CN112874818B - Finite time state feedback control method of spacecraft rendezvous system - Google Patents
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Abstract
The invention discloses a finite time state feedback control method of a spacecraft rendezvous system. The invention designs a finite time state feedback controller aiming at a spacecraft rendezvous system with saturated actuators based on low-gain feedback and event trigger control, saves the computing resources of the spacecraft rendezvous system and enables two spacecrafts to complete rendezvous tasks in finite time.
Description
Technical Field
The invention belongs to the technical field of space control, and designs a finite time state feedback control method based on event triggering aiming at a spacecraft rendezvous system. The spacecraft rendezvous system with the saturated actuator is controlled within limited time, so that the purpose that the spacecraft rendezvous in limited time is achieved.
Background
Spacecraft rendezvous is an important technology for current and future space flight missions. Therefore, it is very important to design an effective control method for a spacecraft rendezvous system so as to realize that two spacecrafts complete rendezvous tasks in a limited time.
The spacecraft orbit rendezvous is that a target spacecraft running on a space orbit meets another tracking spacecraft which tracks the target spacecraft, and the tracking spacecraft finally rendezvouss with the target spacecraft by adjusting the running orbit of the tracking spacecraft. In consideration of the real situation, since the acceleration generated by the thruster of the spacecraft is limited, if the actually designed controller does not consider the upper limit of the acceleration, the instability of the orbit intersection system can be caused, and finally the intersection failure of the spacecraft can be caused. Moreover, due to the development of the spacecraft technology, the realization that two spacecrafts complete the rendezvous mission in limited time also becomes an important index in spacecraft rendezvous, so that the limited time control of a spacecraft rendezvous system is particularly important.
With the development of modern networked control systems, in order to reduce sampling update of the system in data transmission and reduce the data transmission amount and data calculation amount of the system, an event trigger mechanism is introduced to achieve the purpose of saving system resources. Therefore, it is necessary to design an effective method for meeting two spacecrafts in a limited time and saving computing resources for a spacecraft meeting system with saturated actuators.
Disclosure of Invention
The invention provides a finite time state feedback control method based on event triggering to realize finite time control of a spacecraft rendezvous system.
The invention designs a finite time state feedback controller based on low gain feedback and event trigger control in consideration of the influence of the saturation of an actuator of a spacecraft rendezvous system. The invention establishes a relative motion equation of the spacecraft orbit intersection system with actuator saturation, and the designed controller realizes the effective control of the spacecraft intersection system.
The method comprises the following specific steps:
step 1, establishing a relative motion equation of a spacecraft rendezvous system
The target spacecraft is assumed to be running on a circular orbit with radius R and the relative distance between the two is R. And establishing a coordinate system o-xyz of the orbit of the target spacecraft, wherein a coordinate axis x is the direction of the radius R of the circular orbit, a coordinate axis y is the running direction of the tracking spacecraft, the coordinate axis z is perpendicular to the plane of the target spacecraft moving relative to the earth mass center, the direction and the coordinate axes x and y form a right-hand coordinate system, and the origin o is the mass center of the target spacecraft.
Let the gravity constant μ be GM, where M is the mass of the surrounded planet and G is the gravity constant. Can calculate the orbit angular velocity of the target spacecraft
The relative motion equation between the target spacecraft and the tracking spacecraft can be deduced according to the Newton's theory of motion:
wherein,
and x, y and z respectively represent the relative distance between the tracking spacecraft and the target spacecraft in the directions of the x axis, the y axis and the z axis. a is x ，a y ，a z Acceleration components in the directions of coordinate axes x, y, and z, respectively. Omega x ，ω y ，ω z The maximum accelerations in the directions of the three coordinate axes are respectively. sat (-) represents a unit saturation function.
Step 2, establishing a spacecraft rendezvous system state space model
Taylor expansion of γ at the origin and retention to the first order differential term yields a linearized differential equation:
by selecting a state vector
Obtaining a state space model
Wherein
M＝diag{ω x ,ω y ,ω z }，
Matrix A and matrix B are as follows:
step 3, designing time-varying parameters
The design time varying parameter ξ (t) is as follows,
wherein,
β 2 ＝6δ，
limited time
ξ 0 ＝ξ(t 0 )，t 0 Indicating the initial moment of the system.
θ c ＝θ c (ξ 0 ) ≧ 1 is a constant. And a scalar theta c (ξ 0 ) Can be obtained by the following formula
θ c (ξ 0 )＝6ξ 0 λ max (U(ξ 0 )W(ξ 0 ) -1 ),
Wherein W (ξ) 0 ) And U (xi) 0 ) Are unique symmetric positive solutions of the following Lyapunov equation, respectively
Step 4, designing event trigger conditions
The event triggering conditions are designed as follows:
wherein,
alpha is an event triggering parameter. 0 < delta < 1/6 is an alternative positive scalar quantity such that 0 < alpha < 1
The smaller δ, the higher the sampling frequency of the system. And time t e [ t ∈ [ [ t ] k ,t k+1 ) K ∈ N is the time of event trigger, and N represents a natural number set. Xi 0 ＝ξ(t 0 ),ξ Δ ＝ξ(T Δ )，t 0 Indicating the initial time of the system, T Δ Indicating the moment when the system is stable.
Is the variable of the error, and is,
represents the current state X (t) and the last sampling state X (t) of the system k ) The error between.
Step 5, designing a finite time state feedback controller
The finite time state feedback controller is designed as follows
U(t)＝-B T P(ξ(t))X(t k ),t∈[t k ,t k+1 ),
P(ξ(t))∈R 6×6 Is a solution of the following parametric Lyapunov equation
A T P(ξ(t))+P(ξ(t))A-P(ξ(t))BB T P(ξ(t))＝-ξ(t)P(ξ(t))
Step 6, designing an ellipsoid set
Defining two sets
ε(t)＝{X:6ξ(t)X T P(ξ(t))X≤1},
And
|' represents a 2-norm of the matrix or vector, and epsilon (t) is a set of ellipsoids. When X (t) belongs to the set
When the actuator is not saturated.
The calculation results in that,
‖B T P(ξ(t))X‖ 2 ＝X T P(ξ(t))BB T P(ξ(t))X≤6ξ(t)X T P(ξ(t))X,
that is, when X (t) k ) E ε (t), the actuator will not saturate, i.e.
sat(B T P(ξ(t))X(t k ))＝B T P(ξ(t))X(t k )
Step 7, establishing a closed loop system state space model
Substituting the designed finite time state feedback controller into a state space model of a spacecraft rendezvous system to obtain the following closed-loop system state space model
Considering for arbitrary
The actuator does not saturate. Further simplified to obtain the following closed-loop system state space model
Step 8, stability analysis of closed loop system
According to the Lyapunov stability theory, the following Lyapunov function is selected
V(X,t)＝6ξ(t)X T P(ξ(t))X
V (X, t) vs. time t e [ t ] k ,t k+1 ) Is a derivative of
Substituting the event trigger condition can further obtain
That is to say that the temperature of the molten steel,
substituting the designed time-varying parameter xi (t) into the above formula to obtain
This also indicates that the closed loop system is stable for a finite time T.
Aiming at a spacecraft rendezvous system with actuator saturation, the invention designs a finite time state feedback controller based on low-gain feedback and event triggering conditions, thereby avoiding the occurrence of actuator saturation, saving the computing resources of the system and realizing that two spacecrafts complete rendezvous tasks in finite time.
Detailed Description
Step 1, establishing a relative motion equation of a spacecraft rendezvous system
The target spacecraft is assumed to run on a circular orbit with radius R, and the relative distance between the two orbits is R. Establishing a coordinate system o-xyz of the orbit of the target spacecraft, wherein the x direction of a coordinate axis is the direction of the radius R of a circular orbit, the y direction of the coordinate axis is the direction of tracking the running of the spacecraft, the z direction of the coordinate axis is perpendicular to the plane of the target spacecraft moving relative to the earth mass center, the direction and the coordinate axes x and y form a right-hand coordinate system, and an origin o is the mass center of the target spacecraft.
Let the gravity constant μ be GM, where M is the mass of the surrounded planet and G is the gravity constant. The orbit angular velocity of the target spacecraft can be calculated
The relative motion equation between the target spacecraft and the tracking spacecraft can be deduced according to the Newton's theory of motion:
wherein,
x, y, z represent the relative distance of the tracking spacecraft from the target spacecraft in the x, y, z directions, respectively. a is a x ，a y ，a z Acceleration components in the directions of coordinate axes x, y, and z, respectively. Omega x ，ω y ，ω z The maximum accelerations in the directions of the three coordinate axes are respectively. sat (-) represents a unit saturation function.
Step 2, establishing a spacecraft rendezvous system state space model
Taylor expansion of γ at the origin and retention to the first order derivative term can be obtained
Linearized differential equation:
by selecting a state vector
Obtaining a state space model
Wherein
M＝diag{ω x ,ω y ,ω z }，
Matrix A and matrix B are as follows:
step 3, designing time-varying parameters
The design time varying parameter ξ (t) is as follows,
wherein,
β 2 ＝6δ，
limited time
ξ 0 ＝ξ(t 0 )，t 0 Indicating the initial moment of the system.
θ c ＝θ c (ξ 0 ) ≧ 1 is a constant. And a scalar theta c (ξ 0 ) Can be obtained by the following formula
θ c (ξ 0 )＝6ξ 0 λ max (U(ξ 0 )W(ξ 0 ) -1 ),
Wherein W (ξ) 0 ) And U (xi) 0 ) Are unique symmetric positive solutions of the following Lyapunov equation, respectively
Step 4, designing event trigger conditions
The event triggering conditions are designed as follows:
wherein,
alpha is an event triggering parameter. 0 < delta < 1/6 is an alternative positive scalar quantity such that 0 < alpha < 1
The smaller δ, the higher the sampling frequency of the system. And time t e [ t ∈ [ [ t ] k ,t k+1 ) And k ∈ N is the time of the event trigger. Xi 0 ＝ξ(t 0 ),ξ Δ ＝ξ(T Δ )，t 0 Indicating the initial time of the system, T Δ Indicating the moment when the system is stable.
Is the variable of the error, and is,
represents the current state X (t) and the last sampling state X (t) of the system k ) The error between.
Step 5, designing a finite time state feedback controller
A finite time state feedback controller is designed,
U(t)＝-B T P(ξ(t))X(t k ),t∈[t k ,t k+1 ),
P(ξ(t))∈R 6×6 is a solution of the following parametric Lyapunov equation
A T P(ξ(t))+P(ξ(t))A-P(ξ(t))BB T P(ξ(t))＝-ξ(t)P(ξ(t))
Step 6, designing an ellipsoid set
Defining two sets
ε(t)＝{X:6ξ(t)X T P(ξ(t))X≤1},
And
II denotes a 2-norm matrix or vectorAnd ε (t) is a set of ellipsoids. When X (t) belongs to the set
When this happens, the actuator is not saturated.
The calculation results show that the method has the advantages of high efficiency,
‖B T P(ξ(t))X‖ 2 ＝X T P(ξ(t))BB T P(ξ(t))X≤6ξ(t)X T P(ξ(t))X,
that is, when X (t) k ) E ε (t), the actuator will not saturate, i.e.
sat(B T P(ξ(t))X(t k ))＝B T P(ξ(t))X(t k )
Step 7, establishing a closed loop system state space model
Substituting the designed finite time state feedback controller into a state space model of a spacecraft rendezvous system to obtain the following closed-loop system state space model
Considering for arbitrary
The actuator does not saturate. Further simplified to obtain the following closed-loop system state space model
Step 8, stability analysis of closed loop system
According to the Lyapunov stability theory, the following Lyapunov function is selected
V(X,t)＝6ξ(t)X T P(ξ(t))X
V (X, t) versus time t ∈ [ t ] k ,t k+1 ) Is a derivative of
Substituting the event trigger condition can further obtain
substituting the designed time-varying parameter xi (t) into the above formula to obtain
This also indicates that the closed loop system is stable for a finite time T.
Claims (1)
1. A finite time state feedback control method of a spacecraft rendezvous system is characterized by comprising the following steps:
the method comprises the following steps: establishing a relative motion equation of a spacecraft rendezvous system
Assuming that the target spacecraft runs on a circular orbit with the radius of R, and the relative distance between the target spacecraft and the circular orbit is R; establishing a coordinate system o-xyz of the orbit of the target spacecraft, wherein the x direction of a coordinate axis is the direction of the radius R of a circular orbit, the y direction of the coordinate axis is the direction of tracking the running of the spacecraft, the z direction of the coordinate axis is perpendicular to the plane of the target spacecraft moving relative to the earth mass center, the direction and the coordinate axes x and y form a right-hand coordinate system, and an origin o is the mass center of the target spacecraft;
making the gravity constant mu equal to GM, wherein M is the mass of a surrounded planet, and G is a universal gravity constant; determining the orbital angular velocity of a target spacecraft
The relative motion equation between the target spacecraft and the tracking spacecraft can be deduced according to the Newton's theory of motion:
wherein,
x, y and z respectively represent the relative distance between the tracking spacecraft and the target spacecraft in the x, y and z directions; a is x ，a y ，a z Acceleration components in the directions of coordinate axes x, y and z are respectively; omega x ，ω y ，ω z The maximum acceleration in the directions of three coordinate axes are respectively; sat (·) represents a unit saturation function;
step two: establishing a spacecraft rendezvous system state space model
Taylor expansion of γ at the origin and retention to the first order differential term yields a linearized differential equation:
by selecting a state vector
Obtaining a state space model
Wherein
Matrix A and matrix B are as follows:
step three: time-varying parametric design
The design time-varying parameter ξ (t) is as follows,
wherein,
β 2 ＝6δ，
limited time
ξ 0 ＝ξ(t 0 )，t 0 Representing the initial moment of the system;
θ c ＝θ c (ξ 0 ) 1 or more is a constant; and a scalar theta c (ξ 0 ) Can be obtained by the following formula
θ c (ξ 0 )＝6ξ 0 λ max (U(ξ 0 )W(ξ 0 ) -1 ),
Wherein W (ξ) 0 ) And U (xi) 0 ) Are unique symmetric positive solutions of the following Lyapunov equation, respectively
Step four: event trigger condition design
The event triggering conditions are designed as follows:
wherein,
α is an event triggering parameter; 0 < delta < 1/6 is an alternative positive scalar such that 0 < alpha < 1
The smaller the delta, the higher the sampling frequency of the system; and time t e [ t ∈ [ [ t ] k ,t k+1 ) K belongs to N and is the time of event triggering, and N represents a natural number set; xi 0 ＝ξ(t 0 ),ξ Δ ＝ξ(T Δ )，t 0 Indicating the initial time of the system, T Δ Indicating the moment when the system is stable;
is the variable of the error, and is,
represents the current state X (t) and the last sampling state X (t) of the system k ) An error therebetween;
step five: finite time state feedback controller design
A finite-time state feedback controller for controlling the output of the inverter,
U(t)＝-B T P(ξ(t))X(t k ),t∈[t k ,t k+1 ),
P(ξ(t))∈R 6×6 is a solution of the following parametric Lyapunov equation
A T P(ξ(t))+P(ξ(t))A-P(ξ(t))BB T P(ξ(t))＝-ξ(t)P(ξ(t))
Step six: design ellipsoid set
Defining two sets
ε(t)＝{X:6ξ(t)X T P(ξ(t))X≤1},
And
|' represents a 2-norm of the matrix or vector, epsilon (t) is a set of ellipsoids; when X (t) belongs to the set
When the actuator is not saturated;
the calculation results in that,
‖B T P(ξ(t))X‖ 2 ＝X T P(ξ(t))BB T P(ξ(t))X≤6ξ(t)X T P(ξ(t))X,
that is, when X (t) k ) E ε (t), the actuator will not saturate, i.e.
sat(B T P(ξ(t))X(t k ))＝B T P(ξ(t))X(t k )
Step seven: establishing a closed-loop system state space model
Substituting the designed finite time state feedback controller into a state space model of a spacecraft rendezvous system to obtain the following closed-loop system state space model
Considering for arbitrary
The actuator is not saturated; further simplified to obtain the following closed-loop system state space model
Step eight: stability analysis of closed loop systems
According to the Lyapunov stability theory, the following Lyapunov function is selected
V(X,t)＝6ξ(t)X T P(ξ(t))X
V (X, t) versus time t ∈ [ t ] k ,t k+1 ) Is a derivative of
Substituting the event trigger condition can further obtain
substituting the designed time-varying parameter xi (t) into the above formula to obtain
This also indicates that the closed loop system is stable for a finite time T.
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