CN113885563A - Spacecraft formation orbit coordination and connectivity maintenance control method - Google Patents

Spacecraft formation orbit coordination and connectivity maintenance control method Download PDF

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CN113885563A
CN113885563A CN202111214638.1A CN202111214638A CN113885563A CN 113885563 A CN113885563 A CN 113885563A CN 202111214638 A CN202111214638 A CN 202111214638A CN 113885563 A CN113885563 A CN 113885563A
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spacecraft
formation
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orbit
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靳锴
雷耀麟
刘厦
宗茂
李思男
何文志
薛向宏
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CETC 54 Research Institute
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    • G05D1/10—Simultaneous control of position or course in three dimensions
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Abstract

The invention discloses a spacecraft formation orbit coordination and connectivity maintenance control method, and belongs to the technical field of aerospace. Aiming at a spacecraft formation control cooperative control task, a spacecraft formation dynamics model is constructed through spacecraft relative motion modeling; constructing a virtual potential field in space through potential function design, so that a potential function obtains a minimum value at an expected configuration position, thereby ensuring orbital coordination and connectivity among spacecrafts and keeping the spacecrafts in an expected relative state; and controlling the spacecraft to a desired relative position with a predetermined accuracy within a predetermined time by adopting a preset performance control method. The invention adopts the idea of artificial potential function and preset performance control, and accurately and effectively completes spacecraft formation and control.

Description

Spacecraft formation orbit coordination and connectivity maintenance control method
Technical Field
The invention belongs to the technical field of aerospace, and particularly relates to a spacecraft formation orbit coordination and connectivity maintenance control method.
Background
The micro spacecraft has the advantages of low cost, good flexibility, high reliability and the like, and is widely applied to the fields of ground detection, fault satellite maintenance, space debris cleaning, deep space detection and the like. In order to realize complex functions, formation cooperation needs to be completed through distributed cooperative control when a communication network is always in a connected state. The conventional spacecraft formation cooperative control method mainly comprises the following steps: a multiple-input multiple-output method, a cyclic method, a behavioral method, a virtual structure method, a master-slave method, and the like [1 ]. The main idea of behavior-based formation control is to break up the whole task into a number of different behaviors, such as collision avoidance, formation reconstruction, formation maintenance and object tracking etc. [2 ]. The document [3] decomposes the whole task into different behaviors such as collision avoidance, formation maintenance, speed coordination, expected configuration and the like, and realizes the coordinated control of deep space exploration spacecraft formation. The document implements the assignment of the individual tasks by weighting, and has a limited consideration for situations where conflicts occur between different activities. To solve the above problem, document [4] ensures that a high-priority subtask is completed preferentially when a plurality of subtasks collide, by projecting a low-priority task onto a null space of a high-priority subtask. The virtual structure method is another important formation control method, and the main idea thereof is to view the formation shape as a virtual structure or a rigid body, and to maintain the formation shape by minimizing a position error between the virtual structure and an actual formation position. Document [5] proposes a multi-robot formation cooperative control method based on a virtual structure. In the virtual structure method, strict geometric relationships are maintained between all robots and with a reference frame [6 ]. Document [7] firstly generates a relative orbit and attitude reference instruction of a spacecraft by using a virtual structure method, and then tracks the reference instruction by using a self-adaptive sliding mode control method, thereby realizing attitude and orbit cooperative control of spacecraft formation. The basic idea of master-slave type formation control is to select one pilot from formation spacecrafts, and other spacecrafts receive the instruction of the pilot or track the pilot according to a certain mode, so as to realize formation cooperation. Wang [8] proposes a series of master-slave type formation control methods for robot formation. The authors then introduced the idea of master-slave formation into spacecraft formation and proposed master-slave control strategies for near-earth formation and deep-space formation, respectively [9 ]. Document [10] realizes master-slave spacecraft formation cooperative control by using graph theory and linear matrix inequality. Aiming at the problem of uncertainty of the quality of the spacecraft, documents [11-12] provide a master-slave spacecraft formation self-adaptive cooperative control method based on an equivalence certainty principle. In consideration of the influence of spatial interference on formation, a document [13] combines a potential function with a nonlinear interference observer to realize high-precision distributed satellite cooperative control. However, current research has limitations. In the existing spacecraft formation orbit cooperative control method, it is generally assumed that the connectivity of a communication network meets certain assumed conditions. In actual formation, these assumptions cannot be met due to the effect of relative motion between the spacecraft on the connectivity of the communication network. Therefore, it is necessary to consider connectivity maintenance constraints of the communication network in the process of queuing for collaboration.
The references referred to in the background art mentioned above are as follows:
[1]D.P.Scharf,F.Y.Hadaegh,S.R.Ploen.A survey of spacecraft formation flying guidance and control.Part II:Control[C].American Control Conference,2004,2976-2985.
[2]J.R.Lawton,R.W.Beard.Synchronized multiple spacecraft rotations[J].Automatica,2002,38(8):13591364.
[3]M.Sabatini,G.B.Palmerini.Collective control of spacecraft swarms for space exploration[J].Celestial Mechanics and Dynamical Astronomy,2009,105(13):229.
[4]G.Antonelli,S.Chiaverini.Kinematic control of platoons of autonomous vehicles[J].IEEE Transactions on Robotics,2006,22(6):12851292.
[5]K.Tan,M.A.Lewis.Virtual structures for highprecision cooperative mobile robotic control[C].IEEE/RSJ International Conference on Intelligent Robots and Systems,1996,132139.
[6]M.A.Lewis,K.Tan.High precision formation control of mobile robots using virtual structures[J].Autonomous Robots,1997,4(4):387403.
[7]C.Ahn,Y.Kim.Point targeting of multisatellites via a virtual structure formation flight scheme[J].Journal of Guidance,Control,and Dynamics,2009,32(4):13301344.
[8]P.K.Wang.Navigation strategies for multiple autonomous mobile robots moving in formation[J].Journal of Robotic Systems,1991,8(2):177195.
[9]P.K.Wang,F.Hadaegh.Coordination and control of multiple microspaceecraft moving in formation[J].The Journal of the Astronautical Sciences,1996,44:315355.
[10]M.Mesbahi,F.Y.Hadaegh.Formation flying control of multiple spacecraft via graphs,matrix inequalities,and switching[J].Journal ofGuidance,Control,and Dynamics,2001,24(2):369377.
[11]H.Liu,J.Shan,D.Sun.Adaptive synchronization control of multiple spacecraft formation flying[J].Journal of Dynamic Systems,Measurement,and Control,2007,129(3):337–342.
[12]H.Yoon,Y.Eun,C.Park.Adaptive tracking control of spacecraft relative motion with mass and thruster uncertainties[J].Aerospace Science and Technology,2014,34:7583.
[13]D.Lee.Nonlinear disturbance observer based robust control for spacecraft formation flying[J].Aerospace Science and Technology,2018,76:8290.
disclosure of Invention
In view of the above, the invention provides a spacecraft formation orbit coordination and connectivity maintenance control method, which is based on preset performance control, fully considers orbit coordination and connectivity maintenance, and can effectively realize spacecraft formation control.
In order to achieve the purpose, the invention adopts the technical scheme that:
a spacecraft formation orbit coordination and connectivity maintenance control method comprises the following steps:
firstly, establishing a spacecraft formation dynamic model;
designing an artificial potential function;
and step three, performing formation cooperative control by using preset performance control.
Further, the specific manner of the first step is as follows:
in the geocentric inertial coordinate system, the orbit dynamics equations of the service spacecraft and the target spacecraft are as follows:
Figure BDA0003310154120000041
Figure BDA0003310154120000042
Figure BDA0003310154120000043
Figure BDA0003310154120000044
wherein r is a spacecraft position vector, v is a spacecraft velocity vector, μ is an earth gravity parameter, F represents acceleration caused by all forces acting on the spacecraft except thrust and gravity, F is a control force on the service spacecraft, m is a spacecraft mass, subscript c represents a reference spacecraft, and subscript i represents a spacecraft number;
and (3) combining the formula (1) and the formula (3) to obtain an expression of a relative motion dynamics equation of the spacecraft i and the reference spacecraft in an inertial coordinate system:
Figure BDA0003310154120000045
wherein ρ is a relative position vector;
converting the equation into a target spacecraft orbit coordinate system, approximating the gravitational field in a first order, and converting the gravitational field into a scalar form to obtain
Figure BDA0003310154120000051
Figure BDA0003310154120000052
Figure BDA0003310154120000053
Wherein θ is a true proximal angle, [ x, y, z [ ]]TAs the projection of the relative position vector in the LVLH coordinate system, fx、fy、fzThe gravitational force is induced by an external force in [ x, y, z ]]TA component of;
assuming that the spacecraft runs on a near-circular orbit, the above formula is converted into
Figure BDA0003310154120000054
Figure BDA0003310154120000055
Figure BDA0003310154120000056
Wherein n is the angular velocity of the target spacecraft orbit, and u is [ u ═ u [ n [ ]x，uy，uz]TAs a control input for the service spacecraft.
Further, the specific mode of the second step is as follows:
will artificially potential function psiijDefined as the distance p between the spacecraft i and jijThe function, | | is a continuously differentiable, non-negative function, which definesThe following were used:
when | | | pijWhen the absolute value is less than or equal to delta,
Figure BDA0003310154120000057
when | | | pijWhen the ratio is greater than delta, the ratio is,
Figure BDA0003310154120000058
wherein, Delta is the maximum communication distance of the spacecraft, DeltaijFor the safe distance of the spacecraft i from j,
Figure BDA0003310154120000059
for the desired distance of the spacecraft i from j,
Figure BDA00033101541200000510
respectively representing a repulsion function, an attraction potential function and a memory formation cooperative potential function, and satisfying the following four properties:
(1)ψij＝ψjiand is and
Figure BDA0003310154120000061
wherein
Figure BDA0003310154120000062
As partial derivative operator, pi、pjThe positions of the spacecrafts i and j respectively;
(2)
Figure BDA0003310154120000063
in the interval
Figure BDA0003310154120000064
The upper one is monotonically decreased in the upper one,
Figure BDA0003310154120000065
and
Figure BDA0003310154120000066
in the interval
Figure BDA0003310154120000067
The upper monotonic increase;
(3)
Figure BDA0003310154120000068
are all at
Figure BDA0003310154120000069
Take the minimum value, and
Figure BDA00033101541200000610
if true;
(4) when | | | pij||→δijWhen the temperature of the water is higher than the set temperature,
Figure BDA00033101541200000611
when | | | pijThe time | → | > Δ,
Figure BDA00033101541200000612
the artificial potential function is designed as follows:
Figure BDA00033101541200000613
Figure BDA00033101541200000614
Figure BDA00033101541200000615
wherein k isr、ka、kfAre design parameters.
Further, the specific manner of the third step is as follows:
when cooperative control is performed, the spacecraft is expected to form a formation at a preset time and with a preset precision; defining control state quantity r ═ x-xd，y-yd，y-yd]TWhere the subscript d represents a desired state, the relative orbital motion models shown in equations (9) - (11) are converted to affine form as follows:
Figure BDA00033101541200000616
Figure BDA00033101541200000617
wherein
Figure BDA0003310154120000071
Figure BDA0003310154120000072
According to the preset performance control theory, a positive preset performance function which is strictly reduced as follows is designed and defined:
ρ(t)＝(e0-ereq)exp(-l·t)+ereq (21)
wherein l is a normal number determined by the convergence speed requirement of the formation task, t is time, e0As initial error of task, ereqThe task precision requirement is met;
designing an error mapping function
Figure BDA0003310154120000073
Wherein epsilon is an error under an equivalent mapping space, and delta is a constant number between 0 and 1;
defining an error matrix as shown below
E(t)＝∈r+λ∈v (23)
Wherein e isrAnd evRespectively position and velocity errors in the peer-to-peer mapping space,the constant matrix lambda is a design value;
design control rate of
u＝ua-sign(G)(ηa|ua|+ηb|ub|)RTE (24)
Figure BDA0003310154120000074
ub＝F+R-tV+kR-tE+nfRTE (26)
Wherein sign (. eta.) is symbol calculation, Adj (. eta.) and Det (. eta.) are matrix adjoint matrix and determinant, respectively, and deltaD、k、nf、ηa、ηbFor adjustable positive values, R is the design matrix.
The invention has the beneficial effects that:
1. aiming at a spacecraft formation control cooperative control task, a spacecraft formation dynamics model is constructed through spacecraft relative motion modeling; constructing a virtual potential field in space through potential function design, so that a potential function obtains a minimum value at an expected configuration position, thereby ensuring orbital coordination and connectivity among spacecrafts and keeping the spacecrafts in an expected relative state; and controlling the spacecraft to a desired relative position with a predetermined accuracy within a predetermined time by adopting a preset performance control method.
2. The invention adopts the idea of artificial potential function and preset performance control, and accurately and effectively completes spacecraft formation and control.
Drawings
Fig. 1 is an initial communication network topology of an embodiment of the present invention.
Fig. 2 is a graph of relative position between spacecraft.
Fig. 3 is a graph of relative velocity between spacecraft.
FIG. 4 is a spacecraft control force diagram.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the following detailed description of the method for implementing the complex steps of the present invention with reference to the accompanying drawings is provided.
A spacecraft formation orbit coordination and connectivity maintenance control method establishes a spacecraft formation dynamics model and designs a potential energy function for describing orbit coordination and connectivity, thereby realizing a formation coordination control method based on preset performance control. The method comprises the following specific steps:
step one, establishing a spacecraft formation dynamics model.
In the geocentric inertial coordinate system, the orbit dynamics equations of the service spacecraft and the target spacecraft are
Figure BDA0003310154120000091
Figure BDA0003310154120000092
Figure BDA0003310154120000093
Figure BDA0003310154120000094
Wherein r is a spacecraft position vector, v is a spacecraft velocity vector, and μ is an earth gravity parameter. F represents the acceleration of the spacecraft caused by all the forces except the propulsive force and the medium-force gravitational force, F is the control force on the service spacecraft, and m is the spacecraft mass. Subscript c denotes a reference spacecraft and subscript i denotes a spacecraft i.
The expression of the spacecraft i and the reference spacecraft relative motion kinetic equation in an inertial coordinate system can be obtained by combining the formula (1) and the formula (3)
Figure BDA0003310154120000095
Where ρ is a relative position vector. The equation is converted into a target spacecraft orbit coordinate system, the gravity field is approximated to the first order and is converted into a scalar form, and the target spacecraft orbit coordinate system can be obtained
Figure BDA0003310154120000096
Figure BDA0003310154120000097
Figure BDA0003310154120000098
Where θ is the true paraxial angle, [ x, y, z]TIs the projection of the relative position vector in the LVLH coordinate system. Assuming that the spacecraft runs on a near-circular orbit, the above formula can be converted into
Figure BDA0003310154120000099
Figure BDA00033101541200000910
Figure BDA00033101541200000911
Where n is the angular velocity of the target spacecraft orbit, u ═ ux，uy，uz]TAs a control input for the service spacecraft.
And step two, designing an artificial potential energy function.
Artificial potential function psiijDefined as the distance p between the spacecraft i and jijThe function, | | is a continuously differentiable, non-negative function, which is defined as follows
(1) When | | | pijWhen the absolute value is less than or equal to delta,
Figure BDA0003310154120000101
(2) when | | | pijWhen the ratio is greater than delta, the ratio is,
Figure BDA0003310154120000102
wherein
Figure BDA0003310154120000103
Respectively representing the repulsive potential function and the attractive potential function memory formation cooperative potential function, and satisfying the following properties:
(1)ψij＝ψjiand is and
Figure BDA0003310154120000104
wherein
Figure BDA0003310154120000105
As partial derivative operator, pi、pjThe positions of the spacecrafts i and j respectively;
(2)
Figure BDA0003310154120000106
in the interval
Figure BDA0003310154120000107
The upper one is monotonically decreased in the upper one,
Figure BDA0003310154120000108
and
Figure BDA0003310154120000109
in the interval
Figure BDA00033101541200001010
The upper monotonic increase;
(3)
Figure BDA00033101541200001011
are all at
Figure BDA00033101541200001012
Take the minimum value, and
Figure BDA00033101541200001013
if true;
(4) when | | | pij||→δijWhen the temperature of the water is higher than the set temperature,
Figure BDA00033101541200001014
when | | | pijThe time | → | > Δ,
Figure BDA00033101541200001015
satisfying the above definition, the following potential functions are designed
Figure BDA00033101541200001016
Figure BDA00033101541200001017
Figure BDA0003310154120000111
And step three, performing formation cooperative control by using preset performance control.
When cooperative control is carried out, the spacecraft is expected to be formed into a team with preset precision in preset time, and the method proposes that the team formation cooperative control is realized by adopting a preset performance control theory. For the convenience of controller design, the relative orbit motion models shown in equations (9) to (11) are defined to control the state quantity r ═ x-xd，y-yd，y-yd]TWhere the subscript d denotes the desired state, the relative orbital motion model is converted to affine form
Figure BDA0003310154120000112
Figure BDA0003310154120000113
Wherein
Figure BDA0003310154120000114
Figure BDA0003310154120000115
According to the preset performance control theory, a positive preset performance function which is strictly reduced as follows is designed and defined:
ρ(t)＝(e0-ereq)exp(-l·t)+ereq (21)
where l is a normal number determined by the convergence speed requirement of the formation task, e0As initial error of task, ereqThe task precision requirement is met.
Designing an error mapping function
Figure BDA0003310154120000116
Defining an error matrix as shown below
E(t)＝∈r+λ∈v (23)
Where the constant matrix λ ═ diag (λ)1 λ2 λ3) Is a design value.
Design control rate of
u＝ua-sign(G)(ηa|ua|+ηb|ub|)RTE (24)
Figure BDA0003310154120000121
ub＝F+R-tV+kR-tE+nfRTE (26)
Wherein sign (. eta.) is symbol calculation, Adj (. eta.) and Det (. eta.) are matrix adjoint matrix and determinant, respectively, and deltaDIs a small adjustable positive value, k, nf、ηa、ηbFor adjustable positive values, R is the design matrix.
The validity of the method is verified below by way of example:
as shown in fig. 1, in the task of spacecraft formation, the positions of three spacecraft in the reference system are:
p1(0)＝[-40，-30，0]Tm,p2(0)＝[-15，-40，5]Tm,p3(0)＝[0，-40，0]Tm; initial velocity of
Figure BDA0003310154120000122
i＝1，2，3；
The desired distances between the spacecraft are respectively
Figure BDA0003310154120000123
Figure BDA0003310154120000124
All spacecraft safety distances δijThe communication distance Δ is 35m at 10 m.
According to the method, potential energy functions and controller parameters are designed: deltaij＝10、
Figure BDA0003310154120000125
Figure BDA0003310154120000126
Δ＝35、kr＝20、ka＝5、kf＝10；λ＝diag(12 35 12)、k＝30、nf＝0.25、ηa＝0.25、ηb＝0.25。
As can be seen from fig. 2, no collision occurs between all the spacecrafts, and the distances between the spacecrafts 1 and 2 and the spacecrafts 2 and 3 are always smaller than the communication distance between the spacecrafts, so that the communication networks between the spacecrafts are always connected, and the formation spacecrafts can be ensured to realize the final expected configuration. Fig. 3 shows a graph of the velocity error over time, and it can be seen that the velocities of all spacecraft tend to be uniform. Fig. 4 shows that the control force eventually goes to zero.
In a word, aiming at a cooperative control task for spacecraft formation control, a spacecraft formation dynamics model is constructed through spacecraft relative motion modeling; constructing a virtual potential field in space through potential function design, so that a potential function obtains a minimum value at an expected configuration position, thereby ensuring orbital coordination and connectivity among spacecrafts and keeping the spacecrafts in an expected relative state; and controlling the spacecraft to a desired relative position with a predetermined accuracy within a predetermined time by adopting a preset performance control method. The invention adopts the idea of artificial potential function and preset performance control, and accurately and effectively completes spacecraft formation and control.

Claims (4)

1. A spacecraft formation orbit coordination and connectivity maintenance control method is characterized by comprising the following steps:
firstly, establishing a spacecraft formation dynamic model;
designing an artificial potential function;
and step three, performing formation cooperative control by using preset performance control.
2. The spacecraft formation orbit coordination and connectivity maintenance control method according to claim 1, characterized in that the first step is specifically as follows:
in the geocentric inertial coordinate system, the orbit dynamics equations of the service spacecraft and the target spacecraft are as follows:
Figure FDA0003310154110000011
Figure FDA0003310154110000012
Figure FDA0003310154110000013
Figure FDA0003310154110000014
wherein r is a spacecraft position vector, v is a spacecraft velocity vector, μ is an earth gravity parameter, F represents acceleration caused by all forces acting on the spacecraft except thrust and gravity, F is a control force on the service spacecraft, m is a spacecraft mass, subscript c represents a reference spacecraft, and subscript i represents a spacecraft number;
and (3) combining the formula (1) and the formula (3) to obtain an expression of a relative motion dynamics equation of the spacecraft i and the reference spacecraft in an inertial coordinate system:
Figure FDA0003310154110000015
wherein ρ is a relative position vector;
converting the equation into a target spacecraft orbit coordinate system, approximating the gravitational field in a first order, and converting the gravitational field into a scalar form to obtain
Figure FDA0003310154110000021
Figure FDA0003310154110000022
Figure FDA0003310154110000023
Wherein θ is a true proximal angle, [ x, y, z [ ]]TAs the projection of the relative position vector in the LVLH coordinate system, fx、fy、fzThe gravitational force is induced by an external force in [ x, y, z ]]TA component of;
assuming that the spacecraft runs on a near-circular orbit, the above formula is converted into
Figure FDA0003310154110000024
Figure FDA0003310154110000025
Figure FDA0003310154110000026
Wherein n is the angular velocity of the target spacecraft orbit, and u is [ u ═ u [ n [ ]x，uy，uz]TAs a control input for the service spacecraft.
3. The spacecraft formation orbit coordination and connectivity maintenance control method according to claim 2, characterized in that the second step is specifically as follows:
will artificially potential function psiijDefined as the distance p between the spacecraft i and jijThe function of | l, ψ is a continuously differentiable, non-negative function, which is defined as follows:
when | | | pijWhen the absolute value is less than or equal to delta,
Figure FDA0003310154110000027
when | | | pijWhen the ratio is greater than delta, the ratio is,
Figure FDA0003310154110000028
wherein, Delta is the maximum communication distance of the spacecraft, DeltaijFor the safe distance of the spacecraft i from j,
Figure FDA0003310154110000029
for the desired distance of the spacecraft i from j,
Figure FDA0003310154110000031
respectively representing a repulsion function, an attraction potential function and a memory formation cooperative potential function, and satisfying the following four properties:
(1)ψij＝ψjiand is and
Figure FDA0003310154110000032
wherein
Figure FDA0003310154110000033
As partial derivative operator, pi、pjThe positions of the spacecrafts i and j respectively;
(2)
Figure FDA0003310154110000034
in the interval
Figure FDA0003310154110000035
The upper one is monotonically decreased in the upper one,
Figure FDA0003310154110000036
and
Figure FDA0003310154110000037
in the interval
Figure FDA0003310154110000038
The upper monotonic increase;
(3)
Figure FDA0003310154110000039
are all at
Figure FDA00033101541100000310
Take the minimum value, and
Figure FDA00033101541100000311
if true;
(4) when | | | pij||→δijWhen the temperature of the water is higher than the set temperature,
Figure FDA00033101541100000312
when | | | pijThe time | l → Δ,
Figure FDA00033101541100000313
the artificial potential function is designed as follows:
Figure FDA00033101541100000314
Figure FDA00033101541100000315
Figure FDA00033101541100000316
wherein k isr、ka、kfAre design parameters.
4. The spacecraft formation orbit coordination and connectivity maintenance control method according to claim 3, characterized in that the third step is specifically as follows:
when cooperative control is performed, the spacecraft is expected to form a formation at a preset time and with a preset precision; defining control state quantity r ═ x-xd，y-yd，y-yd]TWhere the subscript d represents a desired state, the relative orbital motion models shown in equations (9) - (11) are converted to affine form as follows:
Figure FDA00033101541100000317
Figure FDA00033101541100000318
wherein
Figure FDA0003310154110000041
Figure FDA0003310154110000042
According to the preset performance control theory, a positive preset performance function which is strictly reduced as follows is designed and defined:
ρ(t)＝(e0-ereq)exp(-l·t)+ereq (21)
wherein l is a normal number determined by the convergence speed requirement of the formation task, t is time, e0As initial error of task, ereqThe task precision requirement is met;
designing an error mapping function
Figure FDA0003310154110000043
Wherein epsilon is an error under an equivalent mapping space, and delta is a constant number between 0 and 1;
defining an error matrix as shown below
E(t)＝∈r+λ∈v (23)
Wherein e isrAnd euRespectively representing the position and speed errors in the peer-to-peer mapping space, wherein the constant matrix lambda is a design value;
design control rate of
u＝ua-sign(G)(ηa|ua|+ηb|ub|)RTE (24)
Figure FDA0003310154110000044
ub＝F+R-1V+kR-1E+nfRTE (26)
Wherein sign (. eta.) is symbol calculation, Adj (. eta.) and Det (. eta.) are matrix adjoint matrix and determinant, respectively, and deltaD、k、nf、ηa、ηbFor adjustable positive values, R is the design matrix.
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