阅读说明：本技术 一种基于绳长变化的绳系卫星混沌控制方法 (Tethered satellite chaotic control method based on rope length change ) 是由 余本嵩 金栋平 于 2020-04-01 设计创作，主要内容包括：本发明公开一种基于绳长变化的绳系卫星混沌控制方法,包括如下步骤：步骤1,建立面内绳系卫星系统的动力学方程,将所述动力学方程转化为状态方程形式；步骤2,基于步骤1的状态方程,将控制器u设计为等效控制器u<Sub>eq</Sub>和切换控制器u<Sub>sw</Sub>的组合,即u＝u<Sub>eq</Sub>+u<Sub>sw</Sub>；步骤3,对控制器进行积分计算,得到绳长变化控制律。此种方法可将不规则的混沌运动控制到一个稳定的周期运动。(The invention discloses a tethered satellite chaotic control method based on rope length change, which comprises the following steps: step 1, establishing a kinetic equation of an in-plane tethered satellite system, and converting the kinetic equation into a state equation form; step 2, designing the controller u as an equivalent controller u based on the state equation of the step 1 eq And a switching controller u sw I.e. u-u eq +u sw (ii) a And 3, carrying out integral calculation on the controller to obtain a rope length change control law. The method can control the irregular chaotic motion to a stable periodic motion.)

1. A tethered satellite chaotic control method based on rope length change is characterized by comprising the following steps:

step 1, establishing a kinetic equation of an in-plane tethered satellite system, and converting the kinetic equation into a state equation form;

step 2, designing the controller u as an equivalent controller u based on the state equation of the step 1_eqAnd a switching controller u_swI.e. u-u_eq+u_sw；

And 3, carrying out integral calculation on the controller to obtain a rope length change control law.

2. The tethered satellite chaotic control method based on rope length variation as recited in claim 1, wherein: in the step 1, the in-plane tethered satellite system comprises a main satellite M, a sub satellite S and a space tether connecting the main satellite M and the sub satellite S, wherein the main satellite M and the sub satellite S respectively have mass M_mAnd m_sThe mass and current length of the spatial tether are m_tAnd l;

selecting an in-plane pitch angle and the current rope length as generalized coordinates, wherein a dimensionless kinetic equation of the in-plane tether satellite system is as follows:

wherein θ represents an in-plane pitch angle, R_cRepresenting the distance of the earth's centroid to the system's centroid,representing the first derivative of the in-plane pitch angle to the true anomaly v,representing the two derivatives of the in-plane pitch angle with respect to the true proximal angle v, ξ ═ l/L_rRepresenting length of dimensionless rope, L_rA reference rope length is shown which is,representing the first derivative, Q, of the dimensionless rope length to the true anomaly v_θThe generalized force of the system corresponding to theta is expressed, T represents the tether tension, mu_EWhich represents the constant of the earth's gravity, all represent quality parameters related to the main star M, the sub-star S and the tether mass;

introducing state vectorsMeanwhile, taking the dimensionless rope length ξ as a control variable, the equation (1) is converted into the following equation of state form:

wherein the content of the first and second substances,p is less than or equal to P in | P (v) |, and P is a normal number;

the controller is as follows:

its length ξ and its rate of changeIt is related.

3. The tethered satellite chaotic control method based on rope length variation as recited in claim 2, wherein: in the step 2, the equivalent controller u_eqThe design method comprises the following steps:

first, the slip-form face of the system is defined:

s＝Ce^T(4)

wherein C ═ C₁c₂]Represents a constant vector, and the state error vector is:

wherein (theta)_1d,θ_2d) Represents a desired state and has

Taking the derivative of equation (4) and substituting a second equation of equation (2) into it yields:

at this time, letAnd substituting the formula (6) to obtain:

the equivalent controller can ensure that the system state moves in the sliding mode surface, namely the system state is always kept in a desired state.

4. The tethered satellite chaotic control method based on rope length variation as recited in claim 3, wherein: in said step 2, the switching controller u_swThe design is as follows:

wherein the parameter K is P + λ and the parameter λ > 0.

5. The tethered satellite chaotic control method based on rope length variation as recited in claim 4, wherein: before the step 3, the stability of the controller u is proved by utilizing a Lyapunov function.

6. The tethered satellite chaotic control method based on rope length variation as recited in claim 5, wherein: the specific process of proving the stability of the controller u by using the lyapunov function is as follows:

first, the lyapunov function is selected:

it is clear that V ≧ 0, V ═ 0 if and only if s ═ 0, then V is a positive definite function; changing the controller u to u_eq+u_swSubstituting the derivative of the sliding mode surface to obtain:

and (3) carrying out derivation on the Lyapunov candidate function V and substituting the formula (10) into the derivation to obtain:

it is obvious thatIf and only if s is 0, thenThis is therefore a negative definite function; due to V andis a fixed sign function of the opposite sign, and thus, the controller is asymptotically stable.

7. The tethered satellite chaotic control method based on rope length variation as recited in claim 2, wherein: in step 3, the expression (3) of the controller is integrated to obtain:

ξ＝ξ₀e^∫udν(12)

in the formula, ξ₀Representing the initial dimensionless rope length.

Technical Field

The invention belongs to the technical field of spacecraft flight, and particularly relates to a tethered satellite chaotic control method based on rope length change.

Background

Due to the nonlinear structure of the space tethered satellite, nonlinear phenomena such as chaos and the like often occur during the orbital flight, which has attracted great attention of researchers. For example, Williams studied the release/recovery problem of tethered satellite systems by optimal control methods based on tether tension control [1 ]. Nakanishi et al use time-lag feedback tension control to maintain a cyclic motion of a tethered satellite system operating in an elliptical orbit [2 ]. Steindl successfully applies optimal time control to stably release a tethered satellite system running in an elliptical orbit with small eccentricity through tension control [3 ]. And a self-adaptive sliding mode controller and a fractional order sliding mode controller are respectively designed for Ma and the like, and the load at the tail end of the spatial rope system is successfully and stably released through tension control [4,5 ]. Zhong et al propose a slip-form controller with positive tension restraint to control a tether detection device around a pseudo-stable equilibrium position [6 ]. Razzaghi et al in order to enable a tethered piece towing system to move stably in and out of a plane, an adaptive sliding mode controller and a state dependent ricaciati equation controller were designed and compared for the pulling force control effect [7 ].

Previous researches show that tension control is common for adjustment of release, recovery, attitude and the like of a tethered satellite system, but the tension control is difficult to implement due to high precision requirement of a tension sensor in orbit practice; meanwhile, no learner has conducted intensive research on the chaotic control of the spatial tether system.

Information on the documents involved:

[1]Williams P.Deployment/retrieval optimization for flexible tetheredsatellite systems.Nonlinear Dynamics,2008,52(1-2):159-179.

[2]Nakanishi K.,Kojima H.,Watanabe T.Trajectories of in-planeperiodic solutions of tethered satellite system projected on van der Polplanes.Acta Astronautica,2011,68(7-8):1024-1030.

[3]Steindl A.Optimal deployment of a tethered satellite using tensioncontrol.International Federation of Automatic Control-Papersonline.2015,48(1):53-54.

[4]Ma Z.Q.,Sun G.H.Adaptive sliding mode control of tetheredsatellite deployment with input limitation.Acta Astronautica,2016,127(1):67-75.

[5]Ma Z.Q.,Zhu Z.H.,Sun G.H.Fractional-order sliding mode control fordeployment of tethered spacecraft system.Proceedings of the Institution ofMechanical Engineers,Part G:Journal of Aerospace Engineering,2019,233(13):4721-4734.

[6]Zhong R.,Wang Y.Dynamics and control of a probe tethered to anasteroid.Journal of Guidance,Control,and Dynamics,2018,41(7):1583-1588.

[7]Razzaghi P.,Al Khatib E.,Bakhtiari S.Sliding mode and SDRE controllaws on a tethered satellite system to de-orbit space debris.Advances inSpace Research,2019,64(1):18-27.

disclosure of Invention

The invention aims to provide a rope system satellite chaotic control method based on rope length change, which can control irregular chaotic motion to stable periodic motion.

In order to achieve the above purpose, the solution of the invention is:

a tethered satellite chaotic control method based on rope length change comprises the following steps:

step 1, establishing a kinetic equation of an in-plane tethered satellite system, and converting the kinetic equation into a state equation form;

step 2, designing the controller u as an equivalent controller u based on the state equation of the step 1_eqAnd a switching controller u_swI.e. u-u_eq+u_sw；

And 3, carrying out integral calculation on the controller to obtain a rope length change control law.

In the step 1, the in-plane tethered satellite system comprises a main satellite M, a sub satellite S and a space tether connecting the main satellite M and the sub satellite S, wherein the main satellite M and the sub satellite S have mass M respectively_mAnd m_sThe mass and current length of the spatial tether are m_tAnd l;

selecting an in-plane pitch angle and the current rope length as generalized coordinates, wherein a dimensionless kinetic equation of the in-plane tether satellite system is as follows:

wherein θ represents an in-plane pitch angle, R_cRepresenting the distance of the earth's centroid to the system's centroid,representing the first derivative of the in-plane pitch angle to the true anomaly v,representing the two derivatives of the in-plane pitch angle with respect to the true proximal angle v, ξ ═ l/L_rRepresenting length of dimensionless rope, L_rA reference rope length is shown which is,representing the first derivative, Q, of the dimensionless rope length to the true anomaly v_θThe generalized force of the system corresponding to theta is expressed, T represents the tether tension, mu_EWhich represents the constant of the earth's gravity, all represent quality parameters related to the main star M, the sub-star S and the tether mass;

introducing state vectorsMeanwhile, taking the dimensionless rope length ξ as a control variable, the equation (i1) is converted into the following equation of state form:

wherein the content of the first and second substances,p is less than or equal to P in | P (v) |, and P is a normal number;

the controller is as follows:

its length ξ and its rate of changeIt is related.

In the step 2, the equivalent controller u_eqThe design method comprises the following steps:

first, the slip-form face of the system is defined:

s＝Ce^T(i4)

wherein C ═ C₁c₂]Represents a constant vector, and the state error vector is:

wherein (theta)_1d,θ_2d) Represents a desired state and has

Taking the derivative of equation (i4) and substituting a second equation of equation (i2) therein yields:

at this time, letAnd substituted into formula (i6) to obtain:

the equivalent controller can ensure that the system state moves in the sliding mode surface, namely the system state is always kept in a desired state.

In the above step 2, the switching controller u_swThe design is as follows:

wherein the parameter K is P + λ and the parameter λ > 0.

Before the step 3, the stability of the controller u is proved by utilizing the Lyapunov function.

The specific process of proving the stability of the controller u by using the lyapunov function is as follows:

first, the lyapunov function is selected:

it is clear that V ≧ 0, V ═ 0 if and only if s ═ 0, then V is a positive definite function; changing the controller u to u_eq+u_swSubstituting the derivative of the sliding mode surface to obtain:

and (3) carrying out derivation on the Lyapunov candidate function V and substituting the formula (i10) into the derivation result:

it is obvious thatIf and only if s is 0, thenThis is therefore a negative definite function; due to V andis a fixed sign function of the opposite sign, and thus, the controller is asymptotically stable.

In step 3, the expression (i3) of the controller is integrated to obtain:

ξ＝ξ₀e^∫udν(i12)

in the formula, ξ₀Representing the initial dimensionless rope length.

After the scheme is adopted, the invention provides a method for completing a control task by using rope length change aiming at chaotic motion of a space rope system satellite system, firstly, a system dynamics equation is converted into a state equation form, then an equivalent controller and a switching controller are respectively designed based on a sliding mode control method, then the stability of the controller is proved by a Lyapunov function, finally, a rope length control law is obtained by integral calculation, and the irregular chaotic motion can be controlled to be stable periodic motion.

Drawings

FIG. 1 is a flow chart of the present invention;

FIG. 2 is a schematic view of an in-plane tethered satellite system;

FIG. 3 is a dynamic simulation result of chaotic motion generated by oscillation in a system plane;

fig. 4 is a diagram showing the control result using the present invention.

Detailed Description

The technical solution and the advantages of the present invention will be described in detail with reference to the accompanying drawings.

Figure 2 shows an in-plane tethered satellite system operating in a circumferential orbit around the earth. The system consists of a main star M, a sub-star S and a space tether connecting the main star M and the sub-star S. Because the main star M and the sub-star S have little influence on the system response by the rigid body attitude in the orbit running process, the main star M and the sub-star S are regarded as particles, and the mass of the particles is recorded as M_mAnd m_s(ii) a And since the spatial tether is usually in a tense state, the spatial tether is regarded as a rigid rod, and the mass and the current length of the rigid rod are m_tAnd l. In fig. 1, θ represents an in-plane pitch angle, R_cRepresenting the distance of the earth's centroid to the system centroid.

Selecting an in-plane pitch angle and the current rope length as generalized coordinates, and obtaining a dimensionless kinetic equation of the following system by using a second Lagrange equation:

whereinRepresenting the first derivative of the in-plane pitch angle to the true anomaly v,representing the two derivatives of the in-plane pitch angle to the true anomaly angle v，ξ＝l/L_rRepresenting length of dimensionless rope, L_rA reference rope length is shown which is,representing the first derivative, Q, of the dimensionless rope length to the true anomaly v_θThe generalized force of the system corresponding to theta is expressed, T represents the tether tension, mu_EWhich represents the constant of the earth's gravity, all represent quality parameters related to the main star M, the sub-star S and the tether mass.

Introducing state vectorsMeanwhile, with the dimensionless rope length ξ as a control variable, equation (1) can be converted into the following equation of state form:

the control inputs in the formula are:

its length ξ and its rate of changeIt is related. We design this control input as being from an equivalent controller u_eqAnd a switching controller u_swComposition, i.e. u ═ u_eq+u_swThe two controllers will be designed in sequence later. In addition, in the formula (2)And | P (nu) | is less than or equal to P, wherein P is a normal number.

First, the sliding-mode surface of controlled system expression (2) is defined:

s＝Ce^T(4)

wherein C ═ C₁c₂]Represents a constant vector, and the state error vector is:

wherein (theta)_1d,θ_2d) Represents a desired state and has

The equivalent controller u is designed first_eq. Derivation of the sliding mode surface (4) and substitution of a second expression in the form of a system state equation into it can result:

at this time, only need to orderAnd substituting the formula (6) to obtain:

the equivalent controller can ensure that the system state moves in the sliding mode surface, namely the system state is always kept in a desired state.

Secondly, we can design the switch controller as follows:

wherein the parameter K is P + λ and the parameter λ > 0. The switching controller can make the system state far away from the sliding mode surface approach to the sliding mode surface continuously, even if the system state approaches to the expected state continuously.

Then, the stability of the controller u designed above was verified using the lyapunov function. Selecting a Lyapunov candidate function:

it is clear that V ≧ 0, V ═ 0 if and only if s ═ 0, V is a positive definite function. For the convenience of researchLet us say controller u as u_eq+u_swSubstituting the derivative of the sliding mode surface to obtain:

now, by taking the derivative of the lyapunov candidate function V and substituting equation (10) therein, we can obtain:

it is obvious thatIf and only if s is 0, thenThis is therefore a negative definite function. Due to V andis a fixed sign function of the opposite sign, and thus, the controller is asymptotically stable.

Finally, only the expression (3) of the control input needs to be integrated, so that:

ξ＝ξ₀e^∫udν(12)

formula (III) ξ₀Representing the initial dimensionless rope length. The in-plane chaos generated by the system can be effectively controlled through the dimensionless rope length variation formula (12).

The flow chart of the invention is shown inFig. 1. The rope length control strategy proposed in this embodiment is verified by the numerical simulation result. A set of system parameters is defined as follows, the masses of the main star and the sub-star being m respectively_m500kg and m_s50 kg; the current length, linear density and damping coefficient of the spatial tether are l ═ 1km and rho respectively_t＝5×10^-3kg/m and C_d2; meanwhile, the influence of atmospheric damping and earth oblateness on the system is considered, and the height of the orbit of the system is 650km, and the inclination angle of the orbit is i pi/6.

Firstly, the rope length control is not applied, the system surface oscillation generates chaotic motion, and the dynamic simulation result is shown in fig. 3. Wherein fig. 3(a) is the variation of the system pitch angle with dimensionless time. Fig. 3(b) and 3(c) are poincare sections and enlarged views of the pitch motion, as can be clearly seen from the enlarged views, where a large number of cross-sectional heterodips are present. FIG. 3(d) is the power spectral density of the system, which presents a dense power spectrum in the (0,0.25Hz) frequency band; FIG. 3(e) is the maximum Lyapunov exponent for a system, which ultimately remains a positive number as a function of dimensionless time. The result is integrated to judge that the system generates chaotic motion at the moment.

Now, the rope length control method proposed in this embodiment is applied, and the effect of the control method is examined, assuming that the initial rope length of the system is ξ₀0.8, and let the desired frequency be ω_dThe control result is shown in fig. 4, when the value is 0.5. Fig. 4(a) shows the variation of the system pitch angle with dimensionless time. Fig. 4(b) is a poincare section of the pitch motion, and it can be clearly seen that the system exhibits a steady periodic 4 motion. Fig. 4(c) shows the control rope length as a function of dimensionless time v.

The comparison calculation shows that the rope length control method provided by the invention is feasible and can effectively control the chaotic motion of the system.

The above embodiments are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modifications made on the basis of the technical scheme according to the technical idea of the present invention fall within the protection scope of the present invention.