Decomposition of Systems and Point-to-Point Control

The paper studies the point-to-point steering problems, which consists in determining the program motion that transfers a dynamic system from a given initial state to a given final state. We examined the possibility of reducing this problem to two boundary problems of smaller dimension. The approach is based on the transformation of the system into a decomposable form. In this case, the most general type of transformation is used, where the dependent and independent variables of one system can depend not only on those of the second system, but also on the derivatives of up to some finite order of dependent variables over independent ones. Findings of the research show that the decompositions of control systems under consideration are determined by Lie algebras of vector fields on an infinitedimensional manifold. We obtained the conditions on algebras of vector fields that determine the decomposition of the systems and the decomposition of the point-to-point steering problems. Two analyzed examples demonstrate the possibility of applying the proposed approach to solving specific point-to-point steering problems

References

[1] Murrey R.M., Hauser J., Jadbabie A., Milam M.B., Petit N., Dunbar W.B., Franz R. Online control customization via optimization-based control. In: Software-enabled control: Information technology for dynamical systems. John Wiley & Sons, 2003, pp. 1–21.

[2] Mahadevan C.R., Doyle III F.G. Efficient optimization approaches to nonlinear model predictive control. International Journal of Robust and Nonlinear Control, 2003, vol. 13, pp. 309–329.

[3] Faulwasser T., Hagenmeyer V., Findeisen R. Optimal exact path-following for constrained differentially flat systems. IFAC Proceedings Volumes, 2011, vol. 44, no. 1, pp. 9875–9880.

[4] Faulwasser T., Hagenmeyer V., Findeisen R. Constrained reachability and trajectory generation for flat systems. Automatica, 2014, vol. 50, iss. 4, pp. 1151–1159. DOI: 10.1016/j.automatica.2014.02.011

[5] Bocharov A.V., Verbovetskiy A.M., Vinogradov A.M., et al. Simmetrii i zakony sokhraneniya uravneniy matematicheskoy fiziki [Simmetry and conservation law for the equations of mathematical physics]. Moscow, Faktorial Publ., 2005. 474 p.

[6] Krener A.J. A decomposition theory for differentiable systems. SIAM J. Control Optim., 1977, vol. 15, iss. 5, pp. 813–829. DOI: 10.1137/0315052

[7] Isidori A., Krener A.J., Gori-Giorgi C., Monaco S. Nonlinear decoupling via feedback: A differential-geometric approach. IEEE Transactions on Automatic Control, 1981, vol. 26, no. 2, pp. 331–345. DOI: 10.1109/TAC.1981.1102604

[8] Hirshorn R.M. (A, B)-invariant distributions and disturbance decoupling of nonlinear systems. SIAM J. Control Optim., 1982, vol. 19, iss. 1, pp. 1–19. DOI: 10.1137/0319001

[9] Respondek W. On decomposition of nonlinear control systems. Systems & Control Letters, 1982, vol. 1, iss. 5, pp. 301–308. DOI: 10.1016/S0167-6911(82)80027-3

[10] Nijmeijer H. Feedback decomposition of nonlinear control systems. IEEE Transactions on Automatic Control, 1983, vol. 28, no. 8, pp. 861–862. DOI: 10.1109/TAC.1983.1103330

[11] Nijmeijer H., Van der Schaft A.J. Partial symmetries for nonlinear systems. Math. Systems Theory, 1985, vol. 18, iss. 1, pp. 79–96. DOI: 10.1007/BF01699462`

[12] Fliess M. Cascade decomposition of nonlinear systems, foliations and ideals of transitive Lie algebras. Systems & Control Letters, 1985, vol. 5, iss. 4, pp. 263–265. DOI: 10.1016/0167-6911(85)90019-2

[13] Grizzle G.W., Markus S.I. The structure of nonlinear control systems possessing symmetries. IEEE Transactions on Automatic Control, 1985, vol. 30, no. 3, pp. 248–258. DOI: 10.1109/TAC.1985.1103927

[14] Kanatnikov A.N., Krishchenko A.P. Symmetries and the decomposition of nonlinear systems. Differential Equations, 1994, vol. 30, no. 11, pp. 1735–1745.

[15] Isidori A. Nonlinear control systems. Berlin, Springer, 1995. 549 p.

[16] Fliess M., Levine J., Martin Ph., Rouchon P. A Lie − Backlund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Contr., 1999, vol. 44, no. 5, pp. 922–937. DOI: 10.1109/9.763209

[17] Martin P., Murray R.M., Rouchon P. Flat systems. Plenary Lectures and Minicourses. 4th European Control Conference, 1997, pp. 211–264.

[18] Chetverikov V.N. On the structure of integrable C-fields. Differential Geom. Appl., 1991, vol. 1, iss. 4, pp. 309–325. DOI: 10.1016/0926-2245(91)90011-W

[19] Chetverikov V.N. Higher symmetries and the Brunovskii infinitesimal form of controlled systems. Differential Equations, 2002, vol. 38, iss. 11, pp. 1619–1627. DOI: 10.1023/A:1023645207154

[20] Chetverikov V.N. Dynamically linearizable control systems and coverings. Nauka i obrazovanie: nauchnoe izdanie [Science and Education: Scientific Publication], 2013, no. 9, pp. 251–264 (in Russ.). DOI: 10.7463/0913.0601455 Available at: http://old.technomag.edu.ru/doc/601455.html

[21] Sira-Ramirez H., Castro-Linares R., Liceaga-Castro E. A Liouvillian systems approach for the trajectory planning-based control of helicopter models. Int. J. Robust Nonlinear Control, 2000, vol. 10, iss. 4, pp. 301–320. DOI: 10.1002/(SICI)1099-1239(20000415)10:4<301::AID-RNC474>3.0.CO;2-Q

[22] Belinskaya Yu.S., Chetverikov V.N., Tkachev S.B. Automatic synthesis of the helicopter programmed motion along the horizontal line. Nauka i obrazovanie: nauchnoe izdanie [Science and Education: Scientific Publication], 2013, no. 10, pp. 285–298 (in Russ.).DOI: 10.7463/1013.0660675 Available at: http://technomag.bmstu.ru/doc/660675.html