Backstepping Method for Stabilizing System of 2×2 Riccati Matrix Differential Equations
Keywords:
Backstepping method, Riccati differential equations, Lyapunov functions (Lf), Asymptotically stableAbstract
In this research paper, the backstepping method (BSM) will be proposed for stabilizing and solving system of 2×2 Riccati matrix ordinary differential equations. Such equations have many difficulties in the studying their solutions and stability. The basic idea behind of this approach is to use the BSM as a transformation method for transforming the original system into an equivalent one which is stabilizable and solvable based on constructing the Lyapunov function.
References
[1] Krstic, M.; Smyshlyaev, A.; “Boundary Control of PDEs: A Course on BS Designs”, SIAM, USA, 2008.
[2] Krstic, M.; Kanellakopoulos, I.; Kokotovic, P. V.; “Non-linear and Adaptive Control Design”, Wiley and Sons, Inc., New York, 1995.
[3] Sepulchre, R.; Jankovic, M.; Kokotovic, P.; “Constructive Non-linear Control”, Springer-Verlag Berlin, 1997.
[4] Fossen, T. I.; Berge, S. P.; “Non-linear Vectorial BS Design for Global Exponential Tracking of Marine Vessels in the Presence of Actuator Dynamics”, In: Proceedings of the IEEE Conference on Decision and Control (CDC’97), San Diego, CA., 4237–4242, 1997.
[5] Krstic, M.; Deng, H.; “Stabilization of Non-linear Uncertain Systems”, Springer-Verlag. Berlin, 1998.
[6] Loria, A.; Fossen, T. I.; Teel, A.; “UGAS and ULES of Non Autonomous Systems: Applications to Integral Action Control of Ships and Manipulators”, In: Proceedings of the 5th European Control Conference (ECC’99), Karlsruhe, Germany, 1999.
[7] Fossen, T. I.; Loria, A.; Teel, A.; “A Theorem for UGAS and ULES of (Passive) Nonautonomous Systems: Robust Control of Mechanical Systems and Ships”, Int. J. Rob. Nonlin. Cont. 11, 95–108, 2001.
[8] Smyshlyaev, A.; Krstic, M.; “Closed form boundary state feedbacks for a class of 1D partial integro-differential equations”, IEEE Trans. Auto. Cont. 49(12), pp. 2185–2202, 2004.
[9] Vazquez, R.; Krstic, M.; “A closed-form feedback controller for stabilization of the linearized 2D Navier-Stokes Poiseuille flow”, IEEE Trans. Auto. Cont. 52, 2298–2312, 2007.
[10] Krstic, M.; “Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch”, Automatic 44, 2930-2935, 2008.
[11] Krstic, M.; “On using least-squares updates without regress or filtering in identification and adaptive control of non-linear systems”, Automatic, Vol. 45, pp. 731-735, 2009.
[12] Krstic, M.; “Input delay compensation for forward complete and feed forward non-linear systems”, IEEE Trans. Auto. Cont. 55, 287-303, 2010.
[13] Krstic, M.; “Lyapunov stability of linear predictor feedback for time-varying input delay”, IEEE Trans. Auto. Cont. 55, 554-559, 2010.
[14] Krstic, M.; Bekiaris-Liberis, N.; “Non-linear stabilization in infinite dimension”, Ann. Rev. Cont. 37, 220-231, 2013.
[2] Krstic, M.; Kanellakopoulos, I.; Kokotovic, P. V.; “Non-linear and Adaptive Control Design”, Wiley and Sons, Inc., New York, 1995.
[3] Sepulchre, R.; Jankovic, M.; Kokotovic, P.; “Constructive Non-linear Control”, Springer-Verlag Berlin, 1997.
[4] Fossen, T. I.; Berge, S. P.; “Non-linear Vectorial BS Design for Global Exponential Tracking of Marine Vessels in the Presence of Actuator Dynamics”, In: Proceedings of the IEEE Conference on Decision and Control (CDC’97), San Diego, CA., 4237–4242, 1997.
[5] Krstic, M.; Deng, H.; “Stabilization of Non-linear Uncertain Systems”, Springer-Verlag. Berlin, 1998.
[6] Loria, A.; Fossen, T. I.; Teel, A.; “UGAS and ULES of Non Autonomous Systems: Applications to Integral Action Control of Ships and Manipulators”, In: Proceedings of the 5th European Control Conference (ECC’99), Karlsruhe, Germany, 1999.
[7] Fossen, T. I.; Loria, A.; Teel, A.; “A Theorem for UGAS and ULES of (Passive) Nonautonomous Systems: Robust Control of Mechanical Systems and Ships”, Int. J. Rob. Nonlin. Cont. 11, 95–108, 2001.
[8] Smyshlyaev, A.; Krstic, M.; “Closed form boundary state feedbacks for a class of 1D partial integro-differential equations”, IEEE Trans. Auto. Cont. 49(12), pp. 2185–2202, 2004.
[9] Vazquez, R.; Krstic, M.; “A closed-form feedback controller for stabilization of the linearized 2D Navier-Stokes Poiseuille flow”, IEEE Trans. Auto. Cont. 52, 2298–2312, 2007.
[10] Krstic, M.; “Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch”, Automatic 44, 2930-2935, 2008.
[11] Krstic, M.; “On using least-squares updates without regress or filtering in identification and adaptive control of non-linear systems”, Automatic, Vol. 45, pp. 731-735, 2009.
[12] Krstic, M.; “Input delay compensation for forward complete and feed forward non-linear systems”, IEEE Trans. Auto. Cont. 55, 287-303, 2010.
[13] Krstic, M.; “Lyapunov stability of linear predictor feedback for time-varying input delay”, IEEE Trans. Auto. Cont. 55, 554-559, 2010.
[14] Krstic, M.; Bekiaris-Liberis, N.; “Non-linear stabilization in infinite dimension”, Ann. Rev. Cont. 37, 220-231, 2013.
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2020-06-04
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Backstepping Method for Stabilizing System of 2×2 Riccati Matrix Differential Equations. ANJS 2020, 23 (2), 52-59.
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