A. Appendix

respectively. In the first 140 s the trajectory of L\_

reduces continuously to 3 m/s. After that, <sup>L</sup>\_

between its lower bound and upper bound.

4. Conclusions

Optimization Algorithms - Examples

transfer period. As a reflection of the control input, L\_

Figure 11. Control inputs for the closed-loop control with changing climber speed.

<sup>c</sup> increases continuously until reaches its upper

<sup>c</sup> fluctuates around 3 m/s by the end of the

also shows fluctuation during the

bound. Then, it keeps at 10 m/s by 270 s with some slight fluctuations. From 270 s to 355 s, it

transfer phase with obvious small-scale fluctuations appear in the period of 120 s – 260 s and 360 s – 750 s. This impacts during the whole transfer period, the changeable speed of the climber has the ability to help the suppression of the libration angles and states trajectory tracking. This time history of the control inputs is shown in Figure 11 with frequent changes

This chapter investigated a piecewise parallel onboard optimal control algorithm to solve the optimal control issues in complex nonlinear dynamic systems in aerospace engineering. To test the validity of the proposed two-phase optimal control scheme, the long-term tether libration stability and fast nano-satellite deorbit under complex environmental perturbations and the libration suppression for PSE system are considered. For EDT system, instead of optimizing the control of fast and stable nano-satellite deorbit over the complete process, the current approach divides the deorbit process into a set of intervals. For the PSE system, each time interval is set depends on the minimize transfer time for equal transfer length interval. Within each interval, the predicting phase simplifies significantly the optimal control problem. The dynamic equations of libration motion are further simplified to reduce computational loads using the simple dynamic models. The trajectory of the stable libration states and current control input is then optimized for the fast deorbit within the interval based on the simplified dynamic equations. The tracking optimizes the trajectory tracking control using the finite receding horizon control theory within the time interval corresponding to the open-loop control state trajectory with the same interval number. By applying the close-loop control modification, the system motions are integrated without any simplification of the dynamics or environmental perturbations and the instantaneous states of the orbital and libration motions. The i-th time interval's closed-loop tracking is processed in tracking phase while the The detailed expressions for the matrixes C and M are shown as followed,

C ¼ ½ � C<sup>1</sup> C<sup>2</sup> C<sup>3</sup> , M ¼ M<sup>11</sup> M<sup>12</sup> 0 M<sup>T</sup> M<sup>22</sup> 0 0 0 M<sup>33</sup> C<sup>1</sup> ¼ E 0 χ1 χ<sup>2</sup> χ1 χ<sup>2</sup> ⋱ ⋱ χ1 <sup>n</sup>�<sup>1</sup> <sup>χ</sup><sup>2</sup> n�1 E , C<sup>2</sup> ¼ 0 0 χ3 χ<sup>4</sup> χ3 χ<sup>4</sup> ⋱ ⋱ χ3 <sup>n</sup>�<sup>1</sup> <sup>χ</sup><sup>4</sup> n�1 0 0 , C<sup>3</sup> <sup>¼</sup> <sup>2</sup> γ 0 0 B<sup>0</sup>:<sup>5</sup> ⋱ B<sup>n</sup>�0:<sup>5</sup> 0 0 χ1 <sup>j</sup> <sup>¼</sup> <sup>E</sup> <sup>þ</sup> <sup>γ</sup> <sup>A</sup><sup>j</sup> <sup>þ</sup> <sup>γ</sup> <sup>A</sup><sup>j</sup>þ0:<sup>5</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>A</sup><sup>j</sup>þ0:<sup>5</sup>Aj, <sup>χ</sup><sup>2</sup> <sup>j</sup> ¼ �<sup>E</sup> <sup>þ</sup> <sup>γ</sup> <sup>A</sup><sup>j</sup>þ<sup>1</sup> <sup>þ</sup> <sup>γ</sup> <sup>A</sup><sup>j</sup>þ0:<sup>5</sup> � <sup>γ</sup><sup>2</sup> <sup>A</sup><sup>j</sup>þ0:<sup>5</sup>A<sup>j</sup>þ<sup>1</sup> χ3 <sup>j</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup> <sup>A</sup><sup>j</sup>þ0:<sup>5</sup>B<sup>j</sup> <sup>þ</sup> <sup>γ</sup> <sup>B</sup>j, <sup>χ</sup><sup>4</sup> <sup>j</sup> ¼ � <sup>γ</sup><sup>2</sup> <sup>A</sup><sup>j</sup>þ0:<sup>5</sup>B<sup>j</sup>þ<sup>1</sup> <sup>þ</sup> <sup>γ</sup> <sup>B</sup><sup>j</sup>þ<sup>1</sup>

$$\mathbf{M}\_{11} = \begin{bmatrix} \sigma\_{00}^{11} & \sigma\_{01}^{11} & & & 0 \\\\ \left(\sigma\_{01}^{11}\right)^{T} & \sigma\_{11}^{11} & & & \\ & & \ddots & & \\ & & & \sigma\_{n-1n-1}^{11} & \sigma\_{n-1n}^{11} \\\\ 0 & & \left(\sigma\_{n-1n}^{11}\right)^{T} & \sigma\_{nn}^{11} \end{bmatrix} \quad \mathbf{M}\_{22} = \begin{bmatrix} \sigma\_{00}^{22} & \sigma\_{01}^{22} & & & 0 \\\\ \left(\sigma\_{01}^{22}\right)^{T} & \sigma\_{11}^{22} & & & \\ & & \ddots & & \\ & & & \sigma\_{n-1n-1}^{22} & \sigma\_{n-1n}^{22} \\\\ 0 & & \left(\sigma\_{n-1n}^{22}\right)^{T} & \sigma\_{nn}^{22} \end{bmatrix}$$

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$$\mathbf{M}\_{12} = \begin{bmatrix} \sigma\_{00}^{12} & \sigma\_{01}^{12} & & & 0 \\ \sigma\_{10}^{12} & \sigma\_{11}^{12} & & & \\ & & \ddots & & \\ & & & \sigma\_{n-1n-1}^{12} & \sigma\_{n-1n}^{12} \\ 0 & & & \sigma\_{nn-1}^{12} & \sigma\_{nn}^{12} \end{bmatrix} \qquad \qquad \mathbf{M}\_{33} = \frac{2}{3} \overline{\mathcal{V}} \begin{bmatrix} R & & & 0 \\ & \ddots & \\ 0 & & R \end{bmatrix}$$

ϖ<sup>11</sup> jj <sup>¼</sup> <sup>γ</sup> <sup>6</sup> <sup>4</sup><sup>Q</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> 8 AT <sup>j</sup> QA<sup>j</sup> � �, <sup>ϖ</sup><sup>11</sup> jjþ<sup>1</sup> <sup>¼</sup> <sup>γ</sup> <sup>6</sup> <sup>Q</sup> � <sup>γ</sup><sup>2</sup> <sup>16</sup> <sup>A</sup><sup>T</sup> <sup>j</sup> QA<sup>j</sup>þ<sup>1</sup> <sup>þ</sup> <sup>γ</sup> 4 AT <sup>j</sup> <sup>Q</sup> � <sup>γ</sup> <sup>4</sup> QA<sup>j</sup>þ<sup>1</sup> � � ϖ<sup>11</sup> <sup>00</sup> <sup>¼</sup> <sup>γ</sup> <sup>6</sup> <sup>2</sup><sup>Q</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>16</sup> <sup>A</sup><sup>T</sup> <sup>0</sup> QA<sup>0</sup> <sup>þ</sup> <sup>γ</sup> 4 AT <sup>0</sup> <sup>Q</sup> <sup>þ</sup> <sup>γ</sup> <sup>4</sup> QA<sup>0</sup> � �, <sup>ϖ</sup><sup>11</sup> nn <sup>¼</sup> <sup>γ</sup> <sup>6</sup> <sup>2</sup><sup>Q</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>16</sup> <sup>A</sup><sup>T</sup> <sup>n</sup>QA<sup>n</sup> � <sup>γ</sup> 4 AT <sup>n</sup><sup>Q</sup> � <sup>γ</sup> <sup>4</sup> QA<sup>n</sup> � � þ S<sup>f</sup> ϖ<sup>22</sup> jj <sup>¼</sup> <sup>γ</sup> <sup>6</sup> <sup>2</sup><sup>R</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>8</sup> <sup>B</sup><sup>T</sup> <sup>j</sup> QB<sup>j</sup> � �, <sup>ϖ</sup><sup>22</sup> jjþ<sup>1</sup> ¼ �γB<sup>T</sup> <sup>j</sup> QB<sup>j</sup>þ<sup>1</sup> ϖ<sup>22</sup> <sup>00</sup> <sup>¼</sup> <sup>γ</sup> <sup>6</sup> <sup>R</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>16</sup> <sup>B</sup><sup>T</sup> <sup>0</sup> QB<sup>0</sup> � �, <sup>ϖ</sup><sup>22</sup> nn <sup>¼</sup> <sup>γ</sup> <sup>6</sup> <sup>R</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>16</sup> <sup>B</sup><sup>T</sup> <sup>n</sup>QB<sup>n</sup> � � ϖ<sup>12</sup> jj <sup>¼</sup> <sup>γ</sup><sup>3</sup> <sup>48</sup> <sup>A</sup><sup>T</sup> <sup>j</sup> QB<sup>j</sup> , ϖ<sup>12</sup> jjþ<sup>1</sup> ¼ � <sup>γ</sup><sup>2</sup> <sup>24</sup> QB<sup>j</sup>þ<sup>1</sup> <sup>þ</sup> <sup>γ</sup> 4 AT <sup>j</sup> QB<sup>j</sup>þ<sup>1</sup> � � ϖ<sup>12</sup> <sup>j</sup>þ1<sup>j</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup> <sup>24</sup> QB<sup>j</sup> � <sup>γ</sup> 4 AT <sup>j</sup>þ<sup>1</sup>QB<sup>j</sup> � �, <sup>ϖ</sup><sup>12</sup> <sup>00</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup> <sup>24</sup> QB<sup>0</sup> <sup>þ</sup> <sup>γ</sup> 4 AT <sup>0</sup> QB<sup>0</sup> � �, <sup>ϖ</sup><sup>12</sup> nn ¼ � <sup>γ</sup><sup>2</sup> <sup>24</sup> QB<sup>N</sup> � <sup>γ</sup> 4 AT <sup>n</sup>QB<sup>n</sup> � �

where E is the unit matrix which has the same dimension as Aj, and j = 0,1,2,…,n-1.
