{"648721":{"#nid":"648721","#data":{"type":"event","title":"PhD Defense by Gidado-Yisa Immanuel","body":[{"value":"\u003Cp\u003EStudent: Gidado-Yisa Immanuel\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Cstrong\u003EDefense Announcement\u003C\/strong\u003E\u003C\/p\u003E\r\n\r\n\u003Cp\u003EHere\u0026rsquo;s the BlueJeans link for the meeting:\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Ca href=\u0022https:\/\/bluejeans.com\/4832389748\u0022 target=\u0022_blank\u0022\u003Ehttps:\/\/bluejeans.com\/4832389748\u003C\/a\u003E\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003ETitle:\u0026nbsp;Methods of Analysis and Design of Dynamical Systems Using homogeneous Polynomial Lyapunov Functions\u003C\/p\u003E\r\n\r\n\u003Cp\u003EAdvisor: Prof. Eric Feron\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDate: Monday July 26 at 11am\u003C\/p\u003E\r\n\r\n\u003Cp\u003EAbstract:\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003ELyapunov functions are the mainstay for systems analysis and\u0026nbsp;control, and in particular quadratic Lyapunov\u0026nbsp;functions have been used\u0026nbsp;successfully for many classes of problems. When using quadratic Lyapunov\u0026nbsp;functions for\u0026nbsp;analysis and design of nonlinear systems, there is a measure of\u0026nbsp;conservatism due to the inherent limitations of the\u0026nbsp;associated ellipsoid as a\u0026nbsp;covering for the stability region of the system, and of course, there are classes\u0026nbsp;of systems\u0026nbsp;where quadratic Lyapunov forms yield no results at all. This is the\u0026nbsp;case for switched linear systems, where there\u0026nbsp;may not exist a common quadratic\u0026nbsp;Lyapunov function for each of the switched modes, even though they system is\u0026nbsp;stable. However, Mason\u0026nbsp;et al.\u0026nbsp;have shown that there exists homogeneous polynomial\u0026nbsp;Lyapunov functions that\u0026nbsp;certify stability of the system even though the degree\u0026nbsp;of the homogeneous polynomial may not be known\u0026nbsp;a priori.\u0026nbsp;Quadratic\u0026nbsp;Lyapunov functions are amenable to energy-based problems because energy type bounds\u0026nbsp;and\u0026nbsp;constraints are captured efficiently by ellipsoids. On the other hand,\u0026nbsp;analysis of peak-input bounded types of\u0026nbsp;problems lack closed-form solutions, often\u0026nbsp;utilize approximations and relaxations, in addition to being\u0026nbsp;computationally\u0026nbsp;expensive to compute due to the norm expressing the bounds.\u003Cbr \/\u003E\r\n\u003Cbr \/\u003E\r\nThis research investigates generalizations of quadratic Lyapunov\u0026nbsp;functions, specifically homogeneous Lyapunov\u0026nbsp;functions that are constructed\u0026nbsp;through a process called lifting. The state vector x is lifted via\u0026nbsp;a recursive Kronecker\u0026nbsp;product to a higher degree, homogeneous form resident in\u0026nbsp;a higher-dimensional space. Linear dynamical systems\u0026nbsp;can similarly be\u0026nbsp;constructed in the lifted space with this process. This research demonstrates\u0026nbsp;a method of\u0026nbsp;constructing homogeneous Lyapunov functions that provide good\u0026nbsp;estimates of system characteristics such as\u0026nbsp;domain of attraction and inescapable\u0026nbsp;sets. This method is demonstrated for stability of switched linear systems\u0026nbsp;and\u0026nbsp;implicit systems, as well as for the analysis of the\u0026nbsp;L1 problem. We show that using higher-order homogeneous\u0026nbsp;Lyapunov\u0026nbsp;functions improves estimates of the domain of attraction and inescapable sets. The\u0026nbsp;main contribution of\u0026nbsp;this research is applying this methodology to the\u0026nbsp;L1 problem and improving upper-bounds to the 1-norm of a linear\u0026nbsp;time-invariant system. Moreover, this method is accessible through linear matrix inequalities (LMI) constructions\u0026nbsp;and computationally solvable\u0026nbsp;with standard semidefinite programming.\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n\r\n\u003Cp\u003ECommittee:\u003C\/p\u003E\r\n\r\n\u003Cp\u003EProf. Eric Feron - School of Aerospace Engineering\u003C\/p\u003E\r\n\r\n\u003Cp\u003EProf. Tsiotras Panagiotis - School of Aerospace Engineering\u003C\/p\u003E\r\n\r\n\u003Cp\u003EProf. Kyriakos Vamvoudakis - School of Aerospace Engineering\u003C\/p\u003E\r\n\r\n\u003Cp\u003EProf. Yongxin Chen - School of Aerospace Engineering\u003C\/p\u003E\r\n\r\n\u003Cp\u003EProf. Jeff S. Shamma - Chair of Industrial \u0026amp;\u0026nbsp;Ent. Systems Engineering, University of Illinois Urbana-Champaign\u003C\/p\u003E\r\n\r\n\u003Cp\u003EProf. Eric N. Johnson - School of Aerospace Engineering, Pennsylvania State University\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"Methods of Analysis and Design of Dynamical Systems Using homogeneous Polynomial Lyapunov Functions"}],"uid":"27707","created_gmt":"2021-07-13 16:31:40","changed_gmt":"2021-07-19 19:00:16","author":"Tatianna Richardson","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2021-07-27T12:00:00-04:00","event_time_end":"2021-07-27T14:00:00-04:00","event_time_end_last":"2021-07-27T14:00:00-04:00","gmt_time_start":"2021-07-27 16:00:00","gmt_time_end":"2021-07-27 18:00:00","gmt_time_end_last":"2021-07-27 18:00:00","rrule":null,"timezone":"America\/New_York"},"extras":[],"groups":[{"id":"221981","name":"Graduate Studies"}],"categories":[],"keywords":[{"id":"100811","name":"Phd Defense"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[{"id":"1788","name":"Other\/Miscellaneous"}],"invited_audience":[{"id":"78761","name":"Faculty\/Staff"},{"id":"78771","name":"Public"},{"id":"174045","name":"Graduate students"},{"id":"78751","name":"Undergraduate students"}],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}