A large flexible spacecraft distributed fault-tolerant attitude and vibration control method

Abstract

The application belongs to the technical field of satellite attitude control, and relates to a large flexible spacecraft distributed fault-tolerant attitude and vibration control method, a large flexible spacecraft rigid-flexible coupling dynamics model with distributed actuators is established by using the Lagrange method, a hierarchical Stackelberg game control framework is established by taking an attitude control subsystem as a leader layer and a vibration suppression subsystem and a fault-tolerant control subsystem as follower layers, performance functions of the three subsystems are defined, and through game of spacecraft distributed fault-tolerant attitude and vibration control, strategy cooperation and dynamic balance between at least two control targets are realized; the application realizes three-axis attitude stabilization and low-frequency vibration suppression of a target spacecraft, and shows stronger fault-tolerant capability under node failure.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of aerospace, and relates to a large flexible spacecraft distributed fault-tolerant attitude and vibration control method. BACKGROUND

[0002] With the wide application of large flexible spacecraft in space science, deep space exploration and on-orbit assembly, high-precision attitude control and vibration suppression have become the core technology to ensure the success of the task. The dynamic behavior of large flexible spacecraft shows obvious rigid-flexible coupling characteristics and multi-modal vibration characteristics. Although the redundant actuator design can improve the system reliability, in the case of partial node failure or actuator saturation, local control failure is easy to spread to the whole, leading to the accumulation of attitude deviation and affecting the execution of the load precision task. The existing control method lacks the ability of system-level collaborative optimization, and it is difficult to ensure the attitude precision and vibration attenuation effect at the same time under the conditions of multi-modal vibration interference and node failure. Therefore, a distributed collaborative control method is needed to realize the integrated control of large flexible spacecraft attitude and vibration, and to consider the actuator fault tolerance. This method should be able to use multi-node collaborative optimization, taking into account the global performance of attitude precision, vibration suppression and redundancy allocation, so as to meet the strict requirements of future high-precision space observation, deep space exploration and long-term on-orbit operation on attitude stability and task reliability. SUMMARY

[0003] The purpose of the application is to overcome the shortcomings in the prior art and provide a large flexible spacecraft distributed fault-tolerant attitude and vibration control method.

[0004] To achieve the purpose of the application, the following technical solutions are adopted.

[0005] A large flexible spacecraft distributed fault-tolerant attitude and vibration control method, the large flexible spacecraft is a composite system composed of a rigid body, at least two flexible structures and actuators distributed on the at least two flexible structures, comprising the following steps:

[0006] S1, under the condition of comprehensively considering the coupling effect between the rigid body attitude motion and the flexible structure deformation, the output torque of the actuator is mapped to the generalized coordinate space, and the Lagrange method is used to construct a Lagrange dynamic model containing the actuator node layout and output characteristics;

[0007] S2, based on the characteristics of the composite system having at least two flexible structures and the distribution characteristics of the actuators, taking the flexible structures as the control targets, establishing a hierarchical Stackelberg game control framework with the attitude control subsystem as the leader layer and the vibration suppression subsystem and the fault-tolerant control subsystem as the follower layer, and defining the performance functions of the three subsystems, through the game of distributed fault-tolerant attitude and vibration control of the spacecraft, to realize the strategy coordination and dynamic balance between at least two control targets; wherein:

[0008] The attitude control subsystem has an attitude control law designed based on a particle swarm optimization method, and the global optimization of the control gain parameters is used to stabilize the attitude of the control target;

[0009] The vibration suppression subsystem introduces a saturated sliding mode control strategy, and through the joint action of the linear term and the saturation term, low-frequency vibration is suppressed;

[0010] The fault-tolerant control subsystem has a fault-tolerant control strategy constructed based on a distributed alternating direction multiplier method; when a certain actuator fails or performance degrades, under the condition of satisfying the global consistency constraint, the optimal torque distribution of the remaining actuators is solved to adaptively redistribute the output torque of the remaining actuators.

[0011] Further, the construction process of the Lagrange dynamics model is as follows:

[0012] The attitude motion of the rigid body is described by the rotation of the body coordinate system about the - - The order of Euler angles is used to represent the rotation relationship from the orbital coordinate system to the body coordinate system, and the transformation form is represented as:

[0013] , ,

[0014] In the formula: Corresponding to the , , Three-axis angular velocity of the satellite; Indicates the rotation matrix; Indicates the attitude angle; And Indicate the roll angle and the pitch angle; the upper subscript Indicates the matrix transpose;

[0015] Then, the bending displacement of the flexible structure is modal-discretized, and combined with the boundary conditions to obtain:

[0016] ,

[0017] ,

[0018] In the formula: Indicates the order of the dominant vibration mode, taking... , For the curved shape function, For modal generalized coordinate vectors; Represents the bending displacement of a flexible structure, and its relationship with displacement. and time Simultaneously related; Represents partial derivatives; Indicates the length of the structure.

[0019] The spacecraft's own kinetic energy and the kinetic energy of distributed actuators They are represented as follows:

[0020] ,

[0021] ,

[0022] In the formula: Let be the rotational inertia matrix of the central rigid body. For the first Modal mass matrix of a flexible structure The rigid-flexible coupling matrix is ​​obtained by calculation using the finite element method; Indicates the control node number; The total number of nodes is represented by ; the moment of inertia matrix of the actuator is denoted as . Distributed executors in the first... Flexible structure The angular momentum vector at the control node is Its scalar size is ; angular momentum vector The relationship with the actuator dynamics is as follows:

[0023] ,

[0024] In the formula: For the actuator configuration matrix, The angular velocity vector of the actuator flywheel determines the control torque generated by the actuator; the flexible structure in the first... Angular deformation at control nodes for:

[0025] , ,

[0026] In the formula: Let be the modal rotation matrix at the node; define For the first Given the modal stiffness matrix of a flexible structure, the potential energy of the system can be written as:

[0027] ,

[0028] Combining the kinetic and potential energies of the rigid body, flexible structure, and distributed actuators, we obtain the Lagrangian function of the system, which is the total mechanical energy of the system:

[0029] ,

[0030] To fully describe the system dynamics, non-conservative force terms must also be considered, including:

[0031] 1) External disturbance torque ;2) Flexible modal damping term 3) Additional items caused by distributed actuator failure:

[0032] ,

[0033] In the formula: For actuator proportional failure factor matrix; The bias fault matrix represents the additional error; together, they describe the actuator fault characteristics. This represents the output torque of node i; this fault is represented in both the attitude and control channels as follows:

[0034] ,

[0035] ,

[0036] In the formula: symbol This represents the antisymmetric matrix used to perform the vector cross product operation; Indicates by The resulting angular momentum perturbation For the first The position vector of each control node relative to the origin; Indicates the magnitude of angular momentum; Let represent the angular momentum at node i. Therefore, the dynamic equations of the spacecraft system are derived using the Lagrange method as follows:

[0037] ,

[0038] ,

[0039] In the formula: Indicates the satellite's moment of inertia; and The first The modal damping ratio matrix and modal circular frequency matrix of a flexible structure; , ; is the modal generalized force acting on the first flexible structure; let be the attitude control moment of the central rigid body, then and and are respectively expressed as:

[0040] ,

[0041] .

[0042] Further, the method of the game comprises the following steps:

[0043] S31, game initialization:

[0044] The game is initialized by setting the parameters of the three players of the attitude control subsystem, the vibration suppression subsystem and the fault-tolerant control subsystem, including the dynamic model, the weight factor and the control strategy space; the control law of the Leader is defined as , the vibration controller Follower1 has a control law of , the fault-tolerant controller Follower2 is composed of and ; the system state is expressed as:

[0045] ,

[0046] In the formula: , , respectively correspond to the attitude, the mode and the actuator state.

[0047] The overall strategy variable of the three players is set as follows:

[0048] 1) Strategy space: (Cartesian product),

[0049] 2) Strategy of the Leader: ,

[0050] 3) Response of the Followers: , ,

[0051] 4) Stackelberg equilibrium: ,

[0052] 5) Internal and global weight factors: , ;

[0053] Definition of Stackelberg equilibrium: Leader optimizes its strategy, while Followers 1, 2 choose their optimal responses according to the strategy of Leader;

[0054] S32, Performance function design for three players:

[0055] Performance function includes for attitude stabilization, for vibration suppression, and for fault-tolerant control; these functions represent the weighted objectives assigned to each player, as follows:

[0056] ,

[0057] ,

[0058] ,

[0059] where: denotes the control time; is the node form of ; is the internal weight factor between the controlled variable and the energy consumption related to ; is the minimum value for singularity avoidance; the attitude angle error is denoted as ;

[0060] S33, Distributed control implementation:

[0061] The distributed control process includes the following steps:

[0062] S331, Initialize system state, including attitude, vibration mode, and actuator state;

[0063] S332, Implement stable control for attitude and vibration respectively using attitude and vibration control laws, and generate expected torque;

[0064] S333, In the case of actuator failure, distribute redundant torque among available nodes in a distributed manner;

[0065] S334, Update system state and performance function after fault tolerance evaluation;

[0066] S34, Results are fed back to Leader:

[0067] The results of control actions, as well as the corresponding values, are fed back to the leader for evaluation;

[0068] S35, Leader updates:

[0069] Based on the feedback information, the leader updates the weights of the targets, including control parameters, performance indicators; evaluates the quality of the current strategy, and decides whether to enter a new round of iteration;

[0070] S36, optimal solution output:

[0071] The process is iterated until the convergence condition is met: all control targets reach the specified threshold or reach the maximum number of iterations; once the condition is met, the optimal control parameters are determined and output.

[0072] Further, the specific design process of the attitude and vibration control law includes the following steps:

[0073] S41, give the spacecraft attitude control law based on optimal feedback:

[0074] Generate a set of Leader's strategy , and determine the next optimal strategy through particle swarm optimization algorithm PSO ; optimization target and iterative strategy are as follows:

[0075] ,

[0076] ,

[0077] In the formula: represents the iteration index, is the acceleration coefficient for controlling the moving speed of particles to their individual optimal and global optimal positions, represents a random number distributed in the interval [0, 1], which is used to enhance search diversity, is the individual optimal solution found by the current particle, is the global optimal solution;

[0078] Subsequently, the optimized Leader strategy is passed to the two Followers, which will obtain and in the control process;

[0079] The state of the compound system, including three-axis attitude, vibration mode and actuator state, is initialized as ; the attitude angle error is represented as ; in order to ensure the stability of attitude control, the classical control law is adopted:

[0080] ,

[0081] In the formula: correspond to the diagonal matrix of proportional, integral and derivative gain, is the integral variable;

[0082] The integral term is defined as In view of the over-accumulation of integral action in the presence of actuator saturation, an anti-windup mechanism is introduced to alleviate the integral windup effect as follows:

[0083] ,

[0084] where is the anti-windup feedback gain matrix; thus the actual control law is obtained; according to Lyapunov theory, it is proved that the system under the action of the classical control law is asymptotically stable;

[0085] S42, a saturated sliding mode control law is designed for the Follower1 to quickly attenuate the modal response; the saturated sliding mode control law not only ensures the asymptotic convergence of the modal state, but also alleviates the chattering effect caused by input saturation;

[0086] The sliding surface is defined as:

[0087] ,

[0088] where is the diagonal matrix of the sliding surface coefficient; by replacing the traditional sign function with the saturation function sat, the saturated sliding mode control law is designed as:

[0089] , ,

[0090] where , is the boundary layer thickness matrix; represents the element corresponding to the th vibration modal ; is the diagonal matrix of the switching gain; represents the sign function; represents the compensation term:

[0091] ,

[0092] where and are the modal damping ratio matrix and the modal circular frequency matrix of the th flexible structure, respectively; is the modal generalized force acting on the th flexible structure; is the modal generalized coordinate vector of the th flexible structure; represents the rigid-flexible coupling matrix of the th flexible structure; represents the angular momentum amplitude; denotes the modal rotation matrix; denotes the angular momentum at node i; denotes the angular deformation at node i; denotes the actuator failure term of the vibration control channel.

[0093] To prove the reachability of the sliding surface under the saturated sliding mode control law, the Lyapunov candidate function and its derivative are given as follows:

[0094] ,

[0095] ,

[0096] Obviously, and are always true, and the equality holds only when ; this guarantees that the system trajectory can reach the sliding surface, and then ; along the sliding surface , , thus proving the asymptotic stability of the control system.

[0097] Further, the design process of the fault-tolerant control strategy includes the following steps:

[0098] S51, if the complete failure of the actuator occurs in a certain control node , distributed fault-tolerant control is implemented in Follower 2, and the implementation steps are as follows:

[0099] S511, set the control torque to zero;

[0100] S512, adaptively redistribute the total torque of the original control system to the remaining nodes to meet the global consistency;

[0101] S513, ensure that the overall control goal of the system is achieved by each player;

[0102] Since ADMM is used for distributed control allocation at the Follower level, and the particle swarm optimization algorithm PSO is used to optimize the parameters of the Leader; although ADMM considers the contributions of attitude and vibration control inputs, it strictly operates as a Follower-level optimal response to the current Leader strategy; therefore, the two optimizations work in coordination under the Stackelberg game framework;

[0103] S52, in order to implement distributed optimization based on ADMM in Follower 2, the goal is to minimize the global quadratic cost function of the entire system while satisfying the global consistency constraint:

[0104] ,

[0105] wherein: denotes the positive definite weight matrix in the quadratic cost of the th control node; is the linear cost weight vector of the th control node; denotes the control allocation matrix mapping the global objective to the th node; may be specifically expressed as and , corresponding to the attitude control and vibration control respectively; these inputs aggregate all players' contributions for sequential optimization;

[0106] To enforce consistency among the control nodes, introduce global variable to represent the common control objective enforced among all nodes:

[0107] , ,

[0108] wherein: essentially represents the desired total control moment;

[0109] S53, introduce Lagrange multiplier to construct the augmented Lagrangian function as follows:

[0110] ,

[0111] wherein: is the penalty parameter in ADMM; to realize distributed optimization, the local variable and the consistency variable are updated respectively, and each control node performs computation in parallel, as follows:

[0112] ,

[0113] It can be found that this formula is solved by local quadratic programming; in addition, the strategy update of the Leader belongs to global iteration, while the distributed ADMM of the Followers performs internal iteration; at the th iteration, the global variable and the Lagrange multiplier are updated as follows:

[0114] ,

[0115] ,

[0116] When the consistency constraint and the absolute tolerance are satisfied, the iteration will stop, i.e., the maximum allowed deviation between each local solution and the global consistent solution; at this time, Followers1 and Followers2 reach the optimal for the given Leader strategy respectively:

[0117] ,

[0118] .

[0119] Compared with the prior art, the present application has the beneficial effects:

[0120] The present application realizes the three-axis attitude stabilization and low-frequency vibration suppression of the target spacecraft, and shows stronger fault tolerance capability in the case of node failure, and realizes high-precision attitude pointing and high-stable operation of the large flexible spacecraft system. BRIEF DESCRIPTION OF DRAWINGS

[0121] Figure 1 is a flow chart of the method of the present application;

[0122] Figure 2 is a system model and a space coordinate system of the method of the present application;

[0123] Figure 3 is a spacecraft three-axis attitude angle control effect curve diagram of the method of the present application; wherein: (a) the graph is an attitude angle control effect curve diagram using the traditional method, and (b) the graph is an attitude angle control effect curve diagram using the game control method;

[0124] Figure 4 is a spacecraft three-axis angular velocity control effect curve diagram of the method of the present application; wherein: (a) the graph is an angular velocity control effect curve diagram using the traditional method, and (b) the graph is an angular velocity control effect curve diagram using the game control method;

[0125] Figure 5 is a spacecraft front three-order modal control effect curve diagram of the method of the present application; wherein: (a) the graph is a modal displacement control effect curve diagram using the traditional method, and (b) the graph is a modal displacement control effect curve diagram using the game control method;

[0126] Figure 6 is a spacecraft three-axis control torque curve diagram of the method of the present application; wherein: (a) the graph is a control torque curve diagram using the traditional method, and (b) the graph is a control torque curve diagram using the game control method. DETAILED DESCRIPTION

[0127] The present application will be further described below in conjunction with the drawings. The following examples are only used to more clearly illustrate the technical solutions of the present application, and cannot be used to limit the protection scope of the present application.

[0128] As an embodiment of the present invention, such as Figure 1 As shown, a distributed fault-tolerant attitude and vibration control method for a large flexible spacecraft is presented. The large flexible spacecraft is a composite system consisting of a rigid body, at least two flexible structures, and actuators distributed on at least two flexible structures. The method includes the following steps:

[0129] S1. Taking into account the coupling effect between rigid body attitude motion and flexible structure deformation, the output torque of the actuator is mapped to the generalized coordinate space, and a Lagrangian dynamic model including the actuator node layout and output characteristics is constructed using the Lagrangian method.

[0130] S2. Based on the characteristics of a composite system having at least two flexible structures and the distributed characteristics of actuators, a hierarchical Stackelberg game control framework is established with the flexible structure as the control objective. The attitude control subsystem is the leader layer, and the vibration suppression subsystem and fault-tolerant control subsystem are the followers. Performance functions for the three subsystems are defined. Through the game between the spacecraft's distributed fault-tolerant attitude and vibration control, policy coordination and dynamic equilibrium among at least two control objectives are achieved.

[0131] The attitude control subsystem has an attitude control law designed based on the particle swarm optimization method. It stabilizes the attitude of the target by globally optimizing the control gain parameters.

[0132] The vibration suppression subsystem introduces a saturated sliding mode control strategy, which suppresses low-frequency vibrations through the combined action of linear and saturated terms.

[0133] The fault-tolerant control subsystem has a fault-tolerant control strategy based on the distributed alternating direction multiplier method. When an actuator fails or its performance degrades, under the condition of satisfying the global consistency constraint, the system adaptively redistributes the output torque of the remaining actuators by solving the optimal torque allocation of the remaining actuators.

[0134] As an embodiment of the present invention, such as Figures 1-2 The construction process of the Lagrange dynamics model is shown below:

[0135] The satellite's attitude motion uses orbital motion. - - Sequential Euler angles, used to represent the rotational relationship from the orbital coordinate system to the body coordinate system, can be expressed as follows:

[0136] , ;

[0137] In the formula: The three-axis angular velocity of the satellite. Then, the bending displacement of the flexible structures is modal-discretized and combined with the boundary conditions to obtain

[0138] ,

[0139] ,

[0140] where is the order of the dominant vibration mode, and ; is the bending shape function, is the modal generalized coordinate vector; for the whole system, respectively represent the left and right flexible structures; the kinetic energy of the spacecraft itself and the distributed actuators can be expressed as

[0141] ,

[0142] ,

[0143] where is the rotational inertia matrix of the central rigid body, is the modal mass matrix of the th flexible structure, is the rigid-flexible coupling matrix calculated by the finite element method. is the control node number. The rotational inertia matrix of the actuator is denoted as . The angular momentum vector of the distributed actuator at the th control node of the th flexible structure is , and its scalar size is . The relationship between the angular momentum and the actuator dynamics is

[0144] ,

[0145] where is the actuator configuration matrix, is the angular velocity vector of the actuator flywheel, which determines the control torque generated by the actuator. The angular deformation of the flexible structure at the th control node is

[0146] , ,

[0147] where is the modal rotation matrix at the node. Define as the modal stiffness matrix of the th flexible structure, then the potential energy of the system can be written as

[0148] ,

[0149] Combining the kinetic and potential energies of the rigid body, flexible structure, and distributed actuators, we can obtain the Lagrangian function of the system, i.e., the total mechanical energy of the system is:

[0150] ,

[0151] To fully describe the system dynamics, non-conservative force terms must also be considered, including:

[0152] 1) External disturbance torque ;2) Flexible modal damping term 3) Additional items caused by distributed actuator failure:

[0153] ,

[0154] In the formula: For actuator proportional failure factor matrix; The offset fault matrix represents the additional error; both together describe the actuator fault characteristics. This fault can be represented separately in the attitude and control channels:

[0155] ,

[0156] ,

[0157] In the formula: symbol This represents the antisymmetric matrix used to perform the vector cross product operation. Indicates by The resulting angular momentum perturbation For the first The position vectors of each control node relative to the origin. Therefore, the dynamic equations of the spacecraft system can be derived using the Lagrange method as follows:

[0158] ,

[0159] ,

[0160] In the formula: Indicates the satellite's moment of inertia; and The first The modal damping ratio matrix and modal circular frequency matrix of a flexible structure; , ; To act on the first Modal generalized force of a flexible structure. Let... For the attitude control torque of the central rigid body, then and may be expressed as:

[0161] ,

[0162] .

[0163] As an embodiment of the present application, as shown in Figures 1-2 , the game method, the game is carried out in a hierarchical Stackelberg game control framework, which is used to realize fault-tolerant attitude stabilization and vibration suppression of large flexible spacecraft. In view of the strong coupling relationship between attitude dynamics, structural vibration and actuator failure, the control problem is decomposed into three interrelated subtasks and assigned to different control subjects, which are modeled as participants in Stackelberg differential game, distributed on each execution node. In this hierarchical game structure:

[0164] 1) the attitude controller as the leader (Leader);

[0165] 2) the vibration controller and the fault-tolerant controller as the follower (Followers).

[0166] The specific implementation steps of the game are:

[0167] Step 1: game initialization:

[0168] The game is initialized by setting the parameters of the three players, including the dynamic model, the weight factor and the control strategy space. The control law of the Leader is defined as , the vibration controller is Follower1, and its control law is , the fault-tolerant controller is Follower2, which is composed of and . The system state is represented as:

[0169] ,

[0170] The overall strategy variable of the three players is set as follows:

[0171] 1) strategy space: (Cartesian product),

[0172] 2) strategy of the Leader: ,

[0173] 3) response of the Followers: , ,

[0174] 4) Stackelberg equilibrium: ,

[0175] 5) Internal and global weight factors: , .

[0176] Definition of Stackelberg equilibrium: Leader optimizes its strategy, while Followers 1, 2 choose their optimal responses according to the strategy of Leader.

[0177] Step 2: Performance function design for three players:

[0178] The performance function includes for attitude stabilization, for vibration suppression, and for fault-tolerant control. These functions represent the weighted objectives assigned to each player, as follows:

[0179] ,

[0180] ,

[0181] ,

[0182] where: denotes the control time, is the nodal form of , is the internal weight factor between the controlled variable and the energy consumption associated with , is the minimum value for singularity avoidance; the attitude angle error is denoted as .

[0183] Step 3: Distributed control implementation:

[0184] The distributed control process is as follows:

[0185] 1) Initialize the system state, including attitude, vibration mode, and actuator state.

[0186] 2) Implement stabilization control for attitude and vibration respectively, and generate the desired torque.

[0187] 3) In the case of actuator failure, distribute the redundant torque among the available nodes in a distributed manner.

[0188] 4) Update the system state and performance function after fault tolerance evaluation.

[0189] The specific content of the control law design will be given in the method of the game.

[0190] Step 4: Results feedback to Leader:

[0191] The results of the control action, and the corresponding values, are fed back to the leader for evaluation.

[0192] Step 5: Leader update:

[0193] Based on the feedback information, the leader updates the weights of the objectives, including control parameters, performance indices, etc. The quality of the current strategy is evaluated, and it is decided whether to enter a new round of iteration.

[0194] Step 6: Optimal solution output:

[0195] The process iterates until the convergence criteria are met, such as all control objectives reaching the specified threshold or reaching the maximum number of iterations. Once the conditions are met, the optimal control parameters are determined and output.

[0196] As an embodiment of the present application, as shown in Figures 1-2 , a spacecraft attitude and vibration control law design:

[0197] An optimal feedback-based spacecraft attitude control law is given:

[0198] A set of Leader strategies is generated, and the next optimal strategy is determined by the particle swarm optimization algorithm (PSO). The optimization objectives and iterative strategies are as follows:

[0199] ,

[0200] ,

[0201] In the formula: represents the iteration index. is the acceleration coefficient used to control the movement rate of particles to their individual optimal and global optimal positions. represents a random number distributed in the interval [0, 1], which is used to enhance search diversity. is the individual optimal solution found by the current particle, is the global optimal solution. Subsequently, the optimized Leader strategy is passed to the two Followers, who will obtain and during the control process.

[0202] The spacecraft system state, including three-axis attitude, vibration mode, and actuator state, is initialized as . The attitude angle error is represented as . To ensure the stability of the attitude control, the classical control law is as follows:

[0203] ,

[0204] where are diagonal matrices of proportional, integral and derivative gains, respectively. is the integral variable.

[0205] Define the integral term as , in view of the possible over-accumulation of the integral action under actuator saturation, an anti-windup mechanism is introduced to alleviate the integral windup effect, as follows:

[0206] ,

[0207] where is the anti-windup feedback gain matrix. The actual control law is obtained. According to Lyapunov theory, it can be proved that the system under the above control law is asymptotically stable.

[0208] For Follower 1, a saturated sliding mode control law is designed to quickly attenuate the modal response. The control law guarantees the asymptotic convergence of the modal states while alleviating the chattering effect caused by input saturation.

[0209] The sliding surface can be defined as

[0210] ,

[0211] where is the diagonal matrix of the sliding surface coefficients. By replacing the traditional sign function with a saturation function sat, the modal control law can be designed as

[0212] , ,

[0213] where , is the boundary layer thickness matrix, denotes the element corresponding to the th vibration mode, is the diagonal matrix of switching gains, denotes the sign function, denotes the compensation term:

[0214] ,

[0215] To prove the reachability of the sliding surface under the above control law, the Lyapunov candidate function and its derivative are given as follows:

[0216] ,

[0217] ,

[0218] Obviously, and is always true, and the equality holds only when This guarantees that the system trajectory can reach the sliding surface, and then has Along the sliding surface can be obtained , thus proving the asymptotic stability of the control system.

[0219] As an embodiment of the present application, as shown in Figures 1-2 , the design process of the fault-tolerant control strategy is as follows:

[0220] If the actuator of a certain control node fails completely, the distributed fault-tolerant control in Follower2 is implemented as follows:

[0221] 1) Set the control torque to zero and remove it from the formula;

[0222] 2) Adaptively redistribute the original target torque to the remaining nodes to meet the global consistency;

[0223] 3) Ensure that the overall control target of the system is achieved by each player.

[0224] Note that ADMM is used for distributed control allocation at the Follower level, while the above-mentioned PSO is used to optimize the parameters of the Leader. Although ADMM takes into account the contributions of attitude and vibration control inputs, it strictly operates as a Follower-level optimal response to the current Leader strategy. The two optimizations work in coordination under the Stackelberg game framework.

[0225] In order to implement the ADMM-based distributed optimization in Follower2, the goal is to minimize the global quadratic cost function of the entire system while satisfying the global consistency constraint:

[0226] ,

[0227] In the formula: represents the positive definite weight matrix in the quadratic cost of the th control node; is the linear cost weight vector of the th control node; represents the control allocation matrix that maps the global target to the th node; can be specifically represented as and , corresponding to attitude control and vibration control respectively. These inputs aggregate all players' contributions for sequential optimization.

[0228] To enforce consistency among control nodes, a global variable is introduced to represent the common control objective enforced among all nodes:

[0229] , ,

[0230] where essentially represents the desired total control moment.

[0231] A Lagrange multiplier is introduced to construct the augmented Lagrangian function as follows:

[0232] ,

[0233] where is the penalty parameter in ADMM. To realize distributed optimization, local variables and consistency variables are updated respectively, and each control node performs computation in parallel, as follows:

[0234] ,

[0235] It can be found that this equation can be solved by local quadratic programming. In addition, the strategy update of Leader belongs to global iteration, while the distributed ADMM of Followers performs internal iteration. At the th iteration, the global variable and the Lagrange multiplier are updated as follows:

[0236] ,

[0237] ,

[0238] When the consistency constraint and absolute tolerance are satisfied, the iteration will stop, i.e. the maximum allowed deviation between each local solution and the global consistent solution. At this time, Followers reach optimality respectively for a given Leader strategy, as follows:

[0239] ,

[0240] .

[0241] To verify the effectiveness of the proposed robust attitude stabilization method, a numerical simulation is carried out with a super large flexible satellite as the research object, and the simulation time is set to 150 s with a sampling time of 0.1 s. The key simulation parameters of the target satellite dynamics and control are as follows:

[0242]

[0243] In the verification example of the present application, numerical simulation verification is carried out on distributed fault-tolerant attitude and vibration control of large flexible spacecraft. The simulation object is a satellite model with a central rigid body and a large flexible antenna structure. The three-axis attitude and flexible mode of the satellite are taken as the controlled quantities, and the space environmental torque applied to the satellite is taken as the disturbance input. The finite time attitude control method described in the present application is adopted, and the control parameters given in Table 1 are adopted. The simulation time is 150 s, and the time step is 0.1 s. By comparing with the traditional sliding mode control method, important indicators such as attitude angle, angular velocity, modal displacement, convergence time, fault recovery time, etc. are comprehensively evaluated. The simulation results show that the proposed control method effectively improves the three-axis attitude stabilization and vibration suppression performance of large flexible spacecraft, Figure 3 The simulation results of (a) to (b) in FIG. and Table 2 show that the present application can realize attitude accuracy of 1.93×10 -4 deg, which is improved by 97.2% compared with the traditional method; Figure 4 The simulation results of (a) to (b) in FIG. and Table 2 show that the attitude stability is 1.52×10 -4 deg / s, which is improved by 75.8% compared with the traditional method; Figure 5 The simulation results of (a) to (b) in FIG. and Table 2 show that the modal displacement accuracy is 0.0428, which is improved by 69.5%. Figure 6 The simulation results of (a) to (b) in FIG. and Table 2 show that the control torque amplitude after actuator failure is 0.204 N·m, which is improved by 49.8%. In summary, the present application can meet the main performance requirements of large flexible spacecraft in-orbit pointing tasks, and has good engineering applicability.

[0244]

[0245] The preferred embodiments of the embodiments of the present application are described above with reference to the accompanying drawings, and are not limited to the scope of the embodiments of the present application. Any modifications, equivalent replacements and improvements made by those skilled in the art without departing from the scope and essence of the embodiments of the present application shall be within the scope of the rights of the embodiments of the present application.

Claims

1. A large flexible spacecraft distributed fault-tolerant attitude and vibration control method, the large flexible spacecraft is a composite system composed of a rigid body and at least two flexible structures and actuators distributed on the at least two flexible structures, characterized in that: The method comprises the following steps: S1, mapping the output torque of the actuator to the generalized coordinate space by comprehensively considering the coupling effect between the rigid body attitude motion and the flexible structure deformation, and constructing a Lagrange dynamic model containing the actuator node layout and output characteristics by using the Lagrange method; S2, based on the characteristics of the composite system having at least two flexible structures and the distributed characteristics of the actuator, taking the flexible structure as the control target, establishing a hierarchical Stackelberg game control framework with the attitude control subsystem as the leader layer and the vibration suppression subsystem and the fault-tolerant control subsystem as the follower layer, defining the performance functions of the three subsystems, and realizing the strategy cooperation and dynamic balance between at least two control targets through the game of the spacecraft distributed fault-tolerant attitude and vibration control; wherein: The attitude control subsystem has an attitude control law designed based on the particle swarm optimization method, and the global optimization of the control gain parameter is used to stabilize the attitude of the control target; The vibration suppression subsystem introduces a saturated sliding mode control strategy, and the linear term and the saturated term are combined to suppress low-frequency vibration; The fault-tolerant control subsystem has a fault-tolerant control strategy constructed based on the distributed alternating direction multiplier method; when a fault or performance degradation occurs in an actuator, the optimal torque distribution of the remaining actuators is solved by the distributed alternating direction multiplier method under the condition of meeting the global consistency constraint, so as to adaptively redistribute the output torque of the remaining actuators.

2. The distributed fault-tolerant attitude and vibration control method for large flexible spacecraft according to claim 1, wherein: The construction process of the Lagrange dynamic model is as follows: The attitude motion of the rigid body is described by the rotation of the body-fixed frame about the body-fixed axes - - The Euler angles are described in the order to represent the rotation relationship from the orbital coordinate system to the body coordinate system, and the transformation form is represented as: , ， wherein: corresponds to the satellite , , the three angular velocities; denotes the rotation matrix; denotes the attitude angle; and denote the roll angle and the pitch angle, respectively; the upper index denotes the matrix transposition; Then, the bending displacement of the flexible structure is discretized and combined with the boundary conditions to obtain: ， ， wherein: n represents the order of the dominant vibration mode, taken as , is the bending shape function, is the modal generalized coordinate vector; represents the bending displacement of the flexible structure, and the displacement is simultaneously related to the displacement and time represents the partial derivative; represents the structure length; kinetic energy of the spacecraft itself and of the distributed actuators are respectively denoted ， ， In the formula: is the moment of inertia matrix of the central rigid body, is the modal mass matrix of the flexible structure, is the rigid-flexible coupling matrix calculated by the finite element method; represents the control node number; represents the total number of nodes; the moment of inertia matrix of the actuator is denoted as ; the angular momentum vector of the distributed actuator at the flexible structure at the control node is , and the scalar size is ; the angular momentum vector is related to the actuator dynamics: ， wherein: is the actuator configuration matrix, is the angular velocity vector of the actuator flywheel, which determines the control torque generated by the actuator; the flexible structure is in the first angular deformation at the control node is: , ， where: is the modal rotation matrix at the node; defined as is the modal stiffness matrix of the th flexible structure, then the potential energy of the system is written as: ， The kinetic energy and potential energy of the rigid body, the flexible structure and the distributed actuator are integrated to obtain the Lagrange function of the system, that is, the total mechanical energy of the system: ， In order to completely describe the system dynamics, the non-conservative force term needs to be considered, including: 1) External disturbance torques 2) Flexible mode damping terms 3) Additional terms due to distributed actuator faults: ， wherein: is the actuator proportional fault factor matrix; is the bias fault matrix, representing additional errors; Both together describe the actuator failure characteristics; The output torque of node i is denoted; the actuator failure is denoted in the attitude and control channels, respectively: ， ， In the formula: the symbol represents the skew-symmetric matrix for implementing the vector cross multiplication operation; represents the skew-symmetric matrix for implementing the vector cross multiplication operation; caused by the angular momentum disturbance, is the position vector of the first control node relative to the origin; therefore, the dynamics equation of the spacecraft system is derived by using the Lagrange method as follows: ， ， wherein: represents the moment of inertia of the satellite; and are the modal damping ratio matrix and the modal circular frequency matrix of the th flexible structure, respectively; , ; is the modal generalized force acting on the th flexible structure; let be the attitude control moment of the central rigid body, then and are represented as: ， 。 3. The distributed fault-tolerant attitude and vibration control method for large flexible spacecraft according to claim 2, wherein: The method of the game comprises the following steps: S31, game initialization: The game is initialized by setting the parameters of the three players, including the dynamic model, weight factor and control strategy space; the actual control law of Leader is defined as , the vibration controller of Follower1 is , the fault-tolerant controller of Follower2 is composed of and ; the system state is represented as ， wherein: , , correspond to the pose, the modality and the effector state, respectively; The overall strategy variable of the three players is set as follows: 1) Policy space: , 2) Leader's strategy: , 3) Response of Followers: , , 4) Stackelberg equilibrium: , 5) Internal and global weight factors: , ; Definition of Stackelberg equilibrium: the leader optimizes its own strategy, and the followers 1 and 2 select their optimal responses according to the strategy of the leader; S32, performance function design of the three players: Performance functions include those for attitude stabilization. Used for vibration suppression and for fault-tolerant control These functions represent the weighted targets assigned to each player, as shown below: ， ， ， wherein: denotes the control time; is in the form of a node; is an internal weighting factor between the controlled variable and the related energy consumption; is a minimum value for avoiding singularity; the attitude angle error is denoted as ; S33, distributed control implementation: The distributed control process comprises the following steps: S331, initializing the system state, including the attitude, vibration mode and actuator state; S332, using the attitude and vibration control law to stabilize the attitude and vibration respectively, and generating the expected torque; S333, in the case of actuator failure, distributing the redundant torque among the available nodes in a distributed manner; S334, updating the system state and performance function after fault tolerance evaluation; S34, the result is fed back to the leader: The results of the control actions, and corresponding values, are fed back to the leader for evaluation; S35, leader update: Based on the feedback information, the leader updates the weight of the target, including the control parameter and the performance index; evaluates the quality of the current strategy, and decides whether to enter a new round of iteration; S36, optimal solution output: The process is iterated until the convergence condition is met: all control objectives reach the specified threshold or reach the maximum number of iterations; once the condition is met, the optimal control parameters are determined and output.

4. The distributed fault-tolerant attitude and vibration control method for large flexible spacecraft according to claim 3, wherein: The specific design process of the attitude and vibration control law includes the following steps: S41, give the spacecraft attitude control law based on optimal feedback: Strategy for generating a set of leaders and the next optimal strategy is determined by a particle swarm optimization algorithm PSO ; the optimization objective and the iterative strategy are as follows: ， ， wherein: denotes the iteration index, is an acceleration coefficient used to control the rate of movement of the particles towards their individual and global optimal positions, denotes a random number distributed in the interval [0, 1] used to enhance search diversity, is the individual optimal solution found by the current particle, is the global optimal solution; Subsequently, the optimized Leader strategy is passed to the two Followers, who will obtain and ; The composite system states, including three-axis attitude, vibration modes and actuator states, are initialized as ; the attitude angle error is expressed as ; to ensure the stability of the attitude control, the classical control law is adopted: ， In the formula: corresponding to the proportional, integral, and derivative gains, respectively, is the integral variable; The integral term is defined as In view of the possible over-accumulation of the integral action in the presence of actuator saturation, an anti-windup mechanism is introduced to mitigate the integral windup effect as follows: ， where: is the anti-windup feedback gain matrix; from which the actual control law is derived; according to Lyapunov theory, it is proved that the system under the action of the classical control law is asymptotically stable; S42, design a saturated sliding mode control law for Follower1 to quickly attenuate modal response; the saturated sliding mode control law ensures the asymptotic convergence of modal state while reducing the chattering effect caused by input saturation; The sliding surface is defined as: ， where: is a diagonal matrix of the sliding surface coefficients; by replacing the traditional sign function with a saturation function sat, the saturated sliding mode control law is designed as: , ， wherein: , is a matrix of boundary layer thicknesses; denotes an element corresponding to the vibration mode of order ; is a diagonal matrix of switching gains; denotes a sign function; denotes a compensation term: ， In the formula: and are the modal damping ratio matrix and the modal circular frequency matrix of the first flexible structure, respectively; is the modal generalized force acting on the first flexible structure; is the modal generalized coordinate vector of the first flexible structure; represents the rigid-flexible coupling matrix of the first flexible structure; represents the angular momentum amplitude; represents the modal rotation matrix; represents the angular momentum at node i; represents the angular deformation at node i; represents the actuator fault term of the vibration control channel; In order to prove the reachability of the sliding surface under the action of the saturated sliding mode control law, the Lyapunov candidate function and its derivative are given as follows: ， ， Obviously, and is constant and the equality holds only when ; this guarantees that the system trajectory can reach the sliding surface and then has ; along the sliding surface is derived , thus proving the asymptotic stability of the control system.

5. The distributed fault-tolerant attitude and vibration control method for large flexible spacecraft according to claim 4, wherein: The design process of the fault-tolerant control strategy, when the actuator fails completely, the system needs to start distributed fault-tolerant control, so the control resources of the remaining nodes must be redistributed through distributed optimization; Follower2 uses an ADMM-based optimization framework to minimize the global quadratic cost function and maintain consistency constraints; In order to convert this distributed optimization problem into an iterative form that can be solved, it is necessary to further introduce Lagrange multipliers and construct an augmented Lagrangian function as the basis for ADMM solution; including the following specific steps: S51, if a certain control node If the actuator fails completely, distributed fault-tolerant control is implemented in Follower2, and the implementation steps are as follows: S511, set the control torque to zero; S512, adaptively redistribute the total torque of the original control system to the remaining nodes to meet global consistency; S513, ensure that the overall control objectives of the system are achieved by each player; Since ADMM is used for distributed control allocation at the Follower level, and the particle swarm optimization algorithm PSO is used to optimize the parameters of the Leader; Although ADMM takes into account the contribution of attitude and vibration control inputs, it strictly operates as a Follower-level optimal response to the current Leader strategy; Therefore, the two optimizations work in coordination under the Stackelberg game framework; S52, in order to implement ADMM-based distributed optimization in Follower2, the goal is to minimize the global quadratic cost function of the entire system while satisfying global consistency constraints: ， wherein: represents a positive definite weight matrix in the quadratic cost of the th control node; is a linear cost weight vector of the th control node; represents a control allocation matrix mapping the global objective to the th node; may be specifically represented as and corresponding to the pose control and the vibration control, respectively; these inputs aggregate all players' contributions for sequential optimization; To enforce consistency between control nodes, a global variable is introduced to represent the common control objective that is enforced across all nodes: , ， In the formula: Essentially represents the desired total control moment; S53, introduce Lagrange multiplier , construct the augmented Lagrangian function as follows: ， where: is a penalty parameter in ADMM; to realize distributed optimization, local variables and consensus variables are updated respectively, and each control node performs the calculation in parallel, as follows: ， It can be found that this formulation can be solved by local quadratic programming; in addition, Leader's strategy update is global iteration, while Followers' distributed ADMM performs inner iteration; at the first iteration, global variables and Lagrange multipliers are updated as follows, respectively: ， ， When the consistency constraints and absolute tolerance are met, the iteration will stop, that is, the maximum allowed deviation between each local solution and the global consistent solution; At this time, Follower1 and Follower2 reach optimality for the given Leader strategy: ， 。

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