Earth-formation satellite formation flying PID passive control method

Abstract

The present application relates to a kind of near-earth formation satellite PID passive control method, comprising the following steps: step S1, establish near-earth orbit formation port Hamilton system dynamics model;Step S2, establish passive PID controller according to dynamics model, additional system is constructed, and according to Lyapunov function demand, partial differential equation is solved;Step S3, according to the requirement of equilibrium point, design Lyapunov function, and determine additional system integral item initial value;Step S4, establish PID passive controller based on additional system integral item initial value.The present application effectively avoids the problem that PID controller parameter selection is difficult, and large-scale system performance is poor, and passive PID control method can be simply and effectively established for near-earth orbit satellite formation Hamilton dynamics system, which is beneficial to the design of controller.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of spacecraft formation technology, and particularly relates to a near-earth formation satellite PID passive control method. BACKGROUND

[0002] Spacecraft formation technology disperses the function of a single spacecraft into a group of smaller spacecrafts, thereby reducing risk and cost, and improving reliability and adaptability. The ability of a spacecraft to reshape or relocate from one formation to another can greatly improve the flexibility of formation missions. We call this change of formation configuration formation reconfiguration.

[0003] The core problem of formation reconfiguration is to change the relative position and velocity between spacecrafts by using control methods according to the relative dynamics between spacecrafts. For the study of relative dynamics of spacecrafts, the linearized model has certain requirements for the relative distance of satellite formation, and the accuracy of the dynamic model is reduced when linearized. The nonlinear relative dynamics model is relatively complex and is not conducive to the design of formation flight control law. PID is the most widely used control method in engineering. As we all know, PID controllers are very successful when the main control objective is to drive a given output signal to a constant value. However, the success of the PID controller depends on the proper selection of its gain. The gain is usually selected from the linearized approximation around its fixed operating point. However, when the operating range of the system is large, the linear approximation is not valid, so it is a challenging task to adjust the gain of the PID regulator. The passive PID control method (PID-PBC) can effectively avoid such problems, making the gain adjustment task very easy.

[0004] At present, the research on PID-PBC in near-earth orbit formation is still in a blank state. SUMMARY

[0005] In order to solve the technical problems existing in the prior art, the purpose of the present application is to provide a near-earth formation satellite PID passive control method, which can control the port Hamiltonian dynamics model of the near-earth orbit formation system, avoid the difficulty of gain adjustment, and the problem of invalidation of the PID control method in a large operating range.

[0006] In order to achieve the above application purpose, the present application provides a near-earth formation satellite PID passive control method, comprising the following steps:

[0007] Step S1, establishing a near-earth orbit formation port Hamiltonian system dynamics model;

[0008] Step S2, establishing a passive PID controller according to the dynamics model, constructing an additional system, and solving a partial differential equation according to the Lyapunov function requirement;

[0009] Step S3, according to the balance point demand, design Lyapunov function, and determine the additional system integral item initial value;

[0010] Step S4, based on the additional system integral item initial value to establish PID passive controller.

[0011] According to one aspect of the present application, in the step S1, the dynamics model of the port Hamilton system of the near-earth orbit formation is derived by following the Lagrange function of the spacecraft, and the specific derivation process is as follows:

[0012] The principal star coordinate in the ECI coordinate system is defined as Q0=[X0, Y0, Z0] T The satellite coordinate is Q=[X, Y, Z] j =[X j , Y j , Z j ] T The principal star coordinate in the LVLH coordinate system is q0=[r0, 0, 0] T The satellite coordinate is q=[x, y, z] j =[x j , y j , z j ] T ;

[0013] The satellite kinetic energy is expressed as:

[0014]

[0015] The potential energy of the satellite under the influence of J2 perturbation is expressed as:

[0016]

[0017] The Lagrange function is:

[0018]

[0019] The conversion from the LVLH coordinate to the ECI coordinate system is expressed as:

[0020]

[0021] Wherein, s o =sin(o), c o =cos(o), Omega is the ascending node right ascension, theta is the perigee amplitude angle, i is the principal star s0 orbit inclination; C is an orthogonal matrix, that is, C T C=I3, by derivation on both sides of the formula, we get

[0022] The skew-symmetric matrix function is defined as Then

[0023]

[0024] In the undisturbed orbit, the orbit parameters are constant, so the last two terms of matrix (5) are equal to zero;

[0025] The conversion relationship between the ECI coordinate system and the LVLH coordinate system is:

[0026] Q j = C(q j + q0) (6)

[0027] In the LVLH coordinate system, the spacecraft kinetic energy is:

[0028]

[0029] The spacecraft potential energy is:

[0030]

[0031] Where, R e is the Earth radius, R j = ||q0+q j ||2,

[0032] Then in the LVLH coordinate system, the Lagrange equation is:

[0033]

[0034] Select q j as the base variable, according to the Legendre transformation The relative dynamics between the primary and secondary stars is obtained by the Euler-Lagrange equation, and the formula is:

[0035]

[0036] The port Hamilton system near-earth formation satellite dynamics is calculated, and the formula is:

[0037]

[0038]

[0039]

[0040] Where, g = [0, I3] T , and the Hamilton function is obtained by the Lagrange equation:

[0041]

[0042] Formula (12) is the near-earth orbit formation port Hamilton system dynamics model.

[0043] According to one aspect of the present application, the step S1 specifically comprises:

[0044] The PID controller is defined as:

[0045]

[0046] Wherein, K P , K I > 0, K P > 0, K I > 0 are gain parameters of the PID controller. D

[0047] The initial condition of the controller state quantity is defined as The initial condition is used to set the equilibrium point of the Lyapunov stable closed loop distribution, specifically comprising:

[0048] In order to ensure the closed loop stability of the Lyapunov function, the function V is defined as:

[0049]

[0050] The function H d is defined as: Then

[0051] H d (x)≡V(x, x c ) (16)

[0052] According to the formula (15) and the formula (16), H d (x(t)) is a non-increasing function, if it can be guaranteed to be positive definite, then H d (x) can be obtained as a true Lyapunov function.

[0053] In order to satisfy the identity (16), x c is expressed as a function of x, and it is assumed that there is a function x c = γ(x) + κ, wherein is the initial value correction term,

[0054] Since the following PDE equation is obtained,

[0055]

[0056] According to the dynamic equation (12) and the partial differential equation (17), it can be solved that

[0057] γ(x) = q j (18)

[0058] Then​

[0059]

[0060] where x represents the function variable, and for near-earth formation dynamics, q j , p j ).

[0061] According to an aspect of the present application, the step S3 specifically comprises:

[0062] Based on formula (14), we have:

[0063]

[0064] Assuming the equilibrium point of the closed-loop function is At the equilibrium point, y = 0, and the controller at the equilibrium point has:

[0065]

[0066] The calculation results are:

[0067] Since We have:

[0068]

[0069] Substituting formula (20), we have:

[0070]

[0071] Then the initial value of the additional system integral term is:

[0072]

[0073] According to an aspect of the present application, the step S4 is represented as:

[0074]

[0075] According to an aspect of the present application, the step S3 further comprises Lyapunov stability verification of the initial value of the additional system integral term, specifically comprising:

[0076]

[0077] Since Therefore, we only need to verify

[0078]

[0079] Thus, we can ensure

[0080] ​x * = arg min H d (x) (28)

[0081] LyaPunov stability is achieved.

[0082] Compared with the prior art, the present application has the following beneficial effects:

[0083] The present application provides a low-earth formation satellite PID passive control method, which models the low-earth formation dynamics through a port Hamilton system, retains the nonlinear terms of the system, embodies the inherent dissipation characteristics of the system, avoids the reduction of the accuracy of the system caused by linearization, and guarantees the passivity of the system; the PID-PBC control method is designed for the low-earth formation dynamics system of the port Hamilton system, the explicit solution of the partial differential equation is given, the PID-PBC controller is designed based on the passive output of the system, which is beneficial to solve the formation reconstruction problem in the low-earth spacecraft formation flight, and the PID-PBC controller has higher accuracy in the presence of nonlinearities and disturbances.

[0084] The PID control method obtained by the low-earth formation satellite PID passive control method of the present application can provide a reference for formation design and control law design, and the parameter adjustment requirement of the PID control method is low, and the PID control method has good engineering implementation value in a larger system range. BRIEF DESCRIPTION OF DRAWINGS

[0085] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments will be briefly introduced below. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor on the basis of these drawings.

[0086] Figure 1 The flow chart of the low-earth formation satellite PID passive control method provided in one embodiment of the present application is schematically shown;

[0087] Figure 2 The coordinate system schematic diagram according to one embodiment of the present application is schematically shown;

[0088] Figure 3 The additional system configuration diagram of the controller according to one embodiment of the present application is schematically shown;

[0089] Figure 4 and Figure 5 The simulation experiment result diagram according to one embodiment of the present application is schematically shown. DETAILED DESCRIPTION

[0090] The description of the embodiments of the disclosure should be combined with the corresponding drawings, which should be part of the complete description. In the drawings, the shape or thickness of the embodiment can be exaggerated and simplified or facilitated. Furthermore, parts of the structures in the drawings will be described separately, and it should be noted that the elements not shown or not described by text in the drawings are in the form known to those skilled in the art.

[0091] The description of the embodiments herein, any reference to direction and position, is only for the convenience of description, and cannot be understood as any limitation on the scope of protection of the present application. The following description of the preferred embodiments will involve a combination of features, which can exist independently or in combination, and the present application is not particularly limited to the preferred embodiments. The scope of the present application is defined by the claims.

[0092] As shown in Figures 1 to 5 , a near-earth formation satellite PID passive control method of the present application comprises the following steps:

[0093] Step S1, establishing a near-earth orbit formation port Hamilton system dynamics model;

[0094] Step S2, establishing a passive PID controller according to the dynamics model, constructing an additional system, and solving a partial differential equation according to the Lyapunov function requirement;

[0095] Step S3, designing a Lyapunov function according to the equilibrium point requirement, and determining the initial value of the integral term of the additional system;

[0096] Step S4, establishing a PID passive controller based on the initial value of the integral term of the additional system.

[0097] In an embodiment of the present application, preferably, in the step S1, the near-earth orbit formation port Hamilton system dynamics model is derived by following the spacecraft Lagrange function, and the specific derivation process is as follows:

[0098] Define the main star coordinates in the ECI coordinate system as Q0=[X0,Y0,Z0] T , the satellite coordinates as Q j =[X j ,Y j ,Z j ] T , and the main star coordinates in the LVLH coordinate system as q0=[r0,0,0] T , the satellite coordinates as q j =[x j ,y j ,z j ]T;

[0099] The satellite kinetic energy is expressed as:

[0100]

[0101] The potential energy of the spacecraft under the influence of J2 perturbation is expressed as:

[0102]

[0103] The Lagrange function is:

[0104]

[0105] The transformation from LVLH coordinate to ECI coordinate is expressed as:

[0106]

[0107] where s o = sin(Ω), c o = cos(Ω), Ω is the right ascension of the ascending node, θ is the argument of perigee, i is the inclination of the primary star s0 orbit; C is an orthogonal matrix, i.e. C T C = I3, by taking the derivative of both sides, we get

[0108] Define the skew-symmetric matrix function S: then

[0109]

[0110] In the undisturbed orbit, the orbit parameters are constant, so the last two terms of matrix (5) are equal to zero;

[0111] The transformation relationship between ECI coordinate and LVLH coordinate is:

[0112] Q j = C(q j + q0) (6)

[0113] In the LVLH coordinate system, the spacecraft kinetic energy is:

[0114]

[0115] The spacecraft potential energy is:

[0116]

[0117] where R e is the radius of the earth, R j = ||q0+q j || 2 ,

[0118] Then in the LVLH coordinate system, the Lagrange equation is:

[0119]

[0120] Select q j As the base variable, according to Legendre transformation The relative dynamics between the primary star and the secondary star is obtained by Euler-Lagrange equation, and the formula is:

[0121]

[0122] The port Hamilton system near-earth formation satellite dynamics is calculated, and the formula is:

[0123]

[0124]

[0125]

[0126] Wherein, g = [0, I3] T The Hamilton function is obtained by Lagrange equation:

[0127]

[0128] The formula (12) is the near-earth orbit formation port Hamilton system dynamics model.

[0129] In one embodiment of the present application, preferably, in the step S1, specifically comprising:

[0130] The form of the PID controller is defined, and the formula is:

[0131]

[0132]

[0133] Wherein, K P , K I K P > 0, K I > 0, K D ≥ 0 are gain parameters of the PID controller;

[0134] The initial condition is defined for the controller state quantity The initial condition is used to set the equilibrium point of the Lyapunov function of the closed loop distribution, and specifically comprises:

[0135] In order to ensure the closed loop stability of the Lyapunov function, the function V is defined:

[0136]

[0137] Definition of function H d : Then

[0138] H d (x) = V(x, x c ) (16)

[0139] According to formula (15) and formula (16), H d (x(t) is a non-increasing function, if it can be guaranteed to be positive definite, then H d (x) can be obtained as a true Lyapunov function.

[0140] In order to satisfy the identity (16), x c is expressed as a function of x, and it is assumed that there is a function x c = γ(x) + κ, where is the initial value correction term,

[0141] Since the following PDE equation is obtained,

[0142]

[0143] According to the dynamic equation (12) and the partial differential equation (17), can be solved

[0144] γ(x) = q j (18)

[0145] Then

[0146]

[0147] Where x represents the function variable, and for near-earth formation dynamics, it represents (q j , p j ).

[0148] In an embodiment of the present application, preferably, in the step S3, specifically comprises:

[0149] Based on formula (14), the following is obtained:

[0150]

[0151] Assuming that the equilibrium point of the closed-loop function is At the equilibrium point, y = 0, and the controller at the equilibrium point has:

[0152]

[0153] The calculation result is:

[0154] Since Obtain:

[0155]

[0156] Substitute into equation (20), obtain:

[0157]

[0158] The initial value of the additional system integral term is:

[0159]

[0160] In an embodiment of the present application, preferably, in the step S4, the PID passive controller is represented as:

[0161]

[0162] In an embodiment of the present application, preferably, in the step S3, the Lyapunov stability verification of the initial value of the additional system integral term is further included, specifically comprising:

[0163]

[0164] Since Therefore, only the verification of

[0165]

[0166] It can be guaranteed that

[0167] x * = arg min H d (x) (28)

[0168] Lyapunov stability is achieved.

[0169] Embodiment 1

[0170] In the step S3, the orbit parameters are set as r0=6878.137km, e=0, i=60°, w=90°, Ω=0°, v=0°, and the initial state of the system is set as the position 100 meters away from the main star in the LVLH coordinate system, i.e. The end state is set as The PID-PBC parameters are set as K I =I3, K P =0.001I3, K D =I3, and according to equation (23), it can be obtained that

[0171] κ= [-529.1935-500.0008-10.7780] T (29)

[0172] The initial value of the additional system should be set as

[0173]

[0174] Embodiment 2

[0175] On the basis of embodiment one, Lyapunov stability verification of the initial value of the integral term of the additional system is completed:

[0176]

[0177]

[0178] Embodiment 3

[0179] The above parameters are simulated to obtain the satellite formation trajectory and control input.

[0180] Figure 4 The motion trajectory on the x, y and z axes of the satellite is shown, and it can be seen that the slave satellite starts from [100 0 0] T and finally stabilizes at [500 500 0] T The system completes convergence within 8000s, and each slave satellite gradually approaches the target point under the action of the thrust, and finally forms a new formation configuration. At the terminal time, the x-axis accuracy is about 0.0001m, the y-axis accuracy is about 0.0001m, and the z-axis accuracy is about 1e-8m.

[0181] Figure 5 The control input on the x, y and z axes of the satellite is shown, which meets the satellite thrust range.

[0182] The PID passive control method for the low-earth formation satellite of the application comprises the following steps: step S1, establishing a low-earth orbit formation port Hamilton system dynamics model; step S2, establishing a passive PID controller according to the dynamics model, constructing an additional system, and solving a partial differential equation according to the Lyapunov function requirement; step S3, designing a Lyapunov function according to the equilibrium point requirement, and determining the initial value of the integral term of the additional system; step S4, establishing a PID passive controller based on the initial value of the integral term of the additional system, modeling the low-earth formation dynamics through the port Hamilton system, retaining the nonlinear term of the system, embodying the inherent dissipation characteristics of the system, avoiding the reduction of the accuracy of the system caused by linearization, and ensuring the passivity of the system; the passive PID (PID-PBC) control method is designed for the port Hamilton system of the low-earth formation dynamics system, an explicit solution of the partial differential equation is given, the PID-PBC controller is designed based on the passive output of the system, which is beneficial to solve the formation reconstruction problem in the low-earth spacecraft formation flight, and the PID-PBC controller has higher accuracy in the presence of nonlinearities and disturbances.

[0183] The PID control method obtained based on the near-earth formation satellite PID passive control method of the present application can provide a reference for formation design and control law design, and the parameter adjustment requirement of the PID control method is low, and the PID control method has good engineering implementation value in a large system range.

[0184] In this document, the content not described in detail in the specification is the known technology of those skilled in the art, the term "includes", "contains" or any other variant thereof is intended to cover non-exclusive inclusion, so that the process, method, article or terminal device including a series of elements not only includes those elements, but also includes other elements not explicitly listed, or also includes the elements inherent to such process, method, article or terminal device. Without more limitations, the element defined by the statement "including a" does not exclude the presence of another identical element in the process, method, article or terminal device including the element.

[0185] It should be noted that the above is the preferred embodiment of the present application, and it should be pointed out that although the preferred embodiments of the present application have been described, once the basic creative concept of the present application is known, those skilled in the art can make several improvements and refinements without departing from the principles of the present application. These improvements and refinements should also be considered as the protection scope of the present application. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all modifications and modifications falling within the scope of the embodiments of the present application.

Claims

1. A method for PID passive control of a near-earth formation flying satellite, characterized in that, The method comprises the following steps: Step S1, establishing a near-earth orbit formation port Hamilton system dynamics model; Step S2, establishing a passive PID controller according to the dynamics model, constructing an additional system, and solving a partial differential equation according to a Lyapunov function requirement; Step S3, designing a Lyapunov function according to a balance point requirement, and determining an additional system integral initial value; Step S4, establishing a PID passive controller based on the additional system integral initial value; In the step S1, the near-earth orbit formation port Hamilton system dynamics model is derived through a Lagrange function of a following spacecraft, and a specific derivation process is as follows: The main star coordinates in the ECI coordinate system are defined as The slave star coordinates are The main star coordinates in the LVLH coordinate system are The slave star coordinates are ; The expression of the star kinetic energy is as follows: （1） From the stars The potential energy under perturbation is given by: （2） The Lagrange function is as follows: （3） The conversion from the LVLH coordinate system to the ECI coordinate system is as follows: （4） wherein , is the right ascension of the ascending node, is the argument of perigee, is the primary star is the orbit inclination; is the orthogonal matrix, i.e. , by derivation of both sides of the equation, we obtain ; Defining skew-symmetric matrix functions ; then （5） In the undisturbed orbit, the orbit parameters are constant, so the last two items of the matrix (5) are equal to zero; The conversion relationship between the ECI coordinate system and the LVLH coordinate system is as follows: （6） In the LVLH coordinate system, the spacecraft kinetic energy is as follows: （7） The spacecraft potential energy is as follows: （8） wherein is the radius of the earth, , In the LVLH coordinate system, the Lagrange equation is as follows: （9） Selecting As a base variable, according to the Legendre transformation The relative dynamics between the primary and the secondary are obtained by the Euler-Lagrange equation, which is given by （10） The near-earth formation satellite dynamics of the port Hamilton system is calculated, and the formula is as follows: （11） （12） wherein The Hamiltonian function is derived from the Lagrange equation: （13） The formula (12) is the near-earth orbit formation port Hamilton system dynamics model.

2. The near-earth formation flying satellite PID passive control method of claim 1, wherein, In the step S2, the following is specifically included: The formula of the PID controller is defined as follows: （14） wherein , is a gain parameter of the PID controller; An initial condition is defined for the controller state variables The initial condition is used to set the equilibrium point of the Lyapunov-stabilized closed loop assignment, in particular comprising: To ensure the closed-loop stability of the Lyapunov function, define the function , （15） Definition function then （16） According to equation (15) and equation (16), is a non-increasing function; To satisfy the identity (16), let be a function of , assuming the existence of a function such that is a first-order correction term, Due to the following PDE equation is obtained, （17） According to the dynamics equation (12) and the partial differential equation (17), the following can be solved （18） The step S3 specifically includes the following: （19） wherein representing the function variable, refers to the near-Earth formation dynamics .

3. The near-earth formation flying satellite PID passive control method of claim 2, wherein, Substituted into the formula (20), the following is obtained: Based on equation (14), , we obtain: Assume the equilibrium point of the closed loop function is At the equilibrium point, The controller at the equilibrium point then has: （20） The following were calculated: (21) As a result , it is obtained: （22） The additional system integral initial value is as follows: （23） In the step S4, the PID passive controller is expressed as follows: （24）。 4. The near-earth formation flying satellite PID passive control method of claim 3, wherein, The step S3 further includes Lyapunov stability verification of the additional system integral initial value, and specifically includes the following: （25） 。 5. The near-earth formation flying satellite PID passive control method of claim 4, wherein, It is ensured that （26） Since , it is only necessary to verify （27） Lyapunov stability is achieved. （28） ​

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