﻿ 基于浸入与不变理论的航天器姿态跟踪自适应控制
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1. 西北工业大学 航天飞行动力学技术国家级重点实验室, 西安 710072;
2. 西北工业大学 航天学院, 西安 710072

Immersion and invariance based attitude adaptive tracking control for spacecraft
XIA Dongdong1,2, YUE Xiaokui1,2
1. National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University, Xi'an 710072, China;
2. School of Astronautics, Northwestern Polytechnical University, Xi'an 710072, China
Abstract: In this paper, the spacecraft attitude tracking control with inertia uncertainty is addressed, and a novel Immersion and Invariance (I&I) based adaptive tracking controller is proposed. The results show that the parametric regression matrix is not integrable when I&I methodology is applied to the attitude dynamic systems, which leads to non-analytical solution of partial differential equations in the I&I controller design. To overcome this problem, this paper presented a new I&I adaptive tracking controller for the spacecraft attitude via the dynamic scaling technique. A rigorous Lyapunov analysis is provided to guarantee the globally asymptotic stability of the closed-loop systems. A key feature in this paper is that the controller implementation is no need of the scaling factor and the prior knowledge of inertia matrix by virtue of the innovative scaling factor design. Finally, the effectiveness and superiority of the proposed controller are illustrated by numerical simulations compared with the certainty-equivalence-based controller.
Keywords: immersion and invariance    inertia uncertainty    attitude tracking control    regression matrix    dynamic scaling    Lyapunov analysis

Karagiannis等[11-12]开创性地将动态放缩法(Dynamic Scaling)运用到I&I控制器设计当中，该方法考虑参数回归矩阵不满足可积条件，通过引入状态滤波器，按照一定方式替换掉回归矩阵中的积分变量使得其可积并得到其近似解，然后运用动态放缩技术将近似解和真解之间误差部分消除。由于动态放缩法相比于Seo和Akella[8-9]提出的增广滤波法，只需对被积状态设计滤波器，因此明显地降低了闭环系统的阶数，优势比较突出，吸引了一大批研究者的关注[13-18]，极大推进了I&I理论的发展。但是，基于动态放缩法的控制器设计过程中，动态放缩因子为单调递增函数，尽管能够证明有界，但是事先并不知会增大到多大; 而控制器和滤波器动态反馈增益系数与放缩因子的平方呈线性关系，这些因素会导致控制器反馈增益很大，可能会出现不希望的瞬态特性。

Yang等[17]首次将动态放缩法的I&I理论运用到航天器的姿态控制模型中。针对参数回归矩阵不可积的困难，通过添加一个补偿矩阵使其可积，然后利用构造的角速度滤波器或者参考角速度信号来抵消补偿矩阵的影响，并用动态放缩技术将其影响消除。并且针对动态放缩因子单调递增所带来“高增益”控制现象，通过使用“三标量动态(three scalar dynamics)”方法构造一个动态调节系数，使递减的调节系数中和放缩因子的增长。虽然该文章也提出了基于动态放缩法I&I控制器，但是需要额外设计标量动态，并且需要事先知道惯性矩阵的最小特征值的下界，这给实际应用带来了困难。

Wen等[18]同样将动态放缩法和I&I方法应用在姿态跟踪模型上，采用的是文献[11]中的回归矩阵改造技巧，但是创新地提出了修正缩放因子和附加调节系数动态，使得控制器中不需要惯量矩阵最小特征值，也能约束缩放因子在一个事先确定的上界内。但是该方法只是将反馈增益与缩放因子从平方线性关系修正到呈线性关系，仍需要调节系数来中和缩放因子的增长。

1 问题描述 1.1 坐标系定义

 图 1 坐标系示意图 Fig. 1 Schematic diagram of coordinate system
1.2 动力学模型

 $\mathit{\boldsymbol{\dot q}} = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} { - \mathit{\boldsymbol{q}}_{\rm{v}}^{\rm{T}}}\\ {{q_0}{\mathit{\boldsymbol{I}}_3} + \mathit{\boldsymbol{q}}_{\rm{v}}^ \times } \end{array}} \right]\mathit{\boldsymbol{\omega }}$ （1）
 $\mathit{\boldsymbol{J\dot \omega }} = - {\mathit{\boldsymbol{\omega }}^ \times }\mathit{\boldsymbol{J\omega }} + \mathit{\boldsymbol{u}}$ （2）

 ${\mathit{\boldsymbol{v}}^ \times } = \left[ {\begin{array}{*{20}{c}} 0&{ - {v_3}}&{{v_2}}\\ {{v_3}}&0&{ - {v_1}}\\ { - {v_2}}&{{v_1}}&0 \end{array}} \right]$

 $\mathit{\boldsymbol{R}} = {\mathit{\boldsymbol{I}}_3} - 2{q_0}\mathit{\boldsymbol{q}}_{\rm{v}}^ \times + 2\mathit{\boldsymbol{q}}_{\rm{v}}^ \times \mathit{\boldsymbol{q}}_{\rm{v}}^ \times$ （3）

 ${\mathit{\boldsymbol{q}}_{\rm{e}}} = \left[ {\begin{array}{*{20}{c}} {{q_{{\rm{e}}0}}}\\ {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{q_{{\rm{do}}}}{q_0} + \mathit{\boldsymbol{q}}_{{\rm{dv}}}^{\rm{T}}{\mathit{\boldsymbol{q}}_{\rm{v}}}}\\ {{q_{{\rm{d0}}}}{\mathit{\boldsymbol{q}}_{\rm{v}}} - {q_0}{\mathit{\boldsymbol{q}}_{{\rm{dv}}}} - \mathit{\boldsymbol{q}}_{{\rm{dv}}}^ \times {\mathit{\boldsymbol{q}}_{\rm{v}}}} \end{array}} \right]$ （4）

FDFB的坐标变换矩阵为

 ${\mathit{\boldsymbol{R}}_{\rm{e}}} = {\mathit{\boldsymbol{I}}_3} - 2{q_{{\rm{e}}0}}\mathit{\boldsymbol{q}}_{{\rm{ev}}}^ \times + 2\mathit{\boldsymbol{q}}_{{\rm{ev}}}^ \times \mathit{\boldsymbol{q}}_{{\rm{ev}}}^ \times$ （5）

 ${\mathit{\boldsymbol{\omega }}_{\rm{e}}} = \mathit{\boldsymbol{\omega }} - {\mathit{\boldsymbol{R}}_{\rm{e}}}{\mathit{\boldsymbol{\omega }}_{\rm{d}}} = \mathit{\boldsymbol{\omega }} - \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}$ （6）

 ${{\dot q}_{{\rm{e}}0}} = - \frac{1}{2}\mathit{\boldsymbol{q}}_{{\rm{ev}}}^{\rm{T}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}}$ （7）
 ${{\mathit{\boldsymbol{\dot q}}}_{{\rm{ev}}}} = \frac{1}{2}\left( {{q_{{\rm{e}}0}}{\mathit{\boldsymbol{I}}_3} + \mathit{\boldsymbol{q}}_{{\rm{ev}}}^ \times } \right){\mathit{\boldsymbol{\omega }}_{\rm{e}}}$ （8）
 $\mathit{\boldsymbol{J}}{{\mathit{\boldsymbol{\dot \omega }}}_{\rm{e}}} = - {\mathit{\boldsymbol{\omega }}^ \times }\mathit{\boldsymbol{J\omega }} + \mathit{\boldsymbol{u}} + \mathit{\boldsymbol{J}}{\mathit{\boldsymbol{\omega }}^ \times }\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{J}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{\rm{d}}}$ （9）

1.3 参数线性化

 $\mathit{\boldsymbol{J}} = \left[ {\begin{array}{*{20}{c}} {{J_{11}}}&{{J_{12}}}&{{J_{13}}}\\ {{J_{12}}}&{{J_{22}}}&{{J_{23}}}\\ {{J_{13}}}&{{J_{23}}}&{{J_{33}}} \end{array}} \right]$

J包含6个未知参数，可以设未知参数向量为

 $\mathit{\boldsymbol{\theta }} = {\left[ {\begin{array}{*{20}{c}} {{J_{11}}}&{{J_{12}}}&{{J_{13}}}&{{J_{22}}}&{{J_{23}}}&{{J_{33}}} \end{array}} \right]^{\rm{T}}}$

 ${{\mathit{\boldsymbol{\dot \omega }}}_{\rm{e}}} = - {k_\mathit{\boldsymbol{q}}}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} - {k_\mathit{\boldsymbol{\omega }}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}} + {\mathit{\boldsymbol{J}}^{ - 1}}\left( {\mathit{\boldsymbol{W\theta }} + \mathit{\boldsymbol{u}}} \right)$ （10）

 $\mathit{\boldsymbol{W\theta }} = - {\mathit{\boldsymbol{\omega }}^ \times }\mathit{\boldsymbol{J\omega }} + \mathit{\boldsymbol{J}}{\mathit{\boldsymbol{\omega }}^ \times }\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{J}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{\rm{d}}} + \mathit{\boldsymbol{J}}\left( {{k_\mathit{\boldsymbol{q}}}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} + {k_\mathit{\boldsymbol{\omega }}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right)$ （11）

2 基于动态放缩法的I&I控制器设计

 $\mathit{\boldsymbol{u}} = - \mathit{\boldsymbol{W\hat \theta }}$ （12）

 ${{\mathit{\boldsymbol{\dot \omega }}}_{\rm{e}}} = - {k_\mathit{\boldsymbol{q}}}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} - {k_\mathit{\boldsymbol{\omega }}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}} - {\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{W\tilde \theta }}$ （13）

2.1 参数估计

 $\mathit{\boldsymbol{\hat \theta }} = \mathit{\boldsymbol{\alpha }} + \mathit{\boldsymbol{\beta }}\left( {\mathit{\boldsymbol{\omega }},\mathit{\boldsymbol{\varphi }}} \right)$ （14）

 $\mathit{\boldsymbol{\dot \alpha }} = \frac{{\partial \mathit{\boldsymbol{\beta }}}}{{\partial \mathit{\boldsymbol{\omega }}}}\left( {{k_\mathit{\boldsymbol{q}}}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} + {k_\mathit{\boldsymbol{\omega }}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}} - \mathit{\boldsymbol{ \boldsymbol{\dot \varOmega} }}} \right) - \sum\limits_i {\frac{{\partial \mathit{\boldsymbol{\beta }}}}{{\partial {\mathit{\boldsymbol{\varphi }}_i}}}{{\mathit{\boldsymbol{\dot \varphi }}}_i}}$ （15）

 $\mathit{\boldsymbol{\dot {\tilde \theta} }} = \mathit{\boldsymbol{\dot \alpha }} + \frac{{\partial \mathit{\boldsymbol{\beta }}}}{{\partial \mathit{\boldsymbol{\omega }}}}\left( {{{\mathit{\boldsymbol{\dot \omega }}}_{\rm{e}}} + \mathit{\boldsymbol{ \boldsymbol{\dot \varOmega} }}} \right) + \sum\limits_i {\frac{{\partial \mathit{\boldsymbol{\beta }}}}{{\partial {\mathit{\boldsymbol{\varphi }}_i}}}{{\mathit{\boldsymbol{\dot \varphi }}}_i}} = - \frac{{\partial \mathit{\boldsymbol{\beta }}}}{{\partial \mathit{\boldsymbol{\omega }}}}{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{W\tilde \theta }}$ （16）

 $\frac{{\partial \mathit{\boldsymbol{\beta }}}}{{\partial \mathit{\boldsymbol{\omega }}}} = \gamma {\mathit{\boldsymbol{W}}^{\rm{T}}}$ （17）

 $\begin{array}{l} \mathit{\boldsymbol{\beta }} = \gamma \int\limits_\mathit{\boldsymbol{\omega }} {\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{w}}_1^{\rm{T}}}&{\mathit{\boldsymbol{w}}_2^{\rm{T}}}&{\mathit{\boldsymbol{w}}_3^{\rm{T}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\rm{d}}{\omega _1}}\\ {{\rm{d}}{\omega _2}}\\ {{\rm{d}}{\omega _3}} \end{array}} \right]} = \\ \;\;\;\;\;\gamma \int\limits_\mathit{\boldsymbol{\omega }} {\mathit{\boldsymbol{w}}_1^{\rm{T}}{\rm{d}}{\omega _1} + \mathit{\boldsymbol{w}}_2^{\rm{T}}{\rm{d}}{\omega _2} + \mathit{\boldsymbol{w}}_3^{\rm{T}}{\rm{d}}{\omega _3}} \end{array}$ （18）

 $\frac{{\partial \mathit{\boldsymbol{w}}_1^{\rm{T}}}}{{\partial {\omega _2}}} = \frac{{\partial \mathit{\boldsymbol{w}}_2^{\rm{T}}}}{{\partial {\omega _1}}},\frac{{\partial \mathit{\boldsymbol{w}}_1^{\rm{T}}}}{{\partial {\omega _3}}} = \frac{{\mathit{\boldsymbol{w}}_3^{\rm{T}}}}{{\partial {\omega _1}}},\frac{{\partial \mathit{\boldsymbol{w}}_2^{\rm{T}}}}{{\partial {\omega _3}}} = \frac{{\partial \mathit{\boldsymbol{w}}_3^{\rm{T}}}}{{\partial {\omega _2}}}$ （19）

 $\begin{array}{l} \mathit{\boldsymbol{W\theta }} = \mathit{\boldsymbol{J}}\left( {{k_\mathit{\boldsymbol{q}}}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} - {k_\mathit{\boldsymbol{\omega }}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{\rm{d}}}} \right) + {k_\mathit{\boldsymbol{\omega }}}\mathit{\boldsymbol{J\omega }} + \\ \;\;\;\;\;\;\;\left( { - {\mathit{\boldsymbol{\omega }}^ \times }\mathit{\boldsymbol{J\omega }} + \mathit{\boldsymbol{J}}{\mathit{\boldsymbol{\omega }}^ \times }\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} \right) = \\ \;\;\;\;\;\;\;\left( {{\mathit{\boldsymbol{W}}_1} + {\mathit{\boldsymbol{W}}_2} + {\mathit{\boldsymbol{W}}_3}} \right)\mathit{\boldsymbol{\theta }} \end{array}$ （20）

 $\mathit{\boldsymbol{M}}\left( \mathit{\boldsymbol{x}} \right) = \left[ {\begin{array}{*{20}{c}} {{x_1}}&{{x_2}}&{{x_3}}&0&0&0\\ 0&{{x_1}}&0&{{x_2}}&{{x_3}}&0\\ 0&0&{{x_1}}&0&{{x_2}}&{{x_3}} \end{array}} \right]$ （21）

 ${\mathit{\boldsymbol{W}}_1} = \mathit{\boldsymbol{M}}\left( {{k_\mathit{\boldsymbol{q}}}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} - {k_\mathit{\boldsymbol{\omega }}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{\rm{d}}}} \right),{\mathit{\boldsymbol{W}}_2} = {k_\mathit{\boldsymbol{\omega }}}\mathit{\boldsymbol{M}}\left( \mathit{\boldsymbol{\omega }} \right)$ （22）

 ${\mathit{\boldsymbol{\beta }}_1} = \mathit{\boldsymbol{W}}_1^{\rm{T}}\mathit{\boldsymbol{\omega }}$ （23）
 ${\mathit{\boldsymbol{\beta }}_2} = {k_\mathit{\boldsymbol{\omega }}}{\left[ {\begin{array}{*{20}{c}} {\frac{1}{2}\omega _1^2}&{{\omega _1}{\omega _2}}&{{\omega _1}{\omega _3}}&{\frac{1}{2}\omega _2^2}&{{\omega _2}{\omega _3}}&{\frac{1}{2}\omega _3^2} \end{array}} \right]^{\rm{T}}}$ （24）

 $\begin{array}{l} {\mathit{\boldsymbol{W}}_3} = \left( {\mathit{\boldsymbol{\omega }},\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} \right) = \left[ {\begin{array}{*{20}{c}} 0&{{\omega _1}{\omega _3}}&{ - {\omega _1}{\omega _2}}&{{\omega _2}{\omega _3}}&{\omega _3^2 - \omega _2^2}&{ - {\omega _2}{\omega _3}}\\ { - {\omega _1}{\omega _3}}&{ - {\omega _2}{\omega _3}}&{\omega _1^2 - \omega _3^2}&0&{{\omega _1}{\omega _2}}&{{\omega _1}{\omega _3}}\\ {{\omega _1}{\omega _2}}&{\omega _2^2 - \omega _1^2}&{{\omega _2}{\omega _3}}&{ - {\omega _1}{\omega _2}}&{ - {\omega _1}{\omega _3}}&0 \end{array}} \right] + \\ \;\;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}} {{\omega _2}{\mathit{\Omega }_3} - {\omega _3}{\mathit{\Omega }_2}}&{{\omega _3}{\mathit{\Omega }_1} - {\omega _1}{\mathit{\Omega }_3}}&{{\omega _1}{\mathit{\Omega }_2} - {\omega _2}{\mathit{\Omega }_1}}&0&0&0\\ 0&{{\omega _2}{\mathit{\Omega }_3} - {\omega _3}{\mathit{\Omega }_2}}&0&{{\omega _3}{\mathit{\Omega }_1} - {\omega _1}{\mathit{\Omega }_3}}&{{\omega _1}{\mathit{\Omega }_2} - {\omega _2}{\mathit{\Omega }_1}}&0\\ 0&0&{{\omega _2}{\mathit{\Omega }_3} - {\omega _3}{\mathit{\Omega }_2}}&0&{{\omega _3}{\mathit{\Omega }_1} - {\omega _1}{\mathit{\Omega }_3}}&{{\omega _1}{\mathit{\Omega }_2} - {\omega _2}{\mathit{\Omega }_1}} \end{array}} \right] \end{array}$ （25）

 ${\mathit{\boldsymbol{\beta }}_3} = \mathit{\boldsymbol{W}}_3^{\rm{T}}\left( {\mathit{\boldsymbol{\hat \omega }},\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} \right)\mathit{\boldsymbol{\omega }}$ （26）

 $\mathit{\boldsymbol{\dot {\hat \omega} }} = - {k_\mathit{\boldsymbol{q}}}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} - {k_\mathit{\boldsymbol{\omega }}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}} + \mathit{\boldsymbol{ \boldsymbol{\dot \varOmega} }} - {k_{\rm{f}}}\left( {\mathit{\boldsymbol{\hat \omega }} - \mathit{\boldsymbol{\omega }}} \right)$ （27）

 $\mathit{\boldsymbol{\beta }}\left( {\mathit{\boldsymbol{\omega }},\mathit{\boldsymbol{\varphi }}} \right) = \gamma \left( {{\mathit{\boldsymbol{\beta }}_1} + {\mathit{\boldsymbol{\beta }}_2} + {\mathit{\boldsymbol{\beta }}_3}} \right)$ （28）

 $\begin{array}{l} \frac{{\partial \mathit{\boldsymbol{\beta }}}}{{\partial \mathit{\boldsymbol{\omega }}}} = \gamma {\left( {{\mathit{\boldsymbol{W}}_1} + {\mathit{\boldsymbol{W}}_2} + {\mathit{\boldsymbol{W}}_3}\left( {\mathit{\boldsymbol{\hat \omega }},\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} \right)} \right)^{\rm{T}}} = \\ \;\;\;\;\gamma {\left( {\mathit{\boldsymbol{W}} - {\mathit{\boldsymbol{W}}_3}\left( {\mathit{\boldsymbol{\omega }},\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} \right) + {\mathit{\boldsymbol{W}}_3}\left( {\mathit{\boldsymbol{\hat \omega }},\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} \right)} \right)^{\rm{T}}} = \gamma {\left( {\mathit{\boldsymbol{W}} + \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}} \right)^{\rm{T}}} \end{array}$ （29）

 $\mathit{\boldsymbol{\varphi }} = \left\{ {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}},\mathit{\boldsymbol{ \boldsymbol{\varOmega} }},{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{\rm{d}}},\mathit{\boldsymbol{\hat \omega }}} \right\}$

 ${\left\| {\mathit{\boldsymbol{f}}\left( t \right)} \right\|_2} = \sqrt {\int_0^\infty {{\mathit{\boldsymbol{f}}^{\rm{T}}}\left( \tau \right)\mathit{\boldsymbol{f}}\left( \tau \right){\rm{d}}\tau } } < \infty$

 ${\left\| {\mathit{\boldsymbol{f}}\left( t \right)} \right\|_\infty } = \mathop {\sup }\limits_t \left| {{f_i}\left( t \right)} \right| < \infty$

 ${k_\mathit{\boldsymbol{q}}} = 1 + {\delta _\mathit{\boldsymbol{q}}},{k_\mathit{\boldsymbol{\omega }}} = 1.5 + {\delta _\mathit{\boldsymbol{\omega }}}$ （30）
 ${k_{\rm{f}}} = 1 + {\delta _{\rm{d}}} + {\delta _{\rm{f}}}$ （31）

 ${{\dot \delta }_{\rm{d}}} = \lambda {\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2} = \lambda {\left\| {\mathit{\boldsymbol{\hat \omega }} - \mathit{\boldsymbol{\omega }}} \right\|^2}$ （32）

2.2 动态放缩法

 $\mathit{\boldsymbol{z}} = \frac{{\mathit{\boldsymbol{\tilde \theta }}}}{R}$ （33）

 $R = \frac{{\sqrt {{j_{\rm{m}}}} }}{{\exp \left( {\frac{{1 + C}}{2}} \right)}}\exp \left( {\sqrt {\ln f\left( r \right) + C} } \right)$ （34）

 $\dot r = \frac{\gamma }{{{j_{\rm{m}}}}} \cdot \frac{{f\left( r \right)\sqrt {\ln f\left( r \right) + C} }}{{{f^\prime }\left( r \right)}}{\left\| \Delta \right\|^2}$ （35）

 $\frac{{\dot R}}{R} = \frac{{{f^\prime }\left( r \right)\dot r}}{{2f\left( r \right)\sqrt {\ln f\left( r \right) + C} }} = \frac{\gamma }{{2{j_{\rm{m}}}}}{\left\| \Delta \right\|^2}$ （36）
 $\begin{array}{l} {R^2} \le {\left[ {\frac{{\sqrt {{j_{\rm{m}}}} }}{{\exp \left( {\frac{{1 + C}}{2}} \right)}}\exp \left( {\frac{1}{2} + \frac{{\ln f\left( r \right) + C}}{2}} \right)} \right]^2} = \\ \;\;\;\;{j_{\rm{m}}}f\left( r \right) \end{array}$ （37）

 $\mathit{\boldsymbol{\dot z}} = \frac{{\mathit{\boldsymbol{\dot {\tilde \theta} }}}}{R} - \frac{{\dot R}}{R}\mathit{\boldsymbol{z}} = - \gamma {\left( {\mathit{\boldsymbol{W}} + \Delta } \right)^{\rm{T}}}{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}} - \frac{\gamma }{{2{j_{\rm{m}}}}}{\left\| \Delta \right\|^2}\mathit{\boldsymbol{z}}$ （38）

 ${V_\mathit{\boldsymbol{z}}} = \frac{1}{{2\gamma }}{\mathit{\boldsymbol{z}}^{\rm{T}}}\mathit{\boldsymbol{z}}$ （39）

 $\begin{array}{l} {{\dot V}_\mathit{\boldsymbol{z}}} = - {\mathit{\boldsymbol{z}}^{\rm{T}}}{\mathit{\boldsymbol{W}}^{\rm{T}}}{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}} - {\mathit{\boldsymbol{z}}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}^{\rm{T}}}{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}} - \\ \;\;\;\;\;\;\;\frac{1}{{2{j_{\rm{m}}}}}{\mathit{\boldsymbol{z}}^{\rm{T}}}{\left\| \Delta \right\|^2}\mathit{\boldsymbol{z}} \le {j_{\rm{m}}}{\left\| {{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}}} \right\|^2} - \\ \;\;\;\;\;\;\;{\mathit{\boldsymbol{z}}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}^{\rm{T}}}{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}} - \frac{1}{{2{j_{\rm{m}}}}}{\left\| {\Delta \mathit{\boldsymbol{z}}} \right\|^2} \le - \frac{{{j_{\rm{m}}}}}{2}{\left\| {{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}}} \right\|^2} \end{array}$ （40）
2.3 闭环系统稳定性分析

 ${V_{\mathit{\boldsymbol{\tilde \omega }}}} = \frac{1}{2}{{\mathit{\boldsymbol{\tilde \omega }}}^{\rm{T}}}\mathit{\boldsymbol{\tilde \omega }}$ （41）
 $\begin{array}{l} {V_{\rm{e}}} = \left( {{k_\mathit{\boldsymbol{q}}} + {k_\mathit{\boldsymbol{\omega }}}} \right)\left[ {{{\left( {1 - {q_{{\rm{e0}}}}} \right)}^2} + \mathit{\boldsymbol{q}}_{{\rm{ev}}}^{\rm{T}}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}}} \right] + \\ \;\;\;\;\;\;\;\frac{1}{2}\mathit{\boldsymbol{\omega }}_{\rm{e}}^{\rm{T}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}} + \mathit{\boldsymbol{q}}_{{\rm{ev}}}^{\rm{T}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}} \end{array}$ （42）

 ${{\dot V}_{\mathit{\boldsymbol{\tilde \omega }}}} = {{\mathit{\boldsymbol{\tilde \omega }}}^{\rm{T}}}\mathit{\boldsymbol{\dot {\tilde \omega} }} = R{{\mathit{\boldsymbol{\tilde \omega }}}^{\rm{T}}}{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}} - {k_{\rm{f}}}{\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2}$ （43）
 $\begin{array}{l} {{\dot V}_{\rm{e}}} = \left( {{k_\mathit{\boldsymbol{q}}} + {k_\mathit{\boldsymbol{\omega }}}} \right)\mathit{\boldsymbol{q}}_{{\rm{ev}}}^{\rm{T}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}} + \mathit{\boldsymbol{\omega }}_{\rm{e}}^{\rm{T}}\left( { - {k_\mathit{\boldsymbol{q}}}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} - {k_\omega }{\mathit{\boldsymbol{\omega }}_{\rm{e}}} - } \right.\\ \;\;\;\;\;\left. {R{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}}} \right) + \mathit{\boldsymbol{q}}_{{\rm{ev}}}^{\rm{T}}\left( { - {k_q}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} - {k_\mathit{\boldsymbol{\omega }}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}} - } \right.\\ \;\;\;\;\;\left. {R{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}}} \right) + \mathit{\boldsymbol{\omega }}_{\rm{e}}^{\rm{T}}\left( {\frac{1}{2}\left( {{q_{{\rm{e0}}}}{\mathit{\boldsymbol{I}}_3} + \mathit{\boldsymbol{q}}_{{\rm{ev}}}^ \times } \right){\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right) \le \\ \;\;\;\;\; - {k_\mathit{\boldsymbol{q}}}{\left\| {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}}} \right\|^2} - {k_\mathit{\boldsymbol{\omega }}}{\left\| {{\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right\|^2} - R{\left( {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} + {\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right)^{\rm{T}}} \cdot \\ \;\;\;\;\;{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}} + \frac{{\left\| {{q_{{\rm{e0}}}}{\mathit{\boldsymbol{I}}_3} + \mathit{\boldsymbol{q}}_{{\rm{ev}}}^ \times } \right\|}}{2}{\left\| {{\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right\|^2} \le \\ \;\;\;\;\; - {k_\mathit{\boldsymbol{q}}}{\left\| {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}}} \right\|^2} - \left( {{k_\mathit{\boldsymbol{\omega }}} - \frac{1}{2}} \right){\left\| {{\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right\|^2} - \\ \;\;\;\;\;R{\left( {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} + {\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}} \end{array}$ （44）

 ${V_{\rm{c}}} = {V_\mathit{\boldsymbol{z}}} + {V_{\mathit{\boldsymbol{\tilde \omega }}}} + {V_{\rm{e}}}$ （45）

 $\begin{array}{l} {{\dot V}_{\rm{c}}} \le - \frac{{{j_{\rm{m}}}}}{2}{\left\| {{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}}} \right\|^2} + R{{\mathit{\boldsymbol{\tilde \omega }}}^{\rm{T}}}{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}} - {k_{\rm{f}}}{\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2} - \\ \;\;\;\;\;\;{k_\mathit{\boldsymbol{q}}}{\left\| {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}}} \right\|^2} - \left( {{k_\mathit{\boldsymbol{\omega }}} - \frac{1}{2}} \right){\left\| {{\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right\|^2} - \\ \;\;\;\;\;R{\left( {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} + {\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right)^{\rm{T}}}{\mathit{\boldsymbol{J}}^{ - 1}}\mathit{\boldsymbol{Wz}} \le \\ \;\;\;\;\; - \left( {{k_\mathit{\boldsymbol{q}}} - \frac{{{R^2}}}{{{j_{\rm{m}}}}}} \right){\left\| {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}}} \right\|^2} - \left( {{k_\mathit{\boldsymbol{\omega }}} - \frac{1}{2} - \frac{{{R^2}}}{{{j_{\rm{m}}}}}} \right) \cdot \\ \;\;\;\;\;{\left\| {{\omega _{\rm{e}}}} \right\|^2} - \left( {{k_{\rm{f}}} - \frac{{{R^2}}}{{{j_{\rm{m}}}}} - {\delta _{\rm{d}}}} \right){\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2} \le \\ \;\;\;\;\; - {\delta _\mathit{\boldsymbol{q}}}{\left\| {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}}} \right\|^2} - {\delta _\mathit{\boldsymbol{\omega }}}{\left\| {{\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right\|^2} - \left( {{\delta _{\rm{f}}} + {\delta _{\rm{d}}}} \right){\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2} \le 0 \end{array}$ （46）

 $\begin{array}{l} {\left\| {\mathit{\boldsymbol{W}}_2^{\rm{T}}\left( {\mathit{\boldsymbol{\hat \omega }},\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} \right) - \mathit{\boldsymbol{W}}_2^{\rm{T}}\left( {\mathit{\boldsymbol{\omega }},\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} \right)} \right\|^2} = \\ \;\;\;\;\;\;\left\| {L\left( {\mathit{\boldsymbol{\hat \omega }},\mathit{\boldsymbol{\omega }},\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} \right)} \right\|{\left\| {\mathit{\boldsymbol{\hat \omega }} - \mathit{\boldsymbol{\omega }}} \right\|^2} \le L{\left\| {\mathit{\boldsymbol{\hat \omega }} - \mathit{\boldsymbol{\omega }}} \right\|^2} \end{array}$ （47）

 ${V_{\rm{f}}} = \frac{1}{{2\lambda }}{\left( {{\delta _{\rm{d}}} - L} \right)^2}$ （48）
 ${V_{\rm{R}}} = \frac{1}{\gamma }{R^2}$ （49）

 ${{\dot V}_{\rm{f}}} = \left( {{\delta _{\rm{d}}} - L} \right){\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2}$ （50）
 $\begin{array}{l} {{\dot V}_{\rm{R}}} = \frac{2}{\gamma }R\dot R = \frac{2}{\gamma } \cdot \frac{{\dot R}}{R}{R^2} = \\ \;\;\;f\left( r \right){\left\| \Delta \right\|^2} \le {\left\| \Delta \right\|^2} \le L{\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2} \end{array}$ （51）

 $V = {V_{\rm{c}}} + {V_{\rm{f}}} + {V_{\rm{R}}}$ （52）

 $\begin{array}{l} \dot V \le - {\delta _\mathit{\boldsymbol{q}}}{\left\| {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}}} \right\|^2} - {\delta _\mathit{\boldsymbol{\omega }}}{\left\| {{\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right\|^2} - \left( {{\delta _{\rm{f}}} + {\delta _{\rm{d}}}} \right){\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2} + \\ \;\;\;\;\;L{\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2} + \left( {{\delta _{\rm{d}}} - L} \right){\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2} \le \\ \;\;\;\;\; - {\delta _\mathit{\boldsymbol{q}}}{\left\| {{\mathit{\boldsymbol{q}}_{{\rm{ev}}}}} \right\|^2} - {\delta _\mathit{\boldsymbol{\omega }}}{\left\| {{\mathit{\boldsymbol{\omega }}_{\rm{e}}}} \right\|^2} - {\delta _{\rm{f}}}{\left\| {\mathit{\boldsymbol{\tilde \omega }}} \right\|^2} \le 0 \end{array}$ （53）

 ${{\mathit{\boldsymbol{\dot \omega }}}_{\rm{e}}} = - {k_\mathit{\boldsymbol{q}}}{\mathit{\boldsymbol{q}}_{{\rm{ev}}}} - {k_\mathit{\boldsymbol{\omega }}}{\mathit{\boldsymbol{\omega }}_{\rm{e}}}$ （54）

3 数值仿真

 $\mathit{\boldsymbol{J}} = \left[ {\begin{array}{*{20}{c}} {20}&{1.2}&{0.9}\\ {1.2}&{17}&{1.4}\\ {0.9}&{1.4}&{15} \end{array}} \right]\;{\rm{kg}} \cdot {{\rm{m}}^2}$

 $\begin{array}{l} {\mathit{\boldsymbol{\omega }}_{\rm{d}}}\left( t \right) = \left[ {0.3\left( {1 - {{\rm{e}}^{ - 0.01{t^2}}}} \right)\cos t + t{{\rm{e}}^{ - 0.01{t^2}}}\left( {0.08{\rm{ \mathsf{ π} }} + } \right.} \right.\\ \;\;\;\;\;\;\;\left. {\left. {0.006\sin t} \right)} \right] \cdot {\left[ {\begin{array}{*{20}{l}} 1&1&1 \end{array}} \right]^{\rm{T}}}{\rm{rad}}/{\rm{s}} \end{array}$

 $\mathit{\boldsymbol{q}}\left( 0 \right) = {\left[ {\begin{array}{*{20}{l}} {0.5}&{0.5}&{0.5}&{0.5} \end{array}} \right]^{\rm{T}}}$
 ${\mathit{\boldsymbol{q}}_{\rm{d}}}\left( 0 \right) = {\left[ {\begin{array}{*{20}{c}} 1&0&0&0 \end{array}} \right]^{\rm{T}}}$
 $\mathit{\boldsymbol{\omega }}\left( 0 \right) = {\left[ {\begin{array}{*{20}{l}} {1.5}&{ - 2}&1 \end{array}} \right]^{\rm{T}}},\mathit{\boldsymbol{\hat \omega }}\left( 0 \right) = \mathit{\boldsymbol{\omega }}\left( 0 \right)$
 ${\delta _{\rm{d}}}\left( 0 \right) = \left\| {\mathit{\boldsymbol{\hat \omega }}\left( 0 \right) - \mathit{\boldsymbol{\omega }}\left( 0 \right)} \right\| = 0$
 $\mathit{\boldsymbol{\hat \theta }}\left( 0 \right) = \mathit{\boldsymbol{\alpha }}\left( 0 \right) + \mathit{\boldsymbol{\beta }}\left( 0 \right) = {\left[ {\begin{array}{*{20}{l}} 0&0&0&0&0&0 \end{array}} \right]^{\rm{T}}}$

3.1 调节系数γλ的影响

 图 2 各项参数随时间变化曲线 Fig. 2 Time histories of each parameters

3.2 控制器性能的比较

 图 3 跟踪性能随时间变化曲线 Fig. 3 Time histories of tracking performance

3.3 控制器鲁棒性验证

 $\begin{array}{l} \mathit{\boldsymbol{d}} = \\ {\left[ {\begin{array}{*{20}{c}} {0.15\sin t}&{ - 0.05\cos \left( {t + \frac{{\rm{ \mathsf{ π} }}}{4}} \right)}&{0.1\sin \left( {t - \frac{{\rm{ \mathsf{ π} }}}{6}} \right)} \end{array}} \right]^{\rm{T}}} \end{array}$

 图 4 控制误差范数‖qev‖、‖ωe‖和‖$\mathit{\boldsymbol{\tilde \omega }}$‖随时间变化曲线 Fig. 4 Time histories of control errors norms ‖qev‖, ‖ωe‖ and ‖$\mathit{\boldsymbol{\tilde \omega }}$‖
 图 5 控制力矩范数‖u‖随时间变化曲线 Fig. 5 Time histories of control torque norms ‖u‖
 图 6 估计误差范数‖$\mathit{\boldsymbol{\tilde \theta }}$‖、‖$\mathit{\boldsymbol{W\tilde \theta }}$‖随时间变化曲线 Fig. 6 Time histories of estimated errors norms ‖$\mathit{\boldsymbol{\tilde \theta }}$‖, ‖$\mathit{\boldsymbol{W\tilde \theta }}$‖

4 结论

http://dx.doi.org/10.7527/S1000-6893.2019.23428

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#### 文章信息

XIA Dongdong, YUE Xiaokui

Immersion and invariance based attitude adaptive tracking control for spacecraft

Acta Aeronautica et Astronautica Sinica, 2020, 41(2): 323428.
http://dx.doi.org/10.7527/S1000-6893.2019.23428