Analysis and Control of Linear Time Periodic System using Normal Forms
Tóm tắt
Multiple techniques have been developed in the past towards stability and control of linear time periodic systems. Though the method of normal forms was predominantly applied to nonlinear equations, in this work, it is utilized to transform a linear time varying system with periodic coefficients to a time-invariant system (similar to a Lyapunov-Floquet transformation). The direct application of time independent normal forms is facilitated by a combination of an intuitive state augmentation technique and modal transformation. Additionally, this approach yields a closed form analytical expression for the Lyapunov–Floquet (L-F) transformation and state transition matrix. The transition curves and stability bounds are identified and multiple feedback control strategies are also discussed in this work. Furthermore, the authors demonstrate the control of an unstable periodic system to a stable point, desired periodic orbit and optimally controlled system states using the normal forms approach. The theoretical framework and controller implementation are illustrated using numerical simulations for the case of a linear Mathieu equation.
Tài liệu tham khảo
Rega G (2019) Nonlinear dynamics in mechanics and engineering: 40 years of developments and Ali H. Nayfeh’s legacy. Nonlinear Dyn 99:11–34. doi: 10.1007/s11071-019-04833-w.
Sinha SC, Butcher EA (1997) Symbolic computation of fundamental solution matrices for linear time-periodic dynamical systems. J Sound Vib 206(1), 61–85. doi: 10.1006/jsvi.1997.1079.
Yakubovich VA, Starzhinskiĭ VM (1975) Linear differential equations with periodic coefficients. Wiley, New York.
Nayfeh AH (2011) Introduction to perturbation techniques. Wiley, New York.
Sanders JA, Verhulst F, Murdock JA (2007) Averaging methods in nonlinear dynamical systems. Springer, New York.
Sinha SC, Pandiyan R, Bibb J (1996) Liapunov-Floquet transformation: computation and applications to periodic systems. J Vib Acoust 118(2), 209–219. doi: 10.1115/1.2889651.
Sharma A (2021) Approximate Lyapunov-Perron transformations: computation and applications to quasi-periodic systems. J Comput Nonlinear Dyn 16(5):051005. doi: 10.1115/1.4050614.
Sinha S, Wu D-H, Juneja V, Joseph P (1993) Analysis of dynamic systems with periodically varying parameters via Chebyshev polynomials. J Vib Acoust 115(1), 96–102. doi: 10.1115/1.2930321.
Sinha SC, Gourdon E, Zhang Y (2005) Control of time-periodic systems via symbolic computation with application to chaos control. Commun Nonlinear Sci Numer Simul 10(8), 835–854. doi: 10.1016/j.cnsns.2004.06.001.
Sharma A, Sinha S (2020) Control of nonlinear systems exhibiting chaos to desired periodic or quasi-periodic motions. Nonlinear Dyn 99(1), 559–574. doi: 10.1007/s11071-019-04843-8.
Luo AC (2013) Analytical solutions for periodic motions to chaos in nonlinear systems with/without time-delay. Int J Dyn Control 1(4), 330–359. doi: 10.1007/s40435-013-0024-y.
Follinger O (1978) Design of time-varying system by pole assignment. Automatisierungstechnik 26:189–196. doi: 10.1524/auto.1978.26.112.189.
Kwakernaak H, Sivan R (1972) Linear optimal control systems. Wiley, New York.
Sinha S, Joseph P (1994) Control of general dynamic systems with periodically varying parameters via Liapunov-Floquet transformation. J Dyn Syst Meas Contr 116(4), 650–658. doi: 10.1115/1.2899264.
Nazari M, Butcher EA, Bobrenkov OA (2014) Optimal feedback control strategies for periodic delayed systems. Int J Dyn Control 2(1), 102–118. doi: 10.1007/s40435-013-0053-6.
Nazari M, Butcher EA (2016) Fuel efficient periodic gain control strategies for spacecraft relative motion in elliptic chief orbits. Int J Dyn Control 4(1), 104–122. doi: 10.1007/s40435-014-0126-1.
Nazari M, Butcher EA, Anthony W (2017) Earth-Moon L1 libration point orbit continuous stationkeeping control using time-varying LQR and backstepping. Int J Dyn Control 5(4), 1089–1102. doi: 10.1007/s40435-016-0256-8.
Leonov GA, Moskvin AV (2018) Stabilizing unstable periodic orbits of dynamical systems using delayed feedback control with periodic gain. Int J Dyn Control 6(2), 601–608. doi: 10.1007/s40435-017-0316-8.
Cong C (2019) Observer-based robust control of uncertain systems via an integral quadratic constraint approach. Int J Dyn Control 7(3), 926–939. doi: 10.1007/s40435-018-00507-4.
Polisetty VG, Varanasi SK, Jampana P (2021) Stochastic state-feedback control using homotopy optimization and particle filtering. Int J Dyn Control. doi: 10.1007/s40435-021-00853-w.
Poincaré H (1899) Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars et fils
Birkhoff GD (1927) Dyn Syst. American Mathematical Society, New York.
Moser J, Saari D (1975) Stable and random motions in dynamical systems. Phys Today 28:47. doi: 10.1063/1.3068879.
Arnold V (1989) Mathematical methods of classical mechanics. Springer, New York.
Chua LO, Kokubu H (1988) Normal forms for nonlinear vector fields. I. Theory and algorithm. IEEE Trans Circuits Syst 35(7):863–880
Nayfeh AH (2011) The method of normal forms. Wiley, New York.
Murdock J (2006) Normal forms and unfoldings for local dynamical systems. Springer, New York.
Sinha S, Butcher E, Dávid A (1998) Construction of dynamically equivalent time-invariant forms for time-periodic systems. Nonlinear Dyn 16(3), 203–221. doi: 10.1023/A:1008072713385.
Gabale AP, Sinha SC (2009) A direct analysis of nonlinear systems with external periodic excitations via normal forms. Nonlinear Dyn 55(1–2), 79–93. doi: 10.1007/s11071-008-9346-2.
Jezequel L, Lamarque C-H (1991) Analysis of non-linear dynamical systems by the normal form theory. J Sound Vib 149(3), 429–459. doi: 10.1016/0022-460X(91)90446-Q.
Smith HL (1986) Normal forms for periodic systems. J Math Anal Appl 113(2), 578–600.
Zhang Y, Sinha S (2007) Development of a feedback linearization technique for parametrically excited nonlinear systems via normal forms. J Comput Nonlinear Dyn 2(2), 124–131. doi: 10.1115/1.2447190.
Cherangara Subramanian S, Redkar S (2020) Comparison of poincare normal forms and floquet theory for analysis of linear time periodic systems. J Comput Nonlinear Dyn 16(1):014502. https://doi.org/10.1115/1.4048715.
Cherangara Subramanian S, Dye M, Redkar S (2020) Dynamic analysis of suction stabilized floating platforms. J Mar Sci Eng 8(8):587. doi: https://doi.org/10.3390/jmse8080587.
Aburn M (2016) Critical fluctuations and coupling of stochastic neural mass models. The University of Queensland, Australia.
Kovacic I, Rand R, Mohamed Sah S (2018) Mathieu’s equation and its generalizations: overview of stability charts and their features. Appl Mech Rev 70(2):020802. doi: 10.1115/1.4039144.