{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:05:54Z","timestamp":1760238354259,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,7,28]],"date-time":"2020-07-28T00:00:00Z","timestamp":1595894400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009409","name":"Russian Science Support Foundation","doi-asserted-by":"publisher","award":["16-19-00046-P"],"award-info":[{"award-number":["16-19-00046-P"]}],"id":[{"id":"10.13039\/100009409","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The paper considers the problem of invariance with respect to the unknown input for discrete-time nonlinear dynamic systems. To solve the problem, the algebraic approaches, called algebra of functions and logic\u2013dynamic approach, are used. Such approaches assume that description of the system may contain non-differentiable functions. Necessary and sufficient conditions of solvability the problem are obtained. Moreover, procedures which find the appropriate functions and matrices are developed. Some applications of such invariance in fault detection and isolation, disturbance decoupling problem, and fault-tolerant control are considered.<\/jats:p>","DOI":"10.3390\/sym12081241","type":"journal-article","created":{"date-parts":[[2020,7,30]],"date-time":"2020-07-30T12:15:38Z","timestamp":1596111338000},"page":"1241","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Problem of Invariance in Nonlinear Discrete-Time Dynamic Systems"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5927-7117","authenticated-orcid":false,"given":"Alexey","family":"Zhirabok","sequence":"first","affiliation":[{"name":"Department of Automation and Control, Far Eastern Federal University, 690095 Vladivostok, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ding, S. (2014). Data-driven Design of Fault Diagnosis and Fault-tolerant Control Systems, Springer.","DOI":"10.1007\/978-1-4471-6410-4"},{"key":"ref_2","unstructured":"Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M. (2006). Diagnosis and Fault-Tolerant Control, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Alwi, H., Edwards, C., and Tan, C. (2011). Fault Detection and Fault-Tolerant Control Using Sliding Modes, Springer.","DOI":"10.1007\/978-0-85729-650-4"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1002\/rnc.3194","article-title":"Fault-tolerant control of discrete-time LPV systems using virtual actuators","volume":"25","author":"Tabatabaeipour","year":"2015","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"619","DOI":"10.2478\/v10006-010-0046-y","article-title":"Fault diagnosis and fault tolerant control using set-membership approaches: Application to real case studies","volume":"20","author":"Puig","year":"2010","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1683","DOI":"10.1109\/9.544009","article-title":"Output feedback disturbance decoupling in nonlinear systems","volume":"41","author":"Andiarti","year":"1996","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_7","unstructured":"Battilotti, S. (1997, January 10\u201312). A sufficient condition for nonlinear disturbance decoupling with stability via measurement feedback. Proceedings of the 36th Conference on Decision & Control, San Diago, CA, USA."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1109\/TAC.1981.1102604","article-title":"Nonlinear decoupling via feedback: A differential gemetric approach","volume":"26","author":"Isidori","year":"1981","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1425","DOI":"10.1109\/9.774115","article-title":"Disturbance decoupling by measurement feedback for SISO nonlinear systems","volume":"44","author":"Xia","year":"1999","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Wonham, W. (1979). Linear Multivariable Control: A Geometric Approach, Springer.","DOI":"10.1007\/978-1-4684-0068-7"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Isidori, A. (1995). Nonlinear Control Systems, Springer.","DOI":"10.1007\/978-1-84628-615-5"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1109\/TAC.1985.1104079","article-title":"Controlled invariance for discrete-time nonlinear systems with an application to the disturbance decoupling problem","volume":"30","author":"Grizzle","year":"1985","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_13","first-page":"197","article-title":"A geometric solution to the dynamic disturbance decoupling for discrete-time nonlinear systems","volume":"40","author":"Kotta","year":"2004","journal-title":"Kybernetika"},{"key":"ref_14","unstructured":"Kaldmae, A., and Kotta, U. (2011, January 14\u201317). Disturbance decoupling for discrete-time nonlinear systems by static measurement feedback. Proceedings of the 18th International Conference on Process Control, Tatranska Lomnica, Slovakia."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1599","DOI":"10.1137\/0331075","article-title":"Discrete-time transitivity and accessibility: Analytical systems","volume":"31","author":"Albertini","year":"1993","journal-title":"SIAM J. Control Optim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1080\/0020717011092280","article-title":"Nonlinear non-interactive control with stability in discrete-time framework","volume":"75","author":"Califano","year":"2002","journal-title":"Int. J. Control"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Conte, G., Moog, C.H., and Perdon, A.M. (2007). Algebraic Methods for Nonlinear Control Systems. Theory and Applications, Springer.","DOI":"10.1007\/978-1-84628-595-0"},{"key":"ref_18","unstructured":"Zhirabok, A., and Shumsky, A. (2008). The Algebraic Methods for Analysis of Nonlinear Dynamic Systems, Dalnauka. (In Russian)."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"313","DOI":"10.3166\/ejc.16.313-325","article-title":"Unified approach to the problem of full decoupling via output feedback","volume":"16","author":"Shumsky","year":"2010","journal-title":"Eur. J. Control"},{"key":"ref_20","unstructured":"Hartmanis, J., and Stearns, R. (1966). The Algebraic Structure Theory of Sequential Machines, Prentice-Hall."},{"key":"ref_21","unstructured":"(2014, July 01). Available online: http:\/\/webmathematica.cc.ioc.ee\/webmathematica\/NLControl\/main\/index.html."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1515\/amcs-2017-0019","article-title":"Fault detection in nonlinear systems via linear methods","volume":"27","author":"Zhirabok","year":"2017","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2887","DOI":"10.1016\/j.automatica.2013.06.013","article-title":"Measurement feedback disturbance decoupling in discrete-time nonlinear systems","volume":"49","author":"Kaldmae","year":"2013","journal-title":"Automatica"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"858","DOI":"10.1002\/asjc.1185","article-title":"Faulty plant reconfiguration based on disturbance decoupling methods","volume":"8","author":"Kaldmae","year":"2016","journal-title":"Asian J. 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Symmetry, 11.","DOI":"10.3390\/sym11040555"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1134\/S1064230716050038","article-title":"Method of fault accommodation in technical systems","volume":"55","author":"Bobko","year":"2016","journal-title":"J. Comput. Syst. Sci. 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