CN108536014B - Model prediction control method for spacecraft attitude avoidance considering dynamic characteristics of flywheel - Google Patents

Model prediction control method for spacecraft attitude avoidance considering dynamic characteristics of flywheel Download PDF

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CN108536014B
CN108536014B CN201810298997.1A CN201810298997A CN108536014B CN 108536014 B CN108536014 B CN 108536014B CN 201810298997 A CN201810298997 A CN 201810298997A CN 108536014 B CN108536014 B CN 108536014B
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flywheel
spacecraft
attitude
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time
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CN108536014A (en
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胡庆雷
杨昊旸
郭雷
王陈亮
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Beihang University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B64—AIRCRAFT; AVIATION; COSMONAUTICS
    • B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
    • B64G1/00—Cosmonautic vehicles
    • B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
    • B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
    • B64G1/244—Attitude control

Abstract

A model prediction control method for spacecraft attitude evasion considering dynamic characteristics of a flywheel comprises the following steps: establishing a prediction model based on attitude dynamics of a spacecraft and dynamics of a flywheel; secondly, establishing a constrained mathematical model according to the dynamic characteristics of the flywheel and the instrument line-of-sight angle; then designing performance index functions facing different task requirements, and converting the control problem into an extreme value problem of the objective function under the constraint conditions of equality and inequality; finally, the problem is solved quickly through an optimization method based on real-time iteration, the method can well process the attitude avoidance problem of the spacecraft under the constraint condition of an execution mechanism, the comprehensive optimization of energy and time is achieved through the design of a target function, and the calculated amount for solving the optimization problem can be reduced through the processing of the real-time iteration and hot start method.

Description

Model prediction control method for spacecraft attitude avoidance considering dynamic characteristics of flywheel
Technical Field
The invention relates to a model prediction control method for spacecraft attitude evasion, which belongs to the technical field of spacecraft control and is mainly applied to evasion control of spacecraft attitude maneuver controlled by using a reaction flywheel, in particular to a model prediction control method for spacecraft attitude evasion considering dynamic characteristics of a flywheel.
Background
In recent years, due to rapid development of space technology, the task requirements of in-orbit spacecrafts are more and more, so that various optical instruments such as a CCD camera, an infrared interferometer and the like are mounted on the spacecrafts, and during the working process of the instruments, the direct strong light facing to the sight line of the instruments needs to be avoided so as to protect the components sensitive to light and temperature in the instruments. Therefore, it is required that the spacecraft should point the instruments away from the direction of the strong light during attitude maneuver. Meanwhile, actuating mechanisms in most spacecraft attitude control systems are reaction flywheel combinations, and the reaction flywheels have the advantages of high control precision, only electric energy consumption for outputting torque and the like. However, the reaction flywheel has the problems that the reaction moment is relatively small and the saturation is easy to achieve in the use process, so that the attitude control of the spacecraft is influenced. Therefore, the significance of researching the attitude avoidance problem of the spacecraft under the constraint condition of the performance of the actuating mechanism is very important.
The method for avoiding the attitude can be a potential function method mentioned in patent 201710521561.X, and the potential function method can effectively avoid a constraint area, but the method for using model predictive control can not only consider the attitude avoiding constraint, but also consider the performance constraint of an actuator, and can realize the optimization of energy and time according to the performance function, so that the method is more suitable for attitude maneuver control under the performance constraint of the actuator.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, provides a model prediction control method for spacecraft attitude evasion considering dynamic characteristics of a flywheel, solves the problems of performance constraint and attitude constraint of an executing mechanism in the attitude maneuver process of an in-orbit spacecraft, and provides the model prediction control method for spacecraft attitude evasion considering the dynamic characteristics of the flywheel. The method is a prediction control method capable of simultaneously processing performance constraint of an execution mechanism and an attitude avoidance model, and the method improves the solving efficiency of solving the optimization problem in MPC by using real-time iteration and hot start methods in the optimization process, and designs a performance function to realize comprehensive optimization of energy consumption and time in the attitude maneuver process.
The invention provides a model prediction control method for spacecraft attitude evasion considering dynamic characteristics of a flywheel, which comprises the following steps of:
(1) establishing a spacecraft attitude model containing an actuating mechanism as a prediction model of the MPC according to the attitude kinematics and the dynamic characteristics of the spacecraft and the rotation dynamic characteristics of the flywheel;
(2) establishing a mathematical model of attitude avoidance constraint according to a line-of-sight angle and an avoidance vector of a spacecraft carrying instrument, and establishing a mathematical model of actuating mechanism constraint according to angular momentum saturation and moment saturation characteristics of a flywheel;
(3) designing a corresponding optimization objective function according to task requirements, wherein the optimization objective function comprises a quadratic form input by an execution mechanism and a quadratic form of system state errors so as to comprehensively consider execution time and energy consumption;
(4) converting the control problem into a problem of solving an extreme value of an objective function under the constraint conditions of an equation and a state of a system dynamic equation and an inequality with limited input, then quickly solving the problem by using a real-time iterative optimization processing method, and outputting a solution obtained by optimization as a control quantity of the system.
The spacecraft attitude dynamics model in the step (1) is obtained by combining the kinetic equation of the flywheel into the spacecraft attitude dynamics equation and discretizing, and the representation form is as follows:
wherein ω is [ ω ═ ω [ [ ω ]1,ω2,ω3]TRepresenting the attitude angular velocity vector omega of the spacecraft relative to an inertial coordinate system under a body coordinate system1,ω2,ω3Angular velocity components of the spacecraft about the roll axis, yaw axis and pitch axis in the system, respectively;represents the derivative of ω with time; j is the total inertia matrix of the spacecraft, simplified as the diagonal matrix J ═ diag (J)1,J2,J3)，J1,J2,J3Is the moment of inertia around the inertia main shaft; s (omega) is a skew symmetric matrix of the formτ represents the output torque of the actuator; q ═ q0,q1,q2,q3]TRepresents a unit quaternion of the attitude of the spacecraft,represents the scalar part of the quaternion of the attitude unit of the spacecraft, theta represents an angle of rotation about the euler axis,ex,ey,ezrepresents the rotation axis in three directions of the Euler axis and satisfies Represents the derivative of q with respect to time; Ω (ω) is a diagonally symmetric matrix of the form:
the model of the reaction flywheel combination is as follows:
wherein HrwIs the angular momentum of the flywheel combination,the derivative of the angular momentum of a flywheel assembly with respect to time is given by the following relationship between the angular momentum and the rotational speed in a flywheel assembly consisting of four flywheels:
Hrw＝CJrwN
where C is a 3 × 4 flywheel mounting matrix and N ═ N1,n2,n3,n4]TIs the angular velocity vector of the flywheel, n1,n2,n3,n4Respectively representing the angular velocity of each flywheel; j. the design is a squarerwA moment of inertia matrix, of the form J, representing a flywheel assemblyrw＝JαI4×4，JαRepresenting the moment of inertia of a single flywheel, I4×4Is an identity matrix of 4 orders;
since the actuator controls attitude by exchanging angular momentum with the spacecraft, the total angular momentum of the system is conserved:
H＝Hrw+Jω
h is the total angular momentum of the system, and is a constant in the absence of external moment.
Integrating the model of the actuating mechanism and the model of the spacecraft attitude dynamics to obtain:
discretizing the model, setting a sampling interval to be delta t, and obtaining the model at the kth moment by:
ωk+1＝J-1ΔtS(ωk)H+J-1JαCΔNk+ωk
where the index k denotes the value of the corresponding variable at the k-th instant, Δ Nk＝Nk-Nk-1，I3×3Is a third order identity matrix.
In step (2):
(a) constraint of maximum output torque of reaction flywheel:
the torque output of the reaction flywheel is achieved by changing the angular momentum of the flywheel, so it is expressed in the form:
wherein T ismaxThe maximum output torque vector. Discretizing the formula to obtain:
after finishing, obtaining:
(b) constraint of maximum angular momentum of reaction flywheel:
the angular momentum saturation of the flywheel is reflected in that the rotational speed of the flywheel rotor reaches an upper limit, so the angular momentum saturation constraint can be expressed by the flywheel angular speed constraint:
-Nmax≤ΔNk+Nk-1≤Nmax
wherein N ismaxIs the flywheel maximum angular velocity vector.
After finishing, obtaining:
(c) and (3) restricting the view angle of the spacecraft attitude pointing:
considering the pointing direction of the spacecraft to avoid certain conical sight areas, the designed attitude is constrained in the following form:
α represents a unit vector of the spacecraft pointed in the body coordinate system, β represents a unit vector of the spacecraft in the direction needing to be avoided in the body coordinate system, and theta represents the size of a line-of-sight angle of an avoidance area.
Optimizing the objective function V (x) in step (3)k,uk) Expressed as:
wherein xk＝[qk,ωk]TRepresenting the state quantity of the system, wherein the physical meaning is the attitude and the angular speed of the spacecraft; using flywheel combined speed variation as input, i.e. uk＝ΔNk；NPA prediction range for MPC; q and P are weight matrixes of the state variables and the input variables, if Q is larger than P, the optimization target is more focused on stable time, and if Q is larger than P, the optimization target is more focused on energy consumption;
arranging into a concise form:
wherein
In the step (4), the control problem is converted into an extreme value problem of solving an objective function under the constraint condition, and the control quantity is solved by a real-time iteration method. Converting the problem of solving the control input into the following mathematical problem:
xi,k+1＝F(xi,k,ui,k)
L(xi,k,ui,k)≤017×1k＝0,1,…,NP-1
whereinRepresenting the optimization problem at the time i,indicating the state feedback of the system at the current time i, xi,kIndicating the state x from the current momenti,0The estimated system state at the k-th time; u. ofi,kIndicating the expected input at time k, 017×1Represents a zero matrix of 17 × 1; f (x)i,k,ui,k) The system state equation is specifically expressed as:
L(xi,k,ui,k) For system constraints, it is specifically expressed as:
processing using a method of real-time iterationThe process is as follows: knowing the optimization problem at the previous momentSolved forAndand then solve the optimization problemThe method is specifically divided into two stages: preparation phase and response phase:
the preparation stage comprises the following processes: x is to bei-1And ui-1Shifting a sampling moment, and keeping the last element unchanged to obtainAndaccording toAndcalculating the sensitivity matrix Ai,k,Bi,k,Ci,k,Di,kAnd error li,k,ri,kThe specific expression is as follows:
converting a nonlinear constrained optimization problem into a linear constrained quadratic programming problem through linearization:
Δui,k＝ui,k-ui-1,k
Δxi,k+1＝Ai,kΔxi,k+Bi,kΔui,k+ri,k
Ci,kΔxi,k+Di,kΔui,k+li,k≤0 k＝0,1,…,NP-1
the response phase process is as follows: obtaining state feedback of a systemCarry-over solving problemTo obtain Δ xi,kAnd Δ ui,kBy the formula:
get a problemSolution of (x)i,k,ui,k) Finally, u is again addedi＝ui,0As a control input to the system.
The model prediction control method for spacecraft attitude avoidance considering the dynamic characteristics of the flywheel can enable the spacecraft to perform attitude avoidance under the condition of limited performance of an executing mechanism in the attitude maneuver process, and can meet the comprehensive optimization of energy and time. The optimization method provided by the invention can improve the solution efficiency of the MPC, the speed of solving the optimization problem is accelerated by utilizing the hot start thought, the thought of real-time iteration divided into two stages can output aiming at the state feedback at the latest moment, and the timeliness of the system execution control is improved. In conclusion, the invention has the following advantages:
(1) the model of the spacecraft is integrated with the model of the actuating mechanism, the change quantity of the rotating speed of the flywheel is used as input, and the control method is combined with the traditional control method that the control torque is firstly obtained and then distributed to each flywheel, the control quantity directly acts on the actuating mechanism, the distribution process of the control quantity is omitted, and the whole system process is directly optimized.
(2) Compared with the existing potential function method for avoiding the spacecraft attitude, the model predictive control method can simultaneously consider the performance constraint of the executing mechanism and the attitude pointing constraint in the satellite attitude maneuver process, and can set a corresponding performance function according to the task requirement, so that the comprehensive optimization of energy and time is realized in the attitude maneuver process.
(3) Compared with general attitude avoidance optimal control, the method using model predictive control is closed-loop control, and can make a new trajectory optimization according to state feedback after each execution action is made, so that the control has robustness.
(4) Compared with the common nonlinear model predictive control method, the optimization thought of real-time iteration and hot start used by the model predictive control can reduce the operation amount and the optimization efficiency, can respond to the system state at the latest moment, and has more timeliness in the control output.
Drawings
FIG. 1 is a flow chart of a model predictive control method for spacecraft attitude avoidance considering flywheel dynamic characteristics;
FIG. 2 is a block diagram of a process for solving for control inputs by the control system;
Detailed Description
Reference will now be made in detail to the embodiments of the present invention, the following examples of which are intended to be illustrative only and are not to be construed as limiting the scope of the invention.
The invention provides a model prediction control method for spacecraft attitude evasion considering dynamic characteristics of a flywheel, which comprises the following specific steps as shown in figure 1: firstly, establishing a spacecraft attitude model containing an actuating mechanism as a prediction model of the MPC according to the attitude kinematics and the dynamic characteristics of the spacecraft and the rotation dynamic characteristics of the flywheel; and secondly, establishing a mathematical model of attitude avoidance constraint according to the line-of-sight angle and the avoidance vector of the spacecraft carrying instrument, and establishing a mathematical model of actuating mechanism constraint according to the angular momentum saturation and the moment saturation characteristics of the flywheel. And then designing a corresponding optimization objective function according to task requirements, wherein the optimization objective function comprises a quadratic form input by an execution mechanism and a quadratic form of system state quantity errors so as to comprehensively consider the problems of execution time and energy consumption. Finally, converting the control problem into a mathematical problem of solving an extreme value of the objective function under the constraint conditions of an equation and a state of a system dynamic equation and an inequality with limited input; and (4) rapidly solving by using a real-time iterative optimization processing method, and outputting the solution obtained by optimization as the control quantity of the system. A block diagram of the process by which the control system solves for control inputs is shown in fig. 2.
The specific implementation steps are as follows:
firstly, a mathematical model of the attitude control system is established. The kinetic equation of the flywheel is combined into the spacecraft attitude kinetic equation:
wherein ω is [ ω ═ ω [ [ ω ]1,ω2,ω3]TRepresenting the attitude angular velocity vector omega of the spacecraft relative to an inertial coordinate system under a body coordinate system1,ω2,ω3Angular velocity components of the spacecraft about the roll axis, yaw axis and pitch axis in the system, respectively;represents the derivative of ω with respect to time; j is the total inertia matrix of the spacecraft, simplified as the diagonal matrix J ═ diag (J)1,J2,J3)，J1＝30kg/m2,J2＝50kg/m2,J3＝40kg/m2Is the moment of inertia around the inertia main shaft; s (omega) is a skew symmetric matrix of the formτ represents the output torque of the actuator; q ═ q0,q1,q2,q3]TRepresents a unit quaternion of the attitude of the spacecraft,the scalar part of the quaternion representing the attitude unit of the spacecraft is related to the angle of rotation around the Euler axis, theta represents an angle of rotation around the Euler axis,ex,ey,ezrepresents the rotation axis in three directions of the Euler axis and satisfies Represents the derivative of q with respect to time; omega (omega) is a skew-symmetric matrix of the form
The model of the reaction flywheel combination is as follows:
wherein HrwIs the angular momentum of the flywheel combination,is HrwThe derivative with respect to time, the angular momentum of a flywheel combination consisting of four flywheels as a function of the rotational speed is as follows:
Hrw＝CJrwN
whereinMounting matrices for flywheels, N ═ N1,n2,n3,n4]TIs the angular velocity vector of the flywheel, n1,n2,n3,n4Respectively representing the angular velocity of each flywheel; j. the design is a squarerwA moment of inertia matrix, of the form J, representing a flywheel assemblyrw＝JαI4×4，Jα＝0.01608kg/m2Representing the moment of inertia of a single flywheel, I4×4Is an identity matrix of order 4.
Since the actuator controls attitude by exchanging angular momentum with the spacecraft, the total angular momentum of the system is conserved:
H＝Hrw+Jω
h is the total angular momentum of the system and is a constant when no external moment interference exists.
Integrating the model of the actuating mechanism and the model of the spacecraft attitude dynamics to obtain:
discretizing the model, and setting the sampling interval to be Δ t equal to 0.2s, wherein at the kth moment:
ωk+1＝J-1ΔtS(ωk)H+J-1JαCΔNk+ωk
where the index k denotes the value of the corresponding variable at the k-th instant, Δ Nk＝Nk-Nk-1，I3×3Is a 3 rd order identity matrix.
The initial attitude, the initial angular velocity and the initial angular velocity of the flywheel assembly of the spacecraft are qintial＝[-0.9524,-0.3048,0,0]Tωintial＝[0,0,0]TAnd Nintial＝[0,0,0,0]TThe desired attitude and angular velocity are controlled to be q respectivelyd＝[1,0,0,0]Tωd＝[0,0,0]T. So the total initial angular momentum H of the system is 0.
Secondly, establishing attitude avoidance constraint of the spacecraft in the attitude maneuver process and performance constraint of an actuating mechanism:
(1) constraint of maximum output torque of reaction flywheel:
the torque output of the reaction flywheel is achieved by changing the angular momentum of the flywheel, so it is expressed in the form:
wherein T ismax＝[1 1 1 1]TNm is the maximum output torque vector. Discretizing the formula to obtain:
after finishing, obtaining:
(2) constraint of maximum angular momentum of reaction flywheel:
the angular momentum saturation of the flywheel is reflected in that the rotational speed of the flywheel rotor reaches an upper limit, so the angular momentum saturation constraint can be expressed by the flywheel angular speed constraint:
-Nmax≤ΔNk+Nk-1≤Nmax
wherein N ismax＝[200π 200π 200π 200π]Trad/s is the maximum angular velocity vector.
After finishing, obtaining:
(3) and (3) restricting the view angle of the spacecraft attitude pointing:
considering the pointing direction of the spacecraft to avoid certain conical sight areas, the designed attitude is constrained in the following form:
wherein α ═ 001]TIndicating that the spacecraft points to a unit vector under the body coordinate system,a unit vector which represents the direction of the spacecraft to be avoided under a body coordinate system,indicating the magnitude of the line of sight angle of the avoidance area.
And the third step is to design an optimization objective function based on the model predictive controller:
wherein xk＝[qk,ωk]TRepresenting the state quantity of the system, wherein the physical meaning is the attitude and the angular speed of the spacecraft; using flywheel combined speed variation as input, i.e. uk＝ΔNk；NP5 is the prediction range of MPC; q ═ I7×7And P ═ I4×4Is a weight matrix of state variables and input variables,I7×7the 7 th order identity matrix indicates that the optimization goal is more focused on the stable time if Q is larger than P, and the optimization goal is more focused on the energy consumption if Q is larger than P. Arranging into a concise form:
wherein
Step four, converting the control problem into an extreme value problem of solving an objective function under a constraint condition, solving the control quantity by using a real-time iteration method, integrating the system model in the step one, the constraint model in the step two and the optimized objective function in the step three, and converting the problem of solving the control input into the following mathematical problem:
xi,k+1＝F(xi,k,ui,k)
L(xi,k,ui,k)≤017×1k＝0,1,…,NP-1
whereinRepresenting the optimization problem at the time i,indicating the state feedback of the system at the current time i, xi,kIndicating the state x from the current momenti,0Presume the system state at the kth moment; u. ofi,kRepresenting the predicted input at time k; f (x)i,k,ui,k) Specifically, the system state equation in the first step is expressed as:
L(xi,k,ui,k) The system constraint in the second step is specifically expressed as:
the process of using the real-time iterative method is as follows: knowing the previous optimization problemSolved forAndand then solve the optimization problemThe process of (2) is divided into two stages: preparation phase and response phase
The preparation stage comprises the following processes: x is to bei-1And ui-1Shifting a sampling moment, and keeping the last position unchanged to obtainAndaccording toAndcalculating the sensitivity matrix Ai,k,Bi,k,Ci,k,Di,kAnd error li,k,ri,kConcrete watchThe expression is as follows:
by linearization as above, this transforms a non-linearly constrained optimization problem into a linearly constrained quadratic programming problem:
Δui,k＝ui,k-ui-1,k
Δxi,k+1＝Ai,kΔxi,k+Bi,kΔui,k+ri,k
Ci,kΔxi,k+Di,kΔui,k+li,k≤0 k＝0,1,…,NP-1
the response phase process is as follows: obtaining state feedback of a systemCarry-over solving problemTo obtain Δ xi,kAnd Δ ui,kBy the formula:
get a problemSolution of (x)i,k,ui,k). Finally, u is again combinedi＝ui,0As a control input to the system.
By using the control method, the attitude avoidance of the spacecraft can be carried out under the condition that the performance of an actuating mechanism is limited in the attitude maneuver process, and meanwhile, the comprehensive optimization of energy and time can be met. The provided optimization method can improve the solution efficiency of the MPC, the speed of solving the optimization problem is accelerated by utilizing the hot start idea, the idea of real-time iteration divided into two stages can make control output aiming at the state feedback of the latest moment, the timeliness of the system execution control is improved, and the optimization flow is shown in figure 2.
Although exemplary embodiments of the present invention have been described for illustrative purposes, those skilled in the art will appreciate that various modifications, additions, substitutions and the like can be made in form and detail without departing from the scope and spirit of the invention as disclosed in the accompanying claims, all of which are intended to fall within the scope of the claims, and that various steps in the various sections and methods of the claimed product can be combined together in any combination. Therefore, the description of the embodiments disclosed in the present invention is not intended to limit the scope of the present invention, but to describe the present invention. Accordingly, the scope of the present invention is not limited by the above embodiments, but is defined by the claims or their equivalents.

Claims (1)

1. A model prediction control method for spacecraft attitude evasion considering dynamic characteristics of a flywheel is characterized by comprising the following steps:
(1) establishing a spacecraft attitude model containing an actuating mechanism as a prediction model of the MPC according to the attitude kinematics and the dynamic characteristics of the spacecraft and the rotation dynamic characteristics of the flywheel;
(2) establishing a mathematical model of attitude avoidance constraint according to a line-of-sight angle and an avoidance vector of a spacecraft carrying instrument, and establishing a mathematical model of actuating mechanism constraint according to angular momentum saturation and moment saturation characteristics of a flywheel;
(3) designing a corresponding optimization objective function according to task requirements, wherein the optimization objective function comprises a quadratic form input by an execution mechanism and a quadratic form of system state errors so as to comprehensively consider execution time and energy consumption;
(4) converting the control problem into an extreme value of an objective function under the constraint conditions of an equality and a state of a system dynamic equation and an inequality with limited input, rapidly solving by using a real-time iterative optimization processing method, and outputting a solution obtained by optimization as a control quantity of the system;
combining a kinetic equation of the flywheel into a spacecraft attitude kinetic equation, performing discretization processing, and establishing a prediction model of the MPC;
the spacecraft attitude dynamics model in the step (1) is expressed as follows:
wherein ω is [ ω ═ ω [ [ ω ]1,ω2,ω3]TRepresenting the attitude angular velocity vector omega of the spacecraft relative to an inertial coordinate system under a body coordinate system1,ω2,ω3Angular velocity components of the spacecraft about the roll axis, yaw axis and pitch axis in the system, respectively;represents the derivative of ω with time; j is the total inertia matrix of the spacecraft, simplified as the diagonal matrix J ═ diag (J)1,J2,J3)，J1,J2,J3Is the moment of inertia around the inertia main shaft; s (omega) is a skew symmetric matrix of the formτ represents the output torque of the actuator; q ═ q0,q1,q2,q3]TRepresents a unit quaternion of the attitude of the spacecraft,represents the scalar part of the quaternion of the attitude unit of the spacecraft, theta represents an angle of rotation about the euler axis,ex,ey,ezrepresents the rotation axis in three directions of the Euler axis and satisfies Represents the derivative of q with respect to time; Ω (ω) is a diagonally symmetric matrix of the form:
the model of the reaction flywheel combination is as follows:
wherein HrwIs the angular momentum of the flywheel combination,for its derivative with respect to time, the angular momentum is related to the rotational speed in a flywheel combination consisting of four flywheels as follows:
Hrw＝CJrwN
where C is a 3 × 4 flywheel mounting matrix and N ═ N1,n2,n3,n4]TIs the angular velocity vector of the flywheel, n1,n2,n3,n4Respectively representing the angular velocity of each flywheel; j. the design is a squarerwA moment of inertia matrix, of the form J, representing a flywheel assemblyrw＝JαI4×4，JαRepresenting the moment of inertia of a single flywheel, I4×4Is an identity matrix of 4 orders;
since the actuator controls attitude by exchanging angular momentum with the spacecraft, the total angular momentum of the system is conserved:
H＝Hrw+Jω
h is the total angular momentum of the system and is a constant when no external moment interference exists;
integrating the model of the actuating mechanism and the model of the spacecraft attitude dynamics to obtain:
discretizing the model, setting a sampling interval to be delta t, and obtaining the model at the kth moment by:
ωk+1＝J-1ΔtS(ωk)H+J-1JαCΔNk+ωk
where the index k denotes the value of the corresponding variable at the k-th instant, Δ Nk＝Nk-Nk-1，I3×3Is a third-order identity matrix;
in the step (2):
(1) constraint of maximum output torque of reaction flywheel:
the torque output of the reaction flywheel is achieved by changing the angular momentum of the flywheel, so it is expressed in the form:
wherein T ismaxIs the maximum output torque vector; discretizing the formula to obtain:
after finishing, obtaining:
(2) constraint of maximum angular momentum of reaction flywheel:
the angular momentum saturation of the flywheel is reflected in that the rotational speed of the flywheel rotor reaches an upper limit, so the angular momentum saturation constraint can be expressed by the flywheel angular speed constraint:
-Nmax≤ΔNk+Nk-1≤Nmax
wherein N ismaxIs the maximum angular velocity vector of the flywheel;
after finishing, obtaining:
(3) and (3) restricting the view angle of the spacecraft attitude pointing:
considering the pointing direction of the spacecraft to avoid certain conical sight areas, the designed attitude is constrained in the following form:
α represents a unit vector pointed by the spacecraft in a body coordinate system, β represents a unit vector of a direction to be avoided of the spacecraft in the body coordinate system, and theta represents the size of a line-of-sight angle of an avoidance area;
optimizing the objective function V (x) in the step (3)k,uk) Expressed as:
wherein xk＝[qk,ωk]TRepresenting the state quantity of the system, wherein the physical meaning is the attitude and the angular speed of the spacecraft; using flywheel combined speed variation as input, i.e. uk＝ΔNk；NPA prediction range for MPC; q and P are weight matrixes of the state variables and the input variables, if Q is larger than P, the optimization target is more focused on stable time, and if Q is larger than P, the optimization target is more focused on energy consumption;
arranging into a concise form:
wherein
Converting the control problem into an extreme value problem of solving an objective function under a constraint condition, and solving a control quantity by using a real-time iteration method;
converting the problem of solving the control input into the following mathematical problem:
xi,k+1＝F(xi,k,ui,k)
L(xi,k,ui,k)≤017×1k＝0,1,…,NP-1
whereinIs shown at the timeThe optimization problem of the moment i is solved,indicating the state feedback of the system at the current time i, xi,kIndicating the state x from the current momenti,0The estimated system state at the k-th time; u. ofi,kIndicating the expected input at time k, 017×1Represents a zero matrix of 17 × 1; f (x)i,k,ui,k) The system state equation is specifically expressed as:
L(xi,k,ui,k) For system constraints, it is specifically expressed as:
the process of using the real-time iterative method is as follows: knowing the optimization problem at the previous momentSolved forAndand then solve the optimization problemThe method is specifically divided into two stages: preparation phase and response phase:
the preparation stage comprises the following processes: x is to bei-1And ui-1Shifting and keeping the last element unchanged to obtainAndaccording toAndcalculating the sensitivity matrix Ai,k,Bi,k,Ci,k,Di,kAnd error li,k,ri,kThe specific expression is as follows:
converting a nonlinear constrained optimization problem into a linear constrained quadratic programming problem through linearization:
Δui,k＝ui,k-ui-1,k
Δxi,k+1＝Ai,kΔxi,k+Bi,kΔui,k+ri,k
Ci,kΔxi,k+Di,kΔui,k+li,k≤0k＝0,1,…,NP-1
the response phase process is as follows: of the acquisition systemState feedbackCarry-over solving problemTo obtain Δ xi,kAnd Δ ui,kBy the formula:
get a problemSolution of (x)i,k,ui,k) Finally, u is again addedi＝ui,0As a control input to the system.
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