# Robust State Estimation for Uncertain Discrete Linear Systems with Delayed Measurements

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and the Design of Robust State Estimator

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

## 3. Recursive Procedure and Asymptotic Stability Conditions of the Estimator

**Theorem**

**1.**

**Theorem**

**2.**

## 4. Numerical Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Derivation of the Recursive Procedure

**Lemma**

**A1.**

## Appendix B

**Lemma**

**A2.**

**Proof.**

## Appendix C

## References

- Anderson, B.D.O.; Moore, J.B.; Eslami, M. Optimal Filtering. IEEE Trans. Syst. Man Cybern.
**2007**, 12, 235–236. [Google Scholar] [CrossRef] - Patel, V. Kalman-based Stochastic Gradient Method with Stop Condition and Insensitivity to Conditioning. SIAM J. Optim.
**2016**, 26, 2620–2648. [Google Scholar] [CrossRef] [Green Version] - Bergou, E.H.; Gratton, S.; Mandel, J. On the Convergence of a Non-linear Ensemble Kalman Smoother. Mathematics
**2014**, 137, 151–168. [Google Scholar] [CrossRef] [Green Version] - Belyaev, K.; Kuleshov, A.; Smirnov, I. Generalized Kalman Filter and Ensemble Optimal Interpolation, Their Comparison and Application to the Hybrid Coordinate Ocean Model. Mathematics
**2021**, 9, 2371. [Google Scholar] [CrossRef] - Safa, A.; Baradarannia, M.; Kharrati, H.; Khanmohammadi, S. Global attitude stabilization of rigid spacecraft with unknown input delay. Nonlinear Dyn.
**2015**, 82, 1623–1640. [Google Scholar] [CrossRef] - Keighobadi, J.; Fateh, M.M.; Xu, B. Adaptive fuzzy voltage-based backstepping tracking control for uncertain robotic manipulators subject to partial state constraints and input delay. Nonlinear Dyn.
**2020**, 100, 2609–2634. [Google Scholar] [CrossRef] - He, Q.; Luo, Y.; Mao, Y.; Zhou, X. An acceleration feed-forward control method based on fusion of model output and sensor data. Sens. Actuators A Phys.
**2018**, 284, 186–193. [Google Scholar] [CrossRef] - Zhou, T. Sensitivity penalization based robust state estimation for uncertain linearsystems. IEEE Trans. Autom. Control
**2010**, 55, 1018–1024. [Google Scholar] [CrossRef] - Sun, S.; Ma, J. Linear estimation for networked control systems with random transmission delays and packet dropouts. Inf. Sci.
**2014**, 269, 349–365. [Google Scholar] [CrossRef] - Sun, Y.; Yang, G. Event-triggered state estimation for networked control systems with lossy network communication. Inf. Sci.
**2019**, 492, 1–12. [Google Scholar] [CrossRef] - Geng, H.; Wang, Z.; Cheng, Y. Distributed federated tobit kalman filter fusion over a packet-delaying network: A probabilistic perspective. IEEE Trans. Signal Process.
**2018**, 66, 4477–4489. [Google Scholar] [CrossRef] - Tang, T.; Yang, T.; Qi, B.; Cao, L.; Ren, G.; Hu, H. Error-based plug-in controller of tip-tilt mirror to reject telescope’s structural vibrations. J. Astron. Telesc. Instrum. Syst.
**2018**, 4, 049004. [Google Scholar] [CrossRef] - Tang, T.; Cai, H.; Huang, Y.; Ren, G. Combined line-of-sight error and angular position to generate feedforward control for a charge-coupled device-based tracking loop. Opt. Eng.
**2015**, 54, 105107. [Google Scholar] [CrossRef] [Green Version] - Choi, M.; Choi, J.; Chung, W. State estimation with delayed measurements incorporating time-delay uncertainty. IET Control Theory Appl.
**2012**, 6, 2351–2361. [Google Scholar] [CrossRef] - Bar-Shalom, Y. Update with out-of-sequence measurements in tracking: Exact solution. IEEE Trans. Aerosp. Electron. Syst.
**2002**, 38, 769–778. [Google Scholar] [CrossRef] - Bar-Shalom, Y.; Chen, H.; Mallick, M. One-step solution for the multistep out-of sequence-measurement problem in tracking. IEEE Trans. Aerosp. Electron. Syst.
**2004**, 40, 27–37. [Google Scholar] [CrossRef] - Zhang, K.; Li, X.; Zhu, Y. Optimal update with out-of-sequence measurements for distributed filtering. In Proceedings of the Fifth International Conference on Information Fusion, VII, ISIF, Annapolis, MD, USA, 8–11 July 2002; pp. 1519–1526. [Google Scholar]
- Wang, J.; Mao, Y.; Li, Z. Robust state estimation for uncertain linear discrete systems with d-step state delay. IET Control Theory Appl.
**2021**, 15, 1708–1723. [Google Scholar] [CrossRef] - Challa, S.; Evans, R.J.; Wang, X. A bayesian solution and its approximations to out-of-sequence measurement problems. Inf. Fusion
**2003**, 4, 185–199. [Google Scholar] [CrossRef] - Mu, L.X. Robust finite-time h control of singular stochastic systems via static output feedback. Appl. Math. Comput.
**2012**, 218, 5629–5640. [Google Scholar] - Li, Q.; Wang, Z.; Sheng, W. Dynamic event-triggered mechanism for H-infinity non-fragile state estimation of complex networks under randomly occurring sensor saturations. Inf. Sci.
**2020**, 509, 304–316. [Google Scholar] [CrossRef] - Wang, Y.; Huang, J.; Wu, D. Set-membership filtering with incomplete observations. Inf. Sci.
**2020**, 517, 37–51. [Google Scholar] [CrossRef] - Huang, Y.; Chen, Z.; Wei, C. Least trace extended set-membership filter. Sci. China Inf. Sci.
**2010**, 53, 258–270. [Google Scholar] [CrossRef] [Green Version] - Sayed, A.H. A framework for state-space estimation with uncertain models. IEEE Trans. Autom. Control
**2001**, 46, 998–1013. [Google Scholar] [CrossRef] [Green Version] - Xu, H.; Mannor, S. A kalman filter design based on the performance/robustness tradeoff. IEEE Trans. Autom. Control
**2009**, 54, 1171–1175. [Google Scholar] - Liu, H.; Zhou, T. Robust state estimation for uncertain linear systems with deterministic input signals. Control Theory Technol.
**2014**, 12, 383–392. [Google Scholar] [CrossRef] - Zabari, A.; Tissir, E.H.; Tadeo, F. Delay-dependent robust h-infinity filtering with lossy measurements for discrete-time systems. Arab. J. Sci. Eng.
**2017**, 42, 5263–5273. [Google Scholar] [CrossRef] - Chen, B.; Yu, L.; Zhang, W.A. Robust kalman filtering for uncertain state delay systems with random observation delays and missing measurements. IET Control Theory Appl.
**2011**, 5, 1945–1954. [Google Scholar] [CrossRef] - Feng, J.; Yang, R.; Liu, H.; Xu, B. Robust recursive estimation for uncertain systems with delayed measurements and noises. IEEE Access
**2020**, 8, 14386–14400. [Google Scholar] [CrossRef] - Qian, H.-M.; Huang, W.; Liu, B. Finite-horizon robust kalman filter for uncertain attitude estimation system with star sensor measurement delays. Abstr. Appl. Anal.
**2014**, 2014, 494060. [Google Scholar] [CrossRef] [Green Version] - Ahmad, S.; Rehan, M.; Iqbal, M. Robust generalized filtering of uncertain lipschitz nonlinear systems under measurement delays. Nonlinear Dyn.
**2018**, 92, 1567–1582. [Google Scholar] [CrossRef] - Liu, Y.; Park, J.H.; Guo, B.-Z. Non-fragile H-infinity filtering for nonlinear discrete-time delay systems with randomly occurring gain variations. ISA Trans.
**2016**, 63, 196–203. [Google Scholar] [CrossRef] - Bessaoudi, T.; Hmida, F.B.; Hsieh, C.-S. Robust state and fault estimation for non-linear stochastic systems with unknown disturbances: A multi-step delayed solutions. IET Control Theory Appl.
**2019**, 13, 2556–2570. [Google Scholar] [CrossRef] - Zhou, T. Robust state estimation using error sensitivity penalizing. In Proceedings of the 2008 47th IEEE Conference on Decision and Control, Cancun, Mexico, 9–11 December 2008; pp. 2563–2568. [Google Scholar]
- Kailath, T.; Sayed, A.H.; Hassibi, B. Linear Estimation; Prentice Hall: Hoboken, NJ, USA, 2000. [Google Scholar]
- Rubinstein, R.Y.; Kroese, D.P. Simulation and the Monte Carlo Method; John Wiley and Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Brogan, W.L. Modern Control Theory; Prentice Hall: Hoboken, NJ, USA, 1990. [Google Scholar]
- Hien, L.V.; Phat, V.N. Exponential stability and stabilization of a class of uncertain linear time-delay systems. J. Frankl. Inst.
**2009**, 346, 611–625. [Google Scholar] [CrossRef] - Zhou, K.; Doyle, J.C.; Glover, K. Robust and Optimal Control; Prentice Hall: Hoboken, NJ, USA, 1996. [Google Scholar]
- Zhou, T.; Liang, H.Y. On asymptotic behaviors of a sensitivity penalization based robust state estimator. Syst. Control Lett.
**2011**, 60, 174–180. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Li, Z.; Sun, M.; Duan, Q.; Mao, Y.
Robust State Estimation for Uncertain Discrete Linear Systems with Delayed Measurements. *Mathematics* **2022**, *10*, 1365.
https://doi.org/10.3390/math10091365

**AMA Style**

Li Z, Sun M, Duan Q, Mao Y.
Robust State Estimation for Uncertain Discrete Linear Systems with Delayed Measurements. *Mathematics*. 2022; 10(9):1365.
https://doi.org/10.3390/math10091365

**Chicago/Turabian Style**

Li, Zhijun, Minxing Sun, Qianwen Duan, and Yao Mao.
2022. "Robust State Estimation for Uncertain Discrete Linear Systems with Delayed Measurements" *Mathematics* 10, no. 9: 1365.
https://doi.org/10.3390/math10091365