Abstract
Attitude stabilization of spacecraft using magnetorquers can be achieved by a proportional–derivative-like control algorithm. The gains of this algorithm are usually determined by using a trial-and-error approach within the large search space of the possible values of the gains. However, when finding the gains in this manner, only a small portion of the search space is actually explored. We propose here an innovative and systematic approach for finding the gains: they should be those that minimize the settling time of the attitude error. However, the settling time depends also on initial conditions. Consequently, gains that minimize the settling time for specific initial conditions cannot guarantee the minimum settling time under different initial conditions. Initial conditions are not known in advance. We overcome this obstacle by formulating a min–max problem whose solution provides robust gains, which are gains that minimize the settling time under the worst initial conditions, thus producing good average behavior. An additional difficulty is that the settling time cannot be expressed in analytical form as a function of gains and initial conditions. Hence, our approach uses some derivative-free optimization algorithms as building blocks. These algorithms work without the need to write the objective function analytically: they only need to compute it at a number of points. Results obtained in a case study are very promising.
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References
Sidi, M.J.: Spacecraft Dynamics and Control. Cambridge University Press, New York (1997)
Silani, E., Lovera, M.: Magnetic spacecraft attitude control: a survey and some new results. Control Eng. Pract. 13(3), 357–371 (2005)
Lovera, M., Astolfi, A.: Spacecraft attitude control using magnetic actuators. Automatica 40(8), 1405–1414 (2004)
Lovera, M., Astolfi, A.: Global magnetic attitude control of inertially pointing spacecraft. J. Guid. Control Dyn. 28(5), 1065–1067 (2005)
Celani, F.: Robust three-axis attitude stabilization for inertial pointing spacecraft using magnetorquers. Acta Astronaut. 107, 87–96 (2015)
Conn, A., Vicente, L.: Bilevel derivative-free optimization and its application to robust optimization. Optim. Methods Softw. 27, 561–577 (2012)
Ciccazzo, A., Latorre, V., Liuzzi, G., Lucidi, S., Rinaldi, F.: Derivative-free robust optimization for circuit design. J. Optim. Theory Appl. 164, 842–861 (2015)
Bruni, C., Bruni, R., De Santis, A., Iacoviello, D., Koch, G.: Global Optimal Image Reconstruction from Blurred Noisy Data by a Bayesian Approach. J. Optim. Theory Appl. 115(1), 67–96 (2002)
Jones, D.R.: DIRECT global optimization. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 725–735. Springer, Berlin (2009)
Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993)
Liuzzi, G., Lucidi, S., Piccialli, V.: A DIRECT-based approach exploiting local minimizations for the solution of large-scale global optimization problems. Comput. Optim. Appl. 45, 353–375 (2010)
Lucidi, S., Sciandrone, M.: A derivative-free algorithm for bound constrained optimization. Comput. Optim. Appl. 21, 119–142 (2002)
Kolda, T.G., Lewis, R.M., Torczon, V.: Optimization by direct search: new perspectives on some classical and modern methods. SIAM Rev. 45(3), 385–482 (2003)
Wie, B.: Space Vehicle Dynamics and Control. American Institute of Aeronautics and Astronautics, Reston (2008)
Wertz, J.R. (ed.): Spacecraft Attitude Determination and Control. Kluwer, Dordrecht (1978)
Rodriguez-Vazquez, A.L., Martin-Prats, M.A., Bernelli-Zazzera, F.: Full magnetic satellite attitude control using ASRE method. In: 1st IAA Conference on Dynamics and Control of Space Systems (2012)
Ogata, K.: Modern Control Engineering, 4th edn. PrenticeHall, Upper Saddle River (2002)
Simon, E.: A perspective for optimization in systems and control: from LMIs to derivative-free methods. Ph.D. thesis, Université Catholique de Louvain (2012)
Grippo, L., Lampariello, F., Lucidi, S.: Global convergence and stabilization of unconstrained minimization methods without derivatives. J. Optim. Theory Appl. 56(3), 385–406 (1988)
Acknowledgements
The authors are grateful to Prof. Stefano Lucidi for several essential discussions and suggestions and to Dr. Dennis G. Lucarelli for his help.
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Communicated by Firdaus E. Udwadia.
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Bruni, R., Celani, F. A Robust Optimization Approach for Magnetic Spacecraft Attitude Stabilization. J Optim Theory Appl 173, 994–1012 (2017). https://doi.org/10.1007/s10957-016-1035-6
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DOI: https://doi.org/10.1007/s10957-016-1035-6
Keywords
- Derivative-free optimization
- Spacecraft attitude control
- Robust optimization
- Min–max problems
- Magnetic actuators