{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:53:50Z","timestamp":1760237630474,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,3]],"date-time":"2020-06-03T00:00:00Z","timestamp":1591142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004281","name":"Narodowe Centrum Nauki","doi-asserted-by":"publisher","award":["2017\/27\/B\/ST7\/02443"],"award-info":[{"award-number":["2017\/27\/B\/ST7\/02443"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper the descriptor continuous-time linear systems with the regular matrix pencil     ( E , A )     are investigated using Drazin inverse matrix method. Necessary and sufficient conditions for the stability and superstability of this class of dynamical systems are established. The procedure for computation of the state-feedback gain matrix such that the closed-loop system is superstable is given. The effectiveness of the presented approach is demonstrated on numerical examples.<\/jats:p>","DOI":"10.3390\/sym12060940","type":"journal-article","created":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T03:32:21Z","timestamp":1591327941000},"page":"940","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Superstabilization of Descriptor Continuous-Time Linear Systems via State-Feedback Using Drazin Inverse Matrix Method"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7325-1159","authenticated-orcid":false,"given":"Kamil","family":"Borawski","sequence":"first","affiliation":[{"name":"Faculty of Electrical Engineering, Bialystok University of Technology, Wiejska 45D Street, 15-351 Bialystok, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Guang-Ren, D. (2010). Analysis and Design of Descriptor Linear Systems, Springer.","DOI":"10.1007\/978-1-4419-6397-0"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Dai, L. (1989). Singular Control Systems. Lecture Notes in Control and Information Sciences, Springer.","DOI":"10.1007\/BFb0002475"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1109\/TAC.1981.1102763","article-title":"A Generalized State-Space for Singular Systems","volume":"26","author":"Verghese","year":"1981","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kunkel, P., and Mehrmann, V. (2006). Differential-Algebraic Equations. Analysis and Numerical Solution, EMS Publishing House.","DOI":"10.4171\/017"},{"key":"ref_5","unstructured":"Weierstrass, K. (1868). Zur Theorie der Bilinearen und Quadratischen Formen. Monatsh. Akad. Wiss., 311\u2013338."},{"key":"ref_6","unstructured":"Kronecker, L. (1890). Algebraische Reduktion der Scharen Bilinearer Formen. Sitzungsber. Akad. Wiss., 763\u2013776."},{"key":"ref_7","unstructured":"Gantmacher, F.R. (1959). The Theory of Matrices, Chelsea Publishing Company."},{"key":"ref_8","unstructured":"Kaczorek, T. (1992). Linear Control Systems. Vol. 1, Research Studies Press, J. Wiley."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1080\/00029890.1958.11991949","article-title":"Pseudo-inverses in associative rings and semigroups","volume":"65","author":"Drazin","year":"1958","journal-title":"Am. Math. Mon."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1137\/0131035","article-title":"Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients","volume":"31","author":"Campbell","year":"1976","journal-title":"SIAM J. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1137\/0509054","article-title":"The Laurent Expansion of a Generalized Resolvent with Some Applications","volume":"9","author":"Rose","year":"1978","journal-title":"SIAM J. Math. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"702","DOI":"10.1109\/TAC.1981.1102699","article-title":"Solvability, controllability and observability of continuous descriptor systems","volume":"26","author":"Yip","year":"1981","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1007\/s11071-007-9231-4","article-title":"Control of under actuated mechanical systems with servo-constraints","volume":"50","author":"Blajer","year":"2007","journal-title":"Nonlinear Dyn."},{"key":"ref_14","unstructured":"Campbell, S.L. (1980). Singular Systems of Differential Equations. Research Notes in Mathematics, Pitman Advanced Publishing Program."},{"key":"ref_15","unstructured":"Daoutidis, P. (2012). DAEs in chemical engineering: A survey. Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum, Springer."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kaczorek, T., and Rogowski, K. (2015). Fractional Linear Systems and Electrical Circuits. Studies in Systems, Decision and Control, Springer.","DOI":"10.1007\/978-3-319-11361-6"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/BF01600184","article-title":"A survey of linear singular systems","volume":"5","author":"Lewis","year":"1986","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Marsza\u0142ek, W., and Trzaska, Z.W. (2004, January 14\u201317). DAE models of electrical power systems and their bifurcations around singularities. Proceedings of the 43rd IEEE Conference on Decision and Control, Nassau, Bahamas.","DOI":"10.1109\/CDC.2004.1429557"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2256","DOI":"10.1109\/TII.2016.2619064","article-title":"Modeling Lane Keeping by a Hybrid Open-Closed-Loop Pulse Control Scheme","volume":"12","author":"Zhang","year":"2016","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2668","DOI":"10.1109\/TAC.2012.2190211","article-title":"Characterization and Stability of Autonomous Positive Descriptor Systems","volume":"57","author":"Rami","year":"2012","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2640","DOI":"10.1016\/j.laa.2008.03.002","article-title":"Stability analysis of positive descriptor systems","volume":"429","author":"Virnik","year":"2008","journal-title":"Linear Algebra Appl."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Shafai, B., and Li, C. (2017, January 27\u201330). Positive Stabilization of Singular Systems by Proportional Derivative State Feedback. Proceedings of the IEEE Conference on Control Technology and Applications, Mauna Lani, HI, USA.","DOI":"10.1109\/CCTA.2017.8062612"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1016\/S0005-1098(02)00283-2","article-title":"Structural properties and poles assignability of LTI singular systems under output feedback","volume":"39","author":"Runyi","year":"2003","journal-title":"Automatica"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1137\/0613007","article-title":"Regularization of descriptor systems by derivative and proportional state feedback","volume":"13","author":"Mehrmann","year":"1992","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1742","DOI":"10.1109\/9.310065","article-title":"Regularization of descriptor systems by output feedback","volume":"39","author":"Mehrmann","year":"1994","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1109\/9.746277","article-title":"Necessary and sufficient conditions for the output feedback regularization of descriptor systems","volume":"44","author":"Chu","year":"1999","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Moaaz, O., Chatzarakis, G.E., Chalishajar, D., and Bazighifan, O. (2020). Dynamics of General Class of Difference Equations and Population Model with Two Age Classes. Mathematics, 8.","DOI":"10.3390\/math8040516"},{"key":"ref_28","first-page":"703","article-title":"Superstabilization of positive linear electrical circuit by state-feedbacks","volume":"65","author":"Kaczorek","year":"2017","journal-title":"Bull. Pol. Acad. Sci. Tech."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1239","DOI":"10.1023\/A:1019823208592","article-title":"Superstable Linear Control Systems. I. Analysis","volume":"63","author":"Polyak","year":"2002","journal-title":"Autom. Remote Control"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Talagaev, Y.V. (2014, January 8\u201310). Robust analysis and superstabilization of chaotic systems. Proceedings of the IEEE Conference on Control Applications, Juan Les Antibes, France.","DOI":"10.1109\/CCA.2014.6981525"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Kaczorek, T. (2017, January 28\u201331). Extension of Cayley-Hamilton theorem and a procedure for computation of the Drazin inverse matrices. Proceedings of the 22nd International Conference on Methods and Models in Automation and Robotics (MMAR), Mi\u0119dzyzdroje, Poland.","DOI":"10.1109\/MMAR.2017.8046800"},{"key":"ref_32","unstructured":"Szewczyk, R., Zieli\u0144ski, C., and Kaliczy\u0144ska, M. (2018). Analysis of the positivity of descriptor continuous-time linear systems by the use of Drazin inverse matrix method. Automation 2018: Advances in Automation, Robotics and Measurement Techniques, Springer."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/940\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:35:20Z","timestamp":1760175320000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/940"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,3]]},"references-count":32,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["sym12060940"],"URL":"https:\/\/doi.org\/10.3390\/sym12060940","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,6,3]]}}}