A robust regional eigenvalue assignment problem using rank-one control for undamped gyroscopic systems

Binxin He; Hao Liu; Binxin He; Hao Liu

doi:10.3934/math.2024931

AIMS Mathematics

2024, Volume 9, Issue 7: 19104-19124. doi: 10.3934/math.2024931

Previous Article Next Article

Research article

A robust regional eigenvalue assignment problem using rank-one control for undamped gyroscopic systems

Binxin He ¹,
Hao Liu ^{1,2
,
,}

1.
School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
2.
Shenzhen Research Institute, Nanjing University of Aeronautics and Astronautics, Shenzhen 518063, China

Received: 30 March 2024 Revised: 31 May 2024 Accepted: 04 June 2024 Published: 07 June 2024
MSC : 65F18, 93C15

Considering the advantages of economic benefit and cost reduction by using rank-one control, we investigated the problem of robust regional eigenvalue assignment using rank-one control for undamped gyroscopic systems. Based on the orthogonality relation, we presented a method for solving partial eigenvalue assignment problems to reassign partial undesired eigenvalues accurately. Since it is difficult to achieve robust control by assigning desired eigenvalues to precise positions with rank-one control, we assigned eigenvalues within specified regions to provide the necessary freedom. According to the sensitivity analysis theories, we derived the sensitivity of closed-loop eigenvalues to parameter perturbations to measure robustness and proposed a numerical algorithm for solving robust regional eigenvalue assignment problems so that the closed-loop eigenvalues were insensitive to parameter perturbations. Numerical experiments demonstrated the effectiveness of our method.
- undamped gyroscopic systems,
- regional eigenvalue assignment,
- rank-one control,
- robustness
Citation: Binxin He, Hao Liu. A robust regional eigenvalue assignment problem using rank-one control for undamped gyroscopic systems[J]. AIMS Mathematics, 2024, 9(7): 19104-19124. doi: 10.3934/math.2024931

Related Papers:

Abstract

Considering the advantages of economic benefit and cost reduction by using rank-one control, we investigated the problem of robust regional eigenvalue assignment using rank-one control for undamped gyroscopic systems. Based on the orthogonality relation, we presented a method for solving partial eigenvalue assignment problems to reassign partial undesired eigenvalues accurately. Since it is difficult to achieve robust control by assigning desired eigenvalues to precise positions with rank-one control, we assigned eigenvalues within specified regions to provide the necessary freedom. According to the sensitivity analysis theories, we derived the sensitivity of closed-loop eigenvalues to parameter perturbations to measure robustness and proposed a numerical algorithm for solving robust regional eigenvalue assignment problems so that the closed-loop eigenvalues were insensitive to parameter perturbations. Numerical experiments demonstrated the effectiveness of our method.

References

[1]	F. Tisseur, K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), 235–286. https://doi.org/10.1137/S0036144500381988 doi: 10.1137/S0036144500381988
[2]	B. N. Datta, D. R. Sarkissian, Feedback control in distributed parameter gyroscopic systems: a solution of the partial eigenvalue assignment problem, Mech. Syst. Sig. Process., 16 (2001), 3–17. https://doi.org/10.1006/mssp.2001.1444 doi: 10.1006/mssp.2001.1444
[3]	D. Richiedei, I. Tamellin, A. Trivisani, Unit-rank output feedback control for antiresonance assignment in lightweight systems, Mech. Syst. Sig. Process., 164 (2022), 108250. https://doi.org/10.1016/j.ymssp.2021.108250 doi: 10.1016/j.ymssp.2021.108250
[4]	T. A. M. Euzébio, A. S. Yamashita, T. V. B. Pinto, P. R. Barros, SISO approaches for linear programming based methods for tuning decentralized PID controllers, J. Process Contr., 94 (2020), 75–96. https://doi.org/10.1016/j.jprocont.2020.08.004 doi: 10.1016/j.jprocont.2020.08.004
[5]	Q. Chen, D. Q. Zhu, Z. B. Liu, Attitude control of aerial and underwater vehicles using single-input FUZZY P+ID controller, Appl. Ocean Res., 107 (2021), 102460. https://doi.org/10.1016/j.apor.2020.102460 doi: 10.1016/j.apor.2020.102460
[6]	G. V. Smirnov, Y. Mashtakov, M. Ovchinnikov, S. Shestakov, A. F. B. A. Prado, Tetrahedron formation of nanosatellites with single-input control, Astrophys. Space Sci., 363 (2018), 180. https://doi.org/10.1007/s10509-018-3400-4 doi: 10.1007/s10509-018-3400-4
[7]	N. J. B. Dantas, C. E. T. Dórea, J. M. Araújo, Design of rank-one modification feedback controllers for second-order systems with time delay using frequency response methods, Mech. Syst. Sig. Process., 137 (2020), 106404. https://doi.org/10.1016/j.ymssp.2019.106404 doi: 10.1016/j.ymssp.2019.106404
[8]	N. J. B. Dantas, C. E. T. Dorea, J. M. Araujo, Partial pole assignment using rank-one control and receptance in second-order systems with time delay, Meccanica, 56 (2021), 287–302. https://doi.org/10.1007/s11012-020-01289-w doi: 10.1007/s11012-020-01289-w
[9]	K. V. Singh, Y. M. Ram, Dynamic absorption by passive and active control, J. Vib. Acoust. Trans., 122 (2000), 429–433. https://doi.org/10.1115/1.1311792 doi: 10.1115/1.1311792
[10]	K. V. Singh, B. N. Datta, M. Tyagi, Closed form control gains for zero assignment in the time delayed system, J. Comput. Nonlinear Dyn., 6 (2011), 021002. https://doi.org/10.1115/1.4002340 doi: 10.1115/1.4002340
[11]	Y. M. Ram, J. E. Mottershead, M. G. Tehrani, Partial pole placement with time delay in structures using the receptance and the system matrices, Linear Algebra Appl., 434 (2011), 1689–1696. https://doi.org/10.1016/j.laa.2010.07.014 doi: 10.1016/j.laa.2010.07.014
[12]	Y. Liang, H. Yamaura, H. J. Ouyang, Active assignment of eigenvalues and eigen-sensitivities for robust stabilization of friction-induced vibration, Mech. Syst. Sig. Process., 90 (2017), 254–267. https://doi.org/10.1016/j.ymssp.2016.12.011 doi: 10.1016/j.ymssp.2016.12.011
[13]	J. Q. Teoh, M. G. Tehrani, N. S. Ferguson, S. J. Elliott, Eigenvalue sensitivity minimisation for robust pole placement by the receptance method, Mech. Syst. Sig. Process., 173 (2022), 108974. https://doi.org/10.1016/j.ymssp.2022.108974 doi: 10.1016/j.ymssp.2022.108974
[14]	W. W. Hager, Updating the inverse of a matrix, SIAM Rev., 31 (1989), 221–239. https://doi.org/10.1137/1031049 doi: 10.1137/1031049
[15]	G. H. Golub, C. F. L. Van, Matrix computations, Baltimore: Johns Hopkins University Press, 1983.
[16]	Y. Liang, H. J. Ouyang, H. Yamaura, Active partial eigenvalue assignment for friction-induced vibration using receptance method, J. Phy.: Conf. Ser., 744 (2016), 012008. https://doi.org/10.1088/1742-6596/744/1/012008 doi: 10.1088/1742-6596/744/1/012008
[17]	Y. M. Ram, J. E. Mottershead, Receptance method in active vibration control, AIAA J., 45 (2007), 562–567. https://doi.org/10.2514/1.24349 doi: 10.2514/1.24349
[18]	Y. M. Ram, J. E. Mottershead, Multiple-input active vibration control by partial pole placement using the method of receptances, Mech. Syst. Sig. Process., 40 (2013), 727–735. https://doi.org/10.1016/j.ymssp.2013.06.008 doi: 10.1016/j.ymssp.2013.06.008
[19]	S. K. Zhang, H. J. Ouyang, Receptance-based partial eigenstructure assignment by state feedback control, Mech. Syst. Sig. Process., 168 (2022), 108728. https://doi.org/10.1016/j.ymssp.2021.108728 doi: 10.1016/j.ymssp.2021.108728
[20]	R. Belotti, D. Richiedei, Pole assignment in vibrating systems with time delay: an approach embedding an a-priori stability condition based on Linear Matrix Inequality, Mech. Syst. Sig. Process., 137 (2020), 106396. https://doi.org/10.1016/j.ymssp.2019.106396 doi: 10.1016/j.ymssp.2019.106396
[21]	D. Richiedei, I. Tamellin, A. Trivisani, Pole-zero assignment by the receptance method: multi-input active vibration control, Mech. Syst. Sig. Process., 172 (2022), 108976. https://doi.org/10.1016/j.ymssp.2022.108976 doi: 10.1016/j.ymssp.2022.108976
[22]	H. Liu, B. X. He, X. P. Chen, Partial eigenvalue assignment for undamped gyroscopic systems in control, E. Asian J. Appl. Math., 9 (2019), 831–848. https://doi.org/10.4208/eajam.040718.091218 doi: 10.4208/eajam.040718.091218
[23]	B. N. Datta, S. Elhay, Y. M. Ram, Orthogonality and partial pole assignment for the symmetric definite quadratic pencil, Linear Algebra Appl., 257 (1997), 29–48. https://doi.org/10.1016/S0024-3795(96)00036-5 doi: 10.1016/S0024-3795(96)00036-5
[24]	H. Liu, Y. X. Yuan, A multi-step method for partial quadratic pole assignment problem with time delay, Appl. Math. Comput., 283 (2016), 29–35. https://doi.org/10.1016/j.amc.2016.02.012 doi: 10.1016/j.amc.2016.02.012
[25]	S. Brahma, B. N. Datta, An optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures, J. Sound Vib., 324 (2009), 471–489. https://doi.org/10.1016/j.jsv.2009.02.020 doi: 10.1016/j.jsv.2009.02.020
[26]	Z. J. Bai, J. K. Yang, B. N. Datta, Robust partial quadratic eigenvalue assignment with time delay using the receptance and the system matrices, J. Sound Vib., 384 (2016), 1–14. https://doi.org/10.1016/j.jsv.2016.08.002 doi: 10.1016/j.jsv.2016.08.002
[27]	M. Lu, Z. J. Bai, A modified optimization method for robust partial quadratic eigenvalue assignment using receptances and system matrices, Mech. Syst. Sig. Process., 159 (2021), 73–92. https://doi.org/10.1016/j.apnum.2020.08.018 doi: 10.1016/j.apnum.2020.08.018
[28]	M. G. Tehrani, J. E. Mottershead, A. T. Shenton, Y. M. Ram, Robust pole placement in structures by the method of receptances, Mech. Syst. Sig. Process., 25 (2011), 112–122. https://doi.org/10.1016/j.ymssp.2010.04.005 doi: 10.1016/j.ymssp.2010.04.005
[29]	M. Chen, H. Q. Xie, A receptance method for partial quadratic pole assignment of asymmetric systems, Mech. Syst. Sig. Process., 165 (2022), 108348. https://doi.org/10.1016/j.ymssp.2021.108348 doi: 10.1016/j.ymssp.2021.108348
[30]	L. L. Jia, Robust pole assignment with part parameter perturbation system, Harbin: Harbin Institute of Technology, 2009.
[31]	D. R. Sarkissian, Theory and computations of partial eigenvalue and eigenstructure assignment problems in matrix second-order and distributed-parameter systems, Dekalb: Northern Illinois University, 2001.
[32]	P. Ariyatanapol, Y. P. Xiong, H. J. Ouyang, Partial pole assignment with time delays for asymmetric systems, Acta Mech., 229 (2018), 2619–2629. https://doi.org/10.1007/s00707-018-2118-2 doi: 10.1007/s00707-018-2118-2
[33]	D. J. Ewins, Model testing: theory, practice and application, 2 Eds., British: Research Studies Press, 2000.
[34]	S. Y. Yoon, Z. L. Lin, E. A. Paul, Control of surge in centrifugal compressors by active magnetic bearings: theory and implementation, London: Springer-Verlag, 2013. https://doi.org/10.1007/978-1-4471-4240-9

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)