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

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 for Single Frame Control Moment Gyroscope Group

【Technical field】

The present invention uses Single Gimbal Control Moment Gyros (SGCMGs) as the three-axis stable spacecraft as the actuator, and the attitude fault-tolerant control method (Fault-Tolerant Control, FTC) when the actuator fails partially, In order to realize that the spacecraft has strong robustness to faults, it belongs to the field of spacecraft attitude control.

【Background technique】

With the development of aerospace technology, space missions are becoming more and more complex, which puts forward higher requirements for the safety, stability and control precision of spacecraft. It can be seen from the development history of aerospace technology that many accidents are caused by only a small fault. For example, in 1997, the Lewis satellite launched by NASA failed, causing all thrusters to fail, and finally the satellite fell into the atmosphere, causing huge losses. How to avoid risks and make spacecraft fault-tolerant has become a focus of research by many aerospace experts. At present, fault diagnosis and fault-tolerant control have become an important way to maintain the reliability, maintainability and effectiveness of spacecraft.

The idea of fault-tolerant control was first proposed by Niederlinski in 1971, and then 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. Active fault-tolerant control is to redesign a control system according to the desired characteristics after a fault occurs, and at least make the whole system stable. Passive fault-tolerant control uses a fixed controller to ensure that the closed-loop system is insensitive to specific faults and maintains system stability. Compared with active fault-tolerant control, passive fault-tolerant control does not need to detect or diagnose system faults, and does not require fault reaction time, so the structure is simple, the response speed is fast, and the design difficulty is relatively low.

In the field of attitude fault-tolerant control, the current research results mainly use the control torque as the control variable and the fault modeling object. However, in practical engineering applications, when the angular momentum exchange device is used as the attitude control actuator, the actual control is often the speed of the actuator. For example, for a spacecraft with a flywheel as the actuator, the control torque is determined by the flywheel speed. On the other hand, the torque output of the angular momentum exchange device may also be related to the current attitude of the gyro. For example, when the control moment gyro group (CMG) is used as the actuator, it has the singularity problem and the time-varying transverse matrix of the gyro Therefore, the control torque is affected by the frame angle and the frame speed at the same time.

The existence of the above problems makes it difficult to apply the current research results in actual engineering, especially for the field of spacecraft attitude fault-tolerant control with CMG as the actuator. At present, there is basically no good engineering application method to realize it.

【Content of invention】

The present invention proposes a kind of spacecraft with single-frame control moment gyroscope group SGCMGs as the actuator, through the sliding mode control method and the self-adaptive control method, to realize the partial failure fault of the actuator (each gyroscope of the SGCMGs has torque output) Attitude stabilization control of spacecraft.

For the problems referred to above, the technical scheme of the present invention is as follows:

According to the dynamic equations and kinematic equations of the spacecraft with partial failure of the actuator, the sliding surface is established by using the state quantities such as Euler angle and Euler angular velocity, and the fault information of the spacecraft is estimated online by using the adaptive control method. The sliding mode control strategy and appropriate control parameters are designed so that the spacecraft can achieve attitude stability without failure, and this set of control parameters can also make the spacecraft’s actuators partially fail and still achieve attitude stability. The specific operation steps are as follows

Step 1: Establish the dynamic equation and kinematic equation when the single-frame control moment gyro group SGCMGs has partial failure. Specifically include the following steps:

Step 1.1: Define the Coordinate System

a. Body coordinate system f _b (o _b x _b y _b z _b )

This coordinate system is fixedly connected with the spacecraft, the origin O _b is located at the center of mass of the spacecraft, the O _b x _b axis points to the motion direction of the spacecraft, the O _b z _b axis points to the top of the spacecraft and is perpendicular to the flight orbit plane, and the O by _{y b axis} , The O _b x _b axis and the O _b z _b axis form a right-handed coordinate system.

b. Orbital coordinate system f _o (O _o x _o y _o z _o )

The origin of the orbital coordinate system is at the center of mass of the spacecraft, the O _o z _o axis points to the center of the earth along the local vertical line, the O _o x _o axis is perpendicular to the O _o z _o axis in the orbital plane, and points to the motion direction of the spacecraft, O _o The y _o axis, the O _o x _o axis, and the O _o z _o axis constitute a right-handed coordinate system. The coordinate system rotates around the O _o y _o axis with an angular velocity ω _o in space.

c. Geocentric inertial coordinate system f _i (O _i x _i y _i z _i )

The origin of the earth-centered inertial coordinate system is fixed at the center of the earth O _i , the axis O _i x _i is on the equator plane and points to the vernal equinox, O _i z _i is perpendicular to the equator plane and is consistent with the direction of the earth's rotation angular velocity, and the axis O _i y _i is at In the equatorial plane, and form a Cartesian coordinate system with the O _i x _i axis and the O _i z _i axis.

d. SGCMGs frame coordinate system f _ci (O _ci g _i s _i t _i )

The origin of the frame coordinate system is at the center of mass O _ci of SGCMG, and the unit vectors in each direction of the coordinate system are the unit vectors along the frame axis direction A unit vector along the direction of the rotational speed of the rotor shaft Unit vector along the opposite direction of the gyro torque output

Step 1.2 Establishment of control system state equation

Step 1.2.1 Establish dynamic equations and kinematic equations

Kinetic equations:

Among them, I _b is the inertia matrix of the whole system, and I _b is considered to be a constant value inertia matrix;

ω _b ＝[ω _x ω _y ω _z ] ^T is the component array of the absolute angular velocity of the spacecraft in this system;

h ₀ is the nominal angular momentum of each gyro rotor;

is the derivative of ω _b with respect to time, is defined as follows:

A _s = [s ₁ s ₂ … s _n ] is the SGCMGs rotor speed direction matrix;

I _ws is the axial moment of inertia matrix of the SGCMGs rotor;

Ω is the rotor speed vector; h ₀ is the nominal angular momentum of each gyro rotor;

A _t = [t ₁ t ₂ ... t _n ] is the SGCMGs horizontal matrix;

δ is the gyro frame angle;

T _d is the disturbance torque vector received by the spacecraft;

Kinematic equation:

Among them, the attitude angle θ, ψ are roll angle, pitch angle and yaw angle of the spacecraft; attitude angular velocity respectively θ, ψ derivatives with respect to time; ω _o is the angular velocity of the orbital system rotating around the O _o y _o axis of the system;

Step 1.2.2 Establish failure mode

It should be pointed out that the failure modeling here is formal, and in the actual system, the failure mode equation is implicit in the motion of the spacecraft. Although the specific expressions or specific values of E and f cannot be known with certainty, it is easy to determine whether there is a gyro stuck in the SGCMGs and cannot output torque, which is sufficient in the present invention.

in, is the rotational speed vector of the theoretical framework of the gyroscope; is the actual frame speed vector of the gyroscope;

E=diag(e ₁ e ₂ … e _n ) is the multiplicative fault matrix, and e _i is the failure factor of the i-th gyro;

f＝[f ₁ f ₂ … f _n ] ^T is the influence of additive faults on the rotational speed of the gyro frame,

f _i is the rotational speed deviation of the i-th gyroscope.

Step 1.2.3 Derivation of kinematic equations and dynamic equations under failure mode

Substitute Equation (3) into Equation (1), and define the equivalent disturbance, equivalent moment of inertia matrix and equivalent gyro group angular momentum of the unit nominal angular momentum of the gyro rotor as follows: Equivalent disturbance: d=T _d /h ₀ ;

Equivalent moment of inertia matrix: J＝I _b /h ₀ ; Equivalent gyroscope group angular momentum: h _c ＝A _s I _ws Ω/h ₀ ;

make As the control quantity of the control system, the dynamic equation under fault is obtained:

Under the assumption of small angle, formula (2) can be approximately written as:

in:

ω _o is calculated from the orbital parameters of the spacecraft; Represents the state quantity of the system;

A _s , At _t can be calculated as follows:

s _i0 and t _i0 are determined according to the specific configuration of the gyroscope, and are unit vectors; s _i0 represents the initial value of s _i ; t _i0 represents the initial value of t _i ;

Step 2 is based on the actual characteristics of the spacecraft in orbit, and the application of the present invention is based on the following assumptions:

Assumption 1: The disturbance torque encountered by the spacecraft during operation is bounded, ie: ||d||≤T _d ; and the impact of additive faults on the gyro frame speed is limited, ||A _t f||≤T _f . The convention ||·|| represents the 2-norm of a matrix or vector, and T _d and T _f are unknown constants. Hypothesis 1 can be summarized as the following expression:

||-A _t f+d||≤M _d (7)

Assumption 2: The moment of inertia matrix of the spacecraft is a positive definite symmetric matrix, that is, J is symmetric and positive definite.

Hypothesis 3: The present invention does not consider the situation that the gyroscope fails completely, that is, it is assumed that there is an unknown constant e ₀ that satisfies:

Among them, n is the number of tops in SGCMGs.

Step 3 Sliding mode control law design. Specifically include the following steps:

Step 3.1 Sliding surface design

The selected sliding surface is:

Among them, k>0 is a constant given by the designer. Then when s→0, x→0, is a column vector composed of attitude angles, representing the state quantity of the 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 chosen:

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

A _t ＝[t ₁ t ₂ … t _n ] is the SGCMGs horizontal matrix, is the transpose matrix of At _t ;

J, _hc are the equivalent moment of inertia matrix and equivalent gyroscope group angular momentum defined in step 1.2.3;

Be the derivative of F(x) about time in step 1.2.3; s is the sliding mode surface;

The estimated value of M _d in expression (10), taking the adaptive update law as:

γ(t): a parameter introduced, take

define variable Where c ₀ , c ₁ , ε ₀ are a constant number, n is the number of gyroscopes in a certain configuration, u is the control amount, is the estimated value of ξ, υ is the intermediate variable, for derivative with respect to time.

Take the Lyapunov function as

in The other parameters have been given above, and record ΔE=IE, I is the identity matrix, and E is the multiplicative fault matrix.

Calculate the derivative of the above Lyapunov function with respect to time, and use formulas (10)~(14), we can get:

Equation (16) shows that the function V at least does not increase monotonically, so sup _t≥0 V(t)≤V(0), where sup(·) represents the supremum, namely bounded, therefore, thereby exists and is bounded, according to Barbalat's lemma, we have Hence x→0,

Step 3.3 Control Law Improvement

The above control law actually has chattering and singularity problems, so it needs to be improved on the basis of the above control law.

Chattering problem: Since the sliding mode control has control discontinuity, there is a chattering phenomenon. According to the sliding mode control theory, the present invention adopts s/(||s||+τ) to approximately replace the sign function s/||s||, wherein τ is a small positive number, which is usually given according to the actual situation, generally Selected between 10 ^-3 and 10 ^-1 .

Singularity problem: When the output torques of the gyroscopes are coplanar (or collinear), the normal direction of the torque plane (or the normal plane of the torque direction) cannot output torque. At this time, the SGCMGs transverse matrix A _t is not satisfied with rank. The formula ( 9) cannot be solved, so formula (9) is improved by referring to the method of robust pseudo-inverse solution.

Therefore, the improved control strategy is obtained as:

Among them: λ is a small positive number, which usually needs to be selected according to the actual working conditions, and generally can be taken between 10 ^-3 and 10 ^-1 ; I _3×3 is the third-order unit matrix, and E _3×3 is the pair Angular matrix, in the form:

Each element in the matrix is: ε _j =0.01(0.5πt+φ _j )(j=1,2,3), φ _j =π(j-1)/2.

The other parameters are the same as formulas (11)-(14).

If λ, τ are too large, the above improvement cannot guarantee the stability of the system; theoretically, as long as the values of λ, τ are small enough, the formula (17) can still satisfy the robustness of the fault system. But in fact, if λ, τ are too small, it will not be able to eliminate the singularity and chattering phenomenon. Therefore, the selection of λ and τ must be adjusted according to the actual system parameters. Generally, a parameter from 10 ^-3 to 10 ^-1 can be selected as the basis, and adjusted according to the actual control effect.

In addition, the control law designed by the present invention is also applicable to spacecraft with a flywheel as an angular momentum exchange device, as long as the relative angular momentum h _c of the gyroscope in the dynamic model (4) is replaced by the relative angular momentum h _w of the flywheel, the transverse matrix A If _t is replaced by the installation matrix C of the flywheel, the same control law (17) is used to control the torque of each flywheel, which can also ensure the stability of the fault system.

The present invention has designed a kind of attitude fault-tolerant control method of the spacecraft of actuator fault, and its advantage is mainly as follows:

1) Although the sliding mode control law designed in the present invention is designed with SGCMGs as the background, because the dynamic characteristics of the gyroscope are similar to the flywheel, the control law designed in the present invention is also applicable to the spacecraft fault-tolerant control with the flywheel as the actuator.

2) The present invention does not need to know the prior information of the fault exactly, but estimates the fault information and interference information in real time through adaptive control, so the time-varying fault is allowed, as long as there is no complete failure of a certain gyro.

3) The present invention does not aim at specific SGCMGs configuration or flywheel configuration in the process involved, but only uses the feature that the 2-norm of each column vector of the lateral matrix A _t or the installation matrix C is 1 in the proof of stability , and this is easily satisfied in practical engineering, so the present invention is suitable for partial failure modes of SGCMGs or flywheels with arbitrary configurations.

4) The present invention is used in the spacecraft with SGCMGs, is to take the gyro frame speed as the direct control quantity, conforms to the engineering reality, and is the spacecraft of the executive mechanism to the flywheel, although the output torque of each flywheel is directly controlled, the output of the flywheel The torque is directly proportional to the speed, so it is actually equivalent to controlling the speed of the flywheel, which is also in line with engineering practice.

【Description of drawings】

Figure 1 is a schematic diagram of attitude stability fault-tolerant control.

Figure 2 is a schematic diagram of the spacecraft body coordinate system.

Figure 3 is a schematic diagram of the orbital coordinate system of the spacecraft.

Figure 4 is a schematic diagram of the spacecraft inertial coordinate system.

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

Figure 6 is a schematic diagram of the control law design process.

Fig. 7 is a schematic diagram of SGCMGs in pyramid configuration.

【detailed description】

The implementation flow of the present invention will be described in detail by taking a certain type of spacecraft as an example in conjunction with accompanying drawings 1-7. The parameters of the spacecraft are as follows:

The moment of inertia matrix of the spacecraft is:

SGCMGs with a pyramid configuration are selected, in which the nominal angular momentum of the gyro is 200Nms; the initial attitude angle is: θ(0)＝1.5°, ψ(0)＝1.5°; the initial value of ω _b is ω _b (0)＝[0 0 0] ^T ; The disturbance torque comprehensively considers the earth's gravitational perturbation, solar light pressure moment, solar radiation pressure disturbance, etc., and adopts the following external disturbance form, which is

Among them, A ₀ is the amplitude of the disturbance torque, and A ₀ =1.5×10 ^-5 N·m.

Assume that multiplicative and additive failures occur simultaneously on the spacecraft.

The multiplicative fault parameters are:

Where rand(·) represents a random function with an amplitude of 1, t ₁ =150s, t ₂ =180s, t ₃ =200s, t ₄ =240s

The additive fault parameters are:

f _i (t)=-0.01 (i=1,2,3,4,t≥t _i )

It is only necessary for simulation to give fault parameters and external disturbance expressions.

Next, set up the control law to control the attitude of the spacecraft.

1. Establish the dynamic equation and kinematic equation when SGCMGs have partial failure. Specifically include the following steps:

1.1 Define the coordinate system: Follow step 1.1 to define the relevant coordinate system.

1.2 Establishment of control system state equation

First, according to the relevant parameters of the spacecraft used in the example, the following system parameters can be listed directly:

If SGCMGs with pyramid configuration are selected, the number of gyroscopes is n=4.

The moment of inertia matrix of the spacecraft is:

The nominal angular momentum h ₀ of each gyro rotor = 200Nms;

T _d ＝[T _d1 T _d2 T _d3 ] is the disturbance torque vector received by the spacecraft;

In the following, the expressions of A _s and A _t are calculated according to the schematic diagram of the pyramid configuration.

s ₁₀ ＝[0 -1 0] ^T , g ₁₀ ＝[-sinβ 0 cosβ] ^T

s ₂₀ ＝[-1 0 0] ^T , g ₂₀ ＝[0 sinβ cosβ] ^T

s ₃₀ ＝[0 1 0] ^T , g ₃₀ ＝[sinβ 0 cosβ] ^T

s ₄₀ ＝[1 0 0] ^T , g ₄₀ ＝[0 -sinβ cosβ] ^T

According to formula (6), it can be obtained:

where β can be determined by the following process,

The triaxial angular momentums in this system are:

H _x ＝2h ₀ +2h ₀ cosβ

H _y =2h ₀ +2h ₀ cosβ

H _z =4h ₀ sinβ

In order to make the three-axis angular momentum of the pyramid configuration equal, that is, H _x =H _y =H _z , β=53.1° is obtained. Then h _c =A _s [h ₀ h ₀ h ₀ h ₀ ] ^T .

ω _o is determined based on orbital parameters. Since the spacecraft is in a circular orbit with a radius of 26600km, therefore:

Among them, μ is the gravitational constant of the earth, which is 3.986005×10 ¹⁴ m ³ /s ² , and R is the radius of the orbit.

Calculated: ω ₀ =4.6020×10 ^-4 rad/s.

1.2.1 Establish dynamic equations and kinematic equations

Kinetic equations:

Kinematic equation:

1.2.2 Establish failure mode

E=diag(e ₁ e ₂ e ₃ e ₄ ) is the multiplicative fault factor, and f=[f ₁ f ₂ f ₃ f ₄ ] ^T is the additive fault factor.

1.2.3 Derivation of kinematic equations and dynamic equations under failure mode

Let d=T _d /h ₀ , J=I _b /h ₀ , h _c ＝A _s I _ws Ω/h ₀ , and let As the control quantity of the control system, the failure mode is substituted into the dynamic equation, and the dynamic equation under the fault is obtained:

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

in:

2. Based on the actual characteristics of spacecraft in-orbit operation, the application of the present invention is based on the following assumptions:

Assumption 1: The disturbance torque encountered by the spacecraft during operation is bounded, ie: ||d||≤T _d ; and the impact of additive faults on the gyro frame speed is limited, ||A _t f||≤T _f . The convention ||·|| represents the 2-norm of a matrix or vector, and T _d and T _f are unknown constants. Hypothesis 1 can be summarized as the following expression:

||-A _t f+d||≤M _d (23)

Assumption 2: The moment of inertia matrix of the spacecraft is a positive definite symmetric matrix, that is, J is symmetric and positive definite.

Hypothesis 3: The present invention does not consider the situation that the gyroscope fails completely, that is, it is assumed that there is an unknown constant e ₀ that satisfies:

3. Sliding mode control law design. Specifically include the following sub-steps:

3.1 Sliding surface design

The selected sliding surface is:

Among them, k>0 is a constant.

For convenience, the following parameters are defined:

3.2 Preliminary Design of Control Law

The following sliding mode control law is chosen:

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

The estimated value of M _d in expression (9), taking the adaptive update law as:

γ(t): a parameter introduced, take

Among them, c ₀ , c ₁ , ε ₀ are a normal number.

Because the specific time and parameters of the failure of the spacecraft cannot be known in advance, the control parameters are adjusted when the actuator is working without failure to ensure better attitude control performance. The control parameters are selected as follows:

k=2, c ₀ =0.5, ε ₀ =0.5, c ₁ =10 (31)

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

Unknown, directly set to 0 in this example.

Since it is generally believed that there is no fault in the system at the beginning of the simulation, the selection

Take the Lyapunov function as

Calculate the derivative of the above Lyapunov function with respect to time, and use formulas (27)~(31), we can get:

This formula shows that the function V at least does not increase monotonically, so sup _t≥0 V(t)≤V(0), where sup(·) represents the supremum, that is bounded, therefore, thereby exists and is bounded, according to Barbalat's lemma, we have Hence x→0,

3.3 Improvement of control law

Considering the singularity problem of single-frame control moment gyroscope and the chattering problem inherent in sliding mode control, this paper modifies formula (23) as follows:

Where: λ=0.001; I _3×3 is a third-order identity matrix, E _3×3 is a diagonal matrix, and the form is:

Each element in the matrix is: ε _j =0.01(0.5πt+φ _j )(j=1,2,3), φ _j =π(j-1)/2, τ=0.001.

The other parameters are the same as (27)～(31).

In summary, the choice of control law (34) can ensure that the attitude angle and attitude angular velocity of the system (18) and (19) can still have global asymptotic stability at the origin even in the event of a fault, which also shows that the control law (34 ) is robust to failures of systems (18), (19).

The attitude stability fault-tolerant control method of the spacecraft with SGCMGs as the actuator introduced by the present invention is characterized in that: because the specific parameter information of the fault cannot be determined in advance, the control law cannot be designed according to the prior information of the fault, so this paper adopts the self-adaptive The method estimates the fault information in real time to design the control law, which is more in line with the engineering practice. On the other hand, the reason why the control method of the present invention does not allow the complete failure of the gyroscope is because when the number of gyroscopes in the configuration used is small, if there is a gyroscope that fails completely, then the gyroscope will be singular in configuration. That is, there may always be a direction that cannot output torque. This problem is not among those solved by the present invention.

Claims (1)

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

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 failure modes

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

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 gyroscope;

step 1.2.3: deducing kinematic equation and kinetic equation under 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:

<mrow> <mi>J</mi> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>b</mi> </msub> <mo>+</mo> <msubsup> <mi>&amp;omega;</mi> <mi>b</mi> <mo>&amp;times;</mo> </msubsup> <mrow> <mo>(</mo> <msub> <mi>J&amp;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>&amp;omega;</mi> <mi>b</mi> </msub> <mo>&amp;ap;</mo> <mover> <mi>x</mi> <mo>&amp;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>&amp;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>&amp;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>&amp;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>&amp;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>

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: 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:

<mrow> <mn>0</mn> <mo>&lt;</mo> <msub> <mi>e</mi> <mn>0</mn> </msub> <mo>&amp;le;</mo> <munder> <mi>min</mi> <mrow> <mn>1</mn> <mo>&amp;le;</mo> <mi>i</mi> <mo>&amp;le;</mo> <mi>n</mi> </mrow> </munder> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

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>&amp;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>&amp;omega;</mi> <mi>b</mi> <mo>&amp;times;</mo> </msubsup> <mo>&amp;lsqb;</mo> <msub> <mi>J&amp;omega;</mi> <mi>b</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mi>c</mi> </msub> <mo>&amp;rsqb;</mo> <mo>-</mo> <mi>J</mi> <mi>k</mi> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mi>J</mi> <mover> <mi>F</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;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:

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

<mrow> <mi>&amp;upsi;</mi> <mo>=</mo> <mfrac> <mrow> <mo>|</mo> <mo>|</mo> <mi>u</mi> <mo>|</mo> <mo>|</mo> </mrow> <mover> <mi>&amp;xi;</mi> <mo>^</mo> </mover> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mover> <mover> <mi>&amp;xi;</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <msqrt> <mi>n</mi> </msqrt> <msub> <mi>c</mi> <mn>1</mn> </msub> <mi>&amp;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

<mrow> <mi>V</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>s</mi> <mi>T</mi> </msup> <mi>J</mi> <mi>s</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>c</mi> <mn>0</mn> </msub> </mrow> </mfrac> <msubsup> <mover> <mi>M</mi> <mo>~</mo> </mover> <mi>d</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mfrac> <msub> <mi>e</mi> <mn>0</mn> </msub> <mrow> <mn>2</mn> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> </mfrac> <msup> <mover> <mi>&amp;xi;</mi> <mo>~</mo> </mover> <mn>2</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>

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>&amp;CenterDot;</mo> </mover> <mo>&amp;le;</mo> <mo>-</mo> <msub> <mi>&amp;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>&amp;lsqb;</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <msubsup> <mi>A</mi> <mi>t</mi> <mi>T</mi> </msubsup> <mo>+</mo> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mn>3</mn> <mo>&amp;times;</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>&amp;times;</mo> <mn>3</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>{</mo> <msubsup> <mi>&amp;omega;</mi> <mi>b</mi> <mo>&amp;times;</mo> </msubsup> <mo>&amp;lsqb;</mo> <msub> <mi>J&amp;omega;</mi> <mi>b</mi> </msub> <mo>+</mo> <mi>h</mi> <mo>&amp;rsqb;</mo> <mo>-</mo> <mi>J</mi> <mi>k</mi> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>+</mo> <mi>J</mi> <mover> <mi>F</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;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>&amp;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>&amp;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>&amp;times;</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mn>3</mn> </msub> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&amp;epsiv;</mi> <mn>3</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&amp;epsiv;</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mn>1</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>

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