Abstract
This paper proposes a singular perturbation-based approach to design saturated controllers for a class of nonlinear systems and shows its effectiveness in suppressing wing rock phenomenon in aircraft. Approximate dynamic inversion technique is exploited to design a high-gain dynamic controller which enforces time-scale separation in the closed-loop system. The convergence analysis of closed-loop system is done using the framework of contraction theory. Explicit tracking error bounds are derived both in the presence and absence of disturbances in system dynamics. The derived bounds are valid beyond a restrictive range of perturbation parameter, unlike Lyapunov-based approaches.
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References
Angeli, D.: A lyapunov approach to incremental stability properties. IEEE Trans. Autom. Control 47(3), 410–421 (2002)
Chakrabortty, A., Arcak, M.: Time-scale separation redesigns for stabilization and performance recovery of uncertain nonlinear systems. Automatica 45(1), 34–44 (2009)
Chen, G., Moiola, J.L., Wang, H.O.: Bifurcation control: theories, methods, and applications. Int. J. Bifurc. Chaos 10(3), 511–548 (2000)
Del Vecchio, D., Slotine, J.J.E.: A contraction theory approach to singularly perturbed systems. IEEE Trans. Autom. Control 58(3), 752–757 (2013)
Forni, F., Sepulchre, R.: A differential lyapunov framework for contraction analysis. IEEE Trans. Autom. Control 59(3), 614–628 (2014)
Guglieri, G., Quagliotti, F.: Analytical and experimental analysis of wing rock. Nonlinear Dyn. 24(2), 129–146 (2001)
Hovakimyan, E.L.N., Cao, C.: Adaptive dynamic inversion via time-scale separation. In: Proceedings of 45th IEEE Conference Decision and Control, pp. 1075–1080. San Diego (2006)
Hovakimyan, E.L.N., Sasane, A.: Dynamic inversion for nonaffine-in-control systems via time-scale separation. Part i. J. Dyn. Control Syst. 13(4), 451–465 (2007)
Hovakimyan, N., Lavretsky, E., Cao, C.: Dynamic inversion for multivariable non-affine-in-control systems via time-scale separation. Int. J. Control 81(12), 1960–1967 (2008)
Hsu, C.-H., Lan, C.E.: Theory of wing rock. J. Aircr. 22(10), 920–924 (1985)
Hu, T., Lin, Z.: Control Systems with Actuator Saturation: Analysis and Design. Springer, Berlin (2001)
Jouffroy, J., Lottin, J.: On the use of contraction theory for the design of nonlinear observers for ocean vehicles. Proc. Am. Control Conf. 4, 2647–2652 (2002)
Kaliora, G., Astolfi, A.: Nonlinear control of feedforward systems with bounded signals. IEEE Trans. Autom. Control. 49, 1975–1990 (2004)
Khalil, H.: Nonlinear Systems, 3rd edn. Prentice Hall, New Jersy (2002)
Kokotovic, P.V., O’Reilly, J., Khalil, H.K.: Singular Perturbation Methods in Control: Analysis and Design. Academic Press Inc, Orlando (1986)
Kuperman, A., Zhong, Q.-C.: Ude-based linear robust control for a class of nonlinear systems with application to wing rock motion stabilization. Nonlinear Dyn. 81(1–2), 789–799 (2015)
Lohmiller, W., Slotine, J.-J.E.: On contraction analysis for non-linear systems. Automatica 34(6), 683–696 (1998)
Luo, J., Lan, C.E.: Control of wing-rock motion of slender delta wings. J. Guidance Control Dyn. 16(2), 225–231 (1993)
Nayfeh, A., Elzebda, J., Mook, D.: Analytical study of the subsonic wing-rock phenomenon for slender delta wings. J. Aircr. 26(9), 805–809 (1989)
Rayguru, M.M., Kar, I.N.: Contraction based stabilization of approximate feedback linearizable systems. In: Proceedings of European Control Conference, pp. 587–592. (2015)
Rayguru, M.M., Kar, I.N.: Contraction-based stabilisation of nonlinear singularly perturbed systems and application to high gain feedback. Int. J. Control 90, 1778–1792 (2016). https://doi.org/10.1080/00207179.2016.1221139
Saberi, A., Khalil, H.: Stabilization and regulation of nonlinear singularly perturbed systems-composite control. IEEE Trans. Autom. Control 30(8), 739–747 (1985)
Sepulchre, R.: Slow peaking and low-gain designs for global stabilization of nonlinear systems. IEEE Trans. Autom. Control. 45, 453–461 (2000)
Serajian, R.: Parameters changing influence with different lateral stiffnesses on nonlinear analysis of hunting behavior of a bogie. J. Measurements Eng. 1(4), 195–206 (2013)
Sharma, B., Kar, I.: Adaptive control of wing rock system in uncertain environment using contraction theory. In: American Control Conference, pp. 2963–2968. IEEE (2008)
Sharma, B.B., Kar, I.N.: Adaptive control of wing rock system in uncertain environment using contraction theory. In: Proceedings of American Control Conference, pp. 2963–2968. (2008)
Sharma, B.B., Kar, I.N.: Observer-based synchronization scheme for a class of chaotic systems using contraction theory. Nonlinear Dyn. 63(3), 429–445 (2011)
Sontag, E.D.: Contractive systems with inputs. In: Willems, J., Hara, S., Ohta, Y., Fujioka, H. (eds.) Perspectives in Mathematical System Theory, Control, and Signal Processing, Series Lecture Notes in Control and Information Sciences, pp. 217–228. Springer, Berlin (2010)
Teel, A.R.: Global stabilization and restricted tracking for multiple integrators with bounded controls. Syst. Control Lett. 18, 165–171 (1992)
Teel, A.R.: A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans. Autom. Control 41, 1256–1270 (1996)
Teo, J., How, J.P., Lavretsky, E.: On approximate dynamic inversion and proportional-integral control. In 2009 American Control Conference, pp. 1592–1597. IEEE (2009)
Teo, J., How, J.P.: Equivalence between approximate dynamic inversion and proportional-integral control. In: Proceedings of 47th IEEE Conference Decision and Control, pp. 2179–2183. Cancun, Mexico (2008)
Teo, J., How, J.P., Lavretsky, E.: Proportional-integral controllers for minimum-phase nonaffine-in-control systems. IEEE Trans. Autom. Control 55(6), 1477–1483 (2010)
Wang, W., Slotine, J.-J.E.: On partial contraction analysis for coupled nonlinear oscillators. Biol. Cybern. 92(1), 38–53 (2005)
Yang, C., Sun, J., Ma, X.: Stabilization bound of singularly perturbed systems subject to actuator saturation. Automatica 49(2), 457–462 (2013)
Younesian, D., Jafari, A.A., Serajian, R.: Effects of the bogie and body inertia on the nonlinear wheel-set hunting recognized by the hopf bifurcation theory. Int. J. Auto Eng. 1(3), 186–196 (2011)
Zamani, M., Tabuada, P.: Backstepping design for incremental stability. IEEE Trans. Autom. Control 56(9), 2184–2189 (2011)
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Rayguru, M.M., Kar, I.N. A singular perturbation approach to saturated controller design with application to bounded stabilization of wing rock phenomenon. Nonlinear Dyn 93, 2263–2272 (2018). https://doi.org/10.1007/s11071-018-4323-x
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DOI: https://doi.org/10.1007/s11071-018-4323-x