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The sum of squares (SOS) method can numerically solve nonlinear problems that are not easy to solve analytically, but it still cannot solve HJI inequalities directly. In this paper, an HJI inequality suitable for SOS is firstly derived to solve the problem of nonconvex optimization. Then, the problems of SOS in nonlinear <jats:italic>H<\/jats:italic><jats:sub>\u221e<\/jats:sub> design are analyzed in detail. Finally, a two\u2010step iterative design method for solving nonlinear <jats:italic>H<\/jats:italic><jats:sub>\u221e<\/jats:sub> control is presented. The first step is to design an adjustable nonlinear state feedback of the gain array of the system using SOS. The second step is to solve the <jats:italic>L<\/jats:italic><jats:sub>2<\/jats:sub> gain of the system; the optimization problem is solved by a graphical analytical method. In the iterative design, a diagonally dominant design idea is proposed to reduce the numerical error of SOS. The nonlinear <jats:italic>H<\/jats:italic><jats:sub>\u221e<\/jats:sub> control design of a polynomial system for large satellite attitude maneuvers is taken as our example. Simulation results show that the SOS method is comparable to the LMI method used for linear systems, and it is expected to find a broad range of applications in the analysis and design of nonlinear systems.<\/jats:p>","DOI":"10.1155\/2019\/5124108","type":"journal-article","created":{"date-parts":[[2019,12,2]],"date-time":"2019-12-02T11:35:22Z","timestamp":1575286522000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":96,"title":["Application of Sum of Squares Method in Nonlinear <i>H<\/i><sub>\u221e<\/sub> Control for Satellite Attitude Maneuvers"],"prefix":"10.1155","volume":"2019","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6852-7173","authenticated-orcid":false,"given":"Fanwei","family":"Meng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0168-5921","authenticated-orcid":false,"given":"Dini","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4889-9399","authenticated-orcid":false,"given":"Penghui","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4897-179X","authenticated-orcid":false,"given":"Guanzhou","family":"Xie","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2019,11,30]]},"reference":[{"key":"e_1_2_10_1_2","doi-asserted-by":"publisher","DOI":"10.1109\/9.256331"},{"key":"e_1_2_10_2_2","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2016.2585163"},{"key":"e_1_2_10_3_2","doi-asserted-by":"crossref","unstructured":"PrajnaS. PapachristodoulouA. andParriloP. Introducing SOSTOOLS: a general purpose sum of squares programming solver Proceedings of the IEEE Conference on Decision and Control (CDC) December 2002 Las Vegas NV USA 741\u2013746 https:\/\/doi.org\/10.1109\/cdc.2002.1184594.","DOI":"10.1109\/CDC.2002.1184594"},{"key":"e_1_2_10_4_2","doi-asserted-by":"crossref","unstructured":"Jarvis-WloszekZ. FeeleyR. TanW. SunK. andPackardA. Some controls applications of sum of squares programming Proceedings of the IEEE Conference on Decision and Control (CDC) December 2003 Maui HI USA 4676\u20134681 https:\/\/doi.org\/10.1109\/cdc.2003.1272309 2-s2.0-1542290065.","DOI":"10.1109\/CDC.2003.1272309"},{"key":"e_1_2_10_5_2","doi-asserted-by":"publisher","DOI":"10.1002\/rnc.1541"},{"key":"e_1_2_10_6_2","doi-asserted-by":"publisher","DOI":"10.1109\/tac.2007.914221"},{"key":"e_1_2_10_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2008.03.010"},{"key":"e_1_2_10_8_2","doi-asserted-by":"crossref","unstructured":"GolluN.andRodriguesL. Control of large angle attitude maneuvers for rigid bodies using sum of squares Proceeding of the 2007 American Control Conference July 2007 New York City NY USA 3156\u20133161 https:\/\/doi.org\/10.1109\/acc.2007.4283054 2-s2.0-46449089732.","DOI":"10.1109\/ACC.2007.4283054"},{"key":"e_1_2_10_9_2","doi-asserted-by":"publisher","DOI":"10.1109\/tac.2003.823000"},{"key":"e_1_2_10_10_2","doi-asserted-by":"crossref","unstructured":"KrishnaswamyK. PapageorgiouG. GlavaskiS.et al. Analysis of aircraft pitch axis stability augmentation system using sum of squares optimization Proceedings of the 2005 American Control Conference July 2005 Portland OR USA 1497\u20131502 https:\/\/doi.org\/10.1109\/acc.2005.1470177.","DOI":"10.1109\/ACC.2005.1470177"},{"key":"e_1_2_10_11_2","doi-asserted-by":"publisher","DOI":"10.1109\/tac.2010.2044268"},{"key":"e_1_2_10_12_2","doi-asserted-by":"publisher","DOI":"10.1109\/tac.2009.2017168"},{"key":"e_1_2_10_13_2","doi-asserted-by":"publisher","DOI":"10.1109\/tac.2010.2088590"},{"key":"e_1_2_10_14_2","unstructured":"PrajnaS. PapachristodoulouA. andWuF. Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach Proceedings of the Asian Control Conference ASCC July 2004 Melbourne Victoria Australia 157\u2013165."},{"key":"e_1_2_10_15_2","doi-asserted-by":"crossref","unstructured":"ZhengQ.andWuF. Nonlinear output feedbackH\u221econtrol for polynomial nonlinear systems Proceedings of the 2008 American Control Conference June 2008 Seattle WA USA 1196\u20131201 https:\/\/doi.org\/10.1109\/acc.2008.4586655 2-s2.0-52449110083.","DOI":"10.1109\/ACC.2008.4586655"},{"key":"e_1_2_10_16_2","first-page":"1587","article-title":"Robust H-infinity control for a class of polynomial nonlinear systems","volume":"29","author":"Huang W.","year":"2012","journal-title":"Control Theory and Applications"},{"key":"e_1_2_10_17_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCYB.2016.2643687"},{"key":"e_1_2_10_18_2","doi-asserted-by":"publisher","DOI":"10.1002\/asjc.1625"},{"key":"e_1_2_10_19_2","doi-asserted-by":"publisher","DOI":"10.1049\/iet-cta.2017.0096"},{"key":"e_1_2_10_20_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40998-018-0047-7"},{"key":"e_1_2_10_21_2","doi-asserted-by":"publisher","DOI":"10.1631\/jzus.a071486"},{"key":"e_1_2_10_22_2","first-page":"69","article-title":"SOS design for control of large attitude maneuvers of satellites","volume":"33","author":"He Z.","year":"2013","journal-title":"Chinese Space Science and Technology"},{"key":"e_1_2_10_23_2","doi-asserted-by":"publisher","DOI":"10.1049\/ip-cta:20000342"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2019\/5124108.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2019\/5124108.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2019\/5124108","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T11:25:27Z","timestamp":1723029927000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2019\/5124108"}},"subtitle":[],"editor":[{"given":"Cutberto","family":"Romero-Mel\u00e9ndez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1155\/2019\/5124108"],"URL":"https:\/\/doi.org\/10.1155\/2019\/5124108","archive":["Portico"],"relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"value":"1076-2787","type":"print"},{"value":"1099-0526","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]},"assertion":[{"value":"2019-08-11","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2019-11-12","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2019-11-30","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"5124108"}}