Enlarging the Region of Attraction for Nonlinear Systems through the Sum-of-Squares Programming

Document Type : بنیادی - نظری


1 Department of Mathematics‎, ‎Payame Noor University (PNU), ‎P.O‎. ‎Box‎. ‎19395-3697‎, ‎Tehran‎, ‎Iran

2 Professor of Systems and Control Engineering, KN Toosi University of Technology, Tehran, Iran



In the present study‎, ‎a novel methodology is developed to enlarge the Region of Attraction (ROA) at the point of equilibrium of an input-affine nonlinear control system‎. ‎Enlarging the ROA for non-polynomial dynamical systems is developed by designing a nonlinear state feedback controller through the State-Dependent Riccati Equation (SDRE)‎. ‎Consequently‎, ‎its process is defined in the form of Sum-of-Squares (SOS) optimization problem with control and non-control constraints‎. ‎Of note‎, ‎the proposed technique is effective in estimating the ROA for a nonlinear system functioning on polynomial or non-polynomial dynamics‎. ‎In the present study‎, ‎the application of the proposed scheme are shown by numerical simulations‎.


Main Subjects

‎bibitem{a1-1} Qiang B.‎, ‎Zhang L‎. ‎(2018)‎. ‎lqlq Output feedback control design to enlarge the domain of attraction of a supercavitating vehicle subject to actuator saturationrqrq‎, ‎Transactions of the Institute of Measurement and Control‎, ‎40 (10)‎, ‎3189-3200‎.
‎bibitem{a1-2} Azizi S.‎, ‎Torres L‎. ‎A‎. ‎B.‎, ‎Palhares R‎. ‎M‎. ‎(2018)‎. ‎lqlq Regional Robust Stabilization and Domain of Attraction Estimation for MIMO Uncertain Nonlinear Systems with Input Saturationrqrq‎ , ‎International Journal of Control‎, ‎91 (1)‎, ‎215-229‎.
‎bibitem{a1-3} Jerbi H‎. ‎(2017)‎. ‎lqlq Estimations of the Domains of Attraction for Classes of Nonlinear Continuous Polynomial Systemsrqrq‎, ‎Arabian Journal for Science and Engineering‎ ,42(7), ‎2829-2837‎.
‎bibitem{a1-4} Amato F.‎, ‎Calabrese F‎. ‎et al‎. ‎(2011)‎. ‎qq{Stability analysis of nonlinear quadratic systems via polyhedral Lyapunove finctions}‎, ‎Automatica‎, ‎47‎, ‎614-617‎.
‎bibitem{a1-5} Jarvis-Wloszek Z‎. ‎W.‎, ( ‎2003)‎. ‎qq{Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using Sum-of-Squares Optimization}‎, ‎Berkeley‎, ‎University of California‎.
‎bibitem{a1-6} Tan W.‎, ‎Packard A‎. ‎(2006)‎. ‎qq{Stability Region Analysis using Sum of Squares Programming}‎, ‎In American Control Conference‎, ‎2297-2302‎.
‎bibitem{a1-7} Tan W‎. ‎(2006)‎. ‎qq{Nonlinear Control Analysis and Synthesis using Sum-of-Squares Programming‎, ‎Dissertation (Ph.D)}‎, ‎University of California‎, ‎Berkeley‎.
‎bibitem{a1-8} Rapoport L‎. ‎B.‎, ‎Morozov Yu‎. ‎V‎. ‎(2008)‎. ‎qq{Numerical methods for estimation of the attraction domain in the problem of control of the wheeled robot}‎, ‎Autom‎. ‎Remote Control‎, ‎69(1)‎, ‎13-26‎.
‎bibitem{a1-9} Khodadadi L.‎, ‎Samadi B.‎, ‎Khaloozadeh H‎. ‎(2013)‎. ‎qq{Estimation of region of attraction for polynomial nonlinear systems‎: ‎A numerical method}‎, ‎ISA Transactions‎, ‎53‎, ‎25-32‎.
‎bibitem{a1-10} Moghaddam R‎. ‎Pariz K.‎, ‎N.‎, ‎Shanechi H‎. ‎M.‎, ‎Kamyad A‎. ‎V‎. ‎(2014)‎. ‎qq{Estimating the robust domain of attraction and directional enlargement of attraction domain via Markov models}‎, ‎Optim‎. ‎Control Appl‎. ‎Meth.‎, ‎35‎, ‎21-40‎.
‎bibitem{a1-11} Korda M.‎, ‎Henrion D.‎, ‎Jones C‎. ‎N‎. ‎(2014)‎. ‎qq{Controller design and region of attraction estimation for nonlinear dynamical systems}‎, . ‎arXiv‎: ‎1310.2213‎.
‎bibitem{a1-12} Najafi E.‎, ‎Lopes G‎. ‎A.‎, ‎Babuska R‎. ‎(2014)‎. ‎qq{Balancing a legged robot using state-dependent Riccati equation control}‎, ‎In‎: ‎Proceedings of the 19th IFAC World Congress‎, ‎pp‎. ‎2177-2182‎.
‎bibitem{a1-13} Baier R.‎, ‎Gerdts M‎. ‎(2009)‎. ‎qq{A computational method for nonconvex reachable sets using optimal control}‎, ‎In Proceedings of the European Control Conference‎, ‎23-26‎.
‎bibitem{a1-14} Khlebnikov M‎. ‎V‎. ‎(2015)‎. ‎qq{Estimates of the Attraction Domain of Linear Systems under L2-Bounded Control}‎, ‎Autom‎. ‎Remote Control‎, ‎76 (3)‎, ‎369-376‎.
‎bibitem{a1-15} Lim´on D.‎, ‎Alamo T.‎, ‎Camacho E‎. ‎F‎. ‎(2005)‎. ‎qq{Enlarging the domain of attraction of MPC controllers}‎, ‎Automatica‎, ‎41 (4)‎, ‎629 – 635‎.
‎bibitem{a1-16} Pesterev A‎. ‎V‎. ( ‎2017)‎. ‎qq{Attraction Domain Estimate for Single-Input Affine Systems with Constrained Control}‎, ‎Automation and Remote Control‎, ‎78 (4)‎, ‎581-594‎.
‎bibitem{a1-17} Pesterev A‎. ‎V‎. ‎(2019)‎. ‎qq{Estimation of the Attraction Domain for an Affine System with Constrained Vector Control Closed by the Linearizing Feedback}‎, ‎Automation and Remote Control‎, ‎80 (5)‎, ‎840-855‎.
‎bibitem{a1-18} Haghighatnia S.‎, ‎Moghaddam R‎. ‎K‎. ‎(2012)‎ . ‎qq{Directional Extension of the Domain of Attraction to Increase Critical clearing time of nonlinear systems}‎, ‎Journal of American Science‎, ‎8 444-449‎.
‎bibitem{a1-19} Chesi G‎. ‎(2004)‎. ‎qq{Design of controllers enlarging the domain of attraction‎: ‎a quasi-convex approach}‎, ‎IFAC Symposium on Nonlinear Control Systems‎, ‎Stuttgart‎, ‎Germany‎, ‎September‎.
‎bibitem{a1-20} Hashemzadeh F.‎, ‎Yazdanpanah M‎. ‎J‎. ‎(2006)‎. ‎qq{Semi-Global Enlargement of Domain of Attrction for a Class of Affine Nonlinear Systems}‎, ‎In Proceedings of the IEEE International Conference on Control Applications‎, ‎1-6‎.
‎bibitem{a1-21} Chen X‎. ‎Y.‎, ‎Li C‎. ‎J.‎, ‎Lu J‎. ‎F.‎, ‎Jing Y‎. ‎W‎. ‎(2012)‎. ‎qq{The domain of attraction for aSEIR epidemic model based on sum of square optimization}‎, ‎Bulletin of the Korean mathematical Society‎, ‎49 (3)‎, ‎517-528‎.
‎bibitem{a1-22} Li C‎. , ‎Ryoo C.‎, ‎Li N.‎, ‎Cao L‎. ‎(2009)‎. ‎qq{Estimating the domain of attraction via moment matrices}‎, ‎Bull‎. ‎Korean Math‎. ‎Soc‎. ‎46 (6)‎, ‎1237–1248‎.
‎bibitem{a1-23} Prajna S.‎, ‎Papachristodoulou A.‎, ‎Seiler P.‎, ‎Parrilo P.A‎. ‎(2004)‎. ‎qq{SOSTOOLS‎: ‎Sum of Squares Optimization Toolbox for MATLAB‎, ‎User’s guide}‎, ‎Dynamical Systems‎.
‎bibitem{a1-24} Topcu‎, ‎U.‎, ‎Packard‎, ‎A‎. ‎K.‎, ‎Seiler‎, ‎P.‎, ‎Balas‎, ‎G‎. ‎J‎. ‎(2010)‎. ‎qq{Robust region-ofattraction estimation}‎, ‎IEEE Transactions on Automatic Control‎, ‎55 (1)‎, ‎137–142‎.
‎bibitem{a1-25} Batmani Y.‎, ‎Khaloozadeh H‎. ‎(2013)‎. ‎qq{Optimal chemotherapy in cancer treatment‎: ‎state dependent Riccati equation control and extended Kalman filter}‎, ‎Optimal Control‎, ‎Applications and Methods‎, ‎34 (5)‎, ‎562-577‎.
‎bibitem{a1-26} Batmani Y.‎, ‎Khaloozadeh H‎. ‎(2013)‎. ‎qq{On the design of human immunodeficiency virus treatment based on a non-linear time-delay model}‎, ‎IET Systems Biology‎, ‎10.1049/iet-syb.2013.0012‎.
‎bibitem{a1-27} Batmani Y.‎, ‎Khaloozadeh H‎. ‎(2013)‎. ‎qq{On the design of suboptimal sliding manifold for a class of nonlinear uncertain time-delay systems}‎, ‎International Journal of Systems Science‎.
‎bibitem{a1-28} Batmani Y.‎, ‎Khaloozadeh H‎. ‎(2013)‎. ‎qq{Optimal drug regimens in cancer chemotherapy‎: ‎a multi‎- ‎objective approach}‎, ‎Computers in Biology and Medicine‎, ‎43‎, ‎2089-2095‎.
‎bibitem{a1-29} Batmani Y.‎, ‎Khaloozadeh H‎. ‎(2013)‎. ‎qq{On the design of observer for nonlinear time-delay systems}‎, ‎Asian Journal of Control‎, ‎10.1002/asjc.795‎.
‎bibitem{a1-30} Çimen T‎. ‎(2010)‎. ‎qq{Systematic and effective design of nonlinear feedback controllers via state dependent Riccati equation (SDRE) method}‎, ‎Annual Reviews in control‎, ‎34 (1)‎, ‎3-5‎.
‎bibitem{a1-31} Çimen T‎. ‎(2012)‎. ‎qq{Survey of state-dependent Riccati equation in nonlinear optimal feedback control synthesis}‎, ‎Journal of Guidance‎, ‎Control‎, ‎and Dynamics‎, ‎35 (4)‎, ‎1025-1047‎.
‎bibitem{a1-32} Chen H.‎, ‎Allg"{o}wer F‎. ‎(1998)‎. ‎qq{A Quasi-Infinite Horizon Nonlinear Model Predictive Control Scheme with Guaranteed Stability}‎, ‎Automatica‎, ‎34 (10)‎, ‎1205-1217‎.
‎bibitem{a1-33} Franze G.‎, ‎Casavola A.‎, ‎Famularo D.‎, ‎Garone E‎. ‎(2008)‎. ‎qq{An off-line MPC strategy for nonlinear systems based on SOS programming}‎, ‎IFAC Proceedings Volumes‎, ‎41 (2)‎, ‎10063-10068‎.
‎bibitem{a1-34} Cox D‎. ‎A.‎, ‎Little J.‎, ‎O'Shea D‎. ‎(2004)‎. ‎qq{Using algebraic geometry}‎, ‎Little‎, ‎Springer Verlag‎, ‎Berlin‎.
‎bibitem{a1-35} Najafi E.‎, ‎Babuska R.‎, ‎Lopes G‎. ‎A‎. ‎(2016)‎. ‎qq{A fast sampling method for estimating the domain of attraction}‎, ‎Nonlinear Dyn‎, ‎86‎, ‎823–834‎.