CN117075635A - Multi-spacecraft control method based on trigger strategy - Google Patents
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Abstract
The invention discloses a multi-spacecraft control method based on a trigger strategy, which comprises the following steps: step 1, establishing a spacecraft kinematics and dynamics model by using a quaternion method; step 2, taking interference and uncertainty factors into consideration, and establishing a spacecraft tracking model on the basis of the step 1; step 3, considering the existing fixed time convergence sliding mode surface, and designing a sliding mode variable s by combining the quaternion spacecraft mathematical model established in the step 1; step 4, designing a dynamic event trigger mechanism; step 5, designing a fixed time convergence observer; and 6, designing a spacecraft formation fixed time stable control law and a self-adaptive law based on an event trigger mechanism on the basis of completion of the steps 1-5, and realizing tracking of the spacecraft formation on the expected gesture. The controller designed by the invention can reduce the communication times between spacecraft formations.
Description
Technical Field
The invention belongs to the field of nonlinear system control, relates to a spacecraft attitude control algorithm, and in particular relates to a multi-spacecraft control method based on a trigger strategy.
Background
In a multi-spacecraft formation system, the number of the spacecraft is large, and unified instructions are required to be obeyed, so that the requirement on a communication network is high. But individual spacecraft in a spacecraft formation are typically smaller in size for cost savings. Therefore, the power of the communication device that can be mounted is limited, and the communication capability is insufficient.
In spacecraft formation control, a controller using a trigger mechanism may be considered to reduce the number of communications in spacecraft formation. From the aspect of stable precision, the controller can be designed by utilizing a sliding mode method. The spacecraft formation tracking observer is controlled to observe the desired pose using a control algorithm that is stable in real fixed time. In addition, in consideration of factors such as interference and uncertainty, robustness needs to be improved by using methods such as self-adaption.
Disclosure of Invention
The invention aims to provide a multi-spacecraft control method based on a trigger strategy, which adopts a spacecraft dynamics and kinematics model represented by quaternions, considers external interference and uncertainty, uses an adaptive method to process, selects a sliding mode surface capable of converging at fixed time, designs a fixed time stable posture observer to observe expected postures, and further designs a fixed time stable algorithm to realize tracking of expected signals by each spacecraft in a spacecraft formation.
The invention aims at realizing the following technical scheme:
a multi-spacecraft control method based on a trigger strategy comprises the following steps:
step 1, establishing a spacecraft kinematics and dynamics model by using a quaternion method, wherein the specific form is as follows:
wherein J is i For the moment of inertia, ω, of the ith spacecraft i Is the rotation angular velocity signal of the ith spacecraft,is composed of omega i Derived square matrix, q i And q i0 Is the attitude signal of the ith spacecraft represented by the quaternion representation, u i Is the control signal of the ith spacecraft, w i Is the external disturbance moment of the ith spacecraft, I 3 Is a 3-order identity matrix;
step 2, taking interference and uncertainty factors into consideration, and establishing a spacecraft tracking model based on the step 1, wherein the specific form is as follows:
wherein J is i0 Is the nominal moment of inertia, omega, of the ith spacecraft ei Error value omega of ith spacecraft attitude information and expected attitude d Expected attitude value, q, for spacecraft formation ei And q 0,ei Is the error attitude signal of the ith spacecraft expressed by quaternion representation, R (q ei ) Is a rotation matrix;
step 3, designing a sliding mode variable s by considering the existing fixed time convergence sliding mode surface and combining the quaternion spacecraft mathematical model established in the step 1 i The concrete form is as follows:
s i ＝ω ei +sgn(q 0,ei (0))K 1 S aui
wherein l 1 、l 2 、η、p 1 、K 1 Is a fixed parameter;
step 4, designing a dynamic event triggering mechanism as follows:
wherein e i In order for the event to trigger an error, representing the kth triggering moment of the ith spacecraft, wherein eta is a constant positive variable;
step 5, designing a fixed time convergence observer as follows:
wherein,representing an estimate of the ith spacecraft for the desired information,/->γ 1 ,γ 2 Are all positive and odd numbers of times,and gamma is 1 <γ 2 ，a ij 、b i 、d i Is a spacecraft formation parameter, sig (x) is a special function, assuming x= [ x ] 1 x 2 x 3 ] T Then: sig (sig) α (x)＝[sgn(x 1 )·|x 1 | α sgn*(x 1 )·|x 1 | α sgn(x 1 )·|x 1 | α ] T Sgn is a sign function;
step 6, designing a spacecraft formation fixed time stable control law and a self-adaptive law based on an event trigger mechanism on the basis of completion of the steps 1-5, and tracking expected gestures by the spacecraft formation, wherein the spacecraft formation fixed time stable control law and the self-adaptive law based on the event trigger mechanism are as follows:
wherein,
Π＝αdiag{|ω ei | α-1 }，
s i is the sliding mode variable of the ith spacecraft, s j (t k ) Is t k The sliding mode information, k of the jth spacecraft obtained at moment 1 、k 2 、μ、λ、α、β、z 1 、z 2 Is constant, t=1+||ω ei ||+||ω ei || 2 。
Compared with the prior art, the invention has the following advantages:
1. the posture consistency of the multiple spacecrafts needs to be adjusted through the interaction among the spacecrafts, so that the state that the postures of the spacecrafts are consistent is achieved. This problem has been an important issue in spacecraft formation flight, and has attracted a great deal of attention. The invention designs the fixed time controller based on the event triggering mechanism, so that spacecraft formation can reach a state with consistent gestures.
2. In spacecraft formation, to save costs, the individual spacecraft is smaller in size and fewer communication elements can be loaded, which results in less communication burden that the spacecraft can bear. However, the designed spacecraft formation attitude consistency controller requires a large amount of communication among the spacecraft. The controller designed by the invention can reduce the communication times between spacecraft formations.
Drawings
FIG. 1 is a system block diagram;
FIG. 2 is a graph of the number of dynamic event triggers for spacecraft 1;
FIG. 3 is a graph of the number of dynamic event triggers for spacecraft 2;
FIG. 4 is a graph of the number of dynamic event triggers for spacecraft 3;
FIG. 5 is a graph of the number of dynamic event triggers for spacecraft 4;
fig. 6 is an observation error of the observer 1;
fig. 7 is an observation error of the observer 2;
fig. 8 is an observation error of the observer 3;
fig. 9 is an observation error of the observer 4;
fig. 10 is an angular velocity and attitude of the spacecraft 1;
FIG. 11 is an angular velocity and attitude of the spacecraft 2;
fig. 12 is an angular velocity and attitude of the spacecraft 3;
fig. 13 is an angular velocity and attitude of the spacecraft 4;
Detailed Description
The following description of the present invention is provided with reference to the accompanying drawings, but is not limited to the following description, and any modifications or equivalent substitutions of the present invention should be included in the scope of the present invention without departing from the spirit and scope of the present invention.
The invention provides a multi-spacecraft control method based on a trigger strategy, which adopts a quaternion modeling method of a spacecraft, considers the uncertainty of model parameters and external environment interference of the spacecraft, designs a gesture observer based on an event trigger mechanism and a multi-spacecraft formation gesture control algorithm, and realizes gesture tracking control of spacecraft formation. The method specifically comprises the following steps:
and step 1, establishing a spacecraft kinematics and dynamics model.
Considering that the quaternion representation method of the gesture has the characteristics of no singular, simple calculation and the like, the quaternion method is selected to be used for establishing a spacecraft kinematics and dynamics model, and the specific form is as follows:
wherein omega i Is the rotation angular velocity signal of the ith spacecraft, q i And q i0 Is the attitude signal of the ith spacecraft represented by the quaternion representation, u i Is the control signal of the ith spacecraft, J i Moment of inertia, w, of the ith spacecraft i Is the external disturbance moment of the ith spacecraft.
A quaternion is an extension of a complex number, and consists of a real part and an imaginary part, which is usually represented by three imaginary units i, j, k, i.e. the quaternion can be written in the form of a+ b i + c j + d k, where a, b, c, d are real numbers. q i And q i0 Is the vector and scalar portion of the unit quaternion, and q i And q i0 Satisfy the following requirementsI 3 Is a 3-order identity matrix。/>Is composed of omega i The derived square matrix satisfies: />Where omega i Is 3, written as:
step 2, taking interference and uncertainty factors into consideration, and establishing a spacecraft tracking model based on the step 1, wherein the specific form is as follows:
wherein J is i0 Is the nominal moment of inertia, omega, of the ith spacecraft ei ＝ω i -R(q ei )ω d ，ω ei Error value omega of ith spacecraft attitude information and expected attitude d The desired attitude values for the formation of the spacecraft,like omega × Is omega ei Derived square matrix, q ei And q 0，ei Is the error attitude signal of the ith spacecraft expressed by quaternion representation, R (q ei ) Is a rotation matrix defined as:
considering the influence of uncertainty factors of a spacecraft system, an inertia matrix J i From a nominal inertial matrix J i0 And uncertainty part DeltaJ i Composition, i.e. J i ＝J i0 +ΔJ i . Lumped interference can be written as:to facilitate subsequent analysis, reasonable assumptions are made about the spacecraft systems described above:
suppose 1: let the disturbance be bounded, i.e. ||w i ||≤w max ，w max > 0 and is a known constant.
Suppose 2: let it be assumed that the spacecraft rotates at an angular velocity omega i Bounded, i.e. |omega i ||≤δ 1 ，δ 1 > 0 and is a known constant.
Suppose 3: assuming rotational angular acceleration of spacecraftBounded, i.e.)>δ 2 > 0 and is a known constant.
Step 3, designing a sliding mode variable s by considering the existing fixed time convergence sliding mode surface and combining the quaternion spacecraft mathematical model established in the step 1 i The concrete form is as follows:
s i ＝ω ei +sgn(q 0,ei (0))K 1 S aui
wherein omega ei Is a quaternary representationAngular velocity error in the method, q ei Is the attitude error in the quaternion representation, l 1 、l 2 、η、p 1 、K 1 Is a fixed parameter. For a designed slip-form surface, it can be demonstrated that the slip-form surface eventually can converge in a fixed time.
Taking outWhen S is i When=0, there are:
ω ei ＝-sgn(q 0,ei )K 1 S aui
thus, it is possible to obtain:
q 0,ei (0)≥0
so that:
the system settling time is stable as seen by the following quotients:
and (5) lemma: if the system has Lyapunov equation, the equation is as follows:and alpha, m are positive real numbers, 0<m<1, the system fixed time converges, and the convergence time is T=1/(αm) +2/[ pi α (1-m)]。
Therefore, the designed slip-form surface can converge in a fixed time.
Step 4, designing a dynamic event trigger mechanism, wherein the communication loss in the formation of the spacecraft can be greatly reduced by using the trigger mechanism, and the dynamic event trigger mechanism is designed as follows:
wherein e i The error is taken as the event trigger error Represents the kth trigger moment of the ith spacecraft,/->May represent a series of trigger time points for the ith spacecraft, η being a constant positive variable. The purpose of such design is to enable the spacecraft error signal to update the signal when exceeding a certain interval, thereby avoiding the spacecraft to update the signal from time to time.
The gano phenomenon (Zeno behavir) refers to infinite triggering in a limited time in event triggering control, and for a designed event triggering condition, the triggering condition is continuously met, and a controller cannot effectively adjust triggering. This behavior is therefore to be avoided in the design of trigger conditions, and the following is to be established during the analysis proof process:
where δ is a positive constant. That is, any two trigger moments cannot be equal. This avoids multiple triggers at a certain time. The control method designed by the invention excludes the proof of the zeno phenomenon as follows:
||e i ||-k||s i ||≤η
k||s i ||-||e i ||≥-η
η≥0
furthermore:
at t k i And t must be present between b s So that
And because ofTherefore there is->
Step 5, considering that not all spacecrafts can acquire the desired information in the formation of the spacecrafts, the observer observation signals are designed. The following fixed time convergence observer was designed:
wherein,representing an estimate of the ith spacecraft for the desired information,/->γ 1 ,γ 2 Are all positive odd numbers, and gamma 1 <γ 2 ，a ij 、b i 、d i Is a spacecraft formation parameter, q i Forming desired poses, i.e. q, for spacecraft d ，/>Is a variable in the observer. sig (x) is a special function, assuming x= [ x ] 1 x 2 x 3 ] T Then: sig (sig) α (x)＝[sgn(x 1 )·|x 1 | α sgn(x 1 )·|x 1 | α sgn(x 1 )·|x 1 | α ] T Sgn is a sign function.
Let a be the communication capacity between spacecrafts ij Denoted by a ij =1 indicates that there is communication capability between the ith spacecraft and the jth spacecraft, if a ij =0, then it indicates that there is no communication capability between the ith spacecraft and the jth spacecraft. And communication between spacecrafts is undirected, i.e. a ij ＝a ji And (3) representing the communication capacity of spacecraft formation by using a matrix A, wherein A is a symmetric matrix. In addition, there areWith b i The ability of the ith spacecraft to obtain the desired attitude information is measured. B if the ith spacecraft can acquire the expected information i =1, otherwise b i ＝0。
To demonstrate observer stability, the following Lyapunov equation may be taken:
for V 0 And (3) deriving:
for gamma in the design of the invention, there is x γ sig γ (x)＝x 2γ . Taking outThe method can obtain:
thus, the designed observer fixed time converges.
And 6, designing a spacecraft formation fixed time stable control law and a self-adaptive law based on an event trigger mechanism on the basis of completion of the steps 1-5, and realizing tracking of the spacecraft formation on the expected gesture. The spacecraft formation fixed time stable control law and the self-adaptive law based on the event triggering mechanism are as follows:
for the convenience of calculation, pi and L are taken from the spacecraft model in the step 2 as follows:
Π＝αdiag{|ω ei | α-1 }，
s i is the sliding mode variable of the ith spacecraft, s j (t k ) Is t k And (5) obtaining sliding mode information of the jth spacecraft at the moment. k (k) 1 、k 2 、μ、λ、α、β、z 1 、z 2 Is a constant. T=1+||ω ei ||+||ω ei || 2 。
The following Lyapunov equation was chosen:
V＝V 1 +V 2
wherein:
for V 1 Derivative is obtained by:
the control law proposed above is brought in to obtain:
let a=k 1 sig m (s i )，b＝k 2 sig n (s i ) Then- (k) 1 sig m (s i )+k 2 sig n (s i )) d Can be recorded as- (a+b) d . Then:
so there are:
wherein, kappa 1 、κ 2 、κ 3 All are positive numbers, so:
so that it is possible to obtain:
as will be seen from the following quotation, the control law stabilizes the system for a virtually fixed time.
For nonlinear systemsf(0)＝0,x∈R n Selecting a proper Lyapunov equation V, and if V meets the following formula:
wherein α, β, p, q, k εR + Pk < 1, gk > 1,0 in the range of less than theta < ++ infinity, the system balance point x=0 is stable for a virtually fixed time. The convergence time satisfies:wherein (1)>
Assuming that the number of spacecraft in spacecraft formation is 4, part of parameters are as follows:
rotational inertia of spacecraft:
controller parameters:
α＝1.1,β＝diag(0.250.250.25),λ＝diag(2.492.492.49),γ＝0.5,z 1 ＝0.0001,z 2 ＝0.1,k 1 ＝20,k 2 ＝20
initial state of spacecraft:
q 1 ＝[0 -0.1 0.2] T ，q 10 ＝0.9747，ω 1 ＝[0.1 0.5 -0.1] T
q 2 ＝[0.1 0 0.1] T ,q 20 ＝0.9899，ω 2 ＝[0.7 0.5 -0.5] T
q 3 ＝[0 0.1 -0.1] T ，q 30 ＝0.9899，ω 3 ＝[0.6 -0.3 -0.8] T
q 4 ＝[-0.1 0 -0.2] T ，q 40 ＝0.9747，ω 4 ＝[0.2 0.6 0.1] T
time-varying interference received by the spacecraft:
d 1 ＝[0.005*sin(t) 0.008*cos(t) -0.002*sin(t)] T
d 2 ＝[0.002 0.004*sin(t) -0.002*cos(t)] T
d 3 ＝[0.005*sin(t) -0.008 -0.002*cos(t)] T
d 4 ＝[0 0 -0.005*sin(t)] T
the simulation results are shown in fig. 2-13, and compared with a control algorithm stable in a limited time, the algorithm can reach stability in a faster time and has better effect.
Claims (3)
1. The multi-spacecraft control method based on the trigger strategy is characterized by comprising the following steps of:
step 1, establishing a spacecraft kinematics and dynamics model by using a quaternion method, wherein the specific form is as follows:
wherein J is i For the moment of inertia, ω, of the ith spacecraft i Is the rotation angular velocity signal of the ith spacecraft,is composed of omega i Derived square matrix, q i And q i0 Is the attitude signal of the ith spacecraft represented by the quaternion representation, u i Is the control signal of the ith spacecraft, w i Is the external disturbance moment of the ith spacecraft, I 3 Is a 3-order identity matrix;
step 2, taking interference and uncertainty factors into consideration, and establishing a spacecraft tracking model based on the step 1, wherein the specific form is as follows:
wherein J is i0 Is the nominal moment of inertia, omega, of the ith spacecraft ei Error value omega of ith spacecraft attitude information and expected attitude d Expected attitude value, q, for spacecraft formation ei And q 0,ei Is the error attitude signal of the ith spacecraft expressed by quaternion representation, R (q ei ) Is a rotation matrix;
step 3, designing a sliding mode variable s by considering the existing fixed time convergence sliding mode surface and combining the quaternion spacecraft mathematical model established in the step 1 i The concrete form is as follows:
s i ＝ω ei +sgn(q 0,ei (0))K 1 S aui
wherein l 1 、l 2 、η、p 1 、K 1 Is a fixed parameter;
step 4, designing a dynamic event triggering mechanism as follows:
wherein e i In order for the event to trigger an error,representing the kth triggering moment of the ith spacecraft, wherein eta is a constant positive variable;
step 5, designing a fixed time convergence observer as follows:
wherein,representing an estimate of the ith spacecraft for the desired information,/->γ 1 ,γ 2 Are all positive odd numbers, and gamma 1 <γ 2 ，a ij 、b i 、d i Is a spacecraft formation parameter, sig (x) is a special function, assuming x= [ x ] 1 x 2 x 3 ] T Then: sig (sig) α (x)＝[sgn(x 1 )·|x 1 | α sgn(x 1 )·|x 1 | α sgn(x 1 )·|x 1 | α ] T Sgn is a sign function;
step 6, designing a spacecraft formation fixed time stable control law and a self-adaptive law based on an event trigger mechanism on the basis of completion of the steps 1-5, and tracking expected gestures by the spacecraft formation, wherein the spacecraft formation fixed time stable control law and the self-adaptive law based on the event trigger mechanism are as follows:
wherein,
Π＝αdiag{|ω ei | α-1 }，
s i is the sliding mode variable of the ith spacecraft, s j (t k ) Is t k The sliding mode information, k of the jth spacecraft obtained at moment 1 、k 2 、μ、λ、α、β、z 1 、z 2 Is constant, t=1+||ω ei ||+||ω ei || 2 。
2. The method for controlling a multi-spacecraft based on trigger strategy according to claim 1, characterized in that said R (q ei ) The definition is as follows:
3. the method for controlling a multi-spacecraft based on a trigger strategy according to claim 1, characterized in that said method comprises the steps of
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