White noise estimation for linear discrete fractional order system

Yantong Mu; Huihong Zhao; Zhifang Li; Yantong Mu; Huihong Zhao; Zhifang Li

doi:10.3934/math.2022558

AIMS Mathematics

2022, Volume 7, Issue 6: 10009-10023. doi: 10.3934/math.2022558

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Research article

White noise estimation for linear discrete fractional order system

School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China

Received: 17 January 2022 Revised: 27 February 2022 Accepted: 07 March 2022 Published: 21 March 2022
MSC : 93C55, 93E11

The process white noise (PWN) and observation white noise (OWN) estimation problem for linear discrete fractional order systems (LDFOS) is addressed in this study. By using the Grünwald-Letnikov (G-L) operator as a definition of the discrete fractional calculus (DFC), LDFOS is transformed into a class of linear discrete time-delay systems. However, it is different from the general time-delay system, in which the time-delay part is the cumulative sum from time 0 to the previous time. Based on the orthogonal projection theorem, a suboptimal one-step predictor of LDFOS is designed. Due to the existence of cumulative sum time-delay in system, the Riccati equation has one more cumulative sum state error variance term, which is different from the classical Kalman filter (KF). Moreover, using innovation analysis technology, the filtering and fixed-lag smoothing estimators of PWN and OWN in the form of noise orthogonal projection gain matrices are derived. Finally, two simulation examples are given to verify the effectiveness of PWN and OWN estimators.
- white noise,
- fractional order system,
- one-step predictor,
- filtering,
- fixed-lag smoothing
Citation: Yantong Mu, Huihong Zhao, Zhifang Li. White noise estimation for linear discrete fractional order system[J]. AIMS Mathematics, 2022, 7(6): 10009-10023. doi: 10.3934/math.2022558

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Abstract

The process white noise (PWN) and observation white noise (OWN) estimation problem for linear discrete fractional order systems (LDFOS) is addressed in this study. By using the Grünwald-Letnikov (G-L) operator as a definition of the discrete fractional calculus (DFC), LDFOS is transformed into a class of linear discrete time-delay systems. However, it is different from the general time-delay system, in which the time-delay part is the cumulative sum from time 0 to the previous time. Based on the orthogonal projection theorem, a suboptimal one-step predictor of LDFOS is designed. Due to the existence of cumulative sum time-delay in system, the Riccati equation has one more cumulative sum state error variance term, which is different from the classical Kalman filter (KF). Moreover, using innovation analysis technology, the filtering and fixed-lag smoothing estimators of PWN and OWN in the form of noise orthogonal projection gain matrices are derived. Finally, two simulation examples are given to verify the effectiveness of PWN and OWN estimators.

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