Abstract
This paper presents an adaptive control law for unknown nonlinear switched plants that must follow the trajectory of user-defined linear switched reference models. The effectiveness of the proposed control architecture is proven in two alternative frameworks, that is, analyzing Carathéodory and Filippov solutions of discontinuous differential equations. Numerical and experimental data verify the applicability of the theoretical results to problems of practical interest. The proposed numerical simulation involves the design of a model reference adaptive control law to regulate the roll dynamics of a reconfigurable delta-wing aircraft. The proposed flight tests involve an aerial robot tasked with autonomously mounting a camera of unknown inertial properties to a vertical surface.
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Alexis, K., Darivianakis, G., Burri, M., Siegwart, R.: Aerial robotic contact-based inspection: Planning and control. Auton. Robot. 40(4), 631–655 (2016). https://doi.org/10.1007/s10514-015-9485-5
Anderson, R.B.: Flight experiment of a tilt-rotor quadcopter installing a camera. https://youtu.be/f6wG7IdPXoc. Last accessed 11/29/2019 (2019)
Anderson, R.B., Burke, J.P., Marshall, J.A., L’Afflitto, A.: Robust Adaptive Control for Constrained Tilt-Rotor Quadcopters of Unknown Inertial Properties. In: American Control Conference, pp. 2922–2927. https://doi.org/10.23919/ACC.2019.8814780 (2019)
Ang, K.Z.Y., Cui, J., Pang, T., Li, K., Wang, K., Ke, Y., Chen, B.M.: Development of an Unmanned Tail-Sitter with Reconfigurable Wings: U-Lion. In: IEEE International Conference on Control Automation, pp. 750–755. https://doi.org/10.1109/ICCA.2014.6871015 (2014)
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (2008)
Brogliato, B.: Nonsmooth Mechanics: Models, Dynamics and Control. Communications and Control Engineering. Springer, London (1999)
Bryson, A.E.: Applied Optimal Control: Optimization, Estimation and Control. Taylor & Francis, New York (1975)
Carathéodory, C.: Vorlesungen Über Reelle Funktionen. Teubner, Berlin (1918)
Cheng, H., Dong, C., Jiang, W., Wang, Q., Hou, Y.: Non-fragile switched \(h_{\infty }\) control for morphing aircraft with asynchronous switching. Chin. J. Aeronaut. 30(3), 1127–1139 (2017). https://doi.org/10.1016/j.cja.2017.01.008
Chiang, M.L., Fu, L.C.: Adaptive stabilization of a class of uncertain switched nonlinear systems with backstepping control. Automatica 50(8), 2128–2135 (2014)
Clarke, F.H.: Optimization and nonsmooth analysis. Society of Industrial and Applied Mathematics, Philadelphia (1989)
Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth analysis and control theory. Graduate texts in mathematics. springer, New York NY (1997)
Cortés, J.: Discontinuous dynamical systems. IEEE Control. Syst. Mag. 28(3), 36–73 (2008). https://doi.org/10.1109/MCS.2008.919306
Edwards, C., Spurgeon, S.: Sliding Mode Control: Theory And Applications. Series in Systems and Control. Taylor & Francis, New York (1998)
Federer, H.: Geometric measure theory. Springer, Berlin (2014)
Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Springer, Amesterdam (1988)
Fischer, N., Kamalapurkar, R., Dixon, W.E.: LaSalle-Yoshizawa corollaries for nonsmooth systems. IEEE Trans. Autom. Control 58(9), 2333–2338 (2013). https://doi.org/10.1109/TAC.2013.2246900
Gong, L., Wang, Q., Dong, C.: Disturbance rejection control of morphing aircraft based on switched nonlinear systems. Nonlinear Dyn. 96(2), 975–995 (2019). https://doi.org/10.1007/s11071-019-04834-9
Haddad, W.M., Chellaboina, V.: Nonlinear Dynamical Systems and Control: A Lyapunov-based Approach. Princeton University Press, Princeton (2008)
Hájek, O.: Discontinuous differential equations, i. J. Differ. Equ. 32(2), 149–170 (1979). https://doi.org/10.1016/0022-0396(79)90056-1
Heredia, G., Duran, A., Ollero, A.: Modeling and simulation of the HADA reconfigurable UAV. J. Intell. Robot. Syst. 65(1), 115–122 (2012). https://doi.org/10.1007/s10846-011-9561-9
Hespanha, J.P., Liberzon, D., Morse, A.S.: Overcoming the limitations of adaptive control by means of logic-based switching. Syst. Control Lett. 49(1), 49–65 (2003). https://doi.org/10.1016/S0167-6911(02)00342-0
Hu, S.: Differential equations with discontinuous right-hand sides. J. Math. Anal. Appl 154(2), 377–390 (1991). https://doi.org/10.1016/0022-247X(91)90044-Z
Jiang, H.B.: Hybrid adaptive and impulsive synchronisation of uncertain complex dynamical networks by the generalised Barbalat’s lemma. IET Control Theory Appl. 3(10), 1330–1340 (2009). https://doi.org/10.1049/iet-cta.2008.0335
Jiao, X., Jiang, J.: Design of adaptive switching control for hypersonic aircraft. Adv. Mech. Eng. 7(10), 1687814015610, 465 (2015). https://doi.org/10.1177/1687814015610465
Kamalapurkar, R., Rosenfeld, J.A., Parikh, A., Teel, A.R., Dixon, W.E.: Invariance-like results for nonautonomous switched systems. IEEE Trans. Autom. Control 64(2), 614–627 (2019). https://doi.org/10.1109/TAC.2018.2838055
Khalil, H.K.: Nonlinear systems. Prentice Hall, Princeton (2002)
Kosmatopoulos, E.B., Ioannou, P.A.: A switching adaptive controller for feedback linearizable systems. IEEE Trans. Autom. Control 44(4), 742–750 (1999). https://doi.org/10.1109/9.754811
Koutsoukos, X.D., Antsaklis, P.J., Stiver, J.A., Lemmon, M.D.: Supervisory control of hybrid systems. Proc. IEEE 88(7), 1026–1049 (2000). https://doi.org/10.1109/5.871307
Krasovskii, N.N.: Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations With Delay. Stanford University Press, Stanford (1963)
L’Afflitto, A.: A mathematical perspective on flight dynamics and control. Springer, London (2017)
L’Afflitto, A., Anderson, R.B., Mohammadi, K.: An introduction to nonlinear robust control for unmanned quadrotor aircraft: How to design control algorithms for quadrotors using sliding mode control and adaptive control techniques. IEEE Control. Syst. Mag. 38(3), 102–121 (2018). https://doi.org/10.1109/MCS.2018.2810559
Lavretsky, E., Wise, K.: Robust and adaptive control: With aerospace Spplications. springer, London (2012)
Levant, A.: Introduction to High-Order Sliding Modes. In: Perruquetti, W., Barbot, J.P. (eds.) Sliding Mode Control in Engineering, Chapter 1. Marcel Dekker, Inc., Basel (2002)
Li, Y., Sui, S., Tong, S.: Adaptive fuzzy control design for stochastic nonlinear switched systems with arbitrary switchings and unmodeled dynamics. IEEE Trans. Cybern. 47(2), 403–414 (2016)
Liberzon, D.: Switching in systems and control. Springer, Boston (2003)
Marconi, L., Naldi, R.: Control of aerial robots: Hybrid force and position feedback for a ducted fan. IEEE Control. Syst. Mag. 32(4), 43–65 (2012). https://doi.org/10.1109/MCS.2012.2194841
Marconi, L., Naldi, R., Gentili, L.: Modelling and control of a flying robot interacting with the environment. Automatica 47(12), 2571–2583 (2011). https://doi.org/10.1016/j.automatica.2011.09.020
Miller, D.E., Davison, E.J.: An adaptive controller which provides an arbitrarily good transient and steady-state response. IEEE Trans. Autom. Control 36(1), 68–81 (1991). https://doi.org/10.1109/9.62269
Minyue, F., Barmish, B.: Adaptive stabilization of linear systems via switching control. IEEE Trans. Autom. Control 31(12), 1097–1103 (1986). https://doi.org/10.1109/TAC.1986.1104187
Morse, A.S.: Supervisory control of families of linear set-point controllers – Part 1: Exact matching. IEEE Trans. Autom. Control 41(10), 1413–1431 (1996). https://doi.org/10.1109/9.539424
Morse, A.S.: Supervisory control of families of linear set-point controllers – Part 2: Robustness. IEEE Trans. Autom. Control 42(11), 1500–1515 (1997). https://doi.org/10.1109/9.649687
Narendra, K.S., Balakrishnan, J.: Adaptive control using multiple models. IEEE Trans. Autom. Control 42(2), 171–187 (1997). https://doi.org/10.1109/9.554398
Paden, B., Sastry, S.: A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulators. IEEE Trans. Circ. Syst. 34(1), 73–82 (1987). https://doi.org/10.1109/TCS.1987.1086038
Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. Springer, Vienna (2000)
Seigler, T.M., Neal, D.A., Bae, J.S., Inman, D.J.: Modeling and flight control of large-scale morphing aircraft. AIAA J. Aircraft 44(4), 1077–1087 (2007). https://doi.org/10.2514/1.21439
Shorten, R.N., Narendra, K.S.: On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form. IEEE Trans. Autom. Control 48(4), 618–621 (2003). https://doi.org/10.1109/TAC.2003.809795
Sofla, A., Meguid, S., Tan, K., Yeo, W.: Shape morphing of aircraft wing: Status and challenges. Mater. Des. 31(3), 1284–1292. https://doi.org/10.1016/j.matdes.2009.09.011 (2010)
Spraker, J.S., Biles, D.C.: A comparison of the Carathéodory and Filippov solution sets. J. Math. Anal. Appl. 198(2), 571–580 (1996). https://doi.org/10.1006/jmaa.1996.0099
Stein, E.M., Shakarchi, R.: Real analysis. Princeton, Princeton (2009)
Tan, C., Tao, G., Qi, R., Yang, H.: A direct MRAC based multivariable multiple-model switching control scheme. Automatica 84, 190–198 (2017). https://doi.org/10.1016/j.automatica.2017.07.020
Utkin, V.I.: Sliding Mode Control: Mathematical Tools, Design and Applications, pp. 289–347. Springer, Berlin. https://doi.org/10.1007/978-3-540-77653-6_5 (2008)
Vasista, S., Tong, L., Wong, K.: Realization of morphing wings: A multidisciplinary challenge. AIAA J. Aircraft 49(1), 11–28 (2012). https://doi.org/10.2514/1.C031060
Vu, L., Liberzon, D.: Supervisory control of uncertain linear time-varying systems. IEEE Trans. Autom. Control 56(1), 27–42 (2011). https://doi.org/10.1109/TAC.2010.2060244
Wang, Q., Hou, Y., Dong, C.: Model reference robust adaptive control for a class of uncertain switched linear systems. Int. J. Robust Nonlinear Control 22(9), 1019–1035 (2012). https://doi.org/10.1002/rnc.1744
Wang, X., Zhao, J., Tang, Y.: State tracking model reference adaptive control for switched nonlinear systems with linear uncertain parameters. J. Control Theory Appl. 10(3), 354–358 (2012). https://doi.org/10.1007/s11768-012-1018-6
Wu, C., Zhao, J., Sun, X.M.: Adaptive tracking control for uncertain switched systems under asynchronous switching. Int. J. Robust Nonlinear Control 25(17), 3457–3477 (2015). https://doi.org/10.1002/rnc.3275
Xie, J., Li, S., Yan, H., Yang, D.: Model reference adaptive control for switched linear systems using switched multiple models control strategy. J. Frankl. Inst. 356(5), 2645–2667 (2019). https://doi.org/10.1016/j.jfranklin.2018.10.036
Xie, J., Zhao, J.: Model reference adaptive control for switched LPV systems and its application. IET Control Theory Appl. 10(17), 2204–2212 (2016). https://doi.org/10.1049/iet-cta.2015.1332
Xie, J., Zhao, J.: Model reference adaptive control for nonlinear switched systems under asynchronous switching. Int. J. Adapt. Control Signal Process. 31(1), 3–22 (2017). https://doi.org/10.1002/acs.2666
Xie, J., Zhao, J.: \(h_{\infty }\) model reference adaptive control for switched systems based on the switched closed-loop reference model. Nonlinear Anal. Hybrid Syst. 27, 92–106 (2018). https://doi.org/10.1016/j.nahs.2017.07.003
Xie, W., Wen, C., Li, Z.: Input-to-state stabilization of switched nonlinear systems. IEEE Trans. Autom. Control 46(7), 1111–1116 (2001). https://doi.org/10.1109/9.935066
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This work was supported in part by the US Army Research Lab (ARL), DARPA, and ONR under Grants no. 40304747, D18AP00069, and N000141912422, respectively.
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Anderson, R.B., Marshall, J.A., L’Afflitto, A. et al. Model Reference Adaptive Control of Switched Dynamical Systems with Applications to Aerial Robotics. J Intell Robot Syst 100, 1265–1281 (2020). https://doi.org/10.1007/s10846-020-01260-7
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DOI: https://doi.org/10.1007/s10846-020-01260-7