An approximate solution of the affine-quadratic control problem based on the concept of optimal damping

Veremey, Evgeny I.

International Journal of Applied Mathematics and Computer Science, Tome 31 (2021) no. 1, pp. 5-15.

Résumé

The paper is devoted to a particular case of the nonlinear and nonautonomous control law design problem based on the application of the optimization approach. Close attention is paid to the controlled plants, which are presented by affine-control mathematical models characterized by integral quadratic functionals. The proposed approach to controller design is based on the optimal damping concept firstly developed by V.I. Zubov in the early 1960s. A modern interpretation of this concept allows us to construct effective numerical procedures of control law synthesis initially oriented to practical implementation. The main contribution is the proposition of a new methodology for selecting the functional to be damped. The central idea is to perform parameterization of a set of admissible items for this functional. As a particular case, a new method of this parameterization has been developed, which can be used for constructing an approximate solution to the classical optimization problem. Applicability and effectiveness of the proposed approach are confirmed by a practical numerical example.

Détail

Keywords: feedback, stability, damping control, functional, optimization
Mots-clés : sprzężenie zwrotne, kontrola tłumienia, optymalizacja

Bibliographie

[1] [1] Aguilar-Ibanez, C. and Suarez-Castanon, M.S. (2019). A trajectory planning based controller to regulate an uncertain overhead crane system, International Journal of Applied Mathematics and Computer Science 29(4): 693–702, DOI: 10.2478/amcs-2019-0051.

[2] [2] Balakrishnan, A. (1966). On the controllability of a nonlinear system, Proceedings of the National Academy of Sciences of the USA 55(3): 465–468.

[3] [3] Collado, J., Lozano, R. and Fantoni, I. (2000). Control of convey-crane based on passivity, Proceedings of the American Control Conference, ACC 2000, Chicago, USA, pp. 1260–1264.

[4] [4] Do, K. and Pan, J. (2009). Control of Ships and Underwater Vehicles. Design for Underactuated and Nonlinear Marine Systems, Springer-Verlag, London.

[5] [5] Fantoni, I. and Lozano, R. (2002). Non-linear Control for Underactuated Mechanical Systems, Springer-Verlag, London.

[6] [6] Fossen, T.I. (1994). Guidance and Control of Ocean Vehicles, John Wiley and Sons, New York.

[7] [7] Geering, H.P. (2007). Optimal Control with Engineering Applications, Springer-Verlag, Berlin/Heidelberg.

[8] [8] Hahn, W. and Baartz, A.P. (1967). Stability of Motion, Springer, London.

[9] [9] Khalil, H. (2002). Nonlinear Systems, Prentice-Hall, Englewood Cliffs.

[10] [10] Lewis, F.L., Vrabie, D.L. and Syrmos, V.L. (2012). Optimal Control, John Wiley and Sons, Hoboken.

[11] [11] Lukes, D.L. (1969). Optimal regulation of nonlinear dynamic systems, SIAM Journal on Control and Optimization 7(1): 75–100.

[12] [12] Sepulchre, R., Jankovic, M. and Kokotovic, P. (1997). Constructive Nonlinear Control, Springer, New York.

[13] [13] Slotine, J. and Li, W. (1991). Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs.

[14] [14] Sontag, E.D. (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd Edition, Springer, New York.

[15] [15] Sotnikova, M.V. and Veremey, E.I. (2013). Dynamic positioning based on nonlinear MPC, IFAC Proceedings Volumes 9(1): 31–36.

[16] [16] Veremey, E.I. (2017). Separate filtering correction of observer-based marine positioning control laws, International Journal of Control 90(8): 1561–1575.

[17] [17] Veremey, E.I. (2019). Special spectral approach to solutions of SISO LTI h-optimization problems, International Journal of Automation and Computing 16(1): 112–128.

[18] [18] Veremey, E.I. and Sotnikova, M.V. (2019). Optimization approach to guidance and control of marine vehicles, WIT Transactions on the Built Environment 187(1): 45–56.

[19] [19] Wasilewski, M., Pisarski, D., Konowrocki, R. and Bayer, C.I. (2019). A new efficient adaptive control of torsional vibrations induced by switched nonlinear disturbances, International Journal of Applied Mathematics and Computer Science 29(2): 285–303, DOI: 10.2478/amcs-2019-0021.

[20] [20] Zubov, V.I. (1962). Oscillations in Nonlinear and Controlled Systems, Sudpromgiz, Leningrad, (in Russian).

[21] [21] Zubov, V.I. (1966). Theory of Optimal Control of Ships and Other Moving Objects, Sudpromgiz, Leningrad, (in Russian).

[22] [22] Zubov, V.I. (1978). Théorie de la Commande, Mir, Moscow.