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Region of attraction analysis with integral quadratic constraints. (English) Zbl 1429.93352

Summary: A general framework is presented to estimate the region of attraction of attracting equilibrium points. The system is described by a feedback connection of a nonlinear (polynomial) system and a bounded operator. The input/output behavior of the operator is characterized using an integral quadratic constraint. This allows to analyze generic problems including, for example, hard-nonlinearities and different classes of uncertainties, adding to the state of practice in the field which is typically limited to polynomial vector fields. The IQC description is also nonrestrictive, with the main result given for both hard and soft factorizations. Optimization algorithms based on sum of squares techniques are then proposed, with the aim to enlarge the inner estimates of the ROA. Numerical examples are provided to show the applicability of the approaches. These include a saturated plant where bounds on the states are exploited to refine the sector description, and a case study with parametric uncertainties for which the conservativeness of the results is reduced by using soft IQCs.

MSC:

93D99 Stability of control systems
93C41 Control/observation systems with incomplete information
93C10 Nonlinear systems in control theory
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[1] Anderson, J.; Papachristodoulou, A., Robust nonlinear stability and performance analysis of an F/A-18 aircraft model using sum of squares programming, International Journal of Robust and Nonlinear Control, 23, 10, 1099-1114 (2017) · Zbl 1286.93135
[2] Aylward, E. M.; Parrilo, P. A.; Slotine, J. J.E., Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming, Automatica, 44, 8, 2163-2170 (2008) · Zbl 1283.93217
[3] Balakrishnan, V., Lyapunov functionals in complex \(\mu\) analysis, IEEE Transactions on Automatic Control, 47, 9, 1466-1479 (2002) · Zbl 1364.93585
[4] Balas, G. J., Packard, A. K., Seiler, P., & Topcu, U. Robustness analysis of nonlinear systems, (n.d.)..; Balas, G. J., Packard, A. K., Seiler, P., & Topcu, U. Robustness analysis of nonlinear systems, (n.d.)..
[5] Chakraborty, A.; Seiler, P.; Balas, G. J., (Local performance analysis of uncertain polynomial systems with applications to actuator saturation (2010), IEEE CDC)
[6] Chakraborty, A.; Seiler, P.; Balas, G., Nonlinear region of attraction analysis for flight control verification and validation, Control Engineering Practice, 19, 4, 335-345 (2011)
[7] Chakraborty, A.; Seiler, P.; Balas, G., Susceptibility of F/A-18 flight controllers to the falling-leaf mode: nonlinear analysis, Journal of Guidance, Control and Dynamics, 34, 1, 73-85 (2011)
[8] Chesi, G., Estimating the domain of attraction for uncertain polynomial systems, Automatica, 40, 1981-1986 (2004) · Zbl 1067.93055
[9] da Silva, J. M.G.; Tarbouriech, S., Antiwindup design with guaranteed regions of stability: an lmi-based approach, IEEE Transactions on Automatic Control, 50, 1, 106-111 (2005) · Zbl 1365.93443
[10] Desoer, C.; Vidyasagar, M., Feedback systems: Input-output properties (1975), Academic Press: Academic Press New York · Zbl 0327.93009
[11] Fetzer, M.; Scherer, C. W.; Veenman, J., Invariance with dynamic multipliers, IEEE Transactions on Automatic Control, 63, 1929-1942 (2018) · Zbl 1423.93291
[12] Genesio, R.; Tartaglia, M.; Vicino, A., On the estimation of asymptotic stability regions: State of the art and new proposals, IEEE Transactions on Automatic Control, 30, 8, 747-755 (1985) · Zbl 0568.93054
[13] Heath, W. P.; Wills, A. G., Zames-falb multipliers for quadratic programming, (44th IEEE conference on decision and control (CDC) (2005)) · Zbl 1366.90155
[14] Henrion, D.; Korda, M., Convex computation of the region of attraction of polynomial control systems, IEEE Transactions on Automatic Control, 59, 2, 297-312 (2014) · Zbl 1360.93601
[15] Hu, T.; Teel, A. R.; Zaccarian, L., Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov functions, IEEE Transactions on Automatic Control, 51, 11, 1770-1786 (2006) · Zbl 1366.93439
[16] Iannelli, A.; Marcos, A.; Lowenberg, M., Robust estimations of the region of attraction using invariant sets, Journal of The Franklin Institute, 356, 4622-4647 (2019) · Zbl 1412.93091
[17] Iannelli, A.; Seiler, P.; Marcos, A., An equilibrium-independent region of attraction formulation for systems with uncertainty-dependent equilibria (2018), IEEE CDC
[18] Iannelli, A.; Seiler, P.; Marcos, A., Estimating the region of attraction of uncertain systems with integral quadratic constraints, ((2018), IEEE CDC)
[19] Khalil, H. K., Nonlinear systems (1996), Prentice Hall
[20] Megretski, A.; Rantzer, A., System analysis via integral quadratic constraints, IEEE Transactions on Automatic Control, 42, 6, 819-830 (1997) · Zbl 0881.93062
[21] Parrilo, P. A., Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming, 96, 2, 293-320 (2003) · Zbl 1043.14018
[22] Pfifer, H.; Seiler, P., Less conservative robustness analysis of linear parameter varying systems using integral quadratic constraints, International Journal of Robust and Nonlinear Control, 26, 16, 3580-3594 (2015) · Zbl 1351.93044
[23] Seiler, P., Stability analysis with dissipation inequalities and integral quadratic constraints, IEEE Transactions on Automatic Control, 60, 6, 1704-1709 (2015) · Zbl 1360.93523
[24] Seiler, P., An iterative algorithm to estimate invariant sets for uncertain systems, (IEEE ACC (2018))
[25] Seiler, P.; Balas, G., Quasiconvex sum-of-squares programming, (IEEE CDC (2010))
[26] Summers, E.; Packard, A., L2 gain verification for interconnections of locally stable systems using integral quadratic constraints, (IEEE CDC (2010))
[27] Topcu, U.; Packard, A., Local stability analysis for uncertain nonlinear systems, IEEE Transactions on Automatic Control, 54, 5, 1042-1047 (2009) · Zbl 1367.93483
[28] Topcu, U.; Packard, A. K.; Seiler, P.; Balas, G. J., Robust region-of-attraction estimation, IEEE Transactions on Automatic Control, 55, 1, 137-142 (2010) · Zbl 1368.93510
[29] Valmorbida, G.; Anderson, J., Region of attraction estimation using invariant sets and rational Lyapunov functions, Automatica, 75, 37-45 (2017) · Zbl 1351.93109
[30] Valmorbida, G.; Tarbouriech, S.; Garcia, G., Region of attraction estimates for polynomial systems, (IEEE CDC (2009))
[31] Vannelli, A.; Vidyasagar, M., Maximal lyapunov functions and domains of attraction for autonomous nonlinear systems, Automatica, 21, 69-80 (1985) · Zbl 0559.34052
[32] Veenman, J.; Scherer, C. W., IQC-Synthesis with general dynamic multipliers, International Journal of Robust and Nonlinear Control, 23, 17, 3027-3056 (2014) · Zbl 1305.93074
[33] Veenman, J.; Scherer, C. W.; Köroğlu, H., Robust stability and performance analysis based on integral quadratic constraints, European Journal of Control, 31, 1-32 (2016) · Zbl 1347.93199
[34] Yakubovich, V., S-procedure in nonlinear control theory, Vol. 1, 62-77 (1971), Vestnik Leningrad University · Zbl 0232.93010
[35] Zhou, K.; Doyle, J. C.; Glover, K., Robust and optimal control (1996), Prentice-Hall, Inc. · Zbl 0999.49500
[36] Zubov, V., Methods of A. M. Lyapunov and their application (1964), noordhoff · Zbl 0115.30204
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