CN105867401B

CN105867401B - The spacecraft attitude fault tolerant control method of single-gimbal control moment gyros

Info

Publication number: CN105867401B
Application number: CN201610272881.1A
Authority: CN
Inventors: 张福桢; 金磊
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2016-04-28
Filing date: 2016-04-28
Publication date: 2017-12-05
Anticipated expiration: 2036-04-28
Also published as: CN105867401A

Abstract

A kind of spacecraft attitude fault tolerant control method of single-gimbal control moment gyros, comprises the following steps：1st, the kinetics equation and kinematical equation when single-gimbal control moment gyros SGCMGs has partial failure failure are established；Including：Define coordinate system and control system state equation is established；2nd, based on spacecraft in orbit the characteristics of, using the spacecraft attitude fault tolerant control method of single-gimbal control moment gyros；3rd, sliding formwork control ratio designs；Include the improvement of sliding-mode surface design, control law Preliminary design and control law.Advantage is：1st, design control law is equally applicable to the spacecraft faults-tolerant control using flywheel as executing agency；2nd, the prior information of failure need not be had any actual knowledge of, fault message and interference information are estimated in real time by Self Adaptive Control, it is allowed to failure time-varying.3rd, it is suitable for the SGCMGs of arbitrary configuration or the partial failure pattern of flywheel.4th, in the spacecraft with SGCMGs, using gyro gimbal rotating speed as direct controlled quentity controlled variable, meeting engineering reality.

Description

Spacecraft attitude fault-tolerant control method of single-frame control moment gyro group

[ technical field ] A method for producing a semiconductor device

The invention discloses a three-axis stable spacecraft which adopts a Single-frame Control Moment gyro group (SGCMGs) as an execution mechanism, and relates to a Fault-Tolerant Control method (FTC) for the attitude of the execution mechanism when a partial failure Fault occurs, so as to realize stronger robustness of the spacecraft to the Fault, belonging to the field of spacecraft attitude Control.

[ background of the invention ]

With the development of aerospace technology, aerospace missions are becoming more complex, and thus higher requirements are put forward on the safety, stability and control accuracy of the spacecraft. As can be seen from the development history of aerospace technology, many accidents are caused by only a tiny fault, for example, the fault of a Lewis satellite launched by NASA in 1997 causes the failure of all thrusters, and finally the satellite falls into the atmosphere, thus causing huge loss. How to avoid risks and enable spacecrafts to have a fault-tolerant function becomes a key point of research of many spaceflight experts at present. Fault diagnosis and fault tolerant control have become an important way to maintain the reliability, maintainability and availability of spacecraft today.

The idea of fault-tolerant control was first proposed by Niederlinski in 1971, after which the theory of fault-tolerant control developed rapidly. According to the characteristics of the design method, fault-tolerant control is generally divided into active fault-tolerant control and passive fault-tolerant control. The active fault-tolerant control is to redesign a control system according to expected characteristics after a fault occurs, and at least to stabilize the whole system. The passive fault-tolerant control adopts a fixed controller to ensure that a closed-loop system is insensitive to specific faults and keep the stability of the system. Compared with active fault-tolerant control, passive fault-tolerant control has the advantages of simple structure, high response speed and lower design difficulty because system faults do not need to be detected or diagnosed and fault reaction time is also not needed.

In the field of attitude fault-tolerant control, the current research result mainly takes control moment as a control quantity and a fault modeling object. However, in practical engineering applications, when an angular momentum exchange device is used as an attitude control actuator, the actual control is often the rotation speed of the actuator. For example, in a spacecraft using a flywheel as an actuator, the control torque is determined by the rotation speed of the flywheel. On the other hand, the moment output of the angular momentum exchange device may also be related to the current attitude of the gyroscope, for example, when a control moment gyroscope group (CMG) is used as an actuator, the control moment is affected by the frame angle and the frame rotation speed due to the singularity problem and the time-varying characteristic of the gyroscope transverse matrix.

The existing problems make the current research results difficult to apply in practical engineering, and particularly for the field of spacecraft attitude fault-tolerant control with CMG as an execution mechanism, no better engineering application method is basically realized at present.

[ summary of the invention ]

The invention provides a spacecraft which takes a single-frame control moment gyro group SGCMGs as an execution mechanism, and the attitude stability control of the spacecraft with partial failure faults (each gyro of the SGCMGs has moment output) of the execution mechanism is realized by a sliding mode control method and an adaptive control method.

Aiming at the problems, the technical scheme of the invention is as follows:

according to a dynamics equation and a kinematics equation of the spacecraft with partial failure faults of an execution mechanism, a sliding mode surface is established by using state quantities such as Euler angles, Euler angular velocities and the like, fault information of the spacecraft is estimated on line by using a self-adaptive control method, and attitude stabilization of the spacecraft can be realized under the condition of no fault by designing a sliding mode control strategy and appropriate control parameters, so that the attitude stabilization can still be realized under the condition of partial failure faults of the spacecraft execution mechanism by using the set of control parameters. The specific operation steps are as follows

Step 1: and establishing a dynamic equation and a kinematic equation when the single-frame control moment gyro group SGCMGs have partial failure faults. The method specifically comprises the following steps:

step 1.1: defining a coordinate system

a. Body coordinate system f_b(o_bx_by_bz_b)

The coordinate system is fixedly connected with the spacecraft and has an origin O_bLocated in the center of mass of the spacecraft, O_bx_bThe axis pointing in the direction of motion of the spacecraft, O_bz_bThe axis pointing above the spacecraft perpendicular to the plane of the flight path, O_by_bShaft, O_bx_bShaft and O_bz_bThe axes constitute a right-hand coordinate system.

b. Orbital coordinate system f_o(O_ox_oy_oz_o)

Origin of orbital coordinate system is at center of mass, O, of spacecraft_oz_oThe shaft points to the ground along the vertical lineHeart, O_ox_oAxis perpendicular to O in the plane of the track_oz_oAn axis pointing in the direction of motion of the spacecraft, O_oy_oShaft, O_ox_oShaft and O_oz_oThe axes constitute a right-hand coordinate system. The coordinate system is in space at an angular velocity ω_oAround O_oy_oThe shaft rotates.

c. Centre of earth inertial coordinate system f_i(O_ix_iy_iz_i)

The origin of the earth center inertial coordinate system is fixedly connected with the earth center O_iO of_ix_iThe axis being in the equatorial plane and pointing towards the vernal equinox, O_iz_iPerpendicular to the equatorial plane and in the direction of the rotational angular velocity of the earth, O_iy_iThe axis being in the equatorial plane and being co-incident with_ix_iShaft, O_iz_iThe axes form a rectangular coordinate system.

SGCMGs frame coordinate system f_ci(O_cig_is_it_i)

The origin of the frame coordinate system is at the center of mass O of the SGCMG_ciThe unit vectors in each direction of the coordinate system are unit vectors along the frame axis directionUnit vector in the direction of rotation speed of the rotor shaftUnit vector in the direction opposite to gyro moment output

Step 1.2 control System State equation establishment

Step 1.2.1 establishing a kinetic equation and a kinematic equation

Kinetic equation:

wherein, I_bIs the entire system inertia matrix, consider I_bIs a constant inertia matrix;

ω_b＝[ω_xω_yω_z]^Tthe component array of the spacecraft absolute angular velocity under the system is provided;

h₀the nominal angular momentum of each gyro rotor;

is omega_bWith respect to the derivative of time,is defined as follows:

A_s＝[s₁s₂… s_n]is a rotor rotation speed direction matrix of SGCMGs;

I_wsan SGCMGs rotor axial rotation inertia array;

omega is a rotor speed vector; h is₀The nominal angular momentum of each gyro rotor;

A_t＝[t₁t₂… t_n]is a transverse matrix of SGCMGs;

is a gyro frame angle;

T_dthe interference moment vector of the spacecraft;

kinematic equation:

wherein the attitude angleTheta and psi are a rolling angle, a pitch angle and a yaw angle of the spacecraft; attitude angular velocity Are respectively asThe derivative of θ, ψ with respect to time; omega_oOrbiting a body system O_oy_oAngular velocity of shaft rotation;

step 1.2.2 establishing a failure mode

It should be noted that the fault modeling here is formal, and in a real system, the fault mode equation is implicit in the motion of the spacecraft. Although specific expressions or specific numerical values of E and f cannot be ascertained, it is easy to determine whether there is gyro jamming in the SGCMGs and torque cannot be output, which is sufficient in the present invention.

Wherein,a gyroscope theoretical frame rotating speed vector is obtained;the actual frame rotating speed vector of the gyroscope is obtained;

E＝diag(e₁e₂… e_n) For multiplicative fault matrix, e_iThe failure factor of the ith gyro;

f＝[f₁f₂… f_n]^Tto account for the effect of additive faults on the gyro frame rotation speed,

f_ithe rotation speed deviation of the ith gyro is obtained.

Step 1.2.3 deducing a kinematic equation and a kinetic equation under a fault mode

Formula (3) is substituted for formula (1), and the equivalent interference, the equivalent moment of inertia matrix and the equivalent gyro group angular momentum of the gyro rotor unit nominal angular momentum are defined as follows: equivalent interference: d ═ T_d/h₀；

An equivalent moment of inertia matrix: j ═ I_b/h₀(ii) a Equivalent gyro group angular momentum: h is_c＝A_sI_wsΩ/h₀；

Order toAnd as the control quantity of the control system, obtaining a dynamic equation under the fault:

under the assumption of a small angle, equation (2) can be approximately written as:

wherein:

ω_ocalculating the orbit parameters of the spacecraft to obtain;representing the state quantity of the system;

A_s,A_tthe following can be calculated:

s_i0，t_i0determining the gyroscope as a unit vector according to the specific configuration of the gyroscope; s_i0Denotes s_iAn initial value of (d); t is t_i0Represents t_iAn initial value of (d);

step 2 is based on the actual characteristics of the on-orbit operation of the spacecraft, and the application of the method is based on the following assumptions:

assume that 1: the disturbance moment suffered by the spacecraft in the operation process is bounded, namely: d | | | is less than or equal to T_d(ii) a And the influence of additive faults on the rotation speed of the gyro frame is limited, | | A_tf||≤T_f. Where the convention | · | |, represents the 2-norm of a matrix or vector, T_d,T_fAre unknown constants. Suppose 1 can be integrated as the following expression:

||-A_tf+d||≤M_d(7)

assume 2: the rotational inertia matrix of the spacecraft is a positive definite symmetric matrix, namely J is symmetric and positive definite.

Assume that 3: the invention does not consider the situation that the gyroscope completely fails, namely, the unknown constant e is assumed to exist₀Satisfies the following conditions:

wherein n is the number of gyros in the SGCMGs.

And step 3, sliding mode control law design. The method specifically comprises the following steps:

step 3.1 slip form surface design

The sliding mode surface is selected as follows:

where k > 0, constant for the designer. Then when s → 0, x → 0, is a column vector composed of attitude angles, represents a state quantity of the system,representing the derivative of the state vector with respect to time.

Step 3.2 control law initial design

The following sliding mode control law is selected:

the values and meanings of the parameters in the control law are as follows:

A_t＝[t₁t₂… t_n]is an SGCMGs transverse matrix,is A_tThe transposed matrix of (2);

J、h_cthe equivalent moment of inertia matrix and the equivalent gyro group angular momentum defined in the step 1.2.3;

derivative with respect to time for f (x) in step 1.2.3; s is slidingA die face;

m in the formula (10)_dThe self-adaptive updating law is taken as follows:

γ (t): a parameter is introduced, take

Defining variablesWherein c is₀,c₁,₀Is a normal number, n is the number of gyros under a certain configuration, u is a control quantity,ξ, v is an intermediate variable,is composed ofDerivative with respect to time.

Taking the Lyapunov function as

Wherein The remaining parameters are given above and let Δ E ═ I-E, I is the unit matrix and E is the multiplicative fault matrix.

The derivative of the Lyapunov function with respect to time is obtained by using equations (10) to (14):

equation (16) indicates that the function V does not increase at least monotonically, and therefore can be obtained_t≥0V (t). ltoreq.V (0), wherein sup (. cndot.) denotes supremum, i.e.There is a limit to the amount of space that, therefore,thereby to obtainExisting and bounded, according to the Barbalt theorem, there areTherefore, there is a x → 0 region,

step 3.3 improvement of control laws

The above control law actually has a buffeting problem and a singularity problem, and therefore, it is necessary to improve the above control law.

The problem of buffeting: since the sliding mode control has discontinuity of control, there is a chattering phenomenon. According to the sliding mode control theory, the invention adopts s/(| s | + tau) approximation to replace the symbolic function s/| s |, wherein tau is a small positive number, usually given according to the actual situation, and generally selected to be 10^-3～10^-1In the meantime.

The singularity problem is as follows: when the output moments of all the gyros are coplanar (or collinear), the normal direction of the moment plane (or the normal plane of the moment direction) cannot output the moments, and at the moment, the SGCMGs transverse matrix A_tWith less than full rank, equation (9) cannot be solved, so equation (9) is improved by referring to a robust pseudo-inverse solving method.

The improved control strategy is therefore:

wherein: λ is a small positive number, and usually needs to be taken through actual working conditions, and can be generally taken as 10^-3～10^-1To (c) to (d); i is_3×3Is a third order identity matrix, E_3×3Is a diagonal matrix, and has the form:

the elements in the matrix are:_j＝0.01(0.5πt+φ_j)(j＝1,2,3)，φ_j＝π(j-1)/2。

the remaining parameters are the same as those in equations (11) to (14).

λ, τ are too large, and the above improvement cannot guarantee the stability of the system; theoretically, equation (17) can still satisfy the robustness of the fault system as long as the values of λ, τ are ensured to be small enough. However, in practice, if λ, τ are too small, they cannot play a role in eliminating singularity and chatter. Thus, λ, τMust be adjusted according to the parameters of the actual system, and can generally be from 10^-3～10^-1And selecting a parameter as a basis, and adjusting according to the actual control effect.

In addition, the control law designed by the invention is also suitable for a spacecraft taking a flywheel as an angular momentum exchange device, and only the gyro relative angular momentum h of the dynamic model (4) is used_cRelative angular momentum h converted to flywheel_wTransverse matrix A_tAnd the moment of each flywheel is controlled by adopting the same control law (17) by replacing the installation matrix C of the flywheel, so that the stability of a fault system can be ensured.

The invention designs an attitude fault-tolerant control method of a spacecraft with a fault executing mechanism, which has the following advantages:

1) although the sliding mode control law designed by the invention is designed on the background of SGCMGs, the dynamic characteristic of the gyroscope is similar to that of a flywheel, so the sliding mode control law designed by the invention is also suitable for the fault-tolerant control of a spacecraft by taking the flywheel as an actuating mechanism.

2) According to the method, the fault information and the interference information are estimated in real time through self-adaptive control without exactly knowing the prior information of the fault, so that the time variation of the fault is allowed, and as long as the condition that a certain gyro is completely invalid is ensured to be absent.

3) The invention relates to processes that are not specific to SGCMGs configurations or flywheel configurations, but rather use the transverse matrix A in proving stability_tOr the 2-norm of each column vector of the installation matrix C is 1, which is easily met in practical engineering, so that the method is suitable for SGCMGs or flywheel partial failure modes with any configuration.

4) The invention is used in the spacecraft using SGCMGs, takes the rotation speed of a gyroscope frame as a direct control quantity, accords with the engineering practice, and for the spacecraft using flywheels as an actuating mechanism, although the output torque of each flywheel is directly controlled, the output torque of the flywheel is in direct proportion to the rotation speed, so the invention is also equivalent to the control of the rotation speed of the flywheel in practice, and also accords with the engineering practice.

[ description of the drawings ]

FIG. 1 is a schematic diagram of attitude stabilization fault-tolerant control.

Fig. 2 is a schematic diagram of a spacecraft body coordinate system.

Fig. 3 is a schematic diagram of a spacecraft orbit coordinate system.

Fig. 4 is a schematic view of an inertial coordinate system of a spacecraft.

FIG. 5 is a schematic diagram of the SGCMG frame coordinate system.

FIG. 6 is a schematic diagram of a control law design flow.

FIG. 7 is a schematic diagram of SGCMGs in a pyramidal configuration.

[ detailed description ] embodiments

The following describes the implementation process of the present invention specifically by taking a spacecraft of a certain model as an example, as shown in fig. 1 to 7. The parameters of the spacecraft are as follows:

the spacecraft moment of inertia matrix is:

SGCMGs in a pyramid configuration are selected, wherein the nominal angular momentum of the gyroscope is 200 Nms; the initial attitude angle is:θ(0)＝1.5°,ψ(0)＝1.5°；ω_bhas an initial value of ω_b(0)＝[0 0 0]^T(ii) a The spacecraft has a circular flight orbit with a radius of 26600km, and the environmental disturbance moment comprehensively considers the earth gravity perturbation, the sunlight pressure moment and the solar radiation pressureDisturbances, etc., in the form of external disturbances, of

Wherein A is₀For disturbing the torque amplitude, take A₀＝1.5×10^-5N·m。

It is assumed that the spacecraft experiences multiplicative faults and additive faults at the same time.

Multiplicative fault parameters are:

where rand (-) denotes a random function of amplitude 1, t₁＝150s,t₂＝180s,t₃＝200s,t₄＝240s

The additive fault parameters are:

f_i(t)＝-0.01 (i＝1,2,3,4,t≥t_i)

giving fault parameters and external interference expressions is only needed for simulation.

And starting to set a control law to control the attitude of the spacecraft.

1. And establishing a dynamic equation and a kinematic equation when partial failure faults exist in the SGCMGs. The method specifically comprises the following steps:

1.1 defines the coordinate system: the relevant coordinate system is defined as in step 1.1.

1.2 control System State equation establishment

First, according to the spacecraft-related parameters used for example, the following system parameters can be listed directly:

and 4, selecting SGCMGs in a pyramid configuration, wherein the number n of the gyros is equal to 4.

The spacecraft moment of inertia matrix is:

nominal angular momentum h of individual gyro rotors₀＝200Nms；

T_d＝[T_d1T_d2T_d3]The interference moment vector of the spacecraft;

a is calculated according to the pyramid configuration reference diagram_s,A_tIs described in (1).

s₁₀＝[0 -1 0]^T,g₁₀＝[-sinβ 0 cosβ]^T

s₂₀＝[-1 0 0]^T,g₂₀＝[0 sinβ cosβ]^T

s₃₀＝[0 1 0]^T,g₃₀＝[sinβ 0 cosβ]^T

s₄₀＝[1 0 0]^T,g₄₀＝[0 -sinβ cosβ]^T

From equation (6), the following can be found:

wherein beta can be determined by the following procedure,

the three-axis angular momentum under the system is respectively as follows:

H_x＝2h₀+2h₀cosβ

H_y＝2h₀+2h₀cosβ

H_z＝4h₀sinβ

in order to equalize the three-axis angular momentum of the pyramid configuration, i.e. H_x＝H_y＝H_zWhen β is found to be 53.1 DEG, h is obtained_c＝A_s[h₀h₀h₀h₀]^T。

ω_oBased on the track parameters. Since the spacecraft is a circular orbit with the radius of 26600km, the following components are adopted:

wherein mu is the gravitational constant of the earth, 3.986005 × 10¹⁴m³/s²And R is the radius of the track.

And calculating to obtain: omega₀＝4.6020×10^-4rad/s。

1.2.1 establishing a kinetic equation and a kinematic equation

Kinetic equation:

kinematic equation:

1.2.2 establishing failure modes

E＝diag(e₁e₂e₃e₄) For multiplicative fault factors, f ═ f₁f₂f₃f₄]^TIs an additive failure factor.

1.2.3 deducing kinematic equation and kinetic equation under fault mode

Let d be T_d/h₀,J＝I_b/h₀,h_c＝A_sI_wsΩ/h₀And make an orderSubstituting the control quantity of the control system into the dynamic equation of the fault mode to obtain the dynamic equation under the fault:

under the assumption of small angles, the kinematic equation (2) can be approximately written as:

wherein:

2. based on the actual characteristics of the on-orbit operation of the spacecraft, the method is based on the following assumptions:

||-A_tf+d||≤M_d(23)

3. and (5) sliding mode control law design. The method specifically comprises the following substeps:

3.1 slip form face design

The sliding mode surface is selected as follows:

wherein k is more than 0 and is a constant.

For convenience of representation, the following parameters are defined:

3.2 preliminary design of control Law

The following sliding mode control law is selected:

the values and meanings of the parameters in the control law are as follows:

m in the formula (9)_dThe self-adaptive updating law is taken as follows:

γ (t): a parameter is introduced, take

Wherein c is₀,c₁,₀Is a normal number.

Because the specific time and parameters of the spacecraft during the fault can not be known in advance, the control parameters of the spacecraft can be adjusted when the executing mechanism works without faults, so that the better attitude control performance is ensured, and the control parameters are selected as follows:

k＝2,c₀＝0.5,₀＝0.5,c₁＝10 (31)

the initial values of the two adaptive parameters are selected as follows:

unknown, and in this example is set directly to 0.

Since it is generally assumed that there is no fault in the system at the start of the simulation, the selection is madeGet

Taking the Lyapunov function as

The derivative of the Lyapunov function with respect to time is obtained by using equations (27) to (31):

this equation shows that the function V does not increase at least monotonically, and therefore can be given_t≥0V (t). ltoreq.V (0), wherein sup (. cndot.) denotes supremum, i.e.There is a limit to the amount of space that, therefore,thereby to obtainExisting and bounded, according to the Barbalt theorem, there areTherefore, there is a x → 0 region,

3.3 improvement of control laws

In view of the singularity problem of single frame control moment gyros and the buffeting problem inherent to sliding mode control, equation (23) is modified as follows:

wherein: λ is 0.001; i is_3×3Is a third order identity matrix, E_3×3Is a diagonal matrix, and has the form:

the elements in the matrix are:_j＝0.01(0.5πt+φ_j)(j＝1,2,3)，φ_j＝π(j-1)/2，τ＝0.001。

the remaining parameters are the same as those in equations (27) to (31).

In conclusion, the control law (34) is selected to ensure that the attitude angles and the attitude angular velocities of the systems (18) and (19) have global progressive stability at the origin even in the case of a fault, which also indicates that the control law (34) is robust to the fault of the systems (18) and (19).

The invention discloses a method for controlling attitude stability and fault tolerance of a spacecraft by taking SGCMGs as an execution mechanism, which is characterized by comprising the following steps: because specific parameter information of the fault cannot be predetermined, the control law cannot be designed according to the prior information of the fault, and therefore the control law is designed by estimating the fault information in real time by adopting a self-adaptive method, and the method is more in line with engineering practice. On the other hand, the reason why the control method of the invention does not allow the existence of the complete failure of the gyros is that when the number of the gyros in the used configuration is small, if a certain gyro completely fails, the gyro is strange in configuration, namely, the gyro can always have one direction and cannot output the torque. This problem is not the one solved by the present invention.

Claims

1. A spacecraft attitude fault-tolerant control method of a single-frame control moment gyro group is characterized by comprising the following steps:

step 1: establishing a dynamic equation and a kinematic equation when the single-frame control moment gyro group SGCMGs has partial failure faults; the method specifically comprises the following steps:

step 1.1: defining a coordinate system

a. Body coordinate system f_b(o_bx_by_bz_b)

The body coordinate system is fixedly connected with the spacecraft,origin O_bLocated in the center of mass of the spacecraft, O_bx_bThe axis pointing in the direction of motion of the spacecraft, O_bz_bThe axis pointing above the spacecraft perpendicular to the plane of the flight path, O_by_bShaft, O_bx_bShaft and O_bz_bThe axes form a right-hand coordinate system;

b. orbital coordinate system f_o(O_ox_oy_oz_o)

Origin of orbital coordinate system is at center of mass, O, of spacecraft_oz_oThe axis pointing to the center of the earth along the vertical line, O_ox_oAxis perpendicular to O in the plane of the track_oz_oAn axis pointing in the direction of motion of the spacecraft, O_oy_oShaft, O_ox_oShaft and O_oz_oThe axes form a right-hand coordinate system; the orbital coordinate system is at an angular velocity ω in space_oAround O_oy_oRotating the shaft;

c. centre of earth inertial coordinate system f_i(O_ix_iy_iz_i)

The origin of the earth center inertial coordinate system is fixedly connected with the earth center O_iO of_ix_iThe axis being in the equatorial plane and pointing towards the vernal equinox, O_iz_iPerpendicular to the equatorial plane and in the direction of the rotational angular velocity of the earth, O_iy_iThe axis being in the equatorial plane and being co-incident with_ix_iShaft, O_iz_iThe axes form a rectangular coordinate system;

SGCMGs frame coordinate system f_ci(O_cig_is_it_i)

The origin of the frame coordinate system is at the center of mass O of the SGCMG_ciThe unit vectors in each direction of the frame coordinate system are unit vectors along the frame axis directionUnit vector in the direction of rotation speed of the rotor shaftMoment along the topOutput unit vector of reverse direction

Step 1.2: control system state equation establishment

Step 1.2.1: establishing a kinetic equation and a kinematic equation

Kinetic equation:

<mrow> <msub> <mi>I</mi> <mi>b</mi> </msub> <msub> <mover> <mi>&omega;</mi> <mo>&CenterDot;</mo> </mover> <mi>b</mi> </msub> <mo>+</mo> <msubsup> <mi>&omega;</mi> <mi>b</mi> <mo>&times;</mo> </msubsup> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>b</mi> </msub> <msub> <mi>&omega;</mi> <mi>b</mi> </msub> <mo>+</mo> <msub> <mi>A</mi> <mi>s</mi> </msub> <msub> <mi>I</mi> <mrow> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mi>&Omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>h</mi> <mn>0</mn> </msub> <msub> <mi>A</mi> <mi>t</mi> </msub> <msub> <mover> <mi>&delta;</mi> <mo>&CenterDot;</mo> </mover> <mi>r</mi> </msub> <mo>+</mo> <msub> <mi>T</mi> <mi>d</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>

wherein, I_bIs the entire control system inertia matrix, consider I_bIs a constant inertia matrix;

ω_b＝[ω_xω_yω_z]^Tis a component array of the absolute angular velocity of the spacecraft;

h₀for each one isA nominal angular momentum of the gyro rotor;

is omega_bWith respect to the derivative of time,is defined as follows:

<mrow> <msubsup> <mi>&omega;</mi> <mi>b</mi> <mo>&times;</mo> </msubsup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&omega;</mi> <mi>z</mi> </msub> </mrow> </mtd> <mtd> <msub> <mi>&omega;</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&omega;</mi> <mi>z</mi> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&omega;</mi> <mi>x</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&omega;</mi> <mi>y</mi> </msub> </mrow> </mtd> <mtd> <msub> <mi>&omega;</mi> <mi>x</mi> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow>

A_s＝[s₁s₂… s_n]is a rotor rotation speed direction matrix of SGCMGs;

I_wsan SGCMGs rotor axial rotation inertia array;

A_t＝[t₁t₂… t_n]is a transverse matrix of SGCMGs;

is a gyro frame angle;

T_dthe interference moment vector of the spacecraft;

kinematic equation:

step 1.2.2: establishing failure modes

<mrow> <msub> <mover> <mi>&delta;</mi> <mo>&CenterDot;</mo> </mover> <mi>r</mi> </msub> <mo>=</mo> <mi>E</mi> <mover> <mi>&delta;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mi>f</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

f_ithe rotation speed deviation of the ith gyroscope;

step 1.2.3: deducing kinematic equation and kinetic equation under fault mode

<mrow> <mi>J</mi> <msub> <mover> <mi>&omega;</mi> <mo>&CenterDot;</mo> </mover> <mi>b</mi> </msub> <mo>+</mo> <msubsup> <mi>&omega;</mi> <mi>b</mi> <mo>&times;</mo> </msubsup> <mrow> <mo>(</mo> <msub> <mi>J&omega;</mi> <mi>b</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mi>E</mi> <mi>u</mi> <mo>-</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mi>f</mi> <mo>+</mo> <mi>d</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

under the assumption of a small angle, equation (2) is written as:

<mrow> <msub> <mi>&omega;</mi> <mi>b</mi> </msub> <mo>&ap;</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

wherein:

ω_ocalculating the orbit parameters of the spacecraft to obtain;representing a state quantity of the control system;

A_s,A_tthe calculation is as follows:

<mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>s</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <msub> <mi>&delta;</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <msub> <mi>&delta;</mi> <mi>i</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <msub> <mi>&delta;</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>s</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <msub> <mi>&delta;</mi> <mi>i</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>

step 2: based on the on-orbit operation characteristic of the spacecraft, the spacecraft attitude fault-tolerant control method of the single-frame control moment gyro group is applied, and is based on the following assumptions:

assume that 1: navigation deviceThe disturbance moment suffered by the antenna during operation is bounded, namely: d | | | is less than or equal to T_d(ii) a And the influence of additive faults on the rotation speed of the gyro frame is limited, | | A_tf||≤T_f(ii) a Where the convention | · | |, represents the 2-norm of a matrix or vector, T_d,T_fIs an unknown constant; suppose 1 is integrated as the following expression:

||-A_tf+d||≤M_d(7)

assume 2: the rotational inertia matrix of the spacecraft is a positive definite symmetric matrix, namely J is symmetric and positive definite;

assume that 3: in the case of a complete failure of the gyro, i.e. assuming the presence of an unknown constant e₀Satisfies the following conditions:

wherein n is the number of gyros in the SGCMGs;

and step 3: designing a sliding mode control law; the method specifically comprises the following steps:

step 3.1: slip form face design

The sliding mode surface is selected as follows:

<mrow> <mi>s</mi> <mo>=</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mi>k</mi> <mi>x</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>

wherein k is greater than 0, and a constant is given to a designer; then when s → 0, x → 0, is a column vector composed of attitude angles, represents a state quantity of the control system,represents the derivative of the state vector with respect to time;

step 3.2: preliminary design of control law

The following sliding mode control law is selected:

<mrow> <mi>u</mi> <mo>=</mo> <mo>-</mo> <msubsup> <mi>A</mi> <mi>t</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <msubsup> <mi>A</mi> <mi>t</mi> <mi>T</mi> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>{</mo> <msubsup> <mi>&omega;</mi> <mi>b</mi> <mo>&times;</mo> </msubsup> <mo>&lsqb;</mo> <msub> <mi>J&omega;</mi> <mi>b</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mi>c</mi> </msub> <mo>&rsqb;</mo> <mo>-</mo> <mi>J</mi> <mi>k</mi> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mi>J</mi> <mover> <mi>F</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>&gamma;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mfrac> <mi>s</mi> <mrow> <mo>|</mo> <mo>|</mo> <mi>s</mi> <mo>|</mo> <mo>|</mo> </mrow> </mfrac> <mo>-</mo> <msub> <mover> <mi>M</mi> <mo>^</mo> </mover> <mi>d</mi> </msub> <mfrac> <mi>s</mi> <mrow> <mo>|</mo> <mo>|</mo> <mi>s</mi> <mo>|</mo> <mo>|</mo> </mrow> </mfrac> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>

the values and meanings of the parameters in the control law are as follows:

derivative with respect to time for f (x) in step 1.2.3; s is a slip form surface;

m in the formula (10)_dThe self-adaptive updating law is taken as follows:

<mrow> <msub> <mover> <mover> <mi>M</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>=</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>|</mo> <mo>|</mo> <mi>s</mi> <mo>|</mo> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>

γ (t): a parameter is introduced, take

<mrow> <mi>&gamma;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msqrt> <mi>n</mi> </msqrt> <mi>&upsi;</mi> <mo>+</mo> <msqrt> <mi>n</mi> </msqrt> <mi>&upsi;</mi> <mover> <mi>&xi;</mi> <mo>^</mo> </mover> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>0</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mover> <mover> <mi>&xi;</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msqrt> <mi>n</mi> </msqrt> <msub> <mi>c</mi> <mn>1</mn> </msub> <mi>&upsi;</mi> <mo>|</mo> <mo>|</mo> <mi>s</mi> <mo>|</mo> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>

Defining variablesWherein c is₀,c₁,₀Is a normal number, n is the number of gyros under a certain configuration, u is a control quantity,ξ, v is an intermediate variable,is composed ofA derivative with respect to time;

taking the Lyapunov function as

WhereinThe rest parameters are given above, and let Δ E be I-E, I is a unit matrix, and E is a multiplicative fault matrix;

the derivative of the Lyapunov function with time is obtained by using the formulas (10) to (14):

<mrow> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mo>&le;</mo> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mn>0</mn> </msub> <mo>|</mo> <mo>|</mo> <mi>s</mi> <mo>|</mo> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>

equation (16) indicates that the function V does not increase at least monotonically, and therefore is sup_t≥0V (t). ltoreq.V (0), wherein sup (. cndot.) denotes supremum, i.e.There is a limit to the amount of space that, therefore,thereby to obtainExisting and bounded, according to the Barbalt theorem, there areTherefore, there is a x → 0 region,

step 3.3: improvements in control laws

The control law has the problems of buffeting and singularity, so improvement needs to be made on the basis of the control law;

the problem of buffeting: because the sliding mode control has discontinuity of control, a buffeting phenomenon exists; according to the sliding mode control theory, s/(| s | + tau) is adopted to approximately replace a symbolic function s/| s |, wherein tau is a smaller positive number and is selected to be 10^-3～10^-1To (c) to (d);

the singularity problem is as follows: when all the gyros output moment are coplanar, the moment can not be output in the normal direction of the moment plane, and at the moment, the SGCMGs transverse matrix A_tIf the rank is not full, the formula (9) cannot be solved, so the formula (9) is improved by referring to a robust pseudo-inverse solving method;

the improved control strategy is therefore:

<mrow> <mi>u</mi> <mo>=</mo> <mo>-</mo> <msubsup> <mi>A</mi> <mi>t</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>&lsqb;</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <msubsup> <mi>A</mi> <mi>t</mi> <mi>T</mi> </msubsup> <mo>+</mo> <mi>&lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mn>3</mn> <mo>&times;</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>&times;</mo> <mn>3</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>{</mo> <msubsup> <mi>&omega;</mi> <mi>b</mi> <mo>&times;</mo> </msubsup> <mo>&lsqb;</mo> <msub> <mi>J&omega;</mi> <mi>b</mi> </msub> <mo>+</mo> <mi>h</mi> <mo>&rsqb;</mo> <mo>-</mo> <mi>J</mi> <mi>k</mi> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mi>J</mi> <mover> <mi>F</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>&gamma;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mfrac> <mi>s</mi> <mrow> <mo>|</mo> <mo>|</mo> <mi>s</mi> <mo>|</mo> <mo>|</mo> <mo>+</mo> <mi>&tau;</mi> </mrow> </mfrac> <mo>-</mo> <msub> <mover> <mi>M</mi> <mo>^</mo> </mover> <mi>d</mi> </msub> <mfrac> <mi>s</mi> <mrow> <mo>|</mo> <mo>|</mo> <mi>s</mi> <mo>|</mo> <mo>|</mo> <mo>+</mo> <mi>&tau;</mi> </mrow> </mfrac> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>

wherein: λ is a small positive number, taken at 10^-3～10^-1To (c) to (d); i is_3×3Is a third order identity matrix, E_3×3Is a diagonal matrix, and has the form:

<mrow> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>&times;</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&epsiv;</mi> <mn>3</mn> </msub> </mtd> <mtd> <msub> <mi>&epsiv;</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&epsiv;</mi> <mn>3</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&epsiv;</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&epsiv;</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>&epsiv;</mi> <mn>1</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>