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Reliable sampled-data vibration control for uncertain flexible spacecraft with frequency range limitation

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Abstract

This paper deals with the problem of reliable finite frequency vibration control for flexible spacecraft subject to torque constraint, actuator failure and linear fractional transformation (LFT) uncertainty. The practical sampled-data control signal is converted into a continuous-time input with time-varying delay. Since the main vibration energy of flexible spacecraft is dominated by low-frequency vibration modes lying in a specific frequency band, a novel reliable robust \(H_\infty \) output feedback controller with frequency constraint is employed here to suppress these resonance modes. Compared with classic full frequency scheme, finite frequency algorithm achieves a lower upper bound of vibration reduction performance even under the circumstance of torque constraint, actuator failure and LFT uncertainty. By convex optimization techniques, the problem of seeking admissible controller is transformed into the feasibility of linear matrix inequalities. The merits and effectiveness of proposed control algorithm are confirmed by an illustrative design example.

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Authors and Affiliations

  1. Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, 150080, China

    Shidong Xu, Guanghui Sun & Weichao Sun

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  1. Shidong Xu
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  2. Guanghui Sun
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  3. Weichao Sun
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Correspondence to Guanghui Sun.

Appendix

Appendix

Case 1 The parameters of finite frequency controller with known actuator faults are given as,

$$\begin{aligned}&A_{kf}=\begin{bmatrix} {\begin{matrix} -1.9084 &{}\quad -0.5485&{}\quad -2.2618&{}\quad 4.7793&{}\quad -5.3205&{}\quad 5.5982 \times 10^4\\ -0.3644&{}\quad -1.1119&{}\quad 0.2949&{}\quad -5.7501&{}\quad -1.6501&{}\quad -1.4542 \times 10^5\\ -2.1981&{}\quad -1.0246&{}\quad -4.4853&{}\quad -4.1386&{}\quad -9.3016&{}\quad -1.6853 \times 10^5\\ -2.3774&{}\quad -0.9291&{}\quad -7.8587&{}\quad 4.4295&{}\quad 6.8004&{}\quad 2.9858 \times 10^5\\ 1.5242&{}\quad 0.4157&{}\quad 8.5532&{}\quad -11.4473&{}\quad -20.4866&{}\quad -6.0065 \times 10^5\\ 0.1042&{}\quad -0.0427&{}\quad 2.2619&{}\quad -11.2116&{}\quad -3.7486&{}\quad -3.1226 \times 10^5 \end{matrix}} \end{bmatrix}\\&B_{kf}=\begin{bmatrix} {\begin{matrix} -0.2614 &{}\quad 0.4596 &{}\quad 0.4904 &{}\quad 0.2835\\ 0.1238 &{}\quad 0.4374 &{}\quad -1.3896 &{}\quad 0.1913\\ -0.3950 &{}\quad 0.5624 &{}\quad -2.4488 &{}\quad -0.2903\\ 0.8451 &{}\quad 0.5588 &{}\quad 0.5419 &{}\quad -0.5421\\ 0.5398 &{}\quad -0.2855 &{}\quad -0.3518 &{}\quad 0.6883\\ -0.0000 &{}\quad -0.0000 &{}\quad -0.8193 &{}\quad 0.1680 \end{matrix}} \end{bmatrix}\times 10^6\\&C_{kf}= \begin{bmatrix} {\begin{matrix} 0&\quad 0&\quad 0&\quad 0.0014&\quad 0&\quad 39.0605 \end{matrix}} \end{bmatrix},\\&A_{\tau f}=\begin{bmatrix} {\begin{matrix} 0.1002 &{}\quad -0.6474 &{}\quad -0.5648 &{}\quad 36.8033 &{}\quad -1.2979 &{}\quad 1.012 \times 10^6\\ -0.0860 &{}\quad 0.5556 &{}\quad 0.4847 &{}\quad -31.5829 &{}\quad 1.1138 &{}\quad -8.6859 \times 10^5\\ -0.0392 &{}\quad 0.2535 &{}\quad 0.2212 &{}\quad -14.4102 &{}\quad 0.5082 &{}\quad -3.9631 \times 10^5\\ 0.0083 &{}\quad -0.0535 &{}\quad -0.0466 &{}\quad 3.0396 &{}\quad -0.1072 &{}\quad 8.3593 \times 10^4\\ -0.0015 &{}\quad 0.0097 &{}\quad 0.0085 &{}\quad -0.5509 &{}\quad 0.0194 &{}\quad -1.5149 \times 10^4\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -0.0004 &{}\quad 0 &{}\quad -12.0251 \end{matrix}} \end{bmatrix}. \end{aligned}$$

The parameters of entire frequency controller with known actuator faults are given as,

$$\begin{aligned}&A_{ke}=\begin{bmatrix} {\begin{matrix} -7.5595 &{}\quad -4.0848 &{}\quad -6.5670 &{}\quad -4.0654 &{}\quad 30.6147 &{}\quad -2.0082 \times 10^8\\ 1.8485 &{}\quad -1.2923 &{}\quad 0.1648 &{}\quad -0.3071 &{}\quad 15.1398 &{}\quad -5.8785 \times 10^7\\ 9.1490 &{}\quad 4.5842 &{}\quad 0.4777 &{}\quad 0.6544 &{}\quad 76.4976 &{}\quad -2.7089 \times 10^8\\ 0.5217 &{}\quad 0.0866 &{}\quad 3.8286 &{}\quad -1.2898 &{}\quad 35.4406 &{}\quad -1.3394 \times 10^8\\ 1.5002 &{}\quad -1.9475 &{}\quad -9.6072 &{}\quad -7.3491 &{}\quad 141.2726 &{}\quad -6.6249 \times 10^8\\ 0.1372 &{}\quad -0.4602 &{}\quad 0.7063 &{}\quad -2.0482 &{}\quad 61.8560 &{}\quad -2.6860 \times 10^8 \end{matrix}} \end{bmatrix}\\&B_{ke}=\begin{bmatrix} {\begin{matrix} -0.6489 &{}\quad 0.1281 &{}\quad -1.6354 &{}\quad -0.2057\\ -0.9687 &{}\quad 0.7333 &{}\quad -0.7875 &{}\quad 0.4680\\ -0.4744 &{}\quad -1.1976 &{}\quad -0.0590 &{}\quad 0.3968\\ 0.8568 &{}\quad 0.3328 &{}\quad -0.7629 &{}\quad -0.0094\\ 0.5501 &{}\quad 0.0532 &{}\quad -0.0631 &{}\quad 0.9923\\ 0.0195 &{}\quad 0.0011 &{}\quad -0.5158 &{}\quad 0.2205 \end{matrix}} \end{bmatrix}\times 10^8\\&C_{ke}= \begin{bmatrix} {\begin{matrix} 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 32.7613 \end{matrix}} \end{bmatrix},\\&A_{\tau e}=\begin{bmatrix} {\begin{matrix} -0.0596 &{}\quad -0.0237 &{}\quad 0.7586 &{}\quad -0.1098 &{}\quad 8.5522 &{}\quad -3.3663 \times 10^7\\ -0.0245 &{}\quad -0.0098 &{}\quad 0.3122 &{}\quad -0.0452 &{}\quad 3.5196 &{}\quad -1.3854 \times 10^7\\ -0.0479 &{}\quad -0.0191 &{}\quad 0.6097 &{}\quad -0.0883 &{}\quad 6.8740 &{}\quad -2.7057 \times 10^7\\ -0.1229 &{}\quad -0.0490 &{}\quad 1.5653 &{}\quad -0.2266 &{}\quad 17.6463 &{}\quad -6.9458 \times 10^7\\ 0.0681 &{}\quad 0.0271 &{}\quad -0.8674 &{}\quad 0.1256 &{}\quad -9.7785 &{}\quad 3.8489 \times 10^7\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 8.7615 \end{matrix}} \end{bmatrix}. \end{aligned}$$

The parameters of finite frequency controller with unknown actuator faults are exhibited as,

$$\begin{aligned}&A_{kf}=\begin{bmatrix} {\begin{matrix} -0.8201 &{}\quad 0.06506 &{}\quad 6.3185 &{}\quad 28.3590 &{}\quad -1.3034 \times 10^6 &{}\quad -5.1594 \times 10^6\\ -0.0111 &{}\quad -0.8366 &{}\quad 6.2913 &{}\quad 10.0456 &{}\quad -2.5424 \times 10^5 &{}\quad -1.0060 \times 10^6\\ 0.09703 &{}\quad -0.6153 &{}\quad -194.9919 &{}\quad -312.5761 &{}\quad 1.7059 \times 10^6 &{}\quad 6.7569 \times 10^6\\ -0.0579 &{}\quad 0.5034 &{}\quad 106.2091 &{}\quad 67.2846 &{}\quad -1.3178 \times 10^6 &{}\quad -5.1872 \times 10^6\\ -0.5429 &{}\quad 1.4823 &{}\quad 275.8638 &{}\quad 613.6358 &{}\quad -1.6773 \times 10^7 &{}\quad -6.6399 \times 10^7\\ -0.3123 &{}\quad 0.8539 &{}\quad 158.9534 &{}\quad 352.9242 &{}\quad -9.6451 \times 10^6 &{}\quad -3.8182 \times 10^7 \end{matrix}} \end{bmatrix}\\&B_{kf}=\begin{bmatrix} {\begin{matrix} -1.0100 &{}\quad -0.0040 &{}\quad -0.9836 &{}\quad 0.0506\\ 0.0171 &{}\quad 0.4343 &{}\quad 0.0161 &{}\quad 0.4809\\ -0.0185 &{}\quad -0.1543 &{}\quad 2.0560 &{}\quad 0.9519\\ 0.0129 &{}\quad 0.0004 &{}\quad -0.9999 &{}\quad -0.4948\\ -0.0224 &{}\quad 0.0015 &{}\quad 1.3474 &{}\quad -0.5009\\ -0.0128 &{}\quad 0.0009 &{}\quad 0.7706 &{}\quad -0.2898 \end{matrix}} \end{bmatrix}\times 10^8\\&C_{kf}= \begin{bmatrix} {\begin{matrix} 0&\quad 0&\quad 0&\quad 0&\quad -0.9191&\quad -7.7810 \end{matrix}} \end{bmatrix},\quad A_{\tau f}=\begin{bmatrix} {\begin{matrix} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0.0001 &{}\quad -13.2195 &{}\quad -35.2674\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -0.0001 &{}\quad 3.8839 &{}\quad 9.9619\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0.0008 &{}\quad -108.3447 &{}\quad -317.7968\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -0.0002 &{}\quad 38.3949 &{}\quad 116.3583\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -0.9020 &{}\quad -4.8692\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0.0392 &{}\quad -0.3660 \end{matrix}} \end{bmatrix}. \end{aligned}$$

The parameters of entire frequency controller with unknown actuator faults are exhibited as,

$$\begin{aligned}&A_{ke}=\begin{bmatrix} {\begin{matrix} -1.9176 &{}\quad 0.0570 &{}\quad 45.1981 &{}\quad 200.8648 &{}\quad 1.0985 \times 10^6 &{}\quad 1.4919 \times 10^7\\ -0.4629 &{}\quad -1.5166 &{}\quad 1.0054 &{}\quad -147.6034 &{}\quad -6.9664 \times 10^5 &{}\quad -9.4316 \times 10^6\\ -0.4488 &{}\quad -3.1324 &{}\quad -245.4182 &{}\quad 190.8722 &{}\quad -1.5363 \times 10^6 &{}\quad -2.0839 \times 10^7\\ -0.3115 &{}\quad -1.0115 &{}\quad -91.9749 &{}\quad -103.9126 &{}\quad -7.6268 \times 10^5 &{}\quad -1.0421 \times 10^7\\ 1.0608 &{}\quad 3.3786 &{}\quad 290.7369 &{}\quad 825.19449 &{}\quad 6.0406 \times 10^6 &{}\quad 8.1933 \times 10^7\\ -1.0339 &{}\quad -3.2930 &{}\quad -283.4022 &{}\quad -803.6034 &{}\quad -5.8825 \times 10^6 &{}\quad -7.9789 \times 10^7 \end{matrix}} \end{bmatrix}\\&B_{ke}=\begin{bmatrix} {\begin{matrix} -0.8598 &{}\quad 1.1179 &{}\quad -0.2875 &{}\quad 0.4324\\ 1.6197 &{}\quad 0.6221 &{}\quad 0.5496 &{}\quad 0.1125\\ 0.0186 &{}\quad -0.1750 &{}\quad 2.2413 &{}\quad 1.1957\\ 0.0069 &{}\quad 0.0000 &{}\quad 0.6529 &{}\quad 0.4051\\ -0.0115 &{}\quad 0.0046 &{}\quad 1.2581 &{}\quad -0.5546\\ 0.0113 &{}\quad -0.0044 &{}\quad -1.2211 &{}\quad 0.5418 \end{matrix}} \end{bmatrix}\times 10^8\\&C_{ke}= \begin{bmatrix} {\begin{matrix} 0&0&0&0&0.3667&5.5185 \end{matrix}} \end{bmatrix},\\&A_{\tau e}=\begin{bmatrix} {\begin{matrix} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -0.0465 &{}\quad -8.2361\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -0.0001 &{}\quad -0.2861 &{}\quad -26.9659\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0.0001 &{}\quad 0.3826 &{}\quad 14.9930\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -0.0001 &{}\quad -0.2783 &{}\quad -38.2843\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0.1485 &{}\quad 1.6116\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -0.0729 &{}\quad -1.0502 \end{matrix}} \end{bmatrix}. \end{aligned}$$

Case 2 The parameters of finite frequency controller for system with LFT uncertainty are given as,

$$\begin{aligned}&A_{kf}=\begin{bmatrix} {\begin{matrix} -0.1999 &{}\quad -0.4158 &{}\quad -0.0543 &{}\quad 0.1084 &{}\quad -0.2977 &{}\quad 2.1492\\ -0.1525 &{}\quad -1.1674 &{}\quad 0.0162 &{}\quad 0.9697 &{}\quad -1.0464 &{}\quad 7.7376\\ -0.0268 &{}\quad -0.4417 &{}\quad -0.4588 &{}\quad 0.5287 &{}\quad -0.6117 &{}\quad 4.5844\\ -0.1758 &{}\quad 1.0414 &{}\quad 0.0412 &{}\quad -1.4142 &{}\quad 1.2317 &{}\quad -8.9301\\ 0.0319 &{}\quad -0.8673 &{}\quad -0.0185 &{}\quad 0.9488 &{}\quad -1.5780 &{}\quad 8.2205\\ 0.0071 &{}\quad 0.2740 &{}\quad 0.0568 &{}\quad -0.2898 &{}\quad 0.3284 &{}\quad -2.2539 \end{matrix}} \end{bmatrix}\times 10^5\\&B_{kf}=\begin{bmatrix} {\begin{matrix} -0.4041 &{}\quad 0.5281 &{}\quad -0.2757 &{}\quad 0.0836\\ 0.2327 &{}\quad 0.4674 &{}\quad -0.3559 &{}\quad 0.5663\\ 0.5950 &{}\quad 0.1351 &{}\quad -1.9036 &{}\quad -0.3664\\ -1.1911 &{}\quad 0.3667 &{}\quad 0.7553 &{}\quad -0.5119\\ -1.7558 &{}\quad -0.2613 &{}\quad -1.1571 &{}\quad 0.2813\\ -0.0758 &{}\quad -0.0297 &{}\quad 0.3656 &{}\quad -0.0565 \end{matrix}} \end{bmatrix}\times 10^7\\&C_{kf}= \begin{bmatrix} {\begin{matrix} 0.3934&\quad 0.3355&\quad -0.0688&\quad -1.2613&\quad 1.2271&\quad -8.2417 \end{matrix}} \end{bmatrix},\\&A_{\tau f}=\begin{bmatrix} {\begin{matrix} 0.6842 &{}\quad 0.5841 &{}\quad -0.1187 &{}\quad -2.1935 &{}\quad 2.1336 &{}\quad -14.2853\\ -1.8527 &{}\quad -1.5776 &{}\quad 0.3232 &{}\quad 5.9354 &{}\quad -5.7756 &{}\quad 38.8423\\ -0.9534 &{}\quad -0.8114 &{}\quad 0.1665 &{}\quad 3.0538 &{}\quad -2.9718 &{}\quad 20.0023\\ 0.8113 &{}\quad 0.6891 &{}\quad -0.1423 &{}\quad -2.5974 &{}\quad 2.5283 &{}\quad -17.0758\\ -1.2700 &{}\quad -1.0804 &{}\quad 0.2220 &{}\quad 4.0676 &{}\quad -3.9586 &{}\quad 26.6640\\ -0.1456 &{}\quad -0.1249 &{}\quad 0.0252 &{}\quad 0.4677 &{}\quad -0.4545 &{}\quad 3.0247 \end{matrix}} \end{bmatrix}. \end{aligned}$$

The parameters of entire frequency controller for system with LFT uncertainty are given as,

$$\begin{aligned}&A_{ke}=\begin{bmatrix} {\begin{matrix} -0.0058 &{}\quad 0.0147 &{}\quad 0.0157 &{}\quad -0.0620 &{}\quad 0.7735 &{}\quad -3.6935\\ 0.0083 &{}\quad -0.0232 &{}\quad 0.0146 &{}\quad -0.0553 &{}\quad 0.6648 &{}\quad -3.2361\\ -0.0012 &{}\quad -0.0003 &{}\quad -0.0343 &{}\quad 0.0008 &{}\quad -0.0113 &{}\quad 0.0506\\ 0.0088 &{}\quad 0.0139 &{}\quad 0.0185 &{}\quad -0.1039 &{}\quad 0.8365 &{}\quad -4.0756\\ 0.0054 &{}\quad -0.0189 &{}\quad -0.0289 &{}\quad 0.1077 &{}\quad -1.2886 &{}\quad 6.1717\\ 0.0013 &{}\quad 0.0015 &{}\quad 0.0073 &{}\quad -0.0332 &{}\quad 0.3452 &{}\quad -1.6683 \end{matrix}} \end{bmatrix}\times 10^5\\&B_{ke}=\begin{bmatrix} {\begin{matrix} -1.7257 &{}\quad -0.3345 &{}\quad -4.6848 &{}\quad 0.2467\\ -1.9493 &{}\quad -0.1454 &{}\quad 1.7504 &{}\quad 2.6243\\ 2.4390 &{}\quad -2.5322 &{}\quad 0.2752 &{}\quad 0.0822\\ 6.3127 &{}\quad 0.7547 &{}\quad -3.0940 &{}\quad 1.1231\\ 2.1842 &{}\quad 0.2484 &{}\quad 6.6670 &{}\quad -0.8867\\ 0.3408 &{}\quad 0.0253 &{}\quad -1.0382 &{}\quad 0.5539 \end{matrix}} \end{bmatrix}\times 10^5\\&C_{ke}= \begin{bmatrix} {\begin{matrix} 0.0170&\quad -0.0481&\quad -0.0442&\quad 0.1454&\quad -1.8322&\quad 7.8578 \end{matrix}} \end{bmatrix},\\&A_{\tau e}=\begin{bmatrix} {\begin{matrix} 0.0089 &{}\quad -0.0256 &{}\quad -0.0220 &{}\quad 0.0972 &{}\quad -0.8880 &{}\quad 4.7125\\ -0.0092 &{}\quad 0.0262 &{}\quad 0.0239 &{}\quad -0.0812 &{}\quad 0.9865 &{}\quad -4.3344\\ -0.0010 &{}\quad 0.0028 &{}\quad 0.0025 &{}\quad -0.0097 &{}\quad 0.1012 &{}\quad -0.4902\\ -0.0129 &{}\quad 0.0367 &{}\quad 0.0334 &{}\quad -0.1147 &{}\quad 1.3768 &{}\quad -6.0978\\ 0.0362 &{}\quad -0.1030 &{}\quad -0.0926 &{}\quad 0.3370 &{}\quad -3.8046 &{}\quad 17.5208\\ 0.0067 &{}\quad -0.0190 &{}\quad -0.0170 &{}\quad 0.0627 &{}\quad -0.6982 &{}\quad 3.2470 \end{matrix}} \end{bmatrix}. \end{aligned}$$

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Xu, S., Sun, G. & Sun, W. Reliable sampled-data vibration control for uncertain flexible spacecraft with frequency range limitation. Nonlinear Dyn 86, 1117–1135 (2016). https://doi.org/10.1007/s11071-016-2952-5

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