American Institute of Mathematical Sciences

November  2020, 16(6): 2799-2812. doi: 10.3934/jimo.2019081

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

*Corresponding author: Lei Zhang

Received  October 2018 Revised  March 2019 Published  November 2020 Early access  July 2019

Fund Project: This work is supported by the Programs of National Natural Science Foundation of China (Nos. U1604148, 11501200, 61402149), Innovative Talents of Higher Learning Institutions of Henan (No. 17HASTIT023), Central China thousand talents program(No.ZYQR201810138)

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.

Citation: Lingling Lv, Wei He, Xianxing Liu, Lei Zhang. A robust reduced-order observers design approach for linear discrete periodic systems. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2799-2812. doi: 10.3934/jimo.2019081
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] 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. [3] 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. [4] R. Kalman, A new approach to linear filtering and prediction problems, J. Basic Eng., 82 (1960), 35-45.  doi: 10.1115/1.3662552. [5] B. Li, Y. Rong and J. Sun, et al., A distributionally robust minimum variance beamformer design, IEEE Signal Processing Letters, 25 (2018), 105-109. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, Springer, Berlin Heidelberg, 2014. doi: 10.1007/978-3-642-54206-0. [22] 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. [23] 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. [24] 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.

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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] 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. [3] 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. [4] R. Kalman, A new approach to linear filtering and prediction problems, J. Basic Eng., 82 (1960), 35-45.  doi: 10.1115/1.3662552. [5] B. Li, Y. Rong and J. Sun, et al., A distributionally robust minimum variance beamformer design, IEEE Signal Processing Letters, 25 (2018), 105-109. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, Springer, Berlin Heidelberg, 2014. doi: 10.1007/978-3-642-54206-0. [22] 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. [23] 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. [24] 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.
The trajectories of observed state errors by $L_{t}^{\mathrm{rand}}$ and $L_{t}^{\mathrm{robu}}$
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