{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:27:59Z","timestamp":1760146079550,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T00:00:00Z","timestamp":1727740800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Jet Propulsion Laboratory (JPL)","award":["1659175","N00014-19-1-2435","HM0476-19-1-2015"],"award-info":[{"award-number":["1659175","N00014-19-1-2435","HM0476-19-1-2015"]}]},{"name":"Office of Naval Research","award":["1659175","N00014-19-1-2435","HM0476-19-1-2015"],"award-info":[{"award-number":["1659175","N00014-19-1-2435","HM0476-19-1-2015"]}]},{"DOI":"10.13039\/100000266","name":"National Geospatial Intelligence Agency (NGA)","doi-asserted-by":"publisher","award":["1659175","N00014-19-1-2435","HM0476-19-1-2015"],"award-info":[{"award-number":["1659175","N00014-19-1-2435","HM0476-19-1-2015"]}],"id":[{"id":"10.13039\/100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper investigates state estimation methods for dynamical systems when model evaluations are performed on resource-constrained embedded systems with finite precision compute elements. Minimum mean square estimation algorithms are reformulated to incorporate finite-precision numerical errors in states, inputs, and measurements. Quantized versions of least squares batch estimation, sequential Kalman, and square-root filtering algorithms are proposed for fixed-point implementations. Numerical simulations are used to demonstrate performance improvements over standard filter formulations. Steady-state covariance analysis is employed to capture the performance trade-offs with numerical precision, providing insights into the best possible filter accuracy achievable for a given numerical representation. A low-latency fixed-point acceleration state estimation architecture for optomechanical sensing applications is realized on Field Programmable Gate Array System on Chip (FPGA-SoC) hardware. The hardware implementation results of the estimator are compared with double-precision MATLAB implementation, and the performance metrics are reported. Simulations and the experimental results underscore the significance of modeling quantization errors into state estimation pipelines for fixed-point embedded implementations.<\/jats:p>","DOI":"10.3390\/s24196381","type":"journal-article","created":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T11:08:47Z","timestamp":1727780927000},"page":"6381","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Quantized State Estimation for Linear Dynamical Systems"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9191-2130","authenticated-orcid":false,"given":"Ramchander Rao","family":"Bhaskara","sequence":"first","affiliation":[{"name":"Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8425-8687","authenticated-orcid":false,"given":"Manoranjan","family":"Majji","sequence":"additional","affiliation":[{"name":"Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9136-929X","authenticated-orcid":false,"given":"Felipe","family":"Guzm\u00e1n","sequence":"additional","affiliation":[{"name":"Wyant College of Optical Sciences, The University of Arizona, Tucson, AZ 85721, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1115\/1.3662552","article-title":"A New Approach to Linear Filtering and Prediction Problems","volume":"82","author":"Kalman","year":"1960","journal-title":"J. Basic Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1109\/MCS.2010.936465","article-title":"Applications of Kalman filtering in aerospace 1960 to the present [historical perspectives]","volume":"30","author":"Grewal","year":"2010","journal-title":"IEEE Control. Syst. Mag."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hall, E.C. (1996). Journey to the Moon: The History of the Apollo Guidance Computer, AIAA.","DOI":"10.2514\/4.868023"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1109\/TAC.1971.1099816","article-title":"Discrete square root filtering: A survey of current techniques","volume":"16","author":"Kaminski","year":"1971","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1109\/TAC.1975.1100994","article-title":"Square-root algorithms for least-squares estimation","volume":"20","author":"Morf","year":"1975","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1080\/00207178908953360","article-title":"Systolic approach to square root information Kalman filtering","volume":"50","author":"Gaston","year":"1989","journal-title":"Int. J. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0005-1098(94)00069-U","article-title":"A distributed and iterative method for square root filtering in space-time estimation","volume":"31","author":"Chin","year":"1995","journal-title":"Automatica"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/S0141-9331(97)00040-9","article-title":"High-performance FPGA-based implementation of Kalman filter","volume":"21","author":"Lee","year":"1997","journal-title":"Microprocess. Microsystems"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6900","DOI":"10.1007\/s00034-023-02437-9","article-title":"Low-Complexity Square-Root Unscented Kalman Filter Design Methodology","volume":"42","author":"Dutt","year":"2023","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Liu, K., and Skelton, R. (1990, January 23\u201325). Optimal controllers for finite wordlength implementation. Proceedings of the 1990 American Control Conference, San Diego, CA, USA.","DOI":"10.23919\/ACC.1990.4791066"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1294","DOI":"10.1109\/9.623096","article-title":"Systems with finite communication bandwidth constraints. I. State estimation problems","volume":"42","author":"Wong","year":"1997","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1177\/0142331209342213","article-title":"Quantized filtering of linear stochastic systems","volume":"33","author":"You","year":"2011","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4782","DOI":"10.1109\/TSP.2006.882059","article-title":"SOI-KF: Distributed Kalman filtering with low-cost communications using the sign of innovations","volume":"54","author":"Ribeiro","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1109\/LSP.2004.826634","article-title":"An efficient fixed-point implementation of residual resampling scheme for high-speed particle filters","volume":"11","author":"Hong","year":"2004","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_15","unstructured":"Soh, J. (2017). A Scalable, Portable, FPGA-Based Implementation of the Unscented Kalman Filter. [Ph.D. Thesis, The University of Sydney]."},{"key":"ref_16","first-page":"101084","article-title":"FPGA implementation of multi-dimensional Kalman filter for object tracking and motion detection","volume":"33","author":"Babu","year":"2022","journal-title":"Eng. Sci. Technol. Int. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"221111","DOI":"10.1063\/1.4881936","article-title":"High sensitivity optomechanical reference accelerometer over 10 kHz","volume":"104","author":"Kumanchik","year":"2014","journal-title":"Appl. Phys. Lett."},{"key":"ref_18","unstructured":"Cervantes, F.G., Flatscher, R., Gerardi, D., Burkhardt, J., Gerndt, R., Nofrarias, M., Reiche, J., Heinzel, G., Danzmann, K., and Bot\u00e9, L.G. (2013, January 7\u201312). LISA technology package flight hardware test campaign. Proceedings of the ASP Conference Series, Seattle, WA, USA."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"931","DOI":"10.2514\/1.A34326","article-title":"GRACE-FO: The Gravity Recovery and Climate Experiment Follow-On Mission","volume":"56","author":"Kornfeld","year":"2019","journal-title":"J. Spacecr. Rocket."},{"key":"ref_20","unstructured":"Amaro-Seoane, P., Audley, H., Babak, S., Baker, J., Barausse, E., Bender, P., Berti, E., Binetruy, P., Born, M., and Bortoluzzi, D. (2017). Laser interferometer space antenna. arXiv."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"076901","DOI":"10.1088\/0034-4885\/72\/7\/076901","article-title":"LIGO: The laser interferometer gravitational-wave observatory","volume":"72","author":"Abbott","year":"2009","journal-title":"Rep. Prog. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Iturbe, X., Keymeulen, D., Ozer, E., Yiu, P., Berisford, D., Hand, K., and Carlson, R. (2015, January 5\u20137). An integrated SoC for science data processing in next-generation space flight instruments avionics. Proceedings of the 2015 IFIP\/IEEE International Conference on Very Large Scale Integration (VLSI-SoC), Daejeon, Republic of Korea.","DOI":"10.1109\/VLSI-SoC.2015.7314405"},{"key":"ref_23","unstructured":"Ramchander Rao, B. (2021). Hardware Implementation of Navigation Filters for Automation of Dynamical Systems. [Master\u2019s Thesis, Texas A&M University]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"282","DOI":"10.2514\/3.55779","article-title":"Analytic steady-state accuracy solutions for two common spacecraft attitude estimators","volume":"1","author":"Farrenkopf","year":"1978","journal-title":"J. Guid. Control"},{"key":"ref_25","unstructured":"Oppenheim, A.V., and Schafer, R.W. (1975). Digital Signal Processing, Prentice-Hall, Inc.. Research Supported by the Massachusetts Institute of Technology, Bell Telephone Laboratories, and Guggenheim Foundation."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1109\/TCS.1976.1084254","article-title":"Synthesis of minimum roundoff noise fixed point digital filters","volume":"23","author":"Mullis","year":"1976","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Kelly, P., Majji, M., and Guzm\u00e1n, F. (2021). Estimation and Error Analysis for Optomechanical Inertial Sensors. Sensors, 21.","DOI":"10.3390\/s21186101"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"G167","DOI":"10.1364\/AO.393061","article-title":"Optomechanical inertial sensors","volume":"59","author":"Hines","year":"2020","journal-title":"Appl. Opt."},{"key":"ref_29","unstructured":"Rasras, M., Elfadel, I.A.M., and Ngo, H.D. (2019). MEMS Accelerometers, MDPI."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"19510","DOI":"10.1364\/OE.457499","article-title":"Intrinsically accurate sensing with an optomechanical accelerometer","volume":"30","author":"Reschovsky","year":"2022","journal-title":"Opt. Express"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1088\/0026-1394\/52\/5\/654","article-title":"Optomechanical reference accelerometer","volume":"52","author":"Gerberding","year":"2015","journal-title":"Metrologia"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1109\/TASSP.1976.1162802","article-title":"Roundoff noise in state-space digital filtering: A general analysis","volume":"24","author":"Hwang","year":"1976","journal-title":"IEEE Trans. Acoust. Speech, Signal Process."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1218","DOI":"10.1109\/9.40767","article-title":"Optimal finite wordlength linear quadratic regulation","volume":"34","author":"Williamson","year":"1989","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1109\/TASSP.1977.1162977","article-title":"A necessary and sufficient condition for quantization errors to be uniform and white","volume":"25","author":"Sripad","year":"1977","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Crassidis, J.L., and Junkins, J.L. (2004). Optimal Estimation of Dynamic Systems, Chapman and Hall\/CRC.","DOI":"10.1201\/9780203509128"},{"key":"ref_36","unstructured":"Elmenreich, W. (2002). Sensor Fusion in Time-Triggered Systems. [Ph.D. Thesis, The Pennsylvania State University]."},{"key":"ref_37","unstructured":"Bierman, G.J. (2006). Factorization Methods for Discrete Sequential Estimation, Courier Corporation."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"B87","DOI":"10.1364\/JOSAA.396774","article-title":"Optomechanical lasers for inertial sensing","volume":"37","author":"Wisniewski","year":"2020","journal-title":"JOSA A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1109\/MC.1982.1653825","article-title":"Why systolic architectures?","volume":"15","author":"Kung","year":"1982","journal-title":"Computer"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Golub, G.H., and Van Loan, C.F. (2013). Matrix Computations, JHU Press.","DOI":"10.56021\/9781421407944"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1093\/imamat\/12.3.329","article-title":"Least squares computations by Givens transformations without square roots","volume":"12","author":"Gentleman","year":"1973","journal-title":"IMA J. Appl. Math."},{"key":"ref_42","unstructured":"Bhaskara, R.R., Sung, K., and Majji, M. (2022, January 18\u201322). An FPGA framework for Interferometric Vision-Based Navigation (iVisNav). Proceedings of the 2022 IEEE\/AIAA 41st Digital Avionics Systems Conference (DASC), Portsmouth, VA, USA."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/0020-0255(73)90015-7","article-title":"Exponential data weighting in the Kalman-Bucy filter","volume":"5","author":"Anderson","year":"1973","journal-title":"Inf. Sci."},{"key":"ref_44","unstructured":"Jazwinski, A.H. (2007). Stochastic Processes and Filtering Theory, Courier Corporation."},{"key":"ref_45","unstructured":"Van Der Merwe, R., and Wan, E.A. (2001, January 7\u201311). The square-root unscented Kalman filter for state and parameter-estimation. Proceedings of the 2001 IEEE International Conference on  Acoustics, Speech, and Signal Processing, Proceedings (Cat. No. 01CH37221), Salt Lake City, UT, USA."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/19\/6381\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:08:53Z","timestamp":1760112533000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/19\/6381"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,1]]},"references-count":45,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["s24196381"],"URL":"https:\/\/doi.org\/10.3390\/s24196381","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2024,10,1]]}}}