Document Zbl 1455.65064

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Franklin Inst.</a> <a href="/?q=in%3A459326" title="Articles in this Issue">358, No. 3, 2039-2059 (2021)</a>. </div> <div class="abstract">Summary: This paper studies the numerical solutions of a class of periodic coupled matrix equations. Based on the least square method, a finite iterative algorithm for a class of periodic coupled matrix equations is proposed, and the convergence of the algorithm is proved by theoretical derivation. For any initial value, the algorithm can converge to the solution in finite iterations. Since the equations considered in paper contain many variants, the proposed algorithm has a wide range of applications. Finally some numerical examples in practical systems are given to prove the effectiveness and efficiency of the algorithm.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7315763">Cited in <strong>22</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65F45" title="MSC2020">65F45</a> </td> <td class="space"> Numerical methods for matrix equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A15A24" title="MSC2020">15A24</a> </td> <td class="space"> Matrix equations and identities </td> </tr></table> </div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1455.65064" data-ciurl="/ci/07315763" data-biburl="/bibtex/07315763.bib" data-amsurl="/amsrefs/07315763.bib" data-xmlurl="/xml/07315763.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07315763.pdf" title="Zbl 1455.65064 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jfranklin.2020.11.022" aria-label="DOI for “A numerical solution of a class of periodic coupled matrix equations”" title="10.1016/j.jfranklin.2020.11.022">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">De Souza, C. E.; Coutinho, D., Robust stability and control of uncertain linear discrete-time periodic systems with time-delay, Automatica, 50, 2, 431-441 (2014) · <a href="/1364.93726" class="nowrap">Zbl 1364.93726</a></td> </tr><tr> <td>[2]</td> <td class="space">Longhi, S.; Monteri, A., Fault detection and isolation of linear discrete-time periodic systems using the geometric approach, IEEE Trans. Autom. Control, 62, 3, 1518-1523 (2016) · <a href="/1366.93329" class="nowrap">Zbl 1366.93329</a></td> </tr><tr> <td>[3]</td> <td class="space">Song, X.; Zheng, W. X., Linear estimation for discrete-time periodic systems with unknown measurement input and missing measurements, ISA Trans., 95, 164-172 (2019)</td> </tr><tr> <td>[4]</td> <td class="space">Lv, L.; Zhang, Z.; Zhang, L., A periodic observers synthesis approach for LDP systems based on iteration, IEEE Access, 6, 8539-8546 (2018)</td> </tr><tr> <td>[5]</td> <td class="space">Zhou, B., Finite-time stabilization of linear systems by bounded linear time-varying feedback, Automatica, 113, 108760 (2020) · <a href="/1440.93229" class="nowrap">Zbl 1440.93229</a></td> </tr><tr> <td>[6]</td> <td class="space">Zhou, B., Finite-time stability analysis and stabilization by bounded linear time-varying feedback, Automatica, 121, 109191 (2020) · <a href="/1448.93289" class="nowrap">Zbl 1448.93289</a></td> </tr><tr> <td>[7]</td> <td class="space">Wu, A. G.; Zhang, W. X.; Zhang, Y., An iterative algorithm for discrete periodic Lyapunov matrix equations, Autom. A J. IFAC Int. Fed. Autom. Control, 87, 395-403 (2018) · <a href="/1378.93075" class="nowrap">Zbl 1378.93075</a></td> </tr><tr> <td>[8]</td> <td class="space">Wu, A. G.; Sun, H. J.; Zhang, Y., An SOR implicit iterative algorithm for coupled Lyapunov equations, Automatica, 97, 38-47 (2018) · <a href="/1406.93310" class="nowrap">Zbl 1406.93310</a></td> </tr><tr> <td>[9]</td> <td class="space">Zhou, B., On asymptotic stability of linear time-varying systems, Automatica, 68, 266-276 (2016) · <a href="/1334.93152" class="nowrap">Zbl 1334.93152</a></td> </tr><tr> <td>[10]</td> <td class="space">Zhou, B.; Duan, G. R.; Lin, Z., A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation, Automatica, 47, 2, 316-325 (2011) · <a href="/1207.93061" class="nowrap">Zbl 1207.93061</a></td> </tr><tr> <td>[11]</td> <td class="space">Varga, A., On solving periodic differential matrix equations with applications to periodic system norms computation [c] decision and control, 2005 and 2005 european control conference, Proceedings of the 44th IEEE Conference on CDC-ECC ’05 (2006), IEEE</td> </tr><tr> <td>[12]</td> <td class="space">Lv, L.; Zhang, L., On the periodic Sylvester equations and their applications in periodic Luenberger observers design, J. Frankl. Inst., 353, 5, 1005-1018 (2016) · <a href="/1336.93075" class="nowrap">Zbl 1336.93075</a></td> </tr><tr> <td>[13]</td> <td class="space">Hajarian, M., Extending the CGLS method for finding the least squares solutions of general discrete-time periodic matrix equations, Filomat, 30, 9, 2503-2520 (2016) · <a href="/1474.65111" class="nowrap">Zbl 1474.65111</a></td> </tr><tr> <td>[14]</td> <td class="space">Benner, P.; Hossain, M. S.; Stykel, T., Low rank iterative methods of periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems, Numer. Algorithms, 67, 3, 669-690 (2014) · <a href="/1305.65158" class="nowrap">Zbl 1305.65158</a></td> </tr><tr> <td>[15]</td> <td class="space">Lv, L. L.; Zhang, L., Parametric solutions to the discrete periodic regulator equations, J. Frankl. Inst., 353, 5, 1089-1101 (2016) · <a href="/1336.93102" class="nowrap">Zbl 1336.93102</a></td> </tr><tr> <td>[16]</td> <td class="space">Lv, L. L.; Zhang, Z., Finite iterative solutions to periodic Sylvester matrix equations, J. Frankl. Inst., 354, 5, 2358-2370 (2017) · <a href="/1398.93365" class="nowrap">Zbl 1398.93365</a></td> </tr><tr> <td>[17]</td> <td class="space">Wu, A. G.; Chang, M. F., Current-estimation-based iterative algorithms for solving periodic Lyapunov matrix equations, IET Control Theory Appl., 10, 15, 1928-1936 (2016)</td> </tr><tr> <td>[18]</td> <td class="space">Zhang, L.; Tang, S.; Lv, L., An finite iterative algorithm for solving periodic Sylvester bimatrix equations, J. Frankl. Inst., 357, 15, 10757-10772 (2020) · <a href="/1450.93010" class="nowrap">Zbl 1450.93010</a></td> </tr><tr> <td>[19]</td> <td class="space">Lv, L. L.; Tang, S. Y.; Zhang, L., Parametric solutions to generalized periodic Sylvester bimatrix equations, J. Frankl. Inst., 357, 6, 3601-3621 (2020) · <a href="/1437.93039" class="nowrap">Zbl 1437.93039</a></td> </tr><tr> <td>[20]</td> <td class="space">Ding, F.; Zhang, H., Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems, IET Control Theory Appl., 8, 15, 1588-1595 (2014)</td> </tr><tr> <td>[21]</td> <td class="space">Hajarian, M., Matrix GPBiCG algorithms for solving the general coupled matrix equations, IET Control Theory Appl., 9.1, 74-81 (2014)</td> </tr><tr> <td>[22]</td> <td class="space">Sheng, X., A relaxed gradient based algorithm for solving generalized coupled Sylvester matrix equations, J. Frankl. Inst., 355, 10 (2018) · <a href="/1390.93305" class="nowrap">Zbl 1390.93305</a></td> </tr><tr> <td>[23]</td> <td class="space">Wu, A. G.; Li, B.; Zhang, Y., Finite iterative solutions to coupled Sylvester-conjugate matrix equations, Appl. Math. Model., 35, 3, 1065-1080 (2011) · <a href="/1211.15024" class="nowrap">Zbl 1211.15024</a></td> </tr><tr> <td>[24]</td> <td class="space">Hajarian, M., Developing the CGLS algorithm for the least squares solutions of the general coupled matrix equations, Math. Methods Appl. Sci., 37, 17, 2782-2798 (2014) · <a href="/1323.65041" class="nowrap">Zbl 1323.65041</a></td> </tr><tr> <td>[25]</td> <td class="space">Zhou, B.; Duan, G. R.; Li, Z. Y., Gradient based iterative algorithm for solving coupled matrix equations, Syst. Control Lett., 58, 5, 327-333 (2009) · <a href="/1159.93323" class="nowrap">Zbl 1159.93323</a></td> </tr><tr> <td>[26]</td> <td class="space">Zhang, H.; Ding, F., A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled sylvester matrix equations, J. Frankl. Inst., 351, 1, 340-357 (2014) · <a href="/1293.15006" class="nowrap">Zbl 1293.15006</a></td> </tr><tr> <td>[27]</td> <td class="space">Zhang, H., A finite iterative algorithm for solving the complex generalized coupled sylvester matrix equations by using the linear operators, J. Frankl. Inst., 354, 4, 1856-1874 (2017) · <a href="/1378.93047" class="nowrap">Zbl 1378.93047</a></td> </tr><tr> <td>[28]</td> <td class="space">Ding, F.; Chen, T., On iterative solutions of general coupled matrix equations, SIAM J. Control Optim., 44, 6, 2269-2284 (2006) · <a href="/1115.65035" class="nowrap">Zbl 1115.65035</a></td> </tr><tr> <td>[29]</td> <td class="space">Wu, A.; Duan, G., New iterative algorithms for solving coupled Markovian jump Lyapunov equations, Autom. Control IEEE Trans., 60, 1, 289-294 (2015) · <a href="/1360.65105" class="nowrap">Zbl 1360.65105</a></td> </tr><tr> <td>[30]</td> <td class="space">Xiaochen, X.; James, L., Guaranteed cost control of periodic piecewise linear time-delay systems, Automatica, 94, 274-282 (2018) · <a href="/1401.93178" class="nowrap">Zbl 1401.93178</a></td> </tr><tr> <td>[31]</td> <td class="space">Xie, X.; Lam, J., A novel scheme of non-fragile controller design for periodic piecewise LTV systems, IEEE Trans. Ind. Electron., 67, 12, 10766-10775 (2020)</td> </tr><tr> <td>[32]</td> <td class="space">Hajarian, M., A Finite Iterative Method for Solving the General Coupled Discrete-Time Periodic Matrix Equations (2015) · <a href="/1341.93055" class="nowrap">Zbl 1341.93055</a></td> </tr><tr> <td>[33]</td> <td class="space">Hajarian, M., Developing biCOR and CORS methods for coupled Sylvester-transpose and periodic Sylvester matrix equations, Appl. Math. Modell., 39, 19, 6073-6084 (2015) · <a href="/1443.65055" class="nowrap">Zbl 1443.65055</a></td> </tr><tr> <td>[34]</td> <td class="space">M. Hajarian, A finite iterative method for solving the general coupled discrete-time periodic matrix equations, Circuits Syst. Signal Process. 34(1), 105-125. · <a href="/1341.93055" class="nowrap">Zbl 1341.93055</a></td> </tr><tr> <td>[35]</td> <td class="space">He, Z.; Wang, Q., A system of periodic discrete-time coupled Sylvester quaternion matrix equations, Algebra Colloq., 24, 01, 169-180 (2017) · <a href="/1361.15015" class="nowrap">Zbl 1361.15015</a></td> </tr><tr> <td>[36]</td> <td class="space">Hajarian, M., Solving the general coupled and the periodic coupled matrix equations via the extended QMRCGSTAB algorithms, Comput. Appl. Math., 33, 2, 349-362 (2014) · <a href="/1307.65054" class="nowrap">Zbl 1307.65054</a></td> </tr><tr> <td>[37]</td> <td class="space">Hajarian, M., Convergence analysis of the MCGNR algorithm for the least squares solution group of discrete-time periodic coupled matrix equations, Trans. Inst. Meas. Control, 39, 1, 29-42 (2017)</td> </tr><tr> <td>[38]</td> <td class="space">Varga; Andras, On computing minimal realizations of periodic descriptor systems[C], 175-180 (2007)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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