A predefined time fault-tolerant control method for rigid spacecraft systems

Abstract

The application discloses a predefined time fault-tolerant control method for a rigid spacecraft system and relates to the technical field of aerospace flight control, and comprises the following steps: S1, a kinematics and dynamics model of a fault-tolerant control system of a rigid spacecraft attitude tracking error is established, and system states and control parameters are initialized; S2, a non-singular predefined time sliding mode surface is designed based on an attitude error tracking control system of the rigid spacecraft; and S3, a non-singular adaptive fault-tolerant predefined time controller is designed. The application designs a predefined time preset performance function and a non-singular adaptive fault-tolerant predefined time control method, effectively solves a potential chattering problem in a controller design process, guarantees the steady-state performance and state constraints of the attitude tracking error and guarantees the predefined time convergence of the system in the case that the controller fault, inertia uncertainty and external disturbance exist simultaneously.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of aerospace flight control, and particularly relates to a predefined time fault-tolerant control method for a rigid spacecraft system. BACKGROUND

[0002] In recent years, rigid spacecraft have attracted extensive attention in the development of control algorithms and practical task applications, and the attitude control performance is crucial to ensure the stability and safety of rigid spacecraft, and the safety and reliability of the spacecraft are particularly important, and a small fault in the rigid spacecraft system can lead to significant performance degradation or even instability.

[0003] For the problem of fault-tolerant control in the attitude tracking of the rigid spacecraft system, there are many control methods, among which the sliding mode control is particularly widely used in various fields, and the sliding mode control is considered to be an effective robust control method in solving system uncertainty and external disturbance, and the sliding mode control method has the advantages of simple algorithm, fast response, strong robustness to external disturbance and parameter perturbation, therefore, the sliding mode control method is widely used in various fields, however, the singularity problem in the sliding mode control method becomes an obstacle to the application of the sliding mode control in actual systems, in order to solve this problem, many methods for avoiding singularity have been proposed, such as virtual control input, introduction of a segmented function, etc., in these methods, the virtual control input increases the complexity of the control algorithm, and the introduction of the segmented function destroys the integrity of the controller, resulting in an increase in the uncertain factors of the system; a non-singular sliding surface control method is proposed, which can directly avoid the singularity problem of the controller and ensure the integrity of the controller.

[0004] The attitude tracking control of the rigid spacecraft system also needs to consider the rapid convergence of the system, for some spacecraft tasks, such as earth observation, near-space monitoring and on-orbit service, the spacecraft needs to achieve rapid attitude stabilization, and the finite time and fixed time theories can achieve rapid attitude stabilization, but the finite time theory is heavily dependent on the initial conditions of the system state, and the fixed time theory is usually difficult to calculate the accurate convergence time upper bound, which limits their application in practical applications, therefore, a predefined time fault-tolerant control method for a rigid spacecraft system is proposed. SUMMARY

[0005] The present application relates to the technical field of aerospace flight control, and particularly relates to a predefined time fault-tolerant control method for a rigid spacecraft system.

[0006] A predefined time fault-tolerant control method for a rigid spacecraft system, comprising the following steps:

[0007] S1, establishing a kinematics and dynamics model of a rigid spacecraft attitude tracking error fault-tolerant control system, initializing system states and control parameters;

[0008] S2, designing a non-singular predefined time sliding mode surface based on the attitude error tracking control system of the rigid spacecraft;

[0009] S3, designing a non-singular adaptive fault-tolerant predefined time controller.

[0010] Preferably, in the step S1, the dynamics model of the rigid spacecraft angular velocity error tracking control system is expressed as:

[0011]

[0012] wherein ω and are the angular velocity and angular acceleration of the rigid spacecraft, respectively, ω e and are the angular velocity error and time derivative of the angular velocity error of the rigid spacecraft, respectively, ω d and are the expected value of the angular velocity and time derivative of the expected value of the angular velocity of the rigid spacecraft, respectively, Ω is the angular velocity of the reaction flywheel, × is an operator symbol, the operator symbol × is applied to ω × and ω e × , and is represented as follows: ω × = [0, -ω3, ω2; ω3, 0, -ω1; -ω2, ω1, 0] and ω e × = [0, -ω e3 , ω e2 ; ω e3 , 0, -ω e1 ; -ω e2 , ω e1 , 0], J ω = diag([J ω1 , J ω2 , J ω3 , J ω4 ]) is the moment of inertia matrix of the reaction flywheel, J0 and ΔJ represent the nominal part and the uncertain part of the moment of inertia matrix, respectively, D is the control torque distribution matrix of the reaction flywheel, C represents the corresponding rotation matrix, u is the input of the reaction flywheel, E(t) = diag([e1(t), e2(t), e3(t), e4(t)]) is the actuator control efficiency matrix, 0 ≤ e i (t) ≤ 1, i = 1, 2, 3, 4 represents the effective coefficient of the i-th reaction flywheel, is the additional fault input of the reaction flywheel, d(t) is the external disturbance;

[0013] The kinematic model of the rigid spacecraft attitude tracking error system is expressed as:

[0014]

[0015] where, I3=diag([1,1,1]),q ve =[q 1e ,q 2e ,q 3e ] T denotes the vector part of the attitude tracking error, q 4e denotes the scalar part of the attitude tracking error, and are the time derivatives of q ve and q 4e , respectively.

[0016] Preferably, in the step S2, the design process is as follows:

[0017] The performance function is designed as:

[0018]

[0019] where, 0 T < 1, ρ0and ρ i are the initial value and the final value of ρ T (t), respectively, is the pre-defined upper bound of the convergence time of the performance function;

[0020] The normalized tracking error is designed as:

[0021]

[0022] The non-singular pre-defined time sliding surface s is designed as:

[0023]

[0024] where, ε = [ε1, ε2, ε3] T , sign(·) is the sign function, sig(·) is the vector form of sign(·), and the function sig 1+α (ε) = [|ε1| 1+α sign(ε1), |ε2| 1+α sign(ε2), |ε3| 1+α sign(ε3)] T , is the first order derivative of ε; 0 < a < 1; is a predefined nonsingular predefined time sliding surface convergence time upper bound;

[0025] Time derivative of e is:

[0026]

[0027] where, γ = diag([γ1, γ2, γ3]), is the first derivative of e * , ρ(t) = diag([ρ1, ρ2, ρ3]), ρ -1 (t) is the inverse matrix of ρ(t).

[0028] Preferably, in the step S2, the design process is as follows:

[0029] The design function φ is:

[0030]

[0031] where, μ = (1 - a) / (1 + a);

[0032] The nonsingular adaptive fault-tolerant predefined time controller is designed as:

[0033]

[0034] where, ||b|| is the norm of b, ||D|| is the norm of D, is a predefined attitude tracking error convergence time upper bound, λ max is the maximum value of the characteristic value of the inertia matrix, and is the estimated parameter, and is and the adaptive law of Φ = 1 + ||ω| + ||ω|| 2 , 0 < β < 1;

[0035] The update law of the adaptive parameter is designed as:

[0036]

[0037] where, p0> 0, p1> 0, ||g|| is the norm form of g;

[0038] The Lyapunov functions V1 and V2 are designed as:

[0039]

[0040] where s T is the transpose of s, and is the error value of the estimated parameter;

[0041] Taking the derivative of equation (6) gives:

[0042]

[0043] Substituting the rigid spacecraft dynamics model equations (1), (2) and (3) into equation (7) gives:

[0044]

[0045] where is the second derivative of ε, B(t, q ve ) = bDE(t),

[0046] Taking the derivative of equation (13) gives:

[0047]

[0048] Substituting equation (14) into equation (16) gives:

[0049]

[0050] Substituting equation (15) into equation (17) gives:

[0051]

[0052] Substituting equations (9), (10) and (11) into equation (18) gives:

[0053]

[0054] Since there are two cases of 0 < g < 1 and g > 1 for g, if equation (19) is written in the form of , then the system is pre-defined time convergent; where

[0055] It can be concluded that the attitude tracking error q ve in the rigid spacecraft system can converge to a small enough region, and the convergent region is:

[0056]

[0057] wherein, T max is the time upper bound of T P .

[0058] Compared with the prior art, the present application has the advantages of:

[0059] The present application is based on a preset performance function, a non-singular pre-defined time sliding surface and an adaptive parameter, and designs a pre-defined time preset performance function and a non-singular adaptive fault-tolerant pre-defined time control method, effectively solving the potential chattering problem in the controller design process, ensuring the steady-state performance of the attitude tracking error and the state constraint, and ensuring the pre-defined time convergence of the system in the case of simultaneous existence of controller faults, inertia uncertainty and external disturbances. BRIEF DESCRIPTION OF DRAWINGS

[0060] Figure 1 is a control flow diagram of the present application.

[0061] Figure 2 is an attitude tracking error effect diagram of the present application.

[0062] Figure 3 is an angular velocity error effect diagram of the present application.

[0063] Figure 4 is a non-singular pre-defined time sliding surface diagram of the present application.

[0064] Figure 5 is a controller input diagram of the present application.

[0065] Figure 6 is a comparison diagram of the attitude tracking error of the present application and the attitude tracking error under the finite time controller scheme.

[0066] Figure 7 is a comparison diagram of the angular velocity error of the present application and the angular velocity error under the finite time controller scheme.

[0067] Figure 8 is a comparison diagram of the controller performance of the present application and the performance of the finite time controller. DETAILED DESCRIPTION

[0068] In order to make the technical means, creative features, purposes and effects achieved by the present application easy to understand, the present application is further described below in conjunction with specific embodiments.

[0069] Referring to Figure 1 Fig. 1, a pre-defined time fault-tolerant control method for a rigid spacecraft system includes the following steps:

[0070] S1, establish the kinematics and dynamics model of the rigid spacecraft attitude tracking error fault-tolerant control system, initialize the system state and control parameters;

[0071] S2, considering the existence of rotational inertia uncertainty and external disturbance, based on the attitude error tracking control system of rigid spacecraft, a non-singular predefined time sliding mode surface is designed, which can avoid the singular problem in the controller, and the integrity of the controller is guaranteed through a design function;

[0072] S3, design a non-singular adaptive fault-tolerant pre-defined time controller.

[0073] In step S1, the dynamics model of the rigid spacecraft angular velocity error tracking control system is expressed as:

[0074]

[0075] Where, ω and are the angular velocity and angular acceleration of the rigid spacecraft, ω e and are the angular velocity error and the time derivative of the angular velocity error of the rigid spacecraft, ω d and are the expected value of the angular velocity and the time derivative of the expected value of the angular velocity of the rigid spacecraft, Ω is the angular velocity of the reaction flywheel, × is the operator symbol, and the operator symbol × is applied to ω × and ω e × , which is expressed as: ω × =[0,-ω3,ω2;ω3,0,-ω1;-ω2,ω1,0] and ω e × =[0,-ω e3 ,ω e2 ;ω e3 ,0,-ω e1 ;-ω e2 ,ω e1 ,0], J ω =diag([J ω1 ,J ω2 ,J ω3 ,J ω4 ]) is the rotational inertia matrix of the reaction flywheel, J0 and ΔJ represent the nominal part and the uncertain part of the rotational inertia matrix respectively, D is the control torque distribution matrix of the reaction flywheel, C represents the corresponding rotation matrix, u is the input of the reaction flywheel, E(t)=diag([e1(t),e2(t),e3(t),e4(t)]) is the actuator control efficiency matrix, 0≤e i(t)≤ 1, i = 1, 2, 3, 4 denote the effective coefficients of the ith reaction flywheel, is the additional failure input of the reaction flywheel, d(t) is the external disturbance;

[0076] The kinematic model of the rigid spacecraft attitude tracking error system is expressed as:

[0077]

[0078]

[0079] where, I3= diag([1, 1, 1]), q ve = [q 1e , q 2e , q 3e ] T denotes the vector part of the attitude tracking error, q 4e denotes the scalar part of the attitude tracking error, and are the time derivatives of q ve and q 4e , respectively.

[0080] In the step S2, the design process is as follows:

[0081] The performance function is designed as:

[0082]

[0083] where 0 < c < 1, p0and p T are the initial value and the final value of p i (t), respectively, is the pre-defined upper bound of the convergence time of the performance function;

[0084] The normalized tracking error is designed as:

[0085]

[0086] The nonsingular pre-defined time sliding surface s is designed as:

[0087]

[0088] where, ε = [ε1, ε2, ε3] T , sign(·) is the sign function, sig(·) is the vector form of sign(·), and the function sig 1+α (ε) = [|ε1| 1+α sign(ε1), |ε2| 1+αsign(ε2), |ε3| 1+α sign(ε3) T , is the first derivative of ε with respect to time; 0 < α < 1; is a predefined nonsingular predefined time sliding surface convergence time upper bound;

[0089] Taking the time derivative of ε:

[0090]

[0091] wherein, γ = diag([γ1, γ2, γ3]), is the first derivative of e * , ρ(t) = diag([ρ1, ρ2, ρ3]), ρ -1 (t) is the inverse matrix of ρ(t).

[0092] In the step S2, the design process is as follows:

[0093] The design function φ is:

[0094]

[0095] wherein, μ = (1 - α) / (1 + α);

[0096] The nonsingular adaptive fault-tolerant predefined time controller is designed as:

[0097]

[0098] wherein, ||b|| is the norm of b, ||D|| is the norm of D, is a predefined attitude tracking error convergence time upper bound, λ max is the maximum value of the characteristic value of the inertia matrix, and is an estimated parameter, and is and is the adaptive law of 2 , 0 < β < 1;

[0099] The update law of the adaptive parameter is designed as:

[0100]

[0101] wherein, p0 > 0, p1 > 0, ||g|| is the norm form of g;

[0102] Lyapunov functions V1 and V2 are designed:

[0103]

[0104] where s T is the transpose of s, and is the error value of the estimated parameter;

[0105] Differentiating equation (6) gives:

[0106]

[0107] Substituting the rigid spacecraft dynamics model equations (1), (2) and (3) into equation (7) gives:

[0108]

[0109] where is the second derivative of ε, B(t, q ve ) = bDE(t),

[0110] Differentiating equation (13) gives:

[0111]

[0112] Substituting equation (14) into equation (16) gives:

[0113]

[0114] Substituting equation (15) into equation (17) gives:

[0115]

[0116] Substituting equations (9), (10) and (11) into equation (18) gives:

[0117]

[0118] Since there are two cases of 0 < g < 1 and g > 1 for g, if equation (19) is written in the form of , then the system is pre-defined time convergent; where

[0119] It can be concluded that the attitude tracking error q ve in the rigid spacecraft system can converge to a sufficiently small region, and the convergence region is:

[0120]

[0121] wherein, T max is the upper bound of time. P

[0122] Based on the above analysis, the attitude tracking error, angular velocity error, and non-singular predefined time sliding mode surface of the rigid spacecraft system converge in the predefined time.

[0123] To verify the effectiveness of the method, the present application gives a comparison between the non-singular adaptive fault-tolerant predefined time control (NPFPTC) method and the finite time control (FTC) method:

[0124] In order to compare more effectively, all parameters of the system are consistent, that is: q(0)=[0.3,-0.2,-0.3,0.8832] T , ω(0)=[1,0,-1] T rad / s, J ω =0.015I4 kg·m 2 , E=diag([0.6,0.8,0.7,0.5]), J0=[20,1.2,0.9;1.2,17,1.4;0.9,1.4,15] kg·m 2 , Ω=[50,50,50,50] T kg·m 2 , q d (0)=[0,0,0,1] T , ω d (0)=[0,0,0] T rad / s, the given system inertia uncertainty is: ΔJ=0.1J0 kg·m 2 , and the given controller fault input is: occurs at 15 seconds, occurs at 16 seconds, The given system external disturbance is: d=0.3[sin(0.8t),cos(0.5t),cos(0.3t)] N·m, and the system control signal parameters are: α=0.015, β=0.015, λ max =20, ρ0(t)=5, ρ ∞ (t)=0.05, c=0.4, p0=0.01,p1=0.01.

[0125] From Figure 2 and Figure 3 ​We show that the attitude tracking error and angular velocity error of the rigid spacecraft system converge in predefined time, and the attitude tracking error is constrained under a pre-defined performance function.

[0126] From Figure 4 It can be seen that the nonsingular predefined time sliding mode surface converges in predefined time.

[0127] From Figure 5 It can be seen that the nonsingular adaptive fault-tolerant predefined time controller converges in predefined time in the presence of controller faults, external disturbances and inertia uncertainty matrix.

[0128] From Figure 6 And Figure 7 We show that the NPFPTC method and the FTC method can achieve similar tracking performance in attitude error and angular velocity error, but in Figure 8 By comparison, it can be seen that the NPFPTC method requires less input than the FTC method, and the FTC method has obvious chattering phenomenon when the controller is in the steady state stage, while the NPFPTC method effectively solves the potential chattering problem in the controller design process.

[0129] In summary, compared with the FTC method, the NPFPTC method has stronger stability in the presence of controller faults, external disturbances and inertia uncertainty matrix, and ensures the steady-state performance and state constraint of the attitude tracking error.

[0130] From the technical common sense, the present application can be realized by other embodiments without departing from the spirit or essential characteristics thereof. Therefore, the above disclosed embodiments, in all aspects, are only illustrative and not the only. All changes within the scope of the present application or within the scope equivalent to the present application are included in the present application.

Claims

1. A predefined time fault-tolerant control method for a rigid spacecraft system, characterized by: The following steps are involved: S1. Establish the kinematic and dynamic models of the fault-tolerant control system for the attitude tracking error of the rigid spacecraft and initialize the system state and control parameters. S2. Design of non-singular predefined time sliding surface based on attitude error tracking control system of rigid spacecraft; S3. Design a non-singular adaptive fault-tolerant predefined time controller; In step S1, the dynamic model of the rigid spacecraft angular velocity error tracking control system is expressed as: (1) in, and are the angular velocity and angular acceleration of the rigid spacecraft, and are the angular velocity error and the time derivative of the angular velocity error of the rigid spacecraft, and are the expected value of the angular velocity of the rigid spacecraft and the time derivative of the expected value of the angular velocity, is the angular velocity of the reaction flywheel, Is an operator, the operator Applicable to and , which is expressed as follows: and , is the rotational inertia matrix of the reaction flywheel, and denote the nominal part and the uncertain part of the rotational inertia matrix, respectively. is the control torque distribution matrix of the reaction flywheel, represents the corresponding rotation matrix, , is the reaction flywheel input, is the actuator control efficiency matrix, Indicates the The effective coefficient of a reaction flywheel, is the additional fault input of the reaction flywheel, is an external disturbance; The kinematic model of the rigid spacecraft attitude tracking error system is expressed as: (2) (3) in, , , represents the vector part of the attitude tracking error, , represents the scalar part of the attitude tracking error, and They are and The time derivative of In step S2, the design process is as follows: The design performance function is: (4) in, , and yes The initial and final values ​​of is the upper bound of the convergence time of the predefined performance function; The designed standardized tracking error is: (5) Design of non-singular predefined time sliding surface for: (6) in, , , is a symbolic function, yes The vector form of the function , for The first derivative of ; , ; is the upper bound of the convergence time of the predefined non-singular predefined time sliding surface; right Take the time derivative: (7) in, , , , , , , yes The first derivative of , yes The inverse matrix of In step S3, the design process is as follows: Design Function for: (8) in, ; Design a non-singular adaptive fault-tolerant predefined time controller as: (9) in, for The norm of for The norm of is the upper bound of the convergence time of the predefined attitude tracking error, , , is the maximum value of the eigenvalue of the inertia matrix, and are the estimated parameters, and yes and The adaptive law of , ; The update law of the designed adaptive parameters is: (10) (11) in, , , , , , , yes The norm form of ; Design of Lyapunov functions and : (12) (13) in, yes The transpose of and is the error value of the estimated parameter; Derivative of formula (6) yields: (14) Substituting the rigid spacecraft dynamics model equations (1), (2) and (3) into equation (7) yields: (15) in, for The second derivative of , , ; Differentiating formula (13) yields: (16) Substituting formula (14) into formula (16) yields: (17) Substituting formula (15) into formula (17) yields: (18) Substituting equations (9), (10) and (11) into equation (18), we obtain: (19) because exist and If we write Equation (19) as , then the system is determined to be convergent in a predefined time; where, , , , , ; It can be concluded that the attitude tracking error in the rigid spacecraft system It can converge to a sufficiently small area, and the area of ​​convergence is: (20) in, , , for The time upper bound of .

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