CN103439975A - Distributed index time varying slip mode posture cooperation tracking control method - Google Patents

Abstract

The invention relates to a distributed index time varying slip mode posture cooperation tracking control method and belongs to the technical field of spacecraft formation flight. The method comprises: by establishing a posture kinetic equation in a form of Euler-langrage, selecting index time varying slip mode surface function, calculating a synchronization expectation posture for each spacecraft, and according to the design idea of a time varying slip mode controller, designing a distributed posture cooperation tracking controller so as to enable aircrafts to perform cooperation tracking on the expectation postures and have higher robustness in terms of extraneous interference and inertia uncertainties. Compared to conventional slip mode control, a time varying slip mode enables the phase trajectory of a system to always remain at a slip mode surface, so that in the posture cooperation process of the formation spacecrafts, even if communication is limited and a topology structure needs to be changed, the posture cooperation tracking of the spacecrafts can still be realized.

Description

Become the collaborative tracking and controlling method of Sliding Mode Attitude during a kind of distributed index

Technical field

Become the collaborative tracking and controlling method of Sliding Mode Attitude while the present invention relates to a kind of distributed index, belong to the Spacecraft Formation Flying technical field.

Background technology

The spacecraft attitude Research on Interactive Problem, as a kind of important technology of formation flight, has become a hot issue in the spacecraft field.According to the difference in the place that produces control program, attitude Collaborative Control control mode can be divided into two kinds of centralized control and distributed controls.Than centralized control mode, it is easy that distributed control has control law, better fault-tolerant ability.In distributed control, each single star carrys out to determine control strategy jointly by the information that itself is observed and the data message that communication obtains alternately.Even single satellite lost efficacy, it is stable that entire system still can keep.In addition, colony's spacecraft is when being formed into columns, and the constraint of being communicated by letter, need to change correspondence sometimes, and how guaranteeing still can complete attitude collaborative after changing correspondence is also to need the problem of considering in actual formation process.

In recent decades, many scholars have launched research widely for spacecraft attitude Collaborative Control problem, have proposed many attitude control algolithms, mainly comprise the control based on the lyapunov method, adaptive control, non-linear H ₂/ H _∞and mix and control, Sliding mode variable structure control and ANN (Artificial Neural Network) Control, fuzzy control etc.Wherein, sliding mode control theory has been subject to the generally attention of domestic and international control circle, and sliding mode control theory has been applied to nonlinear system, discrete system, the distributed parameters system, generalized ensemble, the numerous complicated controll plants such as time lag system and incomplete mechanical system.Although sliding mode control theory has become a rounded system and method, and is successfully applied in many fields, still exist many to be studiedly, solve perfect problem.For the performance index of satellites formation system, how CONTROLLER DESIGN and sliding mode, make the appointment of system dynamic, and it is also the problem that should consider in reality that the steady-state behaviour index can be met.

Summary of the invention

The present invention is directed to how CONTROLLER DESIGN is to accelerate the collaborative dynamic response of following the tracks of of Space Vehicle System attitude, and can after changing the communication topology relation, still realize the problem such as harmonious of attitude, in conjunction with sliding mode control theory, become the collaborative tracking and controlling method of Sliding Mode Attitude while proposing a kind of distributed index, become sliding mode controller during design, spacecraft can obtain the expectation attitude information and realize collaborative tracking of attitude under variable topological structure, and disturbs global robustness is arranged to external world.

The inventive method is applicable to meet the following Spacecraft formation required: 1, formation spacecrafts is rigid body, the equal bounded of the uncertain part of the external interference that system is suffered and inertia; 2, formation spacecrafts is expected to attitude is made as one " virtual leader ", there is a directed spanning tree in the communication topology that comprises the Spacecraft formation system of " virtual leader ", all or part of spacecraft can obtain " virtual leader " attitude, even the change Communication topology should " virtual leader " be the root node of communication topology all the time; 3, all variablees have all passed through coordinate transform in the same coordinate system, the relative attitude between spacecraft and the subtraction direct representation of relative attitude angular velocity.

The technical solution used in the present invention is: the attitude dynamic equations of setting up the Euler-langrage form, become the sliding-mode surface function while choosing index, ask for its synchronous expectation attitude for each spacecraft, according to the time become the design philosophy of sliding mode controller, design the collaborative tracking control unit of distributed attitude and make that spacecraft is collaborative follows the tracks of the expectation attitude, and disturb to external world and inertia is uncertain has a stronger robustness.

Specifically comprise the following steps:

Step 1, take the rigid spacecraft formation as object, sets up Euler-Lagrange (Eular-langrage) attitude dynamic equations under the spacecraft body coordinate system.Obtain the attitude information of each spacecraft, ask for the corresponding synchronous expectation attitude of each spacecraft.Concrete grammar is:

Step 1.1, set up the attitude motion model of individual spacecraft.

Formation comprises n spacecraft, and it is numbered 1,2 ... n, represent that " the virtual leader " of Spacecraft formation expectation attitude is numbered n+1, for i spacecraft wherein (i ∈ 1,2 ... n), its attitude dynamics and kinematical equation are as follows:

$J_i{\stackrel{\·}{\mathrm{\ω}}}_i+{\left[{\mathrm{\ω}}_i\right]}^{\×}{J}_i{\mathrm{\ω}}_i=u_i+d_i---\left(1\right)$

${\stackrel{\·}{\mathrm{\σ}}}_i=G\left({\mathrm{\σ}}_i\right){\mathrm{\ω}}_i---\left(2\right)$

J _i∈ R ^{3 * 3}be the inertia matrix of i spacecraft, and the nominal value that means the inertia battle array, Δ J _ithe uncertainty that means the inertia battle array.ω _i∈ R ³be the attitude angular velocity expression in body coordinate system of i rigid body spacecraft with respect to inertial coordinates system, σ _i∈ R ³be the correction Douglas Rodríguez parameter (MRP) of i spacecraft attitude, u _i, d _i∈ R ³mean respectively control moment and disturbance torque that i spacecraft is subject to, () ^*mean vectorial antisymmetric matrix operator.Wherein:

$G\left({\mathrm{\σ}}_i\right)=\frac{1}{2}[\left(\frac{1-{\mathrm{\σ}}_i^T{\mathrm{\σ}}_i}{2}\right)I_3+{{\mathrm{\σ}}_i}^{\×}+{\mathrm{\σ}}_i{\mathrm{\σ}}_i^T]---\left(3\right)$

The attitude dynamic equations of its corresponding Lagrangian Form is:

$H_i\left({\mathrm{\σ}}_i\right){\stackrel{\·\·}{\mathrm{\σ}}}_i+Q_i({\mathrm{\σ}}_i,{\stackrel{\·}{\mathrm{\σ}}}_i){\stackrel{\·}{\mathrm{\σ}}}_i={\mathrm{\τ}}_i---\left(4\right)$

In formula $H_i\left({\mathrm{\σ}}_i\right)=G^{-T}\left({\mathrm{\σ}}_i\right)J_i{G}^{-1}\left({\mathrm{\σ}}_i\right),Q_i({\mathrm{\σ}}_i,{\stackrel{\·}{\mathrm{\σ}}}_i)=-G^{-T}\left({\mathrm{\σ}}_i\right)J_i{G}^{-1}\left({\mathrm{\σ}}_i\right)\stackrel{\·}{G}{\left({\mathrm{\σ}}_i\right)G}^{-1}\left({\mathrm{\σ}}_i\right)-G^{-T}\left({\mathrm{\σ}}_i\right)$

τ _i=G ^-1(σ _i) (u _i+ d _i), H wherein _i(σ _i) be a positive definite symmetric matrices, and

for antisymmetric matrix.Below respectively with G _i, H _i, Δ H _i, Q _i,

Δ Q _imean G (σ _i), H _i(σ _i),

Δ H _i(σ _i),

Step 1.2, for i spacecraft, synchronously expect attitude

be defined as follows:

${\mathrm{\σ}}_i^d=\frac{b_{i(n+1)}{\mathrm{\σ}}_d+\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{a}_{\mathrm{ij}}{\mathrm{\σ}}_j}{b_{i(n+1)}+\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{a}_{\mathrm{ij}}}---\left(5\right)$

σ in formula _ithe attitude that means i spacecraft, a _ijspacecraft i means the correspondence between spacecraft i and spacecraft j, if can obtain the attitude information of j, a _ij=1, otherwise, a _ij=0; If i spacecraft can obtain expectation attitude information σ _d, b _{i (n+1)}=1, otherwise b _{i (n+1)}=0;

Step 2, to each spacecraft, become the sliding-mode surface function while choosing index, ask for the correlation parameter in the sliding-mode surface function, makes at initial time, and the initial error of system is positioned on sliding-mode surface.

For i spacecraft, the time become sliding-mode surface variable s _i(t) be:

$s_i\left(t\right)=({\stackrel{\·}{\mathrm{\σ}}}_i-{\stackrel{\·}{\mathrm{\σ}}}_i^d)+{\mathrm{\Λ}}_i({\mathrm{\σ}}_i-{\mathrm{\σ}}_i^d)-\exp \left({-\mathrm{\Λ}}_{i}t\right)A_i---\left(6\right)$

Λ in formula _i∈ R ^{3 * 3}and be positive definite matrix, A _i∈ R ³, t is time variable.

At initial time, systematic error is positioned on sliding-mode surface, i.e. s _i(0)=0.Obtain:

$A_i=[{\stackrel{\·}{\mathrm{\σ}}}_i\left(0\right)-{\stackrel{\·}{\mathrm{\σ}}}_i^d\left(0\right)]+{\mathrm{\Λ}}_i[{\mathrm{\σ}}_i\left(0\right)-{\mathrm{\σ}}_d\left(0\right)]---\left(7\right)$

Step 3, to the sliding-mode surface function differentiate of step 2 design, then premultiplication H _i:

$H_i{\stackrel{\·}{s}}_i=H_i({\stackrel{\·\·}{\mathrm{\σ}}}_i-{\stackrel{\·\·}{\mathrm{\σ}}}_i^d)+H_i{\mathrm{\Λ}}_i({\stackrel{\·}{\mathrm{\σ}}}_i-{\stackrel{\·}{\mathrm{\σ}}}_i^d)+H_i{\mathrm{\Λ}}_i{A}_i\exp \left({-\mathrm{\Λ}}_{i}t\right)$

$=G_i^{-T}(u_i+d_i)-Q_i{\stackrel{\·}{\mathrm{\σ}}}_i-H_i{\stackrel{\·\·}{\mathrm{\σ}}}_i^d+H_i{\mathrm{\Λ}}_i({\stackrel{\·}{\mathrm{\σ}}}_i-{\stackrel{\·}{\mathrm{\σ}}}_i^d)+H_i{\mathrm{\Λ}}_i{A}_i\exp \left({-\mathrm{\Λ}}_{i}t\right)---\left(8\right)$

$=G_i^{-T}(u_i+d_i)-Q_i^0{\stackrel{\·}{\mathrm{\σ}}}_i+H_i^0{\mathrm{\Λ}}_i({\stackrel{\·}{\mathrm{\σ}}}_i-{\stackrel{\·}{\mathrm{\σ}}}_i^d)+{H_i}^0{\mathrm{\Λ}}_i{A}_i\exp \left({-\mathrm{\Λ}}_{i}t\right)-H_i^0{\stackrel{\·\·}{\mathrm{\σ}}}_i^d+Y_i(\·)$

Wherein

$Y_i(\·)=-{\mathrm{\ΔQ}}_i{\stackrel{\·}{\mathrm{\σ}}}_i-{\mathrm{\ΔH}}_i{\stackrel{\·\·}{\mathrm{\σ}}}_i^d+{\mathrm{\ΔH}}_i{\mathrm{\Λ}}_i({\stackrel{\·}{\mathrm{\σ}}}_i-{\stackrel{\·}{\mathrm{\σ}}}_i^d)+{\mathrm{\ΔH}}_i{\mathrm{\Λ}}_i{A}_i\exp \left(-{\mathrm{\Λ}}_{i}t\right)$

Step 4, become the sliding formwork control law during design index.Control law comprises two parts: a part is in order to the nonlinear terms in bucking-out system dynamics, to reach the expectation index of system; Another part is for offsetting the uncertain part of inertia and external interference, to guarantee the stability of attitude control system.By the uncertain part of inertia and external interference as a whole, ask for its upper bound.

For i spacecraft, become the sliding formwork control law during design as follows:

$u_i=u_{\mathrm{ieq}}+u_{\mathrm{isw}}$ (9)

$=G_i^T[Q_i^0({\stackrel{\·}{\mathrm{\σ}}}_i-s_i)+H_i^0{\stackrel{\·\·}{\mathrm{\σ}}}_i^d-H_i^0{\mathrm{\Λ}}_i({\stackrel{\·}{\mathrm{\σ}}}_i-{\stackrel{\·}{\mathrm{\σ}}}_i^d)+H_i^0{\mathrm{\Λ}}_i{A}_i\exp \left({-\mathrm{\Λ}}_{i}t\right)]-({\mathrm{\γ}}_i+{\mathrm{\λ}}_i)G_i^T\mathrm{sgn}\left(s_i\right)$

Wherein sgn () is sign function, is defined as follows:

$\mathrm{sgn}\left(x\right)=\left\{\begin{array}{c}1,x>0\\ 0,x=0\\ -1,x<0\end{array}\right.$

Sgn (s _i)=[sgn (s _i1); Sgn (s _i2); Sgn (s _i3)], λ in formula _i>0, λ _i+ γ _ifor the switching controls flow gain, and Δ H _i, Δ Q _i, d _ibounded, and || F _i|| _∞bounded, establish γ _i=|| F _i|| _∞, F _ibe expressed as:

$F_i=G_i^T{Y}_i(\&CenterDot;)+{\mathrm{\&Delta;Q}}_i{s}_i+d_i$

Step 5, in formation, spacecraft obtains the attitude information of expectation attitude and neighbours' spacecraft according to the correspondence of self and other spacecrafts, become the input of sliding formwork control law when distributed in using attitude information as step 4, calculate each spacecraft and carry out the needed control moment of the collaborative tracking of attitude, the topworks of each spacecraft will produce these control moments and act on respectively corresponding spacecraft, and the attitude dynamic equations obtained by step 1 is tried to achieve ω _i, and then make σ _ifollow the tracks of the synchronous expectation attitude of each spacecraft, finally realize that the attitude of this spacecraft colony is consistent.

Beneficial effect

The inventive method can solve the attitude stationary problem of spacecraft effectively, and advantage compared with prior art is:

1) design that becomes the sliding formwork control law time has reduced the response time of system, has improved the robustness of system.Control and compare with conventional sliding formwork, in time, become sliding formwork the phase path of system be positioned on sliding-mode surface all the time, cancelled the arrival stage, and then changed conventional sliding formwork and be controlled at the arrival stage and there is no the shortcoming of robustness, reduced the response time of control moment and system, made system there is global robustness.

2) at formation spacecrafts, carry out in the attitude collaborative processes, if communication is restricted while needing to change topological structure, still can realize collaborative tracking of attitude of spacecraft, become the sliding formwork control law when therefore designed and can reduce the communication need between formation spacecrafts.

The accompanying drawing explanation

Fig. 1 is Scheme of Attitude Control schematic diagram of the present invention;

Fig. 2 is the communication topology figure between formation spacecrafts in specific embodiment;

The attitude change curve that Fig. 3 is formation spacecrafts in specific embodiment, wherein (1) is spacecraft 1,3,5 and the expectation attitude the MRP component

simulation curve figure, (2) are corresponding MRP component

simulation curve figure, (3) are corresponding MRP component

simulation curve figure;

Fig. 4 is formation spacecrafts attitude angular velocity change curve in specific embodiment, and wherein (1) is spacecraft 1,3,5 and expectation attitude angular velocity component

simulation curve figure, (2) are the corresponding angular velocity component

simulation curve figure, (3) are the corresponding angular velocity component

simulation curve figure;

Fig. 5 is formation spacecrafts Attitude Tracking error change curve map in specific embodiment, and wherein (1) is spacecraft 1,3,5 Attitude Tracking error components

simulation curve figure, (2) are corresponding error component simulation curve figure, (3) are corresponding error component

simulation curve figure;

The control moment curve map that Fig. 6 is formation spacecrafts in specific embodiment, wherein (1) is spacecraft 1,3,5 control moment components

simulation curve figure, (2) are corresponding control moment component

simulation curve figure, (3) are corresponding control moment component

simulation curve figure;

The change curve of the sliding-mode surface variable that Fig. 7 is formation spacecrafts in specific embodiment, wherein (1) is spacecraft 1,3,5 sliding-mode surface variable components

simulation curve figure, (2) are corresponding sliding-mode surface variable component

simulation curve figure, (3) are corresponding sliding-mode surface variable component

simulation curve figure.

Embodiment

For objects and advantages of the present invention are described better, below in conjunction with drawings and Examples, further set forth.

Spacecraft Attitude Control process flow diagram of the present invention as shown in Figure 1.

Step 1, take the rigid spacecraft formation as object, sets up Euler-Lagrange (Eular-langrage) attitude dynamic equations under the spacecraft body coordinate system.Obtain the attitude information of each spacecraft, ask for the corresponding synchronous expectation attitude of each spacecraft.Concrete grammar is:

Step 1.1, set up the attitude motion model of individual spacecraft.

Formation comprises n spacecraft, and it is numbered 1,2 ... n, represent that " the virtual leader " of Spacecraft formation expectation attitude is numbered n+1, for i spacecraft wherein (i ∈ 1,2 ... n), its attitude dynamics and kinematical equation are as follows:

$J_i{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_i+{\left[{\mathrm{\&omega;}}_i\right]}^{\&times;}{J}_i{\mathrm{\&omega;}}_i=u_i+d_i---\left(1\right)$

${\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i=G\left({\mathrm{\&sigma;}}_i\right){\mathrm{\&omega;}}_i---\left(2\right)$

J _i∈ R ^{3 * 3}be the inertia matrix of i spacecraft, and

the nominal value that means the inertia battle array, Δ J _ithe uncertainty that means the inertia battle array.ω _i∈ R ³be the attitude angular velocity expression in body coordinate system of i rigid body spacecraft with respect to inertial coordinates system, σ _i∈ R ³be the correction Douglas Rodríguez parameter (MRP) of i spacecraft attitude, u _i, d _i∈ R ³mean respectively control moment and disturbance torque that i spacecraft is subject to, () ^*mean vectorial antisymmetric matrix operator.Wherein:

$G\left({\mathrm{\&sigma;}}_i\right)=\frac{1}{2}[\left(\frac{1-{\mathrm{\&sigma;}}_i^T{\mathrm{\&sigma;}}_i}{2}\right)I_3+{\mathrm{\&sigma;}}_i^{\&times;}+{\mathrm{\&sigma;}}_i{\mathrm{\&sigma;}}_i^T]---\left(3\right)$

The attitude dynamic equations of its corresponding Lagrangian Form is:

$H_i\left({\mathrm{\&sigma;}}_i\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&sigma;}}}_i+Q_i({\mathrm{\&sigma;}}_i,{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i){\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i={\mathrm{\&tau;}}_i---\left(4\right)$

In formula $H_i\left({\mathrm{\&sigma;}}_i\right)=G^{-T}\left({\mathrm{\&sigma;}}_i\right)J_i{G}^{-1}\left({\mathrm{\&sigma;}}_i\right),Q_i({\mathrm{\&sigma;}}_i,{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i)=-G^{-T}\left({\mathrm{\&sigma;}}_i\right)J_i{G}^{-1}\left({\mathrm{\&sigma;}}_i\right)\stackrel{\&CenterDot;}{G}\left({\mathrm{\&sigma;}}_i\right)G^{-1}\left({\mathrm{\&sigma;}}_i\right)-G^{-T}\left({\mathrm{\&sigma;}}_i\right)$

τ _i=G ^-1(σ _i) (u _i+ d _i), H wherein _i(σ _i) be a positive definite symmetric matrices, and

for antisymmetric matrix.Below respectively with G _i, H _i,

Δ H _i, Q _i,

Δ Q _imean G (σ _i), H _i(σ _i),

Δ H _i(σ _i),

2), for individual spacecraft, define its corresponding synchronous expectation attitude as follows:

${\mathrm{\&sigma;}}_i^d=\frac{b_{i(n+1)}{\mathrm{\&sigma;}}_d+\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{\mathrm{ij}}{\mathrm{\&sigma;}}_j}{b_{i(n+1)}+\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{\mathrm{ij}}}---\left(5\right)$

In formula the synchronous expectation attitude that means i spacecraft, σ _ithe attitude that means i spacecraft, a _ijmean the correspondence between spacecraft i and spacecraft j, if a _ij=1, mean that spacecraft i can obtain the attitude information of j, otherwise, a _ij=0; If b _{i (n+1)}=1 means that i spacecraft can obtain expectation attitude information σ _d, otherwise b _{i (n+1)}=0;

3) for each spacecraft, choose following form the time become the sliding-mode surface function:

$s_i=({\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i^d)+{\mathrm{\&Lambda;}}_i({\mathrm{\&sigma;}}_i-{\mathrm{\&sigma;}}_i^d)-\exp (-{\mathrm{\&Lambda;}}_{i}t)A_i---\left(6\right)$

Λ in formula _i∈ R ^{3 * 3}and be positive definite matrix, A _i∈ R ³, t ∈ [0, ∞).

In order to make at initial time, systematic error is positioned on sliding-mode surface, i.e. s _i(0)=0.Therefore:

$A_i=\&lsqb;{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i\left(0\right)-{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i^d\left(0\right)\&rsqb;+{\mathrm{\&Lambda;}}_i\&lsqb;{\mathrm{\&sigma;}}_i\left(0\right)-{\mathrm{\&sigma;}}_d\left(0\right)\&rsqb;---\left(7\right)$

4) convolution (1) is to formula (4) differentiate, then premultiplication H _i:

$H_i{\stackrel{\&CenterDot;}{s}}_i=H_i({\stackrel{..}{\mathrm{\&sigma;}}}_i+{\stackrel{..}{\mathrm{\&sigma;}}}_i^d)+H_i{\mathrm{\&Lambda;}}_i({\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i^d)+H_i{\mathrm{\&Lambda;}}_i{A}_i\exp (-{\mathrm{\&Lambda;}}_{i}t)$

$G_i^{-T}(u_i+d_i)-Q_i{\stackrel{.}{\mathrm{\&sigma;}}}_i-H_i{\stackrel{..}{\mathrm{\&sigma;}}}_i^d{H}_i{\mathrm{\&Lambda;}}_i({\stackrel{.}{\mathrm{\&sigma;}}}_i-{\stackrel{.}{\mathrm{\&sigma;}}}_i^d)H_i{\mathrm{\&Lambda;}}_i{A}_i\exp (-{\mathrm{\&Lambda;}}_{i}t)---\left(8\right)$

$=G_i^{-T}(u_i+d_i)-{\mathrm{<figure-callout\; id="0"\; label="Q\; i"\; filenames="HDA0000378941490000012.png,HDA0000378941490000022.png"\; state="\{\{state\}\}">Q</figure-callout>}}_i^{}{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i+{{\mathrm{<figure-callout\; id="0"\; label="H\; i"\; filenames="HDA0000378941490000012.png,HDA0000378941490000022.png"\; state="\{\{state\}\}">H</figure-callout>}}_i}^0{\mathrm{\&Lambda;}}_i({\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i^d)+{{\mathrm{<figure-callout\; id="0"\; label="H\; i"\; filenames="HDA0000378941490000012.png,HDA0000378941490000022.png"\; state="\{\{state\}\}">H</figure-callout>}}_i}^0{\mathrm{\&Lambda;}}_i{A}_i\exp (-{\mathrm{\&Lambda;}}_{i}t)-{\mathrm{<figure-callout\; id="0"\; label="H\; i"\; filenames="HDA0000378941490000012.png,HDA0000378941490000022.png"\; state="\{\{state\}\}">H</figure-callout>}}_i^{}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&sigma;}}}_i^d+Y_i(\&CenterDot;)$

Wherein

$Y_i(\&CenterDot;)=-\mathrm{\&Delta;}{Q}_i{\stackrel{.}{\mathrm{\&sigma;}}}_i-\mathrm{\&Delta;}{H}_i{\stackrel{..}{\mathrm{\&sigma;}}}_i^d+\mathrm{\&Delta;}{H}_i{\mathrm{\&Lambda;}}_i({\stackrel{.}{\mathrm{\&sigma;}}}_i-{\stackrel{.}{\mathrm{\&sigma;}}}_i^d)+\mathrm{\&Delta;}{H}_i{\mathrm{\&Lambda;}}_i{A}_i\exp (-{\mathrm{\&Lambda;}}_{i}t)$

5), for individual spacecraft, on the basis of step (4), design Attitude Tracking control law is as follows:

$u_i=u_{\mathrm{ieq}}+u_{\mathrm{isw}}$

（9）

$=G_i^T[Q_i^0({\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-s_i)+{\mathrm{<figure-callout\; id="0"\; label="H\; i"\; filenames="HDA0000378941490000012.png,HDA0000378941490000022.png"\; state="\{\{state\}\}">H</figure-callout>}}_i^{}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&sigma;}}}_i^d-{\mathrm{<figure-callout\; id="0"\; label="H\; i"\; filenames="HDA0000378941490000012.png,HDA0000378941490000022.png"\; state="\{\{state\}\}">H</figure-callout>}}_i^{}{\mathrm{\&Lambda;}}_i({\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i^d)+{\mathrm{<figure-callout\; id="0"\; label="H\; i"\; filenames="HDA0000378941490000012.png,HDA0000378941490000022.png"\; state="\{\{state\}\}">H</figure-callout>}}_i^{}{\mathrm{\&Lambda;}}_i{A}_i\exp (-{\mathrm{\&Lambda;}}_{i}t)]-({\mathrm{\&gamma;}}_i+{\mathrm{\&lambda;}}_i)G_i^T\mathrm{sgn}\left(s_i\right)$

Wherein sgn () is sign function, is defined as follows:

$\mathrm{sgn}\left(x\right)=\left\{\begin{array}{c}1,x>0\\ 0,x=0\\ -1,x<0\end{array}\right.$

Sgn (s _i)=[sgn (s _il); Sgn (s _i2); Sgn (s _i3)], λ in formula _i>0, λ _i+ γ _ifor the switching controls flow gain, and Δ H _i, Δ Q _ibounded, establish γ _i=|| F _i|| _∞, F _ias follows:

$F_i=G_i^T{Y}_i(\&CenterDot;)+\mathrm{\&Delta;}{Q}_i{s}_i+d_i$

Now in the situation that only there is the part spacecraft can obtain the expectation attitude, becomes sliding mode controller when designed and carry out stability analysis and formation spacecrafts attitude compliance check

Before the proof consistance, first briefly prove following theorem:

Theorem: make L ∈ R ^{n * n}mean the Laplacian matrix between n formation spacecrafts,

mean to comprise the Laplacian matrix between the formation spacecrafts of " virtual leader ",

matrix P=L+diag (b) ∈ R wherein ^{n * n}.If matrix

there is a directed spanning tree in the communication topology characterized, and node n+1 (expectation attitude) is a root node of this spanning tree, and matrix P is positive definite matrix and P so ^-1b=-1 _n, b=-[b wherein _{1 (n+1)}, b _{2 (n+1)}..., b _{n (n+1)}] ^t, 1 _nfor element is the column vector of 1 n * 1 entirely.

Proof: establish the communication network topology laplacian matrix between the spacecraft colony that comprises this " virtual leader "

therefore

can prove P1 _n+ b=0, thus vectorial b can be gone out by the column vector linear list of matrix P, thereby rank ([P b])=rank (P), again because comprise that this virtual leader's communication network topology has a directed spanning tree,

, be also rank (P)=n, so matrix P is positive definite matrix and P ^-1b=-1 _n

Below become sliding formwork control law (7) when designed and carry out stability analysis

Consider the Lyapunov function of following positive definite for individual spacecraft i:

$V_i=\frac{1}{2}{s}_i^T{H}_i{s}_i\&GreaterEqual;0---\left(10\right)$

H in formula _ifor positive definite matrix, and if only if s _i=0 o'clock, V _i=0; Ask V _ito the first order derivative of time, arrange:

${\stackrel{.}{V}}_i=s_i^T{H}_i{\stackrel{.}{s}}_i+\frac{1}{2}{s}_i^T{\stackrel{.}{H}}_i{s}_i=s_i^T(H_i{\stackrel{.}{s}}_i+\frac{1}{2}{\stackrel{.}{H}}_i{s}_i)=s_i^T(H_i{\stackrel{.}{s}}_i+Q_i{s}_i)---\left(11\right)$

In formula, applied to

the character of an antisymmetric matrix.Will the time become the letter of sliding formwork control law (7) substitution above formula into:

${\stackrel{.}{V}}_i\&le;-{\mathrm{\&lambda;}}_i{s}_i^T\mathrm{sgn}\left(s_i\right)\&le;0---\left(12\right)$

While therefore when employing is designed suc as formula (7), becoming the sliding formwork control law, sliding variable s _i(t) ≡ 0,

Formula (8) is write as to the form of column vector:

$V=\frac{1}{2}{S}^T\mathrm{HS}---\left(13\right)$

In formula:

h=diag (H ₁, H ₂... H _n), easily know that H is positive definite matrix, to formula (11), differentiate obtains convolution (10):

$\stackrel{.}{V}=S^T(H\stackrel{\&CenterDot;}{S}+\frac{1}{2}\stackrel{\&CenterDot;}{H}S)=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\stackrel{.}{V}}_i\&le;0---\left(14\right)$

Therefore work as S (t) ≡ 0, define the attitude error e of i spacecraft _i=σ _i-σ _d; Write sliding variable S as matrix form:

$S={(M\&CircleTimes;I_3)}^{-1}(P\&CircleTimes;I_3)(\stackrel{\&CenterDot;}{e}+\mathrm{\&Lambda;e})-\exp (-\mathrm{\&Lambda;t})A=0---\left(15\right)$

In formula $M=\mathrm{diag}(b_{1(n+1)}+\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{1j},b_{2(n+1)}+\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{2j},\&CenterDot;\&CenterDot;\&CenterDot;b_{n(n+1)}+\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{\mathrm{nj}}),$ Λ=diag (Λ ₁, Λ ₂... Λ _n), A=diag (A ₁, A ₂... A _n), e=diag (e ₁, e ₂... e _n), verified matrix P is positive definite matrix, so matrix $P{\&CircleTimes;I}_3$ Reversible, after (13) are arranged:

$\stackrel{\&CenterDot;}{e}+\mathrm{\&Lambda;e}-{(P\&CircleTimes;I_3)}^{-1}(M\&CircleTimes;I_3)\exp (-\mathrm{\&Lambda;t})A=0---\left(16\right)$

Order

the solution of above formula is:

I wherein _3nfor element is all the column vector of 1 3n * 1, therefore e → 0 when t → ∞, that is:

σ _i→σ _j→σ _d，ω _i→ω _j→ω _d

Therefore in this situation, designed controller can be realized the collaborative tracking of attitude, and finally makes the formation spacecrafts attitude consistent.

Become in sliding formwork control law (7) and used sign function when designed, this will make system produce chattering phenomenon at equilibrium point, therefore in emulation, adopt saturation function to be similar to, and sliding-mode surface is approximated to the sliding formwork boundary layer

$\mathrm{sat}(s_i,{\mathrm{\&delta;}}_i)=\left\{\begin{array}{cc}\mathrm{sgn}\left(s_i\right)& \left|s_i\right|>{\mathrm{\&delta;}}_i\\ s_i/{\mathrm{\&delta;}}_i& \left|s_i\right|\&le;{\mathrm{\&delta;}}_i\end{array}\right.---\left(18\right)$

Due to the error that the approximate sign function of saturation function brings, system is no longer progressive stable, but progressive bounded.Embodiment

The present embodiment carries out simulating, verifying by Matlab/simulink, verifies respectively and is only having the part spacecraft can obtain the attitude consistance in expectation attitude information and two kinds of situations of flexible letter topological structure.Each spacecraft parameter is as follows

Spacecraft inertia parameter and initial attitude

The expectation attitude angular velocity:

${\mathrm{\&omega;}}_d=\left[\begin{array}{c}0.1\cos \left(0.1t\right)\\ -0.1\sin \left(0.1t\right)\\ -0.1\cos \left(0.1t\right)\end{array}\right]$

The expectation initial attitude is σ _d(0)=[0 0 0], λ in controller parameter _i=0.1, γ _i=0.01, δ _i=0.01, i=1,2 ... n, Λ _i=0.5diag (1,1,1), and the uncertain part Δ of spacecraft inertia J _i<0.2J _i, be subject to external environment condition and disturb in operational process:

${\mathrm{\&tau;}}_d=\left\{\begin{array}{c}0.002\sin \left(0.4t\right)\\ 0.003\cos \left(0.3t\right)\\ 0.004\sin \left(0.2t\right)\end{array}\right.$

Simulation result is set to 30s, for simplicity, only provides spacecraft 1,3,5 simulation result figure, and wherein two line represent spacecraft 1, and dotted line represents spacecraft 3, and dot-and-dash line represents spacecraft 5, realizes representative expectation attitude.

Provided the communication network topology structural drawing that only has a spacecraft can obtain the expectation attitude information in Fig. 2, the attitude of spacecraft and attitude angular velocity change curve are as Fig. 3, shown in Fig. 4, from Fig. 3 and Fig. 4, can find out that colony's spacecraft realized the collaborative tracking to the expectation attitude.From Fig. 5, the error partial enlarged drawing can find out that the Attitude Tracking error precision of each spacecraft is 2 * 10 ^-4;

Fig. 6 has gone out the control moment change curve of three spacecrafts, the suffered control moment part of spacecraft is used for following the tracks of the expectation attitude as can be seen from Figure 6, another part is used for offsetting the uncertain part of inertia and the caused moment of external interference, and therefore designed controller has stronger robustness.The amplitude of control moment is ± 0.5N.m that this also relatively is consistent with actual conditions.

Fig. 7 has provided the sliding variable change curve of three spacecrafts, the sliding formwork letter face numerical value of three spacecrafts approaches zero all the time as seen from the figure, owing to having used saturation function to be disappeared, trembles, therefore when initial time, little fluctuation has all appearred in each sliding-mode surface functional value

By above experimental result, can be found out, the present invention design the time become sliding mode controller and can finely complete the collaborative task of following the tracks of of attitude, and can reach very high precision, effect is fine.

Claims (3)

1. become the collaborative tracking and controlling method of Sliding Mode Attitude during distributed index, it is characterized in that: specifically comprise the following steps:

Step 1, take the rigid spacecraft formation as object, sets up the Lagrangian attitude dynamic equations of Euler one under the spacecraft body coordinate system; Obtain the attitude information of each spacecraft, ask for the corresponding synchronous expectation attitude of each spacecraft; Concrete grammar is:

Step 1.1, set up the attitude motion model of individual spacecraft;

Formation comprises n spacecraft, and it is numbered 1,2 ... n, represent that " the virtual leader " of Spacecraft formation expectation attitude is numbered n+1, for i spacecraft wherein, and i ∈ 1,2 ... n, its attitude dynamics and kinematical equation are as follows:

$J_i{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_i+{\&lsqb;{\mathrm{\&omega;}}_i\&rsqb;}^{\&times;}{J}_i{\mathrm{\&omega;}}_i=u_i+d_i---\left(1\right)$

${\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i=G\left({\mathrm{\&sigma;}}_i\right){\mathrm{\&omega;}}_i---\left(2\right)$

J _i∈ R ^{3 * 3}be the inertia matrix of i spacecraft, and

,

the nominal value that means the inertia battle array, Δ J _ithe uncertainty that means the inertia battle array; ω _i∈ R ³be the attitude angular velocity expression in body coordinate system of i rigid body spacecraft with respect to inertial coordinates system, σ _i∈ R ³be the correction Douglas Rodríguez parameter of i spacecraft attitude, u _i, d _i∈ R ³mean respectively control moment and disturbance torque that i spacecraft is subject to, () ^*mean vectorial antisymmetric matrix operator; Wherein:

$G\left({\mathrm{\&sigma;}}_i\right)=\frac{1}{2}\&lsqb;\left(\frac{1-{\mathrm{\&sigma;}}_i^T{\mathrm{\&sigma;}}_i}{2}\right)I_3+{\mathrm{\&sigma;}}_i^{\&times;}+{\mathrm{\&sigma;}}_i{\mathrm{\&sigma;}}_i^T---\left(3\right)$

The attitude dynamic equations of its corresponding Lagrangian Form is:

$H_i\left({\mathrm{\&sigma;}}_i\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&sigma;}}}_i+Q_i({\mathrm{\&sigma;}}_i,{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i){\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i={\mathrm{\&tau;}}_i---\left(4\right)$

In formula $H_i\left({\mathrm{\&sigma;}}_i\right)=G^{-T}\left({\mathrm{\&sigma;}}_i\right)J_i{G}^{-1}\left({\mathrm{\&sigma;}}_i\right),Q_i({\mathrm{\&sigma;}}_i,{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i)=-G^{-T}\left({\mathrm{\&sigma;}}_i\right)J_i{G}^{-1}\left({\mathrm{\&sigma;}}_i\right)\stackrel{\&CenterDot;}{G}\left({\mathrm{\&sigma;}}_i\right)G^{-1}\left({\mathrm{\&sigma;}}_i\right)-G^{-T}\left({\mathrm{\&sigma;}}_i\right)$

τ _i=G ^-1(σ _i) (u _i+ d _i), H wherein _i(σ _i) be a positive definite symmetric matrices, and

for antisymmetric matrix; With G _i, H _i,

Δ H _i, Q _i, Δ Q _imean respectively G (σ _i), H _i(σ _i),

Δ H _i(σ _i),

Step 1.2, for i spacecraft, synchronously expect attitude

be defined as follows:

${\mathrm{\&sigma;}}_i^d=\frac{b_{i(n+1)}{\mathrm{\&sigma;}}_d+\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{\mathrm{ij}}{\mathrm{\&sigma;}}_j}{b_{i(n+1)}+\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{\mathrm{ij}}}---\left(5\right)$

σ in formula _ithe attitude that means i spacecraft, a _ijspacecraft i means the correspondence between spacecraft i and spacecraft j, if can obtain the attitude information of j, a _ij=1, otherwise, a _ij=0; If i spacecraft can obtain expectation attitude information a _d, b _{i (n+1)}=1, otherwise b _{i (n+1)}=0;

Step 2, to each spacecraft, become the sliding-mode surface function while choosing index, ask for the correlation parameter in the sliding-mode surface function, makes at initial time, and the initial error of system is positioned on sliding-mode surface;

For i spacecraft, the time become sliding-mode surface variable s _i(t) be:

$s_i\left(t\right)=({\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i^d)+{\mathrm{\&Lambda;}}_i({\mathrm{\&sigma;}}_i-{\mathrm{\&sigma;}}_i^d)-\exp (-{\mathrm{\&Lambda;}}_{i}t)A_i---\left(6\right)$

Λ in formula _i∈ R ^{3 * 3}and be positive definite matrix, A _i∈ R ³, t is time variable;

At initial time, systematic error is positioned on sliding-mode surface, i.e. s _i(0)=0; Obtain:

$A_i=\&lsqb;{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i\left(0\right)-{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i^d\left(0\right)\&rsqb;+{\mathrm{\&Lambda;}}_i\&lsqb;{\mathrm{\&sigma;}}_i\left(0\right)-{\mathrm{\&sigma;}}_d\left(0\right)\&rsqb;---\left(7\right)$

Step 3, to the sliding-mode surface function differentiate of step 2 design, then premultiplication H _i:

$\begin{array}{c}{H}_i{\stackrel{\&CenterDot;}{s}}_i=H_i({\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&sigma;}}}_i-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&sigma;}}}_i^d)+H_i{\mathrm{\&Lambda;}}_i({\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i^d)+H_i{\mathrm{\&Lambda;}}_i{A}_i\exp (-{\mathrm{\&Lambda;}}_{i}t)\\ =G_i^{-T}(u_i+d_i)-Q_i{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-H_i{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&sigma;}}}_i^d+H_i{\mathrm{\&Lambda;}}_i({\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i^d)+H_i{\mathrm{\&Lambda;}}_i{A}_i\exp (-{\mathrm{\&Lambda;}}_{i}t)\\ =G_i^{-T}(u_i+d_i)-Q_i^0{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i+H_i^0{\mathrm{\&Lambda;}}_i({\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i^d)+H_i^0{\mathrm{\&Lambda;}}_i{A}_i\exp \left({-\mathrm{\&Lambda;}}_{i}t\right)-H_i^0{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&sigma;}}}_i^d+Y_i(\&CenterDot;)\end{array}---\left(8\right)$

Wherein, $Y_i(\&CenterDot;)=-{\mathrm{\&Delta;Q}}_i{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-\mathrm{\&Delta;}{H}_i{\stackrel{..}{\mathrm{\&sigma;}}}_i^d+\mathrm{\&Delta;}{H}_i{\mathrm{\&Lambda;}}_i({\stackrel{.}{\mathrm{\&sigma;}}}_i-{\stackrel{.}{\mathrm{\&sigma;}}}_i^d)+\mathrm{\&Delta;}{H}_i{\mathrm{\&Lambda;}}_i{A}_i\exp (-{\mathrm{\&Lambda;}}_{i}t)$

Step 4, become the sliding formwork control law during design index; Control law comprises two parts: a part is in order to the nonlinear terms in bucking-out system dynamics; Another part is for offsetting the uncertain part of inertia and external interference;

For i spacecraft, become the sliding formwork control law during design as follows:

$\begin{array}{c}{u}_i=u_{\mathrm{ieq}}+u_{\mathrm{isw}}\\ =G_i^T\&lsqb;Q_i^0({\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_i-s_i)+H_i^0{\stackrel{..}{\mathrm{\&sigma;}}}_i^d-H_i^0{\mathrm{\&Lambda;}}_i({\stackrel{.}{\mathrm{\&sigma;}}}_i-{\stackrel{.}{\mathrm{\&sigma;}}}_i^d)+H_i^0{\mathrm{\&Lambda;}}_i{A}_i\exp (-{\mathrm{\&Lambda;}}_{i}t)\&rsqb;-({\mathrm{\&gamma;}}_i+{\mathrm{\&lambda;}}_i)G_i^T\mathrm{sgn}\left(s_i\right)\end{array}----\left(9\right)$

Wherein sgn () is sign function, is defined as follows:

$\mathrm{sgn}\left(x\right)=\left\{\begin{array}{c}1,x>0\\ 0,x=0\\ -1,x<0\end{array}\right.$

Sgn (s _i)=[sgn (s _i1); Sgn (s _i2); Sgn (s _i3)], λ in formula _i0, λ _i+ γ _ifor the switching controls flow gain, and Δ H _i, Δ Q _i, d _ibounded, and || F _i|| _∞bounded, establish γ _i=|| F _i|| _∞, F _ibe expressed as:

$F_i=G_i^T{Y}_i(\&CenterDot;)+\mathrm{\&Delta;}{Q}_i{s}_i+d_i$

Step 5, in formation, spacecraft obtains the attitude information of expectation attitude and neighbours' spacecraft according to the correspondence of self and other spacecrafts, become the input of sliding formwork control law when distributed in using attitude information as step 4, calculate each spacecraft and carry out the needed control moment of the collaborative tracking of attitude, the topworks of each spacecraft will produce these control moments and act on respectively corresponding spacecraft, and the attitude dynamic equations obtained by step 1 is tried to achieve ω _i, and then make σ _ifollow the tracks of the synchronous expectation attitude of each spacecraft, finally realize that the attitude of this spacecraft colony is consistent.

2. become the collaborative tracking and controlling method of Sliding Mode Attitude during distributed index according to claim 1, it is characterized in that: formation spacecrafts is rigid body, the equal bounded of the uncertain part of the external interference that system is suffered and inertia.

3. become the collaborative tracking and controlling method of Sliding Mode Attitude during distributed index according to claim 1, it is characterized in that: formation spacecrafts expectation attitude is made as " virtual leader ", there is a directed spanning tree in the communication topology that comprises the Spacecraft formation system of " virtual leader ", all or part of spacecraft can obtain " virtual leader " attitude, even the change Communication topology, " virtual leader " is the root node of communication topology all the time.

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