A distributed model predictive control method and system applied to large flexible spacecraft attachment vibration suppression

Abstract

The application provides a distributed model predictive control method and system applied to large flexible spacecraft accessory vibration suppression, and belongs to the automatic control field. In order to solve the problems that the existing flexible accessory is sensitive to noise and the online solution calculation is complex in the vibration suppression process due to the high conservativeness of the controller and the low robustness caused by the strong dynamic coupling between the subsystems. The application is based on the space interconnection Markov jump system theory to model and analyze the large flexible spacecraft accessory; the dissipation index is introduced into the cost function, and the subsystem state is substituted into the quadratic Lyapunov function to determine the analytical expression form of the random performance upper bound; the concept of robust positive invariance set is extended to the space interconnection subsystem state space to solve the constraint satisfaction problem in the infinite time domain; and the dissipation type performance upper bound and the robust positive invariance set are integrated. The application provides a more relaxed robustness condition and reduces the conservativeness, and reduces the online solution calculation burden of the controller.

Description

Technical Field

[0001] This invention relates to the field of automatic control technology, and more specifically, to a distributed model predictive control method and system for vibration suppression of large flexible spacecraft attachments. Background Technology

[0002] To address the high energy consumption and complex mission requirements of spacecraft, large flexible spacecraft attachments (LSFAS), such as large solar panels, ultra-large aperture telescopes, and long flexible robotic arms, are increasingly used in modern spacecraft. However, due to their large size and low damping characteristics, these attachments are prone to vibration, which significantly impacts the spacecraft's dynamic characteristics and control systems. To address this issue, large flexible spacecraft typically employ vibration control units, such as momentum wheels and piezoelectric ceramics, for active suppression. However, the large number of vibration control units required for large flexible attachments results in a massive system size, leading to excessively high computational complexity for traditional vibration suppression algorithms. Furthermore, these algorithms suffer from high communication costs and susceptibility to packet loss and latency, limiting their application in large flexible attachments. Space interconnection system models, by discretizing a continuous spatial model into a series of interacting subsystems and assigning each subsystem a communication network containing only local neighboring nodes, can effectively reduce communication costs and improve system resilience while considering the coupling effects between subsystems.

[0003] Furthermore, extreme temperature variations during on-orbit operation can significantly alter the mechanical properties of flexible attachments, manifesting as mode transitions between multiple systems. In practical applications, thermally induced mode transitions are not instantaneous but are related to the thermal inertia and thermal conductivity of the material, thus allowing analysis using Markov jump theory with a lower bound residence time. External disturbances in the space environment, such as solar radiation pressure and gravity gradients, can also induce vibrations in flexible attachments and reduce vibration suppression performance. Existing methods generally treat interactions between subsystems as equal disturbances to ensure system robustness, but this approach is overly conservative in cases of strong dynamic coupling between subsystems. In recent years, dissipative system theory has been proposed as an effective solution to ensure the stability and robustness of large-scale interconnected systems. Dissipative conditions assess the stability of the global system by analyzing the relationship between input and stored energy, thus providing more relaxed robustness conditions and reducing conservatism.

[0004] To overcome the aforementioned technical bottlenecks, this invention proposes a modeling and analysis method for highly flexible spacecraft attachments based on the theory of space-interconnected Markov jump systems, and develops a switching distributed model predictive control (MPC) algorithm to ensure that subsystem physical constraints are met and to perform online rolling optimization of system performance. The key technical problems addressed by this invention include:

[0005] i) The space interconnected Markov jump modeling method can accurately describe the vibration dynamics of highly flexible spacecraft attachments and provides a minimally sized communication network to reduce the computational burden of online controller solving;

[0006] ii) The distributed mode switching model prediction algorithm fully considers the dynamic coupling and random mode switching between subsystems, ensures that physical constraints such as system state / control input are met, and strictly guarantees consistent convergence and continuous solvability of online optimization;

[0007] iii) A dissipative cost function is introduced to ensure robustness, fully explore the model's input-output structure, and reduce conservatism. Summary of the Invention

[0008] The technical problem to be solved by this invention is:

[0009] To address the problems of excessively conservative and low robust controllers, resulting in noise sensitivity and complex online solution calculations caused by strong dynamic coupling between factor systems in existing flexible attachments during vibration suppression.

[0010] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0011] This invention provides a distributed model predictive control method for vibration suppression of large flexible spacecraft attachments, comprising the following steps:

[0012] S100. Establish a space interconnected Markov jump system model for the flexible spacecraft attachments, including modeling the vibration dynamics of the large flexible spacecraft attachments based on the Euler-Bernoulli beam model, approximating the position derivative using the central difference method, and approximating the time derivative using the Euler discretization method; perform dynamic analysis on the system modes at different temperatures, thereby transforming the multimodal dynamic equations of the large flexible spacecraft attachments into a discrete-time space interconnected Markov jump system model.

[0013] S200, design the performance indicators of the space interconnection system, introduce a dissipative cost function, including the instantaneous performance of the control system in each step of operation, i.e., the stage cost, which includes control accuracy and control cost; and introduce a dissipative cost term in the stage cost to ensure that the increase in energy inside the system does not exceed the energy of the disturbance input.

[0014] S300. Calculate the upper bound of the system performance index and design a robust positive invariant set to obtain the linear matrix inequality conditions. This includes substituting the subsystem state into the Lyapunov function to determine the analytical expression of the upper bound of the stochastic performance, and transforming the optimization of the cost function into the optimization of the upper bound of the Lyapunov function by establishing the relationship between the Lyapunov function and the cost function, thus deriving the linear matrix inequality conditions that guarantee the stability of the closed-loop system. Introduce the concept of a robust positive invariant set, consider the influence of bounded disturbance inputs, substitute them into the disturbance closed-loop subsystem model, and establish the linear matrix inequality conditions that guarantee recursive feasibility and system constraints.

[0015] S400. Establish a rolling time-domain optimization problem, including establishing a rolling time-domain optimization problem based on the above cost function and linear matrix inequality conditions, selecting a solver to solve the problem, obtaining the subsystem state feedback control gain and applying it to the vibration suppression unit.

[0016] Further, in step S100, the following are included:

[0017] Let the total length of the flexible attachment be The vibration suppression system includes multiple vibration control units to... The vibration control units (VCUs) are evenly spaced on the flexible attachments. Each VCU contains a displacement sensor, actuator, and microcontroller, and establishes a communication network with adjacent VCUs. Each VCU is considered a subsystem. The dynamics of the flexible attachments are expressed by the following partial differential equations:

[0018]

[0019] in, and These represent the external force applied perpendicularly to the flexible attachment and the flexible attachment's position on the surface. time Vibration displacement at position and These represent bending stiffness and mass per unit length, respectively. For Young's modulus, For rotational inertia, For the density of flexible attachments, Let be the cross-sectional area, and define . , where the width is The height is ;

[0020] The following simplified representation is used. ,

[0021] Since the Coriolis force does not induce vertical vibration, the vibration only involves centrifugal force. and tangential force :

[0022]

[0023] in, Indicates the rotation angle of a rigid body; This represents the distance from the center of the rigid body to the point of connection with the flexible attachment;

[0024] The load applied to the flexible attachment is:

[0025] in, Indicates the flexible attachment and the spacecraft coordinate system chamfer, Indicates flexible attachments and The included angle between them;

[0026] Considering the inertial load applied to the flexible attachment during vibration, and neglecting the second-order terms of the trigonometric functions, the distributed load along the flexible attachment is expressed as:

[0027]

[0028] in, ;

[0029] The distributed forces of the subsystem are modeled as ,in Represents the Dirichlet function, Represents the position The The control force output by each subsystem, where N represents the total number of subsystems. Representing the The positions of the subsystems; assuming the influence of the mass and volume of the vibration control unit on the dynamics of the flexible attachment is negligible, the dynamics of the flexible attachment considering the influence of the central rigid body and the subsystems are as follows:

[0030]

[0031] The corresponding boundary conditions are:

[0032]

[0033] in, , For the length of the flexible attachment, time , The damping coefficient of the flexible attachment. This includes external disturbances and unmodeled higher-order dynamics; This represents the vibrational displacement at the connection point between the flexible attachment and the rigid body. Indicates the vibration displacement at the end of the flexible attachment;

[0034] The changes in the properties of flexible attachment materials caused by temperature variations are reflected in Young's modulus and damping ratio, i.e. ,in It is a temperature-dependent modal sequence; considering the coupling effect between the flexible attachments and the central rigid body, the attitude dynamics of the central rigid body are:

[0035]

[0036] in, , and Representing the central rigid body, flexible appendages, and the spacecraft as a whole, respectively, along the inertial coordinate system. Moment of inertia of the shaft; It is the sum of the torques acting on the spacecraft; The angular velocity of the satellite;

[0037] By performing intermediate difference approximation on the positional partial derivatives in equation (4) and Euler discretization on the time partial derivatives, the discrete-time spatial interconnected Markov jump system is obtained as shown below:

[0038]

[0039]

[0040] in, Indicates vibration displacement Vectorization, Indicates vibration velocity. This indicates the control force output by the subsystem. Represents spatial coordinates, Indicates the discretization time; Describing the quality of the subsystem, the state-space equation of the discrete-time spatially interconnected Markov jump system is expressed as:

[0041]

[0042] in, and These represent time variables and space variables, respectively. Represents the state of a subsystem, including spatiotemporal variables and time operators. Indicates positive time shift Spatial operators represent bidirectional displacements.

[0043] Combinatorial operators For the subsystem located at s, the states of adjacent subsystems; the physical constraints on the system are:

[0044]

[0045] in, Indicates the system status and the maximum limit value of control inputs; This represents the system state at time k; Represents the control input at time k; to simplify the notation, make All represent system modes;

[0046] System Mode The random variation pattern is as follows:

[0047]

[0048] in, Indicates the first The moment of the next switch. This represents the dwell time in a particular system mode. Indicates from Modal transition to The transition probability.

[0049] Further, in step S200, the following are included:

[0050] The robust optimization control problem can be described as minimizing the worst-case performance upper bound, i.e., the following optimization problem:

[0051]

[0052] To fully account for mode transitions caused by temperature changes, the infinite time-domain cost function is expected to be expressed as... , The instantaneous performance of the control system at each step of its operation is described, and this is called the stage cost. To combat external disturbances, a dissipation index is introduced to limit the energy introduced by noise from exceeding the energy stored in the system. The stage cost is expressed as follows:

[0053]

[0054]

[0055] in, The weight matrix is ​​positive definite. It is a negative definite weight matrix. It is a symmetric matrix. A positive scalar describes the degree of emphasis on robustness.

[0056] Further, in step S300, the following are included:

[0057] We introduce the Lyapunov function to derive its analytical expression for its upper bound, and use this expression to establish a solvable optimization problem. Consider the following form of modally dependent Lyapunov function. :

[0058]

[0059] If the quadratic function satisfies the following condition:

[0060]

[0061]

[0062] in, Represents all modes A set;

[0063] Then the following equation holds true:

[0064]

[0065] Transforming the above equation from time... Accumulate to ,get:

[0066]

[0067] We obtain the upper bound of the infinite time-domain expected cost function. ;because Corresponding state To determine the value, then for The optimization is to optimize the matrix. The existence condition for the performance upper bound is transformed into the following linear inequality form for optimization:

[0068]

[0069] in, , ; A matrix represents the effect of control inputs on the system state; A matrix represents the effect of a disturbance on the system state;

[0070] By minimizing The upper bound of the cost is used to achieve performance optimization;

[0071] Introducing the concept of robust positive invariant sets, constructing forms The modality-dependent ellipsoid set, where Y represents the Lyapunov parameter matrix, and the elements of the set satisfy the following conditions: Then, under the action of the controller, the state at the next moment will still satisfy the condition. If we can establish a modal dependency robust positive invariant set for a generalized system with variable state dimension... Constraint determination for infinite time-domain state / control inputs can be transformed into constraint determination for ellipsoidal boundaries, thereby enabling the solution of optimization problems that satisfy the constraints.

[0072] Construction form The modal-dependent quadratic function, Let be the corresponding modal dependency level set, and its positive invariance condition be directly expressed as: Considering the influence of bounded disturbance input, substituting it into the disturbance closed-loop subsystem model, the condition is expressed as:

[0073] in, , Let represent the maximum value of the external disturbance. To facilitate numerical solution, the S-process is applied to the above equation, yielding the following linear inequality:

[0074]

[0075] in, Indicates the degree of robustness;

[0076] This condition ensures the set The convergence of the system during operation; in addition, the invariant set must also satisfy the initial bounded condition, i.e. This is equivalent to the following linear inequality:

[0077]

[0078] To ensure that both the state and control input within the invariant set satisfy the constraints, the following boundary constraint criteria for the ellipsoid set need to be introduced:

[0079]

[0080] By simultaneously solving the above inequalities, we obtain the robust positive invariant set that satisfies the constraints in the infinite time domain. .

[0081] Further, in step S400, the following is included:

[0082] Based on the results of steps S200 and S300, the following convex optimization problem for solving the control law of the distributed subsystem is established:

[0083] in, These are the matrix variables to be optimized;

[0084] By the Schur complement transformation, the matrix inequality condition in the above equation is equivalent to: , This is the upper bound of the Lyapunov function, i.e., the upper bound of the cost function; therefore, minimizing... This is equivalent to minimizing the upper bound of the cost function of the original optimization problem in order to achieve system performance optimization.

[0085] A distributed model predictive control system for vibration suppression of large flexible spacecraft attachments is provided. The system has program modules corresponding to the steps described above, and executes the steps in the distributed model predictive control method for vibration suppression of large flexible spacecraft attachments during operation.

[0086] A computer-readable storage medium storing a computer program configured to, when invoked by a processor, implement steps of a distributed model predictive control method for vibration suppression of large flexible spacecraft attachments.

[0087] Compared with the prior art, the beneficial effects of the present invention are:

[0088] This invention addresses the problems of vibration suppression and thermally induced modal changes in the attachments of highly flexible spacecraft. It applies a space-interconnected Markov jump system for modeling and analysis, and designs a switching distributed model predictive control framework, effectively reducing control errors in vibration suppression and improving convergence speed. The beneficial effects of this invention are:

[0089] ① This invention achieves accurate modeling of the vibration dynamics of highly flexible spacecraft attachments through a space interconnection system, effectively solving the problems of high dimensionality, high computational complexity, and high communication pressure in traditional control systems. Furthermore, by integrating Markov jump processes, it achieves unified modeling of the mechanical properties of highly flexible spacecraft attachments in different temperature ranges, effectively supporting the design and performance analysis of vibration suppression controllers.

[0090] ②This invention develops a distributed switching model predictive control algorithm, in which the robust positive invariant set is extended to include random mode switching and dynamic coupling between subsystems, while strictly guaranteeing the uniform convergence and recursive feasibility of the subsystems.

[0091] ③ To overcome the influence of external disturbances on vibration suppression, a dissipation term is introduced into the cost function of model predictive control. By making full use of the mode switching attribute and the input-stored energy relationship, the conservatism is effectively reduced and the solvability of online optimization is improved.

[0092] In summary, this invention addresses the practical engineering problem of vibration suppression for highly flexible spacecraft accessories by proposing a distributed model predictive controller design method based on a space interconnected Markov jump system. This method can effectively improve global vibration suppression performance, reduce online computation, and enhance system toughness and robustness, thus possessing high engineering application value. Attached Figure Description

[0093] Figure 1 This is a flowchart of a distributed model predictive control method for vibration suppression of large flexible spacecraft attachments, as described in an embodiment of the present invention.

[0094] Figure 2 This is a schematic diagram of the deployment of the highly flexible spacecraft accessories and active vibration suppression unit in an embodiment of the present invention;

[0095] Figure 3 This is a schematic diagram illustrating the deformation of the flexible attachment during the on-orbit attitude maneuver of a spacecraft, as described in an embodiment of the present invention.

[0096] Figure 4 This is a schematic diagram of the lightweight local communication network corresponding to each subsystem in this embodiment of the invention;

[0097] Figure 5 This is a vibration offset curve of a highly flexible spacecraft accessory in scenario one of the embodiments of the present invention;

[0098] Figure 6 This is a comparison curve of vibration offset of the deformation center subsystem in scenario one of the embodiments of the present invention;

[0099] Figure 7 This is a vibration offset curve of a highly flexible spacecraft accessory in scenario two of this embodiment of the invention;

[0100] Figure 8 This is a comparison curve of vibration offset of the deformation center subsystem in scenario two of the present invention.

[0101] Figure 9 The above are satellite attitude control comparison curves for scenario two in this embodiment of the invention, where (a) is a satellite attitude angle comparison curve for scenario two, and (b) is a satellite attitude angle control error comparison curve for scenario two. Detailed Implementation

[0102] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0103] Specific Implementation Plan 1: Combining Figures 1 to 4 As shown, this invention provides a distributed model predictive control method for vibration suppression of large flexible spacecraft attachments, comprising the following steps:

[0104] S100. Establish a space interconnected Markov jump system model for the flexible spacecraft attachments, including modeling the vibration dynamics of the large flexible spacecraft attachments based on the Euler-Bernoulli beam model, approximating the position derivative using the central difference method, and approximating the time derivative using the Euler discretization method; perform dynamic analysis on the system modes at different temperatures, thereby transforming the multimodal dynamic equations of the large flexible spacecraft attachments into a discrete-time space interconnected Markov jump system model.

[0105] Specifically, including,

[0106] like Figure 2 As shown, the total length of the flexible attachment is The vibration damping system includes a series of vibration control units (VCUs) to The VCUs are evenly spaced on the flexible attachments. Each VCU contains a displacement sensor, actuator, and microcontroller, and establishes a communication network with adjacent VCUs. Each VCU is considered a subsystem. The dynamics of the flexible attachments are expressed by the following partial differential equations:

[0107] in, and These represent the external force applied perpendicularly to the flexible attachment and the flexible attachment's position on the surface. time Vibration displacement at position and These represent bending stiffness and mass per unit length, respectively. For Young's modulus, For rotational inertia, For the density of flexible attachments, Let be the cross-sectional area, and define . , where the width is The height is ;

[0108] Note that in this invention, for ease of representation, the following simplified representation is used. , , , ; ;

[0109] The motion of flexible attachments during spacecraft attitude control is as follows: Figure 3 As shown in the figure, according to theoretical calculations, the Coriolis force along the t-t direction will not cause vertical vibration; therefore, only centrifugal force is involved. and tangential force The main cause of vibration:

[0110]

[0111] in, This represents the rotation angle of a rigid body. This represents the distance from the center of the rigid body to the point of connection with the flexible attachment;

[0112] Through the Figure 3 Force analysis was performed, and the load applied to the flexible attachment was obtained as follows:

[0113]

[0114] in, Indicates the flexible attachment and the spacecraft coordinate system chamfer, Indicates flexible attachments and The included angle between them;

[0115] Considering the inertial load applied to the flexible attachment during vibration, and neglecting the second-order terms of the trigonometric functions, the distributed load along the flexible attachment can be expressed as:

[0116]

[0117] in, The distributed forces of the subsystem are modeled as ,in Represents the Dirichlet function, Represents the position The The control force output by each subsystem, where N represents the total number of subsystems. Representing the The positions of the subsystems; here it is assumed that the mass and volume of the VCU are extremely small relative to the flexible attachment, so their influence on the dynamics of the flexible attachment can be ignored. The dynamics of the flexible attachment considering the influence of the central rigid body and the subsystems are as follows:

[0118]

[0119]

[0120] The corresponding boundary conditions are:

[0121]

[0122] in, , For the length of the flexible attachment, time , The damping coefficient of the flexible attachment. It includes external disturbances and unmodeled higher-order dynamics. This represents the vibrational displacement at the connection point between the flexible attachment and the rigid body. Indicates the vibration displacement at the end of the flexible attachment;

[0123] In addition, the changes in the properties of flexible attachment materials caused by temperature variations are mainly reflected in Young's modulus and damping ratio, i.e. , ,in It is a temperature-dependent modal sequence; since the vibration of the flexible attachment in this invention is mainly along the spacecraft coordinate system. Since the axis occurs, the analysis in this step focuses on the inertial coordinate system. Axis; Considering the coupling effect between the flexible attachment and the central rigid body, the attitude dynamics of the central rigid body are as follows:

[0124] in, , and Representing the central rigid body, flexible appendages, and the spacecraft as a whole, respectively, along the inertial coordinate system. Moment of inertia of the shaft; The angular velocity of the satellite; It is the sum of the torques acting on the spacecraft;

[0125] By performing intermediate difference approximation on the positional partial derivatives in equation (4) and Euler discretization on the time partial derivatives, the discrete-time spatial interconnected Markov jump system (SIMJS) is obtained as follows:

[0126]

[0127]

[0128] in, Indicates vibration displacement Vectorization, Indicates vibration velocity. This indicates the control force output by the subsystem. Represents spatial coordinates, Indicates the discretization time; Describing the quality of the subsystem, the state-space equation of SIMJS is expressed as:

[0129]

[0130] in, and These represent time variables and space variables, respectively. Represents the state of a subsystem, including spatiotemporal variables, and is determined by the central subsystem located at point s. Figure 4 The time operator is obtained from the communication network shown. Indicates positive time shift Spatial operators represent bidirectional displacements. Combinatorial operators For the subsystem located at s, the states of adjacent subsystems; the physical constraints on the system are:

[0131]

[0132] in, This indicates the system state and the maximum achievable value of the control input; This represents the system state at time k; This represents the control input at time k; for simplicity, it will be referred to below. All represent system modes;

[0133] System modes considered in this invention The random variation pattern is as follows:

[0134]

[0135] in, Indicates the first The moment of the next switch. This represents the dwell time in a particular system mode. Indicates from Modal transition to The transition probability;

[0136] S200. Design the performance indicators of the space interconnection system and introduce a dissipative cost function, including the instantaneous performance of the control system in each step of operation described by the cost function, which is called the stage cost, generally including control accuracy and control cost; In this invention, in order to overcome the interference caused by external environmental factors, including solar radiation pressure and gravity gradient, a QSR dissipative cost term (i.e., dissipative term) is introduced into the stage cost to ensure that the increase in energy inside the system does not exceed the energy of the disturbance input;

[0137] Specifically, including,

[0138] The robust optimization control problem can be described as minimizing the worst-case performance upper bound, i.e., the following optimization problem:

[0139]

[0140] To fully account for mode transitions caused by temperature changes, the infinite time-domain cost function is expected to be expressed as... , The instantaneous performance of the control system at each step of its operation is described, referred to as the stage cost. In this invention, to combat external disturbances, a dissipation index is introduced to limit the energy introduced by noise from exceeding the energy stored in the system. The stage cost is expressed as follows:

[0141]

[0142]

[0143] in, The weight matrix is ​​positive definite. It is a negative definite weight matrix. It is a symmetric matrix. Positive scalars describe the degree of bias in robustness;

[0144] S300. Calculate the upper bound of the system performance index and design a robust positive invariant set to obtain the linear matrix inequality conditions. This includes: First, substituting the subsystem state into the Lyapunov function to determine the analytical expression of the upper bound of the stochastic performance, and by establishing the relationship between the Lyapunov function and the cost function, transforming the optimization of the cost function into the optimization of the upper bound of the Lyapunov function, thus deriving the linear matrix inequality conditions that guarantee the stability of the closed-loop system; Second, introducing the concept of a robust positive invariant set, considering the influence of bounded disturbance inputs, substituting them into the disturbed closed-loop subsystem model, and establishing linear matrix inequality conditions that can guarantee recursive feasibility and system constraints.

[0145] Specifically, including,

[0146] The cumulative control cost of the control system in the infinite time domain is described, which is an infinite summation and cannot be directly calculated numerically. Therefore, this step introduces the Lyapunov function to derive its analytical expression for the upper bound. Using this expression, a solvable optimization problem is established, considering the following form of modal-dependent Lyapunov function. :

[0147]

[0148] If the quadratic function satisfies the following condition:

[0149]

[0150]

[0151] in, Represents all modes For the set of , the following equation holds:

[0152]

[0153] Transforming the above equation from time... Accumulate to We can obtain:

[0154]

[0155] Therefore, an upper bound can be obtained for the infinite time-domain expected cost function. ;because Corresponding state To determine the value, therefore for The optimization is to optimize the matrix. With optimization, the existence condition of the performance upper bound is further transformed into the form of the following linear inequality:

[0156] in, , ; A matrix represents the effect of control inputs on the system state; A matrix represents the effect of a disturbance on the system state;

[0157] Therefore, the upper bound of the system performance index can be obtained by solving the above inequality. By incorporating this equation into the optimization problem, and minimizing it... The upper bound of the cost allows for performance optimization;

[0158] Model predictive control not only requires optimization of system performance but also includes constraints on the state / control inputs. These constraints must be satisfied not only at the current time step but also at all future time steps up to infinity. This step introduces the concept of robust positive invariant sets, constructing sets of the form... The modality-dependent ellipsoid set, where Y represents the Lyapunov parameter matrix, and the elements of the set satisfy the following conditions: Then, under the action of the controller, the state at the next moment will still satisfy the condition. Clearly, if we can establish a modal dependency robust positive invariant set for a generalized system with variable state dimension, Constraint determination for infinite time-domain state / control inputs can be transformed into constraint determination for ellipsoidal boundaries, thereby enabling the solution of optimization problems that satisfy the constraints.

[0159] The structure is like The modal-dependent quadratic function, This can be represented as the corresponding modal dependency level set, and its positive invariance condition can be directly expressed as: Considering the influence of bounded disturbance inputs, substituting this into the disturbance closed-loop subsystem model, the condition can be further expressed as:

[0160] in , Let represent the maximum value of the external disturbance. To facilitate numerical solution, the S-process is applied to the above equation, yielding the following linear inequality:

[0161]

[0162] in, Indicates the degree of robustness. A higher value indicates greater robustness and stronger resistance to noise.

[0163] This condition ensures the set The convergence of the system during operation; in addition, the invariant set must also satisfy the initial bounded condition, i.e. This is equivalent to the following linear inequality:

[0164]

[0165] The above conditions determine the robust positive invariant set of a space interconnection subsystem studied in this invention. To ensure that the state and control input within the invariant set satisfy the constraints, the following ellipsoidal set boundary constraint discrimination conditions need to be introduced:

[0166]

[0167]

[0168] By simultaneously solving the above inequalities, we can obtain the robust positive invariant set that satisfies the constraints in the infinite time domain. ;

[0169] S400. Establish a rolling time-domain optimization problem and perform simulation verification. This includes establishing a rolling time-domain optimization problem based on the aforementioned cost function and linear matrix inequality conditions, selecting a suitable solver to solve the problem, obtaining the subsystem state feedback control gain and applying it to the vibration suppression unit. In the simulation, all subsystems share state information within their respective local communication networks, then perform parallel optimization solutions, and apply the control input to the flexible attachment through the vibration suppression unit.

[0170] Specifically, including,

[0171] Based on the results of steps S200 and S300, the following convex optimization problem for solving the control law of the distributed subsystem is established:

[0172]

[0173]

[0174] in, These are the matrix variables to be optimized;

[0175] By the Schur complement transformation, the matrix inequality condition in the above equation is equivalent to: , This is the upper bound of the Lyapunov function, which is also the upper bound of the cost function; therefore, minimizing... This is equivalent to minimizing the upper bound of the cost function of the original optimization problem. Minimizing this value is equivalent to minimizing the cost, thus achieving system performance optimization.

[0176] Specific Implementation Method Two: The present invention provides a distributed model predictive control system for vibration suppression of large flexible spacecraft attachments. This system has a program module corresponding to the above steps, and executes the steps in the above-described distributed model predictive control method for vibration suppression of large flexible spacecraft attachments during operation.

[0177] The other combinations and connections in this implementation scheme are the same as in Specific Implementation Scheme 1.

[0178] Specific Implementation Method 3: The present invention provides a computer-readable storage medium storing a computer program configured to, when called by a processor, implement the steps of a distributed model predictive control method for vibration suppression of large flexible spacecraft attachments.

[0179] The other combinations and connections in this implementation scheme are the same as in Specific Implementation Scheme 1.

[0180] Controller Deployment and Simulation

[0181] The process for online controller solution is as follows:

[0182] ① Set performance weight parameters for each subsystem ;

[0183] ② Obtain subsystem state and modal information and adjacent subsystems (such as Figure 4 ) state

[0184] ③ Solve the optimization problem described by formula (23)

[0185] ④ Controller of the computing subsystem

[0186] ⑤ The vibration control unit corresponding to the subsystem applies control force to the flexible attachment.

[0187] ⑥ Return to step ② and start the loop.

[0188] To test the vibration suppression effect of the rolling optimization controller, a series of numerical simulations were conducted. The simulation object was a single solar panel satellite, with the solar panel being an 8m long flexible attachment. Vibration suppression units were deployed at 0.5m intervals along the center, totaling 16 subsystems. The dynamics of the temperature-affected flexible attachment system were divided into three modes. The figures represent the dynamics of the attachment system under low, medium, and high temperatures, respectively. Vibration of the flexible attachment can be caused by both external and internal factors. External factors are typically vibrations resulting from large deformation caused by space debris impacts; internal factors are mainly due to the torque exerted on the flexible attachment by the satellite's violent maneuvers. Therefore, this invention demonstrates the effectiveness and practicality of the proposed vibration suppression method through simulations of two scenarios.

[0189] Scenario 1: Set the initial vibration displacement of the 16 subsystems in the appendix as follows: The initial vibration velocity is This is used to simulate the large initial deformation of flexible attachments after being impacted by debris. Additionally, noise is used to simulate external disturbances such as solar radiation pressure and gravitational gradient forces. It is also applied to the attachments. Figure 5The overall control effect of vibration suppression is demonstrated, showing that the proposed controller achieves vibration suppression within 3 seconds. Figure 6 The proposed method is compared with decentralized MPC and spatial interconnected MPC that does not consider mode transitions. It can be seen that the proposed method exhibits higher stability and smoothness.

[0190] Scenario 2: Designing an aggressive attitude maneuver trajectory for the satellite, such as... Figure 9 As shown in (a), the maximum angular velocity reaches . Figure 7 The overall vibration suppression effect was demonstrated, with the satellite recovering to stability within an average of 2 seconds after a sharp rotation causing significant deformation. The vibration suppression effect was compared with that of other controllers. Figure 8 As shown, the proposed controller exhibits high convergence rate and vibration suppression capability. Most existing satellites have extremely high requirements for attitude pointing accuracy, which is a crucial indicator of the quality of their control systems. Therefore, to further illustrate the practicality of this method, Figure 9 (b) in the paper compares the satellite's attitude control accuracy. By comparing it with two other traditional methods, it is demonstrated that the method proposed in this invention significantly improves the satellite's pointing accuracy, further proving its effectiveness and great potential in practical applications.

[0191] While the present invention has been disclosed above, its scope of protection is not limited thereto. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the present invention, and all such changes and modifications will fall within the scope of protection of the present invention.

Claims

1. A distributed model predictive control method for vibration suppression of attachments to large flexible spacecraft, characterized in that, Includes the following steps: S100. Establish a space interconnected Markov jump system model for the flexible spacecraft attachments, including modeling the vibration dynamics of the large flexible spacecraft attachments based on the Euler-Bernoulli beam model, approximating the position derivative using the central difference method, and approximating the time derivative using the Euler discretization method; perform dynamic analysis on the system modes at different temperatures, thereby transforming the multimodal dynamic equations of the large flexible spacecraft attachments into a discrete-time space interconnected Markov jump system model. S200, design the performance indicators of the space interconnection system, introduce a dissipative cost function, including the instantaneous performance of the control system in each step of operation, i.e., the stage cost, which includes control accuracy and control cost; and introduce a dissipative cost term in the stage cost to ensure that the increase in energy inside the system does not exceed the energy of the disturbance input. S300. Calculate the upper bound of the system performance index and design a robust positive invariant set to obtain the linear matrix inequality conditions. This includes substituting the subsystem state into the Lyapunov function to determine the analytical expression of the upper bound of the stochastic performance, and transforming the optimization of the cost function into the optimization of the upper bound of the Lyapunov function by establishing the relationship between the Lyapunov function and the cost function, thus deriving the linear matrix inequality conditions that guarantee the stability of the closed-loop system. Introduce the concept of a robust positive invariant set, consider the influence of bounded disturbance inputs, substitute them into the disturbance closed-loop subsystem model, and establish the linear matrix inequality conditions that guarantee recursive feasibility and system constraints. S400. Establish a rolling time-domain optimization problem, including establishing a rolling time-domain optimization problem based on the above cost function and linear matrix inequality conditions, selecting a solver to solve the problem, obtaining the subsystem state feedback control gain and applying it to the vibration suppression unit.

2. The distributed model predictive control method for vibration suppression of large flexible spacecraft attachments according to claim 1, characterized in that: In step S100, the following are included: Let the total length of the flexible attachment be The vibration suppression system includes multiple vibration control units to... The vibration control units (VCUs) are evenly spaced on the flexible attachments. Each VCU contains a displacement sensor, actuator, and microcontroller, and establishes a communication network with adjacent VCUs. Each VCU is considered a subsystem. The dynamics of the flexible attachments are expressed by the following partial differential equations: in, and These represent the external force applied perpendicularly to the flexible attachment and the flexible attachment's position on the surface. time Vibration displacement at position and These represent bending stiffness and mass per unit length, respectively. For Young's modulus, For rotational inertia, For the density of flexible attachments, Let be the cross-sectional area, and define . , where the width is The height is ; The following simplified representation is used. , Since the Coriolis force does not induce vertical vibration, the vibration only involves centrifugal force. and tangential force : in, Indicates the rotation angle of a rigid body; This represents the distance from the center of the rigid body to the point of connection with the flexible attachment; The load applied to the flexible attachment is: in, Indicates the flexible attachment and the spacecraft coordinate system chamfer, Indicates flexible attachments and The included angle between them; Considering the inertial load applied to the flexible attachment during vibration, and neglecting the second-order terms of the trigonometric functions, the distributed load along the flexible attachment is expressed as: in, ; The distributed forces of the subsystem are modeled as ,in Represents the Dirichlet function, Represents the position The The control force output by each subsystem, where N represents the total number of subsystems. Representing the The positions of the subsystems; assuming the influence of the mass and volume of the vibration control unit on the dynamics of the flexible attachment is negligible, the dynamics of the flexible attachment considering the influence of the central rigid body and the subsystems are as follows: The corresponding boundary conditions are: in, , For the length of the flexible attachment, time , The damping coefficient of the flexible attachment. This includes external disturbances and unmodeled higher-order dynamics; This represents the vibrational displacement at the connection point between the flexible attachment and the rigid body. Indicates the vibration displacement at the end of the flexible attachment; The changes in the properties of flexible attachment materials caused by temperature variations are reflected in Young's modulus and damping ratio, i.e. ,in It is a temperature-dependent modal sequence; considering the coupling effect between the flexible attachments and the central rigid body, the attitude dynamics of the central rigid body are: in, , and Representing the central rigid body, flexible appendages, and the spacecraft as a whole, respectively, along the inertial coordinate system. Moment of inertia of the shaft; It is the sum of the torques acting on the spacecraft; The angular velocity of the satellite; By performing intermediate difference approximation on the positional partial derivatives in equation (4) and Euler discretization on the time partial derivatives, the discrete-time spatial interconnected Markov jump system is obtained as shown below: in, Indicates vibration displacement Vectorization, Indicates vibration velocity. This indicates the control force output by the subsystem. Represents spatial coordinates, Indicates the discretization time; Describing the quality of the subsystem, the state-space equation of the discrete-time spatially interconnected Markov jump system is expressed as: in, and These represent time variables and space variables, respectively. Represents the state of a subsystem, including spatiotemporal variables and time operators. Indicates positive time shift Spatial operators represent bidirectional displacements. Combinatorial operators For the subsystem located at s, the states of adjacent subsystems; the physical constraints on the system are: in, Indicates the system status and the maximum limit value of control inputs; This represents the system state at time k; Represents the control input at time k; to simplify the notation, make All represent system modes; System Mode The random variation pattern is as follows: in, Indicates the first The moment of the next switch. This represents the dwell time in a particular system mode. Indicates from Modal transition to The transition probability.

3. The distributed model predictive control method for vibration suppression of large flexible spacecraft attachments according to claim 2, characterized in that: Step S200 includes, The robust optimization control problem can be described as minimizing the worst-case performance upper bound, i.e., the following optimization problem: To fully account for mode transitions caused by temperature changes, the infinite time-domain cost function is expected to be expressed as... , The instantaneous performance of the control system at each step of its operation is described, and this is called the stage cost. To combat external disturbances, a dissipation index is introduced to limit the energy introduced by noise from exceeding the energy stored in the system. The stage cost is expressed as follows: in, The weight matrix is ​​positive definite. It is a negative definite weight matrix. It is a symmetric matrix. A positive scalar describes the degree of emphasis on robustness.

4. The distributed model predictive control method for vibration suppression of large flexible spacecraft attachments according to claim 3, characterized in that: Step S300 includes, We introduce the Lyapunov function to derive its analytical expression for its upper bound, and use this expression to establish a solvable optimization problem. Consider the following form of modally dependent Lyapunov function. : If the quadratic function satisfies the following condition: in, Represents all modes A set; Then the following equation holds true: Transforming the above equation from time... Accumulate to ,get: We obtain the upper bound of the infinite time-domain expected cost function. ;because Corresponding state To determine the value, then for The optimization is to optimize the matrix. The existence condition for the performance upper bound is transformed into the following linear inequality form for optimization: in, , ; A matrix represents the effect of control inputs on the system state; A matrix represents the effect of a disturbance on the system state; By minimizing The upper bound of the cost is used to achieve performance optimization; Introducing the concept of robust positive invariant sets, constructing forms The modality-dependent ellipsoid set, where Y represents the Lyapunov parameter matrix, and the elements of the set satisfy the following conditions: Then, under the action of the controller, the state at the next moment will still satisfy the condition. If we can establish a modal dependency robust positive invariant set for a generalized system with variable state dimension... Constraint determination for infinite time-domain state / control inputs can be transformed into constraint determination for ellipsoidal boundaries, thereby enabling the solution of optimization problems that satisfy the constraints. Construction form The modal-dependent quadratic function, Let be the corresponding modal dependency level set, and its positive invariance condition be directly expressed as: Considering the influence of bounded disturbance input, substituting it into the disturbance closed-loop subsystem model, the condition is expressed as: in, , Let represent the maximum value of the external disturbance. To facilitate numerical solution, the S-process is applied to the above equation, yielding the following linear inequality: in, Indicates the degree of robustness; This condition ensures the set The convergence of the system during operation; in addition, the invariant set must also satisfy the initial bounded condition, i.e. This is equivalent to the following linear inequality: To ensure that both the state and control input within the invariant set satisfy the constraints, the following boundary constraint criteria for the ellipsoid set need to be introduced: By simultaneously solving the above inequalities, we obtain the robust positive invariant set that satisfies the constraints in the infinite time domain. .

5. The distributed model predictive control method for vibration suppression of large flexible spacecraft attachments according to claim 4, characterized in that: In step S400, the following are included: Based on the results of steps S200 and S300, the following convex optimization problem for solving the control law of the distributed subsystem is established: in, These are the matrix variables to be optimized; By the Schur complement transformation, the matrix inequality condition in the above equation is equivalent to: , This is the upper bound of the Lyapunov function, i.e., the upper bound of the cost function; therefore, minimizing... This is equivalent to minimizing the upper bound of the cost function of the original optimization problem in order to achieve system performance optimization.

6. A distributed model predictive control system for vibration suppression of large flexible spacecraft attachments, characterized in that: The system has a program module corresponding to the steps described in any one of claims 1-5, and executes the steps in the above-described distributed model predictive control method for vibration suppression of large flexible spacecraft attachments when running.

7. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program configured to, when invoked by a processor, implement the steps of any one of claims 1-5, a distributed model predictive control method for vibration suppression of large flexible spacecraft attachments.

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