# American Institute of Mathematical Sciences

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, Zhang Lei. A robust reduced-order observers design approach for linear discrete periodic systems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019081
##### References:

show all references

##### References:
The trajectories of observed state errors by $L_{t}^{\mathrm{rand}}$ and $L_{t}^{\mathrm{robu}}$
 [1] Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo, Rohanin Ahmad. An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 109-125. doi: 10.3934/naco.2013.3.109 [2] Chuandong Li, Fali Ma, Tingwen Huang. 2-D analysis based iterative learning control for linear discrete-time systems with time delay. Journal of Industrial & Management Optimization, 2011, 7 (1) : 175-181. doi: 10.3934/jimo.2011.7.175 [3] Zhendong Luo. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1189-1212. doi: 10.3934/dcdsb.2015.20.1189 [4] Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Addendum to "Optimal control of multiscale systems using reduced-order models". Journal of Computational Dynamics, 2017, 4 (1&2) : 167-167. doi: 10.3934/jcd.2017006 [5] Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Optimal control of multiscale systems using reduced-order models. Journal of Computational Dynamics, 2014, 1 (2) : 279-306. doi: 10.3934/jcd.2014.1.279 [6] Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734 [7] Shaohong Fang, Jing Huang, Jinying Ma. Stabilization of a discrete-time system via nonlinear impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020106 [8] Luigi C. Berselli, Tae-Yeon Kim, Leo G. Rebholz. Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1027-1050. doi: 10.3934/dcdsb.2016.21.1027 [9] Ran Dong, Xuerong Mao. Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020017 [10] Lingling Lv, Zhe Zhang, Lei Zhang, Weishu Wang. An iterative algorithm for periodic sylvester matrix equations. Journal of Industrial & Management Optimization, 2018, 14 (1) : 413-425. doi: 10.3934/jimo.2017053 [11] Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393 [12] Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489 [13] Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial & Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016 [14] Victor Kozyakin. Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3537-3556. doi: 10.3934/dcdsb.2018277 [15] Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023 [16] Hongbiao Fan, Jun-E Feng, Min Meng. Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1535-1556. doi: 10.3934/jimo.2016.12.1535 [17] Jingang Zhao, Chi Zhang. Finite-horizon optimal control of discrete-time linear systems with completely unknown dynamics using Q-learning. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020030 [18] Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650 [19] Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121 [20] Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure & Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457

2018 Impact Factor: 1.025