{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:25Z","timestamp":1740134545683,"version":"3.37.3"},"reference-count":21,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51305321","51035006","K5051304021","K5051304026"],"award-info":[{"award-number":["51305321","51035006","K5051304021","K5051304026"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51305321","51035006","K5051304021","K5051304026"],"award-info":[{"award-number":["51305321","51035006","K5051304021","K5051304026"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["51305321","51035006","K5051304021","K5051304026"],"award-info":[{"award-number":["51305321","51035006","K5051304021","K5051304026"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["51305321","51035006","K5051304021","K5051304026"],"award-info":[{"award-number":["51305321","51035006","K5051304021","K5051304026"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>Gram matrix is an important tool in system analysis and design as it provides a description of the input-output behavior for system; its partial derivative matrix is often required in some numerical algorithms. It is essential to study computation of these matrices. Analytical methods only work in some special circumstances; for example, the system matrix is diagonal matrix or Jordan matrix. In most cases, numerical integration method is needed, but there are two problems when compute using traditional numerical integration method. One is low accuracy: as high accuracy requires extremely small integration step, it will result in large amount of computation; and another is stability and stiffness issues caused by the dependence on the property of system matrix. In order to overcome these problems, this paper proposes an efficient numerical method based on the key idea of precise integration method (PIM) for the Gram matrix and its partial derivative of linear time-invariant systems. Since matrix inverse operation is not required in this method, it can be used with high precision no matter the system is normal or singular. The specific calculation algorithm and block diagram are also given. Finally, numerical examples are given to demonstrate the correctness and validity of this method.<\/jats:p>","DOI":"10.1155\/2014\/134604","type":"journal-article","created":{"date-parts":[[2014,4,22]],"date-time":"2014-04-22T21:05:07Z","timestamp":1398200707000},"page":"1-10","source":"Crossref","is-referenced-by-count":0,"title":["Computation of Gram Matrix and Its Partial Derivative Using Precise Integration Method for Linear Time-Invariant Systems"],"prefix":"10.1155","volume":"2014","author":[{"given":"Sulan","family":"Li","sequence":"first","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, 2 Taibai Road, Xi\u2019an 710071, China"}]},{"given":"Yuanhao","family":"Ren","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, 2 Taibai Road, Xi\u2019an 710071, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2092-238X","authenticated-orcid":true,"given":"Hong","family":"Bao","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, 2 Taibai Road, Xi\u2019an 710071, China"}]},{"given":"Wei","family":"Zhang","sequence":"additional","affiliation":[{"name":"Wuhan Second Ship Design and Research Institute, 450 Zhongshan Road, Wuhan 430064, China"}]}],"member":"311","reference":[{"year":"2005","key":"1"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1080\/00207177008931858"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1109\/TPWRS.2011.2126290"},{"issue":"9","key":"6","first-page":"902","volume":"6","year":"2011","journal-title":"IET Generation, Transmission, and Distribution"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1109\/TPWRS.2007.913731"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1109\/TPWRS.2008.2004741"},{"issue":"12","key":"11","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1049\/el:19910663","volume":"27","year":"1991","journal-title":"Electronics Letters"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1109\/9.241566"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/j.sigpro.2007.06.012"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1016\/j.sigpro.2006.06.007"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1155\/2012\/453230"},{"issue":"6","key":"18","first-page":"427","volume":"208","year":"1994","journal-title":"Proceedings of the Institution of Mechanical Engineers C: Journal of Mechanical Engineering Science"},{"issue":"1","key":"19","first-page":"1","volume":"12","year":"1995","journal-title":"Computational Structural Mechanics and Applications"},{"issue":"7","key":"20","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1016\/S0141-0296(97)83154-9","volume":"19","year":"1997","journal-title":"Engineering Structures"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7949(94)00537-D"},{"issue":"4","key":"22","first-page":"447","volume":"32","year":"2000","journal-title":"Acta Mechanica Sinica"},{"issue":"3","key":"23","first-page":"374","volume":"39","year":"2007","journal-title":"Chinese Journal of Theoretical and Applied Mechanics"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2003.08.053"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1016\/j.compstruc.2006.02.001"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1016\/j.compstruc.2008.11.004"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.1016\/j.euromechsol.2009.11.006"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/134604.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/134604.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/134604.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,11]],"date-time":"2020-05-11T22:25:14Z","timestamp":1589235914000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/134604\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":21,"alternative-id":["134604","134604"],"URL":"https:\/\/doi.org\/10.1155\/2014\/134604","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2014]]}}}