , d​​x , d y , d z Let \(d\) be the coordinate components of \(d\) in the orbital coordinate system of the target spacecraft. Then, Equation (6) can be expressed as:

[0057]

[0058] Where, are the first-order derivatives of \(x\), \(y\), and \(z\), respectively, that is, the components of the velocity of the chaser spacecraft relative to the target spacecraft. are the second-order derivatives of \(x\), \(y\), and \(z\), respectively, that is, the components of the acceleration of the chaser spacecraft relative to the target spacecraft. is the first-order derivative of \(\omega\), that is, the angular acceleration of the target spacecraft.

[0059] Step 2: Obtain the linearized orbital dynamics model according to the non-cooperative target spacecraft elliptical orbit rendezvous system dynamics model established in Step 1, which specifically includes the following steps:

[0060] Since \(\rho_2 = \rho_1 + \rho\), we have:

[0061]

[0062] Since the distance between the target spacecraft and the chaser spacecraft is much smaller than the distance from the target spacecraft to the Earth's center of mass, that is, \(|\rho| \ll |\rho_1|\), the following approximation can be made:

[0063]

[0064] Substituting Equation (9) into Equation (6) gives:

[0065]

[0066] Where, is the first-order derivative of \(\omega\);

[0067] Substituting \(\omega = [0\ 0\ \omega]\) T , \(\rho_1 = [\rho_1\ 0\ 0]\) T , \(\rho = [x\ y\ z]\) T and \(a\) f = [a x a y a z T into Equation (10), the linearized spacecraft orbital dynamics model is obtained:

[0068]

[0069] The relationship between the magnitude \(\omega\) of the orbital angular velocity of the target spacecraft at a certain point and the magnitude \(\rho_1\) of the distance from the Earth's center to the target spacecraft's center of mass is:

[0070]

[0071] Among them, h is the momentum matrix of the target spacecraft moving around the gravitational body.

[0072] Substitute Equation (12) into Equation (11), and denote the intermediate variable to obtain:

[0073]

[0074] Respectively take and U = [a x , a y , a z T as the state variable and the control vector, then Equation (13) can be written in the following form:

[0075]

[0076] Among them, the matrix A S , takes the value of:

[0077]

[0078]

[0079]

[0080] In the non-cooperative elliptical orbit rendezvous problem, A S is time-varying and the parameters cannot be measured.

[0081] Note that the motion in the orbital plane (x, y) and the motion perpendicular to the orbital direction, i.e., the z-axis direction, are decoupled from each other. Therefore, the model can be split into a model in the orbital plane and a model perpendicular to the orbit.

[0082] In the orbital plane, there is:

[0083]

[0084] In the direction perpendicular to the orbit, there is:

[0085]

[0086] Step 3: Based on the linearized orbital dynamics model obtained in Step 2, use a data-driven method to design a non-cooperative target spacecraft elliptical orbit rendezvous H ∞ optimal controller, specifically:

[0087] Step 1: Establish the dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system;

[0088] Step 2: Obtain a linearized orbital dynamics model based on the non-cooperative target spacecraft elliptical orbit rendezvous system dynamics model established in Step 1;

[0089] Step 3: Based on the linearized orbital dynamics model obtained in Step 2, use a data-driven method to design a non-cooperative target spacecraft elliptical orbit rendezvous H ∞ optimal controller:

[0090] Step 3-1: Take u(t) = u0(t) + u e (t) as the input of the closed-loop system. In the time interval [t0, t s , collect u(t), d(t), χ(t), and obtain the intermediate matrix Θ corresponding to the Fourier series of the system state equation and

[0091] where u0(t) is an arbitrary controller that makes the closed-loop system state bounded, and u e (t) is the detection noise, is the input variable of the closed-loop system, is the disturbance of the closed-loop system, is the state variable of the closed-loop system, and n, m, p are the numbers of the state variable, input variable, and disturbance variable of the closed-loop system respectively;

[0092] Step 3-2: Based on the intermediate matrix Θ and obtained in Step 3-1, after integrating both sides of the Fourier series of the system state equation with respect to t and adding the error term caused by the truncation error, obtain the augmented equation of the Fourier series of the system state equation, and then obtain the estimated value of the coefficient matrix in the matrix differential equation corresponding to the augmented equation

[0093] Step 3-3: Select l0 < l1 <... < l q << 0, and use the obtained in Step 3-2 to define the intermediate matrix Y, and use to define the intermediate matrix X, and then use the intermediate matrices X and Y to obtain

[0094] where l0, l1,..., l q are all negative numbers, is an intermediate variable, T is the system period, l is a variable, and n2 is a positive integer;

[0095] Step 3-4: Use the obtained in Step 3-3 to obtain the H ∞ optimal controller approximate feedback gain K ap (t), and use the approximate feedback gain K ap(t) Obtain H ∞ Optimal controller

[0096] The specific derivation process of Step 3 is as follows:

[0097] Obtain the system state equation:

[0098]

[0099] where is the system state, is the system input, is the system disturbance, n, m, p are the numbers of system state variables, input variables, and disturbance variables respectively, A(t), B1(t), B2(t) are matrices with period T, T is the system period, and represent the set of a-dimensional real vectors and the set of b×c-dimensional real matrices respectively

[0100] Design a linear periodic state feedback controller u(t) = -K(t)χ(t) according to the system state equation, where K(t) is the state feedback gain matrix, so that the closed-loop system satisfies the following conditions:

[0101] ① When d(t) = 0, the closed-loop system is asymptotically stable;

[0102] ② The L2 gain from the disturbance to the system output is less than or equal to the given constant attenuation parameter γ, that is, the following formula holds:

[0103]

[0104] where Q(t), R(t) are given periodic weight matrices, and Q(t) is positive semi-definite, R(t) is positive definite

[0105] Define the cost function:

[0106]

[0107] where τ is the integration variable, then the problem is transformed into designing a controller such that J(χ(t), u(t), d(t)) ≤ 0. At this time, according to the optimal control principle, we can obtain the desired H ∞ The form of the optimal controller is:

[0108] u * (t) = -R -1 (t)B1 T (t)P * (t)χ(t)(20)

[0109] where u * (t) is the desired H ∞ Optimal controller

[0110] where P * (t) is the unique symmetric positive definite periodic solution of the periodic game Riccati equation (21) with respect to P(t):

[0111]

[0112] To obtain this solution and further construct the desired optimal controller, a basic property is first introduced. Only consider the cost function in a finite range:

[0113]

[0114] where t f is the given terminal time.

[0115] According to the optimal control theory, when the controller is in the form shown in formula (23), the cost function J shown in formula (22) f (χ(t), u(t), d(t)) is minimized:

[0116]

[0117] where P(t, t f ) is the solution of the backward matrix differential equation (24) with respect to P(t):

[0118]

[0119] In addition, the solution of equation (24) will approach the periodic solution of equation (21) as t → -∞. Based on this property, the optimal controller is solved. For simplicity of description, the following operator symbols are defined:

[0120] For a column vector define For a symmetric matrix where denotes the element in the i S -th row and j S -th column of S, define Let vec(·) denote expanding a matrix into a column vector by columns; denotes the Kronecker product, n s is the number of rows and columns of S, and n a is the number of rows of α.

[0121] Define the following intermediate variables:

[0122]

[0123]

[0124]

[0125] Among them, l is a variable.

[0126] Then, according to Equation (17), we have:

[0127]

[0128] Integrating both sides of Equation (25) with respect to t gives:

[0129]

[0130] Among them, Δt is the time interval.

[0131] Based on the Fourier series decomposition, vecs(N(l,t)) and vec(M(l,t)) can be approximated in the following form:

[0132]

[0133] Among them, n N , n M , n O is a positive integer, T is the system period, and has the same form but different numbers of terms, is the coefficient matrix. represents the truncation errors of vecs(N(l,t)), vec(M(l,t)) and vec(O(l,t)).

[0134] Then Equation (26) can be transformed into:

[0135]

[0136]

[0137]

[0138] Among them, e(l,t) represents the error caused by , is an intermediate variable, and is a vector;

[0139] Take the time series {t0, t1, …, t s}, where each term t i = t0 + iΔt, i = 0, 1, 2,... s, and define the following intermediate matrix:

[0140]

[0141]

[0142] Among them, i is the label of time in the time series, and s is a positive integer;

[0143] Then, Equation (28) can be expanded to:

[0144]

[0145] Among them, E(l) represents the error term caused by the truncation error.

[0146] Furthermore, let Φ = (Θ T Θ) -1 Θ T , then we have:

[0147]

[0148] Differentiate both sides of Equation (30) with respect to l, and we get:

[0149]

[0150]

[0151]

[0152] Among them, is the first-order derivative with respect to l, e0(l) is the error term, and H(W n (l), l) is the intermediate variable;

[0153] Ignoring the error term, we can obtain:

[0154]

[0155]

[0156]

[0157]

[0158] Among them, is the estimated value of is the first-order derivative of

[0159] It can be seen that Equation (32) is completely independent of the dynamic model of the system. That is, the optimal controller of the system can be obtained without knowing the specific information of matrices A(t), B1(t), and B2(t).

[0160] According to Equation (27), substituting \(t\) with \(l\) gives:

[0161]

[0162] where is the coefficient matrix, is the truncation error for approximating \(vec(M(l, l))\) using .

[0163] Neglecting the error term, we have:

[0164]

[0165] where is 's estimated value.

[0166] According to experience, take the sequence \(\{l_0, l_1,... l q}\), where \(l_0 \lt l_1 \lt... \lt l q \ll l f = 0\), and define the following intermediate matrices:

[0167]

[0168]

[0169] Then we can obtain:

[0170]

[0171] Then the obtained approximate feedback gain \(K ap (t)\) has the form shown in Equation (36):

[0172]

[0173] The optimal controller \(u(t)= -K ap \chi(t)\) is obtained through Equation (36).

[0174] It is proven through the following analysis that the approximate feedback gain \(K ap (t)\) can achieve arbitrary approximate accuracy through the selection of parameters.

[0175] First, note that i.e., each term in \(W n (l)\) is differentiable with respect to \(l\), and when \(n N , n M , n O takes values large enough, according to the approximation principle, the value of \(W n (l)\) can be approximated with sufficient accuracy by solving (31) on the interval \([l_0, 0]\), that is, for any \(\varepsilon \gt 0\), there exists such that for any in the fixed interval [l0, l f , we can obtain

[0176]

[0177] At this time, we may assume that ε1 = 3ε and consider the interval [l0, 0]. It is not difficult to obtain that there exists such that for all in the fixed interval [l0, 0], we can obtain:

[0178]

[0179] The above equation also shows that:

[0180]

[0181] Similarly, when approximating any periodic function matrix with a Fourier series, as the Fourier series approaches infinity, the truncation error of the approximation approaches 0. Therefore, there exists such that for all in the fixed interval [l0, 0], we can obtain

[0182]

[0183] In addition, according to the property described above, that is, the solution of equation (24) will approach the periodic solution of equation (21) as t → -∞. Therefore, we can obtain that there exists such that for all there exists

[0184] ||M(l, l) - B1 T (l)P * (l)|| < ε (41)

[0185] According to the analysis in (37)-(41), we can further obtain that for all and in the interval [l0, l q , we have

[0186]

[0187] Define

[0188]

[0189] Then for the selected l0 < l1 <... < l q << l f = 0, by letting

[0190]

[0191]

[0192] According to (43), we can obtain

[0193]

[0194] Furthermore, according to the definitions of matrices X and Y, we can obtain

[0195]

[0196] At this time, by appropriately selecting the sequence X to satisfy where α is a positive real number independent of n2. At this time, according to (44), we can obtain

[0197]

[0198] At this time, noting that according to (42) and the definition of Y, it is not difficult to find that ||Y * -Y|| < qε1. Combining with the Fourier series approximation principle, that is, there exists such that for any holds, we can obtain

[0199]

[0200] Therefore, when ε1 → 0, Using the Fourier series approximation principle once again, we can obtain that for any ε2 → 0, by appropriately selecting n N , n M , n O , n2, X, Y and other related parameters, we can always obtain

[0201]

[0202]

[0203] Finally, combining the triangle inequality and equation (34), we can obtain

[0204] ||K ap (t) - R -1 (t)B T (t)P * (t)|| < E1(t) + E2(t) < ε2

[0205] That is, the approximate feedback gain K ap (t) can achieve arbitrary approximate accuracy through the selection of parameters.

[0206] Example:

[0207] According to Equations (15) and (16), the system is divided into the motion within the orbital plane and the motion perpendicular to the orbital plane, and simulations are carried out separately. The parameters of the system are shown in Table 1:

[0208] Table 1 System-related parameters

[0209] The angular momentum (h) of the target spacecraft orbiting the gravitational body <![CDATA[6.762×10 10 m 2 / s]]> The orbital eccentricity (e) of the target spacecraft 0.7304 The semi-major axis (a) of the target spacecraft orbit <![CDATA[2.4616×10 7 m]]>

[0210] Since the above values cannot be measured in actual situations, these parameters are only used in the verification process during simulation and are not used when designing the controller.

[0211] The design process of the controller for the motion within the orbital plane is as follows:

[0212] In the data acquisition stage, the system disturbance is taken as: 0.01[sin(1.5t); -sin(2t)], and the system input is taken as:

[0213]

[0214] where ω i is a two-dimensional column vector, and each element is randomly taken from [-5, 5]. Take s = 300, t0 = 0, the time interval Δt = 300, and the initial state value is set as: x(0), y(0), representing the values of x, y, at the zero moment. The input values and state values are collected on [t0, t s , and is calculated. Take n N = 10. Since B1(t) and B2(t) are constant matrices, take n M = n O = 1, and then the matrix Θ can be obtained using the collected input values and state values.

[0215] The values of the two weight matrices are, for example, Q = 10 -3 diag([0.0001, 0.0001, 1, 1]), R = 10 3 E2. Take the attenuation parameter γ = 100, take l0 = -2×10 5 , and obtain by solving the matrix differential equation (32) on [l0, 0]. Select q = 1000, l q = -16000, n2 = 10, calculate the matrices X and Y, and calculate through Equation (35) to obtain the optimal controller The design process of the controller for the motion in the direction perpendicular to the orbital plane is similar to the above process. The system disturbance is taken as: 0.01sin(t), and the system input is taken as:

[0216]

[0217] where ω k is randomly valued in [-5, 5]. The state initial value is set as: denotes the value of z at the zero moment. The weight matrix Q = 10 -3 diag([0.0001, 1]), R = 10 3 , the attenuation parameter γ = 100, and the selection of the remaining parameters and the design steps are the same as those of the system in the orbital plane.

[0218] Applying the designed controller to the spacecraft elliptical orbit rendezvous system, Figures 3 - 8 shows the change trajectories of the system state variables. It can be seen from the figure that each state tends to 0, that is, the position and velocity of the tracking spacecraft relative to the target spacecraft both tend to 0, which means that the designed controller can enable the tracking spacecraft to reach the position of the target spacecraft. Figures 9 - 11 This is the output of the controller in this process. Figures 12 - 13 They are the disturbance attenuation levels in the orbital plane and in the direction perpendicular to the orbit respectively. It can be seen that the system can satisfy Equation (18), that is, the disturbance rejection ability reaches the expectation.

[0219] The present invention can also have many other embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art can make various corresponding changes and deformations according to the present invention, but these corresponding changes and deformations should all fall within the protection scope of the present invention.

Claims

1. Data-driven H-infinity control method for non-cooperative spacecraft rendezvous in elliptical orbits, characterized in that The specific process of the method is as follows: Step 1: Establish the dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system; Step 2: Obtain the linearized orbit dynamic model according to the dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system established in Step 1; Step 3: Based on the linearized orbital dynamics model obtained in Step 2, use a data-driven method to design a non-cooperative target spacecraft elliptical orbit rendezvous H ∞ optimal controller: Step 3.1: Take \(u(t)=u_0(t)+u(t)\) as the input of the closed-loop system. In the time interval \([t_0,t]\), collect \(u(t)\), \(d(t)\), \(\chi(t)\), and obtain the intermediate matrix \(\Theta\) corresponding to the Fourier series of the system state equation based on the cost function and the system state equation. e (t) as the input of the closed-loop system. In the time interval \([t_0,t s , collect \(u(t)\), \(d(t)\), \(\chi(t)\), and obtain the intermediate matrix \(\Theta\) corresponding to the Fourier series of the system state equation based on the cost function and the system state equation. where \(u_0(t)\) is an arbitrary controller that makes the closed-loop system state bounded, and \(u\) e (t) is the detection noise, is the input variable of the closed-loop system, is the disturbance of the closed-loop system, is the state variable of the closed-loop system, and \(n\), \(m\), \(p\) are the numbers of the state variable, input variable, and disturbance variable of the closed-loop system, respectively; Step 32. Based on the intermediate matrix Θ obtained in Step 31 and After integrating both sides of the Fourier series of the system state equation with respect to t, adding an error term caused by the truncation error, obtaining an extended equation of the Fourier series of the system state equation, and then obtaining an estimated value of the coefficient matrix in the matrix differential equation corresponding to the extended equation Step 3: Select \(l_0 \lt l_1 \lt \cdots \lt l\) q \(\lt \lt 0\), and use the one obtained in Step 3-2 Define the intermediate matrix \(Y\), and use Define the intermediate matrix \(X\), and then use the intermediate matrices \(X\) and \(Y\) to obtain Among them, l0, l1, …, l q are all negative numbers, is an intermediate variable, T is the system period, l is a variable, and n2 is a positive integer; Step 3-4: Using the result obtained in Step 3-3 to obtain the ∞ approximate feedback gain K ap (t) of the optimal controller, and using the approximate feedback gain K ap (t) to obtain the ∞ optimal controller.

2. The data-driven non-cooperative spacecraft elliptical orbit rendezvous H-infinity control method according to claim 1, characterized in that: The establishment of the dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system in Step 1 includes the following steps: Step 1: Establish the geocentric equatorial inertial coordinate system OXYZ and the target spacecraft orbital coordinate system O′X o Y o Z o ; The origin O of the geocentric equatorial inertial coordinate system OXYZ is located at the center of the earth. The OX axis points from the center of the earth to the vernal equinox direction, the OZ axis points from the center of the earth to the north pole, and the OY axis forms a right-handed coordinate system with the OX axis and the OZ axis; The origin O' of the orbital coordinate system O′X o Y o Z o of the target spacecraft is located at the center of mass of the target spacecraft. The direction of the O′X o axis is the direction from the center of the earth to the target spacecraft. The O′Y o axis points to the direction of the velocity of the target spacecraft. The O′Z o axis and the O′X o axis, O′Y o axis form a right-handed coordinate system; Steps 1 and 2: Based on the geocentric equatorial inertial coordinate system OXYZ and the target spacecraft orbital coordinate system O′X o Y o Z o Establish a dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system.

3. The data-driven non-cooperative spacecraft elliptical orbit rendezvous H-infinity control method according to claim 2, characterized in that: In the first and second steps, based on the geocentric equatorial inertial coordinate system OXYZ and the target spacecraft orbital coordinate system O′X o Y o Z o Establish a dynamic model for the non-cooperative target spacecraft elliptical orbit rendezvous system, including the following steps: Step 121: Obtain the motion equation of the tracking spacecraft relative to the target spacecraft in the geocentric equatorial inertial coordinate system OXYZ: First, obtain the motion equation of the target spacecraft in the geocentric equatorial inertial coordinate system: Among them, μ = GM is an intermediate variable, where G is the gravitational constant and M is the mass of the Earth, and the value of μ is 3.986×10 14 m 3 / s 2 , t is the time, and ρ1 is the distance vector from the Earth's center to the center of mass of the target spacecraft; Then, obtain the motion equation of the tracking spacecraft in the geocentric equatorial inertial coordinate system: where a f is the acceleration generated by the thrust acting on the tracking spacecraft, d is the disturbance term, and ρ2 is the distance vector from the earth's center to the centroid of the tracking spacecraft; Finally, subtract Equation (2) from Equation (1) to obtain the motion equation of the tracking spacecraft relative to the target spacecraft in the geocentric equatorial inertial coordinate system: ρ = ρ2 - ρ1 where ρ is the distance vector pointing from the center of mass of the target spacecraft to the tracking spacecraft; Step 122: Based on the motion equation of the tracking spacecraft relative to the target spacecraft obtained in Step 121, obtain the orbital coordinate system O′X of the target spacecraft o Y o Z o In this system, the motion equation of the tracking spacecraft relative to the target spacecraft; Step 123: Obtain the dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system according to the motion equation of the tracking spacecraft relative to the target spacecraft.

4. The data-driven non-cooperative spacecraft elliptical orbit rendezvous H-infinity control method according to claim 3, characterized in that: Based on the motion equation of the tracking spacecraft relative to the target spacecraft obtained in Steps 1 and 2, obtain the target spacecraft orbital coordinate system O′X o Y o Z o In the following, the motion equation of the tracking spacecraft relative to the target spacecraft is as follows: First, obtain the derivative of ρ with respect to time in the orbital coordinate system of the target spacecraft and the derivative of ρ with respect to time in the geocentric equatorial inertial coordinate system The relationship between them is as follows: where ω is the orbital angular velocity vector of the target spacecraft at a certain point; Then, obtain the second derivative of ρ with respect to time in the orbital coordinate system of the target spacecraft and the second derivative of ρ with respect to time in the geocentric equatorial inertial coordinate system The relationship between them is as follows: Finally, substitute Equation (5) into Equation (3) to obtain the motion equation of the tracking spacecraft relative to the target spacecraft in the target spacecraft orbit coordinate system: Among them, is the velocity of the chaser spacecraft relative to the target spacecraft in the target spacecraft's orbital coordinate system, is the acceleration of the chaser spacecraft relative to the target spacecraft in the target spacecraft's orbital coordinate system.

5. The data-driven non-cooperative spacecraft elliptical orbit rendezvous H-infinity control method according to claim 4, characterized in that: The obtaining of the dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system according to the motion equation of the tracking spacecraft relative to the target spacecraft in Step 123 includes the following steps: First, define ω = [00ω] T , ρ1 = [ρ100] T , ρ = [x y z] T , a f = [a x a y a z T , d = [d x d y d z T ;​​ where, ω = [00ω] T is the orbital angular velocity vector of the target spacecraft at a certain point, ρ1 = [ρ100] T is the distance vector from the Earth's center to the center of mass of the target spacecraft, ω is the magnitude of the orbital angular velocity of the target spacecraft at a certain point, ρ1 is the magnitude of the distance from the Earth's center to the center of mass of the target spacecraft, ρ = [x y z] T is the representation in the orbital coordinate system of the target spacecraft, x, y, and z are the respective directional components of the position of the chaser spacecraft relative to the target spacecraft, a f = [a x a y a z T is the representation of a f in the orbital coordinate system of the target spacecraft, a x , a y , a z are the respective directional coordinate components of a f in the orbital coordinate system of the target spacecraft, d = [d x d y d z T is the representation of d in the orbital coordinate system of the target spacecraft, d x , d y , d z are the respective directional coordinate components of d in the orbital coordinate system of the target spacecraft;​​ Then, substitute ω = [00ω] T , ρ1 = [ρ100] T , ρ = [x y z] T , a f = [a x a y a z T , d = [d x d y d z T into formula (6) to obtain the dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system as follows:​​ wherein, are the first-order derivatives of x, y, and z respectively, that is, the respective directional components of the velocity of the tracking spacecraft relative to the target spacecraft, are the second-order derivatives of x, y, and z respectively, that is, the respective directional components of the acceleration of the tracking spacecraft relative to the target spacecraft, is the first-order derivative of ω, that is, the angular acceleration of the target spacecraft.

6. The data-driven non-cooperative spacecraft elliptical orbit rendezvous H-infinity control method according to claim 5, characterized in that: The obtaining of the linearized orbit dynamic model according to the dynamic model of the non-cooperative target spacecraft elliptical orbit rendezvous system established in Step 1 in Step 2 is as follows: In the orbital plane, there is: In the direction perpendicular to the orbit, there is: Among them, is an intermediate variable, and h is the momentum matrix of the target spacecraft orbiting the gravitational body.

7. The data-driven non-cooperative spacecraft elliptical orbit rendezvous H-infinity control method according to claim 6, characterized in that: In step 31, the intermediate matrix Θ corresponding to the Fourier series of the system state equation is obtained based on the cost function and the system state equation, and includes the following steps: Define the cost function: where τ is the integration variable, T is the system period, γ is a constant decay parameter, and Q(t), R(t) are the given periodic weight matrices; Obtain the desired H when J(χ(t), u(t), d(t)) ≤ 0 ∞ Optimize the controller as follows: u * (t) = -R -1 (t)B1 T (t)P * (t)χ(t)(20) where B1(t) is a matrix with a period of T; P * (t) is the value of the unique symmetric positive definite periodic solution of the following equation for P(t): where, u * (t) is the desired H ∞ optimal controller, R(t) is the weight matrix of the given period, and A(t), B1(t), B2(t) are matrices with a period of T; Set the termination time t f , and obtain the cost function within the termination time range: Obtain the cost function J within the termination time range f (χ(t), u(t), d(t)) The expected H when it is minimized ∞ Optimized controller: where P(t, t f ) is the value of the solution of the following equation for P(t): In addition, the solution of equation (24) will approach the periodic solution of equation (21) as t → -∞, and for the cost function J within the termination time range f (χ, u, d) when it is minimized, the desired H ∞ Solve using an optimal controller: Define the following parameters: Column vector Symmetric matrix Among them, represents the element at the \(i\)th S row and \(j\)th S column of \(S\), and \(\text{vec}(\cdot)\) represents expanding the matrix column by column into a column vector; represents the Kronecker product, \(n\) s is the number of rows and columns of \(S\), and \(n\) a is the number of rows of \(\alpha\); Define the following intermediate variables: where l is a variable; Then according to the system state equation, there is: The system state equation is as follows: Integrating both sides of Equation (25) with respect to t gives: where Δt is the time interval; Based on the Fourier series decomposition, approximate vecs(N(l,t)), vec(M(l,t)) in the following form: Among them, n N , n M , n O is a positive integer, T is the system period, and have the same form but different numbers of terms; is the coefficient matrix; represent the truncation errors of vecs(N(l,t)), vec(M(l,t)) and vec(O(l,t)); Then Equation (26) is transformed into: Among them, e(l,t) represents the error caused by and is an intermediate variable; Take the time series {t0, t1,..., t s}, where each term t i = t0 + iΔt, i = 0, 1, 2,... s, and obtain the following intermediate matrix: where i is the label of time in the time series and s is a positive integer.

8. The data-driven non-cooperative spacecraft elliptical orbit rendezvous H-infinity control method according to claim 7, characterized in that: The intermediate matrix Θ obtained based on Step 3-1 in Step 3-2 and After integrating both sides of the Fourier series of the system state equation with respect to t, adding an error term caused by truncation error, obtaining an extended equation of the Fourier series of the system state equation, and then obtaining an estimated value of the coefficient matrix in the matrix differential equation corresponding to the extended equation Specifically: Expand Equation (28) to: where E(l) represents the error term caused by the truncation error; Let Φ = (Θ T Θ) -1 Θ T , then we have: Differentiate both sides of Equation (30) with respect to l to get: Among them, is the first derivative of l, e0(l) is the error term, H(W n (l), l) is the intermediate variable; Ignoring the error term, we get: Among them, is the estimated value of, is the first derivative of; Solve the matrix differential equation (32) on [l0, 0] to obtain the intermediate variable 9. The data-driven non-cooperative spacecraft elliptical orbit rendezvous H-infinity control method according to claim 8, characterized in that: In step 33, select l0 < l1 <... < l q <<0, and use the result obtained in step 32 Define the intermediate matrix Y, and use Define the intermediate matrix X, and then use the intermediate matrices X and Y to obtain It includes the following steps: According to Equation (27), replacing t with l gives: Among them, is the coefficient matrix, is the truncation error of approximating vec(M(l, l)) with ; Ignoring the error term, there is: Among them, is the estimated value; Take the sequence {l0, l1,... l q}, where l0 < l1 <... < l q << l f = 0, and define the following intermediate matrix: Then we get:

10. The data-driven non-cooperative spacecraft elliptical orbit rendezvous H-infinity control method according to claim 9, characterized in that: In Steps 3 and 4, using what is obtained in Step 3-3 Obtain H ∞ Optimal controller approximate feedback gain K ap (t), and use the approximate feedback gain K ap (t) to obtain H ∞ Optimal controller, comprising the following steps: First, according to obtain an approximate feedback gain: Then, using K ap (t) to obtain H ∞ The optimal controller u(t) = -K ap χ(t).

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