# 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  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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. Kalman, A new approach to linear filtering and prediction problems, J. Basic Eng., 82 (1960), 35-45.  doi: 10.1115/1.3662552.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [21] B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, Springer, Berlin Heidelberg, 2014. doi: 10.1007/978-3-642-54206-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. Kalman, A new approach to linear filtering and prediction problems, J. Basic Eng., 82 (1960), 35-45.  doi: 10.1115/1.3662552.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [21] B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, Springer, Berlin Heidelberg, 2014. doi: 10.1007/978-3-642-54206-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar
The trajectories of observed state errors by $L_{t}^{\mathrm{rand}}$ and $L_{t}^{\mathrm{robu}}$
 [1] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [2] Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166 [3] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [4] Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266 [5] Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017 [6] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [7] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [8] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [9] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [10] Mathew Gluck. Classification of solutions to a system of $n^{\rm th}$ order equations on $\mathbb R^n$. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 [11] Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381 [12] Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323 [13] Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029 [14] Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 [15] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055 [16] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302 [17] Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353 [18] Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299 [19] Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168 [20] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

2019 Impact Factor: 1.366

## Metrics

• PDF downloads (96)
• HTML views (591)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]