Математика и математическое моделирование. 2015; : 27-38
О стабилизации аффинных систем второго порядка при наличии возмущений
Аннотация
Список литературы
1. Krstic M., Deng H. Stabilization of Nonlinear Uncertain Systems. London: Springer-Verlag, 1998. 193 p
2. Sontag E.D. A ‘universal’ construction of Artstein's theorem on nonlinear stabilization // Systems & Control Letters. 1989. Vol. 13, no. 2. P. 117-123. DOI: 10.1016/0167-6911(89)90028-5
3. Sontag E.D., Wang Y. On characterizations of the input-to-state stability property // Systems & Control Letters. 1995. Vol. 24, no. 5. P. 351-359. DOI: 10.1016/0167-6911(94)00050-6
4. Дашковский С.Н., Ефимов Д.В., Сонтаг Э.Д. Устойчивость от входа к состоянию и смежные свойства систем // Автоматика и телемеханика. 2011. № 8. С. 3-40
5. Sontag E.D. Input to State Stability: Basic Concepts and Results // Nonlinear and optimal control theory / ed. by P. Nistri, G. Stefani. Springer, 2008. P. 163-220 (Ser. Lecture Notes in Mathematics; vol. 1932.). DOI: 10.1007/978-3-540-77653-6_3
6. Sontag E.D. Further facts about input to state stabilization // IEEE Transactions on Automatic Control. 1990. Vol. 35, no. 4. P. 473-476. DOI: 10.1109/9.52307
7. Freeman R.A., Kokotovic P.V. Robust nonlinear control design. Boston: Birkhauser, 1996. 257 p
8. Краснощёченко В.И., Крищенко А.П. Нелинейные системы: геометрические методы анализа и синтеза. М.: Изд-во МГТУ им. Н.Э. Баумана, 2005. 520 с
9. Isidori A. Nonlinear Control Systems. 3rd ed. London: Springer, 1995. 550 p. DOI: 10.1007/978-1-84628-615-5
10. Khalil H.K. Nonlinear systems. 2nd ed. New York: Prentice-Hall, 1996. 750 p
Mathematics and Mathematical Modeling. 2015; : 27-38
On the Problem of 2D Affine Systems Input to State Stabilization
Abstract
Various statements and a variety of solutions to the problem of input-to-state stabilization of dynamic systems with disturbances are known. Methods based on the use of Lyapunov functions play an important role with regard to non-linear systems. When using these methods, the problem of finding an appropriate Lyapunov function arises. The Lyapunov functions redesign method provides a Lyapunov function for a certain subclass of affine systems with disturbances using transformation of the corresponding affine system without disturbances to the equivalent regular canonical form. The desired Lyapunov function is constructed as a quadratic form of the canonical variables. Further, the found Lyapunov function can be used to construct the input-to-state asymptotically stabilizing control. The limits of applicability of this approach remain unclear: in general, constructed on the basis of the transformation to the equivalent canonical form the Lyapunov function for the system without disturbances can both be and not be the Lyapunov function for the affine system with disturbance.
In the paper, we study the possibility of using the described approach to second-order affine systems with scalar control and scalar disturbances for which the corresponding systems without disturbances are equivalent to regular systems of canonical form in the whole state space. We have obtained the easily verifiable conditions for construction of the Lyapunov function on the basis of the regular canonical form where the Lyapunov function for the system with control will be the function for the system with disturbances. Thus, the class of systems which can be stabilized by using the above method is defined. Examples of applications of the obtained conditions with regard to certain classes of second-order affine systems and the results of numerical simulation of the stabilization process of the zero equilibrium point in the presence of various disturbances for the particular two-dimensional system with disturbances are given.
References
1. Krstic M., Deng H. Stabilization of Nonlinear Uncertain Systems. London: Springer-Verlag, 1998. 193 p
2. Sontag E.D. A ‘universal’ construction of Artstein's theorem on nonlinear stabilization // Systems & Control Letters. 1989. Vol. 13, no. 2. P. 117-123. DOI: 10.1016/0167-6911(89)90028-5
3. Sontag E.D., Wang Y. On characterizations of the input-to-state stability property // Systems & Control Letters. 1995. Vol. 24, no. 5. P. 351-359. DOI: 10.1016/0167-6911(94)00050-6
4. Dashkovskii S.N., Efimov D.V., Sontag E.D. Ustoichivost' ot vkhoda k sostoyaniyu i smezhnye svoistva sistem // Avtomatika i telemekhanika. 2011. № 8. S. 3-40
5. Sontag E.D. Input to State Stability: Basic Concepts and Results // Nonlinear and optimal control theory / ed. by P. Nistri, G. Stefani. Springer, 2008. P. 163-220 (Ser. Lecture Notes in Mathematics; vol. 1932.). DOI: 10.1007/978-3-540-77653-6_3
6. Sontag E.D. Further facts about input to state stabilization // IEEE Transactions on Automatic Control. 1990. Vol. 35, no. 4. P. 473-476. DOI: 10.1109/9.52307
7. Freeman R.A., Kokotovic P.V. Robust nonlinear control design. Boston: Birkhauser, 1996. 257 p
8. Krasnoshchechenko V.I., Krishchenko A.P. Nelineinye sistemy: geometricheskie metody analiza i sinteza. M.: Izd-vo MGTU im. N.E. Baumana, 2005. 520 s
9. Isidori A. Nonlinear Control Systems. 3rd ed. London: Springer, 1995. 550 p. DOI: 10.1007/978-1-84628-615-5
10. Khalil H.K. Nonlinear systems. 2nd ed. New York: Prentice-Hall, 1996. 750 p
