CN104460678A - Spacecraft posture control method based on Sigmoid nonlinear sliding mode function - Google Patents

Abstract

The invention discloses a spacecraft posture control method based on a Sigmoid nonlinear sliding mode function. The method comprises the steps that a relative posture motion equation when a spacecraft is subjected to large-angle posture maneuvering is expressed to be in a cascading mode, then a typical Sigmoid function is introduced into a sliding mode function, and the Sigmoid nonlinear sliding mode function and a sliding mode posture control law are determined, so that the system state achieves expected equivalent system dynamics in the sliding mode section. The sliding mode posture control law is amended, the buffeting of control torque is restrained, and conservatism for switching gain selections is reduced. By the utilization of the method, the problem of weighing of sliding mode function gain selections of a sliding mode posture control law based on a linear sliding mode function in the prior art is effectively solved, and the control performance of the sliding mode posture control law is improved. In addition, the problem of saturation of a sensor is effectively avoided by utilization of boundedness of the Sigmoid function, and high-performance posture control over the spacecraft is achieved when the relative posture angular speed is restricted.

Description

A kind of Spacecraft Attitude Control method based on Sigmoid type nonlinear sliding mode function

Technical field

The present invention relates to a kind of Spacecraft Attitude Control method, particularly a kind of Spacecraft Attitude Control method based on Sigmoid type nonlinear sliding mode function.

Background technology

For rigid body spacecraft, carry out each interchannel coupling in Large Angle Attitude Maneuver process serious, present strong non-linear dynamic characteristic.In addition, the existence of various parameter uncertainty and external disturbance causes its gesture stability to become complex.The key issue that the design of rigid body spacecraft attitude control system will solve suppresses the impact of parameter uncertainty and external disturbance.

At present, the method for the design of rigid body spacecraft attitude control system has had many.Wherein, sliding formwork controls is the robust nonlinear control method be most widely used.Sliding formwork control is a branch of variable-structure control.Variable structure control method deliberately changes the structure of system according to the current state of system.If system state can constrain on certain Manifold of state space by the change of this structure, then variable-structure control is now claimed to be that sliding formwork controls.Correspondingly, claim the stream shape on state space to be sliding-mode surface or sliding mode, the motion of system state on this stream shape is sliding formwork motion.The maximum feature that sliding formwork controls is that the closed-loop system under its effect has insensitivity for the interference of mating and uncertainty, and this feature makes sliding formwork control just to be widely applied to the every field comprising gesture stability at the beginning of generation.

Follow the tracks of motor-driven control task to complete spacecraft attitude, existing Sliding Mode Attitude control law still uses such as eigenwert or Characteristic Structure Configuration in the design of sliding formwork function, quadratic form minimizes and the linear design method such as LMI.Wherein, for the attitude stabilization problem of spacecraft, Vadali [Vadali S.Variable-structure control of spacecraft large-angle maneuvers [J] .Journal of Guidance, Control, and Dynamics, 1986,9 (2): 235-239.] using hypercomplex number as attitude, sign devises Sliding Mode Attitude control law, and utilizes optimal control technique to have studied the synthtic price index of sliding formwork function.By minimizing the hypercomplex number quadratic performance relevant with attitude angular velocity, the sliding formwork function that Vadali obtains is the linear function of attitude angular velocity and hypercomplex number.In follow-up study, Yeh [Yeh F.Sliding-mode adaptive attitude controller design for spacecrafts with thrusters [J] .IET Control Theory Applications, 2010, 4 (7): 1254-1264.], Jorgensen [Jorgensen U, Gravdahl J.Observer based sliding mode attitude control:theoretical and experimental results [J] .Modeling, Identification and Control, 2010, 31 (1): 1-9.] and Zhu [Zhu Z, Xia Y, Fu M.Adaptive sliding mode control for attitude stabilization with actuator saturation [J] .IEEE Transactions on Industrial Electronics, 2011, 58 (10): 4898-4907.], have studied attitude based on linear sliding mode function to be redirected and Attitude tracking control problem.

But for existing linear sliding mode function, in order to accelerate the response speed of relative attitude variable in sliding formwork section, Sliding Mode Attitude control law needs to increase sliding formwork function gain.Because sliding formwork function gain directly affects control moment corresponding to Sliding Mode Attitude control law, therefore when relative attitude variable is larger, carrying out Linear Amplifer to it can may cause control moment amplitude to exceed the saturation limit of actuator.In addition, for given relative attitude initial guess, increase the initial value that sliding formwork function gain can increase sliding formwork function equally, then increase the distance that relative attitude variable arrives sliding-mode surface.Otherwise too small sliding formwork function gain can slow down again the response speed of system.Visible, for existing based on linear sliding mode function design Sliding Mode Attitude control law for, the selection of sliding formwork function gain also exists trade-off problem.Due to the existence of Noncontinuous control item, there is buffeting problem in the attitude control law based on the design of sliding formwork control technology.The high frequency of this control signal switches the easy activating system Unmarried pregnancy of phenomenon, produces less desirable system responses.In addition, the handoff gain that sliding formwork controls on the one hand determines buffeting amplitude, also determine the robustness of attitude control system for parameter uncertainty and external disturbance on the one hand.Typically, in order to ensure the robustness of system, the selection of handoff gain generally adopting conservative approach, namely selecting an abundant large handoff gain value.Then exacerbate the buffeting problem of control moment, also can cause extra control consumption.Finally, the existing attitude control law based on the design of linear sliding mode function does not all consider sensor saturation problem, cannot complete gesture stability task when relative angle limited speed.

Summary of the invention

The object of the invention is to provide a kind of Spacecraft Attitude Control method based on Sigmoid type nonlinear sliding mode function, solves that sliding formwork function gain that the existing single order Sliding Mode Attitude control law based on linear sliding mode function exists selects that trade-off problem, control moment are buffeted, handoff gain conservative property and sensor saturation problem.

Based on a Spacecraft Attitude Control method for Sigmoid type nonlinear sliding mode function, its concrete steps are:

The first step builds the spacecraft attitude control system based on Sigmoid type nonlinear sliding mode function

Based on the spacecraft attitude control system of Sigmoid type nonlinear sliding mode function, comprising: cascade form relative attitude equation of motion module, Sigmoid type nonlinear sliding mode function module, Sliding Mode Attitude control law module and Sliding Mode Attitude control law correcting module.The function of cascade form relative attitude equation of motion module is: the attitude motion rule describing rigid spacecraft, the function of Sigmoid type nonlinear sliding mode function module is: the Sigmoid type nonlinear correspondence relation setting up relative attitude parameter and relative attitude angular velocity, the function of Sliding Mode Attitude control law module for: ensure that the equivalent system that spacecraft attitude control system has corresponding to Sigmoid type nonlinear sliding mode function is dynamic, the function of Sliding Mode Attitude control law correcting module for: eliminate buffeting existing for Sliding Mode Attitude control law and handoff gain selects conservative property.

Second step cascade form relative attitude equation of motion module sets up the cascade form relative attitude equation of motion

To carry out the rigid spacecraft of attitude tracking control for object, cascade form relative attitude equation of motion module to revise Douglas Rodríguez parameter as attitude characterization parameter definition relative attitude variable, sets up the cascade form relative attitude equation of motion in the configuration space of attitude motion under spacecraft body coordinate system.Wherein, relative attitude kinetics equation is:

Relative attitude kinematical equation is:

${\stackrel{.}{\mathrm{\σ}}}_e=M{\mathrm{\ω}}_e---\left(2\right)$

In formula, for the matrix representation of nominal value under body coordinate system of spacecraft inertia battle array tensor, ω _erepresent the vector representation of relative attitude angular velocity vector under body coordinate system between spacecraft body coordinate system and reference frame, T _cfor the vector representation of control moment vector under body coordinate system, T _dfor the vector representation of disturbance torque under body coordinate system that external disturbance moment and systematic parameter uncertainty produce attitude motion of spacecraft, () ^×represent the antisymmetric matrix operator of vector, R represents the transition matrix between spacecraft body coordinate system and reference coordinate, ω _drepresent the vector representation of reference angular velocities vector under reference frame.σ _erepresent the vector representation of correction Douglas Rodríguez parameter vector under body coordinate system that spacecraft body coordinate system is corresponding with the relative attitude between reference frame, M is Jacobi matrix, and above parameter, band point represents the derivative of parameter.

3rd step Sigmoid type nonlinear sliding mode function module determines Sigmoid type nonlinear sliding mode function

For the relative attitude kinematical equation set up, Sigmoid type nonlinear sliding mode function module will revise Douglas Rodríguez parameter relatively as Sigmoid argument of function, dependent variable using relative attitude angular velocity as Sigmoid function, determines a quasi-nonlinear sliding formwork function.For typical Sigmoid function f (x)=arctan (x), Sigmoid type nonlinear sliding mode function is defined as:

s＝ω _e+karctan(Aσ _e) (3)

In formula, k > 0, A=diag (a ₁, a ₂, a ₃) and a _i> 0 (i=1,2,3).In addition, parameter a _iselection also meet work as | σ _ei| when → 0, there is following formula to set up

arctan(a _i|σ _ei|)＞|σ _ei|

In formula, σ _ei(i=1,2,3) are relatively revise the vector representation of Douglas Rodríguez parameter under body coordinate system.

4th step Sliding Mode Attitude control law module determines the attitude control law based on Sigmoid type nonlinear sliding mode function

Based on Sigmoid type nonlinear sliding mode function, Sliding Mode Attitude control law module adds the attitude control law switching control theory and determine as formula (4) according to equivalent control.

In formula, T _eqrepresent equivalent control term, T _swrepresent and switch control item, and || || ₂for 2 norms of vector, η > || T _d|| _∞+ δ and || || _∞for the Infinite Norm of vector, δ > 0 is arbitrarily small constant.

Lyapunov function shown in selecting type (5)

Have along the differentiate of closed loop track Lyapunov function (5):

For Lyapunov function (5), following relation is had to set up:

In formula, for induction 2 norm of nominal inertia battle array.

Formula (6) is substituted in Lyapunov function derivative, can obtain:

According to Lyapunov finite time stability principle, for arbitrarily t ₀represent initial time, represent space, sliding formwork function s is at finite time t _rinside converge to zero.Due to t ∈ [t _r,+∞) and there is s ≡ 0, select Lyapunov function further:

$V=\frac{1}{2}{\mathrm{\σ}}_e^T{\mathrm{\σ}}_e---\left(8\right)$

To the track differentiate that it is determined along s ≡ 0, have:

$\begin{array}{c}\stackrel{.}{V}={\mathrm{\σ}}_e^T{\mathrm{M\ω}}_e\\ =-\frac{k(1+{\mathrm{\σ}}_e^T{\mathrm{\σ}}_e)}{4}{\mathrm{\σ}}_e^T\arctan \left({\mathrm{A\σ}}_e\right)\end{array}---\left(9\right)$

Due to arctan (a _iσ _ei) and σ _eijack per line, above-mentioned Lyapunov functional derivative negative definite.Global consistent asymptotic stability according to the known closed-loop system of Lyapunov stability principle.According to equivalent control measurements, known closed-loop system in the equivalent system of sliding formwork section is dynamically:

${\stackrel{.}{\mathrm{\σ}}}_e=-\mathrm{kM}\arctan \left({\mathrm{A\σ}}_e\right)---\left(10\right)$

5th step Sliding Mode Attitude control law correcting module correction attitude control law

Sliding Mode Attitude control law correcting module utilizes boundary layer strategy and ADAPTIVE CONTROL to be modified to attitude control law:

In formula, T _eqcotype (4), T _sarepresent revised switching control item, and and

$\begin{array}{c}{\hat{c}}_1={\mathrm{\κ}}_1{\left|\right|s\left|\right|}_2-{\mathrm{\κ}}_1{\mathrm{\ψ}}_1{\hat{c}}_1\\ {\hat{c}}_2={\mathrm{\κ}}_2{\left|\right|s\left|\right|}_2{\left|\right|{\mathrm{\σ}}_e\left|\right|}_{\∞}-{\mathrm{\κ}}_2{\mathrm{\ψ}}_2{\hat{c}}_2\\ {\hat{c}}_3={\mathrm{\κ}}_3{\left|\right|s\left|\right|}_2{\left|\right|{\mathrm{\ω}}_e\left|\right|}_{\∞}-{\mathrm{\κ}}_3{\mathrm{\ψ}}_3{\hat{c}}_3\end{array}---\left(12\right)$

In formula, κ _i> 0 (i=1,2,3), ψ _i> 0, and

So far, the Spacecraft Attitude Control based on Sigmoid type nonlinear sliding mode function is completed.

The sliding formwork function gain that this method can solve the existence of existing linear sliding mode function effectively based on the nonlinear sliding mode function that Sigmoid function designs selects trade-off problem, corresponding attitude control law can quickening system response time while inhibitory control moment amplitude, improve the dynamic property of attitude control system.In the present invention, the boundedness of Sigmoid function makes spacecraft relative attitude angular velocity when carrying out attitude tracking control can not exceed the measurement range of sensor, efficiently avoid sensor saturation problem.In addition, Boundary Layer Method combines with adaptive control by the present invention, effectively can weaken control moment and buffet and the conservative property reducing handoff gain selection, be convenient to Project Realization.

Accompanying drawing explanation

The typical Sigmoid function curve of a kind of Spacecraft Attitude Control method based on Sigmoid type nonlinear sliding mode function of Fig. 1;

The relative correction Douglas Rodríguez parameter response curve of a kind of Spacecraft Attitude Control method based on Sigmoid type nonlinear sliding mode function of Fig. 2;

The relative attitude angular velocity response curve of a kind of Spacecraft Attitude Control method based on Sigmoid type nonlinear sliding mode function of Fig. 3;

The control moment curve of a kind of Spacecraft Attitude Control method based on Sigmoid type nonlinear sliding mode function of Fig. 4 and partial enlarged drawing;

The handoff gain self-adaptation curve of a kind of Spacecraft Attitude Control method based on Sigmoid type nonlinear sliding mode function of Fig. 5 and partial enlarged drawing.

Embodiment

Based on a Spacecraft Attitude Control method for Sigmoid type nonlinear sliding mode function, its concrete steps are:

The first step builds the spacecraft attitude control system based on Sigmoid type nonlinear sliding mode function

Based on the spacecraft attitude control system of Sigmoid type nonlinear sliding mode function, comprising: cascade form relative attitude equation of motion module, Sigmoid type nonlinear sliding mode function module, Sliding Mode Attitude control law module and Sliding Mode Attitude control law correcting module four parts.The function of cascade form relative attitude equation of motion module is the attitude motion rule describing rigid spacecraft, the function of Sigmoid type nonlinear sliding mode function module is set up the Sigmoid type nonlinear correspondence relation of relative attitude parameter and relative attitude angular velocity, the function of Sliding Mode Attitude control law module for the equivalent system ensureing spacecraft attitude control system and have corresponding to Sigmoid type nonlinear sliding mode function dynamic, the function of Sliding Mode Attitude control law correcting module is for eliminating buffeting existing for Sliding Mode Attitude control law and handoff gain selects Conservative Property.

Second step cascade form relative attitude equation of motion module sets up the cascade form relative attitude equation of motion

First, as follows at spacecraft body coordinate system relative attitude variable of giving a definition:

$\begin{array}{c}{\mathrm{\σ}}_e={\mathrm{\σ}}_b\⊕(-{\mathrm{\σ}}_d)\\ =\frac{(1-{\left|\right|{\mathrm{\σ}}_d\left|\right|}^2){\mathrm{\σ}}_b-(1-{\left|\right|{\mathrm{\σ}}_b\left|\right|}^2){\mathrm{\σ}}_d+{2\mathrm{\σ}}_b^{\×}{\mathrm{\σ}}_d}{1+{\left|\right|{\mathrm{\σ}}_d\left|\right|}^2{\left|\right|{\mathrm{\σ}}_b\left|\right|}^2+{2\mathrm{\σ}}_d^T{\mathrm{\σ}}_b}\end{array}$

ω _e＝ω _b-Rω _d

In formula, σ _bfor the vector representation of correction Douglas Rodríguez parameter vector under body coordinate system that spacecraft body coordinate system attitude is corresponding, σ _dfor the vector representation of correction Douglas Rodríguez parameter vector under reference frame that reference frame attitude is corresponding, σ _erepresent the vector representation of correction Douglas Rodríguez parameter vector under body coordinate system that spacecraft body coordinate system is corresponding with the relative attitude between reference frame, ω _brepresent the vector representation of spacecraft angular velocity vector under body coordinate system, ω _drepresent the vector representation of reference angular velocities vector under reference frame, ω _erepresent the vector representation of relative attitude angular velocity vector under body coordinate system between spacecraft body coordinate system and reference frame; || || represent the Euclidean norm of vector, () ^×represent the antisymmetric matrix operator of vector, () ^trepresent vector or transpose of a matrix operator, represent the multiplication operator revising Douglas Rodríguez parameter; Transition matrix R between spacecraft body coordinate system and reference coordinate is:

$R=I_3+\frac{{8\mathrm{\σ}}^{\×}-4(1-{\left|\right|{\mathrm{\σ}}_e\left|\right|}^2){\mathrm{\σ}}_e^{\×}}{{(1+{\left|\right|{\mathrm{\σ}}_e\left|\right|}^2)}^2}$

In formula, I ₃represent the unit matrix of 3 × 3.

According to Eulerian angle momentum principle, relative attitude kinetics equation is:

In formula, for the matrix representation of nominal value under body coordinate system of spacecraft inertia battle array tensor, T _cfor the vector representation of control moment vector under body coordinate system, T _dfor the vector representation of disturbance torque under body coordinate system that external disturbance moment and systematic parameter uncertainty produce attitude motion of spacecraft.

On the basis of relative attitude variable-definition, relative attitude kinematical equation is:

${\stackrel{.}{\mathrm{\σ}}}_e={\mathrm{M\ω}}_e---\left(2\right)$

In formula, Jacobi matrix M is:

$M=\frac{1}{4}[(1-{\left|\right|{\mathrm{\σ}}_e\left|\right|}^2)I_3+{2\mathrm{\σ}}_e^{\×}+{2\mathrm{\σ}}_e{\mathrm{\σ}}_e^T]$

3rd step Sigmoid type nonlinear sliding mode function module determines Sigmoid type nonlinear sliding mode function

Sigmoid function is a S type function, and Fig. 1 gives the curve of part typical case's Sigmoid function when independent variable x ∈ [-3,3].Therefrom can find out, when | during x| →+∞, the Sigmoid function described in figure is all tending towards ultimate value, the saturability that this feature has for Sigmoid function and boundedness; When | during x| → 0, Sigmoid function derivative will increase gradually, the variable slope characteristic that this feature has for Sigmoid function.Mentality of designing based on Sigmoid type nonlinear sliding mode function can be described as: with σ _eas Sigmoid argument of function, with ω _eas the dependent variable of Sigmoid function, the restriction relation that Sigmoid function is determined is converted into the restriction relation of sliding formwork function.Without loss of generality, the present invention, for one typical Sigmoid function f (x)=arctan (x), provides the Spacecraft Attitude Control rule design based on Sigmoid type nonlinear sliding mode function.

According to above-mentioned design philosophy, Sigmoid function f (x)=arctan (x) is specifically chosen as

ω _e＝-karctan(Aσ _e)

In formula, k > 0, A=diag (a ₁, a ₂, a ₃) and a _i> 0 (i=1,2,3).In addition, parameter a _iselection also must meet work as | σ _ei| when → 0, there is following formula to set up

arctan(a _i|σ _ei|)＞|σ _ei|

According to above-mentioned Sigmoid function, Sigmoid type nonlinear sliding mode function can be designed to:

s＝ω _e+karctan(Aσ _e) (3)

4th step Sliding Mode Attitude control law module determines the attitude control law based on Sigmoid type nonlinear sliding mode function

Based on Sigmoid type nonlinear sliding mode function (3), Sliding Mode Attitude control law module utilizes equivalent control to add the sliding formwork design of control law method switching and control, and Sliding Mode Attitude design of control law is divided into two steps.Corresponding Sliding Mode Attitude control law is also made up of two parts, i.e. T _c=T _eq+ T _sw.Wherein, T _swbe referred to as switching control item, its design object is: at disturbance torque T _dimpact under, with T _eqensure s together _i(i=1,2,3) are zero at Finite-time convergence; T _eqbe referred to as equivalent control term, its design object is: as s=0, under the impact of disturbance torque, and itself and T _swsynergy s can be made to remain zero (s ≡ 0), and if do not consider the impact of disturbance torque, under its independent role, s ≡ 0.

Based on above-mentioned explanation, first design equivalent control T _eq.For this reason, to sliding formwork function (3) differentiate, can obtain:

When not considering that disturbance torque affects, T _eqneed the unchangeability ensureing sliding formwork function.By the T in above formula _creplace with T _eq, and make T _d=0 He , have:

For switching control T _sw, order:

T _sw＝-ηsgn(s)

In formula, and || || ₂for 2 norms of vector, η > || T _d|| _∞+ δ and || || _∞for the Infinite Norm of vector, δ > 0 is arbitrarily small constant.

So far, can be described as based on attitude control law:

For analyzing the attitude control system stability under attitude control law (4) effect, select Lyapunov function

Have along the differentiate of closed loop track it:

For Lyapunov function (5), following relation is had to set up:

In formula, for induction 2 norm of nominal inertia battle array.

Formula (6) is substituted in Lyapunov function derivative, can obtain:

According to Lyapunov finite time stability principle, known for arbitrarily sliding formwork function s is at finite time t _rinside converge to zero.Due to t ∈ [t _r,+∞) and there is s ≡ 0, select Lyapunov function further:

$V=\frac{1}{2}{\mathrm{\σ}}_e^T{\mathrm{\σ}}_e---\left(8\right)$

To the track differentiate that it is determined along s ≡ 0, have:

$\begin{array}{c}\stackrel{.}{V}={\mathrm{\σ}}_e^T{\mathrm{M\ω}}_e\\ =-\frac{k(1+{\mathrm{\σ}}_e^T{\mathrm{\σ}}_e)}{4}{\mathrm{\σ}}_e^T\arctan \left({\mathrm{A\σ}}_e\right)\end{array}---\left(9\right)$

Due to arctan (a _iσ _ei) and σ _eijack per line, above-mentioned Lyapunov functional derivative negative definite.Global consistent asymptotic stability according to the known closed-loop system of Lyapunov stability principle.

According to equivalent control method, known equivalent system under attitude control law (4) effect is dynamically:

${\stackrel{.}{\mathrm{\σ}}}_e=-\mathrm{kM}\arctan \left({\mathrm{A\σ}}_e\right)---\left(10\right)$

And the existing equivalent system corresponding based on linear sliding mode function is dynamically:

${\stackrel{.}{\mathrm{\σ}}}_e=-k{\mathrm{\σ}}_e$

Contrast two kinds of equivalent systems dynamically can find: for linear sliding mode function, the speed of convergence of relative correction Douglas Rodríguez parameter only depends on parameter k, and k value is larger, and the response speed of system is faster, required control moment amplitude is also larger, and vice versa; And for nonlinear sliding mode function, when the amplitude relatively revising Douglas Rodríguez parameter is larger, due to the saturation characteristic of Sigmoid function, control moment amplitude can be effectively suppressed, and along with relatively revising the convergence of Douglas Rodríguez parameter, the variable slope characteristic of Sigmoid function makes formula (10) corresponding feedback gain increase gradually, thus can accelerate the response speed of system.In summary, Sliding Mode Attitude control law based on the design of Sigmoid type nonlinear sliding mode function can suppress the control moment amplitude in large tracking error situation, simultaneously along with the convergence of tracking error, the speed of system responses is also accelerated thereupon, efficiently solves the balance select permeability of the sliding formwork function gain k that existing linear sliding mode function exists.

5th step Sliding Mode Attitude control law correcting module correction attitude control law

Owing to switching the existence of control item, Sliding Mode Attitude control law can produce chattering phenomenon in engineer applied, is unfavorable for the Project Realization of algorithm.In addition, the handoff gain switching control item in Sliding Mode Attitude control law depends on the prior imformation in the interference upper bound.But as a rule, the prior imformation in this upper bound cannot be measured or estimate, need to adopt conservative approach to select an enough large handoff gain to ensure the stability of system.For solving the problem, Sliding Mode Attitude control law correcting module is revised Sliding Mode Attitude control law (4) in conjunction with boundary layer method and ADAPTIVE CONTROL.First, suppose that the impact that disturbance torque produces spacecraft body is bounded, and meet following inequality:

||T _d|| _∞≤c ₁+c ₂||σ _e||∞+c ₃||ω _e|| _∞

In formula, c _i>=0 (i=1,2,3) are unknown normal scalar,

Sliding Mode Attitude control law (4) is revised as:

In formula, T _eqcotype (4), and $\widehat{\mathrm{\η}}={\hat{c}}_1+{\hat{c}}_2{\left|\right|{\mathrm{\σ}}_e\left|\right|}_{\∞}+{\hat{c}}_3{\left|\right|{\mathrm{\ω}}_e\left|\right|}_{\∞}+\mathrm{\δ},$ And

$\begin{array}{c}{\hat{c}}_1={\mathrm{\κ}}_1{\left|\right|s\left|\right|}_2-{\mathrm{\κ}}_1{\mathrm{\ψ}}_1{\hat{c}}_1\\ {\hat{c}}_2={\mathrm{\κ}}_2{\left|\right|s\left|\right|}_2{\left|\right|{\mathrm{\σ}}_e\left|\right|}_{\∞}-{\mathrm{\κ}}_2{\mathrm{\ψ}}_2{\hat{c}}_2\\ {\hat{c}}_3={\mathrm{\κ}}_3{\left|\right|s\left|\right|}_2{\left|\right|{\mathrm{\ω}}_e\left|\right|}_{\∞}-{\mathrm{\κ}}_3{\mathrm{\ψ}}_3{\hat{c}}_3\end{array}---\left(12\right)$

In formula, κ _i> 0 (i=1,2,3), ψ _i> 0, and

Select Lyapunov function

When time, to this Lyapunov function along the differentiate of closed loop track, have:

When time, to this Lyapunov function along the differentiate of closed loop track, have:

Comprehensive above-mentioned two situations, known for arbitrarily lyapunov functional derivative meets:

According to Lyapunov stability principle, the consistent bounded after all of known closed-loop system.

Embodiment

The present invention carries out simulating, verifying under Matlab2009a environment.The nominal value of spacecraft inertia battle array is

The range of indeterminacy of inertia battle array is ± 20%, and external disturbance is determined by spacecraft place orbit parameter, and spacecraft initial inertia attitude variable is: σ _b(t ₀)=0, ω _b(t ₀)=0 (rad/s).Expect that attitude variable is the σ of target track LVLH system correspondence _dand ω _d.Wherein, target track is circuit orbit, orbit radius a=6899807 (m), and excentricity is 0, orbit inclination i=30 °, right ascension of ascending node Ω=60 °, argument of perigee ω=0 °, initial true anomaly f (t ₀)=90 °.Consider in emulation that attitude sensor is 0.02 (rad/s) at the range of measurement relative attitude angular velocity.

Control law Selecting parameter: k=0.0127, a _i=200 (i=1,2,3), ψ _i=0.01.Simulation result as shown in Figures 2 to 5.

The present invention is based on the σ under the Sliding Mode Attitude control law effect of Sigmoid type nonlinear sliding mode function _eresponse curve as shown in Figure 2, ω _eresponse curve as shown in Figure 3.Can find out, compared with controlling with the existing Sliding Mode Attitude based on linear sliding mode function, the Sliding Mode Attitude control law based on Sigmoid type nonlinear sliding mode function adopting the present invention to propose can carry out relative attitude angular velocity in attitude tracking control process to spacecraft to be planned, the approximate tracing process realizing " peak acceleration accelerates-at the uniform velocity-peak acceleration deceleration ", under the prerequisite suppressing control moment amplitude in big error situation, ensure the response speed of system, and efficiently avoid sensor saturation problem.The present invention is based on the T under the Sliding Mode Attitude control law effect of Sigmoid type nonlinear sliding mode function _ccurve as shown in Figure 4.Can find out, adopt boundary layer method combining adaptive control strategy, effectively can retrain the buffeting problem that control moment exists.Fig. 5 gives the self-adaptation curve of handoff gain, showing that the present invention when disturbing upper bound the unknown, can be regulated handoff gain by adaptive control, significantly reducing the conservative property that handoff gain is selected.

Claims (1)

1., based on a Spacecraft Attitude Control method for Sigmoid type nonlinear sliding mode function, it is characterized in that concrete steps are:

The first step builds the spacecraft attitude control system based on Sigmoid type nonlinear sliding mode function

Based on the spacecraft attitude control system of Sigmoid type nonlinear sliding mode function, comprising: cascade form relative attitude equation of motion module, Sigmoid type nonlinear sliding mode function module, Sliding Mode Attitude control law module and Sliding Mode Attitude control law correcting module; The function of cascade form relative attitude equation of motion module is: the attitude motion rule describing rigid spacecraft, the function of Sigmoid type nonlinear sliding mode function module is: the Sigmoid type nonlinear correspondence relation setting up relative attitude parameter and relative attitude angular velocity, the function of Sliding Mode Attitude control law module for: ensure that the equivalent system that spacecraft attitude control system has corresponding to Sigmoid type nonlinear sliding mode function is dynamic, the function of Sliding Mode Attitude control law correcting module for: eliminate buffeting existing for Sliding Mode Attitude control law and handoff gain selects conservative property;

Second step cascade form relative attitude equation of motion module sets up the cascade form relative attitude equation of motion

To carry out the rigid spacecraft of attitude tracking control for object, cascade form relative attitude equation of motion module to revise Douglas Rodríguez parameter as attitude characterization parameter definition relative attitude variable, sets up the cascade form relative attitude equation of motion in the configuration space of attitude motion under spacecraft body coordinate system; Wherein, relative attitude kinetics equation is:

Relative attitude kinematical equation is:

In formula, for the matrix representation of nominal value under body coordinate system of spacecraft inertia battle array tensor, ω _erepresent the vector representation of relative attitude angular velocity vector under body coordinate system between spacecraft body coordinate system and reference frame, T _cfor the vector representation of control moment vector under body coordinate system, T _dfor the vector representation of disturbance torque under body coordinate system that external disturbance moment and systematic parameter uncertainty produce attitude motion of spacecraft, () ^×represent the antisymmetric matrix operator of vector, R represents the transition matrix between spacecraft body coordinate system and reference coordinate, ω _drepresent the vector representation of reference angular velocities vector under reference frame; σ _erepresent the vector representation of correction Douglas Rodríguez parameter vector under body coordinate system that spacecraft body coordinate system is corresponding with the relative attitude between reference frame, M is Jacobi matrix, and above parameter, band point represents the derivative of parameter;

3rd step Sigmoid type nonlinear sliding mode function module determines Sigmoid type nonlinear sliding mode function

For the relative attitude kinematical equation set up, Sigmoid type nonlinear sliding mode function module will revise Douglas Rodríguez parameter relatively as Sigmoid argument of function, dependent variable using relative attitude angular velocity as Sigmoid function, determines a quasi-nonlinear sliding formwork function; For typical Sigmoid function f (x)=arctan (x), Sigmoid type nonlinear sliding mode function is defined as:

s=ω _e+karctan(Aσ _e) (3)

In formula, k > 0, A=diag (a ₁, a ₂, a ₃) and a _i> 0 (i=1,2,3); In addition, parameter a _iselection also meet work as | σ _ei| when → 0, there is following formula to set up

arctan(a _i|σ _ei|)＞|σ _ei|

In formula, σ _ei(i=1,2,3) are relatively revise the vector representation of Douglas Rodríguez parameter under body coordinate system;

4th step Sliding Mode Attitude control law module determines the attitude control law based on Sigmoid type nonlinear sliding mode function

Based on Sigmoid type nonlinear sliding mode function, Sliding Mode Attitude control law module adds the attitude control law switching control theory and determine as formula (4) according to equivalent control;

In formula, T _eqrepresent equivalent control term, T _swrepresent and switch control item, and || || ₂for 2 norms of vector, η > || T _d|| _∞+ δ and || || _∞for the Infinite Norm of vector, δ > 0 is arbitrarily small constant;

Lyapunov function shown in selecting type (5)

Have along the differentiate of closed loop track Lyapunov function (5):

For Lyapunov function (5), following relation is had to set up:

In formula, for induction 2 norm of nominal inertia battle array;

Formula (6) is substituted in Lyapunov function derivative, can obtain:

According to Lyapunov finite time stability principle, for arbitrarily t ₀represent initial time, represent space, sliding formwork function s is at finite time t _rinside converge to zero; Due to t ∈ [t _r,+∞) and there is s ≡ 0, select Lyapunov function further:

To the track differentiate that it is determined along s ≡ 0, have:

Due to arctab (a _iσ _ei) and σ _eijack per line, above-mentioned Lyapunov functional derivative negative definite; Global consistent asymptotic stability according to the known closed-loop system of Lyapunov stability principle; According to equivalent control measurements, known closed-loop system in the equivalent system of sliding formwork section is dynamically:

5th step Sliding Mode Attitude control law correcting module correction attitude control law

Sliding Mode Attitude control law correcting module utilizes boundary layer strategy and ADAPTIVE CONTROL to be modified to attitude control law:

In formula, T _eqcotype (4), T _sarepresent revised switching control item, and and

In formula, k _i> 0 (i=1,2,3), ψ _i> 0, and

So far, the Spacecraft Attitude Control based on Sigmoid type nonlinear sliding mode function is completed.