CN117262245A - Flexible spacecraft stability control method under unknown harmonic interference - Google Patents

Abstract

The invention relates to the technical field of spacecraft control systems, in particular to a flexible spacecraft stability control method under unknown harmonic interference, which comprises the steps of describing the unknown harmonic interference by using an exogenous system, converting the unknown harmonic interference into an expression with an unknown factor by using an auxiliary filter, constructing a self-adaptive interference observer for estimating the unknown factor to obtain an unknown harmonic interference error dynamic model, constructing a flexible vibration observer for estimating a flexible mode to obtain a flexible mode error dynamic model, combining the flexible mode error dynamic model with a dynamic event triggering mechanism, constructing a Lyapunov function, controlling the derivative of the Lyapunov function to be smaller than a threshold value, ensuring the limitation of all closed-loop signals, and ensuring the stable operation of the flexible spacecraft by gradually converging the state of the system and the error of the flexible vibration observer to a balance point.

Description

A stable control method for flexible spacecraft under unknown harmonic disturbance

Technical Field

The invention relates to the technical field of spacecraft control systems, and in particular to a method for stabilizing a flexible spacecraft under unknown harmonic interference.

Background Art

In recent years, with the diversification of space mission objectives, higher requirements have been placed on the on-orbit service life and mission accuracy of spacecraft. In this case, in order to improve energy efficiency and save launch costs, spacecraft use lightweight flexible materials. However, the increase in flexible structures produces elastic vibrations, and continuous vibrations affect the accuracy of attitude control of flexible spacecraft systems. The inertial parameters of spacecraft are usually not accurately known, and there is uncertainty in the parameters of the dynamic model. In addition, the space environment is complex, and spacecraft will also be affected by solar pressure torque, gravity gradient torque, geomagnetic torque, and aerodynamic torque, and these interference torques are also uncertain. All these uncertainties will affect the performance of the spacecraft attitude control system. Especially when the spacecraft is in operation, harmonic interference will reduce or destroy the working performance of the electronic equipment on the spacecraft, causing system instability.

Common methods for attitude stability control of flexible spacecraft include sliding mode control, H. control, control based on disturbance observer and feedback control. Among them, control based on disturbance observer is widely used in spacecraft systems due to its superior performance and simple structure, and has been extended to composite hierarchical anti-interference control of multiple interference systems. However, in practical applications, it is easily disturbed by unknown information, which increases the error and causes unstable operation of flexible spacecraft.

Summary of the invention

The purpose of the present invention is to provide a method for stabilizing and controlling a flexible spacecraft under unknown harmonic interference.

The technical solution of the present invention is as follows:

A method for stabilizing a flexible spacecraft under unknown harmonic disturbances includes the following operations:

S1. Based on the kinematic model and dynamic model of the flexible spacecraft and unknown harmonic interference, a fuzzy dynamic model of the flexible spacecraft is constructed;

S2. The unknown harmonic interference in the flexible spacecraft fuzzy dynamic model is processed by target conversion to obtain an unknown harmonic interference expression; an adaptive interference observer is preset to estimate the unknown factors in the unknown harmonic interference expression to obtain an unknown factor estimate; based on the unknown factor estimate, an unknown harmonic interference error dynamics model is obtained;

A flexible vibration observer is preset to estimate the flexible mode in the dynamic model to obtain a flexible mode estimation; based on the flexible mode estimation, a flexible mode error dynamic model is obtained;

S3. The unknown harmonic interference error dynamics model and the flexible modal error dynamics model are combined with the event trigger mechanism based on the fuzzy controller and the adaptive law to construct a Lyapunov function, and the derivative of the Lyapunov function is obtained by differentiation; the range of the derivative of the Lyapunov function is controlled not to exceed a threshold value to achieve stable control of the flexible spacecraft.

The unknown harmonic interference in S1 can be obtained by the following formula:

d _j (t) is the component of the unknown harmonic interference d(t), d(t)=[d _j (t), j=1, 2, 3] ^T , is the derivative of the exogenous system state variable υ _j (t), W _j is the first measurable parameter, τ _j is the harmonic interference frequency, V _j is a measurable constant parameter, V _j = [1 0].

The operation of the target conversion process in S2 is specifically as follows: presetting an auxiliary filter, the auxiliary filter is used to convert the unknown harmonic interference into an expression with unknown factors to obtain the unknown harmonic interference expression; the auxiliary filter can be obtained by the following formula:

ξ _j (t) is the filter, is the filter derivative, M _j is the filter matrix, N _j is the filter control matrix, d _j (t) is the component of the unknown harmonic interference d(t), d(t) = [d _j (t), j = 1, 2, 3] ^T ;

The unknown harmonic interference expression is:

ξ _j1 (t) is the first component of the filter, θ _j1 is the unknown factor, ξ _j2 (t) is the second component of the filter, m _j2 is the second parameter of the filter matrix, is the transpose of the uncertain parameters, and δ _dj (t) is the attenuation vector.

The adaptive disturbance observer in S2 can be obtained by the following formula:

is the unknown factor estimate, is the first auxiliary variable of the disturbance observer, ε _j (t) is the second auxiliary variable of the disturbance observer, is the first auxiliary variable derivative of the disturbance observer, is the design parameter of the disturbance observer, m _j1 is the first parameter of the filter matrix, ξ _j1 (t) is the first component of the filter, is the derivative of the second component of the filter, θ _j1 is the unknown factor, is the uncertain parameter transpose, δ _dj (t) is the attenuation vector, and t is the time.

The dynamic model of the unknown harmonic interference error in S2 can be obtained by the following formula:

is the unknown harmonic interference error derivative, is the disturbance observer design parameter, ξ _j1 (t) is the first component of the filter, e _θj (t) is the unknown harmonic interference error, is the uncertain parameter transpose, δ _dj (t) is the attenuation vector, and t is the time.

The flexible modal error dynamics model in S2 can be obtained by the following formula:

is the flexible modal error derivative, e _Ψ (t) is the flexible modal error, e _Ψ (t) = [e _η (t), e _ψ (t)] ^T , e _η (t) is the first component of the flexible modal error, η(t) is the flexible mode, is the flexible modal estimation, e _ψ (t) is the second component of the flexible modal error, ψ(t) is the auxiliary state, is the auxiliary state estimation, J ₂ is the modal matrix, and t is the time.

The fuzzy controller in S3 can be obtained by the following formula:

u _F (t) is the fuzzy control law, is the total control gain, x(t) is the state variable, _ua (t) is the adaptive compensation term, is the unknown harmonic interference estimation, It is the fuzzy auxiliary item estimation;

The adaptive compensation term u _a (t) can be obtained by the following formula:

is the upper bound estimate of the total interference, B ^T is the transpose of the system control matrix, P ₁ is the first component of the positive definite matrix, and x ^T (t) is the transpose of the state variable;

The fuzzy auxiliary term estimation It can be obtained by the following formula:

z _i (x(t)) is the system membership function, S() is a skew-symmetric matrix, x _ωi (t) is the auxiliary angular velocity state variable, δ ^T is the transpose of the coupling matrix, is the auxiliary state estimation, C ₀ is the damping matrix, K ₀ is the stiffness matrix, is the flexible mode estimation, δ is the coupling matrix, ω(t) is the angular velocity, and t is the time.

The adaptive law in S3 can be obtained by the following formula:

is the derivative of the upper bound estimate of the total disturbance, β is the additional parameter of the adaptive law, x ^T (t) is the transpose of the state variable, P ₁ is the first component of the positive definite matrix, B is the system control matrix, κ is the correction parameter, is the upper bound estimate of the total interference, α(t) is the adaptive law integrable function, and t is the time.

The dynamic event triggering mechanism in S3 can be obtained by the following formula:

u(t)＝[u ₁ (t), u ₂ (t), u ₃ (t)] ^T ,

w(t)=[w ₁ (t), w ₂ (t), w ₃ (t)] ^T ,

w(t)＝u _F (t),

u(t) is the control input, w(t) is the event-triggered control signal, u _F (t) is the fuzzy control law, x ^T (t) is the state variable transpose, P ₁ is the first component of the positive definite matrix, B is the system control matrix, e(t) is the measurement error, c ₁ is the first parameter of the trigger mechanism, ρ(t) is the internal dynamic variable, ∈ is the scaling parameter, Q is the auxiliary positive definite matrix, σ(t) is the floor function, The derivative of the internal dynamic variable, c ₂ is the second parameter of the trigger mechanism, and t is the time.

The Lyapunov function in S3 can be obtained by the following formula:

V(t) is the Lyapunov function, is the total state transpose, P is the total positive definite matrix, Φ ₁ (t) is the total state, is the transpose of the first component of the unknown harmonic interference error, P ₃ is the third component of the positive definite matrix, e _θ1 is the first component of the unknown harmonic interference error, is the transpose of the second component of the unknown harmonic interference error, P ₄ is the fourth component of the positive definite matrix, e _θ2 is the second component of the unknown harmonic interference error, is the third component transpose of the unknown harmonic interference error, P ₅ is the fifth component of the positive definite matrix, e _θ3 is the third component of the unknown harmonic interference error, δ _d ^T is the total attenuation vector transpose, P ₆ is the sixth component of the positive definite matrix, δ _d (t) is the total attenuation vector, β is the additional parameter of the adaptive law, is the adaptive error, ρ(t) is the internal dynamic variable, and t is the time.

The beneficial effects of the present invention are:

The present invention provides a method for stable control of a flexible spacecraft under unknown harmonic interference. An exogenous system is used to describe the unknown harmonic interference, and an auxiliary filter is used to convert the unknown harmonic interference into an expression with unknown factors. Then, an adaptive interference observer is constructed to estimate the unknown factors to obtain an unknown harmonic interference error dynamics model. A flexible vibration observer is constructed to estimate the flexible mode to obtain a flexible modal error dynamics model. The method is combined with a dynamic event trigger mechanism to construct a Lyapunov function, and the derivative of the Lyapunov function is controlled to be less than a threshold value, thereby ensuring the boundedness of all closed-loop signals. The state of the system and the error of the flexible vibration observer converge to an equilibrium point asymptotically, thereby ensuring the stable operation of the flexible spacecraft.

BRIEF DESCRIPTION OF THE DRAWINGS

By reading the detailed description of the preferred embodiment below, the scheme and advantages of the present application will become clear to those skilled in the art. The accompanying drawings are only for the purpose of illustrating the preferred embodiment and are not to be considered as limiting the present invention.

In the attached picture:

Figure 1 shows the response curves of angular velocity under two schemes;

Figure 2 shows the response curves of quaternions under two schemes;

Figure 3 shows the change curves of control input under two schemes;

FIG. 4 is a curve of harmonic interference and harmonic interference estimation in this embodiment.

DETAILED DESCRIPTION

Exemplary embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings.

This embodiment provides a method for stabilizing and controlling a flexible spacecraft under the influence of unknown harmonic interference, including the following operations:

S1. Based on the kinematic model and dynamic model of the flexible spacecraft and unknown harmonic interference, a fuzzy dynamic model of the flexible spacecraft is constructed;

S2. The unknown harmonic interference in the flexible spacecraft fuzzy dynamic model is processed by target conversion to obtain an unknown harmonic interference expression; an adaptive interference observer is preset to estimate the unknown factors in the unknown harmonic interference expression to obtain an unknown factor estimate; based on the unknown factor estimate, an unknown harmonic interference error dynamics model is obtained;

A flexible vibration observer is preset to estimate the flexible mode in the dynamic model to obtain a flexible mode estimation; based on the flexible mode estimation, a flexible mode error dynamic model is obtained;

S3. The unknown harmonic interference error dynamics model and the flexible modal error dynamics model are combined with the dynamic event trigger mechanism based on the fuzzy controller and the adaptive law to construct a Lyapunov function, and the derivative of the Lyapunov function is obtained by differentiation; the range of the derivative of the Lyapunov function is controlled not to exceed a threshold value to achieve stable control of the flexible spacecraft.

S1. Based on the kinematic model and dynamic model of the flexible spacecraft and the unknown harmonic interference, a fuzzy dynamic model of the flexible spacecraft is constructed.

The current flexible spacecraft control method does not pay enough attention to the results caused by unpredictable disturbances, resulting in poor stability control of flexible spacecraft. To solve this technical problem, this embodiment considers harmonic disturbances with unknown frequency and amplitude. For flexible spacecraft disturbed by unknown harmonics, the unknown harmonic disturbance is described by an exogenous system by establishing a kinematic model and a dynamic model, and a fuzzy model is used for modeling to construct a fuzzy dynamic model of the flexible spacecraft. To enhance the authenticity of the fuzzy dynamic model of the flexible spacecraft, this embodiment uses the T-S fuzzy model in the fuzzy model for modeling, and the obtained T-S fuzzy dynamic model of the flexible spacecraft can perform the operations in S2 and S3.

The kinematic model can be obtained by the following formula:

in is a quaternion, satisfy is the derivative of q ₀ (t), is the derivative of q _v (t), is the identity matrix, represents angular velocity, and S() is a skew-symmetric matrix.

Kinetic model:

J is the inertia matrix, Δ _J is the uncertainty term of the inertia matrix, Δ _J = 0.1e ^-0.01 t, Δ _J can also be described as Δ _J = Δ _F D(t)Δ _R , where Δ _F and Δ _R are parameter matrices satisfying ^DT (t)D(t)＜I, the time-varying matrix D(t) = diag{e ^-0.01 t, e ^-0.01 t, e ^-0.01t }, and the parameter matrices Δ _R and Δ _F are expressed as:

is the angular velocity derivative, δ ^T is the transpose of the coupling matrix, represents the coupling matrix, is the control input, Flexible mode The second-order derivative of Flexible mode The first-order derivative of is referred to as the flexible modal derivative, d(t) is the unknown harmonic disturbance, d ₀ (t) is the additional disturbance term, is the damping matrix, is the stiffness matrix, r is the number of modes, ξ _k is the damping rate, and Ω _k is the natural frequency.

The inertia matrix J, coupling matrix δ, stiffness matrix K ₀ , and damping matrix C ₀ are expressed as:

For the flexible spacecraft, the initial values of the angular velocity and quaternion are ω(0) = [0.1, -0.3, 0.2] ^T , q ₀ (0) = 0.755, q _v (0) = [0.3, 0.5, -0.3] ^T . The initial value of the flexible mode is η(0) = [-0.001, 0.002, 0.001] ^T .

In order to facilitate the state definition of the flexible spacecraft TS fuzzy dynamic model setting system, the auxiliary state is defined as The initial value is ψ(0) = [0, 0, 0] ^T , and the above dynamic equation (1.2) is converted to:

Wherein, J ₀ =J-δ ^T δ.

In addition, in order to solve the influence of inertial uncertainty on the stability of flexible spacecraft, the angular velocity derivative is calculated. When , the matrix inversion formula is used to transform the inertial uncertainty into the total interference, and then the adaptive law is used to estimate the upper bound of the total interference. According to the matrix inversion formula, the above formula is transformed into:

The total interference Expressed as

Define state variables x ₁ (t) = ω ₁ (t), x ₂ (t) = ω ₂ (t), x ₃ (t) = ω ₃ (t), x ₄ (t) = q ₁ (t), x ₅ (t) = q ₂ (t), x ₆ (t) = q ₃ (t). Therefore, the state variables are expressed as x _ω (t) = [x ₁ (t), x ₂ (t) x ₃ (t)] ^T , x _q (t) = [x ₄ (t), x ₅ (t), x ₆ (t)] ^T . Let q _i (t)∈[-0.55 0.55], ω _i (t)∈[-1 1]rad/s, (i=1, 2, 3). The seven operating points selected are _x1 (t)=[0,0,0,0,0,0] ^T , _x2 (t)=[0,0,0,0.55,0.55,0.55] ^T , _x3 (t)=[0,0,0,-0.55,-0.55,-0.55] ^T , _x4 (t)=[0.5,0.5,0.5,0,0,0] ^T , _x5 (t)=[-0.5,-0.5,-0.5,0,0,0] ^T , _x6 (t)=[1,1,1,0.55,0.55,0.55] ^T , _x7 (t)=[-1,-1,-1,-0.55,-0.55,-0.55] ^T . Based on the TS fuzzy model, modeling rule i is: if _β1 (t) is β ₂ (t) is …, β ₆ (t) is Here β(t) = [β ₁ (t), β ₂ (t), …, β ₆ (t)] ^T and is the antecedent variable and TS fuzzy set, λ is the number of fuzzy rules, then based on the above kinematic model and dynamic model, the TS fuzzy dynamic model of the flexible spacecraft is obtained as follows:

is the derivative of the state variable x(t), _Ai is the system matrix, is the system uncertainty term, B is the system control matrix, u(t) is the control input, s _i (t) is the fuzzy auxiliary term, is the total interference, is the output vector of the fuzzy dynamic model of the flexible spacecraft TS of the system, and _Ci = I is the output matrix.

Among them, _xωi (t) is the auxiliary angular velocity state variable, _xqi (t) is the auxiliary quaternion state variable, R＝[Δ _R 0].

Then the T-S fuzzy dynamic model of the flexible spacecraft can be described as:

in 0≤z _i (β(t))≤1. Therefore, for any β(t),

The antecedent variable β(t) is related to the state variable x(t), so the T-S fuzzy dynamic model of the flexible spacecraft is finally described as:

There exists an upper bound h＞0, which makes

In addition, the unknown harmonic interference d(t)=[d ₁ (t), d ₂ (t), d ₃ (t)] ^T , d _j (t) is the unknown harmonic interference component, which is specifically expressed as follows: Among them, the amplitude is A _dj , the harmonic interference frequency is τ _j , and the phase is an unknown constant.

Therefore, the unknown harmonic interference described by the exogenous system can be obtained by the following formula:

dj(t) is the component of unknown harmonic interference d(t), d(t) = [ _dj (t), j = 1, 2, 3] ^T , is the exogenous system state variable The derivative of , W _j is the first measurable parameter, τ _j is the harmonic interference frequency, V _j is a measurable constant parameter, V _j = [1 0].

S2. The unknown harmonic interference in the fuzzy dynamic model of the flexible spacecraft is processed by target transformation to obtain the unknown harmonic interference expression; an adaptive interference observer is preset to estimate the unknown factors in the unknown harmonic interference expression to obtain the unknown factor estimation; based on the unknown factor estimation, the unknown harmonic interference error dynamic model is obtained; a flexible vibration observer is preset to estimate the flexible mode in the dynamic model to obtain the flexible mode estimation; based on the flexible mode estimation, the flexible mode error dynamic model is obtained.

The operation of the target conversion processing is specifically as follows: presetting an auxiliary filter, the auxiliary filter is used to convert the unknown harmonic interference into an expression with unknown factors, and obtain the unknown harmonic interference expression.

The auxiliary filter can be obtained by the following formula:

ξ _j (t) is the filter, is the filter derivative, M _j is the filter matrix, N _j is the filter control matrix, which is known, d _j (t) is the component of the unknown harmonic interference d (t), d (t) = [d _j (t), j = 1, 2, 3] ^T. Among them, ξ _j1 (t) is the first component of the filter, ξ _j2 (t) is the second component of the filter, m _j1 is the first parameter of the filter matrix, m _j2 is the second parameter of the filter matrix, (M _j , N _j ) is controllable, and M _j and N _j have no common eigenvalues. Therefore, the component d _j (t) (j = 1, 2, 3) of the unknown harmonic interference d(t) is expressed as θ _j is an uncertain parameter, δ _dj (t) is the decay vector, And δ _dj (t) satisfies the condition Uncertain parameter transposition

Based on the above formula, the unknown harmonic interference expression is:

ξ _j1 (t) is the first component of the filter, θ _j1 is the unknown factor, ξ _j2 (t) is the second component of the filter, m _j2 is the second parameter of the filter matrix, is the transpose of the uncertain parameters, and δ _dj (t) is the attenuation vector.

In order to estimate the unknown factor of θ _j1 , the designed adaptive disturbance observer can be obtained by the following formula:

is the unknown factor estimate, is the first auxiliary variable of the disturbance observer, ε _j (t) is the second auxiliary variable of the disturbance observer, is the first auxiliary variable derivative of the disturbance observer, is the design parameter of the disturbance observer, m _j1 is the first parameter of the filter matrix, ξ _j1 (t) is the first component of the filter, is the derivative of the second component of the filter, θ _j1 is an unknown factor, is the transpose of the uncertain parameters, δ _dj (t) is the attenuation vector, and t is time. In addition, the total attenuation vector δ _d (t) = [δ _d1 (t), δ _d2 (t), δ _d3 (t)] ^T , and the derivative of the total attenuation vector is M=diag{M ₁ , M ₂ , M ₃ } is the total filter matrix. The unknown harmonic interference d(t)=[d ₁ (t), d ₂ (t), d ₃ (t)] ^T can be further expressed as d(t)=E ₁ θ _X1 +E ₂ m ₂ +θ ^T δ _d (t). Among them, the first component of the filter state E ₁ =diag{ξ ₁₁ (t), ξ ₂₁ (t), ξ ₃₁ (t)}, the second component of the filter state E ₂ =diag{ξ ₁₂ (t), ξ ₂₂ (t), ξ ₃₂ (t)}, the total unknown factor θ _X1 =[θ ₁₁ ,θ ₂₁ ,θ ₃₁ ] ^T , the total known constant m ₂ =[m ₁₂ ,m ₂₂ ,m ₃₂ ] ^T , the total uncertain parameter θ =diag{θ ₁ ,θ ₂ ,θ ₃ }, therefore, the estimation of the unknown harmonic interference d(t) It can be expressed as in Estimate the total unknown factors.

definition It is an unknown factor estimate, so the unknown harmonic interference error dynamics model can be obtained by the following formula:

is the unknown harmonic interference error derivative, is the disturbance observer design parameter, ξ _j1 (t) is the first component of the filter, e _θj (t) is the unknown harmonic disturbance error, is the uncertain parameter transpose, δ _dj (t) is the attenuation vector, and t is the time.

A flexible vibration observer is constructed (a mature technology in this field, so it will not be described in detail here) to estimate the flexible mode, and the flexible mode estimate is obtained as follows:

is the estimated derivative of the flexible mode, is the auxiliary state estimation derivative, J ₂ is the modal matrix, is the flexible mode estimation, Auxiliary state estimation.

Based on the flexible mode estimation, the obtained flexible mode error dynamic model can be obtained by the following formula:

is the flexible modal error derivative, e _Ψ (t) is the flexible modal error, e _Ψ (t) = [e _η (t), e _ψ (t)] ^T , e _η (t) is the first component of the flexible modal error, η(t) is the flexible mode, is the flexible mode estimation, e _ψ (t) is the second component of the flexible mode error, ψ(t) is the auxiliary state, is the auxiliary state estimation, J ₂ is the modal matrix, and t is the time.

S3. The unknown harmonic interference error dynamics model and the flexible modal error dynamics model are combined with the dynamic event trigger mechanism based on the fuzzy controller and the adaptive law to construct the Lyapunov function, and the derivative of the Lyapunov function is obtained by differentiation; the range of the derivative of the Lyapunov function is controlled not to exceed the threshold value to achieve stable control of the flexible spacecraft.

Based on the fuzzy dynamic model of flexible spacecraft TS, a fuzzy controller with compensation terms is designed by adopting a parallel distributed compensation control method, where the first rule is as follows: Control rule 1: If β ₁ (t) is β ₂ (t) is …, β ₆ (t) is So l = 1, 2, ..., λ,

where β(t)=[β ₁ (t), β ₂ (t), ..., β ₆ (t)] ^T and is the antecedent variable and TS fuzzy set, l is the number of fuzzy rules, u _Fl (t) represents the control law, and the controller gain is u _a (t) is the adaptive compensation term that needs to be designed.

The antecedent variable β _l (t) is defined by the state variable x(t), and the fuzzy controller can be obtained by the following formula:

u _F (t) is the fuzzy control law, is the total control gain, x(t) is the state variable, _ua (t) is the adaptive compensation term, is the unknown harmonic interference estimation, It is the fuzzy auxiliary item estimation;

The adaptive compensation term u _a (t) can be obtained by the following formula:

is the upper bound estimate of the total interference, B ^T is the transpose of the system control matrix, P ₁ is the first component of the positive definite matrix, and x ^T (t) is the transpose of the state variable;

Fuzzy auxiliary term estimation It can be obtained by the following formula:

z _i (x(t)) is the system membership function, S() is a skew-symmetric matrix, x _ωi (t) is the auxiliary angular velocity state variable, δ ^T is the transpose of the coupling matrix, is the auxiliary state estimation, C ₀ is the damping matrix, K ₀ is the stiffness matrix, is the flexible mode estimation, δ is the coupling matrix, ω(t) is the angular velocity, and t is the time.

Based on this, the designed adaptive law can be obtained by the following formula:

is the derivative of the upper bound estimate of the total disturbance, β＞0 is the additional parameter of the adaptive law, x ^T (t) is the transpose of the state variable, P ₁ is the first component of the positive definite matrix, B is the system control matrix, κ＞0 is the correction parameter, is the upper bound estimate of the total interference, α(t) is an adaptive law integrable function, which satisfies the condition is a positive variable, and t is time.

Based on the above conditions, the fuzzy controller is combined with the adaptive law, and the designed dynamic event trigger mechanism can be obtained by the following formula:

u(t)＝[u ₁ (t), u ₂ (t), u ₃ (t)] ^T ,

w(t)=[w ₁ (t), w ₂ (t), w ₃ (t)] ^T ,

w(t)＝u _F (t),

u(t) is the control input, w(t) is the event-triggered control signal, u _F (t) is the fuzzy control law, x ^T (t) is the state variable transpose, P ₁ is the first component of the positive definite matrix, B is the system control matrix, e(t) is the measurement error, c ₁ is the first parameter of the trigger mechanism, ρ(t) is the internal dynamic variable, ∈ is the scaling parameter, Q is the auxiliary positive definite matrix, σ(t) is the floor function, The internal dynamic variable derivative, c ₂ is the second parameter of the trigger mechanism, and t is time. _{t ι} is the triggering moment of the current event. If the event triggering condition is met, this moment is marked as t _ι+1 . At this time, the control input u(t _ι+1 ) acts on the system. At t∈[t _ι , t _ι+1 ), the event triggering control signal remains w(t _ι ). This embodiment is based on the fuzzy controller and the adaptive law. The dynamic event triggering mechanism obtained can effectively extend the triggering time interval, reduce the communication burden, save communication resources, and offset the error of event triggering.

For the stable operation of flexible spacecraft, all closed-loop signals of the flexible spacecraft must be bounded and the state of the system must converge to the equilibrium point asymptotically, that is, Therefore, based on the above-mentioned unknown harmonic interference error dynamics model, flexible modal error dynamics model, and dynamic event triggering mechanism, the construction of the Lyapunov function can be obtained by the following formula:

V(t) is the Lyapunov function, is the total state transpose, P is the total positive definite matrix, Φ ₁ (t) is the total state, is the transpose of the first component of the unknown harmonic interference error, P ₃ is the third component of the positive definite matrix, e _θ1 is the first component of the unknown harmonic interference error, is the transpose of the second component of the unknown harmonic interference error, P ₄ is the fourth component of the positive definite matrix, e _θ2 is the third component of the unknown harmonic interference error, is the third component transpose of the unknown harmonic interference error, P ₅ is the fifth component of the positive definite matrix, e _θ3 is the third component of the unknown harmonic interference error, δ _d ^T is the total attenuation vector transpose, P ₆ is the sixth component of the positive definite matrix, δ _d (t) is the total attenuation vector, β is the additional parameter of the adaptive law, is the adaptive error, ρ(t) is the internal dynamic variable, and t is time.

Design the derivative of the Lyapunov function to be less than the threshold, that is,

z _i (x(t)) is the system membership function, z _l (x(t)) is the controller membership function, Γ is the change parameter, which is a positive constant, P is the total positive definite matrix, Φ ₁ is the total state, σ(s) is the floor function, is the upper bound of the floor function, α(s) is an adaptive law integrable function, is the upper bound of the adaptive law integrable function, β is the additional parameter of the adaptive law, κ is the correction parameter, and h is the upper bound of the total disturbance. In addition, the state variable x(t)∈L ₂ , the flexible modal error e _Ψ (t)∈L ₂ , and the condition is satisfied

By applying Barbalat's lemma, we can deduce Therefore, the control method proposed in this embodiment can achieve stable operation of the flexible spacecraft under unknown harmonic interference.

In order to prove the effect of the stable control method provided in this embodiment, a comparative scheme is introduced. The fuzzy controller of the comparative scheme is designed as follows: t∈[t _k , t _k+1 ), where z(t _k ) is the antecedent variable, and t ₀ , t ₁ , ... , t _k , (k=1, 2, ... , ∞) is a series of sampled values.

The corresponding simulation curves are shown in Figures 1 to 4: Figure 1 is the response curve of angular velocity under the two schemes, and the upper, middle and lower figures respectively represent the change of pitch angle ω ₁ , roll angle ω ₂ , and yaw angle ω ₃ over time. Figure 2 is the response curve of quaternion (q ₀ , q ₁ , q ₂ , q ₃ ) under the two schemes. Figure 3 is the time curve of control input (u ₁ , u ₂ , u ₃ ) under the two schemes. Figure 4 is the time curve of harmonic interference and harmonic interference estimation in this embodiment. In Figures 1 and 2, the horizontal and vertical coordinates are time, angular velocity or quaternion, respectively. After comparison, it can be seen that the convergence speed and control accuracy of the control method provided by this embodiment are better than those of the comparative scheme. In Figure 4, the horizontal and vertical coordinates are time and harmonic interference, respectively. It can be seen that the disturbance observer proposed by the control method provided by this embodiment can effectively capture harmonic disturbances and accurately and quickly estimate interference. In addition, the disturbance estimation error can quickly converge to zero, that is, the disturbance observer proposed by the control method provided by this embodiment can quickly estimate the unknown frequency of the disturbance. Therefore, overall, the control method provided by this embodiment is superior to the comparative solution in terms of control performance.

The present embodiment provides a method for stable control of a flexible spacecraft under unknown harmonic interference, which uses an exogenous system to describe the unknown harmonic interference, and uses an auxiliary filter to convert the unknown harmonic interference into an expression with unknown factors, then constructs an adaptive interference observer to estimate the unknown factors, obtains an unknown harmonic interference error dynamics model, and constructs a flexible vibration observer to estimate the flexible mode, obtains a flexible modal error dynamics model, and combines it with a dynamic event trigger mechanism to construct a Lyapunov function, controls the derivative of the Lyapunov function to be less than a threshold, ensures the boundedness of all closed-loop signals, and the state of the system and the flexible vibration observer error converge to an equilibrium point asymptotically, thereby ensuring the stable operation of the flexible spacecraft.

Claims (10)

1. A method for stabilizing a flexible spacecraft under unknown harmonic interference, characterized by comprising the following operations: S1. Based on the kinematic model and dynamic model of the flexible spacecraft and unknown harmonic interference, a fuzzy dynamic model of the flexible spacecraft is constructed; S2. The unknown harmonic interference in the flexible spacecraft fuzzy dynamic model is processed by target conversion to obtain an unknown harmonic interference expression; an adaptive interference observer is preset to estimate the unknown factors in the unknown harmonic interference expression to obtain an unknown factor estimate; based on the unknown factor estimate, an unknown harmonic interference error dynamics model is obtained; A flexible vibration observer is preset to estimate the flexible mode in the dynamic model to obtain a flexible mode estimation; based on the flexible mode estimation, a flexible mode error dynamic model is obtained; S3. The unknown harmonic interference error dynamics model and the flexible modal error dynamics model are combined with the dynamic event trigger mechanism based on the fuzzy controller and the adaptive law to construct a Lyapunov function, and the derivative of the Lyapunov function is obtained by differentiation; the range of the derivative of the Lyapunov function is controlled not to exceed a threshold value to achieve stable control of the flexible spacecraft. 2. The flexible spacecraft stability control method according to claim 1 is characterized in that the unknown harmonic interference in S1 can be obtained by the following formula: d _j (t) is the component of the unknown harmonic interference d(t), d(t)=[d _j (t), j=1,2,3] ^T , is the derivative of the exogenous system state variable υ _j (t), W _j is the first measurable parameter, τ _j is the harmonic interference frequency, V _j is a measurable constant parameter, V _j = [1 0]. 3. The flexible spacecraft stability control method according to claim 1 is characterized in that the operation of the target conversion processing in S2 is specifically: presetting an auxiliary filter, the auxiliary filter is used to convert the unknown harmonic interference into an expression with unknown factors to obtain the unknown harmonic interference expression; The auxiliary filter can be obtained by the following formula: ξ _j (t) is the filter, is the filter derivative, M _j is the filter matrix, N _j is the filter control matrix, d _j (t) is the component of the unknown harmonic interference d(t), d(t) = [d _j (t), j = 1, 2, 3] ^T ; The unknown harmonic interference expression is: ξ _j1 (t) is the first component of the filter, θ _j1 is the unknown factor, ξ _j2 (t) is the second component of the filter, m _j2 is the second parameter of the filter matrix, is the transpose of the uncertain parameters, and δ _dj (t) is the attenuation vector. 4. The flexible spacecraft stability control method according to claim 1, characterized in that the adaptive disturbance observer in S2 can be obtained by the following formula: is the unknown factor estimate, is the first auxiliary variable of the disturbance observer, ε _j (t) is the second auxiliary variable of the disturbance observer, is the first auxiliary variable derivative of the disturbance observer, is the design parameter of the disturbance observer, m _j1 is the first parameter of the filter matrix, ξ _j1 (t) is the first component of the filter, is the derivative of the second component of the filter, θ _j1 is the unknown factor, is the uncertain parameter transpose, δ _dj (t) is the attenuation vector, and t is the time. 5. The flexible spacecraft stability control method according to claim 1 is characterized in that the unknown harmonic interference error dynamics model in S2 can be obtained by the following formula: is the unknown harmonic interference error derivative, is the disturbance observer design parameter, ξ _j1 (t) is the first component of the filter, e _θj (t) is the unknown harmonic interference error, is the uncertain parameter transpose, δ _dj (t) is the attenuation vector, and t is the time. 6. The flexible spacecraft stability control method according to claim 1 is characterized in that the flexible modal error dynamics model in S2 can be obtained by the following formula: is the flexible modal error derivative, e _Ψ (t) is the flexible modal error, e _Ψ (t)＝[e _η (t),e _ψ (t)] ^T , e _η (t) is the first component of the flexible modal error, η(t) is the flexible mode, is the flexible modal estimation, e _ψ (t) is the second component of the flexible modal error, ψ(t) is the auxiliary state, is the auxiliary state estimation, J ₂ is the modal matrix, and t is the time. 7. The flexible spacecraft stability control method according to claim 1, characterized in that the fuzzy controller in S3 can be obtained by the following formula: u _F (t) is the fuzzy control law, is the total control gain, x(t) is the state variable, _ua (t) is the adaptive compensation term, is the unknown harmonic interference estimation, It is the fuzzy auxiliary item estimation; The adaptive compensation term u _a (t) can be obtained by the following formula: is the upper bound estimate of the total interference, B ^T is the transpose of the system control matrix, P ₁ is the first component of the positive definite matrix, and x ^T (t) is the transpose of the state variable; The fuzzy auxiliary term estimation It can be obtained by the following formula: z _i (x(t)) is the system membership function, S() is a skew-symmetric matrix, x _ωi (t) is the auxiliary angular velocity state variable, δ ^T is the transpose of the coupling matrix, is the auxiliary state estimation, C ₀ is the damping matrix, K ₀ is the stiffness matrix, is the flexible mode estimation, δ is the coupling matrix, ω(t) is the angular velocity, and t is the time. 8. The flexible spacecraft stability control method according to claim 1, characterized in that the adaptive law in S3 can be obtained by the following formula: is the derivative of the upper bound estimate of the total disturbance, β is the additional parameter of the adaptive law, x ^T (t) is the transpose of the state variable, P ₁ is the first component of the positive definite matrix, B is the system control matrix, κ is the correction parameter, is the upper bound estimate of the total interference, α(t) is the adaptive law integrable function, and t is the time. 9. The flexible spacecraft stability control method according to claim 1, characterized in that the dynamic event triggering mechanism in S3 can be obtained by the following formula: u(t)＝[u ₁ (t), u ₂ (t), u ₃ (t)] ^T , w(t)＝[w ₁ (t), w ₂ (t), w ₃ (t)] ^T , w(t)＝u _F (t), u(t) is the control input, w(t) is the event-triggered control signal, u _F (t) is the fuzzy control law, x ^T (t) is the state variable transpose, P ₁ is the first component of the positive definite matrix, B is the system control matrix, e(t) is the measurement error, c ₁ is the first parameter of the trigger mechanism, ρ(t) is the internal dynamic variable, ∈ is the scaling parameter, Q is the auxiliary positive definite matrix, σ(t) is the floor function, The derivative of the internal dynamic variable, c ₂ is the second parameter of the trigger mechanism, and t is the time. 10. The flexible spacecraft stability control method according to claim 1, characterized in that the Lyapunov function in S3 can be obtained by the following formula: V(t) is the Lyapunov function, is the total state transpose, P is the total positive definite matrix, Φ ₁ (t) is the total state, is the transpose of the first component of the unknown harmonic interference error, P ₃ is the third component of the positive definite matrix, e _θ1 is the first component of the unknown harmonic interference error, is the transpose of the second component of the unknown harmonic interference error, P ₄ is the fourth component of the positive definite matrix, e _θ2 is the second component of the unknown harmonic interference error, is the third component transpose of the unknown harmonic interference error, P ₅ is the fifth component of the positive definite matrix, e _θ3 is the third component of the unknown harmonic interference error, δ _d ^T is the total attenuation vector transpose, P ₆ is the sixth component of the positive definite matrix, δ _d (t) is the total attenuation vector, β is the additional parameter of the adaptive law, is the adaptive error, ρ(t) is the internal dynamic variable, and t is the time.