Abstract
The previous Chapter presented a tracking controller of the position and heading of a helicopter based on the linearized helicopter dynamics. The adopted parametric linear model, on which the flight controller is based on, represented the quasi steady state behavior of the helicopter dynamics at hover.
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Notes
- 1.
The override of the f B components in the \(\vec {i}_{{B}}\) and \(\vec{j}_{{B}}\) directions of the body-fixed frame achieves the decoupling of the helicopter external force and moment model. The work reported in [47] indicates that if the complete description of the force vector given in (7.8) is used, then the state space dynamics of the nonlinear helicopter model cannot be input–output linearizable and the zero-dynamics of the system will be unstable. If the system dynamics are not input–output linearizable most of the standard control methodologies will be inapplicable. If the proposed approximation takes place, the helicopter nonlinear model becomes full state linearizable by considering the position and the yaw as outputs. To the authors knowledge, there is not any controller design in the literature that is based on the exact model and in all case studies this approximation is performed. The use of the approximated model also took place in Chap. 6 indicating that for the helicopter control problem only practical stability can be achieved based on the approximated model.
- 2.
The function f(t,s) is Lipschitz in s∈ℝn if it is piecewise continuous in t and satisfies the Lipschitz condition:
$$\|f(t,s_{1})-f(t,s_{2})\|\leq{}^{f}L\|s_{1}-s_{2}\|$$for every s 1,s 2∈ℝn and a positive constant f L (called Lipschitz constant). If the function f(t,s) is Lipschitz in s, then the system \(\dot{s}=f(t,s)\) with s(t 0)=s o has a unique solution for every t [43].
- 3.
Note that ρ 3,3=ρ 1,1 ρ 2,2−ρ 1,2 ρ 2,1.
- 4.
Based on [43] a continuous function \(\alpha_{\mathcal{K}}(s):[0\ \infty )\rightarrow[0\ \infty)\) belongs to the class \(\mathcal {K}\) if it is strictly increasing and \(\alpha_{\mathcal{K}}(0)=0\). A continuous function \(\beta_{\mathcal{KL}}(s_{1},s_{2}):[0\ \infty)\times [0\ \infty)\rightarrow[0\ \infty)\) belongs to the class \(\mathcal{KL}\) if, for each fixed s 2, the mapping \(\beta _{\mathcal{KL}}(s_{1},s_{2})\) belongs to the class \(\mathcal{K}\) with respect to s 1 and for each fixed s 1, the mapping \(\beta_{\mathcal{KL}}(s_{1},s_{2})\) is decreasing with respect to s 2 and \(\beta_{\mathcal{KL}}(s_{1},s_{2})\rightarrow0\) as s 2→∞.
References
M. Bejar, A. Isidori, L. Marconi, R. Naldi, Robust vertical/lateral/longitudinal control of a helicopter with constant yaw-attitude, in 44th IEEE Conference on Decision and Control, and 2005 European Control Conference, CDC-ECC, 2005
A.R.S. Bramwell, G. Done, D. Balmford, Bramwell’s Helicopter Dynamics (Butterworth Heinemann, Stoneham, 2001)
E. Frazzoli, M.A. Dahleh, E. Feron, Trajectory tracking control design for autonomous helicopters using a backstepping algorithm, in Proceedings of the American Control Conference, vol. 6, 2000, pp. 4102–4107
D. Fujiwara, J. Shin, K. Hazawa, K. Nonami, \(\mathcal{H}_{\infty}\) hovering and guidance control for an autonomous small-scale unmanned helicopter, in International Conference on Intelligent Robots and Systems, 2004
J. Gadewadikar, F.L. Lewis, K. Subbarao, K. Peng, B.M. Chen, \(\mathcal{H}_{\infty}\) static output-feedback control for rotorcraft, in AIAA Guidance, Navigation, and Control Conference and Exhibit, 2006
V. Gavrilets, B. Mettler, E. Feron, Dynamical model for a miniature aerobatic helicopter, Technical report, Massachusetts Institute of Technology, 2001
V. Gavrilets, B. Mettler, E. Feron, Nonlinear model for a small-size acrobatic helicopter, in AIAA Guidance, Navigation, and Control Conference and Exhibit, 2001
A. Isidori, L. Marconi, A. Serrani, Robust nonlinear motion control of a helicopter. IEEE Transactions on Automatic Control 48, 413–426 (2003)
W. Johnson, Helicopter Theory (Princeton University Press, Princeton, 1980)
H.K. Khalil, Nonlinear Systems (Prentice Hall, New York, 2002)
T.J. Koo, S. Sastry, Output tracking control design of a helicopter model based on approximate linearization, in Proceedings of the 37th IEEE Conference on Decision and Control, vol. 4, 1998, pp. 3635–3640
M. La Civita, G. Papageorgiou, W.C. Messner, T. Kanade, Integrated modeling and robust control for full-envelope flight of robotic helicopters, in Proceedings of IEEE International Conference on Robotics and Automation, 2003, pp. 552–557
M. La Civita, G. Papageorgiou, W.C. Messner, T. Kanade, Design and flight testing of an \(\mathcal{H}_{\infty}\) controller for a robotic helicopter. Journal of Guidance, Control, and Dynamics, 485–494 (2006)
E.H. Lee, H. Shim, L. Park, K. Lee, Design of hovering attitude controller for a model helicopter, in Proceedings of Society of Instrument and Control Engineers, 1993, pp. 1385–1390
A. Loria, E. Panteley, Advanced Topics in Control Systems Theory: Lecture Notes from FAP 2004, chapter 2 (Springer, Berlin, 2005), pp. 23–64
L. Marconi, R. Naldi, Robust full degree-of-freedom tracking control of a helicopter. Automatica 43, 1909–1920 (2007)
B. Mettler, Identification Modeling and Characteristics of Miniature Rotorcraft (Kluwer Academic Publishers, Norwell, 2003)
B. Mettler, M.B. Tischler, T. Kanade, System identification of small-size unmanned helicopter dynamics, in Presented at the American Helicopter Society 55th Forum, May 1999
R.M. Murray, L. Zexiang, S. Sastry, A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, 1994)
R.W. Prouty, Helicopter Performance, Stability and Control (Krieger Publishing Company, Melbourne, 1995)
H. Shim, T.J. Koo, F. Hoffmann, S. Sastry, A comprehensive study of control design for an autonomous helicopter, in Proceedings of the 37th IEEE Conference on Decision and Control, vol. 4, 1998, pp. 3653–3658
H.D. Shim, H.J. Kim, S. Sastry, Control system design for rotorcraft-based unmanned aerial vehicles using time-domain system identification, in Proceedings of the 2000 IEEE International Conference on Control Applications, 2000, pp. 808–813
H. Sira-Ramirez, M. Zribi, S. Ahmad, Dynamical sliding mode control approach for vertical flight regulation in helicopters, Control Theory and Applications 141, 19–24 (1994)
E.D. Sontag, Remarks on stabilization and input-to-state stability, in Proceedings of the 28th IEEE Conference on Decision and Control, vol. 2, 1989, pp. 1376–1378
A.R. Teel, Global stabilization and restricted tracking for multiple integrators with bounded controls. Systems & Control Letters 18, 165–171 (1992)
A.R. Teel, Using saturation to stabilize a class of single-input partially linear composite systems, in IFAC NOLCOS’92 Symposium, 1992, pp. 379–384
M.B. Tischler, R.K. Remple, Aircraft and Rotorcraft System Identification, AIAA Education Series (AIAA, Washington, 2006)
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Raptis, I.A., Valavanis, K.P. (2011). Nonlinear Tracking Controller Design for Unmanned Helicopters. In: Linear and Nonlinear Control of Small-Scale Unmanned Helicopters. Intelligent Systems, Control and Automation: Science and Engineering, vol 45. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0023-9_7
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