GB2585253A - Dynamic gain control method for multi-spacecraft consensus - Google Patents
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Abstract
A dynamic gain control method for a multi-spacecraft consensus, comprising: building a spacecraft kinematics and dynamics model in an inertial coordinate frame, building a multi-spacecraft interaction model, constructing a distributed finite-time disturbance observer, designing a dynamic gain controller, and implementing a control strategy for the multi-spacecraft consensus.
Description
DYNAMIC GAIN CONTROL METHOD FOR MULTI-SPACECRAFT CONSENSUS
FIELD OF THE INVENTION
[0001] The patent relates to spacecraft control technology, in particular to a spacecraft robust finite-time control method, and to a dynamic gain control method for multi-spacecraft consensus based on a distributed finite-time disturbance observer.
BACKGROUND OF THE INVENTION
[0002] With the deepening of space research and the improvement of space application capability, the demand for on-orbit service technology is increasingly urgent. The research on on-orbit service technology has attracted more and more attention. These research can be used for orbital garbage removal, on-orbit maintenance and other tasks. The space missions have become more and more complex with the deepening of research, resulting incapability of completing the task well for a single spacecraft. Therefore, there are good application prospects for multi-spacecraft system to cooperate in completing complex tasks. The spacecraft structure is a typical nonlinear system with strong coupling. In addition, there are various disturbances in space, and the non-ideal characteristics of satellite-borne actuators will further increase the uncertainties of system. In order to make multi-spacecrafts successfully accomplish space mission, it is necessary to ensure that the designed distributed attitude control algorithm can still achieve multi-spacecrafts consensus in the presence of the above uncertainties.
[0003] At present, most consensus control strategies for multi-spacecrafts can only obtain asymptotic stability results, and the robustness is poor. For multi-spacecrafts with strong coupling, strong nonlinear and external disturbance, it is of great significance to improve the robustness performance, control accuracy and consensus convergence speed of the multi-spacecraft system. In order to further improve the control accuracy and uniform convergence speed, as well as the robustness of the multi-spacecraft system, the dynamic gain control method based on distributed finite time disturbance observer is designed to obtain the finite time stability results.
[0004] For the traditional disturbance observer, the disturbance in the system is estimated and compensated in real time. However, most disturbance observers cannot obtain finite-time stability results and are usually designed for a single spacecraft. At the same time, there is no theoretical guidance for parameter adjustment of third-order distributed disturbance observers, which brings great difficulties to engineering applications.
[0005] In the traditional finite time control, the sliding mode control can make the system, under the sliding mode, has nothing to do with the system parameters uncertainty and external disturbance. Therefore, the sliding mode control can provide quick response. However, the sliding mode control having high frequency chatter not only destroys the precision of system, but also increases the system's energy consumption, which brings a great burden for the spacecraft energy.
SUMMARY OF THE INVENTION
[0006] In order to overcome the shortcomings of the existing technology, a dynamic gain control method for multi-spacecraft consensus is provided in this disclosure. By establishing a spacecraft system model in an inertial coordinate frame, designing a distributed finite-time disturbance observer for a spacecraft, solving a linear matrix inequality to obtain parameters of the observer, designing a dynamic gain controller to control the spacecraft and compensating in real time the uncertainty of the system, it is possible that the control algorithm has strong robustness, and a higher control accuracy and a faster response speed can be achieved to facilitate the engineering implementation.
[0007] In order to achieve the above object, the present invention is implemented by the following technical solutions.
A dynamic gain control method for multi-spacecraft consensus includes the following steps: Step 1: Building a spacecraft kinematics and dynamics model in an inertial coordinate frame; Step 2: Building a multi-spacecraft information interaction model; Step 3: Constructing a distributed finite-time disturbance observer; Step 4: Designing a finite-time dynamic gain controller; Step 5: Implementing a control strategy for the multi-spacecraft consensus. [0008] The further improvement of the present invention is as follows. [0009] The specific method of Step 1 is as follows.
[0010] Equations (1)-(2) of the spacecraft attitude kinematics and dynamics model are considered:
T
el, =12 (q,41,3 q, )a),,q,4= --2q, a), (1) = fp +ii, +1/, (2) where a subscript i indicates the i th spacecraft, and in this disclosure, co, e 143 represents an angular velocity, unit quatemion(q,,q,4) denotes an attitude orientation of the spacecraft i, satisfying (IT q,24 1, and J, e 143' and 11, ER' denote a positive definite inertia matrix and a control torque of the spacecraft i, respectively, where U0 =0, and v, E R3 is unknown external disturbances including environmental disturbances, solar radiation and magnetic effects, etc., and 0' is a skew-symmetric matrix and defined as: 0 -a3 a, a = a3 0 -a, -a, a, 0 [0011] The equations (1)-(2) are transformed to a Lagrange equation: IV,(q)q, + C,(q,, q = ,(0 +12, (3) where, NI,(q,)= (q,)JZ-'(q,) (q,),I,Z 1(4)Z(q)7. 1-(q,)-Z (q)(1,Z 1(q,)q,)' Z 1(q) 11 1-H gill' Z(q)=-q,, +q,q, + 2 z ri(t)= Z (q,)1t,(2,= Z (q,)v, where M; (q,) is a positive definite inertia matrix, and J, is defined as follows: 21.31 0.2 -0.5 0.2 20.37 0.3 -0.5 0.3 19.64 [0012] The specific method of Step 2 is as follows.
[0013] N +1 spacecrafts are considered, where i -0 is regarded as a leader spacecraft and i =1, 2,.. N are regarded as follower spacecrafts. The multi-spacecraft information interaction model is established as a directed topology g = (2,5), where V = va,* * *,v, is a set of the respective spacecrafts, and £c V x V denotes a set of all of transmission. An adjacency matrix of the follower is defined as A = [a R and if the attitude information of the follower spacecraft / is transmitted directly to the spacecraft i, a" > 0 and otherwise cr, = 0. A diagonal element of the adjacency matrix is a, =0. A set of all of neighborhood for the spacecraft i is denoted by N; . A Laplacian matrix 1 =[L,,] RA KA is given, where Li = Li_ if i = 1 and Lil = -a, if i 1. When the spacecraft i can receive the leader's attitude information directly, b, > 0 and otherwise k -0. A matrix B -cliagIbi,12, * * * is defined.
[0014] Each follower spacecraft can receive information from the leader spacecraft directly or indirectly.
[0015] The specific method of Step 3 is as follows.
[0016] A consensus attitude error of the local neighborhood for the i th follower spacecraft is defined as follows: = (q, q1)+ b,(q, go) = (4; -4,)+1),4, (4) where 4, = -q% is a leading-following consensus attitude error.
[0017] A third-order distributed finite-time disturbance observer is constructed as: 1 m rn,, Z = Z. , -p e(5) ±, 2 (0 = -,02,8,3e,7 -114,1(g,)x(C z-,(0) (6) ±0 (I) = where fig, ,8,2, fie, are gains of the observer, e (7) = = 2a -1 and m = 3a -2 are three positive definite constants that are less than 1 and satisfy -2 <a <1, and where e,1= L, 1.1 b. z Jz, I is the observed error, P is a positive definite constant, and z", zi2 and z 3 is the observer state estimation of the leading-following consensus error.
[0018] The parameters of the disturbance observer can be adjusted by solving the following linear matrix inequality (8), so that the disturbance observer can achieve a good estimation performance: 211 (0 0 f)+ (0 P, )21 + Amax(/' +13) (541 (0 0 D,P,)+ (0 RP/)-4) < 0 (8) i, nn) where 0 denotes a Kronecker product, 2",1,,(*) and 2"," (.) are maximum and minimum eigenvalues of a matrix, respectively,'Nis an AT dimensional identity matrix, P, is a symmetric positive definite matrix and p,,L, is defined as an element in a first row and a first column of p, A= 04 -(L+B)0,13/;),A, = D, = diag,{9,1-a,*** ,20 -7)1, A=[,1p1, and 0 =diag{-Y1,* * * ,Y'} is a positive definite diagonal matrix, defining 1= [1,* * * , l]r to obtain: Ni x =[;,* * * , = (I, + 13)-11, y =[yi,*-* , = (I, + 1 [0019] The specific method of Step 4 is as follows. [0020] The following state transformations are given: ail= 41 I '), 0-, 2= el, I (F2471) where F, and u, are adaptive functions, -4, and 4, are pre-transformation states, aLl and 72 are post-transformation states.
[0021] Based on the post-transformed system states a, = [cm the finite-time dynamic gain controller is obtained as: T(I)= -11,1"'s -z,, (9) where K, is a gain vector of the controller, z,;(t) is the unknown disturbance information observed by the disturbance observer for real-time compensation of the internal uncertainty of the system and the changeable external disturbance. The adaptive law is designed as: sign(' 6,11)E, 1(0) > 1 (10) vti --a)tv-,t)'-a +af S,a-,, t < v, = (1-0)o, (11) o-, otherwise where v,2, i are parameters of the adaptive law, and 17,(0), v,o are initial values of the adaptive law, respectively.
[0022] By solving the following linear matrix inequality (12), the controller parameters can be adjusted to achieve a good control effect for the multi-spacecraft system.
S, (A, + B,K,)+ (A, + B,K,)r S, (3,02, + (132,8,) 0 (12) where 5, is a symmetric positive definite matrix, and B, and 0, are defined as 0 0 0 1 432,-ding{et, 2a -1} [0023] The specific method of Step 5 is as follows.
[0024] A control torque r,(t) is obtained and it is brought into the spacecraft system model (4) in the inertial coordinate frame. According to the control strategy, the distributed finite-time disturbance observer and dynamic gain controller are designed for the spacecraft, respectively, so as to control the spacecraft so that the multi-spacecraft consensus can be achieved.
[0025] Compared with the prior art, the invention has the following beneficial advantages: [0026] (1) A distributed finite-time disturbance observer is proposed in this invention. By solving linear matrix inequalities, it is easy to adjust the parameters of the distributed finite-time disturbance observer. The difficulty of adjusting the parameters of third-order distributed finite-time disturbance observer is solving, which is convenient for engineering realization. (2) A finite-time dynamic gain controller is designed, which guarantees the continuity of control input, improves the robustness of the system and achieves better control performance while obtaining the results of finite-time consensus. (3) Based on the distributed control strategy, the multi-spacecraft information exchange in the directed topology is realized, the information transmission is reduced, and the multi-spacecraft application scenarios are greatly expanded.
BRIEF DESCRIPTION OF THE DRAWINGS [0027] Fig. 1 shows the flowchart of the present invention.
DETAILED DESCRIPTION OF THE EMBODIMENTS
[0028] The technical solutions in the embodiments of the present invention are clearly and completely described in the following with reference to the accompanying drawings in the embodiments of the present invention, in order for those skilled in the art better understand the solutions of the present invention. It is apparent that the described embodiments are only a part of the embodiments of the invention, not all of the embodiments, and are not intended to limit the scope of the disclosure. In addition, descriptions of well-known structures and techniques are omitted in the following description in order to avoid unnecessary obscuring the concepts disclosed in the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative efforts shall fall within the scope of the present invention.
[0029] Various structural schematics in accordance with embodiments of the present disclosure are shown in the drawings. The figures are not drawn to scale, and some details are exaggerated for clarity of illustration and some details may be omitted. The various shapes of regions and layers, and the relative sizes and positional relationships therebetween as illustrated in the drawings are merely exemplary, and may vary in practice due to manufacturing tolerances or technical limitations, and those skilled in the art can additionally design regions/layers having different shapes, sizes, and relative positions according to actual needs..
[0030] In the context of the present disclosure, when a layer/element is referred to as being "on" the other layer/element, the layer/element may be located directly on the said other layer/element, or an intermediate layer/element may be present therebetween. In addition, if a layer/element is "on" the other layer/element in an orientation, the layer /element may be "under" the said other layer/element when the orientation is reversed.
[0031] It is to be understood that the terms "first", "second" and the like in the specification and claims of the present invention are used to distinguish similar objects, and are not necessarily used to describe a particular order or order. It is to be understood that the numerals so used may be interchanged where appropriate, so that the embodiments of the invention described herein can be implemented in sequences other than those illustrated or described herein. In addition, the terms "include" and "have" and any variants thereof are intended to cover a non-exclusive inclusion. For example, a process, method, system, product, or device that comprises a series of steps or units is not necessarily limited to those steps or units that are clearly listed, but may include other steps or units that are not clearly listed or that are inherent to the process, method, product, or device.
[0032] The present invention will be further described in detail below with reference to the accompanying drawings.
[0033] As shown in Fig. 1, a dynamic gain control method for multi-spacecraft consensus of the present invention is realized by the following steps.
[0034] Step 1: Building a spacecraft kinematics and dynamics model in an inertial coordinate frame.
[0035] The equations (1)-(2) of the spacecraft attitude kinematics and dynamics model are considered: =' (q +gico =-1 -qT co (1)4 J(O, u, +v, (2) where subscript i indicates the i th spacecraft, and in this disclosure, co, E R3 represents the angular velocity, the unit quaternion (q,q") denotes the attitude orientation of the spacecraft satisfying, and J, E R3" and II, E R3 denote the positive definite inertia matrix and the control torque of spacecraft i, respectively, where a0 = 0, and v, ER: is the unknown external disturbances including environmental disturbances, solar radiation and magnetic effects, etc., and 0 is a skew-symmetric matrix and defined as: 0 a, a3 0 -a, a 0 [0036] The equations (1)-(2) can be transformed to the Lagrange equation: A 1,(T)q, + C,(q" 4)4, = i(i)± 12, (3) where, 111,(q,)= (q,)J,Z-'(q,) C,(q"4,) = -z-T (cf)J,Z1(q3Z(OZ-'(q,)- (q,)(J,Z1(ci)q,)/ Z-'(q,) 1-H q Z(q,)=-q, +q,cf,T + ' I 2 2 y r, (0= ZT (ch)n, , S2, = Z-T (q)v, where M, (q;) is the positive definite inertia matrix.
[0037] The parameters of J, are defined as follows: 21.31 0.2 -0.5 0.2 20.37 0.3 -0.5 0.3 19.64 [0038] Step 2: Building the multi-spacecraft information interaction model.
[0039] Firstly, N +1 spacecrafts are considered in this disclosure. We regard i = 0 as the leader spacecraft and regard =1,2,. N as the follower spacecrafts. The multi-spacecraft information interaction model is established as a directed topology = (V, e) , where V = vo, * * *,1)\. is the set of indexes for the corresponding spacecrafts, and ScVxV denotes the set of all of the transmission. The adjacency matrix of the follower is defined as A =[a,1] e, and if the attitude information of the follower spacecraft / is transmitted directly to the spacecraft i, a,, > 0 and otherwise a" = 0. Note that the diagonal element of the adjacency matrix an = 0. The set of all of neighborhood for the spacecraft i is denoted by = J, = The Laplacian matrix = [L,1] e RN >A is given, where L,2 = - if i =1 and a if i #1. When the spacecraft i can receive the leader's attitude information directly, b, >0 and otherwise k =0. We define the following matrix B = diag{b,,b2} . We assume that each follower spacecraft can receive information from the leader spacecraft directly or indirectly.
[0040] In an example, four spacecrafts are considered, one of which is the leader spacecraft, and the other three are the follower spacecrafts. The related parameters of topology are given as follows. 0 0 0
L= -1 1 0, B = dictg11,0,01 0 -1 1 [0041] Step 3: Constructing the distributed finite-time disturbance observer.
[0042] The consensus attitude error of the local neighborhood for the i th follower is defined as follows: =la"g -q,)+b,(q, -q")= ci" (i); - bi4, (4) where 4; = q; -q, is the leading-following consensus attitude error.
[0043] Considering the second-order equations of spacecraft attitude kinematics and dynamics, the third-order distributed finite-time disturbance observer is constructed to estimate the system uncertainties and the environmental disturbances: P fidedij (5) 42(0= (t)-p2 Ine2,8,e,72 (q,)x (q,,d,)z,2( )-2-, (0) (6) (7) where Po A2, are gains of the observer, = = 2a -1 and m, ", =3a -2 are three positive definite constants that are less than 1 and satisfy -3 <a<1, and where =IL,z,j+b,z is the observed error, P is a positive definite constant, an i=1 and is the observer state estimation of the leading-following consensus error [0044] The parameter adjustment method for the third-order distributed finite-time disturbance observer is given. The parameters of the observer can be adjusted by solving the following linear matrix inequality, so that the disturbance observer can achieve a good estimation performance: Ar (0 0 Pi) + (0 0 p) + Amdx(f " +13) (Ar (0 0 Dia+ (0 0 0P,)-A) < 0 (8) Amdo( 0) where ® denotes the Kronecker product, J.) and &m(*) are the maximum and minimum eigenvalues of a matrix, respectively, I,. is an N dimensional identity matrix, P, is a symmetric positive definite matrix and we define that p,", is the element in the first row and the first column of P, , ° 0 = D, = (hag{ 0,1-a,. * , 2(1-a)/ ,I3, =[1.17). flu, P0,1 and 0 =cliagl-Y1 is a positive definite diagonal matrix, defining 1= , we can -1 xh obtain: x = [xi, * * *, = (L + ,t3) 11, y = bti, * * , = (L + ,6) Ti.
[0045] In an example, p =1.5, a = 0.75, fl", fin are given as A1-tliag(95.8767, 95.8767, 95.87671 /8,2=d/0g-{68.3535, 68.3535,68.3535} 11,,-diagi 15.6706, 15.6706, 15.67061.
[0046] Step 4: Designing the finite-time dynamic gain controller.
[0047] In order to obtain a higher control accuracy and a faster response speed, we consider design finite-time dynamic gain control algorithm to achieve finite-time stability. [0048] The following state transformations are given: co = I = I (Fi2tr'2) where F, and tt, are the adaptive functions, 4, and 4, are the pre-transformation states, a, and 6., are the post-transformation states.
[0049] Based on the post-transformed system states a, = [a2]'the finite-time dynamic gain controller is obtained as: t-,(t)= K zi", (9) where K, is a gain vector of the controller, (t) is the unknown disturbance information observed by the disturbance observer for real-time compensation of the internal uncertainty of the system and the changeable external disturbance. The adaptive law is designed as: sign( F(0) >1 (10) VI' -(1-a)0,01 + < vF -a)0; (11) 0-1TS9-1, otherwise where v", I, are parameters of the adaptive law, and.F (0), u," are the initial values of the adaptive law, respectively.
[0050] The parameter adjustment method for the adaptive finite-time controller is given. By solving the following linear matrix inequality, the controller parameters can be adjusted to achieve a good control effect for the multi-spacecraft system.
S,(A, + 13K,)+ (A, +13,K,)] +17,(S,0 + 02,S,) 0 (12) where 5, is a symmetric positive definite matrix, and B, and (13,, are defined as 13= [0,1] , = diag{ce, 2a -1} [0051] In an example, the adjustable parameters IC,,vil,eue. F, (0) and t);0 are written as: ICI,= [1.5 1.5], o, =0.1,F,(0)= 3, vil + 2.4, + 0.5 [0052] Step 5: Implementing the control strategy for the multi-spacecraft consensus.
[0053] Finally, the control torque r(t) is obtained and it is brought into the spacecraft system model (4) in the inertial coordinate frame. According to the control strategy, the distributed finite-time disturbance observer and dynamic gain controller are designed for the spacecraft, respectively, so as to control the spacecraft so that the multi-spacecraft consensus can be achieved.
[0054] The above content is only for explaining the technical idea of the present invention, and the scope of protection of the present invention cannot be limited thereby. Any changes made on the basis of the technical solutions according to the technical idea of the present invention fall within the protection scope of the claims of the present invention.
I I
Claims (7)
- WHAT IS CLAIMED IS: 1. A dynamic gain control method for multi-spacecraft consensus comprising: Step 1: Building a spacecraft kinematics and dynamics model in an inertial coordinate frame, Step 2: Building a multi-spacecraft information interaction model; Step 3: Constructing a distributed finite-time disturbance observer; Step 4: Designing a finite-time dynamic gain controller; Step 5: Implementing a control strategy for the multi-spacecraft consensus.
- 2. According to the dynamic gain control method of claim 1, wherein in the step 1, Equations (1)-(2) of the spacecraft attitude kinematics and dynamics model are illustrated (q",/, +qi)ot,a,4= , 2 o, (1) = +it +1/, (2) where a subscript i indicates the i th spacecraft, co,ER1 represents an angular velocity, unit quaternion (q, q,4) denotes an attitude orientation of the spacecraft i, satisfying ci,r cL+ ce, =1, and.l,c R3x3 and zt,c R3 denote a positive definite inertia matrix and a control torque of the spacecraft i, respectively, where no -0, and v, e 12' is unknown external disturbances including environmental disturbances, solar radiation and magnetic effects and the like, and ()x is a skew-symmetric matrix, which is defined as: 0 -nt3 0, a. = a, 0 -a 1 -a, a, 0 and wherein the equations (1)-(2) are transformed to a Lagrange equation: A I i(q)q, + C,(q" 4)4, = i(i)± 12, (3) where as: Al,(q,)= (q,)J,Z-I(q,) C,(q,,q,)= -Z (C1,),1 (CI,)7., (CI ?).7., (q,)- (q i)("1,Z, 1(q)q,)) Z 1(q,) 1-H q11 Z(q,)=-q, + ' I ri(t)= Z (q)tt,512, = Z (g)v, where M, (q,) is a positive definite inertia matrix, and J, is defined as: 21.31 0.2 -0.5 0.2 20.37 0.3 -0.5 0.3 19.64
- 3. According to the dynamic gain control method of claim 2, wherein in the step 2, N +1 spacecrafts are considered, where i= 0 is regarded as a leader spacecraft and i =1, 2,... N are regarded as follower spacecrafts, wherein the multi-spacecraft information interaction model is established as a directed topology Q = (V, 8), where V = is a set of the respective spacecrafts, and Ec V x V denotes a set of all of transmission, wherein an adjacency matrix of the follower spacecraft is defined as A =[all] E R \A and if the attitude information of the follower spacecraft I is transmitted directly to the spacecraft,a,10 and otherwise a,1 = 0, where a diagonal element of the adjacency matrix is a" = 0, wherein a set of all of neighborhood for the spacecraft / is denoted by N,, wherein a T.aplacian matrix L =[L,JERN/N is given, where L, = ET, v if i -/ and if i /, wherein h,> 0 when the spacecraft i receives the leader's attitude information directly, and otherwise h, =0, and wherein a matrix 13 = diaglb,,h, * * * is defined.
- 4. According to the dynamic gain control method of claim 3, wherein each follower spacecraft receives information from the leader spacecraft directly or indirectly.
- 5. According to the dynamic gain control method of claim 4, wherein in the step 3, A consensus attitude error of the local neighborhood for the i th follower spacecraft is defined as: 4-0=la,(c4-0+1),(q,-(70)=Lajij,-4,)+1),(1, (4) where 4, = q, -q0 is a leading-following consensus attitude error, wherein a third-order distributed finite-time disturbance observer is constructed as: = p p eu (5) (0-P flue, -A11'((t)x(Ci(qi,ci)z,i,(0-r (0) (6) ±,3 (1) *Beil (7) where ll,,,,8,,,,g,, are gains of the observer, = a, m,,, = 2a -1 and in, 3 =3a-2 are three positive definite constants that are less than 1 and satisfy -<ce <1, and where ed= - is the observed error, P is a positive definite constant, and z,,, z,, and z,3 is the observer state estimation of the 1 eadi ng-following consensus error, wherein the parameters of the disturbance observer is adjusted by solving the linear matrix inequality (8), so that the disturbance observer achieve the estimation performance: 21r(QOP,)-k(OO1',);4+ Amax(L+8c))'()(ArrQ0DP)+COODP)21)< 0 (8) min where 0 denotes a Kronecker product, 2"(.) and 2.11. (.) are maximum and minimum eigenvalues of a matrix, respectively, is an N dimensional identity matrix, is a symmetric positive definite matrix and p,") is defined as an element in a first row and a first column of P, , and wherein A = (IN 0 A,-(I, +13)0 AC,), A, = 00 01,A= D, = cliagf,0,1 -a,. * 2(1-(1)1,71, i,2.* 1371,1 11, and Q = dictg{L, * * * , is a positive definite diagonal matrix, defining 1 = [1,* * *,1]1 to obtain: x = [x1, = (L + .6) 11, Y = Dip ',YN lT = (L *B) T 1 *
- 6. According to the dynamic gain control method of claim 5, wherein in the step 4, the following state transformations are given: = -4, /(F,10:7" ), 012 = -4, I where h and v, are adaptive functions, 4, and 4, are pre-transformation states, all and are post-transformation states, wherein based on the post-transformed system states a, =[a,, the finite-time dynamic gain controller is obtained as: 2,(I) = -z,, (9) where K, is a gain vector of the controller, z,;(0 is the unknown disturbance information observed by the disturbance observer for real-time compensation of the internal uncertainty of the system and the changeable external disturbance, and where the adaptive law is designed as: F; _ sign( 0-11)1"; F,(0) >1 (10) -(-a)tv-,t)i-rx + af S,a, t< 0,0 v, = (1-a)zu, (11) otherwise where v,I, Tv-, are parameters of the adaptive law, and 13;(0), oo are initial values of the adaptive law, respectively, and wherein by solving the linear matrix inequality (12), the controller parameters are adjusted to achieve the control effect for the multi-spacecraft system.S, (A, + B,K,)+ (A, + B,K,)r S i(S 2, + (132,S,) 0 (12) where S, is a symmetric positive definite matrix, and B, and (1),, are defined as: B, =[0, if, W21 = dictg{a,2a -1}
- 7. According to the dynamic gain control method of claim 6, wherein in the step 5, the control torque r,(t) is obtained and it is brought into the spacecraft system model (4) in the inertial coordinate frame, and wherein according to the control strategy, the distributed finite-time disturbance observer and dynamic gain controller are designed for the spacecraft, respectively, so as to control the spacecraft so that the multi-spacecraft consensus is achieved.
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