In this paper, a partial eigenstructure assignment problem for the high-order linear time-invariant (LTI) systems via proportional plus derivative (PD) state feedback is considered. By partitioning the open-loop system into two parts (the altered part and the unchanged part) and utilizing the solutions to the high-order generalized Sylvester equation (HGSE), complete parametric expressions of the feedback gain matrices of the closed-loop system are established. Meanwhile, a group of arbitrary parameters representing the degrees of freedom of the proposed method is provided and optimized to satisfy the stability of the system and robustness criteria. Finally, a numerical example and a three-axis dynamic flight motion simulator system example with the simulation results are offered to illustrate the effectiveness and superiority of the proposed method.
Citation: Da-Ke Gu, Rui-Yuan Wang, Yin-Dong Liu. A parametric approach of partial eigenstructure assignment for high-order linear systems via proportional plus derivative state feedback[J]. AIMS Mathematics, 2021, 6(10): 11139-11166. doi: 10.3934/math.2021647
In this paper, a partial eigenstructure assignment problem for the high-order linear time-invariant (LTI) systems via proportional plus derivative (PD) state feedback is considered. By partitioning the open-loop system into two parts (the altered part and the unchanged part) and utilizing the solutions to the high-order generalized Sylvester equation (HGSE), complete parametric expressions of the feedback gain matrices of the closed-loop system are established. Meanwhile, a group of arbitrary parameters representing the degrees of freedom of the proposed method is provided and optimized to satisfy the stability of the system and robustness criteria. Finally, a numerical example and a three-axis dynamic flight motion simulator system example with the simulation results are offered to illustrate the effectiveness and superiority of the proposed method.
| [1] |
Z. Y. Sun, M. M. Yun, T. Li, A new approach to fast global finite-time stabilization of high-order nonlinear system, Automatica, 81 (2017), 455-463. doi: 10.1016/j.automatica.2017.04.024
|
| [2] | T. D. Abhayapala, D. B. Ward, Theory and design of high order sound field microphones using spherical microphone array, 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2002, II-1949-II-1952. |
| [3] |
R. Hu, A fully-implicit high-order system thermal-hydraulics model for advanced non-LWR safety analyses, Ann. Nucl. Energy, 101 (2017), 174-181. doi: 10.1016/j.anucene.2016.11.004
|
| [4] |
C. J. Damaren, On the dynamics and control of flexible multibody systems with closed loops, Int. J. Robot Res., 19 (2000), 238-253. doi: 10.1177/02783640022066842
|
| [5] |
M. Balas, Trends in large space structure control theory: fondest hopes, wildest dreams, IEEE Trans. Autom. Control, 27 (1982), 522-535. doi: 10.1109/TAC.1982.1102953
|
| [6] |
J. T. Sawicki, G. Genta, Modal uncoupling of damped gyroscopic systems, J. Sound Vib., 244 (2001), 431-451. doi: 10.1006/jsvi.2000.3473
|
| [7] |
S. Tayebi-Haghighi, F. Piltan, J. M. Kim, Robust composite high-order super-twisting sliding mode control of robot manipulators, Robotics, 7 (2018), 13. doi: 10.3390/robotics7010013
|
| [8] | R. W. Clough, S. Mojtahedi, Earthquake response analysis considering non-proportional damping, Earthquake Eng. Struct. Dyn., 4 (1976), 486-496. |
| [9] |
E. B. Kosmatopoulos, M. M. Polycarpou, M. A. Christodoulou, P. A. Ioannou, High-order neural network structures for identification of dynamical systems, IEEE Trans. Neural Networks, 6 (1995), 422-431. doi: 10.1109/72.363477
|
| [10] | X. J. Xie, N. Duan, Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system, IEEE Trans. Autom. Control, 59 (2010), 13-37. |
| [11] | K. M. Sobel, E. Y. Shapiro, A. N. Andry JR, Eigenstructure assignment, Int. J. Control, 55 (1994), 1197-1202. |
| [12] | B. White, Eigenstructure assignment: a survey, Proc. Inst. Mech. Eng., Part I, 209 (1995), 1-11. |
| [13] |
G. R. Duan, G. P. Liu, Complete parametric approach for eigenstructure assignment in a class of second-order linear systems, Automatica, 38 (2002), 725-729. doi: 10.1016/S0005-1098(01)00251-5
|
| [14] |
G. R. Duan, Parametric eigenstructure assignment in second-order descriptor linear systems, IEEE Trans. Autom. Control, 49 (2004), 1789-1794. doi: 10.1109/TAC.2004.835580
|
| [15] |
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. doi: 10.1016/S0024-3795(96)00036-5
|
| [16] |
D. K. Gu, G. P. Liu, G. R. Duan, Parametric control to a type of quasi-linear second-order systems via output feedback, Int. J. Control, 92 (2019), 291-302. doi: 10.1080/00207179.2017.1350885
|
| [17] | G. R. Duan, H. H. Yu, Complete eigenstructure assignment in high-order descriptor linear systems via proportional plus derivative state feedback, 2006 6th World Congress on Intelligent Control and Automation, 2006,500-505. |
| [18] |
H. H. Yu, G. R. Duan, ESA in high-order linear systems via output feedback, Asian J. Control, 11 (2009), 336-343. doi: 10.1002/asjc.111
|
| [19] | G. R. Duan, Parametric approaches for eigenstructure assignment in high-order linear systems, Int. J. Control Autom., 3 (2005), 419-429. |
| [20] |
G. R. Duan, H. H. Yu, Robust pole assignment in high-order descriptor linear systems via proportional plus derivative state feedback, IET Control Theory A., 2 (2008), 277-287. doi: 10.1049/iet-cta:20070164
|
| [21] |
D. K. Gu, D. W. Zhang, G. R. Duan, Parametric control to a type of quasi-linear high-order systems via output feedback, Eur. J. Control, 47 (2019), 44-52. doi: 10.1016/j.ejcon.2018.09.008
|
| [22] | D. K. Gu, D. W. Zhang, Y. D. Liu, Controllability results for quasi-linear systems: standard and descriptor cases, Asian J. Control, (2021), 1-11. |
| [23] |
D. K. Gu, D. W. Zhang, Parametric control to a type of descriptor quasi-linear high-order systems via output feedback, Eur. J. Control, 58 (2021), 223-231. doi: 10.1016/j.ejcon.2020.09.002
|
| [24] |
H. Liu, J. J. Xu, A multi-step method for partial eigenvalue assignment problem of high order control systems, Mech. Syst. Signal Process., 94 (2017), 346-358. doi: 10.1016/j.ymssp.2017.03.002
|
| [25] |
M. Heyouni, F. Saberi-Movahed, A. Tajaddini, On global Hessenberg based methods for solving Sylvester matrix equations, Comput. Math. Appl., 77 (2019), 77-92. doi: 10.1016/j.camwa.2018.09.015
|
| [26] |
S. K. Li, M. X. Wang, G. Liu, A global variant of the COCR method for the complex symmetric Sylvester matrix equation AX + XB = C, Comput. Math. Appl., 94 (2021), 104-113. doi: 10.1016/j.camwa.2021.04.026
|
| [27] | P. Z. Yu, Partial eigenstructure assignment problem for vibration system via feedback control, Asian J. Control, (2020), 1-12. |
| [28] | D. A. Silva, E. Baleeiro, A. José Mário, Damping Power System Oscilations in Multi-Machine System: A Partial Eigenstructure Assignment plus State Observer Approach, Int. J. Innov. Comput. I., 16 (2020), 1559-1578. |
| [29] |
J. F. Zhang, H. J. Ouyang, J. Yang, Partial eigenstructure assignment for undamped vibration systems using acceleration and displacement feedback, J. Sound Vib., 333 (2014), 1-12. doi: 10.1016/j.jsv.2013.08.040
|
| [30] | J. F. Zhang, J. P. Ye, H. J. Ouyang, Static output feedback for partial eigenstructure assignment of undamped vibration systems, Mech. Syst. Signal Process., 68 (2016), 555-561. |
| [31] |
L. Zhang, F. Yu, X. Wang, An algorithm of partial eigenstructure assignment for high‐order systems, Math. Method Appl. Sci., 41 (2018), 6070-6079. doi: 10.1002/mma.5118
|
| [32] | B. N. Datta, W. W. Lin, J. N. Wang, Robust and minimum gain partial pole assignment for a third order system, IEEE Conference on Decision and Control 2003, 2358-2363. |
| [33] |
M. A. Ramadan, E. A. El-Sayed, Partial eigenvalue assignment problem of high order control systems using orthogonality relations, Comput. Math, Appl., 59 (2010), 1918-1928. doi: 10.1016/j.camwa.2009.07.063
|
| [34] |
Y. F. Cai, J. Qian, S. F. Xu, Robust partial pole assignment problem for high order control systems, Automatica, 48 (2012), 1462-1466. doi: 10.1016/j.automatica.2012.05.015
|
| [35] | G. R. Duan, On a type of high-order generalized Sylvester equations, Proceedings of the 32nd Chinese Control Conference, 2013,328-333. |
| [36] | G. R. Duan, Generalized Sylvester equations: unified parametric solutions, Boca Raton, FL, USA: CRC Press, 2014. |
| [37] | H. H. Yu, G. R. Duan, The analytical general solutions to the higher-order Sylvester matrices equation, Control Theory A., 28 (2011), 698-702. |
| [38] | G. R. Duan, Solution to high-order generalized Sylvester matrix equations, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC'05, 2005, 7247-7252. |
| [39] |
G. R. Duan, Circulation algorithm for partial eigenstructure assignment via state feedback, Eur. J. Control, 50 (2019), 107-116. doi: 10.1016/j.ejcon.2019.02.006
|
| [40] | G. R. Duan, G. S. Wang, Partial eigenstructure assignment for descriptor linear systems: A complete parametric approach, 42nd IEEE International Conference on Decision and Control, 4 (2003), 3402-3407. |
| [41] |
X. B. Mao, H. Dai, Minimum norm partial eigenvalue assignment of high order linear system with no spill-over, Linear Algebra Appl., 438 (2013), 2136-2154. doi: 10.1016/j.laa.2012.10.049
|
Figures(15)
Da-Ke Gu, Rui-Yuan Wang, Yin-Dong Liu. A parametric approach of partial eigenstructure assignment for high-order linear systems via proportional plus derivative state feedback[J]. AIMS Mathematics, 2021, 6(10): 11139-11166. doi: 10.3934/math.2021647
DownLoad: