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1. 中国空间技术研究院 北京控制工程研究所, 北京 100190;
2. 中国空间技术研究院 北京空间飞行器总体设计部, 北京 100194;
3. 中山大学 航空航天学院, 广州 510006

Reconfigurability evaluation and autonomous reconfiguration of systems based on operator theory
XU Heyu1, WANG Dayi2, LI Wenbo1, LIU Chengrui1, FU Fangzhou3
1. Beijing Institute of Control Engineering, China Academy of Space Technology, Beijing 100190, China;
2. Beijing Institute of Spacecraft System Engineering, China Academy of Space Technology, Beijing 100194, China;
3. School of Aeronautics and Astronautics, Sun Yat-sen University, Guangzhou 510006, China
Abstract: The spacecraft in orbit resources are severely limited, including computing resources, hardware resources, and energy resources. In this paper, the evaluation of reconfigurability and the autonomous reconfiguration strategy for attitude control system of spacecraft are studied based on the operator theory. Firstly, based on the operator theory of Stable Kernel Representation (SKR) and Stable Image Representation (SIR), the evaluation index of system reconfigurability is given, and the ability of system reconfigurability is described quantitatively. The theory breaks through the limitation of the evaluation method of system reconfigurability based on the theory of coprime decomposition to the linear property of system. At the same time, based on the above results, the maximum boundary of system reconfigurability is given, which provides a clear index for designers to design autonomous reconfiguration strategies. Then, by considering the system reconfigurability in the design stage, the system reconfigurability potential is fully exploited and utilized, providing theoretical guidance for the design of autonomous reconfiguration strategy. Finally, the validity and correctness of the proposed method are verified by a simulation example.
Keywords: spacecraft attitude control system    reconfigurability evaluation    autonomous reconfiguration    stable kernel representation    stable image representation

1 问题描述

 ${\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}^{{\mathit{\boldsymbol{x}}_0}}}:U \to Y:\mathit{\boldsymbol{y}}(t) = {\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}^{{\mathit{\boldsymbol{x}}_0}}}(\mathit{\boldsymbol{u}}(t))$ （1）

2 基于算子理论的可重构性评价方法

 ${R_\varSigma }\left( {\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{u}}\\ \mathit{\boldsymbol{y}} \end{array}} \right]} \right) = 0$ （2）

 $\left( {\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{u}}\\ \mathit{\boldsymbol{y}} \end{array}} \right]} \right) = {I_\varSigma }(\mathit{\boldsymbol{v}})\;\:\forall {\mathit{\boldsymbol{p}}_0} = {\mathit{\boldsymbol{x}}_0}$ （3）

 ${\varDelta _{\rm{f}}}:\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{\alpha }}(\mathit{\boldsymbol{x,u,y}})}\\ {\mathit{\boldsymbol{\omega }} = \mathit{\boldsymbol{\beta }}(\mathit{\boldsymbol{x,u,y}})} \end{array}} \right.$ （4）

 ${\mathit{\boldsymbol{z}}_{{\varSigma _{\rm{f}}}}} = {R_\varSigma }\left( {\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{u}}\\ \mathit{\boldsymbol{y}} \end{array}} \right]} \right) + {\varDelta _{\rm{f}}}\left( {\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{u}}\\ \mathit{\boldsymbol{y}} \end{array}} \right]} \right)$ （5）

 $G = {\tilde M^{ - 1}}\tilde N = N{M^{ - 1}}$ （6）

 $\begin{array}{*{20}{l}} {{G_{\rm{f}}} = {{(\tilde M + {{\tilde M}_{\rm{f}}})}^{ - 1}}(\tilde N + {{\tilde N}_{\rm{f}}}) = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (N + {N_{\rm{f}}}){{(M + {M_{\rm{f}}})}^{ - 1}}} \end{array}$ （7）

 $\begin{array}{l} 0 = [ - \tilde N - {{\tilde N}_{\rm{f}}}\;\:\tilde M + {{\tilde M}_{\rm{f}}}]\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{u}}\\ \mathit{\boldsymbol{y}} \end{array}} \right] = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \underbrace {\left[ {\begin{array}{*{20}{l}} { - \tilde N}&{\tilde M} \end{array}} \right]\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{u}}\\ \mathit{\boldsymbol{y}} \end{array}} \right]}_{{R_\varSigma }(\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{u}}\\ \mathit{\boldsymbol{y}} \end{array}} \right])} + \underbrace {\left[ {\begin{array}{*{20}{l}} { - {{\tilde N}_{\rm{f}}}}&{{{\tilde M}_{\rm{f}}}} \end{array}} \right]\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{u}}\\ \mathit{\boldsymbol{y}} \end{array}} \right]}_{{\Delta _{\rm{f}}}(\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{u}}\\ \mathit{\boldsymbol{y}} \end{array}} \right])} \end{array}$ （8）

 $Q:\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot x}}}_q} = \varphi ({\mathit{\boldsymbol{x}}_q},{\mathit{\boldsymbol{z}}_l})}\\ {{\mathit{\boldsymbol{r}}_k} = \phi ({\mathit{\boldsymbol{x}}_q},{\mathit{\boldsymbol{z}}_l})} \end{array}} \right.$ （9）

 ${{R_{{K_Q}}} = {R_Q} \circ {R_{\{ \varSigma ,K\} }}}$ （10）
 ${{I_{{K_Q}}} = R_\varSigma ^ - + {I_\varSigma } \circ Q}$ （11）

 $\lambda ( - {\varDelta _{\rm{f}}} \circ {I_{{K_Q}}}) \cdot \lambda ({({R_\varSigma } \circ {I_{{K_Q}}})^{ - 1}}) < 1$ （12）

 ${\mathit{\boldsymbol{z}}_{{\varSigma _{\rm{f}}}}} = ({R_\varSigma } + {\varDelta _{\rm{f}}})(\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{u}}\\ \mathit{\boldsymbol{y}} \end{array}} \right]) = ({R_\varSigma } + {\Delta _{\rm{f}}}) \circ {I_{{K_Q}}}({\mathit{\boldsymbol{z}}_\varSigma }) = 0$ （13）

 ${R_\varSigma } \circ {I_{{K_Q}}}({\mathit{\boldsymbol{z}}_\varSigma }) = - {\varDelta _{\rm{f}}} \circ {I_{{K_Q}}}({\mathit{\boldsymbol{z}}_\varSigma })$ （14）

 $\underbrace {\lambda ( - {\varDelta _{\rm{f}}} \circ {I_{{K_O}}})}_{{\rm{故障值}}} < \underbrace {\frac{1}{{\lambda ({{({R_\varSigma } \circ {I_{{K_Q}}})}^{ - 1}})}}}_{{\rm{可重构性评价指标}}}$ （15）

3 基于算子理论的自主重构

 ${\delta _{{\rm{max}}}} = \sqrt {1 - \left\| {{R_\varSigma }} \right\|_{\rm{H}}^2}$ （16）

 ${\delta _{{\rm{max}}}} = \sqrt {1 - \left\| {[\tilde N,\tilde M]} \right\|_{\rm{H}}^2}$ （17）

① 若δ < δmax，则该控制器可以达到重构目的，进行步骤3；

② 若δδmax，则该控制器不可以达到重构目的，需重新设计控制器。

① 若故障值λ(-Δf$\circ$IKQ)≤δ，重构控制器仍能使故障系统恢复原有性能；

② 若故障值λ(-Δf$\circ$IKQ)>δ，则该系统无法通过自主重构恢复原有性能。

4 仿真验证

 $\left\{ {\begin{array}{*{20}{l}} {{{\dot x}_1} = - x_1^2 + {x_3}}\\ {{{\dot x}_2} = - {x_1} + x_3^2 + {u_1}}\\ {{{\dot x}_3} = - {x_2} + {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {x_3}}\\ {{{\dot x}_4} = - {x_1} - x_4^2 + {u_2}} \end{array}} \right.$ （18）

$\left\{ {\begin{array}{*{20}{l}} {{y_1} = {x_1}}\\ {{y_2} = {x_4}} \end{array}} \right.$, $\mathit{\boldsymbol{x}} = {\left[ {\begin{array}{*{20}{c}} {{x_1}}&{{x_2}}&{{x_3}}&{{x_4}} \end{array}} \right]^{\rm{T}}}$, $\mathit{\boldsymbol{y}} = \left[ {\begin{array}{*{20}{c}} {{y_1}}\\ {{y_2}} \end{array}} \right]$, 则式(19)可记为

 $\Sigma :\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) + \mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}})\mathit{\boldsymbol{u}}}\\ {\mathit{\boldsymbol{y}} = \mathit{\boldsymbol{hx}}} \end{array}} \right.$ （19）

 $\begin{array}{l} \mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) = \left[ {\begin{array}{*{20}{c}} { - x_1^2 + {x_3}}\\ { - {x_1} + x_3^2}\\ { - {x_2} + {\rm{sin}}{x_3}}\\ { - {x_1} - x_4^2} \end{array}} \right],\;\:\mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}}) = \left[ {\begin{array}{*{20}{c}} 0&0\\ 1&0\\ 0&0\\ 0&1 \end{array}} \right]\\ \mathit{\boldsymbol{h}} = \left[ {\begin{array}{*{20}{l}} 1&0&0&0\\ 0&0&0&1 \end{array}} \right] \end{array}$ （20）

 ${\mathit{\boldsymbol{z}}_1} = {R_\varSigma }({[{\mathit{\boldsymbol{u}}^{\rm{T}}},{\mathit{\boldsymbol{y}}^{\rm{T}}}]^{\rm{T}}})$ （21）

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) + \mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}})\mathit{\boldsymbol{u}} + \mathit{\boldsymbol{l}}(\mathit{\boldsymbol{x}})(\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{h}}(\mathit{\boldsymbol{x}}))}\\ {{\mathit{\boldsymbol{z}}_1} = \mathit{\boldsymbol{y}} - \mathit{\boldsymbol{hx}}} \end{array}} \right.$ （22）

h=lT(x)WxTWx> 0是Hamilton Jacobi等式(式(23))的解。

 ${\mathit{\boldsymbol{W}}_x}\mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) + \frac{1}{2}{\mathit{\boldsymbol{W}}_x}\mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}}){\mathit{\boldsymbol{g}}^{\rm{T}}}(\mathit{\boldsymbol{x}}){\mathit{\boldsymbol{W}}_x} - \frac{1}{2}{\mathit{\boldsymbol{h}}^{\rm{T}}}(\mathit{\boldsymbol{x}})\mathit{\boldsymbol{h}}(\mathit{\boldsymbol{x}}) = {\bf{0}}$ （23）

 $\left( {\left[ {\begin{array}{*{20}{l}} \mathit{\boldsymbol{y}}\\ \mathit{\boldsymbol{u}} \end{array}} \right]} \right) = {I_\Sigma }{\mathit{\boldsymbol{z}}_{\rm{r}}}$ （24）

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) + \mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}})\mathit{\boldsymbol{k}}(\mathit{\boldsymbol{x}}) + \mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}}){\mathit{\boldsymbol{z}}_{\rm{r}}}}\\ {\mathit{\boldsymbol{y}} = \mathit{\boldsymbol{hx}}}\\ {{\mathit{\boldsymbol{z}}_{\rm{r}}} = \mathit{\boldsymbol{u}} - \mathit{\boldsymbol{k}}(\mathit{\boldsymbol{x}})} \end{array}} \right.$ （25）

k(x)=-gT(x)VxTVx> 0是Hamilton Jacobi等式(式(26))的解。

 ${\mathit{\boldsymbol{V}}_x}\mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) - \frac{1}{2}{\mathit{\boldsymbol{V}}_x}\mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}}){\mathit{\boldsymbol{g}}^{\rm{T}}}(\mathit{\boldsymbol{x}}){\mathit{\boldsymbol{V}}_x} + \frac{1}{2}{\mathit{\boldsymbol{h}}^{\rm{T}}}(\mathit{\boldsymbol{x}})\mathit{\boldsymbol{h}}(\mathit{\boldsymbol{x}}) = {\bf{0}}$ （26）

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) + \mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}})\mathit{\boldsymbol{u}} + \mathit{\boldsymbol{l}}(\mathit{\boldsymbol{x}})(\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{h}}(\mathit{\boldsymbol{x}}))}\\ {\mathit{\boldsymbol{y}} = \mathit{\boldsymbol{hx}} - \mathit{\boldsymbol{\beta }}} \end{array}} \right.$ （27）

 $\begin{array}{*{20}{l}} {{\delta _{{\rm{max}}}} = \sqrt {1 - \left\| {{R_\Sigma }} \right\|_{\rm{H}}^2} = \sqrt {1 - \left\| {{\mathit{\boldsymbol{z}}_1}} \right\|_{\rm{H}}^2} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {1 - \left\| {\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{h}}(\mathit{\boldsymbol{x}})} \right\|_{\rm{H}}^2} } \end{array}$ （28）

 ${K_Q}:\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{h}}(\mathit{\boldsymbol{x}}) + \mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}})\mathit{\boldsymbol{u}} + \mathit{\boldsymbol{l}}(\mathit{\boldsymbol{x}})(\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{h}})}\\ {\mathit{\boldsymbol{u}} = - {\mathit{\boldsymbol{g}}^{\rm{T}}}(\mathit{\boldsymbol{x}}){\mathit{\boldsymbol{V}}^{\rm{T}}}(\mathit{\boldsymbol{x}}) + \mathit{\boldsymbol{f}}} \end{array}} \right.$ （29）

 ${\mathit{\boldsymbol{\phi}}} = \left[ {\begin{array}{*{20}{c}} { - 6x_1^4 + 6x_1^3 - 3x_1^2 + 4{x_1}}\\ { - 3{x_4} + x_4^2} \end{array}} \right]$ （30）

 ${\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {{u_1} = - 6x_1^4 + 8x_1^2{x_3} + 4{x_1} - 3x_3^2 + 2{x_1}{x_2} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2{x_1}{\kern 1pt} {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {x_3} - {x_2}{\kern 1pt} {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {x_3} + {\rm{sin}}{\kern 1pt} {\kern 1pt} {x_3}{\kern 1pt} {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {x_3} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 3( - x_1^2 + {x_3}) + 3(2x_1^3 - 2{x_1}{x_3} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {x_2} + {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {x_3})}\\ {{u_2} = {x_1} - 3{x_4} + x_4^2} \end{array}} \right.$ （31）

 $\begin{array}{*{20}{l}} {\delta = \frac{1}{{\lambda ({{({R_\Sigma } \circ (R_\Sigma ^ - + {I_\Sigma } \circ Q))}^{ - 1}})}} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{\lambda ({{({\mathit{\boldsymbol{z}}_l}(\mathit{\boldsymbol{z}}_l^{ - 1} + {\mathit{\boldsymbol{z}}_r}\phi ))}^{ - 1}})}} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{\lambda ({{((\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{h}})({{(\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{h}})}^{ - 1}} + (\mathit{\boldsymbol{u}} - \mathit{\boldsymbol{k}}(\mathit{\boldsymbol{x}})){\mathit{\boldsymbol{\phi}}} ))}^{ - 1}})}}} \end{array}$ （32）

 $\delta = 0.540{\kern 1pt} {\kern 1pt} {\kern 1pt} 9 < {\delta _{{\rm{max}}}} = 0.763{\kern 1pt} {\kern 1pt} {\kern 1pt} 1$ （33）

 图 1 无故障时系统单位阶跃响应 Fig. 1 Unit step response of system without fault

 $\lambda ( - {\varDelta _{\rm{f}}} \circ {I_{{K_Q}}}) = 0.558{\kern 1pt} {\kern 1pt} {\kern 1pt} 3 > \delta = 0.540{\kern 1pt} {\kern 1pt} {\kern 1pt} 9$ （34）

 图 2 故障时系统单位阶跃响应 Fig. 2 Unit step response of system with fault

 ${{\mathit{\boldsymbol{\phi}}} ^\prime } = \left[ {\begin{array}{*{20}{c}} { - 6x_1^4 + 10x_1^3 - 5x_1^2 + 6{x_1}}\\ { - 5{x_4} + x_4^2} \end{array}} \right]$ （35）

 $\left\{ {\begin{array}{*{20}{l}} {u_1^\prime = - 6x_1^4 + 8x_1^2{x_3} + 6{x_1} - 3x_3^2 + 2{x_1}{x_2} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2{x_1}{\kern 1pt} {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {x_3} - {x_2}{\kern 1pt} {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {x_3} + {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {x_3}{\kern 1pt} {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {x_3} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 5( - x_1^2 + {x_3}) + 5(2x_1^3 - 2{x_1}{x_3} - {x_2} + {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {x_3})}\\ {u_2^\prime = {x_1} - 5{x_4} + x_4^2} \end{array}} \right.$ （36）

 $\delta < \lambda ( - {\varDelta _{\rm{f}}} \circ {I_{{K_Q}}}) < {\delta ^\prime } < {\delta _{{\rm{max}}}}$ （37）

 图 3 自主重构后系统单位阶跃响应 Fig. 3 Unit step response of system after autonomous reconfiguration

5 结论

1) 基于系统核表示与像表示的技术手段针对非线性系统定量地给出了可重构性评价指标，使基于算子理论的可重构性评价方法更加完备化。

2) 通过求取可重构性最大边界挖掘系统重构潜力，从根本上提高了航天器姿态控制系统的重构能力。

3) 通过本文所提可重构性评价方法为自主重构提供必要的理论依据，在该理论指导下对系统进行自主重构，并通过仿真验证了该方法的正确性和有效性。

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http://dx.doi.org/10.7527/S1000-6893.2019.23747

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#### 文章信息

XU Heyu, WANG Dayi, LI Wenbo, LIU Chengrui, FU Fangzhou

Reconfigurability evaluation and autonomous reconfiguration of systems based on operator theory

Acta Aeronautica et Astronautica Sinica, 2020, 41(S1): 723747.
http://dx.doi.org/10.7527/S1000-6893.2019.23747