CN117572780B - Self-adaptive fault-tolerant control method for flexible spacecraft faults - Google Patents
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Abstract
The invention relates to a self-adaptive fault-tolerant control method for flexible spacecraft faults, which comprises the steps of constructing an interconnected ODE-PDE ordinary differential equation-partial differential equation model of a flexible spacecraft system under the conditions of boundary executors and structural composite faults; parameterizing the boundary actuator offset fault and the structure fault to obtain an unknown constant vector caused by the actuator and the structure composite fault; designing a fault-tolerant controller, and compensating the composite fault through boundary control signals and distributed control signals; and designing an adaptive parameter updating law for the fault-tolerant controller to dynamically adjust the parameters of the fault-tolerant controller, and verifying the progressive stability of the flexible spacecraft attitude control closed-loop system through a Lyapunov function. The invention combines the self-adaptive control and the fault-tolerant control, and solves the problem of difficult control of the interconnected ODE-PDE system under fault.
Description
Technical Field
The invention relates to the technical field of spacecraft control, in particular to a self-adaptive fault-tolerant control method for flexible spacecraft faults.
Background
The flexible spacecraft is a type of spacecraft with large flexible accessories and complex structures in the field of modern aerospace. Compared with a rigid structure, the flexible structure has light weight and good ductility, and can adapt to various complex application environments. In recent years, flexible spacecraft have been widely used in the fields of communication, remote sensing, and the like. However, these characteristics of the flexible structure also raise a non-negligible problem, namely the problem of elastic vibration of the flexible attachment. Elastic vibration may adversely affect the attitude of the spacecraft, resulting in failure of scientific tasks and even crashes of the spacecraft. Therefore, studies on vibration suppression and attitude control of flexible spacecraft have received a great deal of attention.
In order to simplify the research problem, some students often use the ordinary differential equation ODE to approximately describe the flexible vibration characteristics of the spacecraft system. Although their research results bring about a certain inspiring, ODE only considers the first-order coupling characteristic of the vibration mode, but ignores the higher-order coupling term, so that the model cannot accurately describe the flexible vibration characteristic of the spacecraft and can bring about overflow instability.
At present, the control method of the flexible spacecraft mainly aims at the problems of external interference, input constraint and the like of a system, and the problem of system faults is freshly researched. The flexible spacecraft works in an outer space environment with a severe environment for a long time, and the system failure occurrence rate is high. Therefore, a control method with tolerance capability to system faults is needed to be designed, so that stable operation of the spacecraft can be ensured under the conditions of system faults and anomalies, and serious accidents are avoided.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a self-adaptive fault-tolerant control method for faults of a flexible spacecraft, which solves the problems that the traditional method cannot accurately describe the flexible vibration characteristics of the spacecraft, overflow instability is caused, and the fault occurrence rate of an external space environment system with a severe environment of the flexible spacecraft is high, so that the control of the fault system is difficult. According to the invention, the flexible spacecraft attitude control system is modeled by adopting the interconnected ODE-PDE, so that the limitation of most of the existing achievements based on ODE modeling is broken, the model is more accurate, and the overflow instability problem is avoided; the invention researches the fault-tolerant control problem of the flexible spacecraft system under the composite fault of the actuator and the structure, combines the self-adaptive control and the fault-tolerant control, solves the problem of difficult control of the interconnected ODE-PDE system under the fault, provides a theoretical basis for safe and reliable operation of the flexible spacecraft, and also provides references and ideas for the fault-tolerant control research of other complex system engineering.
In order to solve the technical problems, the invention provides the following technical scheme: an adaptive fault-tolerant control method for flexible spacecraft faults comprises the following steps:
S1, based on Hamiltonian principle, respectively designing flexible accessories at boundary and inside the PDE system in time Boundary actuator offset failure/>、/>And in space/>And time/>Structural failure/>Thus constructing an interconnected ODE-PDE ordinary differential equation-partial differential equation model of the flexible spacecraft system under the composite fault of the boundary actuator and the structure;
S2, offsetting the boundary actuator 、/>And structural failure/>Parameterizing to obtain an unknown constant vector/>, which is caused by the composite faults of the actuator and the structure、/>And/>；
S3, passing through unknown constant vector，/>And/>To design a fault tolerant controller via boundary control signals/> 、/>Distributed control signal/>Compensating the composite fault;
S4, designing an adaptive parameter update law for the fault-tolerant controller to dynamically adjust the parameters of the fault-tolerant controller, thereby improving the robustness of the system;
S5, verifying progressive stability of the flexible spacecraft attitude control closed-loop system through the Lyapunov function.
Further, the flexible attachment is designed at step S1 at the boundary of the PDE system and inside the PDE system in time by the dynamics equation and the boundary equationBoundary actuator offset failure/>、/>And in space/>And time/>Structural failure/>Wherein the dynamic equation and the boundary equation are expressed as follows:
Kinetic equation:
；
；
Boundary equation:
；
；
Wherein, Space/>, for flexible accessoriesAnd time/>Flexible vibration in (a); /(I)Uniform mass per unit length; /(I)Is bending rigidity; /(I)Is tension; /(I)Is a viscous damping coefficient; /(I)The radius of the rigid body hub is; /(I)Is the attitude angle of the spacecraft; /(I)The rotational inertia of the hub is the rotational inertia of the hub; /(I)Is flexible accessory length; /(I)Is the end mass; /(I)And/>For actuator offset failure,/>Is a structural failure; /(I)And/>Is boundary control signal,/>Is a distributed control signal.
Further, in step S2, the boundary actuator shifts to failure、/>The parameters can be as follows:
；
Wherein, And/>Representing unknown constant vector,/>Representing a known basis function vector,/>Representing a column vector consisting of n real numbers;
The structural failure may be parameterized as: ；
Wherein, Representing unknown constant vector,/>And/>Representing a known basis function vector,/>Representing an n-th order matrix, i.e. having both rows and columns of n,/>Representing a row vector consisting of n real numbers,/>Representing a column vector consisting of n real numbers.
Further, in step S3, the passing unknown constant vector，/>And/>To design a fault tolerant controller via boundary control signals/> 、/>Distributed control signal/>Compensating the composite fault, wherein the specific process comprises the following steps:
S31, defining an attitude tracking error Let/>For the desired attitude angle of the spacecraft, then/>；
S32, defining total vibration offset of the flexible spacecraft system，Intermediate variable/>；
S33, estimating an unknown constant vector through an estimator，/>And/>Estimate of/>，/>And/>And introduce auxiliary items/>And/>，/>，/>; Thereby constructing an adaptive fault tolerant controller, wherein/>And/>Representing a normal number to be designed;
S34, boundary control signals passing through the fault-tolerant controller 、/>Distributed control signal/>Compensating for a composite fault, wherein the boundary control signal/> 、/>Distributed control signal/>The formula of (2) is as follows:
；
；
；
Wherein, ，/>And/>Respectively representing control gains to be designed;
s35, defining parameter estimation errors as 、/>And/>Wherein/>，，/>。
Further, in step S4, the formula for designing the adaptive parameter update law is as follows:
；
；
。
Further, in step S5, the specific process of verifying the progressive stability performance of the closed loop system for attitude control of the flexible spacecraft by using the lyapunov function includes the following steps:
S51, selecting Lyapunov candidate function Obtain/>Whereby the following quotation 1 is possible: if positive constant/>，/>，/>，/>And/>Satisfy inequality/>Then Lyapunov candidate function/>Is positive;
s52, pair Differentiation is performed, and the system model is combined, and the/>, is known by the quotients 1Constant true, then/>Whereby the following quotation 2 is possible: under the lemma 1, for the flexible spacecraft system under the boundary actuator and structure composite fault in the S1, designing a fault-tolerant controller in the S3 and an adaptive parameter updating law in the S4, and a Lyapunov candidate function/>Is semi-negative;
s53, performing progressive stability analysis according to Poincare inequality and boundary conditions, and further obtaining: When/> Time/>By the following constitutionGet/>The closed loop system is known to be asymptotically stable using the extended LaSalle invariance principle. Aiming at the flexible spacecraft attitude control system under the composite fault, the self-adaptive fault-tolerant control strategy provided by the invention can ensure good vibration suppression and attitude tracking performance. So far, it has been demonstrated that the adaptive fault-tolerant control algorithm is theoretically effective.
By means of the technical scheme, the invention provides a self-adaptive fault-tolerant control method for flexible spacecraft faults, which has at least the following beneficial effects:
In order to accurately describe dynamics of the flexible spacecraft system and simultaneously consider a flexible vibration model and rigid motion, the invention adopts the interconnected ODE-PDE to model the flexible spacecraft attitude control system, breaks the limitation of most of the prior achievements based on ODE modeling, ensures that the model is more accurate, and avoids the problem of overflow instability; the invention researches the fault-tolerant control problem of the flexible spacecraft system under the composite fault of the actuator and the structure, combines the self-adaptive control and the fault-tolerant control, solves the problem of difficult control of the interconnected ODE-PDE system under the fault, provides a theoretical basis for safe and reliable operation of the flexible spacecraft, and also provides references and ideas for the fault-tolerant control research of other complex system engineering.
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The accompanying drawings, which are included to provide a further understanding of the application and are incorporated in and constitute a part of this specification, illustrate embodiments of the application and together with the description serve to explain the application and do not constitute a limitation on the application. In the drawings:
FIG. 1 is a schematic flow diagram of an adaptive fault-tolerant control method for a composite fault of a flexible spacecraft;
FIG. 2 is a simplified model diagram of a flexible spacecraft system of the present invention;
FIG. 3 is a vibration displacement of a spacecraft of the present invention;
fig. 4 is a spacecraft attitude tracking error of the present invention.
Detailed Description
In order that the above-recited objects, features and advantages of the present application will become more readily apparent, a more particular description of the application will be rendered by reference to the appended drawings and appended detailed description. Therefore, the realization process of how to apply the technical means to solve the technical problems and achieve the technical effects can be fully understood and implemented.
Those of ordinary skill in the art will appreciate that all or a portion of the steps in a method of implementing an embodiment described above may be implemented by a program to instruct related hardware, and thus, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present application may 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, and the like) having computer-usable program code embodied therein.
Referring to fig. 1-4, a specific implementation manner of the present embodiment is shown, and the present embodiment considers the flexible vibration model and the rigid motion at the same time, and the present invention uses the interconnected ODE-PDE to model the attitude control system of the flexible spacecraft, so as to break the limitation of the existing achievements based on ODE modeling, make the model more accurate, and avoid the overflow instability problem; the invention researches the fault-tolerant control problem of the flexible spacecraft system under the composite fault of the actuator and the structure, combines the self-adaptive control and the fault-tolerant control, solves the problem of difficult control of the interconnected ODE-PDE system under the composite fault, provides a theoretical basis for safe and reliable operation of the flexible spacecraft, and also provides references and ideas for fault-tolerant control research of other complex system engineering. For convenience of presentation, in the present invention, it is specified that:，/>，/>，/>，/>；
Referring to fig. 1, the embodiment provides an adaptive fault-tolerant control method for a flexible spacecraft fault, which includes the following steps:
S1, based on Hamiltonian principle, respectively designing flexible accessories at boundary and inside the PDE system in time Boundary actuator offset failure/>、/>And in space/>And time/>Structural failure/>Thus constructing an interconnected ODE-PDE ordinary differential equation-partial differential equation model of the flexible spacecraft system under the composite fault of the boundary actuator and the structure;
As a preferred embodiment of step S1, in step S1, the flexible attachment is designed at time by means of the dynamics equation and the boundary equation at the boundary of the PDE system and inside the PDE system Boundary actuator offset failure/>、/>And in space/>And time/>Structural failure/>Wherein the dynamic equation and the boundary equation are expressed as follows:
Kinetic equation:
；
；
Boundary equation:
；
；
Wherein, Space/>, for flexible accessoriesAnd time/>Flexible vibration in (a); /(I)Uniform mass per unit length; /(I)Is bending rigidity; /(I)Is tension; /(I)Is a viscous damping coefficient; /(I)The radius of the rigid body hub is; /(I)Is the attitude angle of the spacecraft; /(I)The rotational inertia of the hub is the rotational inertia of the hub; /(I)Is flexible accessory length; /(I)Is the end mass; /(I)And/>For actuator offset failure,/>Is a structural failure; /(I)And/>Is boundary control signal,/>For distributed control signals, a simplified model diagram of the flexible spacecraft system of the invention is shown in fig. 2.
In this embodiment, the attitude control system of the flexible spacecraft is characterized by an interconnected ODE-PDE, the boundary actuator failure occurs at the boundary of the PDE system, and the structural failure occurs inside the PDE system. In order to accurately describe dynamics of the flexible spacecraft system and consider the flexible vibration model and rigid motion, the invention adopts the interconnected ODE-PDE to model the flexible spacecraft attitude control system, breaks the limitation of most of existing achievements based on ODE modeling, ensures that the model is more accurate, and avoids the problem of overflow instability.
S2, offsetting the boundary actuator、/>And structural failure/>Parameterizing to obtain an unknown constant vector/>, which is caused by the composite faults of the actuator and the structure、/>And/>；
As a preferred embodiment of step S2, in step S2, the boundary actuator shifts to failure、/>The parameters can be as follows:
；
Wherein, And/>Representing unknown constant vector,/>Representing a known basis function vector,/>Representing a column vector consisting of n real numbers;
Structural failure The parameters can be as follows: /(I)；
Wherein,Representing unknown constant vector,/>And/>Representing a known basis function vector,/>Representing an n-th order matrix, i.e. having both rows and columns of n,/>Representing a row vector consisting of n real numbers,/>Representing a column vector consisting of n real numbers.
S3, passing through unknown constant vector，/>And/>To design a fault tolerant controller via boundary control signals/> 、/>Distributed control signal/>Compensating the composite fault;
as a preferred embodiment of step S3, the passing of the unknown constant vector ，/>And/>To design a fault tolerant controller via boundary control signals/> 、/>Distributed control signal/>Compensating the composite fault, wherein the specific process comprises the following steps:
S31, defining an attitude tracking error Let/>For the desired attitude angle of the spacecraft, then/>；
S32, defining total vibration offset of the flexible spacecraft system，/>Intermediate variable/>；
S33, estimating an unknown constant vector through an estimator，/>And/>Estimate of/>，/>And/>And introduce auxiliary items/>And/>，/>，/>; Thereby constructing an adaptive fault tolerant controller, wherein/>And/>Representing a normal number to be designed;
S34, boundary control signals passing through the fault-tolerant controller 、/>Distributed control signal/>Compensating for a composite fault, wherein the boundary control signal/> 、/>Distributed control signal/>The formula of (2) is as follows:
；
；
；
Wherein, ，/>And/>Respectively representing control gains to be designed;
s35, defining parameter estimation errors as 、/>And/>Wherein/>，，/>。
In the embodiment, the invention defines the attitude tracking error, the intermediate variable and the auxiliary item, combines the self-adaptive control and the fault-tolerant control, researches the fault-tolerant control problem of the flexible spacecraft system under the composite fault of the actuator and the structure, solves the control difficulty problem of the interconnected ODE-PDE system under the composite fault, provides a theoretical basis for the safe and reliable operation of the flexible spacecraft, and also provides references and ideas for the fault-tolerant control research of other complex system engineering.
S4, designing an adaptive parameter update law for the fault-tolerant controller to dynamically adjust the parameters of the fault-tolerant controller, thereby improving the robustness of the system;
As a preferred embodiment of step S4, in step S4, the formula for designing the adaptive parameter update law is as follows:
；
；
。
S5, verifying progressive stability of the flexible spacecraft attitude control closed-loop system through the Lyapunov function.
As a preferred embodiment of step S5, in step S5, the specific process of verifying the progressive stability performance of the closed loop system for attitude control of the flexible spacecraft by means of the lyapunov function includes the following steps:
S51, selecting a Lyapunov candidate function: Wherein:
；
；
；
；
According to inequality Know/>Wherein/>Then, the first and second processes, respectively,
；
The following quotation can be made:
lemma 1: if a positive constant ，/>，/>，/>And/>Satisfy inequality/>Then Lyapunov candidate function/>Is positive.
S52, pairDifferentiation is carried out: /(I)Combining the system model to obtain:
；
Known from the quotation mark 1 Constant true, then/>The lemma for realizing the stability analysis of the flexible spacecraft is proposed, and the specific contents are as follows:
And (4) lemma 2: under the lemma 1, for the flexible spacecraft system under the boundary actuator and structure composite fault in the S1, designing a fault-tolerant controller in the S3 and an adaptive parameter updating law in the S4, and a Lyapunov candidate function Is semi-negative;
s53, progressive stability analysis: according to the poincare inequality and boundary conditions, there are:
；
Wherein, Is a positive scalar that needs to be given. Thus, it is possible to further obtain:
；
Selected parameters And/>The following should be satisfied: /(I)，/>. When/>When the method is used, the following steps are included:
；
From the following components Get/>。
In this embodiment, the present invention exploits the extended LaSalle invariance principle, knowing that the closed loop system is asymptotically stable. Therefore, aiming at the flexible spacecraft attitude control system under the composite fault, the self-adaptive fault-tolerant control strategy provided by the invention can ensure good vibration suppression and attitude tracking performance. So far, it has been demonstrated that the adaptive fault-tolerant control algorithm is theoretically effective.
Next, the adaptive fault-tolerant control method for the composite fault of the flexible spacecraft performs a simulation experiment in MATLAB:
Selecting a space step Time step/>. Parameters in the flexible spacecraft attitude control system are respectively as follows: /(I)，/>，/>，/>，/>，/>，，/>. Actuator offset failure/>，/>Wherein/>，/>，/>; Structural failure/>Wherein, the method comprises the steps of, wherein,，/>，/>. Furthermore, the initial conditions of the spacecraft system are as follows: /(I)，，/>，/>. The adaptive initial values of the fault parameters are: /(I)，/>，. The control parameters in the self-adaptive boundary fault-tolerant controller are respectively selected as follows: /(I)，/>，/>，，/>，/>. Desired attitude angle of spacecraft/>。
The self-adaptive fault-tolerant control method for the composite fault of the flexible spacecraft is simulated according to the designed parameters, and simulation results are shown in fig. 3 and 4. Fig. 3 is a vibration displacement of the spacecraft, and fig. 4 is a posture tracking error of the spacecraft. From the two figures, it can be found that the vibration displacement and the attitude tracking error of the flexible spacecraft gradually converge to the original point. This shows that the method can be applied to the flexible spacecraft system under the composite faults of the actuator and the structure, and can achieve the aims of vibration suppression and attitude tracking control. The simulation result verifies the real-time effectiveness of the design method of the invention, and expands a new visual angle for subsequent researches.
In the description of the present specification, a description referring to terms "one embodiment," "some embodiments," "examples," "specific examples," or "some examples," etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the present application. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples. Furthermore, the different embodiments or examples described in this specification and the features of the different embodiments or examples may be combined and combined by those skilled in the art without contradiction.
Logic and/or steps represented in the flowcharts or otherwise described herein, e.g., a ordered listing of executable instructions for implementing logical functions, can be embodied in any computer-readable medium for use by or in connection with an instruction execution system, apparatus, or device, such as a computer-based system, processor-containing system, or other system that can fetch the instructions from the instruction execution system, apparatus, or device and execute the instructions.
Claims (4)
1. The self-adaptive fault-tolerant control method for the flexible spacecraft fault is characterized by comprising the following steps of:
S1, based on Hamiltonian principle, respectively designing a boundary actuator offset fault F 1(t)、f2 (t) of a flexible accessory in a time t and a structural fault F (x, t) in a space x and a time t at the boundary of the PDE system and in the PDE system through a dynamic equation and a boundary equation, so as to construct an interconnected ODE-PDE ordinary differential equation-partial differential equation model of the flexible spacecraft system under the boundary actuator and structural composite fault; the kinetic and boundary equations are expressed as follows:
Kinetic equation:
Boundary equation:
ω(0,t)＝ω′(0,t)＝ω″(l,t)＝0；
Wherein ω (x, t) is the flexible vibration of the flexible accessory in space x and time t; ρ is the uniform mass per unit length; EI is bending stiffness; t is tension; gamma is the viscous damping coefficient; r is the radius of the rigid hub; θ (t) is the attitude angle of the spacecraft; i h is the rotational inertia of the hub; l is the flexible accessory length; m is the end mass; f 1 (t) and F 2 (t) are actuator offset faults, and F (x, t) is a structural fault; mu 1 (t) and mu 2 (t) are boundary control signals, Is a distributed control signal;
S2, parameterizing the boundary actuator offset fault F 1(t)、f2 (t) and the structural fault F (x, t) to obtain an unknown constant vector caused by actuator and structural composite faults And Λ *; wherein, the structural failure F (x, t) can be parameterized as:
F(x,t)＝g0(x)Λ*TΦ0(t)；
where Λ *∈Rn×n represents an unknown constant vector, g 0(x)∈R1×n and Φ 0(t)∈Rn×1 represent known basis function vectors, R n×n represents an n-order square matrix, i.e. the number of rows and columns are n, R 1×n represents a row vector consisting of n real numbers, and R n×1 represents a column vector consisting of n real numbers;
s3, passing through unknown constant vector And Λ * to design a fault tolerant controller that compensates for the composite fault by boundary control signal μ 1(t)、μ2 (t) and distributed control signal τ (x, t); wherein the distributed control signalThe formula of (2) is as follows: /(I)Λ (t) is an estimated value of the unknown constant vector Λ * estimated by an estimator;
S4, designing an adaptive parameter update law for the fault-tolerant controller to dynamically adjust the parameters of the fault-tolerant controller, thereby improving the robustness of the system; the design of the self-adaptive parameter updating law specifically comprises the following steps:
Wherein Φ (t) ∈r n×1 represents a known basis function vector, S 1 (t) and S 2 (t) are auxiliary terms, y (x, t) is a total vibration offset of the flexible spacecraft system, y e (x, t) is an intermediate variable, and α >0 and β >0 represent normal numbers to be designed;
s5, verifying the stability of the attitude control closed-loop system of the flexible spacecraft through the Lyapunov function.
2. The adaptive fault-tolerant control method for flexible spacecraft failure according to claim 1, wherein: in step S2, the boundary actuator offset fault f 1(t)、f2 (t) may be parameterized as:
3. The adaptive fault-tolerant control method for flexible spacecraft failure according to claim 1, wherein: in step S3, the pass through unknown constant vector And Λ * to design a fault tolerant controller by boundary control signal μ 1(t)、μ2 (t) and distributed control signal/>Compensating the composite fault, wherein the specific process comprises the following steps:
S31, defining an attitude tracking error e (t), and setting θ d as a desired attitude angle of the spacecraft, wherein e (t) =θ (t) - θ d;
S32, defining the total vibration offset y (x, t), y (x, t) = (r+x) theta (t) +omega (x, t), and then the intermediate variable y e (x, t) = (r+x) e (t) +omega (x, t);
S33, estimating an unknown constant vector through an estimator And the estimated values θ 1(t),θ2 (t) and Λ (t) of Λ *, and introducing the auxiliary terms S 1 (t) and S 2 (t),/>Thereby constructing an adaptive fault-tolerant controller, wherein alpha > 0 and beta > 0 represent normal numbers to be designed;
s34, compensating for a composite fault through a boundary control signal mu 1(t)、μ2 (t) and a distributed control signal tau (x, t) of the fault-tolerant controller, wherein the formula of the boundary control signal mu 1(t)、μ2 (t) is as follows:
Wherein k m＞0,kp > 0 and k f > 0 respectively represent control gains to be designed;
s35, defining parameter estimation errors as And/>Wherein/>
4. The adaptive fault-tolerant control method for flexible spacecraft failure according to claim 1, wherein: in step S5, the specific process of verifying the progressive stability performance of the closed loop system for attitude control of the flexible spacecraft by using the lyapunov function includes the following steps:
S51, selecting a Lyapunov candidate function E (t) to obtain E (t) > 0, thereby obtaining the following lements 1: if the positive constants delta, alpha, beta, gamma and rho satisfy the inequality The lyapunov candidate function E (t) is positive;
s52, differentiating E (t), and combining a system model, wherein when the quotation 1 shows that alpha gamma-beta rho > 0 is constant, the method is The following quotation 2 can be obtained: under the lemma 1, for the flexible spacecraft system under the boundary executor and structure composite fault in the S1, designing a fault-tolerant controller in the S3 and an adaptive parameter updating law in the S4, wherein the time derivative of the Lyapunov candidate function E (t) is semi-negatively determined;
S53, performing stability analysis according to Poincare inequality and boundary conditions, and further obtaining: When/> Time/>From the following componentsGet/>The closed loop system is known to be stable using the extended LaSalle invariance principle.
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| CN113885547A (en) * | 2021-10-20 | 2022-01-04 | 河北工业大学 | Fault-tolerant attitude control strategy for rigid spacecraft in preset time |
| CN115079574A (en) * | 2022-07-19 | 2022-09-20 | 安徽大学 | Distributed fault compensation method for flexible hypersonic aircraft |
| CN116923730A (en) * | 2023-07-24 | 2023-10-24 | 哈尔滨工业大学 | Spacecraft attitude active fault-tolerant control method with self-adjusting preset performance constraint |
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| CN113885547A (en) * | 2021-10-20 | 2022-01-04 | 河北工业大学 | Fault-tolerant attitude control strategy for rigid spacecraft in preset time |
| CN115079574A (en) * | 2022-07-19 | 2022-09-20 | 安徽大学 | Distributed fault compensation method for flexible hypersonic aircraft |
| CN116923730A (en) * | 2023-07-24 | 2023-10-24 | 哈尔滨工业大学 | Spacecraft attitude active fault-tolerant control method with self-adjusting preset performance constraint |
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