CN106407619B - A kind of linear feedback Stabilization method of Control constraints Spacecraft Rendezvous control system - Google Patents
A kind of linear feedback Stabilization method of Control constraints Spacecraft Rendezvous control system Download PDFInfo
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Abstract
A kind of linear feedback Stabilization method of Control constraints Spacecraft Rendezvous control system, the present invention relates to the controller design methods of Spacecraft Rendezvous control system.Limited thrust can only be provided due to being mounted on the spaceborne thruster of pursuit, which must face Control constraints problem.Control platform can be reduced by ignoring the problem, result even in system unstability.In view of the shortcomings of the prior art, the invention proposes the Stabilization control laws based on Linear State Feedback, and give the optimal selection scheme of control law parameter, guarantee that closed-loop system has most fast convergence rate.Its advantage are as follows: firstly, the control law proposed be it is linear, facilitate design and implementation.Secondly, the global asymptotic stability of closed-loop system is guaranteed.Finally, giving optimal parameter to guarantee that closed-loop system has most fast convergence rate.The present invention is applied to Spacecraft Rendezvous control field.
Description
Technical field
The controller design method of Spacecraft Rendezvous control system of the present invention.
Background technique
Spacecraft Rendezvous docking is to realize spacecraft maintenance, intercept, enter a port, and more spacecrafts assemble on a large scale, multi satellite network
The basis of the related space mission such as networkization cooperation and spacefarer's rescue.The success of Spacecraft Rendezvous docking directly affects above-mentioned
The realization of task.There is important engineering significance for the controller design of Spacecraft Rendezvous system.
Limited thrust can only be provided due to being mounted on the carry-on thruster of pursuit, so that limited control can only be generated
Acceleration processed.If not considering this problem when controller design, thruster needs the acceleration generated to be possible to be greater than thrust
The peak acceleration that device itself is capable of providing.At this point, practical closed-loop system will not rule desirably work, this is not
But the Control platform that can reduce actual control system results even in the unstability of closed-loop system, causes catastrophic consequence.
The controller design problem of Spacecraft Rendezvous docking system under Control constraints situation has been obtained a large amount of domestic and international
The research of scientist achieves some good results.Using the control law of these existing design of control method it is general there are two types of
Form: first is that Nonlinear control law, the disadvantage is that the realization of control law is more complicated, it is difficult to debug；Second is that Linear Control is restrained,
The disadvantage is that can only guarantee the local stability of closed-loop system (partial linear control law can realize half Stabilization).By the end of mesh
Before, realize that the Linear Control rule of Stabilization there is no related ends to report.
Summary of the invention
The present invention is the design problem in order to solve the Spacecraft Rendezvous control system under Control constraints situation, and propose
A kind of linear feedback Stabilization method of Control constraints Spacecraft Rendezvous control system.
A kind of linear feedback Stabilization method of Control constraints Spacecraft Rendezvous control system is realized according to the following steps:
Step 1: introducing passive space vehicle orbital coordinate system o-xyz, and origin o is located at the mass center of passive space vehicle, x-axis
Along the direction that the direction of circular orbit radius, y-axis are flown along pursuit spacecraft, z-axis is directed toward outer orbit plane and x-axis and y-axis
Constitute right-handed coordinate system；The inearized model of Spacecraft Rendezvous system in x-y plane based on the description of C-W equation, as x and y
When thruster in axis direction all works, the linear Feedback Control rule for realizing Stabilization is established:
U=FX (1)
WhereinState vector, wherein x andThe position and speed of x-axis direction respectively, y andPoint
It is not the position and speed in y-axis direction；U=[u1,u2]TIt is input vector, wherein u1And u2It is thruster respectively in x and y-axis side
The normalization acceleration generated upwards；N is the orbit angular velocity of passive space vehicle；δ1And δ2It is thruster respectively in x and y-axis side
The upward peak acceleration that can be generated；Parameter k1,k2,k3,k4,k5Meet following condition:
Wherein
Step 2: to control law obtained in step 1 (1), the item for guaranteeing that closed-loop system has most rapid convergence rate is established
Part；
Step 3: the inearized model of the Spacecraft Rendezvous system on the z-axis direction based on the description of C-W equation is established real
The linear Feedback Control of existing Stabilization；
WhereinIt is state vector, z,It is the position and speed in z-axis direction respectively；u3It is thruster in z-axis
The normalization acceleration generated on direction；δ3It is the peak acceleration that thruster generates in z-axis direction；f1It is any nonnegative constant；
f2It is any normal number.
Invention effect:
The present invention is directed to the most significant excellent of the control method that the Spacecraft Rendezvous control system under Control constraints situation proposes
There are three aspects for point.Firstly, the control law proposed be it is linear, facilitate design and implementation.Secondly, the overall situation of closed-loop system is gradually
Nearly stability is guaranteed.Finally, giving optimal parameter to guarantee that closed-loop system has most fast convergence rate.By right
Typical Spacecraft Rendezvous control system is emulated, and compared with the conventional method compared with, the results showed that the control method proposed
It can be quickly completed intersection task, control performance is far superior to existing control method.In addition, in simulation study, due to set
The linear controller of meter directly acts on true Spacecraft Rendezvous nonlinear model, so simulation result also shows this hair
The control program of bright proposition has stronger robustness to linearized stability.
Detailed description of the invention
Fig. 1 proposes the position signal curve graph under control law effect in the present invention by Spacecraft Rendezvous system；
Fig. 2 proposes the speed signal curve graph under control law effect in the present invention by Spacecraft Rendezvous system；
Fig. 3 mentions the control signal curve under control law effect in the present invention by Spacecraft Rendezvous system；
Fig. 4 is position signal curve graph of the Spacecraft Rendezvous system under the effect of different control laws；
Fig. 5 is speed signal curve graph of the Spacecraft Rendezvous system under the effect of different control laws.
Specific embodiment
Specific embodiment 1: a kind of linear feedback Stabilization method packet of Control constraints Spacecraft Rendezvous control system
Include following steps:
Step 1: introducing passive space vehicle orbital coordinate system o-xyz, and origin o is located at the mass center of passive space vehicle, x-axis
Along the direction that the direction of circular orbit radius, y-axis are flown along pursuit spacecraft, z-axis is directed toward outer orbit plane and x-axis and y-axis
Constitute right-handed coordinate system；The inearized model of Spacecraft Rendezvous system in x-y plane based on the description of C-W equation, as x and y
When thruster in axis direction all works, the linear Feedback Control rule for realizing Stabilization is established:
U=FX (1)
WhereinState vector, wherein x andThe position and speed of x-axis direction respectively, y and
It is the position and speed in y-axis direction respectively；U=[u1,u2]TIt is normalization input vector, wherein u1And u2It is thruster respectively in x
With the normalization acceleration generated on y-axis direction,N is the orbit angular velocity of passive space vehicle；
δ1And δ2It is the peak acceleration that can generate of the thruster on x and y-axis direction respectively；Parameter k1,k2,k3,k4,k5Meet such as
Lower condition:
Wherein
Step 2: to control law obtained in step 1 (1), the item for guaranteeing that closed-loop system has most rapid convergence rate is established
Part:
Step 3: the inearized model of the Spacecraft Rendezvous system on the z-axis direction based on the description of C-W equation is established real
The linear Feedback Control of existing Stabilization；
WhereinIt is state vector, z,It is the position and speed in z-axis direction respectively；u3It is thruster in z-axis
The normalization acceleration generated on direction；δ3It is the peak acceleration that thruster generates in z-axis direction；f1It is any nonnegative constant；
f2It is any normal number.
Specific embodiment 2: the present embodiment is different from the first embodiment in that: it is established in the step 1 real
The detailed process of the linear Feedback Control rule of existing Stabilization are as follows:
Step 1 one: the foundation of system model；
Mathematical model using linear C-W equation as Spacecraft Rendezvous control system, concrete form are as follows:
Wherein ax,ay,azIt is mounted in what the spaceborne thruster of pursuit generated on x, tri- reference axis of y, z respectively
Acceleration,For saturation function；
σδ(): R → [- δ, δ] is the saturation function of standard, and concrete form is
To simplify the explanation, σ (x) is enabled to indicate σ hereinafter1(x)。
Spacecraft Rendezvous process description is state vectorFrom the initial value φ (t of non-zero0) turn
Change to state φ (tfThe process of)=0, t0Indicate intersection task start time, tfIt represents intersection task and completes the moment；By formula
(4) obtain control of the Spacecraft Rendezvous control system in x-y plane with control be in the z-axis direction it is full decoupled, because
This can separately be designed them.
For the Spacecraft Rendezvous control system in x-y plane, equation are as follows:
State vector X is chosen respectively and normalization input vector u is as follows:
Then formula (5) is written as follow the form of state space:
Wherein A, B are constant matrices, and form is as follows:
Step 1 two: the transformation of closed-loop system model；
For the closed-loop system that formula (1) and formula (6) are constituted, linear transformation χ=TX is introduced, wherein matrix T form is such as
Under:
Due to:
Therefore T is a nonsingular square matrix, then herein under transformation, intersection model (6) is rewritten are as follows:
Wherein:
Control law (1) is written as follow form:
Closed-loop system are as follows:
Step 1 three: the inspection of closed-loop system global stability；
Select positive semidefinite matrix P0:
Define following positive definite diagonal matrix:
Choose following Lyapunov function:
Wherein fiRepresent F0The i-th row, V (χ)=0 and if only if following three formulas simultaneously set up when:
χTP0χ=0, ρ1χTf1 Tf1χ=0, ρ1χTf2 Tf2χ=0 (16)
Due to k2> 0, so ρ1>0,ρ2>0；Observation matrix:
The Principal Minor Sequences at different levels of calculating matrix (17) are as follows:
From the above equation, we can see that P0+F0 TF0> 0, the χ for meeting (18) formula is only 0, so V (χ) is positive definite integral form, to be to close
Suitable Lyapunov function.
Lyapunov function V (x) along closed-loop system (12) time-derivative are as follows:
Inequality σT(u)T0(u- σ (u)) >=0, T in formula0For any positive semidefinite matrix, enable:
Then following formula is set up:
It calculates:
Due to k4Meet formula (2), matrix S above0It is positive definite, composite type (21) can be obtained by formula (19):
By LaSalle invariant set principle it is found that state vector can converge to set Ξ={ x:F0X=0 } in, in set Ξ
Closed-loop system (12) becomes:
By
Matrix is to (A0,F0) be to any μ >=0 it is observable, unique element is 0 in set Ξ, therefore, closed-loop system
It (12) is globally asymptotically stable.
Other steps and parameter are same as the specific embodiment one.
Specific embodiment 3: unlike one of present embodiment and specific embodiment one to two: the step 2
It is middle to establish the detailed process for guaranteeing that closed-loop system has the most condition of rapid convergence rate are as follows:
Since closed-loop system (1) and (6) (or (12)) is asymptotically stable in the large, system can work after finite time
Make to become linear system in linear zone.The convergence rate of system depends on the position of the set of poles λ (A+BF) of system at this time:
λ (A+BF)=λ (nA0+nB0F0)=n λ (A0+B0F0) (26)
Wherein λ (A0+B0F0) unrelated with angular speed n, make closed-loop system that there is most fast convergence rate, feedback oscillator F0It is
The optimal solution of following extreme-value problems:
λ(A0+B0F0) related to μ；Even if above-mentioned extreme-value problem is also one and is difficult the non-of solution in the case where μ is given
Linear non-convex optimization problem.Here its locally optimal solution is only provided.
X is identical as the lifting force device configuration of the direction y, then μ=1；In following section:
By linear search, a locally optimal solution of formula (29) is acquired are as follows:
(k1,k2,k3,k4,k5)=(0.61,2.97,2.17,5.7138,1.21) (29)
The characteristic value collection of the linearized system of closed-loop system are as follows:
nλ(A0+B0F0)=n { -1.5015 ± 0.3125i, -1.5079 ± 0.1372i } (30)
Other steps and parameter are identical as one of specific embodiment one to two.
Specific embodiment 4: unlike one of present embodiment and specific embodiment one to three: the step 3
It is middle to establish the detailed process for realizing the linear Feedback Control of Stabilization are as follows:
Step 3 one: Spacecraft Rendezvous system kinetics equation in the z-axis direction is established；
The Spacecraft Rendezvous control system (4) described by C-W equation in x-y plane internal dynamics equation and z-axis it is found that move
Mechanical equation is decoupling；Therefore, Spacecraft Rendezvous system kinetics equation in the z-axis direction is known by formula (4) are as follows:
Take the state vector Z and dominant vector u of system3Form it is as follows:
The then following System describe of formula (31):
The form of Φ, Ψ are as follows:
Step 3 two:
Introduce following state transformations:
Then formula (33) becomes following form:
Concrete form it is as follows:
The form of controller (3) becomes simultaneously:
u3=H0ζ,H0=[- f1 -f2] (38)
Closed-loop system is write as:
Step 3 three: the inspection of closed-loop system global stability；
Choose following positive definite matrix:
Q0Meet equationFollowing Lyapunov function is chosen again:
WhereinIt can be seen that W (ζ) is positive definite, it is suitable Lyapunov function；Enable Π0=1, then:
Lyapunov function W (ζ) along closed-loop system (39) derivative are as follows:
Since matrix is to (Φ0,H0) for arbitrary f1≥0,f2> 0 be it is observable, by LaSalle invariant set principle
Know that closed-loop system (39) are globally asymptotically stable.
Other steps and parameter are identical as one of specific embodiment one to three.
Embodiment one:
Assuming that the orbit altitude of passive space vehicle is 500km, orbit radius is R=6.8781 × 106m.If
δ1=δ2=8 × 10-3m/s2,δ3=6 × 10-3m/s2
That is μ=1.For the controller (3) on z-axis direction, f is taken1=f2=5, the set of poles of closed-loop system is λ at this time
(Φ0+Ψ0H0)={ -2, -3 }.For the controller (1) in x-y plane, parameter selection therein is (29)；
Directly against the primary nonlinear equation of Spacecraft Rendezvous control system
It is emulated, here
If primary condition is
The condition responsive curve and controlling curve of closed-loop system are separately recorded on Fig. 1, Fig. 2 and Fig. 3.
The intersection deadline that can be seen that closed-loop system from Fig. 1 and Fig. 2 is tf1=8000s ≈ 1.4Tp, T herepIt is
The orbital period of passive space vehicle；In addition, being also seen that under the action of the control law from simulation result, carves hold at the beginning
Row device is all saturation.This illustrates that closed-loop system shows the nonlinear characteristic of essence.In addition, for used in controller design
Nonlinear model (44) used in inearized model (4) and emulation is visibly different, so simulation result also illustrates the present invention
The control program proposed has stronger robustness to linearized stability.
Finally, by control law and document [B.Zhou, Q.Wang, Z.Lin and G.Duan, Gain proposed by the invention
scheduled control of linear systems subject to actuator saturation with
application to spacecraft rendezvous,IEEE Transactions on Control Systems
Technology, Vol.22, No.5, pp.2031-2038,2014] in propose linear gain scheduling controlling rule and document
[B.Zhou,N.G.Cui and G.Duan,Circular orbital rendezvous with actuator
saturation and delay:A parametric Lyapunov equation approach,IET Control
Theory&Applications, Vol.6, No.9, pp.1281-1287,2012] in propose linear half Stabilization control law
It compares, simulation result is as shown in Figure 4 and Figure 5.The control effect that can be seen that control law proposed by the present invention from this two figure is remote
It is much better than above-mentioned two kinds of control methods reported in the literature.
Claims (4)
1. a kind of linear feedback Stabilization method of Control constraints Spacecraft Rendezvous control system, which is characterized in that the control
Make the linear feedback Stabilization method of limited Spacecraft Rendezvous control system the following steps are included:
Step 1: introducing passive space vehicle orbital coordinate system o-xyz, and origin o is located at the mass center of passive space vehicle, x-axis along
The direction of circular orbit radius, the direction that y-axis is flown along pursuit spacecraft, z-axis is directed toward outer orbit plane and x-axis and y-axis is constituted
Right-handed coordinate system；The inearized model of Spacecraft Rendezvous system in x-y plane based on the description of C-W equation, as x and y-axis side
When upward thruster all works, the linear Feedback Control rule for realizing Stabilization is established:
U=FX (1)
WhereinState vector, wherein x andThe position and speed of x-axis direction respectively, y andIt is respectively
The position and speed in y-axis direction；U=[u1,u2]TIt is normalization input vector, wherein u1And u2It is thruster respectively in x and y-axis
The normalization acceleration generated on direction,N is the orbit angular velocity of passive space vehicle；δ1And δ2
It is the peak acceleration that can generate of the thruster on x and y-axis direction respectively；Parameter k1,k2,k3,k4,k5Meet following item
Part:
Wherein
Step 2: to control law obtained in step 1 (1), the condition for guaranteeing that closed-loop system has most rapid convergence rate is established；
Step 3: the inearized model of the Spacecraft Rendezvous system on the z-axis direction based on the description of C-W equation is established and is realized entirely
Office's quelling linear Feedback Control；
WhereinIt is state vector, z,It is the position and speed in z-axis direction respectively；u3Be thruster in the z-axis direction
The normalization acceleration of generation；δ3It is the peak acceleration that thruster generates in z-axis direction；f1It is any nonnegative constant；f2It is to appoint
Meaning normal number.
2. a kind of linear feedback Stabilization side of Control constraints Spacecraft Rendezvous control system according to claim 1
Method, which is characterized in that the detailed process for realizing the linear Feedback Control rule of Stabilization is established in the step 1 are as follows:
Step 1 one: the foundation of system model；
Mathematical model using linear C-W equation as Spacecraft Rendezvous control system, concrete form are as follows:
Wherein ax,ay,azIt is mounted in respectively and pursues the acceleration that spaceborne thruster generates on x, tri- reference axis of y, z
Degree,For saturation function；
Spacecraft Rendezvous process description is state vectorFrom the initial value φ (t of non-zero0) be transformed into
State φ (tfThe process of)=0, t0Indicate intersection task start time, tfIt represents intersection task and completes the moment；It is obtained by formula (4)
It is full decoupled to control of the Spacecraft Rendezvous control system in x-y plane and control in the z-axis direction；
For the Spacecraft Rendezvous control system in x-y plane, equation are as follows:
State vector X is chosen respectively and normalization input vector u is as follows:
Then formula (5) is written as follow the form of state space:
Wherein A, B are constant matrices, and form is as follows:
Step 1 two: the transformation of closed-loop system model；
For the closed-loop system that formula (1) and formula (6) are constituted, linear transformation χ=TX is introduced, wherein matrix T form is as follows:
Due to:
Therefore T is a nonsingular square matrix, then herein under transformation, intersection model (6) is rewritten are as follows:
Wherein:
Control law (1) is written as follow form:
Closed-loop system are as follows:
Step 1 three: the inspection of closed-loop system global stability；
Select positive semidefinite matrix P0:
Define following positive definite diagonal matrix:
Choose following Lyapunov function:
Wherein fiRepresent F0The i-th row, V (χ)=0 sets up simultaneously and if only if following three formulas:
Due to k2> 0, so ρ1>0,ρ2>0；Observation matrix:
The Principal Minor Sequences at different levels of calculating matrix (17) are as follows:
From the above equation, we can see thatThe χ for meeting (18) formula is only 0, so V (χ) is positive definite integral form；
Lyapunov function V (χ) along closed-loop system (12) time-derivative are as follows:
Inequality σT(u)T0(u- σ (u)) >=0, T in formula0For any positive semidefinite matrix, enable:
Then following formula is set up:
It calculates:
Due to k4Meet formula (2), matrix S above0It is positive definite, composite type (21) can be obtained by formula (19):
By LaSalle invariant set principle it is found that state vector can converge to set Ξ={ x:F0X=0 } in；The closed loop in set Ξ
System (12) becomes:
By
Matrix is to (A0,F0) be to any μ >=0 it is observable, unique element is 0 in set Ξ；Therefore, closed-loop system (12)
It is globally asymptotically stable.
3. a kind of linear feedback Stabilization side of Control constraints Spacecraft Rendezvous control system according to claim 2
Method, which is characterized in that the detailed process for guaranteeing that closed-loop system has the most condition of rapid convergence rate is established in the step 2 are as follows:
Since closed-loop system (1) and (6) is asymptotically stable in the large, the convergence rate of system depends on the set of poles of system at this time
The position of λ (A+BF):
λ (A+BF)=λ (nA0+nB0F0)=n λ (A0+B0F0) (26)
Wherein λ (A0+B0F0) unrelated with angular speed n, make closed-loop system that there is most fast convergence rate, feedback oscillator F0It is following poles
The optimal solution of value problem:
λ(A0+B0F0) related to μ；
X is identical as the lifting force device configuration of the direction y, then μ=1；In following section:
By linear search, the locally optimal solution of formula (29) is acquired are as follows:
(k1,k2,k3,k4,k5)=(0.61,2.97,2.17,5.7138,1.21) (29)
The characteristic value collection of the linearized system of closed-loop system are as follows:
nλ(A0+B0F0)=n { -1.5015 ± 0.3125i, -1.5079 ± 0.1372i } (30)
4. a kind of linear feedback Stabilization side of Control constraints Spacecraft Rendezvous control system according to claim 3
Method, which is characterized in that the detailed process for realizing the linear Feedback Control of Stabilization is established in the step 3 are as follows:
Step 3 one: Spacecraft Rendezvous system kinetics equation in the z-axis direction is established；
The Spacecraft Rendezvous control system (4) described by C-W equation is it is found that x-y plane internal dynamics equation and dynamics in z-axis
Equation is decoupling；Therefore, Spacecraft Rendezvous system kinetics equation in the z-axis direction is known by formula (4) are as follows:
Take the state vector Z and dominant vector u of system3Form it is as follows:
The then following System describe of formula (31):
The form of Φ, Ψ are as follows:
Step 3 two:
Introduce following state transformations:
Then formula (33) becomes following form:
Concrete form it is as follows:
The form of controller (3) becomes simultaneously:
u3=H0ζ,H0=[- f1 -f2] (38)
Closed-loop system is write as:
Step 3 three: the inspection of closed-loop system global stability；
Choose following positive definite matrix:
Q0Meet equationFollowing Lyapunov function is chosen again:
WhereinIt can be seen that W (ζ) is positive definite, it is suitable Lyapunov function；Enable Π0=1, then:
Lyapunov function W (ζ) along closed-loop system (39) derivative are as follows:
Since matrix is to (Φ0,H0) for arbitrary f1≥0,f2> 0 be it is observable, from LaSalle invariant set principle
Closed-loop system (39) is globally asymptotically stable.
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