A nonlinear anti-windup robust control method for flexible satellite attitude system based on dynamic observer

Abstract

The application relates to a nonlinear anti-saturation robust control method of a flexible satellite attitude system based on a dynamic observer, and comprises the following steps: S1, establishing a flexible satellite attitude system, and converting the flexible satellite attitude system into a corresponding nonlinear state space model; S2, constructing a nonlinear dimension reduction dynamic observer according to the nonlinear state space model, designing a nonlinear state feedback robust controller based on the nonlinear dimension reduction dynamic observer, constructing a nonlinear anti-saturation compensator based on the nonlinear state feedback robust controller, and obtaining a nonlinear closed-loop control system which can simultaneously process state unmeasurability, convex polyhedral uncertainty, external disturbance and actuator saturation problems; S3, respectively deriving SOS solvability conditions under the conditions of ignoring and considering actuator saturation; and S4, respectively solving the nonlinear dimension reduction dynamic observer and the state feedback robust controller, and the nonlinear anti-saturation compensator under the conditions of ignoring and considering actuator saturation.

Description

Technical Field

[0001] This invention relates to the field of flexible satellite attitude control, specifically to a nonlinear anti-saturation robust control method for flexible satellite attitude systems based on dynamic observers. Background Technology

[0002] As the complexity of space missions continues to increase, flexible satellites generally consist of a rigid body and large flexible attachments, such as antennas, solar panels, and flexible robotic arms. This structure, when performing large-angle attitude maneuvers, can cause significant flexible vibrations due to the rigid-flexible coupling effect, severely impacting the stability and pointing accuracy of the attitude control system. Furthermore, various disturbances in the on-orbit environment, including dynamic parameter perturbations caused by temperature differences, fuel consumption, and structural deformation, as well as external disturbances such as gravity gradients, solar radiation pressure, and geomagnetic interference, all pose serious challenges to the control system. Simultaneously, the actuator saturation problem, which is prevalent in the control process due to limitations in the output capacity of physical actuators, can lead to control signal distortion, resulting in a sharp deterioration in system performance.

[0003] In existing technologies, flexible vibration suppression techniques are mainly divided into two categories: active control and passive control. While active control schemes are highly effective, they require additional specialized sensors and measurement devices to detect flexible modes, significantly increasing system complexity and cost, and introducing new reliability risks. In contrast, observer-based passive control methods exhibit a clear advantage due to the absence of additional sensing equipment. Among various observer designs, dynamic observers, with their higher parameter freedom and ability to achieve more flexible control strategy design through dynamic gain, have become a research hotspot. However, current research on dynamic observers mainly focuses on linear systems, and their application in nonlinear systems still faces technical bottlenecks such as imperfect theoretical methods and difficulty in solving problems. Regarding actuator saturation problems, although anti-saturation compensators are a commonly used solution in engineering practice, most existing compensation strategies are based on the framework of linear time-invariant systems, making it difficult to effectively handle the complex saturation characteristics of nonlinear systems. In terms of parameter uncertainty modeling, compared with traditional norm-bounded uncertain system description methods, convex polyhedral models can more accurately characterize multi-parameter independent or coupled uncertain structures, effectively avoiding the over-conservatism problem caused by global constraints in norm-bounded methods. However, existing research based on convex polyhedral models mainly relies on state feedback control methods, which clearly do not adequately consider situations where the state is unmeasurable.

[0004] The purpose of this invention is to design a nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer, addressing the problems existing in the prior art. Summary of the Invention

[0005] To address the problems existing in the prior art, the present invention provides a nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer, which can effectively solve at least one of the problems existing in the prior art.

[0006] The technical solution of this invention is:

[0007] A nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer includes the following steps:

[0008] S1. Establish a flexible satellite attitude system with actuator saturation, some states that are unmeasurable and affected by external disturbances and parameter perturbations, and convert the flexible satellite attitude system into a corresponding nonlinear state-space model.

[0009] S2. Construct a nonlinear dimension-reduced dynamic observer based on the nonlinear state-space model, construct a nonlinear state feedback robust controller based on the nonlinear dimension-reduced dynamic observer, and construct a nonlinear anti-saturation compensator based on the nonlinear state feedback robust controller to obtain a nonlinear closed-loop control system that can simultaneously handle state unmeasurability, convex polyhedral uncertainty, external disturbances and actuator saturation problems.

[0010] S3, based on the nonlinear closed-loop control system, derive the first SOS solvability condition of the state feedback robust controller based on the nonlinear dimension-reduced dynamic observer under the condition of neglecting actuator saturation, and derive the second SOS solvability condition of the anti-saturation compensator based on the first SOS solvability condition under the condition of considering actuator saturation.

[0011] S4. Based on the first SOS solvability condition, the nonlinear reduced-dimensional dynamic observer and the state feedback robust controller are solved using the SOSTOOLS toolbox of Matlab, ignoring actuator saturation. Based on the second SOS solvability condition, the anti-saturation compensator is solved using the SOSTOOLS toolbox of Matlab, considering actuator saturation, based on the nonlinear reduced-dimensional dynamic observer and the state feedback robust controller.

[0012] Further, S1, establishing a flexible satellite attitude system with actuator saturation, partially unmeasurable states, and susceptible to external disturbances and parameter perturbations, and converting the flexible satellite attitude system into a corresponding nonlinear state-space model includes:

[0013] S1.1, a disturbance torque term and a rotational inertia perturbation term are introduced into the kinematic and dynamic equations of the flexible satellite attitude system to construct a flexible satellite attitude system that includes external disturbances and parameter uncertainties; the flexible satellite attitude system that includes external disturbances and parameter uncertainties is represented as follows:

[0014]

[0015] ;

[0016] in, For the Rodrigues parameter vector, These are the Rodriguez parameters. , The attitude angular velocity vector, These are the three orthogonal components of the attitude angular velocity vector. Here is the rotational inertia matrix. For the perturbation term of the rotational inertia matrix, for The corresponding cross product matrix, The rigid-flexible coupling coefficient matrix is... For the flexible modal coordinate vector, Indicates the modal order being intercepted. Here is the modal damping matrix. The natural frequency matrix, and These are the control torque and the disturbance torque, respectively.

[0017] S1.2, by introducing an auxiliary vector and ,limit This imposes control input constraints, retains the nonlinear terms in the flexible satellite attitude system, and imposes different restrictions on the actual control input in different intervals when the actuator input signal exceeds a preset saturation threshold. Simultaneously, the parameter perturbation term is represented as a convex polyhedral uncertainty, thereby reconstructing the flexible satellite attitude system into a corresponding nonlinear state-space model. The nonlinear state-space model is expressed as follows:

[0018] ;

[0019] in, For system status, and They are Measurable and unmeasurable substates; , and These are the actual control input, the controlled output, and the measurement output, respectively. It is external interference and ; and They represent The set of dimensional real vectors and 3D real matrix set; Represents an uncertain vector The Each component; define the following polyhedron set:

[0020] , , To be constrained by vertices , , The vectors that form the convex polyhedral region;

[0021] It is the first of the system state matrix. A matrix of uncertain sub-components, It is the first of the system input matrix and interference matrix. A matrix of uncertain sub-components,

[0022] , , , ; and These represent the identity matrix and the zero matrix, respectively.

[0023] Furthermore, when the actuator's input signal exceeds a preset saturation threshold, different restrictions are imposed on the actual control input in different ranges, including:

[0024] When the actuator input signal Exceeding the preset saturation threshold At that time, actual control input for:

[0025] , ;

[0026] in, express The OK, express The OK, .

[0027] Further, S2, a nonlinear dimensionality-reduced dynamic observer is constructed based on the nonlinear state-space model; a nonlinear state feedback robust controller is designed based on the nonlinear dimensionality-reduced dynamic observer; and a nonlinear anti-saturation compensator is constructed based on the nonlinear state feedback robust controller. This results in a nonlinear closed-loop control system capable of simultaneously handling state unmeasurability, convex polyhedral uncertainty, external disturbances, and actuator saturation problems, including:

[0028] S2.1, Based on the input signal and measurable output signal of the nonlinear state-space model, construct the nonlinear dimensionality-reduced dynamic observer; the constructed nonlinear dimensionality-reduced dynamic observer is as follows:

[0029] ;

[0030] in, and As an auxiliary variable, yes The estimated quantity, State variables representing observer gain , , and This is the observer parameter matrix to be designed; express The OK;

[0031] S2.2, using the state estimation information provided by the nonlinear dimensionality-reduced dynamic observer, a nonlinear state feedback robust controller is designed; the output signal of the designed nonlinear state feedback robust controller is used as the input signal of the actuator. The result is as follows:

[0032] ;

[0033] in, , , This is the observation error; Let be the controller gain matrix to be determined;

[0034] S2.3, construct the nonlinear anti-saturation compensator by measuring the deviation between the actuator input and the actual output; the constructed nonlinear anti-saturation compensator as follows:

[0035] ;

[0036] in, For the output of the compensator, The compensator gain matrix, Dead-zone function;

[0037] S2.4, the nonlinear dimensionality-reduced dynamic observer, the nonlinear state feedback robust controller, and the nonlinear anti-saturation compensator are collaboratively integrated into the nonlinear state-space model to form a nonlinear closed-loop system that combines state estimation, robust stability, and anti-saturation capability; the nonlinear closed-loop system is as follows:

[0038] ;

[0039] in, , , , , , , , , .

[0040] Furthermore, the design of a nonlinear state feedback robust controller using the state estimation information provided by the nonlinear dimensionality-reduced dynamic observer includes: designing the nonlinear dimensionality-reduced dynamic observer to estimate the state of the flexible satellite attitude system in real time, and constructing the nonlinear state feedback robust controller;

[0041] The overall control objective of the nonlinear dimensionality-reduced dynamic observer, the nonlinear state feedback robust controller, and the nonlinear anti-saturation compensator is: under the presence of parameter perturbations and external disturbances, to make the nonlinear closed-loop control system considering actuator saturation asymptotically stable at the zero equilibrium point, and to satisfy the following conditions: Gain This also mitigates the impact of actuator saturation on system performance, among which, The gain is H ∞ Control performance indicators is a given constant.

[0042] Further, S3, based on the nonlinear closed-loop control system, deriving the first SOS solvability condition of the state feedback robust controller based on the nonlinear dimension-reduced dynamic observer under the condition of neglecting actuator saturation, and deriving the second SOS solvability condition of the anti-saturation compensator based on the first SOS solvability condition under the condition of considering actuator saturation, includes:

[0043] S3.1, Based on Lyapunov stability theory, the first H of the nonlinear closed-loop control system is established under the condition of neglecting actuator saturation. ∞ Performance criteria, considering actuator saturation, establish the second H of the nonlinear closed-loop control system. ∞ Performance criteria;

[0044] S3.2, according to the first H ∞ The performance criteria, derived through variable substitution and the SOS method, establish a robust stability criterion for the state feedback controller based on the nonlinear reduced-dimensional dynamic observer. This criterion ensures that the nonlinear closed-loop control system is asymptotically stable at the zero equilibrium point while neglecting actuator saturation, and satisfies a given condition. The first SOS solvability condition for the gain of the nonlinear dimensionality-reduced dynamic observer and the nonlinear state feedback robust controller;

[0045] S3.3, with the second H ∞ Using the performance criterion as a constraint, and based on the first SOS solvability condition, a second SOS solvability condition is established for the nonlinear anti-saturation robust control problem based on the dynamic observer.

[0046] Furthermore, variable substitution includes:

[0047] The first H ∞ The inequality expansion in the performance criterion, applied with Schur's complement lemma, yields the first H. ∞ The equivalent conditions of the performance criteria are decoupled from the coupling terms of the equivalent conditions by means of variable substitution.

[0048] Furthermore, the first H ∞ The performance criterion is: for a given constant If there exists a continuously differentiable function and ,right satisfy:

[0049] ;

[0050] , , The number of vertices of the convex polyhedron;

[0051] Nonlinear closed-loop system where actuator saturation is not determined It is asymptotically stable at its zero equilibrium point and has -Gain ,in ;

[0052] Second H ∞ The performance criterion is: for a given constant If there exists a continuously differentiable function ,satisfy And for All conditions are met:

[0053] ;

[0054] , ;

[0055] Therefore, a nonlinear closed-loop system with actuator saturation is asymptotically stable at its zero equilibrium point and has the following properties: -Gain ,in ;

[0056] The first SOS solvability condition is: for a given constant If a constant exists and symmetric polynomial matrix symmetric constant matrix , and polynomial matrices , , , and The following conditions must be met:

[0057] (I) ;

[0058] (II) ;

[0059] (III) ;

[0060] (IV) ;

[0061] (V) ;

[0062] in, , , , , , , Represents the SOS polynomial set. , , , , , , , , , .

[0063] Therefore, the aforementioned nonlinear dimensionality-reduced dynamic observer and the aforementioned nonlinear state feedback robust controller can make the closed-loop system asymptotically stable at the zero equilibrium point without considering actuator saturation, and it has... -Gain The observer parameter matrix is:

[0064] ;

[0065] The controller gain matrix is: ;

[0066] in, and It is an invertible matrix and satisfies .

[0067] The solvability condition for the second SOS is: if there exists a constant... , and symmetric polynomial matrix diagonal polynomial matrix and polynomial matrices and Make the following conditions true:

[0068] (i) ;

[0069] (ii) ;

[0070] (iii) ;

[0071] (iv) ;

[0072] in, , , , , , The matrix to be found is... yes The OK, and Let be the matrix to be determined; .

[0073] Then there exists an anti-saturation compensator gain matrix. This allows the nonlinear closed-loop system to operate within an ellipsoid, considering actuator saturation. Internal asymptotic stability, in which, for The OK.

[0074] Further, S4, based on the first SOS solvability condition, the nonlinear reduced-dimensional dynamic observer and the state feedback robust controller are solved using the Matlab SOSTOOLS toolbox while ignoring actuator saturation. Based on the second SOS solvability condition, the anti-saturation compensator is solved using the Matlab SOSTOOLS toolbox while considering actuator saturation, including:

[0075] S4.1, neglecting actuator saturation, given the kinematic and dynamic parameters H of the flexible satellite attitude system... ∞Performance metrics are defined by setting the dimensions and polynomial orders of the relevant auxiliary matrices used to construct parameters and gain matrices, and setting the symmetric positive definiteness, dimensions and polynomial orders of the matrices to be solved and the relevant auxiliary matrices in the Lyapunov function. Based on the first SOS solvability condition, the nonlinear dimension-reduced dynamic observer and the state feedback robust controller are solved using the SOSTOOLS toolbox of Matlab.

[0076] S4.2, Considering actuator saturation, based on the nonlinear dimension-reduced dynamic observer and the state feedback robust controller, and based on the second SOS solvability condition, the parameter matrix of the nonlinear anti-saturation compensator is further solved using the SOSTOOLS toolbox of Matlab.

[0077] Therefore, the present invention provides the following effects and / or advantages:

[0078] This application proposes a nonlinear, robust anti-saturation control method based on a dynamic observer for flexible satellite attitude systems characterized by nonlinearity, actuator saturation, partial unmeasurable states, and susceptibility to external disturbances and parameter perturbations. This method enables large-angle attitude maneuver control of the satellite, exhibits good robustness against rotational inertia perturbations and external disturbances, and effectively suppresses the influence of flexible appendage vibrations under input saturation constraints.

[0079] Compared to the linearization process in existing technologies, this application avoids linear approximation of the flexible satellite attitude system. Instead, it directly conducts control design based on the nonlinear system model. This method preserves the nonlinear dynamic characteristics of the system, such as rigid-flexible coupling and large-angle maneuvers, making the model closer to the actual physical system and laying a more solid foundation for improving control accuracy and stability.

[0080] This application achieves an improvement in anti-saturation control strategy, moving from "linear saturation-limited control" to "nonlinear anti-saturation control." Unlike existing technologies that passively avoid saturation by limiting control commands through conservative constraints at the expense of dynamic response, this application designs a nonlinear anti-saturation compensator based on the actuator input-output difference without increasing system dimensionality. This effectively eliminates the adverse effects of input saturation on the control system, achieving more proactive and precise saturation management.

[0081] By introducing a systematic solution method based on SOS theory, this invention effectively overcomes the common problem of solution difficulties in nonlinear system control design. It provides a powerful mathematical tool for the nonlinear collaborative design of dynamic observers, convex polyhedral uncertainties, and anti-saturation compensation, ensuring that the entire control design process is not only theoretically rigorous but also feasible.

[0082] Other features and advantages of the invention will be set forth in the following description, and will be apparent in part from the description, or may be learned by practicing the invention. The objects and other advantages of the invention are realized and obtained through the structures particularly pointed out in the description and the drawings.

[0083] It should be understood that the above summary and the following detailed description of the invention are exemplary and explanatory, and are intended to provide further explanation of the invention as claimed. Attached Figure Description

[0084] Figure 1 A schematic diagram illustrating the uncertainty of the convex polyhedral form described in this application for a flexible satellite attitude system.

[0085] Figure 2 External disturbances for perturbation-uncertainty coupled systems A graph, with units of Newton-meters.

[0086] Figure 3 Control input under anti-saturation strategy A curve graph, which has limitations .

[0087] Figure 4 This shows the Rodrigues parameter vector. The trajectory curve.

[0088] Figure 5 Attitude angular velocity The trajectory.

[0089] Figure 6 Flexible mode , The trajectory.

[0090] Figure 7 Flexible mode , The locus of the first derivative.

[0091] Figure 8 A flowchart is provided for an embodiment of the present invention. Detailed Implementation

[0092] To facilitate understanding by those skilled in the art, the present invention will now be described in further detail with reference to the embodiments:

[0093] refer to Figure 8 A nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer includes the following steps:

[0094] S1. Establish a flexible satellite attitude system with actuator saturation, some states that are unmeasurable and affected by external disturbances and parameter perturbations, and convert the flexible satellite attitude system into a corresponding nonlinear state-space model.

[0095] S2. Construct a nonlinear dimension-reduced dynamic observer based on the nonlinear state-space model, construct a nonlinear state feedback robust controller based on the nonlinear dimension-reduced dynamic observer, and construct a nonlinear anti-saturation compensator based on the nonlinear state feedback robust controller to obtain a nonlinear closed-loop control system that can simultaneously handle state unmeasurability, convex polyhedral uncertainty, external disturbances and actuator saturation problems.

[0096] S3, based on the nonlinear closed-loop control system, derive the first SOS solvability condition of the state feedback robust controller based on the nonlinear dimension-reduced dynamic observer under the condition of neglecting actuator saturation, and derive the second SOS solvability condition of the anti-saturation compensator based on the first SOS solvability condition under the condition of considering actuator saturation.

[0097] S4. Based on the first SOS solvability condition, the nonlinear reduced-dimensional dynamic observer and the state feedback robust controller are solved using the SOSTOOLS toolbox of Matlab, ignoring actuator saturation. Based on the second SOS solvability condition, the anti-saturation compensator is solved using the SOSTOOLS toolbox of Matlab, considering actuator saturation, based on the nonlinear reduced-dimensional dynamic observer and the state feedback robust controller.

[0098] This application focuses on three key areas: dynamic observer control methods, convex polyhedron uncertainty handling techniques, and anti-saturation compensation techniques. Particularly within the framework of nonlinear systems, these three key technologies have not yet achieved organic integration and synergistic optimization. Addressing this technological gap, this invention, based on a flexible satellite attitude control system characterized by actuator saturation, partially unmeasurable states, and susceptibility to external disturbances and parameter perturbations, employs a convex polyhedron model to accurately describe system uncertainties and, based on the SOS sum-of-squares theory, derives a nonlinear anti-saturation robust control method.

[0099] Further, S1, establishing a flexible satellite attitude system with actuator saturation, partially unmeasurable states, and susceptible to external disturbances and parameter perturbations, and converting the flexible satellite attitude system into a corresponding nonlinear state-space model includes:

[0100] S1.1, a disturbance torque term and a rotational inertia perturbation term are introduced into the kinematic and dynamic equations of the flexible satellite attitude system to construct a flexible satellite attitude system that includes external disturbances and parameter uncertainties; the flexible satellite attitude system that includes external disturbances and parameter uncertainties is represented as follows:

[0101]

[0102] ;

[0103] in, For the Rodrigues parameter vector, These are the Rodriguez parameters. , The attitude angular velocity vector, These are the three orthogonal components of the attitude angular velocity vector. Here is the rotational inertia matrix. For the perturbation term of the rotational inertia matrix, for The corresponding cross product matrix, The rigid-flexible coupling coefficient matrix is... For the flexible modal coordinate vector, Indicates the modal order being intercepted. Here is the modal damping matrix. The natural frequency matrix, and These are the control torque and the disturbance torque, respectively.

[0104] Satellite attitude systems typically only obtain direct measurements of attitude angles and angular velocities, which makes some states unmeasurable in the system's dynamic equations. Furthermore, the system itself is susceptible to external disturbances. The impact must also be considered, along with the uncertain perturbation of the moment of inertia. And the input saturation constraint problem of the actuator, at this time Restricted.

[0105] S1.2, In this embodiment, the order of the truncated flexible modes is used. Let's take an example.

[0106] By introducing auxiliary vectors and ,limit This imposes control input constraints, retains the nonlinear terms in the flexible satellite attitude system, and imposes different restrictions on the actual control input in different intervals when the actuator input signal exceeds a preset saturation threshold. At the same time, the parameter perturbation term is represented as a convex polyhedral uncertainty, thereby reconstructing the flexible satellite attitude system into a corresponding nonlinear state-space model.

[0107] The kinematics and dynamics equations of the flexible satellite attitude are equivalently expressed as:

[0108] ;

[0109] in, For system status, and They are Measurable and unmeasurable substates; , and These are the actual control input, the controlled output, and the measurement output, respectively. It is external interference and ;

[0110] , , , ; and For an uncertain matrix, , ,

[0111] , and These represent the identity matrix and the zero matrix, respectively.

[0112] Next, considering the physical limiting characteristics of actuators in engineering practice, different restrictions are imposed on the actual control input in different ranges when the actuator's input signal exceeds a preset saturation threshold, including:

[0113] When the actuator input signal Exceeding the preset saturation threshold At that time, actual control input for:

[0114] , ;

[0115] in, express The OK, express The OK, .

[0116] At the same time, given vector And define the following polyhedron set:

[0117] ;

[0118] Then for , can be set as follows:

[0119] ;

[0120] in, It is the first of the system state matrix. A matrix of uncertain sub-components, It is the first of the system input matrix and interference matrix. A matrix of uncertain sub-components, This represents the number of vertices of a convex polyhedron.

[0121] Obviously, Constrained by vertices , , Within the convex polyhedral region, this indicates that the flexible satellite attitude system possesses the convex polyhedral form uncertainty described in the above equation, one case being... Figure 1 As shown. For ease of explanation, it will be referred to thereafter. All are abbreviated as .

[0122] At this point, the nonlinear state-space model of the flexible satellite attitude system is expressed as follows:

[0123] ;

[0124] Further, S2, a nonlinear dimensionality-reduced dynamic observer is constructed based on the nonlinear state-space model; a nonlinear state feedback robust controller is designed based on the nonlinear dimensionality-reduced dynamic observer; and a nonlinear anti-saturation compensator is constructed based on the nonlinear state feedback robust controller. This results in a nonlinear closed-loop control system capable of simultaneously handling state unmeasurability, convex polyhedral uncertainty, external disturbances, and actuator saturation problems, including:

[0125] S2.1, Based on the input signal and measurable output signal of the nonlinear state-space model, construct the nonlinear dimensionality-reduced dynamic observer; the constructed nonlinear dimensionality-reduced dynamic observer is as follows:

[0126] ;

[0127] in, and As an auxiliary variable, yes The estimated quantity, State variables representing observer gain , , and This is the observer parameter matrix to be designed; express The OK;

[0128] S2.2, using the state estimation information provided by the nonlinear dimensionality-reduced dynamic observer, a nonlinear state feedback robust controller is designed; the output signal of the designed nonlinear state feedback robust controller is used as the input signal of the actuator. The result is as follows:

[0129] ;

[0130] in, , , This is the observation error; Let be the controller gain matrix to be determined;

[0131] S2.3, construct the nonlinear anti-saturation compensator by measuring the deviation between the actuator input and the actual output; the constructed nonlinear anti-saturation compensator as follows:

[0132] ;

[0133] in, For the output of the compensator, The compensator gain matrix, Dead-zone function;

[0134] S2.4, the nonlinear dimensionality-reduced dynamic observer, the nonlinear state feedback robust controller, and the nonlinear anti-saturation compensator are collaboratively integrated into the nonlinear state-space model to form a nonlinear closed-loop system that combines state estimation, robust stability, and anti-saturation capability; the nonlinear closed-loop system is as follows:

[0135] ;

[0136] in, , , , , , , , , .

[0137] Furthermore, the design of a nonlinear state feedback robust controller using the state estimation information provided by the nonlinear dimensionality-reduced dynamic observer includes: designing the nonlinear dimensionality-reduced dynamic observer to estimate the state of the flexible satellite attitude system in real time, and constructing the nonlinear state feedback robust controller;

[0138] The overall control objective of the nonlinear dimensionality-reduced dynamic observer, the nonlinear state feedback robust controller, and the nonlinear anti-saturation compensator is: under the presence of parameter perturbations and external disturbances, to make the nonlinear closed-loop control system considering actuator saturation asymptotically stable at the zero equilibrium point, and to satisfy the following conditions: Gain This also mitigates the impact of actuator saturation on system performance, among which, The gain is H ∞ Control performance indicators is a given constant.

[0139] The specific derivation process of step S2 is as follows.

[0140] In this step, the nonlinear dimension-reduced dynamic observer adopts the following structure:

[0141]

[0142] in, and As an auxiliary variable, yes The estimated quantity, State variables representing observer gain , , and It is the observer parameter matrix to be designed.

[0143] Based on this observer, the output signal of the nonlinear state feedback robust controller The construction is as follows:

[0144]

[0145] in, , , This is the observation error; Let be the gain matrix of the control law to be determined.

[0146] Furthermore, the dynamic equations of the nonlinear anti-saturation compensator are designed as follows:

[0147] ;

[0148] in, For the output of the compensator, The compensator gain matrix, This is a dead-zone function.

[0149] Will Using this as another input signal to the nonlinear dimension-reduced dynamic observer, and combining it with the nonlinear state-space model of the flexible satellite, the nonlinear state feedback robust controller, and the anti-saturation compensator equations, the resulting nonlinear closed-loop system has the following expression:

[0150] ;

[0151] in, , , , , , , , , .

[0152] For the nonlinear state-space model of the flexible satellite attitude system, the nonlinear anti-saturation robust control problem based on a dynamic observer specifically refers to: designing the nonlinear reduced-dimensional dynamic observer to estimate the system state in real time, and constructing the nonlinear state feedback robust controller and anti-saturation compensator, so that the closed-loop system asymptotically stabilizes at the zero equilibrium point under the presence of parameter perturbations and external disturbances, and satisfies... Gain This effectively mitigates the impact of actuator saturation on system performance, ensuring the feasibility of control commands under saturation constraints. The gain is H ∞ Control performance indicators reflect the ability of a closed-loop system to suppress disturbances; is a given constant.

[0153] Further, S3, based on the nonlinear closed-loop control system, deriving the first SOS solvability condition of the state feedback robust controller based on the nonlinear dimension-reduced dynamic observer under the condition of neglecting actuator saturation, and deriving the second SOS solvability condition of the anti-saturation compensator based on the first SOS solvability condition under the condition of considering actuator saturation, includes:

[0154] S3.1, Based on Lyapunov stability theory, the first H of the nonlinear closed-loop control system is established under the condition of neglecting actuator saturation. ∞ Performance criteria, considering actuator saturation, establish the second H of the nonlinear closed-loop control system. ∞ Performance criteria;

[0155] S3.2, according to the first H ∞The performance criteria, derived through variable substitution and the SOS method, establish a robust stability criterion for the state feedback controller based on the nonlinear reduced-dimensional dynamic observer. This criterion ensures that the nonlinear closed-loop control system is asymptotically stable at the zero equilibrium point while neglecting actuator saturation, and satisfies a given condition. The first SOS solvability condition for the gain of the nonlinear dimensionality-reduced dynamic observer and the nonlinear state feedback robust controller;

[0156] S3.3, with the second H ∞ Using the performance criterion as a constraint, and based on the first SOS solvability condition, a second SOS solvability condition is established for the nonlinear anti-saturation robust control problem based on the dynamic observer.

[0157] Furthermore, variable substitution includes:

[0158] The first H ∞ The inequality expansion in the performance criterion, applied with Schur's complement lemma, yields the first H. ∞ The equivalent conditions of the performance criteria are decoupled from the coupling terms of the equivalent conditions by means of variable substitution.

[0159] Furthermore, the first H ∞ The performance criterion is: for a given constant If there exists a continuously differentiable function and ,right satisfy:

[0160] ;

[0161] , , The number of vertices of the convex polyhedron;

[0162] Nonlinear closed-loop system where actuator saturation is not determined It is asymptotically stable at its zero equilibrium point and has -Gain ,in ;

[0163] Second H ∞ The performance criterion is: for a given constant If there exists a continuously differentiable function ,satisfy And for All conditions are met:

[0164] ;

[0165] , ;

[0166] Therefore, a nonlinear closed-loop system with actuator saturation is asymptotically stable at its zero equilibrium point and has the following properties: -Gain ,in ;

[0167] The first SOS solvability condition is: for a given constant If a constant exists and symmetric polynomial matrix symmetric constant matrix , and polynomial matrices , , , and The following conditions must be met:

[0168] (I) ;

[0169] (II) ;

[0170] (III) ;

[0171] (IV) ;

[0172] (V) ;

[0173] in, , , , , , , Represents the SOS polynomial set. , , , , , , , , , .

[0174] Therefore, the aforementioned nonlinear dimensionality-reduced dynamic observer and the aforementioned nonlinear state feedback robust controller can make the closed-loop system asymptotically stable at the zero equilibrium point without considering actuator saturation, and it has... -Gain The observer parameter matrix is:

[0175] ;

[0176] The controller gain matrix is: ;

[0177] in, and It is an invertible matrix and satisfies .

[0178] The solvability condition for the second SOS is: if there exists a constant... , and symmetric polynomial matrix diagonal polynomial matrix and polynomial matrices and Make the following conditions true:

[0179] (i) ;

[0180] (ii) ;

[0181] (iii) ;

[0182] (iv) ;

[0183] in, , , , , , The matrix to be found is... yes The OK, and Let be the matrix to be determined; .

[0184] Then there exists an anti-saturation compensator gain matrix. This makes the nonlinear closed-loop system considering actuator saturation in an ellipsoid... Internal asymptotic stability. Among them, for The OK.

[0185] The specific process of step S3 is as follows.

[0186] First, the first H ∞ Performance criteria and second H ∞ The derivation method of the performance criteria is as follows.

[0187] Specifically, in this step, based on Lyapunov stability theory, the first H can be established.∞ Performance criteria, second H ∞ Performance criteria.

[0188] Second H ∞ Performance criterion: for a given constant If there exists a continuously differentiable function ,satisfy And for All of them are:

[0189] ;

[0190] , ;

[0191] The nonlinear closed-loop system considering actuator saturation is asymptotically stable at its zero equilibrium point and has the following properties: -Gain ,in .

[0192] The proof is as follows: Definition for The Row elements. For a nonlinear closed-loop system considering actuator saturation, we can derive:

[0193] ;

[0194] ;

[0195] Based on the above formula, since and It is easy to know:

[0196] .

[0197] Combined It can be known that when At that time, for and ,have Therefore, the closed-loop system is asymptotically stable at the zero equilibrium point.

[0198] Furthermore, in Under the condition that, apply the above inequality from arrive Integrating, we get:

[0199] ,

[0200] Therefore, it can be concluded that the system has... -Gain Q.E.D.

[0201] First H ∞ Performance criterion: for a given constant If there exists a continuously differentiable function ( ),right satisfy:

[0202] ;

[0203] , ;

[0204] The transformation of the nonlinear closed-loop system without considering actuator saturation is as follows:

[0205] ;

[0206] It is asymptotically stable at its zero equilibrium point and has -Gain ,in .

[0207] The proof is as follows: For a nonlinear closed-loop system that does not consider actuator saturation, it can be seen that:

[0208] ;

[0209] ;

[0210] Subsequent based on the second H ∞ The proof method of the performance criterion leads to the conclusion that this is true.

[0211] Next, in the first H ∞ Based on performance criteria, a robust stability criterion for the state feedback controller based on the reduced-dimensional dynamic observer is derived through variable substitution and the SOS method. This ensures that the closed-loop system, without considering actuator saturation, is asymptotically stable at the zero equilibrium point and satisfies a given condition. Gain observer and controller parameter matrices.

[0212] Then, the derivation process of the first SOS solvability condition and the second SOS solvability condition is as follows.

[0213] The first SOS solvability condition is: for a given constant If a constant exists and symmetric polynomial matrix symmetric constant matrix , and polynomial matrices , , , and satisfy:

[0214] (I) ;

[0215] (II) ;

[0216] (III) ;

[0217] (IV) ;

[0218] (V) ;

[0219] in, , , , , , , Represents the SOS polynomial set. ,

[0220] , ,

[0221] , ,

[0222] , , , , .

[0223] Therefore, the aforementioned nonlinear dimensionality-reduced dynamic observer and the aforementioned nonlinear state feedback robust controller can make the closed-loop system asymptotically stable at the zero equilibrium point without considering actuator saturation, and it has... -Gain The observer parameter matrix is:

[0224] ;

[0225] The controller gain matrix is: ;

[0226] in, and It is an invertible matrix and satisfies .

[0227] The proof is as follows: Since conditions (I)-(IV) hold, it can be concluded that... , , and .

[0228] First, for the closed-loop system that does not consider actuator saturation, the Lyapunov function is selected. ,in and .

[0229] Will and Divide into blocks, denoted as follows:

[0230] , ;

[0231] because Therefore, there is Based on this, we can further define:

[0232] ;

[0233] According to the above formula, it is easy to know Equivalent to:

[0234] .

[0235] Therefore, combining conditions (I)-(IV), we can deduce .

[0236] Next, for a closed-loop system that does not consider actuator saturation, we can conclude that:

[0237] ;

[0238] in , .

[0239] It is easy to see from Schur's supplementary lemma that, in order to make For a statement to be valid, the following conditions must be met:

[0240] ;

[0241] Will Substitute into the above equation and multiply by each on the left. Multiply the right side by its transpose, then let:

[0242] ;

[0243] It can be concluded that Therefore, the fulfillment of condition (V) means... .

[0244] Finally, in summary, it can be concluded that, under the premise of satisfying conditions (I)-(V), there exists a state feedback robust controller based on a reduced-dimensional dynamic observer that enables the closed-loop system without considering actuator saturation to asymptotically stabilize at the zero equilibrium point, and possesses... Gain .

[0245] Furthermore, based on the determined observer and controller parameter matrices, and considering the actuator saturation effect, the second H... ∞ Using performance criteria as constraints, the SOS solvability conditions for the nonlinear anti-saturation robust control problem based on the dynamic observer are established.

[0246] To derive subsequent conclusions, a matrix is ​​defined. and polyhedral sets:

[0247] ;

[0248] in, and They are respectively and The OK.

[0249] The solvability condition for the second SOS is: if there exists a constant... , and symmetric polynomial matrix diagonal polynomial matrix and polynomial matrices and Make the following conditions true:

[0250] (i)

[0251] (ii)

[0252] (iii)

[0253] (iv)

[0254] in, , , , , ,

[0255]

[0256] Then there exists an anti-saturation compensator gain matrix. This makes the nonlinear closed-loop system considering actuator saturation in an ellipsoid... Internal asymptotic stability. Among them, for The OK.

[0257] The proof is as follows: First, since conditions (i)-(iii) hold, it can be seen that... , and ( ).

[0258] right ( Multiply by each side separately and define Then, by Schur's complement lemma, we can conclude that:

[0259] ;

[0260] Therefore, the fulfillment of condition (ii) means .

[0261] Next, from condition (iv), we know that Multiply it by the left and the right respectively. and order and , can be obtained

[0262] ;

[0263] According to Schur's complement lemma, the above equation is equivalent to:

[0264] ;

[0265] Multiply the above equation on the left. Multiplying it by its transpose matrix on the right, we get:

[0266] ;

[0267] Choosing Lyapunov functions It is easy to know For a nonlinear closed-loop system considering actuator saturation, we also have:

[0268] ;

[0269] Finally, combining the two equations above, and by the lemma of the dead-zone function inequality, it is easy to see that:

[0270] .

[0271] In summary, if conditions (i)-(iii) hold, then there exists an anti-saturation compensator such that the nonlinear closed-loop system considering actuator saturation fits within the ellipsoid. Asymptotically stable.

[0272] Further, S4, based on the first SOS solvability condition, the nonlinear reduced-dimensional dynamic observer and the state feedback robust controller are solved using the Matlab SOSTOOLS toolbox while ignoring actuator saturation. Based on the second SOS solvability condition, the anti-saturation compensator is solved using the Matlab SOSTOOLS toolbox while considering actuator saturation, including:

[0273] S4.1, neglecting actuator saturation, given the kinematic and dynamic parameters H of the flexible satellite attitude system... ∞ Performance metrics are defined by setting the dimensions and polynomial orders of the relevant auxiliary matrices used to construct parameters and gain matrices, and setting the symmetric positive definiteness, dimensions and polynomial orders of the matrices to be solved and the relevant auxiliary matrices in the Lyapunov function. Based on the first SOS solvability condition, the nonlinear dimension-reduced dynamic observer and the state feedback robust controller are solved using the SOSTOOLS toolbox of Matlab.

[0274] S4.2, Considering actuator saturation, based on the nonlinear dimension-reduced dynamic observer and the state feedback robust controller, and based on the second SOS solvability condition, the parameter matrix of the nonlinear anti-saturation compensator is further solved using the SOSTOOLS toolbox of Matlab.

[0275] In this step, the specific parameters of the flexible satellite attitude system are as follows:

[0276] , , , ,

[0277] ,

[0278] ( ), , ,

[0279] , Constrained by , ,

[0280] , , and In the convex octahedron formed by these 6 vertices, such as Figure 1 As shown.

[0281] Based on the first SOS solvability condition and the second SOS solvability condition, given H ∞ Performance indicators and simulation parameters ( ; Under the condition of ), set , and Let be a constant matrix, and let... , , , , , and All are second-order polynomial matrices, and this nonlinear anti-saturation robust attitude control problem based on a dynamic observer can be solved using the SOSTOOLS toolbox in Matlab.

[0282] Experimental data

[0283] The partial parameter matrices and gain matrices of the obtained nonlinear dimension-reduced dynamic observer, nonlinear state feedback robust controller, and nonlinear anti-saturation compensator are shown below.

[0284] ;

[0285] ;

[0286] Controller gain matrix ;

[0287] ;

[0288] Observer parameter matrix , , , ;

[0289] ; ;

[0290] ; ;

[0291] Anti-saturation gain matrix ;

[0292] .

[0293] Suppose that the satellite needs to perform an 80° attitude maneuver to complete a specific mission, and its initial state is as follows:

[0294] .

[0295] Simulation studies were conducted for the following three different disturbance and uncertainty conditions:

[0296] i) Nominal system, abbreviated as NS;

[0297] Settings: , ;

[0298] ii) Disturbance-uncertainty coupled system 1, abbreviated as DUCS1;

[0299] Settings ;

[0300] iii) Disturbance-uncertainty coupled system 2, abbreviated as DUCS2;

[0301] Settings ;

[0302] It can be seen that, even in the presence of external interference, model uncertainty, and input saturation, the method provided in this application can still achieve rapid attitude maneuvering of flexible satellites, and their trajectories tend to be consistent with the curves of the nominal system.

[0303] By combining the obtained controller and dynamic observer, a MATLAB / Simulink simulation platform can be built to plot the system trajectory curves under the different conditions described above, as shown in the figure. Figure 2-7 The trajectory.

[0304] Figure 2 External disturbances for perturbation-uncertainty coupled systems A graph, with units of Newton-meters.

[0305] Figure 3 Control input under anti-saturation strategy A curve graph, which has limitations .

[0306] Figure 4 This shows the Rodrigues parameter vector. The trajectory curves demonstrate that the method provided in this application can quickly complete the attitude maneuver of a flexible satellite under external disturbances, uncertainties, and actuator saturation, and the trajectory curves tend to be close to those of the nominal system.

[0307] Figure 5 Attitude angular velocity The trajectory shows that, under the influence of disturbances and uncertainties and with input saturation, the method provided in this embodiment can enable the satellite attitude angular velocity to converge rapidly to a stable state, which is similar to the nominal system trajectory.

[0308] Figure 6 Flexible mode , The trajectory Figure 7 Flexible mode , The trajectory of the first derivative demonstrates that the nonlinear observer provided in this embodiment has a good estimation effect on the system state that cannot be directly measured. Moreover, under the conditions of disturbance, uncertainty and input saturation, it can make the flexible mode of the flexible satellite and its first derivative quickly stabilize and approach the trajectory curve of the nominal system.

[0309] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0310] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0311] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0312] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0313] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., refer to specific features, structures, materials, or characteristics described in connection with that embodiment or example, which are included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms should not be construed as necessarily referring to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

Claims

1. A nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer, characterized in that: Includes the following steps: S1. Establish a flexible satellite attitude system with actuator saturation, some states that are unmeasurable and affected by external disturbances and parameter perturbations, and convert the flexible satellite attitude system into a corresponding nonlinear state-space model. S2. Construct a nonlinear dimension-reduced dynamic observer based on the nonlinear state-space model, construct a nonlinear state feedback robust controller based on the nonlinear dimension-reduced dynamic observer, and construct a nonlinear anti-saturation compensator based on the nonlinear state feedback robust controller to obtain a nonlinear closed-loop control system that can simultaneously handle state unmeasurability, convex polyhedral uncertainty, external disturbances and actuator saturation problems. S3, based on the nonlinear closed-loop control system, derive the first SOS solvability condition of the state feedback robust controller based on the nonlinear reduced-dimensional dynamic observer under the condition of neglecting actuator saturation, and derive the second SOS solvability condition of the anti-saturation compensator based on the first SOS solvability condition under the condition of considering actuator saturation; including: S3.1, Based on Lyapunov stability theory, the first H of the nonlinear closed-loop control system is established under the condition of neglecting actuator saturation. ∞ Performance criteria, considering actuator saturation, establish the second H of the nonlinear closed-loop control system. ∞ Performance criteria; S3.2, according to the first H ∞ The performance criteria, derived through variable substitution and the SOS method, establish a robust stability criterion for the state feedback controller based on the nonlinear reduced-dimensional dynamic observer. This criterion ensures that the nonlinear closed-loop control system is asymptotically stable at the zero equilibrium point while neglecting actuator saturation, and satisfies a given condition. The first SOS solvability condition for the gain of the nonlinear dimensionality-reduced dynamic observer and the nonlinear state feedback robust controller. The gain is H ∞ Control performance indicators; S3.3, with the second H ∞ Using the performance criterion as a constraint, and based on the first SOS solvability condition, a second SOS solvability condition is established for the nonlinear anti-saturation robust control problem based on the dynamic observer. S4. Based on the first SOS solvability condition, the nonlinear reduced-dimensional dynamic observer and the state feedback robust controller are solved using the SOSTOOLS toolbox of Matlab, ignoring actuator saturation. Based on the second SOS solvability condition, the anti-saturation compensator is solved using the SOSTOOLS toolbox of Matlab, considering actuator saturation, based on the nonlinear reduced-dimensional dynamic observer and the state feedback robust controller.

2. The nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer according to claim 1, characterized in that: S1, establishing a flexible satellite attitude system with actuator saturation, partially unmeasurable states, and susceptible to external disturbances and parameter perturbations, and converting the flexible satellite attitude system into a corresponding nonlinear state-space model includes: S1.1, a disturbance torque term and a rotational inertia perturbation term are introduced into the kinematic and dynamic equations of the flexible satellite attitude system to construct a flexible satellite attitude system that includes external disturbances and parameter uncertainties; the flexible satellite attitude system that includes external disturbances and parameter uncertainties is represented as follows: ； in, For the Rodrigues parameter vector, These are the Rodriguez parameters. , The attitude angular velocity vector, These are the three orthogonal components of the attitude angular velocity vector. Here is the rotational inertia matrix. For the perturbation term of the rotational inertia matrix, for The corresponding cross product matrix, The rigid-flexible coupling coefficient matrix is... For the flexible modal coordinate vector, Indicates the modal order being intercepted. Here is the modal damping matrix. The natural frequency matrix, and These are the control torque and the disturbance torque, respectively. S1.2, by introducing an auxiliary vector and ,limit This imposes control input constraints, retains the nonlinear terms in the flexible satellite attitude system, and imposes different restrictions on the actual control input in different intervals when the actuator input signal exceeds a preset saturation threshold. Simultaneously, the parameter perturbation term is represented as a convex polyhedral uncertainty, thereby reconstructing the flexible satellite attitude system into a corresponding nonlinear state-space model. The nonlinear state-space model is expressed as follows: ; in, For system status, and They are Measurable and unmeasurable substates; , and These are the actual control input, the controlled output, and the measurement output, respectively. It is external interference and ; and They represent The set of dimensional real vectors and 3D real matrix set; Represents an uncertain vector The Each component; define the following polyhedron set: , , To be constrained by vertices , , , The vectors that form the convex polyhedral region, for abbreviation; It is the first of the system state matrix. A matrix of uncertain sub-components, It is the first of the system input matrix and interference matrix. A matrix of uncertain sub-components, , , , ; and These represent the identity matrix and the zero matrix, respectively.

3. The nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer according to claim 2, characterized in that: When the actuator's input signal exceeds a preset saturation threshold, different restrictions are applied to the actual control input in different ranges, including: When the actuator input signal Exceeding the preset saturation threshold At that time, actual control input for: ， ； in, express The OK, express The OK, .

4. The nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer according to claim 3, characterized in that: S2, construct a nonlinear dimensionality-reduced dynamic observer based on the nonlinear state-space model, design a nonlinear state feedback robust controller based on the nonlinear dimensionality-reduced dynamic observer, and construct a nonlinear anti-saturation compensator based on the nonlinear state feedback robust controller to obtain a nonlinear closed-loop control system capable of simultaneously handling state unmeasurability, convex polyhedral uncertainty, external disturbances, and actuator saturation problems, including: S2.1, Based on the input signal and measurable output signal of the nonlinear state-space model, construct the nonlinear dimensionality-reduced dynamic observer; the constructed nonlinear dimensionality-reduced dynamic observer is as follows: ； in, and As an auxiliary variable, yes The estimated quantity, State variables representing observer gain , , and This is the observer parameter matrix to be designed; express The OK; S2.2, using the state estimation information provided by the nonlinear dimensionality-reduced dynamic observer, a nonlinear state feedback robust controller is designed; the output signal of the designed nonlinear state feedback robust controller is used as the input signal of the actuator. The result is as follows: ； in, , , This is the observation error; Let be the controller gain matrix to be determined; S2.3, construct the nonlinear anti-saturation compensator by measuring the deviation between the actuator input and the actual output; the constructed nonlinear anti-saturation compensator as follows: ； in, For the output of the compensator, The compensator gain matrix, Dead-zone function; S2.4, the nonlinear dimensionality-reduced dynamic observer, the nonlinear state feedback robust controller, and the nonlinear anti-saturation compensator are collaboratively integrated into the nonlinear state-space model to form a nonlinear closed-loop system that combines state estimation, robust stability, and anti-saturation capability; the nonlinear closed-loop system is as follows: ； in, , , , , , , , , .

5. The nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer according to claim 4, characterized in that: The design of a nonlinear state feedback robust controller using the state estimation information provided by the nonlinear dimensionality-reduced dynamic observer includes: designing the nonlinear dimensionality-reduced dynamic observer to estimate the state of the flexible satellite attitude system in real time, and constructing the nonlinear state feedback robust controller. The overall control objective of the nonlinear dimensionality-reduced dynamic observer, the nonlinear state feedback robust controller, and the nonlinear anti-saturation compensator is: under the presence of parameter perturbations and external disturbances, to make the nonlinear closed-loop control system considering actuator saturation asymptotically stable at the zero equilibrium point, and to satisfy the following conditions: Gain This also mitigates the impact of actuator saturation on system performance, among which, is a given constant.

6. The nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer according to claim 1, characterized in that: Variable substitution includes: The first H ∞ The inequality expansion in the performance criterion, applied with Schur's complement lemma, yields the first H. ∞ The equivalent conditions of the performance criteria are decoupled from the coupling terms of the equivalent conditions by means of variable substitution.

7. The nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer according to claim 6, characterized in that: First H ∞ The performance criterion is: for a given constant If there exists a continuously differentiable function and ,right satisfy: ； , , The number of vertices of the convex polyhedron; Nonlinear closed-loop system where actuator saturation is not determined It is asymptotically stable at its zero equilibrium point and has -Gain ,in ; Second H ∞ The performance criterion is: for a given constant If there exists a continuously differentiable function ,satisfy And for All conditions are met: ； ， ； Therefore, a nonlinear closed-loop system with actuator saturation is asymptotically stable at its zero equilibrium point and has the following properties: -Gain ,in ; The first SOS solvability condition is: for a given constant If a constant exists and symmetric polynomial matrix symmetric constant matrix , and polynomial matrices , , , and The following conditions must be met: (I) ； (II) ; (III) ; (IV) ; (V) ； in, , , , , , , Represents the SOS polynomial set. , , , , , , , , , ; Therefore, the aforementioned nonlinear dimensionality-reduced dynamic observer and the aforementioned nonlinear state feedback robust controller can make the closed-loop system asymptotically stable at the zero equilibrium point without considering actuator saturation, and it has... -Gain The observer parameter matrix is: ； The controller gain matrix is: ; in, and It is an invertible matrix and satisfies ; The solvability condition for the second SOS is: if there exists a constant... , and symmetric polynomial matrix diagonal polynomial matrix and polynomial matrices and Make the following conditions true: (i) ； (ii) ; (iii) ; (iv) ; in, , , , , , The matrix to be found is... yes The OK, and Let be the matrix to be determined; ; Then there exists an anti-saturation compensator gain matrix. This allows the nonlinear closed-loop system to operate within an ellipsoid, considering actuator saturation. Internal asymptotic stability, in which, for The OK.

8. The nonlinear anti-saturation robust control method for a flexible satellite attitude system based on a dynamic observer according to claim 7, characterized in that: S4, based on the first SOS solvability condition, the nonlinear reduced-dimensional dynamic observer and the state feedback robust controller are solved using the Matlab SOSTOOLS toolbox while ignoring actuator saturation. Based on the second SOS solvability condition, the anti-saturation compensator is solved using the Matlab SOSTOOLS toolbox while considering actuator saturation, including: S4.1, neglecting actuator saturation, given the kinematic and dynamic parameters H of the flexible satellite attitude system... ∞ Performance metrics are defined by setting the dimensions and polynomial orders of the relevant auxiliary matrices used to construct parameters and gain matrices, and setting the symmetric positive definiteness, dimensions and polynomial orders of the matrices to be solved and the relevant auxiliary matrices in the Lyapunov function. Based on the first SOS solvability condition, the nonlinear dimension-reduced dynamic observer and the state feedback robust controller are solved using the SOSTOOLS toolbox of Matlab. S4.2, Considering actuator saturation, based on the nonlinear dimension-reduced dynamic observer and the state feedback robust controller, and based on the second SOS solvability condition, the parameter matrix of the nonlinear anti-saturation compensator is further solved using the SOSTOOLS toolbox of Matlab.

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