## Pyung Soo Kim*## |

Parameter | F404 engine system |
---|---|

Noise covariances | [TeX:] $$Q=0.1^{2}, R=0.1^{2}$$ |

Measurement memory length (sec) | [TeX:] $$T=1 \ and \ T=1.5$$ |

Simulation duration (sec) | 80 |

Model uncertainty | |

First scenario | [TeX:] $$\delta(t)=0.6 \text { for } 20 \sec \leq t \leq 25 \text { sec }$$ |

Second scenario | [TeX:] $$\delta(t)=\left\{\begin{array}{ll}0.6 \text { for } 20 \text { sec } \leq t \leq 25 \text { sec } \\ 0.3 \text { for } 30 \text { sec } \leq t \leq 35 \text { sec }\end{array}\right.$$ |

As shown in Figs. 2 and 3, the proposed state estimation filtering algorithm can be superior to an IMS filter, such as the Kalman filter, in terms of error magnitude and error convergence for both scenarios according to different types of temporary uncertainties. The estimation errors of the proposed state estimation filter are shown to be smaller than those of the IMS filter at the modeling uncertainty interval. After temporary modeling uncertainty disappears, the estimation error convergence is shown to be better than that of the IMS filter. In particular, the proposed state estimation filtering can be shown to outperform the IMS filter significantly when temporary uncertainties occur in succession. It is well known that the Kalman filter with the infinite memory structure utilizes all past measurements using a recursive formulation. Hence, estimation errors tend to accumulate over time in the IMS filter and thus in some severe cases can show the error divergence for temporary uncertainties. Thus, as shown in the second plots of Figs. 2 and 3, the estimation error due to the preceding model uncertainty can be propagated, and then the estimation error due to the following model uncertainty worsens although the

following model uncertainty,[TeX:] $$\delta(t)=0.3 \text { for } 30 \sec \leq t \leq 35 \text { sec }$$is smaller than the preceding model uncertainty [TeX:] $$\delta(t)=0.6 \text { for } 20 \sec \leq t \leq 25 \text { sec. }$$

Therefore, when two estimation filters, IMS filter and proposed filter, are applied to temporarily uncertain systems, the proposed filtered estimate can be more robust than the IMS filtered estimate, although the robustness is not considered during the design process of the proposed estimation filter. If the effect of temporary modeling uncertainty completely disappears, the proposed state estimation filter is also shown to be comparable to the IMS filter. Moreover, it is also shown that the noise reduction of the FMS filter used in the proposed state estimation filtering can be greatly affected by the measurement memory length for past measurements. The estimation filter can have greater noise reduction with increasing the measurement memory length. On the other hand, the convergence speed of the estimation error worsens as the measurement memory length increases. This can be verified by the simulation results of Fig. 2 with T = 1 second and Fig. 3 with T = 1.5 seconds.

This paper has developed an alternative state estimation filtering algorithm designed for continuous time systems with noises as well as control input. In the proposed algorithm, two types of estimation filters with different memory structures are operated selectively in order to take full advantage of both IMS and FMS filters. The IMS filter is operated for a certain continuous time system. On the other hand, the FMS filter is operated for temporarily uncertain continuous time system. Thus, one of FMS and IMS filtered estimates is operated selectively to obtain the valid estimate depending on the presence of uncertainty. A couple of test variables and a declaration rule have been developed to detect the presence of uncertainty, to perform the suitable choice from IMS and FMS filters, and to obtain ultimately the valid filtered estimate. Computer simulations for a continuous time aircraft engine system have shown that the proposed state estimation filtering algorithm can work well in a temporarily uncertain continuous time system as well as a certain continuous time system.

The measurement memory length for the FMS filter can be considered as one of useful design parameters for the proposed state estimation filtering algorithm. Hence, it can be interesting issue to choose an appropriate measurement memory length that makes the filtering performance to be better. The noise reduction of the proposed state estimation filtering algorithm can be greatly affected by the measurement memory length, and it can lead to greater noise suppression as the measurement memory length increases, which enhances the filtering performance. In this paper, the measurement memory length has been selected with these observations, which is actually not a systematic method. Therefore, a more systematic method to determine the measurement memory length should be addressed as a future research topic.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2017R1D1A1B03033024). The work reported in this paper was conducted during the sabbatical year of Korea Polytechnic University in 2020.

He received the B.S. degree in Electrical Engineering from Inha University, Incheon, Korea, in 1994. He received the M.S. degree in Control and Instrumentation Engin-eering and the Ph.D. degree at the School of Electrical Engineering and Computer Science from Seoul National University, Seoul, Korea, in 1996 and 2001, respectively. From 2001 to 2005, he was a senior researcher at the Digital Media R&D Center of Samsung Electronics Co. Ltd. Since 2005, he is currently a Professor in the Department of the Electronics Engineering at Korea Polytechnic University, Shiheung, Korea. His main research interests are in the areas of system software solutions, statistical signal processing, wireless mobile networks, next generation network system design, and various industrial applications.

- 1 S. U. Khan, W. Y. Chai, C. S. See, A. Khan, "X-ray image enhancement using a boundary division wiener filter and wavelet-based image fusion approach,"
*Journal of Information Processing Systems*, vol. 12, no. 1, pp. 35-45, 2016.doi:[[[10.3745/JIPS.02.0029]]] - 2 L. Wang, C. Wang, W. Huang, X. Zhou, "Image deblocking scheme for JPEG compressed images using an adaptive-weighted bilateral filter,"
*Journal of Information Processing Systems*, vol. 12, no. 4, pp. 631-643, 2016.doi:[[[10.3745/JIPS.02.0046]]] - 3 M. T. N. Truong, S. Kim, "Parallel implementation of color-based particle filter for object tracking in embedded systems,"
*Human-centric Computing and Information Sciences*, vol. 7, no. 2, 2017.doi:[[[10.1186/s13673-016-0082-1]]] - 4 H. H. Afshari, S. A. Gadsden, S. Habibi, "Gaussian filters for parameter and state estimation: a general review of theory and recent trends,"
*Signal Processing*, vol. 135, pp. 218-238, 2017.doi:[[[10.1016/j.sigpro.2017.01.001]]] - 5 M. S. Grewal, A. P. Andrews, "Applications of Kalman filtering in aerospace 1960 to the present,"
*IEEE Control Systems Magazine*, vol. 30, no. 3, pp. 69-78, 2010.custom:[[[-]]] - 6 F. Auger, M. Hilairet, J. M. Guerrero, E. Monmasson, T. Orlowska-Kowalska, S. Katsura, "Industrial applications of the Kalman filter: a review,"
*IEEE Transactions on Industrial Electronics*, vol. 60, no. 12, pp. 5458-5471, 2013.doi:[[[10.1109/TIE.2012.2236994]]] - 7 M. B. Rhudy, R. A. Salguero, K. Holappa, "A Kalman filtering tutorial for undergraduate students,"
*International Journal of Computer Science & Engineering Survey*, vol. 8, no. 1, pp. 1-18, 2017.custom:[[[-]]] - 8 A. Barrau, S. Bonnabel, "Invariant Kalman filtering,"
*Annual Review of ControlRobotics, and Autonomous Systems*, vol. 2018, pp. 237-257, 2018.custom:[[[-]]] - 9 P. S. Kim, "An alternative FIR filter for state estimation in discrete-time systems,"
*Digital Signal Processing*, vol. 20, no. 3, pp. 935-943, 2010.doi:[[[10.1016/j.dsp.2009.10.033]]] - 10 S. Zhao, Y. S. Shmaliy, B. Huang, F. Liu, "Minimum variance unbiased FIR filter for discrete time-variant systems,"
*Automatica*, vol. 53, pp. 355-361, 2015.doi:[[[10.1016/j.automatica.2015.01.022]]] - 11 Y. S., Shmaliy, S. Zhao, C. K. Ahn, "Unbiased finite impluse response filtering: an iterative alternative to Kalman filtering ignoring noise and initial conditions,"
*IEEE Control Systems Magazine*, vol. 37, no. 5, pp. 70-89, 2017.custom:[[[-]]] - 12 S. S. Yuriy, N. Yrjo, K. Sanowar, "Review of unbiased FIR filters, smoothers, and predictors for polynomial signals,"
*Frontiers in Signal Processing2018*, vol. 2, 2100.doi:[[[10.22606/fsp.2018.1]]] - 13 W. H. Kwon, P. S. Kim, P. Park, "A receding horizon Kalman FIR filter for linear continuous-time systems,"
*IEEE Transactions on Automatic Control*, vol. 44, no. 11, pp. 2115-2120, 1999.doi:[[[10.1109/9.802927]]] - 14 S. H. Han, W. H. Kwon, P. S. Kim, "Receding-horizon unbiased FIR filters for continuous-time state-space models without a priori initial state information,"
*IEEE Transactions on Automatic Control*, vol. 46, no. 5, pp. 766-770, 2001.doi:[[[10.1109/9.920798]]] - 15 P. S. Kim, "Two-stage estimation filtering for temporarily uncertain systems,"
*in Advanced Multimedia and Ubiquitous Engineering. Singapore: Springer*, pp. 303-309, 2016.custom:[[[-]]] - 16 M. Vazquez-Olguin, Y. S. Shmaliy, O. Ibarra-Manzano, "Distributed UFIR filtering over WSNs with consensus on estimates,"
*IEEE Transactions on Industrial Informatics*, vol. 16, no. 3, pp. 1645-1654, 2020.custom:[[[-]]] - 17 Y. Xu, Y. S. Shmaliy, Y. Li, X. Chen, H. Guo, "Indoor INS/LiDAR-based robot localization with improved robustness using cascaded FIR filter,"
*IEEE Access*, vol. 7, pp. 34189-34197, 2019.custom:[[[-]]] - 18 P. S. Kim, "Selective finite memory structure filtering using the chi-square test statistic for temporarily uncertain systems,"
*Applied Sciences*, vol. 9, no. 4257, 2019.custom:[[[-]]] - 19 Y. Zhai, W. Song, X. Liu, L. Liu, X. Zhao, "A chi-square statistics based feature selection method in text classification," in
*Proceedings of 2018 IEEE 9th International Conference on Software Engineering and Service Science (ICSESS)*, Beijing, China, 2018;pp. 160-163. custom:[[[-]]] - 20 W. Xue, Y. Q. Guo, X. D. Zhang, "Application of a bank of Kalman filters and a robust Kalman filter for aircraft engine sensor/actuator fault diagnosis,"
*International Journal of Innovative ComputingInformation and Control*, vol. 4, no. 12, pp. 3161-3168, 2018.custom:[[[-]]]