CN105786008A - Flexible spacecraft attitude control method for flywheel saturation and friction characteristics - Google Patents

Abstract

The invention relates to a flexible spacecraft attitude control method for flywheel saturation and friction characteristics and aims to solve a problem of influence of flywheel saturation and friction characteristics on spacecraft control precision. According to the method, a spacecraft coupling kinetic equation containing flywheel saturation and friction characteristics is firstly constructed; secondly, a friction interference estimator according to flywheel friction characteristics is established; thirdly, a flexible vibration observer is designed according to interference caused by flexible accessory vibration; fourthly, an anti-saturation controller is designed for inhibition according to friction interference estimation errors and flexible vibration interference observation errors; and lastly, through solving control gain of the anti-saturation controller, the friction interference estimator and the flexible vibration observer, a composite layered anti-interference controller is designed, and spacecraft anti-interference attitude control under the action of multi-source interference influence is accomplished. The method is advantaged in that attitude control precision of spacecrafts employing a flywheel execution mechanism can be remarkably improved, and the method can be applied to high precision attitude control on spacecrafts such as high precision earth observation satellites in the aerospace field and space telescopes.

Description

A kind of Flexible Spacecraft control method of saturated for flywheel and frictional behavior

Technical field

The present invention relates to a kind of saturated for flywheel and frictional behavior Flexible Spacecraft control method, it is adaptable to need to carry executive capability Flexible Spacecraft that is limited and that need realization to control in high precision and control system, belong to Spacecraft Attitude Control field.

Background technology

Exploring space action recently as the mankind to enrich constantly, task complexity is increasing, and demand is also increasing, it is desirable to spacecraft carries bigger solaode tabula rasa and is provided that more energy.Additionally, more remote along with spacecraft Mission Operations, also spacecraft communication antenna is proposed requirements at the higher level, it is necessary to structure is bigger, power is higher antenna is so that the faint signal of distance reception；The demands such as power supply and communication all make spacecraft need to carry increasing adnexa, from launch cost and technology implementation difficulty, the quality volume carrying spacecraft all has strict restriction, therefore above-mentioned adnexa generally adopts the flexible structure design of low quality, Low rigidity, substantial amounts of flexible appendage use, when spacecraft body carries out motor-driven, flexible structure can produce for, thus Spacecraft Attitude Control precision can be had a strong impact on, even affect last task.

In addition the reliability of spacecraft attitude control system and long-term working stability are always up the key technology during spacecraft is developed.Flywheel is one of most important execution unit in satellite attitude control system, the long-life launched in recent years, in high precision, multifunctional triaxial stabilized satellite, almost do not utilize flywheel as main execution unit exceptionally.But flywheel has high-definition feature, it is limited to machining accuracy, can there is a degree of friction, thus bringing flywheel to perform error, on the other hand, the output torque size of the flywheel in kind of actual physical system is very critical, therefore needs also exist for considering saturated and frictional behavior problem further.Moment of friction is transferred to spacecraft body by flywheel wheel body on the other hand, causes that shivering occurs in spacecraft body, thus very big trouble can be brought to spacecraft attitude control system.Therefore, in order to complete Spacecraft Attitude Control more accurately, the process of Spacecraft guidance and control must pull against the impact that above-mentioned two classes are mainly disturbed.

Number of patent application is propose a kind of anti-interference attitude control method based on reaction wheel frictional behavior in 201510294341.9, but there are two problems: the vibration interference that (1) this article flexible appendage that consideration Spacecraft is not subsidiary brings, spacecraft precision can be impacted；(2) do not consider the saturated characteristic of flywheel, be subject to a definite limitation in actually used meeting.Number of patent application is propose a kind of Flexible Spacecraft for flywheel low speed friction in 201510303102.5 to control System and method for, and the method wherein proposed equally exists Similar Problems: (1) does not consider the saturated characteristic that flywheel exists；(2) what relate in method is completely different for wheel friction observer and this patent form, does not ensure that finite time quickly follows the tracks of the interference that rubs, therefore can be inferior to context of methods in precision and rapidity.

Summary of the invention

The technology of the present invention solves problem: overcome existing deficiency, a kind of saturated for flywheel and frictional behavior Flexible Spacecraft control method is provided, utilize the method anti-flywheel can be provided saturated and friction, capability for employing flywheel as main actuator Spacecraft system, it is possible to realize the high-precision attitude control of wheel control Spacecraft system.

The technical solution of the present invention is: the Flexible Spacecraft control method of a kind of saturated for flywheel and frictional behavior, and implementation step is as follows:

The first step, sets up saturated containing flywheel and frictional behavior Spacecraft the coupled dynamical equation:

For saturated and frictional behavior common in flywheel actuator, and consider flexible appendage kinetics equation simultaneously, set up saturated with flywheel and frictional behavior flexible spacecraft dynamics system model, be expressed as follows:

${\mathrm{\Σ}}_1:\left\{\begin{array}{c}J\stackrel{\·\·}{\mathrm{\θ}}\left(t\right)+F\stackrel{\·\·}{\mathrm{\η}}\left(t\right)=sat\left(T_c\left(t\right)\right)+M_f\left(t\right)+d_1\left(t\right)\\ \stackrel{\·\·}{\mathrm{\η}}\left(t\right)+2\mathrm{\ξ}\mathrm{\ω}\stackrel{\·}{\mathrm{\η}}\left(t\right)+{\mathrm{\ω}}^2\mathrm{\η}\left(t\right)+F^T\stackrel{\·\·}{\mathrm{\θ}}\left(t\right)=0\end{array}\right.$

In formula, t express time, J is the rotary inertia of spacecraft；For spacecraft attitude angular acceleration, F is the coupling matrix between spacecraft attitude and flexible structure, the mode of oscillation that η (t) is flexible appendage, and ω is the frequency of vibration that flexible appendage mode of oscillation is corresponding, and ξ is the damping of flexible appendage mode, T_cT () represents the control moment that the attitude controller of Space Vehicle System resolves, sat (T_c(t)) it is the saturated control moment after considering flywheel saturated characteristic, M_f(t) wheel friction interference for introducing after considering wheel friction characteristic；d₁T outer space environmental disturbances moment that () is subject to for spacecraft；

Can obtain further:

$(J-{\mathrm{FF}}^T)\stackrel{\·\·}{\mathrm{\θ}}\left(t\right)=d_0\left(t\right)+sat\left(T_c\right(t\left)\right)+M_f\left(t\right)+d_1\left(t\right)$

In formulaRepresent that flexible appendage vibrates the vibration interference brought, d₁T outer space environmental disturbances moment that () is subject to for spacecraft；This equation already has accounted for flywheel is saturated and frictional behavior, and flexible appendage vibrates the interference brought, further saturated with flywheel and frictional behavior flexible spacecraft dynamics system model is transformed into state space form, state space form represent under new system as follows:

$\stackrel{\·}{x}\left(t\right)=Ax\left(t\right)+B\left(sat\right(T_c\left(t\right))+M_f(t)+d_0(t)+d_1(t\left)\right)$

In formula, x (t) is system mode,The attitude angle that θ (t) is spacecraft,For the attitude angular velocity of spacecraft,For sytem matrix,For controlling input matrix；

Second step, the vibration interference caused for flexible appendage is characterized by following interference model:

$\left\{\begin{array}{c}{d}_0\left(t\right)=Vw\left(t\right)\\ \stackrel{\·}{w}\left(t\right)=Ww\left(t\right)+H(sat\left(T_c\left(t\right)\right)+M_f\left(t\right)+d_1\left(t\right))\end{array}\right.$

In formula, V, w (t) and W are expressed as the output matrix of the vibration interference that flexible appendage causes, state variable, sytem matrix, H represents the gain battle array of the output of saturated controller, wheel friction interference and environmental disturbances, wherein the state variable of vibration adnexa interferenceWherein intermediate variable R=(1-F^TI^-1F)^-1；

The flexible vibration observer that flexible appendage vibration interference is constructed as follows:

$\left\{\begin{array}{c}{\hat{d}}_0\left(t\right)=V\widehat{\mathrm{\ω}}\left(t\right)\\ \widehat{\mathrm{\ω}}\left(t\right)=v\left(t\right)-Lx\left(t\right)\\ \stackrel{\·}{v}\left(t\right)=(W+{\mathrm{LB}}_{1}V)\left(v\right(t)-Lx(t\left)\right)+LAx\left(t\right)+(LB+H){\hat{M}}_f\left(t\right)+(LB+H)u\left(t\right)\end{array}\right.$

Wherein v (t) is an auxiliary State Variable of flexible vibration observer,It is that flexible appendage is for interference d₀T the estimated value of (), L is the gain of interference observer to be asked；

Wheel friction is disturbed, the moment of friction M that fly wheel system is subject to_fT () mainly includes the solid friction moment of bearing and lubricates the gentle dynamic resistance square of viscous friction moment brought, above-mentioned two classes are relevant to Speed of Reaction Wheels owing to lubricating the moment brought, in the relatively low situation of Speed of Reaction Wheels comparatively small, therefore the low speed friction characteristic of flywheel mostlys come from solid friction, and the kinetic model state-space representation formula of flywheel is as follows:

$\left[\begin{array}{c}\stackrel{\·}{\mathrm{\Ω}}\left(t\right)\\ {\stackrel{\·}{M}}_f\left(t\right)\end{array}\right]=\left[\begin{array}{c}-{J_w}^{-1}(T_c\left(t\right)+M_f\left(t\right)+D\mathrm{\Ω}\left(t\right))\\ \mathrm{\β}\mathrm{\Ω}\left(t\right){\left(M_f\left(t\right)sign\left(\mathrm{\Ω}\left(t\right)\right)-M_{f0}\right)}^2\end{array}\right]$

In formulaFor the angular acceleration of flywheel, Ω (t) is Speed of Reaction Wheels,For the moment of friction rate of change of flywheel, wherein D represents the damped coefficient of flywheel, Rotary Inertia of Flywheel J_w, bearing static ramp parameter β, Coulomb friction moment M_f0；

For wheel friction, design following wheel friction interference estimator:

$\left[\begin{array}{c}\stackrel{\·}{\widehat{\mathrm{\Ω}}}\left(t\right)\\ {\stackrel{\·}{\hat{M}}}_f\left(t\right)\end{array}\right]=\left[\begin{array}{c}-{J_w}^{-1}(T_c+D\widehat{\mathrm{\Ω}}+{\hat{M}}_f)+k_1{e}_1\left(t\right)+{\mathrm{\α}}_1\mathrm{sgn}{e}_1\left(t\right)\\ k_2{e}_1\left(t\right)+{\mathrm{\α}}_2\mathrm{sgn}(-{\mathrm{\α}}_1{J}_w\mathrm{sgn}\left(e_1\left(t\right)\right))\end{array}\right]$

WhereinWithIt is the estimated value of Speed of Reaction Wheels and moment of friction, parameter k₁And k₂, positive parameter alpha₁With α₂So thatWithCan at Finite-time convergence in Ω (t) and M_f(t), it is ensured that wheel friction tracking error can be cut down at finite time；

3rd step, design anti-saturation controller:

In conjunction with friction interference estimator, flexible appendage Vibration device, and consider flywheel damp constraint characteristic, design following anti-saturation controller further:

$\left\{\begin{array}{c}u\left(t\right)=sat\left(T_c\left(t\right)\right)\\ T_c\left(t\right)=2Kx\left(t\right)-{\hat{d}}_0\left(t\right)-{\hat{M}}_f\left(t\right)\end{array}\right.$

In formula, u (t)=sat (T_c(t)) export for anti-saturation controller, namely consider the saturated control moment after flywheel saturated characteristic, K is that anti-saturation controller controls gain, and saturated owing to considering flywheel, the maximum output torque arranging flywheel is u_max, and u_max> 0 thus having:

$u\left(t\right)=\left\{\begin{array}{cc}{T}_c\left(t\right)& -u_{min}\≤T_c\left(t\right)\≤u_{max}\\ -u_{\max }& T_c\left(t\right)\≤-u_{max}\\ u_{\max }& u_{\max }\≤T_c\left(t\right)\end{array}\right.$

The control moment instruction that u (t) in formula receives for flywheel, not over the maximum of flywheel, also is able to ensure that Space Vehicle System is stable, it is achieved there is spacecraft high-precision attitude under saturated and friction condition and control in addition.

Present invention advantage compared with prior art is in that:

A kind of being mainly directed towards for saturated and frictional behavior Flexible Spacecraft control method related in the present invention adopts flywheel to control system as the Flexible Spacecraft of the Flexible Spacecraft control flywheel low speed friction of main actuator, consider the saturated and frictional behavior that fly wheel system exists simultaneously, what make for actuator error analysis is more comprehensive, has the face that uses widely.In addition the friction interference observer with finite time convergence control is employed for wheel friction, wheel friction interference can be followed the tracks of quickly and accurately, add Flexible Spacecraft and control the capability of fast response of system, significantly improve the speed of Spacecraft Attitude Control method, precision and degree of stability.

Accompanying drawing explanation

Fig. 1 is that the present invention is a kind of saturated for flywheel and the design flow diagram of the Flexible Satellite Attitude control method of frictional behavior.

Detailed description of the invention

For the satellite system of a class generalized ribbon flexible appendage, implementing of system and method being described, attitude control accuracy and degree of stability, in earth observation pattern, are had and have high requirements by satellite operation；

As it is shown in figure 1, the present invention to be embodied as step as follows:

1, saturated containing flywheel and frictional behavior flexible satellite the coupled dynamical equation is set up

For saturated and frictional behavior common in flywheel actuator, and consider flexible appendage kinetics equation simultaneously, set up saturated with flywheel and frictional behavior flexible satellite dynamic system model, be expressed as follows:

${\mathrm{\Σ}}_1:\left\{\begin{array}{c}J\stackrel{\·\·}{\mathrm{\θ}}\left(t\right)+F\stackrel{\·\·}{\mathrm{\η}}\left(t\right)=sat\left(T_c\left(t\right)\right)+M_f\left(t\right)+d_1\left(t\right)\\ \stackrel{\·\·}{\mathrm{\η}}\left(t\right)+2\mathrm{\ξ}\mathrm{\ω}\stackrel{\·}{\mathrm{\η}}\left(t\right)+{\mathrm{\ω}}^2\mathrm{\η}\left(t\right)+F^T\stackrel{\·\·}{\mathrm{\θ}}\left(t\right)=0\end{array}\right.$

In formula, t express time, J is the rotary inertia of satellite；For attitude of satellite angular acceleration, F is the coupling matrix between the attitude of satellite and flexible structure, the mode of oscillation that η (t) is flexible appendage, and ω is the frequency of vibration that flexible appendage mode of oscillation is corresponding, and ξ is the damping of flexible appendage mode, T_cT () represents the control moment that the attitude controller of satellite system resolves, sat (T_c(t)) it is the saturated control moment after considering flywheel saturated characteristic, M_f(t) wheel friction interference for introducing after considering wheel friction characteristic；d₁T outer space environmental disturbances moment that () is subject to for satellite；

Can obtain further:

$(J-{\mathrm{FF}}^T)\stackrel{\·\·}{\mathrm{\θ}}\left(t\right)=d_0\left(t\right)+sat\left(T_c\right(t\left)\right)+M_f\left(t\right)+d_1\left(t\right)$

In formulaRepresent that flexible appendage vibrates the vibration interference brought, d₁T outer space environmental disturbances moment that () is subject to for spacecraft；This equation already has accounted for flywheel is saturated and frictional behavior, and flexible appendage vibrates the interference brought, further saturated with flywheel and frictional behavior flexible satellite dynamic system model is transformed into state space form, state space form represent under new system as follows:

$\stackrel{\·}{x}\left(t\right)=Ax\left(t\right)+B\left(sat\right(T_c\left(t\right))+M_f(t)+d_0(t)+d_1(t\left)\right)$

In formulaThe attitude angle that θ (t) is satellite,For the attitude angular velocity of satellite, x (t) is system mode,For sytem matrix,For controlling input matrix；

2, the vibration interference caused for flexible appendage is characterized by following interference model:

$\left\{\begin{array}{c}{d}_0\left(t\right)=Vw\left(t\right)\\ \stackrel{\·}{w}\left(t\right)=Ww\left(t\right)+H(sat\left(T_c\left(t\right)\right)+M_f\left(t\right)+d_1\left(t\right))\end{array}\right.$

In formula, V, w (t) and W are expressed as the output matrix of the vibration interference that flexible appendage causes, state variable, sytem matrix, H represents the gain battle array of the output of saturated controller, wheel friction interference and environmental disturbances, wherein the state variable of vibration adnexa interferenceWherein intermediate variable R=(1-F^TI^-1F)^-1；

The flexible vibration observer that flexible appendage vibration interference is constructed as follows:

$\left\{\begin{array}{c}{\hat{d}}_0\left(t\right)=V\widehat{\mathrm{\ω}}\left(t\right)\\ \widehat{\mathrm{\ω}}\left(t\right)=v\left(t\right)-Lx\left(t\right)\\ \stackrel{\·}{v}\left(t\right)=(W+{\mathrm{LB}}_{1}V)\left(v\right(t)-Lx(t\left)\right)+LAx\left(t\right)+(LB+H){\hat{M}}_f\left(t\right)+(LB+H)u\left(t\right)\end{array}\right.$

Wherein v (t) is an auxiliary State Variable of flexible vibration observer,It is that flexible appendage is for interference d₀T the estimated value of (), L is the gain of interference observer to be asked；

Wheel friction is disturbed, the moment of friction M that fly wheel system is subject to_fT () mainly includes the solid friction moment of bearing and lubricates the gentle dynamic resistance square of viscous friction moment brought, above-mentioned two classes are relevant to Speed of Reaction Wheels owing to lubricating the moment brought, in the relatively low situation of Speed of Reaction Wheels comparatively small, therefore the low speed friction characteristic of flywheel mostlys come from solid friction, and the kinetic model state-space representation formula of flywheel is as follows:

$\left[\begin{array}{c}\stackrel{\·}{\mathrm{\Ω}}\left(t\right)\\ {\stackrel{\·}{M}}_f\left(t\right)\end{array}\right]=\left[\begin{array}{c}-{J_w}^{-1}(T_c\left(t\right)+M_f\left(t\right)+D\mathrm{\Ω}\left(t\right))\\ \mathrm{\β}\mathrm{\Ω}\left(t\right){\left(M_f\left(t\right)sign\left(\mathrm{\Ω}\left(t\right)\right)-M_{f0}\right)}^2\end{array}\right]$

In formulaFor the angular acceleration of flywheel, Ω (t) is Speed of Reaction Wheels,For the moment of friction rate of change of flywheel, wherein D represents the damped coefficient of flywheel, Rotary Inertia of Flywheel J_w, bearing static ramp parameter β, Coulomb friction moment M_f0；

For wheel friction, design following wheel friction interference estimator:

$\left[\begin{array}{c}\stackrel{\·}{\widehat{\mathrm{\Ω}}}\left(t\right)\\ {\stackrel{\·}{\hat{M}}}_f\left(t\right)\end{array}\right]=\left[\begin{array}{c}-{J_w}^{-1}(T_c+D\widehat{\mathrm{\Ω}}+{\hat{M}}_f)+k_1{e}_1\left(t\right)+{\mathrm{\α}}_1\mathrm{sgn}{e}_1\left(t\right)\\ k_2{e}_1\left(t\right)+{\mathrm{\α}}_2\mathrm{sgn}(-{\mathrm{\α}}_1{J}_w\mathrm{sgn}\left(e_1\left(t\right)\right))\end{array}\right]$

WhereinWithIt is the estimated value of Speed of Reaction Wheels and moment of friction, parameter k₁And k₂, positive parameter alpha₁With α₂So thatWithCan at finite time convergence control in Ω (t) and M_f(t), it is ensured that wheel friction tracking error can be cut down at finite time；

3, design anti-saturation controller

In conjunction with friction interference estimator, flexible appendage Vibration device, and consider flywheel damp constraint characteristic, design following anti-saturation controller further:

$\left\{\begin{array}{c}u\left(t\right)=sat\left(T_c\left(t\right)\right)\\ T_c\left(t\right)=2Kx\left(t\right)-{\hat{d}}_0\left(t\right)-{\hat{M}}_f\left(t\right)\end{array}\right.$

In formula, u (t)=sat (T_c(t)) export for anti-saturation controller, namely consider the saturated control moment after flywheel saturated characteristic, K is that anti-saturation controller controls gain, and saturated owing to considering flywheel, the maximum output torque arranging flywheel is u_max, and u_max> 0 thus having:

$u\left(t\right)=\left\{\begin{array}{cc}{T}_c\left(t\right)& -u_{min}\≤T_c\left(t\right)\≤u_{max}\\ -u_{\max }& T_c\left(t\right)\≤-u_{max}\\ u_{\max }& u_{\max }\≤T_c\left(t\right)\end{array}\right.$

The control moment instruction that u (t) in formula receives for flywheel, not over the maximum of flywheel, also is able to ensure that satellite system is stable, it is achieved there is satellite high-precision gesture stability under saturated and friction condition in addition.

The content not being described in detail in description of the present invention belongs to the known prior art of professional and technical personnel in the field.

Claims (1)

1. one kind saturated for flywheel and the Flexible Spacecraft control method of frictional behavior, it is characterized in that: comprise the following steps: first build saturated containing flywheel and frictional behavior spacecraft the coupled dynamical equation, secondly, set up the friction interference estimator for wheel friction characteristic, again, to in Spacecraft owing to vibrating, for flexible appendage, the interference that brings, design flexible vibration observer；Then, for friction Interference Estimation error and flexible vibration disturbance-observer error, design anti-saturation controller suppresses；Finally by asking for anti-saturation controller, friction interference estimator and the control gain of flexible vibration observer, design composite layered anti-interference controller, complete the anti-interference gesture stability of the spacecraft under multi-source interference effect；Specifically comprise the following steps that

The first step, sets up saturated containing flywheel and frictional behavior Spacecraft the coupled dynamical equation

For saturated and frictional behavior common in flywheel actuator, and consider flexible appendage kinetics equation simultaneously, set up saturated with flywheel and frictional behavior flexible spacecraft dynamics system model, be expressed as follows:

${\mathrm{\Σ}}_1:\left\{\begin{array}{c}J\stackrel{\·\·}{\mathrm{\θ}}\left(t\right)+F\stackrel{\·\·}{\mathrm{\η}}\left(t\right)=sat\left(T_c\right(t\left)\right)+M_f\left(t\right)+d_1\left(t\right)\\ \stackrel{\·\·}{\mathrm{\η}}\left(t\right)+2\mathrm{\ξ}\mathrm{\ω}\stackrel{\·}{\mathrm{\η}}\left(t\right)+{\mathrm{\ω}}^2\mathrm{\η}\left(t\right)+F^T\stackrel{\·\·}{\mathrm{\θ}}\left(t\right)=0\end{array}\right.$

In formula, t express time, J is the rotary inertia of spacecraft；For spacecraft attitude angular acceleration, F is the coupling matrix between spacecraft attitude and flexible structure, the mode of oscillation that η (t) is flexible appendage,For the single order mode of oscillation of flexible appendage,For the second order vibration mode of flexible appendage, ω is the frequency of vibration that flexible appendage mode of oscillation is corresponding, and ξ is the damping of flexible appendage mode, T_cT () represents the control moment that the attitude controller of Space Vehicle System resolves, sat (T_c(t)) it is the saturated control moment after considering flywheel saturated characteristic, M_f(t) wheel friction interference for introducing after considering wheel friction characteristic；d₁T outer space environmental disturbances moment that () is subject to for spacecraft；

Can obtain further:

$(J-{\mathrm{FF}}^T)\stackrel{\·\·}{\mathrm{\θ}}\left(t\right)=d_0\left(t\right)+sat\left(T_c\right(t\left)\right)+M_f\left(t\right)+d_1\left(t\right)$

In formulaRepresent that flexible appendage vibrates the vibration interference brought, d₁T outer space environmental disturbances moment that () is subject to for spacecraft；This equation already has accounted for flywheel is saturated and frictional behavior, and flexible appendage vibrates the interference brought, further saturated with flywheel and frictional behavior flexible spacecraft dynamics system model is transformed into state space form, state space form represent under new system as follows:

$\stackrel{\·}{x}\left(t\right)=Ax\left(t\right)+B\left(sat\right(T_c\left(t\right))+M_f(t)+d_0(t)+d_1(t\left)\right)$

In formula, x (t) is system mode,The attitude angle that θ (t) is spacecraft,For the attitude angular velocity of spacecraft,For sytem matrix,For controlling input matrix；

Second step, the vibration interference caused for flexible appendage is characterized by following interference model:

$\left\{\begin{array}{c}{d}_0\left(t\right)=Vw\left(t\right)\\ \stackrel{\·}{w}\left(t\right)=Ww\left(t\right)+H\left(sat\right(T_c\left(t\right))+M_f(t)+d_1(t\left)\right)\end{array}\right.$

In formula, V, w (t) and W are expressed as the output matrix of the vibration interference that flexible appendage causes, state variable, sytem matrix, H represents the gain battle array of the output of saturated controller, wheel friction interference and environmental disturbances, wherein the state variable of vibration adnexa interferenceWherein intermediate variable R=(1-F^TI^-1F)^-1；

The flexible vibration observer that flexible appendage vibration interference is constructed as follows:

$\left\{\begin{array}{c}{\hat{d}}_0\left(t\right)=V\widehat{\mathrm{\ω}}\left(t\right)\\ \widehat{\mathrm{\ω}}\left(t\right)=v\left(t\right)-Lx\left(t\right)\\ \stackrel{\·}{v}\left(t\right)=(W+{\mathrm{LB}}_{1}V)\left(v\right(t)-Lx(t\left)\right)+LAx\left(t\right)+(LB+H){\hat{M}}_f\left(t\right)+(LB+H)u\left(t\right)\end{array}\right.$

Wherein v (t) is an auxiliary State Variable of flexible vibration observer,It is that flexible appendage is for interference d₀T the estimated value of (), L is the gain of interference observer to be asked；

Wheel friction is disturbed, the moment of friction M that fly wheel system is subject to_fT () mainly includes the solid friction moment of bearing and lubricates the gentle dynamic resistance square of viscous friction moment brought, above-mentioned two classes are relevant to Speed of Reaction Wheels owing to lubricating the moment brought, in the relatively low situation of Speed of Reaction Wheels comparatively small, therefore the low speed friction characteristic of flywheel mostlys come from solid friction, and the kinetic model state-space representation formula of flywheel is as follows:

$\left[\begin{array}{c}\stackrel{\·}{\mathrm{\Ω}}\left(t\right)\\ {\stackrel{\·}{M}}_f\left(t\right)\end{array}\right]=\left[\begin{array}{c}-{J_w}^{-1}\left(T_c\right(t)+M_f(t)+D\mathrm{\Ω}(t\left)\right)\\ \mathrm{\β}\mathrm{\Ω}\left(t\right){\left(M_f\right(t\left)sign\right(\mathrm{\Ω}\left(t\right))-M_{f0})}^2\end{array}\right]$

In formulaFor the angular acceleration of flywheel, Ω (t) is Speed of Reaction Wheels,For the moment of friction rate of change of flywheel, wherein D represents the damped coefficient of flywheel, Rotary Inertia of Flywheel J_w, bearing static ramp parameter β, Coulomb friction moment M_f0；

For wheel friction, design following wheel friction interference estimator:

$\left[\begin{array}{c}\stackrel{\·}{\widehat{\mathrm{\Ω}}}\left(t\right)\\ {\stackrel{\·}{\hat{M}}}_f\left(t\right)\end{array}\right]=\left[\begin{array}{c}-{J_w}^{-1}(T_c+D\widehat{\mathrm{\Ω}}+{\hat{M}}_f)+k_1{e}_1\left(t\right)+{\mathrm{\α}}_1\mathrm{sgn}{e}_1\left(t\right)\\ k_2{e}_1\left(t\right)+{\mathrm{\α}}_2\mathrm{sgn}(-{\mathrm{\α}}_1{J}_w\mathrm{sgn}(e_1\left(t\right)\left)\right)\end{array}\right]$

WhereinWithIt is the estimated value of Speed of Reaction Wheels and moment of friction, parameter k₁And k₂, positive parameter alpha₁With α₂So thatWithCan at finite time convergence control in Ω (t) and M_f(t), it is ensured that wheel friction tracking error can be cut down at finite time；

3rd step, design anti-saturation controller

In conjunction with friction interference estimator, flexible appendage Vibration device, and consider flywheel damp constraint characteristic, design following anti-saturation controller further:

$\left\{\begin{array}{c}u\left(t\right)=sat\left(T_c\right(t\left)\right)\\ T_c\left(t\right)=2Kx\left(t\right)-{\hat{d}}_0\left(t\right)-{\hat{M}}_f\left(t\right)\end{array}\right.$

In formula, u (t)=sat (T_c(t)) export for anti-saturation controller, namely consider the saturated control moment after flywheel saturated characteristic, K is that anti-saturation controller controls gain, and saturated owing to considering flywheel, the maximum output torque arranging flywheel is u_max, and u_max> 0 thus having:

$u\left(t\right)=\left\{\begin{array}{cc}{T}_c\left(t\right)& -u_{\min }\≤T_c\left(t\right)\≤u_{\max }\\ -u_{\max }& T_c\left(t\right)\≤-u_{\max }\\ u_{\max }& u_{\max }\≤T_c\left(t\right)\end{array}\right.$

The control moment instruction that u (t) in formula receives for flywheel, not over the maximum of flywheel, also is able to ensure that Space Vehicle System is stable, it is achieved there is spacecraft high-precision attitude under saturated and friction condition and control in addition.