CN105912005A - Space non-cooperative target attitude joint takeover control method utilizing tether thruster - Google Patents

Abstract

The invention discloses a space non-cooperative target attitude joint takeover control method utilizing tether thrusters. The Space non-cooperative target attitude joint takeover control method comprises the steps of: generating an error instruction according to a control instruction and a state measured value by adopting a formula, updating part of parameters of a controller online in real time, and then generating a pseudo control quantity (control torque) by utilizing an attitude self-adaptive control law; converting an allocation problem of the control quantity into a robust optimization problem, converting the robust optimization problem into a conic quadratic optimization problem, solving the conic quadratic optimization problem by adopting an interior point method to obtain real control quantities (thrust and tension); and finally driving 12 thrusters and tether tension, and achieving attitude joint takeover control of a non-cooperative target.

Description

Space non-cooperative target posture joint takeover control method by using tether thruster

[ technical field ] A method for producing a semiconductor device

The invention belongs to the field of spacecraft attitude control, and relates to a space non-cooperative target attitude joint takeover control method by using a tether thruster.

[ background of the invention ]

The control of taking over refers to that after the operating mechanism of the service spacecraft is fixedly connected with the target spacecraft, the attitude and orbit control system is used for taking over the attitude and orbit control system of the target spacecraft to realize attitude and orbit control. With the development of space technology, on-track service receives more and more attention. Aiming at satellites stranded in waste orbits, satellites losing effectiveness due to attitude rolling, satellites incapable of working normally due to attitude pointing errors and the like, if the service spacecraft can be used for attitude take-over, take-over control such as auxiliary orbit change and auxiliary attitude determination is provided for the satellites, and great economic benefit and social influence are achieved. The DEOS (Deutsche Orbital serving) project of DLR in Germany, the SMART-OLEV (SMART Orbital Life extension vehicle) project of the European Bureau, the FREND (Front-end Robotics engineering Near-term-termDemonstroration) project of the United states, etc. were all studied and prepared to develop on-track tests for such problems.

However, most of the existing research take-over control is to capture a target by using a rigid space robot such as a robot arm carried by a service spacecraft, and then take over the control of the target spacecraft by using an attitude and orbit control system of the service spacecraft. The space rope system robot is a novel rigid-flexible combined space robot system, has the advantages of long operation distance, safety, flexibility and the like, and has attracted much attention in recent years. Due to the need to approach the target spacecraft remotely, the operating mechanism of the space tethered robot has thrusters. Therefore, the target spacecraft can be subjected to joint take-over control by using the thruster and the tether. Wang Dongke et al performed a study of a combination of a tether and a thruster to control a target attitude. However, it regards the target inertia as known, and does not consider the problems of distribution of the control amount, constraint of the control amount, and the like, but considers that the operating mechanism of the space tether robot can output any control torque by using the thruster. In fact, for a non-cooperative target, the moment of inertia of the target satellite is unknown, the output torque of the thruster is limited, and the position of the center of mass of the target satellite is unknown, so that the acting point and the acting direction of the thrust are unknown. These problems greatly increase the difficulty of taking over control of the attitude of the target star using the tether/thruster combination. Aiming at the difficult problem, the invention gives a method for taking over the control of the attitude of the non-cooperative target satellite by fully considering the unknown and limited state quantities and the control quantity.

[ summary of the invention ]

The invention aims to solve the problem of attitude takeover control of a space non-cooperative target satellite and provides a space non-cooperative target attitude joint takeover control method by using a tether thruster.

In order to achieve the purpose, the invention adopts the following technical scheme to realize the purpose:

the space non-cooperative target posture combined takeover control method by using the tether thruster comprises the following steps of:

1) establishing a spatial non-cooperative target satellite attitude takeover control model;

2) designing a posture self-adaptive take-over control law of a non-cooperative target satellite;

3) takes over the robust distribution of the control torque.

The invention further improves the following steps:

in the step 1), a specific method for establishing a space non-cooperative target satellite attitude takeover control model comprises the following steps:

with O_TEstablishing a target satellite body system O for the mass center of a spatial non-cooperative target satellite_TX_TY_TZ_TWith O_GEstablishing an operating mechanism body system O for the center of mass of the operating mechanism_GX_GY_GZ_G，O_CIs the connection point of the tether and the operating mechanism; in order to simplify the modeling process, the mass center O of the operating mechanism is set on the assumption that the coordinate axes of the two coordinate systems are parallel to each other_GIn the target star body system O_TX_TY_TZ_TThe lower coordinate is X_G＝[x_G,y_G,z_G]；

Four groups of operating mechanisms, wherein 12 thrusters are arranged in a cross shape; wherein, the 1 st group of operating mechanisms (4) and the 2 nd group of operating mechanisms (5) comprise 5 thrusters which are orthogonally arranged, and the 3 rd group of operating mechanisms (6) and the 4 th group of operating mechanisms (7) are respectively 1 thruster;

assuming that the thrust range of each thruster is [0a ] N, the thrust generated by the 12 thrusters in the operating mechanism body system and the action positions of the operating mechanism body system are as follows:

a first group:

second group:

third and fourth groups:

to simplify the modeling process, assume that its orientation is unchanged and is along the operating mechanism body system-x direction; therefore, the maximum pulling force of the tying rope is a₅N, the rope tension and the action point are expressed as follows under the system in the operating mechanism:

	tension force	Point of action	Tension restraint
				Tether rope	F13＝[-F13x,0,0]	X5＝[x5,y5,z5]	0N≤F13x≤a5N

Because the operation object is a non-cooperative target, and the measuring device and the executing mechanism are both arranged on the operation mechanism of the space tether robot, under the system in the operation mechanism, the attitude dynamics equation of the space non-cooperative target is established as follows:

$J\stackrel{\·}{\mathrm{\ω}}+{\mathrm{\ω}}^{\×}J\mathrm{\ω}=T+T_d---\left(1\right)$

wherein J is a target star moment of inertia matrix, omega is the angular velocity of the target star,^×as a cross product, T_dFor disturbing torque, T ═ T_c+T_tFor controlling torque, T_cControl torque generated for the thruster:

$T_c=\underset{i=1}{\overset{5}{\mathrm{\Σ}}}(X_G+X_1)\×F_i+\underset{i=6}{\overset{10}{\mathrm{\Σ}}}(X_G+X_2)\×F_i+(X_G+X_3)\×F_{11}+(X_G+X_4)\×F_{12}---\left(2\right)$

F_ithe thrust corresponding to the ith thruster, i is the number of the thruster, T_tControl moment generated for the tether:

T_t＝(X_G+X₅)×F₁₃(3)

the control torque T is simplified to:

$T=T_c+T_t=DF={\left[\begin{array}{ccc}0& (z_G+z_1)& -(y_G+y_1)\\ 0& -(z_G+z_1)& (y_G+y_1)\\ (z_G+z_1)& 0& -(x_G+x_1)\\ (y_G+y_1)& -(x_G+x_1)& 0\\ -(y_G+y_1)& (x_G+x_1)& 0\\ 0& (z_G+z_2)& -(y_G+y_2)\\ 0& -(z_G+z_2)& (y_G+y_2)\\ -(z_G+z_2)& 0& (x_G+x_2)\\ (y_G+y_2)& -(x_G+x_2)& 0\\ -(y_G+y_2)& (x_G+x_2)& 0\\ -(y_G+y_3)& (x_G+x_3)& 0\\ (y_G+y_3)& -(x_G+x_3)& 0\\ 0& -(z_G+z_5)& (y_G+y_5)\end{array}\right]}^T\left[\begin{array}{c}{F}_{1x}\\ F_{2x}\\ F_{3y}\\ F_{4z}\\ F_{5z}\\ F_{6x}\\ F_{7x}\\ F_{8y}\\ F_{9z}\\ F_{10z}\\ F_{11z}\\ F_{12z}\\ F_{13x}\end{array}\right]---\left(4\right)$

wherein D is a control quantity distribution matrix, and F is a column vector formed by actuators;

the kinematic equation of the attitude of the target star described by the modified Rodrigues parameter is as follows:

$\stackrel{\·}{\mathrm{\σ}}=G\left(\mathrm{\σ}\right)\mathrm{\ω}---\left(5\right)$

$G\left(\mathrm{\σ}\right)=\frac{1}{4}\[(1-{\mathrm{\σ}}^T\mathrm{\σ})I_3+2{\mathrm{\σ}}^{\×}+2{\mathrm{\σ\σ}}^T\]---\left(6\right)$

wherein σ is the attitude-corrected Rodrigues parameter of the target star, I₃An identity matrix of 3 × 3;

let the expected attitude of the non-cooperative target star be σ_dDesired angular velocity is ω_dThen the target star attitude error dynamics/kinematics equation is:

$\left\{\begin{array}{c}{\stackrel{\·}{\mathrm{\σ}}}_e=G\left({\mathrm{\σ}}_e\right){\mathrm{\ω}}_e\\ J{\stackrel{\·}{\mathrm{\ω}}}_e=-{\left(\mathrm{\ω}\right)}^{\×}J\left(\mathrm{\ω}\right)-{\mathrm{J\ω}}_d+T+T_d\end{array}\right.---\left(7\right)$

wherein σ_eAs attitude error, ω_eFor angular velocity error, the expressions for both are:

$\left\{\begin{array}{c}{\mathrm{\σ}}_e=\mathrm{\σ}\⊗{\mathrm{\σ}}_d^{-1}=\frac{(1-{\mathrm{\σ}}_d^T{\mathrm{\σ}}_d)\mathrm{\σ}+({\mathrm{\σ}}^T\mathrm{\σ}-1){\mathrm{\σ}}_d-2{\mathrm{\σ}}_d\×\mathrm{\σ}}{1+\left({\mathrm{\σ}}_d^T{\mathrm{\σ}}_d\right)\left({\mathrm{\σ}}^T\mathrm{\σ}\right)+2{\mathrm{\σ}}_d^T\mathrm{\σ}}\\ {\mathrm{\ω}}_e=\mathrm{\ω}-{\mathrm{\ω}}_d\end{array}\right.---\left(8\right)$

in the formula (8), the reaction mixture is,representing MRP multiplication.

In the step 2), the method for designing the attitude adaptive take-over control law of the non-cooperative target satellite comprises the following steps:

first, an auxiliary error variable is defined: s- ω_e+ασ_eα is greater than or equal to 0, then

$\begin{array}{c}J\stackrel{\·}{s}=J{\stackrel{\·}{\mathrm{\ω}}}_e+J\mathrm{\α}{\stackrel{\·}{\mathrm{\σ}}}_e\\ =-{\left(\mathrm{\ω}\right)}^{\×}J\left(\mathrm{\ω}\right)-{\mathrm{J\ω}}_d+T+T_d+\mathrm{\α}JG\left({\mathrm{\σ}}_e\right){\mathrm{\ω}}_e\\ =T+L\end{array}---\left(9\right)$

Wherein L ═ - (ω) × J (ω) - [ J + α JG (σ)_e)]ω_d+T_d+αJG(σ_e) Omega; the euclidean norm of the vector is expressed by | · |, which is analyzed:

due to the fact thatω_dIs bounded; setting external disturbance T_dThe Euclidean norm of (c) satisfies | | T_d||≤c_d0+c_d1||ω||²，c_d0And c_d1Are both unknown and non-negative constants, then:

||L||≤b₀+b₁||ω||+b₂||ω||²(10)

wherein, b₀、b₁And b₂Are all unknown and non-negative constants;

then, on the basis, an attitude adaptive control law is designed:

$T=-k_1{\mathrm{\α\σ}}_e-k_2\frac{\mathrm{sgn}\left(s\right)}{\left|\right|s\left|\right|}-({\hat{b}}_0+{\hat{b}}_1|\left|\mathrm{\ω}\right||+{\hat{b}}_2|\left|\mathrm{\ω}\right|{|}^2)\frac{s}{\left|\right|s\left|\right|}---\left(11\right)$

wherein k is₁And k₂For the normal number of the design, sgn (-) is a sign function,andare respectively the parameter b₀、b₁And b₂The online updating law of the estimated value of (c) is as follows:

$\left\{\begin{array}{c}{\stackrel{\·}{\hat{b}}}_0=\frac{\left|\right|s\left|\right|}{c_0}\\ {\stackrel{\·}{\hat{b}}}_1=\frac{\left|\right|s\left|\right|\left|\right|\mathrm{\ω}\left|\right|}{c_1}\\ {\stackrel{\·}{\hat{b}}}_2=\frac{\left|\right|s\left|\right|\left|\right|\mathrm{\ω}|{|}^2}{c_2}\end{array}\right.---\left(12\right)$

c₀、c₁and c₂Is a designed normal number;

finally, stability verification was performed:

selecting:

$V=\frac{1}{2}{s}^{T}Js+\frac{c_0}{2}{\tilde{b}}_0^2+\frac{c_1}{2}{\tilde{b}}_1^2+\frac{c_2}{2}{\tilde{b}}_2^2---\left(13\right)$

wherein,

and (3) obtaining the derivatives of two sides of the formula (13):

$\stackrel{\·}{V}=s^{T}J\stackrel{\·}{s}+c_0{\tilde{b}}_0{\stackrel{\·}{\hat{b}}}_0+c_1{\tilde{b}}_1{\stackrel{\·}{\hat{b}}}_1+c_2{\tilde{b}}_2{\stackrel{\·}{\hat{b}}}_2---\left(14\right)$

substituting the formulas (9) to (12) into the above formula, and simplifying to obtain:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}=s^{T}J\stackrel{\&CenterDot;}{s}-c_0{\tilde{b}}_0{\stackrel{\&CenterDot;}{\hat{b}}}_0-c_1{\tilde{b}}_1{\stackrel{\&CenterDot;}{\hat{b}}}_1-c_2{\tilde{b}}_2{\stackrel{\&CenterDot;}{\hat{b}}}_2\\ =s^T\&lsqb;-k_1{\mathrm{\&alpha;\&sigma;}}_e-k_2\frac{\mathrm{sgn}\left(s\right)}{\left|\right|s\left|\right|}-({\hat{b}}_0+{\hat{b}}_1|\left|\mathrm{\&omega;}\right||+{\hat{b}}_2|\left|\mathrm{\&omega;}\right|{|}^2)\frac{s}{\left|\right|s\left|\right|}+L\&rsqb;-c_0{\tilde{b}}_0{\stackrel{\&CenterDot;}{\hat{b}}}_0-c_1{\tilde{b}}_1{\stackrel{\&CenterDot;}{\hat{b}}}_1-c_2{\tilde{b}}_2{\stackrel{\&CenterDot;}{\hat{b}}}_2\\ \&le;-k_1\mathrm{\&alpha;}\left|\right|s\left|\right|\left|\right|{\mathrm{\&sigma;}}_e\left|\right|-k_2\\ \&le;-k_2<0\end{array}$

therefore, the system is uniformly and asymptotically stabilized under the control of the control law (11) and the parameter adaptive law (12).

In the step 3), a specific method for taking over robust distribution of the control torque comprises the following steps:

because control moment is realized by thrust and tether tension jointly, and thrust and tension are strict limited, then:

$\left\{\begin{array}{c}0N\&le;F_{1x},F_{2x},F_{6x},F_{7x}\&le;a_{1}N\\ 0N\&le;F_{3y},F_{8y}\&le;a_{2}N\\ 0N\&le;F_{4z},F_{5z},F_{9z},F_{10z}\&le;a_{3}N\\ 0N\&le;F_{11z},F_{12z}\&le;a_{4}N\\ 0N\&le;F_{13x}\&le;a_{5}N\end{array}\right.---\left(15\right)$

let a be ═ a₁a₁a₂a₃a₃a₁a₁a₂a₃a₃a₄a₄a₅]0 is a zero matrix of 13 × 1, and formula (15) is represented by F ≦ a of 0 ≦ 0;

distributing the control torque T calculated in the step 2) to real control execution quantity, namely the thrust of 12 thrusters and the tether tension by using a robust distribution method, wherein the method comprises the following specific steps:

3-1) the control distribution problem is transformed into the following robust optimization problem with the minimum fuel consumption as an objective function.

An objective function: min ([ 1111111111110)]F)＝min(W^TF)；

And (3) constraint: f is more than or equal to 0 and less than or equal to a;

order toConverting equality constraint into inequality constraint;

and (3) constraint: HF is more than or equal to N, F is more than or equal to 0 and less than or equal to a;

using robust optimization theory, the optimization problem is rewritten as:

$\left\{\begin{array}{c}\min \left(W^{T}F\right)\\ s.t.h_{i}F\&GreaterEqual;n_i\\ 0\&le;F\&le;a\end{array}\right.,\&ForAll;h_i\&Element;{\mathrm{\&Xi;}}_i,\&ForAll;i=1...6---\left(16\right)$

wherein h is_iIs the ith row of the matrix H containing uncertainty, and is at the uncertainty XI_iTaking a middle value; uncertain centre xi_iCan be described by an ellipsoidal uncertainty, i.e.:

${\mathrm{\&Xi;}}_i=\left\{h_i\right|{h_i}^T={\stackrel{\&OverBar;}{h}}_i^T+{\mathrm{\&Theta;}}_i{u}_i,\left|\right|u_i\left|\right|\&le;\mathrm{\&rho;}\},i=1,...6---\left(17\right)$

indicating the nominal value, theta, of each line as measured or identified_iFor symmetrical positive or semi-positive definite matrices, u, associated with the distribution of uncertainty_iIs the column vector associated with uncertainty, ρ is the upper bound of the euclidean norm of uncertainty;

3-2) utilizing the characteristics of ellipsoid uncertainty, simplifying formula (16) by utilizing formula (17) and utilizing

min(x^TΘu_i)＝-ρ||Θx||

Converting the robust optimization problem into a cone quadratic optimization problem:

$\left\{\begin{array}{c}min\left(W^{T}F\right)\\ s.t.\stackrel{\&OverBar;}{h_i}F-\mathrm{\&rho;}\left|\right|{\mathrm{\&Theta;}}_{i}F\left|\right|\&GreaterEqual;n_i\\ 0\&le;F\&le;a\end{array}\right.,\&ForAll;i=1...6---\left(18\right)$

and finally, solving the cone optimization problem by using an interior point method.

Compared with the prior art, the invention has the following beneficial effects:

the invention adopts the tether thruster based on the space tether robot to carry out the attitude joint takeover control method for the space non-cooperative target satellite, can fully utilize the tether tension and save the consumption of chemical propellant in the takeover control process. In the research of combining a tether and a thruster at home and abroad on target attitude control, the target inertia is regarded as known, and the problems of control quantity distribution, control quantity constraint and the like are not considered, so that the application range and the reliability of the device are greatly reduced. By adopting the robust distribution method of the self-adaptive attitude control and the moment, the characteristics of unknown inertia, unknown mass center, unknown capture point of an operating mechanism and the like of a non-cooperative target, the limited state quantity and the controlled quantity are fully considered, so that the system can obtain unknown parameters on line in real time, the influence of the limited state quantity and the controlled quantity on the system is avoided, and the actual application range and the reliability of the system are greatly improved.

[ description of the drawings ]

FIG. 1 is a schematic diagram of spatial tethered robot target capture;

FIG. 2 is a distribution diagram of the thruster;

FIG. 3 is a flow chart of a spatial non-cooperative target attitude joint takeover control using a tether/thruster.

Wherein, 1-space non-cooperative target star; 2-an operating mechanism of the spatial tether robot; 3-a tether of the spatial tether robot; 4-group 1 operating mechanism; 5-group 2 operating mechanisms; 6-group 3 operating mechanisms; 7-group 4 operating mechanisms.

[ detailed description ] embodiments

The invention is described in further detail below with reference to the accompanying drawings:

referring to fig. 1-3, the spatial non-cooperative target attitude joint takeover control method using the tether thruster of the present invention includes the following steps:

step 1: establishing space non-cooperative target satellite attitude take-over control model

As shown in fig. 1, 1 is a space non-cooperative target star, 2 is an operating mechanism of a space tethered robot, and 3 is a tether of the space tethered robot. O is_TIs the centroid of a space non-cooperative target star, O_TX_TY_TZ_TIs the target star system, O_GIs the center of mass of the operating mechanism, O_GX_GY_GZ_GFor operating the mechanism body system, O_CTo tie rope and exerciseAs the attachment point of the mechanism. To simplify the modeling process, it is assumed that the coordinate axes of the two coordinate systems are parallel to each other. Setting the center of mass O of the operating mechanism_GIn the target star body system O_TX_TY_TZ_TThe lower coordinate is X_G＝[x_G,y_G,z_G]。

As shown in fig. 2, the 1 st group operating mechanism 4, the 2 nd group operating mechanism 5, the 3 rd group operating mechanism 6 and the 4 th group operating mechanism 7 are twelve thrusters and are installed in a cross shape. The 1 st group operating mechanism 4 and the 2 nd group operating mechanism 5 include five thrusters orthogonally attached, and the 3 rd group operating mechanism 6 and the 4 th group operating mechanism 7 are each a single thruster.

If the thrust range of each thruster is [0a ] N, the thrust generated by the twelve thrusters in the operating mechanism body system and the action positions of the twelve thrusters in the operating mechanism body system are as follows:

a first group:

second group:

third and fourth groups

Since the tether of the space tether robot can reach hundreds of meters, and the direction change of the tether caused by the attitude motion of the target star is small in the control of the take-over, in order to simplify the modeling process, the direction of the tether can be assumed to be unchanged and is along the system-x direction of the operating mechanism. Therefore, the maximum pulling force of the tying rope is a₅N, the tension and the action point of the tether are measured under the operating mechanism body systemShown as follows:

	tension force	Point of action	Tension restraint
				Tether rope	F13＝[-F13x,0,0]	X5＝[x5,y5,z5]	0N≤F13x≤a5N

Because the operation object is a non-cooperative target, the measuring device such as a gyroscope, the actuator such as a thruster/tether and the like are all installed on the operation mechanism of the space tether robot, and therefore, under the system of the operation mechanism, the attitude dynamics equation establishing the space non-cooperative target is as follows:

$J\stackrel{\&CenterDot;}{\mathrm{\&omega;}}+{\mathrm{\&omega;}}^{\&times;}J\mathrm{\&omega;}=T+T_d---\left(1\right)$

wherein J is a target star moment of inertia matrix, omega is the angular velocity of the target star,^×as a cross product, T_dFor disturbing torque, T ═ T_c+T_tTo control the torque. T is_cControl moment for thruster

$T_c=\underset{i=1}{\overset{5}{\mathrm{\&Sigma;}}}(X_G+X_1)\&times;F_i+\underset{i=6}{\overset{10}{\mathrm{\&Sigma;}}}(X_G+X_2)\&times;F_i+(X_G+X_3)\&times;F_{11}+(X_G+X_4)\&times;F_{12}---\left(2\right)$

T_tControl moment generated for tether

T_t＝(X_G+X₅)×F₁₃(3)

The control torque T can be simplified to:

$T=T_c+T_t=DF={\left[\begin{array}{ccc}0& (z_G+z_1)& -(y_G+y_1)\\ 0& -(z_G+z_1)& (y_G+y_1)\\ (z_G+z_1)& 0& -(x_G+x_1)\\ (y_G+y_1)& -(x_G+x_1)& 0\\ -(y_G+y_1)& (x_G+x_1)& 0\\ 0& (z_G+z_2)& -(y_G+y_2)\\ 0& -(z_G+z_2)& (y_G+y_2)\\ -(z_G+z_2)& 0& (x_G+x_2)\\ (y_G+y_2)& -(x_G+x_2)& 0\\ -(y_G+y_2)& (x_G+x_2)& 0\\ -(y_G+y_3)& (x_G+x_3)& 0\\ (y_G+y_3)& -(x_G+x_3)& 0\\ 0& -(z_G+z_5)& (y_G+y_5)\end{array}\right]}^T\left[\begin{array}{c}{F}_{1x}\\ F_{2x}\\ F_{3y}\\ F_{4z}\\ F_{5z}\\ F_{6x}\\ F_{7x}\\ F_{8y}\\ F_{9z}\\ F_{10z}\\ F_{11z}\\ F_{12z}\\ F_{13x}\end{array}\right]---\left(4\right)$

wherein D is a control quantity distribution matrix, and F is a column vector formed by actuators.

The kinematic equation of the attitude of the target star described by the modified Rodrigues parameter is as follows:

$\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}=G\left(\mathrm{\&sigma;}\right)\mathrm{\&omega;}---\left(5\right)$

$G\left(\mathrm{\&sigma;}\right)=\frac{1}{4}\&lsqb;(1-{\mathrm{\&sigma;}}^T\mathrm{\&sigma;})I_3+2{\mathrm{\&sigma;}}^{\&times;}+2{\mathrm{\&sigma;\&sigma;}}^T\&rsqb;---\left(6\right)$

wherein σ is the attitude-corrected Rodrigues parameter of the target star, I₃Is an identity matrix of 3 × 3.

Let the expected attitude of the non-cooperative target star be σ_dDesired angular velocity is ω_dThen, the target star attitude error dynamics/kinematics equation is:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_e=G\left({\mathrm{\&sigma;}}_e\right){\mathrm{\&omega;}}_e\\ J{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_e=-{\left(\mathrm{\&omega;}\right)}^{\&times;}J\left(\mathrm{\&omega;}\right)-{\mathrm{J\&omega;}}_d+T+T_d\end{array}\right.---\left(7\right)$

wherein σ_eAs attitude error, ω_eFor angular velocity error, the expressions for both are:

$\left\{\begin{array}{c}{\mathrm{\&sigma;}}_e=\mathrm{\&sigma;}\&CircleTimes;{\mathrm{\&sigma;}}_d^{-1}=\frac{(1-{\mathrm{\&sigma;}}_d^T{\mathrm{\&sigma;}}_d)\mathrm{\&sigma;}+({\mathrm{\&sigma;}}^T\mathrm{\&sigma;}-1){\mathrm{\&sigma;}}_d-2{\mathrm{\&sigma;}}_d\&times;\mathrm{\&sigma;}}{1+\left({\mathrm{\&sigma;}}_d^T{\mathrm{\&sigma;}}_d\right)\left({\mathrm{\&sigma;}}^T\mathrm{\&sigma;}\right)+2{\mathrm{\&sigma;}}_d^T\mathrm{\&sigma;}}\\ {\mathrm{\&omega;}}_e=\mathrm{\&omega;}-{\mathrm{\&omega;}}_d\end{array}\right.---\left(8\right)$

in the formula,representing MRP multiplication.

Step 2: attitude adaptive take-over control law for designing non-cooperative target satellites

Because the thrust moment and the tether tension moment form a redundant control system, and each thrust and tether tension are constrained quantities, designing the controller by directly taking the thrust and the tension as control quantities is very complicated, and therefore, a method of separately designing a posture take-over control law and a control distribution law is adopted.

In the design of the attitude takeover control law, since the rotational inertia of a non-cooperative target is unknown, a self-adaptive attitude takeover control law needs to be designed.

First, an auxiliary error variable is defined: s- ω_e+ασ_e. Then

$\begin{array}{c}J\stackrel{\&CenterDot;}{s}=J{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_e+J\mathrm{\&alpha;}{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_e\\ =-{\left(\mathrm{\&omega;}\right)}^{\&times;}J\left(\mathrm{\&omega;}\right)-{\mathrm{J\&omega;}}_d+T+T_d+\mathrm{\&alpha;}JG\left({\mathrm{\&sigma;}}_e\right){\mathrm{\&omega;}}_e\\ =T+L\end{array}---\left(9\right)$

Wherein L ═ - (ω) × J (ω) - [ J + α JG (σ)_e)]ω_d+T_d+αJG(σ_e) ω. The euclidean norm of the vector is expressed as | · |, which is analyzed below.

Due to the fact thatω_dIs bounded. Setting external disturbance T_dThe Euclidean norm of (c) satisfies | | T_d||≤c_d0+c_d1||ω||²，c_d0And c_d1Are all unknown and non-negative constants. Then:

||L||≤b₀+b₁||ω||+b₂||ω||²(10)

wherein, b₀、b₁And b₂Are all unknown and non-negative constants.

Then, on the basis, an attitude adaptive control law is designed:

$T=-k_1{\mathrm{\&alpha;\&sigma;}}_e-k_2\frac{\mathrm{sgn}\left(s\right)}{\left|\right|s\left|\right|}-({\hat{b}}_0+{\hat{b}}_1|\left|\mathrm{\&omega;}\right||+{\hat{b}}_2|\left|\mathrm{\&omega;}\right|{|}^2)\frac{s}{\left|\right|s\left|\right|}---\left(11\right)$

wherein k is₁，k₂For the normal number of the design, sgn (-) is a sign function,andare respectively the parameter b₀、b₁And b₂The online updating law of the estimated value of (c) is as follows:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\hat{b}}}_0=\frac{\left|\right|s\left|\right|}{c_0}\\ {\stackrel{\&CenterDot;}{\hat{b}}}_1=\frac{\left|\right|s\left|\right|\left|\right|\mathrm{\&omega;}\left|\right|}{c_1}\\ {\stackrel{\&CenterDot;}{\hat{b}}}_2=\frac{\left|\right|s\left|\right|\left|\right|\mathrm{\&omega;}|{|}^2}{c_2}\end{array}\right.---\left(12\right)$

c₀，c₁and c₂Is a designed normal number.

Finally, stability verification was performed:

selecting:

$V=\frac{1}{2}{s}^{T}Js+\frac{c_0}{2}{\tilde{b}}_0^2+\frac{c_1}{2}{\tilde{b}}_1^2+\frac{c_2}{2}{\tilde{b}}_2^2---\left(13\right)$

wherein,

and (3) obtaining the derivatives of two sides of the formula (13):

$\stackrel{\&CenterDot;}{V}=s^{T}J\stackrel{\&CenterDot;}{s}+c_0{\tilde{b}}_0{\stackrel{\&CenterDot;}{\hat{b}}}_0+c_1{\tilde{b}}_1{\stackrel{\&CenterDot;}{\hat{b}}}_1+c_2{\tilde{b}}_2{\stackrel{\&CenterDot;}{\hat{b}}}_2---\left(14\right)$

substituting the formulas (9) to (12) into the above formula, and simplifying to obtain:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}=s^{T}J\stackrel{\&CenterDot;}{s}-c_0{\tilde{b}}_0{\stackrel{\&CenterDot;}{\hat{b}}}_0-c_1{\tilde{b}}_1{\stackrel{\&CenterDot;}{\hat{b}}}_1-c_2{\tilde{b}}_2{\stackrel{\&CenterDot;}{\hat{b}}}_2\\ =s^T\&lsqb;-k_1{\mathrm{\&alpha;\&sigma;}}_e-k_2\frac{\mathrm{sgn}\left(s\right)}{\left|\right|s\left|\right|}-({\hat{b}}_0+{\hat{b}}_1|\left|\mathrm{\&omega;}\right||+{\hat{b}}_2|\left|\mathrm{\&omega;}\right|{|}^2)\frac{s}{\left|\right|s\left|\right|}+L\&rsqb;-c_0{\tilde{b}}_0{\stackrel{\&CenterDot;}{\hat{b}}}_0-c_1{\tilde{b}}_1{\stackrel{\&CenterDot;}{\hat{b}}}_1-c_2{\tilde{b}}_2{\stackrel{\&CenterDot;}{\hat{b}}}_2\\ \&le;-k_1\mathrm{\&alpha;}\left|\right|s\left|\right|\left|\right|{\mathrm{\&sigma;}}_e\left|\right|-k_2\\ \&le;-k_2<0\end{array}$

therefore, the system is uniformly and asymptotically stabilized under the control of the control law (11) and the parameter adaptive law (12).

And step 3: robust distribution of control torque of connecting pipe

The control moment is realized by the thrust and the tether tension together, and the thrust and the tension are strictly limited.

$\left\{\begin{array}{c}0N\&le;F_{1x},F_{2x},F_{6x},F_{7x}\&le;a_{1}N\\ 0N\&le;F_{3y},F_{8y}\&le;a_{2}N\\ 0N\&le;F_{4z},F_{5z},F_{9z},F_{10z}\&le;a_{3}N\\ 0N\&le;F_{11z},F_{12z}\&le;a_{4}N\\ 0N\&le;F_{13x}\&le;a_{5}N\end{array}\right.---\left(15\right)$

Let a be ═ a₁a₁a₂a₃a₃a₁a₁a₂a₃a₃a₄a₄a₅]0 is a zero matrix of 13 × 1, and the above formula can be expressed as 0 ≦ F ≦ a.

In addition, because of the operating mechanism center of mass O_GThe coordinates under the target satellite system are uncertain, so that the coordinates of the thrust and tether tension action point under the target satellite system are uncertain, therefore, the control moment T calculated in the step 2 is distributed to the real control execution quantity by using a robust distribution method in the step: thrust of the twelve thrusters and tether tension.

First, the control distribution problem is transformed into the following robust optimization problem with the minimum fuel consumption as an objective function.

An objective function: min ([ 1111111111110)]F)＝min(W^TF)；

And (3) constraint: and F is more than or equal to 0 and less than or equal to a.

Order toThe equality constraint is converted into an inequality constraint,

and (3) constraint: HF is more than or equal to N, and F is more than or equal to 0 and less than or equal to a.

Using robust optimization theory, the optimization problem is rewritten as:

$\left\{\begin{array}{c}\min \left(W^{T}F\right)\\ s.t.h_{i}F\&GreaterEqual;n_i\\ 0\&le;F\&le;a\end{array}\right.,\&ForAll;h_i\&Element;{\mathrm{\&Xi;}}_i,\&ForAll;i=1...6---\left(16\right)$

wherein h is_iIs the ith row of the matrix H containing uncertainty, and is at the uncertainty XI_iTaking the value in the step (1). Uncertain centre xi_iCan be described by an ellipsoidal uncertainty, i.e.:

${\mathrm{\&Xi;}}_i=\left\{h_i\right|{h_i}^T={\stackrel{\&OverBar;}{h}}_i^T+{\mathrm{\&Theta;}}_i{u}_i,\left|\right|u_i\left|\right|\&le;\mathrm{\&rho;}\},i=1,...6---\left(17\right)$

indicating the nominal value, theta, of each line as measured or identified_iFor symmetrical positive or semi-positive definite matrices, u, associated with the distribution of uncertainty_iFor the column vector associated with uncertainty, ρ is the upper bound of the euclidean norm of uncertainty.

Then, by using the characteristics of the uncertainty of the ellipsoid, simplified formula (16) is converted by using formula (17), and

min(x^TΘu_i)＝-ρ||Θx||

and converting the robust optimization problem into a cone quadratic optimization problem.

$\left\{\begin{array}{c}min\left(W^{T}F\right)\\ s.t.\stackrel{\&OverBar;}{h_i}F-\mathrm{\&rho;}\left|\right|{\mathrm{\&Theta;}}_{i}F\left|\right|\&GreaterEqual;n_i\\ 0\&le;F\&le;a\end{array}\right.,\&ForAll;i=1...6---\left(18\right)$

And finally, solving the cone optimization problem by using an interior point method.

The detailed steps of the interior point method are described in detail in DAVID G, LUENBERGER, YINYU YE.Linear and nonlinear Programming [ M ]. Third edition.Berlin Springer Verlag,2008: 111-.

The above-mentioned contents are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modification made on the basis of the technical idea of the present invention falls within the protection scope of the claims of the present invention.

Claims (4)

1. The space non-cooperative target posture combined takeover control method by using the tether thruster is characterized by comprising the following steps of:

1) establishing a spatial non-cooperative target satellite attitude takeover control model;

2) designing a posture self-adaptive take-over control law of a non-cooperative target satellite;

3) takes over the robust distribution of the control torque.

2. The space non-cooperative target attitude joint takeover control method using the tether thruster as claimed in claim 1, wherein in the step 1), a specific method for establishing a space non-cooperative target star attitude takeover control model is as follows:

with O_TEstablishing a target satellite body system O for the mass center of a spatial non-cooperative target satellite_TX_TY_TZ_TWith O_GEstablishing an operating mechanism body system O for the center of mass of the operating mechanism_GX_GY_GZ_G，O_CIs the connection point of the tether and the operating mechanism; in order to simplify the modeling process, the mass center O of the operating mechanism is set on the assumption that the coordinate axes of the two coordinate systems are parallel to each other_GIn the target star body system O_TX_TY_TZ_TThe lower coordinate is X_G＝[x_G,y_G,z_G]；

Four groups of operating mechanisms, wherein 12 thrusters are arranged in a cross shape; wherein, the 1 st group of operating mechanisms (4) and the 2 nd group of operating mechanisms (5) comprise 5 thrusters which are orthogonally arranged, and the 3 rd group of operating mechanisms (6) and the 4 th group of operating mechanisms (7) are respectively 1 thruster;

assuming that the thrust range of each thruster is [0a ] N, the thrust generated by the 12 thrusters in the operating mechanism body system and the action positions of the operating mechanism body system are as follows:

a first group:

second group:

third and fourth groups:

to simplify the modeling process, assume that its orientation is unchanged and is along the operating mechanism body system-x direction; therefore, the maximum pulling force of the tying rope is a₅N, the rope tension and the action point are expressed as follows under the system in the operating mechanism:

because the operation object is a non-cooperative target, and the measuring device and the executing mechanism are both arranged on the operation mechanism of the space tether robot, under the system in the operation mechanism, the attitude dynamics equation of the space non-cooperative target is established as follows:

$J\stackrel{\&CenterDot;}{\mathrm{\&omega;}}+{\mathrm{\&omega;}}^{\&times;}J\mathrm{\&omega;}=T+T_d---\left(1\right)$

wherein J is a target star moment of inertia matrix, omega is an angular velocity of the target star, × is a cross product, T_dFor disturbing torque, T ═ T_c+T_tFor controlling torque, T_cControl torque generated for the thruster:

$T_c=\underset{i=1}{\overset{5}{\&Sigma;}}(X_G+X_1)\&times;F_i+\underset{i=6}{\overset{10}{\&Sigma;}}(X_G+X_2)\&times;F_i+(X_G+X_3)\&times;F_{11}+(X_G+X_4)\&times;F_{12}---\left(2\right)$

F_ithe thrust corresponding to the ith thruster, i is the number of the thruster, T_tControl moment generated for the tether:

T_t＝(X_G+X₅)×F₁₃(3)

the control torque T is simplified to:

$T=T_c+T_t=DF={\left[\begin{array}{ccc}0& \left(z_G+z_1\right)& -\left(y_G+y_1\right)\\ 0& -\left(z_G+z_1\right)& \left(y_G+y_1\right)\\ \left(z_G+z_1\right)& 0& -\left(x_G+x_1\right)\\ \left(y_G+y_1\right)& -\left(x_G+x_1\right)& 0\\ -\left(y_G+y_1\right)& \left(x_G+x_1\right)& 0\\ 0& \left(z_G+z_2\right)& -\left(y_G+y_2\right)\\ 0& -\left(z_G+z_2\right)& \left(y_G+y_2\right)\\ -\left(z_G+z_2\right)& 0& \left(x_G+x_2\right)\\ \left(y_G+y_2\right)& -\left(x_G+x_2\right)& 0\\ -\left(y_G+y_2\right)& \left(x_G+x_2\right)& 0\\ -\left(y_G+y_3\right)& \left(x_G+x_3\right)& 0\\ \left(y_G+y_3\right)& -\left(x_G+x_3\right)& 0\\ 0& -\left(z_G+z_5\right)& \left(y_G+y_5\right)\end{array}\right]}^T\left[\begin{array}{c}{F}_{1x}\\ F_{2x}\\ F_{3y}\\ F_{4z}\\ F_{5z}\\ F_{6x}\\ F_{7x}\\ F_{8y}\\ F_{9z}\\ F_{10z}\\ F_{11z}\\ F_{12z}\\ F_{13x}\end{array}\right]---\left(4\right)$

wherein D is a control quantity distribution matrix, and F is a column vector formed by actuators;

the kinematic equation of the attitude of the target star described by the modified Rodrigues parameter is as follows:

$\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}=G\left(\mathrm{\&sigma;}\right)\mathrm{\&omega;}---\left(5\right)$

$G\left(\mathrm{\&sigma;}\right)=\frac{1}{4}\&lsqb;(1-{\mathrm{\&sigma;}}^T\mathrm{\&sigma;})I_3+2{\mathrm{\&sigma;}}^{\&times;}+2{\mathrm{\&sigma;\&sigma;}}^T\&rsqb;---\left(6\right)$

wherein σ is the attitude-corrected Rodrigues parameter of the target star, I₃An identity matrix of 3 × 3;

let the expected attitude of the non-cooperative target star be σ_dDesired angleSpeed of omega_dThen the target star attitude error dynamics/kinematics equation is:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_e=G\left({\mathrm{\&sigma;}}_e\right){\mathrm{\&omega;}}_e\\ J{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_e=-{\left(\mathrm{\&omega;}\right)}^{\&times;}J\left(\mathrm{\&omega;}\right)-{\mathrm{J\&omega;}}_d+T+T_d\end{array}\right.---\left(7\right)$

wherein σ_eAs attitude error, ω_eFor angular velocity error, the expressions for both are:

$\left\{\begin{array}{c}{\mathrm{\&sigma;}}_e=\mathrm{\&sigma;}\&CircleTimes;{\mathrm{\&sigma;}}_d^{-1}=\frac{(1-{\mathrm{\&sigma;}}_d^T{\mathrm{\&sigma;}}_d)\mathrm{\&sigma;}+({\mathrm{\&sigma;}}^T\mathrm{\&sigma;}-1){\mathrm{\&sigma;}}_d-2{\mathrm{\&sigma;}}_d\&times;\mathrm{\&sigma;}}{1+\left({\mathrm{\&sigma;}}_d^T{\mathrm{\&sigma;}}_d\right)\left({\mathrm{\&sigma;}}^T\mathrm{\&sigma;}\right)+2{\mathrm{\&sigma;}}_d^T\mathrm{\&sigma;}}\\ {\mathrm{\&omega;}}_e=\mathrm{\&omega;}-{\mathrm{\&omega;}}_d\end{array}\right.---\left(8\right)$

in the formula (8), the reaction mixture is,representing MRP multiplication.

3. The spatial non-cooperative target attitude joint takeover control method using the tether thruster as claimed in claim 1, wherein in the step 2), the method for designing the attitude adaptive takeover control law of the non-cooperative target star comprises the following steps:

first, an auxiliary error variable is defined: s- ω_e+ασ_eα is greater than or equal to 0, then

$\begin{array}{c}J\stackrel{\&CenterDot;}{s}=J{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_e+J\mathrm{\&alpha;}{\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}}_e\\ =-{\left(\mathrm{\&omega;}\right)}^{\&times;}J\left(\mathrm{\&omega;}\right)-{\mathrm{J\&omega;}}_d+T+T_d+\mathrm{\&alpha;}JG\left({\mathrm{\&sigma;}}_e\right){\mathrm{\&omega;}}_e\\ =T+L\end{array}---\left(9\right)$

Wherein, L ═ - (ω)^×J(ω)-[J+αJG(σ_e)]ω_d+T_d+αJG(σ_e) Omega; the euclidean norm of the vector is expressed by | · |, which is analyzed:

due to the fact thatω_dIs bounded; setting external disturbance T_dThe Euclidean norm of (c) satisfies | | T_d||≤c_d0+c_d1||ω||²，c_d0And c_d1Are both unknown and non-negative constants, then:

||L||≤b₀+b₁||ω||+b₂||ω||²(10)

wherein, b₀、b₁And b₂Are all unknown and non-negative constants;

then, on the basis, an attitude adaptive control law is designed:

$T=-k_1{\mathrm{\&alpha;\&sigma;}}_e-k_2\frac{\mathrm{sgn}\left(s\right)}{\left|\right|s\left|\right|}-({\hat{b}}_0+{\hat{b}}_1|\left|\mathrm{\&omega;}\right||+{\hat{b}}_2|\left|\mathrm{\&omega;}\right|{|}^2)\frac{s}{\left|\right|s\left|\right|}---\left(11\right)$

wherein k is₁And k₂For the normal number of the design, sgn (-) is a sign function,andare respectively the parameter b₀、b₁And b₂The online updating law of the estimated value of (c) is as follows:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\hat{b}}}_0=\frac{\left|\right|s\left|\right|}{c_0}\\ {\stackrel{\&CenterDot;}{\hat{b}}}_1=\frac{\left|\right|s\left|\right|\left|\right|\mathrm{\&omega;}\left|\right|}{c_1}\\ {\stackrel{\&CenterDot;}{\hat{b}}}_2=\frac{\left|\right|s\left|\right|\left|\right|\mathrm{\&omega;}|{|}^2}{c_2}\end{array}\right.---\left(12\right)$

c₀、c₁and c₂Is a designed normal number;

finally, stability verification was performed:

selecting:

$V=\frac{1}{2}{s}^{T}Js+\frac{c_0}{2}{\tilde{b}}_0^2+\frac{c_1}{2}{\tilde{b}}_1^2+\frac{c_2}{2}{\tilde{b}}_2^2---\left(13\right)$

wherein,

and (3) obtaining the derivatives of two sides of the formula (13):

$\stackrel{\&CenterDot;}{V}=s^{T}J\stackrel{\&CenterDot;}{s}+c_0{\tilde{b}}_0{\stackrel{\&CenterDot;}{\hat{b}}}_0+c_1{\tilde{b}}_1{\stackrel{\&CenterDot;}{\hat{b}}}_1+c_2{\tilde{b}}_2{\stackrel{\&CenterDot;}{\hat{b}}}_2---\left(14\right)$

substituting the formulas (9) to (12) into the above formula, and simplifying to obtain:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}=s^{T}J\stackrel{\&CenterDot;}{s}-c_0{\tilde{b}}_0{\stackrel{\&CenterDot;}{\hat{b}}}_0-c_1{\tilde{b}}_1{\stackrel{\&CenterDot;}{\hat{b}}}_1-c_2{\tilde{b}}_2{\stackrel{\&CenterDot;}{\hat{b}}}_2\\ =s^T\&lsqb;-k_1{\mathrm{\&alpha;\&sigma;}}_e-k_2\frac{\mathrm{sgn}\left(s\right)}{\left|\right|s\left|\right|}-\left({\hat{b}}_0+{\hat{b}}_1\left|\right|\mathrm{\&omega;}\left|\right|+{\hat{b}}_2\left|\right|\mathrm{\&omega;}|{|}^2\right)\frac{s}{\left|\right|s\left|\right|}+L\&rsqb;-c_0{\tilde{b}}_0{\stackrel{\&CenterDot;}{\hat{b}}}_0-c_1{\tilde{b}}_1{\stackrel{\&CenterDot;}{\hat{b}}}_1-c_2{\tilde{b}}_2{\stackrel{\&CenterDot;}{\hat{b}}}_2\\ \&le;-k_1\mathrm{\&alpha;}\left|\right|s\left|\right|\left|\right|{\mathrm{\&sigma;}}_e\left|\right|-k_2\\ \&le;-k_2<0\end{array}$

therefore, the system is uniformly and asymptotically stabilized under the control of the control law (11) and the parameter adaptive law (12).

4. The spatial non-cooperative target attitude joint takeover control method using the tether thruster as recited in claim 1, wherein in the step 3), a specific method for taking over robust distribution of the control torque is as follows:

because control moment is realized by thrust and tether tension jointly, and thrust and tension are strict limited, then:

$\left\{\begin{array}{c}0N\&le;F_{1x},F_{2x},F_{6x},F_{7x}\&le;a_{1}N\\ 0N\&le;F_{3y},F_{8y}\&le;a_{2}N\\ 0N\&le;F_{4z},F_{5z},F_{9z},F_{10z}\&le;a_{3}N\\ 0N\&le;F_{11z},F_{12z}\&le;a_{4}N\\ 0N\&le;F_{13x}\&le;a_{5}N\end{array}\right.---\left(15\right)$

let a be ═ a₁a₁a₂a₃a₃a₁a₁a₂a₃a₃a₄a₄a₅]0 is a zero matrix of 13 × 1, and formula (15) is represented by F ≦ a of 0 ≦ 0;

distributing the control torque T calculated in the step 2) to real control execution quantity, namely the thrust of 12 thrusters and the tether tension by using a robust distribution method, wherein the method comprises the following specific steps:

3-1) converting the control distribution problem into the following robust optimization problem by taking the minimum fuel consumption as an objective function;

an objective function: min ([ 1111111111110)]F)＝min(W^TF)；

And (3) constraint: f is more than or equal to 0 and less than or equal to a;

order toConverting equality constraint into inequality constraint;

and (3) constraint: HF is more than or equal to N, F is more than or equal to 0 and less than or equal to a;

using robust optimization theory, the optimization problem is rewritten as:

$\left\{\begin{array}{c}\min \left(W^{T}F\right)\\ s.t.h_{i}F\&GreaterEqual;n_i\\ 0\&le;F\&le;a\end{array}\right.,\&ForAll;h_i\&Element;{\mathrm{\&Xi;}}_i,\&ForAll;i=1...6---\left(16\right)$

wherein h is_iIs the ith row of the matrix H containing uncertainty, and is at the uncertainty XI_iTaking a middle value; uncertain centre xi_iCan be described by an ellipsoidal uncertainty, i.e.:

${\mathrm{\&Xi;}}_i=\left\{h_i\right|{h_i}^T={{\stackrel{\&OverBar;}{h}}_i}^T+{\mathrm{\&Theta;}}_i{u}_i,\left|\right|u_i\left|\right|\&le;\mathrm{\&rho;}\},i=1,...6---\left(17\right)$

indicating the nominal value, theta, of each line as measured or identified_iFor symmetrical positive or semi-positive definite matrices, u, associated with the distribution of uncertainty_iIs the column vector associated with uncertainty, ρ is the upper bound of the euclidean norm of uncertainty;

3-2) utilizing the characteristics of ellipsoid uncertainty, simplifying formula (16) by utilizing formula (17) and utilizing

min(x^TΘu_i)＝-ρ||Θx||

Converting the robust optimization problem into a cone quadratic optimization problem:

$\left\{\begin{array}{c}min\left(W^{T}F\right)\\ s.t.{\stackrel{\&OverBar;}{h}}_{i}F-\mathrm{\&rho;}\left|\right|{\mathrm{\&Theta;}}_{i}F\left|\right|\&GreaterEqual;n_i\\ 0\&le;F\&le;a\end{array}\right.,\&ForAll;i=1...6---\left(18\right)$

and finally, solving the cone optimization problem by using an interior point method.