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A dynamic lot sizing model with production-or-outsourcing decision under minimum production quantities
A robust reduced-order observers design approach for linear discrete periodic systems
1. | Institute of Electric Power, North China University of Water Resources and Electric Power, Zhengzhou 450011, China |
2. | Key Laboratory of Big Data Analysis and Processing of Henan Province, Henan University, Zhengzhou 450011, China |
This paper investigates the problem of designing reduced-order observers for linear discrete-time periodic (LDP) systems. In case that the linear discrete-time periodic system is observable, an algebraic equivalent system is obtained by non-singular linear transformation, and the partial states to be observed are separated simultaneously. Then the considered problem is transformed into the problem of solving a class of periodic Sylvester matrix equation and an iterative algorithm for periodic reduced-order state observers design is derived. In addition, robust consideration based on periodic reduced-order state observers for LDP systems is also conducted. At last, one numerical example is worked out to illustrate the effectiveness of the proposed approaches.
References:
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H. Bourles and U. Oberst,
Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection, Mathematics of Control Signals & Systems, 28 (2016), 1-34.
doi: 10.1007/s00498-016-0171-8. |
[2] |
H. Bourles and U. Oberst,
Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection, Mathematics of Control Signals & Systems, 28 (2016), 1-34.
doi: 10.1007/s00498-016-0171-8. |
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Y. Chen and J. Lam,
Estimation and synthesis of reachable set for discrete-time periodic systems, Optimal Control Applications & Methods, 37 (2016), 885-901.
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O. M. Grasselli and S. Longhi,
Finite zero structure of linear periodic discrete-time systems, International Journal of Systems Science, 22 (1991), 1785-1806.
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R. Kalman,
A new approach to linear filtering and prediction problems, J. Basic Eng., 82 (1960), 35-45.
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[6] |
B. Li, Y. Rong and J. Sun, et al., A distributionally robust minimum variance beamformer design, IEEE Signal Processing Letters, 25 (2018), 105-109. Google Scholar |
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B. Li, J. Sun and H. Xu, et al., A class of two-stage distributionally robust games, Journal of Industrial & Management Optimization, 15 (2019), 387-400. |
[8] |
B. Li, X. Qian and J. Sun, et al., A model of distributionally robust two-stage stochastic convex programming with linear recourse, Applied Mathematical Modelling, 58 (2018), 86-97.
doi: 10.1016/j.apm.2017.11.039. |
[9] |
W. Lin, L. Zhao and K. Dong, Performance analysis of re-adhesion optimization control based on full-dimension state observer, Procedia Engineering, 23 (2011), 531-536. Google Scholar |
[10] |
Y. Liu, Y. Yin and K. L. Teo, et al., Probabilistic control of Markov jump systems by scenario optimization approach, Journal of Industrial & Management Optimization, (2018), 742–753. Google Scholar |
[11] |
L. Lv and Z. Zhang,
Finite iterative solutions to periodic Sylvester matrix equations, Journal of the Franklin Institute, 354 (2017), 2358-2370.
doi: 10.1016/j.jfranklin.2017.01.004. |
[12] |
L. Lv, Z. Zhang and L. Zhang,
A parametric poles assignment algorithm for second-order linear periodic systems, Journal of the Franklin Institute, 354 (2017), 8057-8071.
doi: 10.1016/j.jfranklin.2017.09.029. |
[13] |
R. Sanz, P. Garcia, E. Fridman and P. Albertos, A predictive extended state observer for a class of nonlinear systems with input delay subject to external disturbances, in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), IEEE, 2017, 4345–4350.
doi: 10.1109/CDC.2017.8264300. |
[14] |
H. A. Tehrani and J. Esmaeili,
Stability of fractional-order periodic discrete-time linear systems, IMA Journal of Mathematical Control and Information, 34 (2017), 271-281.
doi: 10.1093/imamci/dnv043. |
[15] |
H. Trinh and M. Aldeen,
A reduced-order state observer for large-scale discrete-time systems, Computers & Electrical Engineering, 23 (1997), 301-309.
doi: 10.1109/9.649721. |
[16] |
L. Y. Wang, C. Li and G. G. Yin, et al., State observability and observers of linear-timeinvariant systems under irregular sampling and sensor limitations, IEEE Transactions on Automatic Control, 56 (2011), 2639-2654.
doi: 10.1109/TAC.2011.2122570. |
[17] |
A. Wu and G. Duan,
Robust fault detection in linear systems based on full-order state observers, Journal of Control Theory and Applications, 5 (2007), 325-330.
doi: 10.1007/s11768-006-6073-4. |
[18] |
L. Yan, H. Qiao and Z. Jiao, et al., Linear motor tracking control based on adaptive robust control and extended state observer, in IEEE International Conference on Cybernetics and Intelligent Systems, IEEE, 2017,704–709. |
[19] |
Y. Yang,
An efficient LQR design for discrete-time linear periodic system based on a novel lifting method, Automatica, 87 (2018), 383-388.
doi: 10.1016/j.automatica.2017.10.019. |
[20] |
Y. Yin, Y. Liu and K. L. Teo, et al. Event-triggered probabilistic robust control of linear systems with input constrains: By scenario optimization approach, International Journal of Robust and Nonlinear Control, 28 (2018), 144-153.
doi: 10.1002/rnc.3858. |
[21] |
B. Zhou, D. Li and Z. Lin,
Control of discrete-time periodic linear systems with input saturation via multi-step periodic invariant sets, International Journal of Robust & Nonlinear Control, 25 (2015), 103-124.
|
[22] |
B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, Springer, Berlin Heidelberg, 2014.
doi: 10.1007/978-3-642-54206-0. |
[23] |
B. Zhou, Z. Y. Li and Z. Lin,
Observer based output feedback control of linear systems with input and output delays, Automatica, 49 (2013), 2039-2052.
doi: 10.1016/j.automatica.2013.03.031. |
[24] |
B. Zhou, C. Xu and G. Duan,
Distributed and truncated reduced-order observer based output feedback consensus of multi-agent systems, IEEE Transactions on Automatic Control, 59 (2014), 2264-2270.
doi: 10.1109/TAC.2014.2301573. |
[25] |
F. Zhu and F. Cen, Full-order observer-based actuator fault detection and reduced-order observer-based fault reconstruction for a class of uncertain nonlinear systems, Journal of Process Control, 20 (2010), 1141-1149. Google Scholar |
show all references
References:
[1] |
H. Bourles and U. Oberst,
Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection, Mathematics of Control Signals & Systems, 28 (2016), 1-34.
doi: 10.1007/s00498-016-0171-8. |
[2] |
H. Bourles and U. Oberst,
Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection, Mathematics of Control Signals & Systems, 28 (2016), 1-34.
doi: 10.1007/s00498-016-0171-8. |
[3] |
Y. Chen and J. Lam,
Estimation and synthesis of reachable set for discrete-time periodic systems, Optimal Control Applications & Methods, 37 (2016), 885-901.
doi: 10.1002/oca.2211. |
[4] |
O. M. Grasselli and S. Longhi,
Finite zero structure of linear periodic discrete-time systems, International Journal of Systems Science, 22 (1991), 1785-1806.
doi: 10.1080/00207729108910751. |
[5] |
R. Kalman,
A new approach to linear filtering and prediction problems, J. Basic Eng., 82 (1960), 35-45.
doi: 10.1115/1.3662552. |
[6] |
B. Li, Y. Rong and J. Sun, et al., A distributionally robust minimum variance beamformer design, IEEE Signal Processing Letters, 25 (2018), 105-109. Google Scholar |
[7] |
B. Li, J. Sun and H. Xu, et al., A class of two-stage distributionally robust games, Journal of Industrial & Management Optimization, 15 (2019), 387-400. |
[8] |
B. Li, X. Qian and J. Sun, et al., A model of distributionally robust two-stage stochastic convex programming with linear recourse, Applied Mathematical Modelling, 58 (2018), 86-97.
doi: 10.1016/j.apm.2017.11.039. |
[9] |
W. Lin, L. Zhao and K. Dong, Performance analysis of re-adhesion optimization control based on full-dimension state observer, Procedia Engineering, 23 (2011), 531-536. Google Scholar |
[10] |
Y. Liu, Y. Yin and K. L. Teo, et al., Probabilistic control of Markov jump systems by scenario optimization approach, Journal of Industrial & Management Optimization, (2018), 742–753. Google Scholar |
[11] |
L. Lv and Z. Zhang,
Finite iterative solutions to periodic Sylvester matrix equations, Journal of the Franklin Institute, 354 (2017), 2358-2370.
doi: 10.1016/j.jfranklin.2017.01.004. |
[12] |
L. Lv, Z. Zhang and L. Zhang,
A parametric poles assignment algorithm for second-order linear periodic systems, Journal of the Franklin Institute, 354 (2017), 8057-8071.
doi: 10.1016/j.jfranklin.2017.09.029. |
[13] |
R. Sanz, P. Garcia, E. Fridman and P. Albertos, A predictive extended state observer for a class of nonlinear systems with input delay subject to external disturbances, in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), IEEE, 2017, 4345–4350.
doi: 10.1109/CDC.2017.8264300. |
[14] |
H. A. Tehrani and J. Esmaeili,
Stability of fractional-order periodic discrete-time linear systems, IMA Journal of Mathematical Control and Information, 34 (2017), 271-281.
doi: 10.1093/imamci/dnv043. |
[15] |
H. Trinh and M. Aldeen,
A reduced-order state observer for large-scale discrete-time systems, Computers & Electrical Engineering, 23 (1997), 301-309.
doi: 10.1109/9.649721. |
[16] |
L. Y. Wang, C. Li and G. G. Yin, et al., State observability and observers of linear-timeinvariant systems under irregular sampling and sensor limitations, IEEE Transactions on Automatic Control, 56 (2011), 2639-2654.
doi: 10.1109/TAC.2011.2122570. |
[17] |
A. Wu and G. Duan,
Robust fault detection in linear systems based on full-order state observers, Journal of Control Theory and Applications, 5 (2007), 325-330.
doi: 10.1007/s11768-006-6073-4. |
[18] |
L. Yan, H. Qiao and Z. Jiao, et al., Linear motor tracking control based on adaptive robust control and extended state observer, in IEEE International Conference on Cybernetics and Intelligent Systems, IEEE, 2017,704–709. |
[19] |
Y. Yang,
An efficient LQR design for discrete-time linear periodic system based on a novel lifting method, Automatica, 87 (2018), 383-388.
doi: 10.1016/j.automatica.2017.10.019. |
[20] |
Y. Yin, Y. Liu and K. L. Teo, et al. Event-triggered probabilistic robust control of linear systems with input constrains: By scenario optimization approach, International Journal of Robust and Nonlinear Control, 28 (2018), 144-153.
doi: 10.1002/rnc.3858. |
[21] |
B. Zhou, D. Li and Z. Lin,
Control of discrete-time periodic linear systems with input saturation via multi-step periodic invariant sets, International Journal of Robust & Nonlinear Control, 25 (2015), 103-124.
|
[22] |
B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, Springer, Berlin Heidelberg, 2014.
doi: 10.1007/978-3-642-54206-0. |
[23] |
B. Zhou, Z. Y. Li and Z. Lin,
Observer based output feedback control of linear systems with input and output delays, Automatica, 49 (2013), 2039-2052.
doi: 10.1016/j.automatica.2013.03.031. |
[24] |
B. Zhou, C. Xu and G. Duan,
Distributed and truncated reduced-order observer based output feedback consensus of multi-agent systems, IEEE Transactions on Automatic Control, 59 (2014), 2264-2270.
doi: 10.1109/TAC.2014.2301573. |
[25] |
F. Zhu and F. Cen, Full-order observer-based actuator fault detection and reduced-order observer-based fault reconstruction for a class of uncertain nonlinear systems, Journal of Process Control, 20 (2010), 1141-1149. Google Scholar |

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