Fixed time spacecraft formation flying state constraint control method

Abstract

The application discloses a fixed-time spacecraft formation flight state constraint control method, solves the problem of how to realize fixed-time position preset performance cooperative control, and belongs to the field of spacecraft formation flight control.The application comprises the following steps: S1, a dynamic model of a controlled spacecraft is established according to a performance constraint function; the dynamic model comprises the relationship between spacecraft position tracking error and the minimum safety distance between the spacecrafts, and the performance constraint function with fixed-time convergence characteristics is used to limit the overshoot and steady-state error of the tracking error; S2, a sliding mode surface of the spacecraft i and a disturbance observer are established, a cooperative control law is established based on the dynamic model, the sliding mode surface and the disturbance observer, and the cooperative control law is used to realize fixed-time cooperative control of the formation spacecraft.

Description

TECHNICAL FIELD

[0001] The present application relates to a fixed-time spacecraft formation flight state constraint control method, belonging to the field of spacecraft formation flight control. BACKGROUND

[0002] In recent decades, spacecraft formation flight (SFF) has become an important technology for advanced space applications such as earth observation, high resolution, deep space exploration, etc. At present, different spacecraft formation flight missions have been proposed, such as TechSat-21, PRISMA, etc. In order to complete the above-mentioned tasks, different members of the spacecraft formation flight must track their positions in the reference coordinate system in order to perform specific formation configurations. However, the high nonlinearity of spacecraft dynamics, time-varying disturbances and uncertainties make controller design more difficult. Therefore, it is still a challenge to design a high-performance controller for spacecraft formation flight.

[0003] The existing controller designed for spacecraft formation position coordination tasks can ensure the asymptotic stability of the controlled system. The main constraints considered are model uncertainty, external disturbance, communication time delay, strong nonlinearity of the controlled system, lack of velocity measurement, communication packet loss, etc. The biggest problem of this kind of controller when meeting the constraints is that the robustness of the control algorithm to the initial value of the state is not strong. When a set of control parameters is selected, if the initial value of the state changes greatly, the actual stabilization time of the controlled system will also change greatly, and even it may not be stable.

[0004] Another constraint of spacecraft formation position coordination control is the collision avoidance problem between members, which is a very important constraint problem that must be considered in practical applications. If this constraint problem is ignored, it may lead to the destruction of member spacecraft due to collision, and ultimately lead to the failure of the entire control task. It is particularly important when in close proximity formation. The artificial potential function (APF) method is one of the most common methods for handling collision avoidance problems. Controllers designed by combining it with other control methods have been widely studied. Using the artificial potential function as a tool, combined with other control methods, the controller can effectively avoid collisions between member spacecraft and between them and obstacles. The artificial potential function can also be combined with sliding mode control, linear control methods, etc. to design a controller to ensure that the state of the controlled system is stable within a limited time.

[0005] In addition, the prescribed performance control (PPC) can also achieve the function of collision avoidance online. It helps to ensure that the output tracking error always remains within the region defined by the desired performance function. That is, it can limit the maximum overshoot of the state. Through this point, the relative position relationship between the members of the formation can be reasonably analyzed, and the mutual collision avoidance between the members can be achieved. The prescribed performance function can be used in combination with other control methods to study the collision avoidance problem between the members of the multi-agent formation and the collision avoidance problem with obstacles. In the problem of cooperative control of multiple agents, due to various disturbances in the environment, the dynamics of the individual members are also uncertain, and the communication between the members is also affected by the medium environment and the actual physical device. These unfavorable factors and unknown quantities will reduce the control accuracy of the designed controller. The reason is that the robust technology, whether it is adaptive control, state observer, disturbance observer, neural network, fuzzy logic and various function approximation methods, cannot completely estimate the unknown quantity, which ultimately makes the accuracy of the controller unknown or difficult to determine. That is, the formation tracking error of the control system based on approximation can be proved to converge to a set, the size of which is related to the upper bound of the controller parameters, the approximation error or the external disturbance. However, there is no systematic method to accurately find its upper bound, so it is impossible to achieve fixed-time position prescribed performance cooperative control. SUMMARY

[0006] In view of how to achieve fixed-time position prescribed performance cooperative control, the present application provides a fixed-time spacecraft formation flight state constraint control method for avoiding collision between spacecrafts.

[0007] The fixed-time spacecraft formation flight state constraint control method of the present application comprises:

[0008] S1, a dynamic model of the controlled spacecraft is established according to a performance constraint function, and the dynamic model is:

[0009]

[0010]

[0011] x 1i represents the error variable of the spacecraft i, the spacecraft serial number i=1, 2, …, n, n represents the total number of formation spacecrafts, x 2i represents the derivative of x 1i , M i ′=[m ix ,m iy ,m iz ] T , m ikdenotes the mass of spacecraft i in each direction in formation spacecrafts, g i denotes the polynomial of the orbit parameter of spacecraft i, d i denotes the external disturbance received by spacecraft i, u i denotes the thrust applied on spacecraft i; x 1i = [ε ix , ε iy , ε iz ] T , subscript k = x, y, z denotes the coordinate axis in three directions, e i denotes the error of spacecraft i from the desired position; e i = [e ix , e iy , e iz ] T , e ik denotes the error of spacecraft i from the desired position in each direction, and ||e ik ||≤ χ ik η ik ; Ξ i ' = [Ξ ix , Ξ iy , Ξ iz ] T , Ψ i = [ψ ix , ψ iy , ψ iz ] T , η ik = η ik (t), the performance constraint function η ik (t) is:

[0012]

[0013] η0 is the initial value of η ik (t), η ∞ is the final value of η ik (t), η0 > η ∞ > 0;

[0014] S2, a sliding mode surface and a disturbance observer of spacecraft i are established, a cooperative control law is established based on the dynamic model, the sliding mode surface and the disturbance observer, and the cooperative control law is used to realize the fixed-time cooperative control of the formation spacecrafts.

[0015] As preferred, the cooperative control law u c is:

[0016]

[0017] wherein, uc = [u c1 ; u c2 ;... ; u cn ] ;

[0018] u = [u1; u2;... ; u n ], u i denotes the thrust applied on the spacecraft i;

[0019] Θ4= [θ 41 ; θ 42 ;... ; θ 4n ], θ 4i denotes the compensated observation of the spacecraft i;

[0020] denotes the saturation compensation;

[0021] B = diag(B i ), B i = diag(1 - tanh(x 2ix ) 2 , 1 - tanh(x 2iy ) 2 , 1 - tanh(x 2iz ) 2 ) ;

[0022] x 2i = [x 2ix , x 2iy , x 2iz ] T ;

[0023] L denotes the Laplace matrix, which is the adjacency matrix of the communication topology;

[0024] N = diag(N i ), N i = diag(N ix , N iy , N iz ), M = diag(M i ), M i = diag(m ix , m iy , m iz ), m ik denotes the mass of the spacecraft i in the three directions in the formation; η = diag(η i ), η i = diag(η ix , η iy , η iz ) ; Ξ = diag(Ξ i ), Ξ i= diag(Ξ ix ,Ξ iy ,Ξ iz ) ;

[0025] S represents a sliding mode surface of the formation spacecraft directed communication topology;

[0026]

[0027]

[0028] virtual control law u d = [u d1 ; u d2 ;... ; u dn ], u di represents a virtual control quantity of the spacecraft i, u d is used to ensure that S converges to 0 within a fixed time;

[0029]

[0030] 0 < γ < 1;

[0031] 0 < γ3 < 1 is a positive number, μ1 > 4;

[0032] k1, k2, k3, k6 are coefficients and are positive constants.

[0033] As preferred, the sliding mode surface S = Θ1s of the formation spacecraft directed communication topology, s = [s1; s2;... ; s n ] of the spacecraft i is: i

[0034] s ik = tanh(x 2ik ) + k s1 P(x 1ik / k s2 )

[0035] wherein s i = [s ix , s iy , s iz ] T , x 1i = [x 1ix , x 1iy , x 1iz ] T ;

[0036]

[0037] The selection of parameters l1 and l2 should ensure that equation P(x 1ik ​ / k s2 ) and its derivative are continuous, k s1 , k s2 , σ3 are coefficients and positive constants.

[0038] As a preference, the virtual control law u d :

[0039]

[0040] where A = diag(A i ), x 2ik denotes the components of x 2i in three directions;

[0041] x1= [x 11 ; x 12 ;... ; x 1n ], x 1ik denotes the components of x 1i in three directions; g = [g1; g2;... ; g n ], θ2= [θ 21 ; θ 22 ;... ; θ 2n ], θ 2i denotes the observation of the disturbance observer;

[0042] 0 < γ2 < 1, k4, k5 are coefficients and positive constants.

[0043] As a preference, the disturbance observer:

[0044]

[0045]

[0046] where, p > 1, ν2 > 0, ν3 > 4L1, L1 is a positive number.

[0047] As a preference, the saturation compensation quantity is:

[0048]

[0049] where Δu ik = sat(u cik ) - u cik , u ci = [u cix , u ciy , uciz T represents the control output of the cooperative control law to spacecraft i in three directions, u max is the maximum value of the control force provided by the actuator.

[0050] As a preferred, the compensation observation θ 4i is:

[0051]

[0052]

[0053] where ω i = θ 3i - Y i , p1>1, ν5>0, ν6>4L2, L2 is a positive number.

[0054] As a preferred, the method for realizing the fixed-time anti-saturation cooperative control of the formation spacecraft by using the cooperative control law is:

[0055]

[0056]

[0057] u i = [u ix , u iy , u iz ] T represents the thrust applied to spacecraft i in three directions, u ci = [u cix , u ciy , u ciz ] T represents the control output of the cooperative control law to spacecraft i in three directions, u max is the maximum value of the control force provided by the actuator, k3>1.

[0058] The beneficial effects of the present application are that the spacecraft formation flight fixed-time cooperative control problem is studied on the basis of considering input saturation and preset performance constraints (safety distance constraints between formation members and steady-state error). By analyzing the relationship between the spacecraft position tracking error and the minimum safety distance between spacecrafts, a performance constraint function with fixed-time convergence characteristics is proposed to limit the overshoot and steady-state error of the tracking error. The present application also designs a new non-singular terminal sliding mode surface, which can make the system state stable in a fixed time after the sliding manifold is established. Combined with sliding mode control, backstepping method, fixed-time observer, saturation compensator and auxiliary system, a fixed-time cooperative tracking controller with input saturation and input rate saturation is proposed. ​BRIEF DESCRIPTION OF DRAWINGS

[0059] Figure 1 For the formation spacecraft relative motion coordinate system;

[0060] Figure 2 For the two member spacecraft position diagram considering safety distance constraint;

[0061] Figure 3 For the communication topology diagram;

[0062] Figure 4 For the disturbance estimation error of spacecraft 1 disturbance observer;

[0063] Figure 5 For the estimation error of ;

[0064] Figure 6 For the spacecraft 1 sliding mode surface change diagram;

[0065] Figure 7 For the state x 11 change diagram;

[0066] Figure 8 For the state x 21 change diagram;

[0067] Figure 9 For the spacecraft 1 position error change diagram;

[0068] Figure 10 For the spacecraft 1 velocity error change diagram;

[0069] Figure 11 For the spacecraft 1 control input change diagram;

[0070] Figure 12 For the spacecraft 1 control input rate of change change diagram;

[0071] Figure 13 For the formation spacecraft relative distance change diagram;

[0072] Figure 14 For the four spacecraft trajectory tracking in the reference coordinate. DETAILED DESCRIPTION

[0073] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative work fall within the scope of protection of the present application.

[0074] It should be noted that the embodiments in the present application and the features in the embodiments can be combined with each other without conflict.

[0075] The present application will be further described below in conjunction with the drawings and specific embodiments, but not as a limitation of the present application.

[0076] The fixed-time spacecraft formation flight state constraint control method of the embodiment comprises:

[0077] Step 1, design a performance constraint function and determine an avoidance strategy, and determine the dynamics model of the controlled spacecraft according to the performance constraint function and the avoidance strategy;

[0078] The relative motion model of the formation spacecraft is shown in the following formula (1). Figure 1 As shown in the formula (1), O I X I Y I Z I is the equatorial inertial coordinate system, denoted as F I . Each spacecraft in the formation is a rigid spacecraft. The reference spacecraft runs on a specific orbit, and o c x c y c z c is the orbit coordinate system of the reference spacecraft, denoted as F c . The origin of the coordinate system F c is located at the centroid of the reference spacecraft, the coordinate axis o c x c is along the local vertical direction, the coordinate axis o c y c is along the local horizontal direction, and the coordinate axis o c z c and the coordinate axis o c x c and the coordinate axis o c y c form a right-handed orthogonal coordinate system. Since we are studying distributed satellite formation cooperative control, the reference spacecraft is virtual in the present embodiment.

[0079] Let ρ i =[x i ,y i ,z i ] T represent the position vector of the spacecraft i in the formation relative to the reference spacecraft in the reference coordinate system. Then, its relative kinematics equation in the inertial coordinate system can be expressed as:

[0080]

[0081]

[0082]

[0083]

[0084]

[0085] where θi c is the true anomaly of the reference spacecraft; ri c = ||ri c || is the distance from the center of the Earth to the center of the reference spacecraft; μ E is the Earth gravitational constant; Mi i is the mass of spacecraft i in the formation; di i is the external disturbance experienced by spacecraft i; typically spacecraft experience external disturbances mainly due to the Earth's oblateness, the Earth's thin atmosphere, and other celestial bodies, etc., but these effects are usually bounded; ui i is the thrust applied to spacecraft i; the distance between the center of spacecraft i and the center of the Earth is denoted as i = 1, 2,..., n, n is the total number of spacecraft in the formation. Since this model does not make any approximation or linearization, it is exact and applicable to any eccentricity orbit when the distance of the formation members from the reference spacecraft is not very large.

[0086] Let ρi id denote the desired position of the i-th spacecraft in the formation, e i = ρi i - ρi id is the error of the i-th spacecraft from the desired position, then the following position error dynamics model can be obtained:

[0087]

[0088] where,

[0089] Lemma 1: If there exists a continuous radially unbounded function V(x): R n → R + ∪ {0} and satisfies the constraint for any state, where a, β, p, q, k > 0 are constants and pk< 1, qk> 1. Then the state converges to the origin in a bounded time and the convergence time equation satisfies

[0090] Lemma 2: If there exists a continuous radially unbounded function V: R n → R + ∪ {0} whose derivative satisfies where a, b, is positive and satisfies μ * > 1, φ > 0 is also a constant. Then the value of the function V will converge to a small neighborhood of 0 in a fixed time, say V < 2θ, where θ satisfies The upper bound of the convergence time is where

[0091] The new performance constraint function is designed as

[0092]

[0093] where k1, k2, α, γ are positive constants, and 0 < α < 1, γ > 1.

[0094] Theorem 1: For the designed performance constraint function (7), if η0> η ∞ > 0, then , η ik (t) = η ∞ . And the upper bound of the time for the function η ik (t) to converge from its initial value η0to its final value η ∞ is bounded, and the upper bound is determined by the four pre-set parameters.

[0095] Proof: The Lyapunov function is chosen as V1= 0.5(η(t)-η ∞ ) 2 . Then its derivative satisfies:

[0096]

[0097] where a1= k12 (α+1) / 2 , b1= k22 (γ+1)2 . According to Theorem 1, the upper bound of the convergence time of the function η(t) = η ∞ is This means that the function η(t) will converge to η ∞ at the latest at t = T.

[0098] In this embodiment, to avoid collision between member spacecrafts during formation flight, the tracking error between spacecraft i and its desired trajectory ρ id is used to describe the collision avoidance strategy.

[0099] The following assumptions must be met.

[0100] Assumption 1: It is assumed that each spacecraft has a physical safety sphere with the center of the spacecraft as the center and a radius of R c . As shown in FIG. 1, if any two members of the formation do not collide, then there is a Figure 2 ​ For members i ≠ j, n ≥ i, j ≥ 1, ||ρ i -ρ j ||>2R c .

[0101] Assumption 2: In spacecraft formation flying, the initial positions of any pair of spacecraft should satisfy the safe distance constraint between them, which means that ||ρ i (0)-ρ j (0)||≥2R c +σ1, where σ1 is a positive number.

[0102] Assumption 3: The minimum distance between the desired trajectories of any pair of spacecraft in formation flying is known. Then, define d... min =min 1≤i,j≤n ||ρ id -ρ jd ||.

[0103] To avoid collisions between spacecraft during formation flight, the position error in equation (6) should meet the following conditions.

[0104]

[0105] Where σ² is a positive number. This provides sufficient space during spacecraft formation flying to prevent collisions. For e ik The applied performance constraints should satisfy the following equation, where k = x, y, z, representing the three coordinate axes of the reference coordinate system.

[0106]

[0107] The conditions in equation (9) are guaranteed by the following equation.

[0108]

[0109] Where 0 < χ ik ≤d0 / η ik (0), η ik The definition is the same as in equation (7), η ik (0) is the function η ik The initial value.

[0110] Note 1: In this embodiment, the calculation of d0 considers a safe sphere in 3D space, and the formation spacecraft maneuvers in 3D space. In reality, d0 can be specifically calculated according to the requirements of a given mission. For example, spacecraft may only be able to perform in-plane maneuvers.

[0111] Note 2: Based on the above constraints on the state, the position tracking accuracy can be inferred in advance to be ||e ik ||≤χ ik η ik (∞), according to equation (11), the upper bound of the time to reach the steady-state error is T.

[0112] Note 3: The constraint in equation (11) also indicates that the method of this embodiment is not globally applicable, but only applies when the initial position error satisfies ||e ik (0) ||≤d0. This implementation method avoids collisions between spacecraft by constraining the upper bound of the maximum overshoot of the tracking error.

[0113] Next, a dynamic model of the controlled spacecraft is established based on the performance constraint function and the collision avoidance strategy, which is the controlled object model in this section. It is worth noting that it is difficult to find a solution for constraint problems like equation (11). This implementation introduces a new error variable ε. ik ∈(-∞,+∞), transforming the constraint error in equation (11) into a general controlled object model, where z ik =e ik / η ik .

[0114]

[0115] Note that if ε ik If the spacecraft is bounded, then the constraints in equation (11) hold, and no collisions will occur between spacecraft. Therefore, the controller needs to ensure that the variable ε... ik It is bounded and converges to 0 in a fixed time.

[0116] To facilitate controller design, the dynamic characteristics of the new error are required. Taking the time derivative of equation (12), we obtain:

[0117]

[0118] Where, χ i =[χ ix ,χ iy ,χ iz ] T , z i =[z ix ,z iy ,z iz ] T , ε i =[ε ix ,ε iy ,ε iz ] T .

[0119] Taking the time derivative of equation (13), we get:

[0120]

[0121] wherein N i ′ = [N ix , N iy , N iz ] T , Ψ i ′ = [ψ ix , ψ iy , ψ iz ] T , z i ′ = z i , Substituting equation (6) into equation (14) gives:

[0122]

[0123] wherein Ξ i ′ = [Ξ ix , Ξ iy , Ξ iz ] T , η i ′ = [η ix , η iy , η iz ] T , M i ′ = [m ix , m iy , m iz ] T , let x 1i = ε i , Finally, the dynamic model of the controlled object is shown in the following equation.

[0124]

[0125]

[0126] Step 2, establish the sliding mode surface and disturbance observer of the spacecraft i, and establish the cooperative control law based on the dynamic model, the sliding mode surface and the disturbance observer, and realize the fixed-time cooperative control of the formation spacecraft using the cooperative control law.

[0127] Before designing the cooperative control law, first define the relative position conversion error between the formation spacecraft member i and member j:

[0128] x 1ij = x 1i -x 1j (17)

[0129] Based on the definitions of directed G and the Laplace matrix, the lumped error with respect to member spacecraft i is defined as:

[0130]

[0131] The derivative is defined as follows:

[0132]

[0133] Among them l ij These are elements of matrix L, where L represents the Laplace matrix, which is the adjacency matrix of the communication topology. Let... The above formula can be written as

[0134]

[0135]

[0136] Where, x1=[x 11 ;x 12 ;...;x 1n ], Note that when the communication topology is a directed connected graph, Θ1 is an asymmetric invertible matrix.

[0137] In a preferred embodiment, for the dynamic system (16), regarding the sliding surface of spacecraft i:

[0138] s ik =tanh(x) 2ik )+k s1 P(x 1ik / k s2 ) (twenty two)

[0139] in The selection of parameters l1 and l2 should ensure that the equation P(x 1ik / k s2 ) and its derivative are continuous. μ > 4, k s1 k s2 It is a positive number, and σ3 is a small positive number.

[0140] Theorem 2: For the dynamic equation of the controlled system described by equation (16), the designed sliding surface equation (22) can guarantee that the system state converges to 0 within a fixed time after its sliding manifold is established. The upper bound of the convergence time is T. s Furthermore, it is independent of the initial state value. It is worth noting that this sliding surface is bounded, with its upper bound being 2k. s1 +1.

[0141] Proof: The Lyapunov function is chosen as Then its derivative satisfies:

[0142]

[0143] According to equation (23), the sliding surface can guarantee that the system state is bounded and asymptotically converges to 0. If the value of parameter k s2 is chosen such that the term is positive, then the following result holds.

[0144]

[0145] where According to Theorem 1, the system state x 1ik converges to 0 within a fixed time, and the upper bound of the convergence time is According to the expression of the sliding surface equation (22), we can get ik |≤|tanh(x 2ik )|+|P(x 1ik )|≤2k s1 +1.

[0146] Consider the sliding surface design with directed communication topology as:

[0147] S = Θ1s (25)

[0148] where s = [s1; s2;...; s n ], s i = [s ix , s iy , s iz ] T . It is worth mentioning that the sliding surface equation (25) is also saturated, and its upper bound is (2k s1 +1)||Θ1||.

[0149] In the preferred embodiment, the fixed-time disturbance observer is:

[0150]

[0151] where, p > 1, ν2> 0, ν3> 4L1.

[0152] For system (16), if assumption 4 is satisfied, the designed observer equation (26) can estimate the variable Δ 2i within a fixed time through the vector θ i . The upper bound of the estimation time is T ob .

[0153]

[0154] where v = (2 1 / 4 v1 / v2) 1 / (p+1 / 2) .

[0155] In practical applications, the output amplitude of the orbit control engine of the spacecraft is bounded, such as |u ik |≤u max , where u max is the maximum value of the control force provided by the actuator. In an embodiment, in order to achieve anti-saturation, the dynamic equation of the actuator of the spacecraft is constrained by the following formula.

[0156]

[0157] where u cik is the collaborative control law to be designed, k3≥1 is a positive number, u ik (0) = 0.

[0158] This embodiment is based on the dynamic model in formula (16), the sliding mode control technology and the backstepping method to design a fixed-time collaborative control law.

[0159] Step 21: design a virtual control law u d is to ensure that the sliding mode surface formula (25) converges to 0 in a fixed time. The design is as follows:

[0160]

[0161] where A = diag(A i ), g = [g1; g2; …; g n ], θ2 = [θ 21 ; θ 22 ; …; θ 2n ], M = diag(M i ), M i = diag(m ix , m iy , m iz ), η = diag(η i ), η i = diag(η ix , η iy , η iz ), N = diag(N i ), N i = diag(N ix , N iy , N iz ), Ξ = diag(Ξ i ), Ξ i = diag(Ξix ,Ξ iy ,Ξ iz );

[0162] τ1 is a reaching law and is designed as k4, k5, r3, r4, μ1 > 4, 0 < γ1, γ2 < 1 are positive constants.

[0163] Step 22: Since the amplitude of the output is limited, there is a mismatch between the control command signal and the actual control signal. In order to compensate for this mismatch Δu ik = sat(u cik )-u cik , a saturation compensator shown in the following equation is introduced.

[0164]

[0165] where is the state of the compensator and is included in the following controller. k6 > 0 is a constant.

[0166] Define the intermediate variable Taking the derivative of it, we get

[0167]

[0168] Assumption 5: The term is bounded and satisfies where L2 is a positive number.

[0169] The observer is designed as follows:

[0170]

[0171] where ω i = θ 3i -Υ i , p1 > 1, ν5 > 0, ν6 > 4L2.

[0172] For the system described by equation (31), if the condition in assumption 5 is satisfied, the designed observer equation (32) can observe 4i in a fixed time The upper bound of this time is T od . The expression of T od is similar to T ob , which is not repeated here.

[0173] Finally, the controller shown in the following equation is designed.

[0174]

[0175] where Y = [Y1; Y2;... ; YN], θ4= [θ1; θ2;... ; θN], n 41 42 4n u = [u1; u2;... ; uN], u = [u1; u2;... ; uN], n i ix iy iz T 0 < γ3< 1 is a positive number, B i = diag(1 - tanh(x1), 1 - tanh(x2),..., 1 - tanh(xN) ), 2i1 2 2i2 2 2i3 2 B = diag(B1), i

[0176] Theorem 3: For the dynamic system (16), if the controller in (33), the observer in (26), and the saturation compensator in (30) are adopted with (29), and assumptions 4 and 5 are satisfied, then the following conclusions are true.

[0177] (1) The sliding surface (25) converges to 0 in a fixed time, and the upper bound of the convergence time is T1, which is independent of the initial values of the states. The system states x 1i and x 2i converge to the origin in a fixed time, and the upper bound of the convergence time is T2, which is also independent of the initial values of the states.

[0178] (2) If the conditions in assumptions 1-3 are satisfied, and the initial positions of each member spacecraft satisfy the constraint in (10), then no collision between any two member spacecrafts will occur during the formation reconfiguration transition.

[0179] (3) The specified steady-state error will be achieved in a fixed time, and the upper bound of the time is T. The desired formation of the formation spacecrafts will be tracked in a fixed time, and the upper bound of the time is T2.

[0180] (4) All the signals in the controller are ultimately uniformly bounded.

[0181] Proof: The whole proof of the closed-loop system is divided into two steps. The first step considers the case that the designed observer has estimated the corresponding variables. For t ≥ To bs T = max(T obs T ob ​​​​​​​​​​​​​​​T od ), the Lyapunov function is chosen as Then its derivative is

[0182]

[0183] Substitute the controller into the above equation, note that θ2= Δ, Δ = [Δ1; Δ2;...; Δ n ], Then the following result holds.

[0184]

[0185] According to Young's inequality, Then the above equation can be written as

[0186]

[0187] According to Theorem 1, the following result holds.

[0188]

[0189] where b3= 2k5. According to Theorem 1, the Lyapunov function V3converges to 0 in a fixed time, and the upper bound of the convergence time is Therefore, the upper bound of the total time for the sliding surface to converge to 0 is T1= T obs + T r . According to the expression of V3, it can be known that the vectors S and Y converge to 0 in a fixed time, so they are bounded. Further, it can be obtained that the states x1, x2are also bounded, and u d is bounded. Therefore, when t≥ T obs , all variables are bounded. S = 0 means that the sliding flow of the sliding surface has been established. Finally, it is obtained that the states x1and x2converge to the origin in a fixed time, and the upper bound of the convergence time is T2= T1+ T s .

[0190] The analysis of the second step is the time period 0≤ t≤ T obs . In this period of time, the designed controller may not completely estimate the corresponding variables, which means that there is an estimation error. Therefore, it is necessary to analyze the influence of the estimation error. According to the proof process of the designed observer, it can be deduced that the estimation error is bounded. Without loss of generality, it is assumed that the upper bound of the error satisfies where Γ1and Γ2are positive numbers, The Lyapunov function is still chosen as V3, and its derivative is obtained by substituting the designed control law into it:​

[0191]

[0192] According to Young's inequality, we get... The final result is:

[0193]

[0194] According to Lemma 2, the Lyapunov function converges to the region V3≤2Ψ, where Therefore, it can be deduced that variables S and Y in time 0≤t≤T obs The inner state is also bounded. Furthermore, all variables are bounded. Therefore, V3 is eventually uniformly bounded, and ultimately all variables are eventually uniformly bounded. In addition, the boundedness of states x1 and x2 means that the constraints in equation (11) are satisfied. If the conditions from assumption 1 to 3 are satisfied, then any pair of spacecraft in the spacecraft formation flight can be guaranteed not to collide during the transient and steady-state processes. According to the description of the performance constraint function (7) above, its final value η ∞ The arrival time is fixed, with an upper bound of T. Therefore, the specified steady-state error can be obtained within a fixed time, with an upper bound of T; the desired configuration can also be tracked within a fixed time, with an upper bound of T2.

[0195] Simulation verification:

[0196] The performance of the proposed control scheme was verified through numerical simulation. The spacecraft formation flight scenario used involved a virtual lead spacecraft and four crew spacecraft performing a formation flight mission. In this scenario, it was assumed that the spacecraft performed a formation reconfiguration task, changing from any four positions in space to a rectangle. The effectiveness of the proposed controller was verified through two numerical examples. The orbital parameters of the virtual spacecraft are: a c =7178km, e c =0.01, Ω c =0 rad, i c =π / 6rad, ω c =0 rad, θ c (0) = 0 rad, mass is m c =100kg. The mass of each member's spacecraft is m. i =100kg. The maximum control force that the orbital control actuator can provide is 5N. The effects of Earth's non-spherical perturbation are considered the main external perturbation. The directed communication topology of the spacecraft is as follows: Figure 3 As shown.

[0197] The initial position and initial velocity of each member spacecraft in the reference coordinate system are: ρ1 = [-7 38 8] Tm, p2 = [56 -6 7] T m, p3 = [8 43 -6] T m, p4 = [-55 6 9] T m, v i = [0 0 0] T m / s, The desired position and desired velocity of each member spacecraft are set as: p 1d = [0 -50+sin(0.1t) 0.1t] T m, p 2d = [50 sin(0.1t) 0.1t] T m, p 3d = [0 50+sin(0.1t) 0.1t] T m, p 4d = [-50 sin(0.1t) 0.1t] T m,

[0198]

[0199] The parameters of the performance constraint function are set as η0 = 1, η ∞ = 0.5, k1 = k2 = 0.01, a = 1.4, g = 0.6. According to the desired position of each member spacecraft, d min is set as Suppose each member spacecraft is surrounded by a sphere with radius R c = 10 m, then d0 = 14.6389 m is obtained. The parameters of the sliding mode surface are set as k s1 = 2, k2 = 1, s3 = 0.1, m = 5. The parameters of the observer are set as v1 = v4 = 1, v3 = v5 = 0.1, v3 = v6 = 0.0001, p = p1 = 1.1. The parameters of the reaching law are set as k4 = 1, k5 = 0.1, m1 = 5, g1 = 0.01, g2 = g3 = 0.1. The remaining parameters are set as k3 = 1, k6 = 0.1, u max = 5.

[0200] The simulation results are shown in Figures 4 to 14 According to the foregoing statement, the time upper bound for the designed observer to completely estimate the corresponding variable is T obs = 231.9837 s. Figure 4 and Figure 5 show the difference between the designed observer and the corresponding variable, and the results show that the time required for the designed observer to approach its estimation target is 180 s, which is less than the calculated upper bound. The time upper bound for the sliding mode surface to converge to 0 is 263.2011 s, and the absolute value upper bound is 14.0232. Figure 6The actual time for the sliding surface to converge to zero is 190s, and its absolute value upper bound is 3.2944, which is also less than the calculated upper bound. The region where the sliding surface actually converges is {s ik ||s ik | < 6 x 10 -8}. The upper bound of the time for the system state to converge to the origin is 265.7252s. Figure 7 and Figure 8 The actual time for the states x 1ik and x 2ik to converge to the regions {x 1ik ||x 1ik | < 6 x 10 -9} and {x 2ik ||x 2ik | < 1.5 x 10 -8}, respectively, is 190s. According to the definition of the switching error, x 1ik = 0 implies e ik = 0. Therefore, the actual time for the position error to converge to zero is also 190s. Figure 9 This is verified, and it is also shown that the collision avoidance constraint is satisfied throughout the control process. The actual time for the performance constraint function η(t) to converge to its terminal value is 160s, which is shorter than the calculated upper bound of 504.8812s. This proves the correctness of Theorem 1. Figure 9 and Figure 10 The position tracking error e ik and the velocity tracking error actually converge to the regions {e ik ||e ik | < 4 x 10 -8}, The simulation results of the control input of spacecraft 1 and its first order derivative are shown in Figure 11 and Figure 12 respectively. These results show that the maximum values of |u ik | and are 5N and 10N / s, respectively, which satisfy the imposed constraints. The time responses of the relative distances between any pair of spacecrafts in the formation are shown in Figure 13 It can be seen that the distances between each pair of spacecrafts in the formation are all greater than the safety distance of 20m. The trajectories of the spacecrafts in the formation flight are shown in Figure 14 From these trajectories, it can be inferred that the required square structure has been established. The analysis of the above results proves the correctness of Theorem 3.

[0201] While the application has been described with reference to particular embodiments, it is to be understood that the application is not limited to the particulars disclosed. Rather, it is a continuation of the principles and applications of the present application. It is therefore to be understood that numerous modifications, both as to the details and embodiments illustrated and the application defined by the appended claims, can be made as forms of the application. It is also to be understood that the features described in connection with one embodiment can be used in conjunction with another embodiment.

Claims

1. A method for fixed-time spacecraft formation flight state constraint control, characterized in that, The method comprises: S1, establishing a dynamics model of the controlled spacecraft according to a performance constraint function, wherein the dynamics model is: x 1i error variable of spacecraft i, spacecraft index i = 1, 2, …, n, n represents the total number of formation spacecrafts, x 2i represents the derivative of x 1i , M i ′ = [m ix , m iy , m iz ] T , m ik represents the mass of spacecraft i in each direction among formation spacecrafts, g i represents the polynomial of the orbit parameter of spacecraft i, d i represents the external disturbance received by spacecraft i, u i represents the thrust applied on spacecraft i; x 1i = [ε ix , ε iy , ε iz ] T , subscript k = x, y, z represents the coordinate axis in three directions, z ik = e ik / η ik , e i represents the error of spacecraft i from the desired position; e i = [e ix , e iy , e iz ] T , e ik represents the error of spacecraft i from the desired position in each direction, and ||e ik || ≤ χ ik η ik ; Ξ i ′ = [Ξ ix , Ξ iy , Ξ iz ] T , Ψ i = [ψ ix , ψ iy , ψ iz ] T , η ik = η ik (t), the performance constraint function η ik (t) is: η0is η ik the initial value of η ∞ (t), η ik the final value of η ∞ > 0; S2, establishing a sliding mode surface and a disturbance observer of the spacecraft i, establishing a cooperative control law based on the dynamics model, the sliding mode surface and the disturbance observer, and realizing the fixed-time cooperative control of the formation spacecraft by using the cooperative control law; The cooperative control law u c is: wherein u c = [u c1 ; u c2 ;... ; u cn ] ; u = [u1; u2;... ; uN], u n i represents the thrust applied on the spacecraft i;​ θ4= [θ 41 ; θ 42 ;... ; θ 4n ], θ 4i denotes the compensated observation of spacecraft i; represents the amount of saturation compensation; B = diag(B i ) i = diag(1-tanh(x 2ix ) 2 ,1-tanh(x 2iy ) 2 ,1-tanh(x 2iz ) 2 ); x 2i = [x 2ix , x 2iy , x 2iz ] T ; L represents the Laplace matrix, which is the adjacency matrix of the communication topology; N = diag(N i ), N i = diag(N ix , N iy , N iz ), M = diag(M i ), M i = diag(m ix , m iy , m iz ), m ik denotes the mass of spacecraft i in three directions in the formation; η = diag(η i ), η i = diag(η ix , η iy , η iz ); Ξ = diag(Ξ i ), Ξ i = diag(Ξ ix , Ξ iy , Ξ iz ); S represents a sliding mode surface of the formation spacecraft with a directed communication topology; virtual control law u d = [u d1 ; u d2 ;... ; u dn ], u di denotes the virtual control of the spacecraft i, u d is used to guarantee that S converges to 0 in a fixed time; 0 < γ1 < 1; 0 < γ3 < 1 is a positive number, μ1 > 4; k1, k2, k3 and k6 are coefficients and are positive constants.

2. The fixed-time spacecraft formation flight state constraint control method according to claim 1, characterized in that, The formation spacecraft has a sliding mode surface S = Θ1s, s = [s1; s2;...; s n ] with the sliding mode surface s i for spacecraft i as: s ik = tanh(x 2ik ) + k s1 P(x 1ik / k s2 ) where s i = [s ix ,s iy ,s iz ] T and x 1i = [x 1ix ,x 1iy ,x 1iz ] T ; The choice of parameters l1and l2should guarantee that the equation P(x 1ik / k s2 ) and its derivatives are continuous, μ > 4, k s1 , k s2 , σ3are coefficients and are positive constants.

3. The fixed-time spacecraft formation flight state constraint control method according to claim 2, characterized in that, virtual control law u d : wherein x 2ik representing x 2i components in three directions; x1= [x 11 ; x 12 ;... ; x 1n ], x 1ik denotes the components of x 1i in three directions; g = [g1; g2;... ; g n ], θ2= [θ 21 ; θ 22 ;... ; θ 2n ], θ 2i denotes the observation of the disturbance observer; 0 < γ2< 1, k4, k5 are coefficients and are positive constants.

4. The fixed-time spacecraft formation flight state constraint control method according to claim 3, characterized in that, The disturbance observer is: wherein p > 1, v2 > 0, v3 > 4L1, L1 is a positive number.

5. The fixed-time spacecraft formation flight state constraint control method according to claim 1, wherein, Saturation compensation amount is: where Δu ik = sat(u cik ) - u cik , u ci = [u cix , u ciy , u ciz ] T denotes the control output of the cooperative control law in three directions for spacecraft i, u max is the maximum value of the control force provided by the actuator.

6. The fixed-time spacecraft formation flight state constraint control method according to claim 1, wherein, Compensated observation θ of spacecraft i 4i is: where ω i = θ 3i - Y i , p1>1, ν5>0, ν6>4L2, L2 is a positive number.

7. The fixed-time spacecraft formation flight state constraint control method of claim 1, wherein, The method for realizing the fixed-time anti-saturation cooperative control of the formation spacecraft by using the cooperative control law is: u i = [u ix , u iy , u iz ] T denotes the thrust applied to spacecraft i in three directions, u ci = [u cix , u ciy , u ciz ] T denotes the control output of the cooperative control law on spacecraft i in three directions, u max is the maximum value of the control force provided by the actuator, k3≥ 1.

8. A storage device readable by a computer, the storage device storing a computer program, characterized in that, The computer program is executed to realize the fixed-time spacecraft formation flight state constraint control method according to any one of claims 1 to 7.

9. A fixed-time spacecraft formation flight state constraint control apparatus comprising a storage device, a processor, and a computer program stored in the storage device and operable on the processor, wherein The processor executes the computer program to realize the fixed-time spacecraft formation flight state constraint control method according to any one of claims 1 to 7.

Images