Decomposition of Equations of Nonlinear Affine Control Systems and its Application to the Synthesis of Regulators

The article deals with the decomposition of nonlinear differential equations based on the group-theoretic approach. At the beginning, the decomposition of differential equations of linear systems using a transition matrix of state is presented, and then, based on the theory of continuous groups (Lie groups), the process of decomposition of differential equations of nonlinear systems is shown. The decomposition approach is based on the isomorphism theorem of the space of vector fields and Lie derivatives, which allows us to consider vector fields as differential operators of smooth functions. A formula is derived about the adjoin representation of a Lie group in its Lie algebra, which actually determines the finding of a vector field that characterizes the interaction of two or more vector fields. The Lie algebra of derivatives makes it possible to determine the infinitesimal action of the Lie group, i.e. the linearization of this action is carried out (transformation of the points of the trajectory space of the original system in a small neighborhood). Decomposition allows, as in the linear case, to separate the finding of an action (only locally) of a group of transformations from the transformed points themselves. For linear systems, this separation is global. It is also shown that the decomposition of linear equations is a particular case of the decomposition of nonlinear equations. An algorithm of the method of model predictive control with Gramian weighting using this decomposition is presented. A practical example of decomposition and application of the model predictive control for stabilization of a nonstationary nonlinear system is considered.

1. Chen K. T. Decomposition of Differential Equations, Math. Annalen, 1962, no. 146, pp. 263—278.

2. Krener A. J. A decomposition theory for differentiable systems, SIAM J. Contr. & Opt., 1977, vol. 15, no. 5, pp. 813—829.

3. Respondek W. On decomposition of nonlinear control systems, SIAM Control Lett., 1982, vol. 1, pp. 301—308.

4. Gruzzle J. W., Marcus S. I. The structure of nonlinear systems possessing symmetries, IEEE Trans. Aut. Control, 1985, vol. 30, no. 3, pp. 773—779.

5. Vidyasagar M. Decomposition techniques for largescale systems with nonadditive interactions: stability and stabilizability, IEEE Trans. Aut.Control, 1980, vol. 25, no. 4, pp. 773—779.

6. Krasnoshchechenko V. I. Group-theoretic approach to the synthesis of regulators of nonlinear affine systems on the example of inverted pendulum control, Mekhatronika, Avtomatizatsiya, Upravlenie. 2010, no. 7, pp. 2—10.

7. Poznjak Je. G, Shikin E. V. Differential geometry, Moscow, Editorial URSS, 2003, 408 p. (In Russian).

8. Polishhuk V. V. Sophus Lie, Moscow, Nauka, 1983, 214 p. (In Russian).

9. Kobayashi S., Nomizu K. Foundations of differential geometry, vol. 1, NY, London, Interscience Publ., 1963, 344 p.

10. Trofimov V. V. Introduction in geometry of manifolds with symmetries, Moscow, Publ. Moscow state University, 1989, 359 p. (in Russian).

11. Postnikov M. M. Lie groups and algebras, Moscow, Nauka, 1982, 448 p. (In Russian).

12. Hall B. C. Lie groups, Lie algebras and representations, NY, Springer, 2003, 350 p.

13. Doolin D. F., Martin C. F. Global differential geometry. An introduction for control engineers, NASA Reference publication 1091, 1982, 65 p.

14. Sontag E. D. Mathematical control theory, NY, Springer, 1998, 531 p.

15. Hunt L. R. Controllability of general nonlinear systems, Math. Systems Theory, 1979, no. 12, pp. 361—370.

16. Sussmann H. A sufficient condition for local controllability, SIAM J. Cont.Optim., 1983, vol. 16, no. 5, pp. 790—802.

17. Petrov N. N. On local controllability, Differencial’nye uravneniya, 1976, vol. 12, no. 12, pp. 2214—2222 (In Russian).

18. Krasnoshchechenko V. I. A approach to get gramian controllability of nonlinear affine control systems, Izvestija TulGU. Serija. Vychislitel’naja tehnika, Informacionnye tehnologii. Sistemy upravlenija, vol. 1, 2006, pp. 239—243 (In Russian).

19. Borisovich Ju. G., Beliznyakov N. M., Izrailevich L. A. Introduction in topology, Moscow, Vysshaya shkola, 1980, 296 p. (in Russian).