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 深空探测学报  2016, Vol. 3 Issue (1): 61-67  DOI: 10.15982/j.issn.2095-7777.2016.01.010 0

### 引用本文 [复制中英文]

[复制中文]
Yan X P, Sun H B, Guo L. Finite Time Anti-Disturbance Guidance Law Design for Mars Entry[J]. Journal of Deep Space Exploration, 2016, 3(1): 61-67. DOI: 10.15982/j.issn.2095-7777.2016.01.010.
[复制英文]

### 文章历史

1. 北京航空航天大学 自动化科学与电气工程学院, 北京 100191;
2. 北京航空航天大学 飞行器控制一体化技术国防重点实验室, 北京 100191

Finite Time Anti-Disturbance Guidance Law Design for Mars Entry
YAN Xiaopeng1, 2, SUN Haibin1, 2, GUO Lei1, 2
1. School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China;
2. Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100191, China
Abstract: This paper considers the guidance design for Mars entry vehicles with disturbance modulation, providing a composite strategy based on drag. First, according to dynamic equations of the vehicle and combining with the definition of drag, the drag dynamic equation contained with disturbance is given. Second, in order to make sure the system obtain a better anti-disturbance performance and more quickly track speed, the finite time feedback guidance law is designed based on drag dynamic equation. With the purpose of further improving the anti-disturbance ability, a disturbance observer is designed to estimate unknown disturbance and the estimated value is used for feed-forward compensation, then a composite law is obtained. In the end, a comparison simulation is carried out to examine the efficiency and superiority of this strategy.
Key words: Mars entry    finite time feedback    disturbance observer    composite control
0 引言

1 大气进入段着陆器模型

 $\dot r = V\sin \gamma$ (1)
 $\dot \theta = \frac{{V\cos \gamma \cos \psi }}{{r\cos \phi }}$ (2)
 $\dot \phi = \frac{{V\cos \gamma \sin \psi }}{r}$ (3)
 $\dot V = - D - g\sin \gamma$ (4)
 $\dot \gamma = \frac{1}{{V\;}}\left[{L\cos \sigma - \left( {g - \frac{{{V^2}}}{r}} \right)\cos \gamma } \right] + d$ (5)
 $\psi = \frac{{ - 1}}{{V\cos \gamma }}\left[{L\sin \sigma + \frac{{{V^2}}}{r}{{\cos }^2}\gamma \cos \psi \tan \Phi } \right]$ (6)
 $\dot S = V\cos \gamma$ (7)

 $D = \frac{1}{2}\frac{{\rho {V^2}{C_D}{S_r}}}{m} = \frac{1}{2}\frac{{\rho {V^2}}}{{{B_f}}}$ (8)
 $L = \frac{1}{2}\frac{{\rho {V^2}{C_L}{S_r}}}{m}$ (9)

 $\rho (r) = {\rho _s}\exp \left( { - \beta \left( {r - {r_s}} \right)} \right)$ (10)

 ${B_f} = \frac{m}{{{C_D}{S_r}}}$ (11)

 $g(r) = \frac{\mu }{{{r^2}}}$ (12)

2 制导控制律

 $\dot D = - \frac{{2D}}{V}(D + g\sin \gamma ) - D\beta V\sin \gamma$ (13)

 $\begin{gathered} \ddot D = [- \dot DV\beta \sin \gamma + D\beta \sin \gamma (D + g\sin \gamma ) - \hfill \\ \frac{{2D\dot D}}{V} - \frac{{2\dot D}}{V}(D + g\sin \gamma ) - \hfill \\ \frac{{2D}}{{{V^2}}}{(D + g\sin \gamma )^2} - D\beta (\frac{{{V^2}}}{r} - g){\cos ^2}\gamma - \hfill \\ \frac{{2Dg}}{{{V^2}}}(\frac{{{V^2}}}{r} - g){\cos ^2}\gamma] + \hfill \\ [- D\cos \gamma (\beta + \frac{{2g}}{{{V^2}}})L]u - \hfill \\ (\frac{{2D}}{V}g\cos \gamma + D\beta V\cos \gamma )d \hfill \\ = a + bu + fd \hfill \\ \end{gathered}$ (14)

 图 1 带有干扰观测器的系统结构图 Fig. 1 Block diagram of system with disturbance observer
2.1 非线性干扰观测器的设计

 $\dot x = f(t,x,u),x \in {R^n},u \in {R^m}$ (15)

1)系统(15)是全局输入状态稳定的

2) $\mathop {\lim }\limits_{t \to \infty } u = 0$

 $\left\{ \begin{gathered} \hat d = z + l\gamma \hfill \\ \dot z = - l\frac{1}{V}\left[{L\cos \sigma - \left( {g - \frac{{{V^2}}}{r}} \right)\cos \gamma } \right] - l\hat d \hfill \\ \end{gathered} \right.$ (16)

 $\begin{gathered} \dot e = \dot d - \hat d = \dot d - (\dot z + l\dot \gamma ) \hfill \\ = \dot d - \left\{ \begin{gathered} - l\frac{1}{V}\left[ {L\cos \sigma - (g - \frac{{{V^2}}}{r})\cos \gamma } \right] - l\hat d \hfill \\ + l\frac{1}{V}[L\cos \sigma - (g - \frac{{{V^2}}}{r})\cos \gamma ] + ld \hfill \\ \end{gathered} \right\} \hfill \\ = \dot d - l(d - \hat d) = \dot d - le \hfill \\ \end{gathered}$ (17)

2.2 有限时间控制器的设计

 $\begin{gathered} {{\dot x}_1} = {x_2} \hfill \\ {{\dot x}_2} = u \hfill \\ \end{gathered}$ (18)

 $u = - {k_1}{\left| {{x_1}} \right|^{{\alpha _1}}}{\text{sign}}({{\text{x}}_{\text{1}}}){\text{ - }}{{\text{k}}_{\text{2}}}{\left| {{{\text{x}}_{\text{2}}}} \right|^{{\alpha _{\text{2}}}}}{\text{sign}}({{\text{x}}_{\text{2}}})$

 ${x_1} = D - {D_r},{x_2} = \dot D - {\dot D_r}$

 $\begin{gathered} {{\dot x}_1} = {x_2} \hfill \\ {{\dot x}_2} = a + bu + fd - {{\ddot D}_r} \hfill \\ \end{gathered}$ (19)

 $u = {\text{ }}[- {k_1}{\left| {{x_1}} \right|^{{\alpha _1}}}{\text{sign}}({{\text{x}}_{\text{1}}}){\text{ - }}{{\text{k}}_{\text{2}}}{\left| {{{\text{x}}_{\text{2}}}} \right|^{{\alpha _{\text{2}}}}}{\text{sign}}({{\text{x}}_{\text{2}}}){\text{ - }}a + {\ddot D_r}]/b$ (20)

 $\begin{gathered} {{\dot x}_1} = {x_2} \hfill \\ {{\dot x}_2} = - {k_1}{\left| {{x_1}} \right|^{{\alpha _1}}}{\text{sign}}({{\text{x}}_{\text{1}}}){\text{ - }}{{\text{k}}_{\text{2}}}{\left| {{{\text{x}}_{\text{2}}}} \right|^{{\alpha _{\text{2}}}}}{\text{sign}}({{\text{x}}_{\text{2}}}) \hfill \\ \end{gathered}$ (21)

2.3 复合制导律设计

 $\begin{gathered} u = [- {k_1}{\left| {{x_1}} \right|^{{\alpha _1}}}{\text{sign}}({{\text{x}}_{\text{1}}}){\text{ - }}{{\text{k}}_{\text{2}}}{\left| {{{\text{x}}_{\text{2}}}} \right|^{{\alpha _{\text{2}}}}}{\text{sign}}({x_{\text{2}}}){\text{ - }} \hfill \\ a - f\hat d - {{\ddot D}_r}]/b \hfill \\ \end{gathered}$ (22)

3 仿真分析

 $u = \frac{1}{b}( - a + {\ddot D_r} - {k_D}(\dot D - {\dot D_r}) - {k_P}(D - {D_r}) - f\hat d)$

 $\left\{ \begin{gathered} V{\text{ > }}3500m/s时,\sigma = 75° \hfill \\ 1500m/s{\text{ < }}V{\text{ < }}3500m/s时,\sigma = 0.0125V + 31.25 \hfill \\ 假定干扰量为正弦,即dr = 0.01\sin \left( t \right) \hfill \\ V{\text{ < }}1500m/s时,\sigma = 50° \hfill \\ \end{gathered} \right.$

 图 2 速度跟踪曲线 Fig. 2 Curve of velocity

 图 3 经度跟踪曲线 Fig. 3 Curve of latitude

 图 4 阻力加速度跟踪曲线 Fig. 4 Curve of drag acceleration

 图 5 阻力加速度变化率跟踪曲线 Fig. 5 Curve of the rate of drag acceleration

 图 6 高度跟踪曲线 Fig. 6 Curve of altitude

 图 7 航程跟踪曲线 Fig. 7 Curve of crossrange

 图 8 干扰真实值与估计值曲线 Fig. 8 Curves of the real value and estimated value of disturbances

 图 9 干扰观测器估计误差曲线 Fig. 9 Curve of error of disturbance observer

4 结论

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