CN102945000B - Based on the planet landing trajectory random optimization method of observability constraint - Google Patents

Abstract

The present invention relates to a kind of planet landing trajectory random optimization method based on observability constraint, belong to survey of deep space device navigation and guidance technical field. Present method is taking the deep space landing guidance control task based on single order vision guided navigation as background, consider the dual problem between effectively control and reliable estimation, by extended mode space, systematic uncertainty is introduced in quadratic form performance index as part cost, thus adopt liner quadratic regulator technology to provide random optimization Feedback Control Laws, substitute into the extended mode spatial description model of the detector system set up, it is achieved to the real-time optimization of planet landing track; Avoid dynamic programming and the computational burden based on searching method, effectively overcome observability in landing mission and lack problem, the navigation estimated performance of raising system, makes planetary detection device navigational guidance control overall performance be protected, reaches the ultimate aim of reliable landing planet.

Description

Based on the planet landing trajectory random optimization method of observability constraint

Technical field

The present invention relates to a kind of planet landing trajectory random optimization method based on observability constraint, belong to survey of deep space device navigation and guidance technical field.

Background technology

In planet landing mission, owing to lander is subject to the impact of various uncertain factor, the design of its Guidance and control rule must be based upon on the basis to system state optimal estimation, and this makes navigation location accurately seem particularly crucial; In addition landing times is shorter, and in deep space environment, communication delay is longer, and the navigational guidance master mode based on terrestrial station is no longer applicable, therefore develops the autonomous accurate navigational guidance control strategy in real time of deep space landing and becomes the focus of current Chinese scholars research.

(see R.R.Sostaric in first technology [1], J.R.Rea.Powereddescentguidancemethodsforthemoonandmars.S anFrancisco, USA:AmericanInstituteofAeronauticsandAstronauticsInc, 2005.), traditional Guidance and control design process assumes based on determinacy equivalence, with navigating, choosing of its Guidance and control strategy estimates that uncertainty is independent of each other, thus the navigational guidance control design case of system can be divided into two independent processes, namely system state is estimated utilizing Navigation Filter to measure from sound pollution, assume that such estimated state is the time of day of system again, utilize traditional Guidance and control technology to provide the manipulated variable arrived needed for target landing region. due to this kind, method is simple and computing amount is low, therefore becomes the selection of conventional landing section task-directed control strategy.

But, for non-linear lander system, between its observability with state track, there is non-linear coupling, therefore its observability can be subject to the impact of system state track. When observing capacity is restricted, the estimated performance that limited observing capacity can cause deficiency state to be observed is not good, and then weaken Guidance and control performance, especially for the asteroid landing guidance control with irregular gravitational field characteristic, a very little controlled deviation can cause the deviation on a large scale of final landing device and target landing point, reaches reliable landing performance and just becomes to have challenge. Therefore, how to utilize the least possible navigation sensor, by optimizing observation track, while meeting landing task requirement, reduce navigation positioning error as much as possible, thus the overall performance improving navigational guidance Controlling System is problem demanding prompt solution in planet landing mission.

Summary of the invention

It is an object of the invention to in planet landing vision guided navigation process due to the not good problem of estimated performance that finite observation ability causes, it is proposed to a kind of planet landing trajectory random optimization method based on observability constraint.

Present method is taking the deep space landing guidance control task based on single order vision guided navigation as background, consider the dual problem between effectively control and reliable estimation, by extended mode space, systematic uncertainty is introduced in quadratic form performance index as part cost, thus adopt liner quadratic regulator technology to provide random optimization Feedback Control Laws. Specifically comprise the steps:

Step 1, sets up the extended mode spatial description model of detector system

$\stackrel{\·}{\mathrm{\χ}}=\left[\begin{array}{c}\stackrel{\·}{x}\\ \stackrel{\·}{s}\end{array}\right]=\mathrm{\Ψ}(\mathrm{\χ},u)---\left(1\right)$

Wherein, detector's status x=[rv]^T, r, v are respectively detector relative to the position vector of expected point of impact and velocity vector, and u is the control inputs of detector; Extended mode variable $\mathrm{\χ}={\left[\begin{array}{cc}x& s\end{array}\right]}_{x=\hat{x}}^T,$ Wherein variable s=[s₁₁s₂₁s₂₂��s₆₆]^TComprising all nonzero elements of error variance square root battle array S, �� is the functional vector of detector's status variable �� and control inputs u.

Step 2, design of feedback control inputs u:

$u=-K(\mathrm{\χ}-{\mathrm{\χ}}_{\mathrm{ss}}){|}_{x=\hat{x}}=-K_x(\hat{x}-x_d)-K_s(s-s_{\mathrm{ss}})---\left(2\right)$

$K=W_u^{-1}{B}_e^T\mathrm{\Γ}=\left[\begin{array}{cc}{K}_x& K_s\end{array}\right]---\left(3\right)$

Make it meet

$u^{*}=\arg \underset{u}{\min }{\∫}_0^{t_f}\{{\mathrm{\χ}}^T{W}_e\mathrm{\χ}+u^T{W}_{u}u\}\mathrm{dt}---\left(4\right)$

Wherein, u^*For adopting without limit steady-state gain state feedback control method, at time interval [0, t_f) on search out optimum control input; ��_ss=[x_ds_ss]^TFor expanding system steady state solution, x_dFor the detector's status expected, s_ssFor the steady state solution of the uncertain state elements of system, K_xFor detector's status deviation control coefficient, lay particular emphasis on stability contorting target, K_sOffer additional feedback controls, uncertain with impair system, it is to increase control performance.

By solving, algebraic riccati equation obtains matrix ��:

$0=A_e^T\mathrm{\Γ}+\mathrm{\Γ}{A}_e+W_e-\mathrm{\Γ}{B}_e{W}_u^{-1}{B}_e^T\mathrm{\Γ}---\left(5\right)$

Wherein, For state estimation; t_fFor detector landing times; W_uFor control burnup weight matrix, $W_e=\left[\begin{array}{cc}{W}_x& 0\\ 0& W_s\end{array}\right],$ W_xFor state deviation weight matrix,For extended mode deviation weight matrix.

Step 3, feedback control step 2 obtained input u substitutes into the extended mode spatial description model of the detector system that step 1 is set up, thus realizes the real-time optimization to planet landing track.

Useful effect

The inventive method is as a kind of planetary detection device random optimization Guidance and control method, navigation is estimated uncertain introducing and expanding system state, provide and comprise the quadratic form performance index evaluation method relevant to estimation variance, avoid dynamic programming and the computational burden based on searching method; Meanwhile, utilize the non-linear coupled characteristic between landing state observability and landing track, reasonably optimizing observation track, effectively overcome observability in landing mission and lack problem; By reasonably optimizing lander decline track, increase substantially the navigation estimated performance of system, planetary detection device navigational guidance control overall performance is protected, reaches the ultimate aim of reliable landing planet.

Accompanying drawing explanation

Fig. 1 is the schema of the inventive method;

Fig. 2 is that the landing point in embodiment is connected system of coordinates schematic diagram.

Embodiment

In order to objects and advantages of the present invention are described further, below in conjunction with drawings and Examples, content of the present invention is described further.

1. under landing point is connected system of coordinates, set up detector kinetic equation

$\stackrel{\·}{r}=\stackrel{\·}{v}$

$\stackrel{\·}{v}=u+g-a_e-a_k+a_{\mathrm{\Δ}}$

Wherein, r, v are respectively detector relative to the position vector of expected point of impact and velocity vector, and u is the control inputs of detector, and g is planet gravitational acceleration, a_e,a_kIt is respectively the centrifugal inertial acceleration caused by planet spins and Coriolis acceleration, a_��For non-modeling acceleration.

The described landing point system of coordinates that is connected is defined as ��^l:o_l-x_ly_lz_l, this coordinate origin o_lIt is positioned at predetermined landing point, o_lz_lAxle and planet barycenter point to landing point vector o_io_lDirection is consistent. o_lx_lSouth pole orientation is pointed to, o in tangent line direction along warp_ly_lWith o_lx_l��o_lz_lBetween meet right hand rule, as shown in Figure 2. Wherein, system of coordinates ��ⁱ:o_i-x_iy_iz_iFor planetocentric inertial coordinates system, its initial point in the mass center of planet, z_iAxle along the maximum inertia direction of principal axis of planet, x_iAxle along the minimum inertia axle direction of moment epoch planet, y_iAxle and x_iAxle, z_iRight hand rule is met between axle.

2. by the acceleration item-a in formula (6)_e-a_k+a_��As systematic procedure noise, obtain system dynamics model:

$\stackrel{\·}{x}=\left[\begin{array}{cc}0& I\\ 0& 0\end{array}\right]x+\left[\begin{array}{c}0\\ I\end{array}\right]\{u+g\left(E_{r}x\right)+\mathrm{\ω}\}$

$=\mathrm{Ax}+B\{u+g\left(E_{r}x\right)+\mathrm{\ω}\}$

Wherein, x=[rv]^TFor detector's status, E_r=[I0] is matrix of coefficients, A=[[00]^T[I0]^T] it is system matrix, B=[0I]^TFor input matrix, ��=-a_e-a_k+a_��For systematic procedure noise.

3. adopting based on single visual relative navigation scheme of optics feeling vector measurement, select position vector that landing point is connected under system of coordinates detector as observed quantity, obtaining observation model is

$z_k=h\left(x_c\left(t_k\right)\right)+{\mathrm{\υ}}_k$

$={\left[\begin{array}{cc}f\frac{r_{\mathrm{cy}}}{r_{\mathrm{cx}}}& f\frac{r_{\mathrm{cz}}}{r_{\mathrm{cx}}}\end{array}\right]}^T+{\mathrm{\υ}}_k$

Wherein, r_c=C_clR=[r_cxr_cyr_cz]^TFor detector is relative to the projection under camera coordinates system of the position vector of target landing point, C_clFor the conversion matrix that landing point is connected between system of coordinates and camera coordinates system, ��_kFor optics navigation observation noise, f is camera focus.

Described camera coordinates is the useful load that detector self loads.

4. detector's status x is described as the particle cloud x being distributed in state space_n(n=1,2 ..., N, N are population). Particle cloud is around the expectation state x in detector flight course_d, random estimation obtains; A corresponding particle cloud of flight expectation state.

Expect state x_dEach particle x in the particle cloud corresponding with it_nSpacing sum of squares as optimize cost function. This cost function reflects the distance between particle and expectation state and its degree of scatter. A certain expectation state x in detector landing mission is described by quadratic form expression formula_dThe control performance in corresponding moment is:

$\mathrm{\δJ}=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\{{(x_n-x_d)}^T{W}_x(x_n-x_d)+u^T{W}_{u}u\}---\left(9\right)$

$E[{(x-x_d)}^T{W}_x(x-\left(x_d\right)+u^T{W}_{u}u)$

Wherein, W_xFor state deviation weight matrix, W_uFor control burnup weight matrix.

DefinitionFor the state estimation of x, evaluated errorAnd meet

$\left\{\begin{array}{c}E\left[\tilde{x}\right]=0\\ E\left[{\tilde{x}}^T\tilde{x}\right]=P\end{array}\right.---\left(10\right)$

Then formula (9) arranges and is

$\mathrm{\δJ}=E[{(\hat{x}+\tilde{x}-x_d)}^T{W}_x(\hat{x}+\tilde{x}-x_d)+u^T{W}_{u}u]$

$={(\hat{x}-x_d)}^T{W}_x(\hat{x}-x_d)+u^T{W}_{u}u+E\left[{\tilde{x}}^T{W}_x\tilde{x}\right]---\left(11\right)$

$={(\hat{x}-x_d)}^T{W}_x(\hat{x}-x_d)+u^T{W}_{u}u+\mathrm{tr}\left[W_{x}P\right]$

In formula (11), first two is typical quadratic form, if Section 3 can be described as quadratic form form, then existing mathematics instrument can be utilized problem to be solved. Notice that error covariance matrix P is non-negative definite matrix, theoretical it will be seen that the non-negative definite matrix P �� R of symmetry by matrix^6��6P=SS can be resolved into^T, wherein S �� R^6��6For inferior triangular flap, uniquely determining by P, namely the lower triangle decomposition square root battle array of matrix P is

If W_xFor pair of horns battle array, and W_x(i, i)=w_ii(i=1,2 ..., 6), then in formula (11), last arrangement is

$\mathrm{tr}\left[W_{x}P\right]=\mathrm{tr}\left[W_x{S}^{T}S\right]=\underset{i=1}{\overset{6}{\mathrm{\Σ}}}\left\{w_{\mathrm{ii}}\underset{j=1}{\overset{i}{\mathrm{\Σ}}}{s}_{\mathrm{ij}}^2\right\}---\left(13\right)$

Formula (11) is rewritten as state estimation deviationThe Quadratic Function Optimization of control inputs u and error variance square root battle array S nonzero element:

By formula (14) to time integral, obtain the performance index of random optimization:

$J={\∫}_0^{t_f}\{{(\hat{x}-x_d)}^T{W}_x(\hat{x}-x_d)+u^T{W}_{u}u+\underset{i=1}{\overset{6}{\mathrm{\Σ}}}\{w_{\mathrm{ii}}\underset{j=1}{\overset{i}{\mathrm{\Σ}}}{s}_{\mathrm{ij}}^2\left\}\right\}\mathrm{dt}---\left(15\right)$

In formula, first two correspond to determinacy equivalence control cost, and Section 3 is corresponding to systematic uncertainty cost.

5. extended mode variable is defined $\mathrm{\χ}={\left[\begin{array}{cc}x& s\end{array}\right]}_{x=\hat{x}}^T,$ Wherein variable s=[s₁₁s₂₁s₂₂��s₆₆]^TComprise all nonzero elements of error variance square root battle array S, for ease of describing, if expecting state x_d=0, then quadratic form performance index described by formula (15) can be simplified shown as

$J={\∫}_0^{t_f}\{{\mathrm{\χ}}^T{W}_e\mathrm{\χ}+u^T{W}_{u}u\}\mathrm{dt}---\left(16\right)$

Wherein, $W_e=\left[\begin{array}{cc}{W}_x& 0\\ 0& W_s\end{array}\right],$ For extended mode deviation weight matrix.

6. the extended mode spatial description model of survey of deep space device decline track optimizing problem is:

$\stackrel{\·}{\mathrm{\χ}}=\left[\begin{array}{c}\stackrel{\·}{x}\\ \stackrel{\·}{s}\end{array}\right]=\mathrm{\Ψ}(\mathrm{\χ},u)---\left(17\right)$

At time interval [0, t_f) upper searching optimum control input u^*, meet

$u^{*}=\arg \underset{u}{\min }{\∫}_0^{t_f}\{{\mathrm{\χ}}^T{W}_e\mathrm{\χ}+u^T{W}_{u}u\}\mathrm{dt}---\left(18\right)$

Owing to relating to the complicacy that covariance is propagated and extending space describes, the optimum solution seeking non-linear control system is very difficult. Here the suboptimal solution obtaining optimization problem without limit steady-state gain STATE FEEDBACK CONTROL technology is adopted.

Detailed process is: solve the matrix �� meeting algebraic riccati equation:

$0=A_e^T\mathrm{\Γ}+\mathrm{\Γ}{A}_e+W_e-\mathrm{\Γ}{B}_e{W}_u^{-1}{B}_e^T\mathrm{\Γ}---\left(19\right)$

Wherein, $A_e=\frac{\∂\mathrm{\Ψ}(\mathrm{\χ},u)}{{\∂}_{\mathrm{\χ}}}{|}_{x=\hat{x}},B_e=\frac{\∂\mathrm{\Ψ}(\mathrm{\χ},u)}{\∂u}{|}_{x=\hat{x}},$ Then feed back input form is:

$u=-K(\mathrm{\χ}-{\mathrm{\χ}}_{\mathrm{ss}}){|}_{x=\hat{x}}=-K_x(\hat{x}-x_d)-K_s(s-s_{\mathrm{ss}})---\left(20\right)$

$K=W_u^{-1}{B}_e^T\mathrm{\Γ}=\left[\begin{array}{cc}{K}_x& K_s\end{array}\right]---\left(21\right)$

Wherein, ��_ss=[x_ds_ss]^TFor expanding system steady state solution, K_xMainly lay particular emphasis on stability contorting target, K_sThere is provided additional feedback control to be used for the uncertainty of impair system, thus improve control performance.

Using asteroid Eros433 as target celestial body, its spin angle velocity is 1639.4 ��/day, and nominal radius is 16km, and gravity constant is 4.4621 �� 10⁵m³/s², adopt its four rank gravitational field model in simulations. Under landing point is connected system of coordinates, detector starting position is [50020-50]^TM, relative asteroid surface velocity is [1.520]^TMs, it is contemplated that landing times 300s. Considering lander Initial state estimation error, it is 100m that its position initial estimation error and speed estimation error obey variance respectively²And 0.1m²/s²Stochastic distribution time, utilize first technology [1] that lander is carried out feed forward control, in expection landing times, lander final landing speed is up to 3ms, is hitting the danger into target celestial body at expectation landing time memory. This is mainly owing to the observability of detector's status is by the impact of system state track, and when observing capacity is restricted, navigation evaluated error is by lasting increase, and its Guidance and control performance significantly can be weakened along with the increase of navigation error. Consider the existence of navigation evaluated error, landing track is carried out random optimization by the feedback control input adding inventive design in planet Landing Control process, the evaluated error of detector is along with the change of decline track is in continuous reduction, the path point that end lander time of day produces with feedforward guidance is consistent substantially, its final landing position deviation is within the scope of 5m, and speed variation is at about 0.05ms. Navigation and guidance control performance is made to obtain overall raising the random optimization effect of decline track by feedback control.

Claims (2)

1. based on the planet landing trajectory random optimization method of observability constraint, it is characterised in that: comprise the following steps:

Step 1, sets up detector kinetic equation under landing point is connected system of coordinates

$\stackrel{\·}{r}=\stackrel{\·}{v}$

$\stackrel{\·}{v}=u+g-a_e-a_k+a_{\mathrm{\Δ}}$

Wherein, r, v are respectively detector relative to the position vector of expected point of impact and velocity vector, and u is the control inputs of detector, and g is planet gravitational acceleration, a_e,a_kIt is respectively the centrifugal inertial acceleration caused by planet spins and Coriolis acceleration, a_��For non-modeling acceleration;

The described landing point system of coordinates that is connected is defined as ��^l:o_l-x_ly_lz_l, this coordinate origin o_lIt is positioned at predetermined landing point, o_lz_lAxle and planet barycenter point to landing point vector o_io_lDirection is consistent; o_lx_lSouth pole orientation is pointed to, o in tangent line direction along warp_ly_lWith o_lx_l��o_lz_lBetween meet right hand rule; Wherein, system of coordinates ��ⁱ:o_i-x_iy_iz_iFor planetocentric inertial coordinates system, its initial point in the mass center of planet, z_iAxle along the maximum inertia direction of principal axis of planet, x_iAxle along the minimum inertia axle direction of moment epoch planet, y_iAxle and x_iAxle, z_iRight hand rule is met between axle;

Step 2, by the acceleration item-a in step 1 kinetic equation_e-a_k+a_��As systematic procedure noise, obtain system dynamics model:

$\begin{array}{c}\stackrel{\·}{x}=\left[\begin{array}{cc}0& I\\ 0& 0\end{array}\right]x+\left[\begin{array}{c}0\\ I\end{array}\right]\left\{u+g\left(E_{r}x\right)+\mathrm{\ω}\right\}\\ =Ax+B\left\{u+g\left(E_{r}x\right)+\mathrm{\ω}\right\}\end{array}$

Wherein, x=[rv]^TFor detector's status, E_r=[I0] is matrix of coefficients, A=[[00]^T[I0]^T] it is system matrix, B=[0I]^TFor input matrix, ��=-a_e-a_k+a_��For systematic procedure noise;

Step 3, adopts based on single visual relative navigation scheme of optics feeling vector measurement, selects position vector that landing point is connected under system of coordinates detector as observed quantity, and obtaining observation model is

$\begin{array}{c}{z}_k=h\left(x_c\left(t_k\right)\right)+{\mathrm{\υ}}_k\\ ={\left[\begin{array}{cc}f\frac{r_{cy}}{r_{cx}}& f\frac{r_{cz}}{r_{cx}}\end{array}\right]}^T+{\mathrm{\υ}}_k\end{array}$

Wherein,For detector is relative to the projection under camera coordinates system of the position vector of target landing point, C_clFor the conversion matrix that landing point is connected between system of coordinates and camera coordinates system, ��_kFor optics navigation observation noise, f is camera focus;

Described camera coordinates is the useful load that detector self loads;

Step 4, is described as the particle cloud x being distributed in state space by detector's status x_n, n=1,2 ..., N, N are population; Particle cloud is around the expectation state x in detector flight course_d, random estimation obtains; A corresponding particle cloud of flight expectation state;

Expect state x_dEach particle x in the particle cloud corresponding with it_nSpacing sum of squares as optimize cost function; This cost function reflects the distance between particle and expectation state and its degree of scatter; A certain expectation state x in detector landing mission is described by quadratic form expression formula_dThe control performance in corresponding moment is:

$\begin{array}{c}\mathrm{\δ}J=\frac{1}{N}\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\left\{{\left(x_n-x_d\right)}^T{W}_x\left(x_n-x_d\right)+u^T{W}_{u}u\right\}\\ =E\[{\left(x-x_d\right)}^T{W}_x\left(x-x_d\right)+u^T{W}_{u}u\]\end{array}$

Wherein, W_xFor state deviation weight matrix, W_uFor control burnup weight matrix;

DefinitionFor the state estimation of x, evaluated errorAnd meet

$\left\{\begin{array}{c}E\[\tilde{x}\]=0\\ E\[{\tilde{x}}^T\tilde{x}\]=P\end{array}\right.$

Then x_dThe control performance in corresponding moment arranges and is

$\begin{array}{c}\mathrm{\δ}J=E\[{\left(\hat{x}+\tilde{x}-x_d\right)}^T{W}_x\left(\hat{x}+\tilde{x}-x_d\right)+u^T{W}_{u}u\]\\ ={\left(\hat{x}-x_d\right)}^T{W}_x\left(\hat{x}-x_d\right)+u^T{W}_{u}u+E\[{\tilde{x}}^T{W}_x\tilde{x}\]\\ ={\left(\hat{x}-x_d\right)}^T{W}_x\left(\hat{x}-x_d\right)+u^T{W}_{u}u+tr\[W_{x}P\]\end{array}$

In upper formula, first two is typical quadratic form, if Section 3 can be described as quadratic form form, then existing mathematics instrument can be utilized problem to be solved; Notice that error covariance matrix P is non-negative definite matrix, theoretical it will be seen that the non-negative definite matrix P �� R of symmetry by matrix^6��6P=SS can be resolved into^T, wherein S �� R^6��6For inferior triangular flap, uniquely determining by P, namely the lower triangle decomposition square root battle array of matrix P is

If W_xFor pair of horns battle array, and W_x(i, i)=w_ii, i=1,2 ..., 6, then the x after arranging_dIn the control performance equation in corresponding moment, last arrangement is

$tr\[W_{x}P\]=tr\[W_x{S}^{T}S\]=\underset{i=1}{\overset{6}{\Σ}}\left\{w_{ii}\underset{j=1}{\overset{i}{\Σ}}{s}_{ij}^2\right\}$

X after arrangement_dThe control performance equation in corresponding moment is rewritten as state estimation deviationThe Quadratic Function Optimization of control inputs u and error variance square root battle array S nonzero element:

$\mathrm{\δ}J={(\hat{x}-x_d)}^T{W}_x(\hat{x}-x_d)+u^T{W}_{u}u+\underset{i=1}{\overset{6}{\Σ}}\left\{w_{ii}\underset{j=1}{\overset{i}{\Σ}}{s}_{ij}^2\right\}$

By upper formula to time integral, obtain the performance index of random optimization:

$J={\∫}_0^{t_f}\left\{{\left(\hat{x}-x_d\right)}^T{W}_x\left(\hat{x}-x_d\right)+u^T{W}_{u}u+\underset{i=1}{\overset{6}{\mathrm{\Σ}}}\left\{w_{ii}\underset{j=1}{\overset{i}{\mathrm{\Σ}}}{s}_{ij}^2\right\}\right\}dt$

In formula, first two correspond to determinacy equivalence control cost, and Section 3 is corresponding to systematic uncertainty cost; t_fFor detector landing times;

Step 5, definition extended mode variable $\mathrm{\χ}={\left[\begin{array}{cc}x& s\end{array}\right]}_{x=\hat{x}}^T,$ Wherein variable s=[s₁₁s₂₁s₂₂��s₆₆]^TComprise all nonzero elements of error variance square root battle array S, for ease of describing, if expecting state x_d=0, then quadratic form performance index described by the performance index of random optimization can be simplified shown as

$J={\∫}_0^{t_f}\{{\mathrm{\χ}}^T{W}_e\mathrm{\χ}+u^T{W}_{u}u\}dt$

Wherein, $W_e=\left[\begin{array}{cc}{W}_x& 0\\ 0& W_s\end{array}\right],$ For extended mode deviation weight matrix;

Step 6, the extended mode spatial description model of survey of deep space device decline track optimizing problem is:

$\stackrel{\·}{\mathrm{\χ}}=\left[\begin{array}{c}\stackrel{\·}{x}\\ \stackrel{\·}{s}\end{array}\right]=\mathrm{\Ψ}(\mathrm{\χ},u)$

Wherein, �� is the functional vector of detector's status variable �� and control inputs u;

At time interval [0, t_f) upper searching optimum control input u^*, meet

$u^{*}=\arg \underset{u}{\min }{\∫}_0^{t_f}\{{\mathrm{\χ}}^T{W}_e\mathrm{\χ}+u^T{W}_{u}u\}dt$

Owing to relating to the complicacy that covariance is propagated and extending space describes, the optimum solution seeking non-linear control system is very difficult, adopts the suboptimal solution obtaining optimization problem without limit steady-state gain STATE FEEDBACK CONTROL technology here;

Detailed process is: solve the matrix �� meeting algebraic riccati equation:

$0={A_e}^T\mathrm{\Γ}+{\mathrm{\ΓA}}_e+W_e-{\mathrm{\ΓB}}_e{W}_u^{-1}{B}_e^T\mathrm{\Γ}$

Wherein, $A_e=\frac{\∂\mathrm{\Ψ}(\mathrm{\χ},u)}{\∂\mathrm{\χ}}{|}_{{}_{x=\hat{x}}},B_e=\frac{\∂\mathrm{\Ψ}(\mathrm{\χ},u)}{\∂u}{|}_{{}_{x=\hat{x}}},$ Then feed back input form is:

$u=-K(\mathrm{\χ}-{\mathrm{\χ}}_{ss}){|}_{x=\hat{x}}=-K_x(\hat{x}-x_d)-K_s(s-s_{ss})$

$K=W_u^{-1}{B}_e^T\mathrm{\Γ}=\left[\begin{array}{cc}{K}_x& K_s\end{array}\right]$

Wherein, ��_ss=[x_ds_ss]^TFor expanding system steady state solution, s_ssFor the steady state solution of the uncertain state elements of system, K_xMainly lay particular emphasis on stability contorting target, K_sThere is provided additional feedback control to be used for the uncertainty of impair system, thus improve control performance.

2. the planet landing trajectory random optimization method based on observability constraint according to claim 1, it is characterised in that: adopt and ask for u without limit steady-state gain state feedback control method^*��