Abstract
A minimax optimal control strategy for quasi-Hamiltonian systems with bounded parametric and/or external disturbances is proposed based on the stochastic averaging method and stochastic differential game. To conduct the system energy control, the partially averaged Itô stochastic differential equations for the energy processes are first derived by using the stochastic averaging method for quasi-Hamiltonian systems. Combining the above equations with an appropriate performance index, the proposed strategy is searching for an optimal worst-case controller by solving a stochastic differential game problem. The worst-case disturbances and the optimal controls are obtained by solving a Hamilton-Jacobi-Isaacs (HJI) equation. Numerical results for a controlled and stochastically excited Duffing oscillator with uncertain disturbances exhibit the efficacy of the proposed control strategy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cheng, H., Zhu, W.Q., Ying, Z.G., 2006. Stochastic optimal semi-active control of hysteretic systems by using a magnetorhelogical damper. Smart Material Structures, 15(3):711–718. [doi:10.1088/0964-1726/15/3/006]
Fleming, W.H., Soner, H.M., 1993. Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York.
Petersen, I.R., James, M.R., 1996. Performance analysis and controller synthesis for nonlinear systems with stochastic uncertainty constraints. Automatica, 32(7):959–972. [doi:10.1016/0005-1098(96)00023-4]
Petersen, I.R., Ugrinovskii, V.A., Savkin, A.V., 2000. Robust Control Design using H-infinity Methods. Springer-Verlag, London.
Ugrinovskii, V.A., Petersen, I.R., 1997. Finite Horizon Minimax Optimal Control of Nonlinear Continuous Time Systems with Stochastic Uncertainty. Proceedings of the 36th IEEE Conference on Decision and Control. San Diego, CA.
Ying, Z.G., Zhu, W.Q., 2003. Stochastic optimal control of hysteretic systems under externally and parametrically random excitations. Acta Mechanica Solida Sinica, 16(1):61–66. [doi:10.1007/PL00013119]
Ying, Z.G., Zhu, W.Q., 2006. A stochastically averaged optimal control strategy for quasi-Hamiltonian systems with actuator saturation. Automatica, 42(9):1577–1582. [doi:10.1016/j.automatica.2006.04.023]
Ying, Z.G., Zhu, W.Q., 2008. A stochastic optimal control strategy for partially observable nonlinear quasi-Hamiltonian systems. Journal of Sound and Vibration, 310(1–2):184–196. [doi:10.1016/j.jsv.2007.07.065]
Ying, Z.G., Zhu, W.Q., Soong, T.T., 2003. A stochastic optimal semi-active control strategy for ER/MR dampers. Journal of Sound and Vibration, 259(1):45–62. [doi:10.1006/jsvi.2002.5136]
Yong, J.M., Zhou, X.Y., 1999. Stochastic Controls, Hamiltonian Systems and HJB Equations. Springer-Verlag, New York.
Zhou, K.M., Doyle, J.C., Glover, K., 1996. Robust and Optimal Control. Prentice-Hall, New Jersey.
Zhu, W.Q., 2004. Feedback stabilization of quasi non-integrable Hamiltonian systems by using Lyapunov exponent. Nonlinear Dynamics, 36(2–4):455–470. [doi:10.1023/B: NODY.0000045517.37421.c9]
Zhu, W.Q., Yang, Y.Q., 1997. Stochastic averaging of quasinonintegrable-Hamiltonian systems. ASME Journal of Applied Mechanics, 64:157–164.
Zhu, W.Q., Ying, Z.G., 1999. Optimal nonlinear feedback control of quasi-Hamiltonian systems. Science in China Series A: Mathematics, 42(11):1213–1219. [doi:10.1007/BF02875989]
Zhu, W.Q., Ying, Z.G., 2002. Nonlinear stochastic optimal control of partially observable linear structures. Engineering Structures, 24(3):333–342. [doi:10.1016/S0141-0296(01)00099-2]
Zhu, W.Q., Huang, Z.L., 2003. Feedback stabilization of quasi integrable Hamiltonian systems. ASME Journal of Applied Mechanics, 70:129–136.
Zhu, W.Q., Deng, M.L., 2004. Optimal bounded control for minimizing the response of quasi integrable Hamiltonian systems. International Journal of Non-Linear Mechanics, 39(9):1535–1546. [doi:10.1016/j.ijnonlinmec.2004.02.014]
Zhu, W.Q., Huang, Z.L., Yang, Y.Q., 1997. Stochastic averaging of quasi-integrable Hamiltonian systems. ASME Journal of Applied Mechanics, 64:975–984.
Zhu, W.Q., Ying, Z.G., Ni, Y.Q., Ko, J.M., 2000. Optimal nonlinear stochastic control of hysteretic systems. ASCE Journal of Engineering Mechanics, 126(10):1027–1032. [doi:10.1061/(ASCE)0733-9399(2000)126:10(1027)]
Zhu, W.Q., Ying, Z.G., Soong, T.T., 2001. An optimal nonlinear feedback control strategy for randomly excited structural systems. Nonlinear Dynamics, 24(1):31–51. [doi:10.1023/A:1026527404183]
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (No. 10772159), the Specialized Research Fund for Doctor Program of Higher Education of China (No. 20060335125), and the Natural Science Foundation of Zhejiang Province (No. Y607087), China
Rights and permissions
About this article
Cite this article
Wang, Y., Ying, Zg. & Zhu, Wq. A minimax optimal control strategy for uncertain quasi-Hamiltonian systems. J. Zhejiang Univ. Sci. A 9, 950–954 (2008). https://doi.org/10.1631/jzus.A0820014
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1631/jzus.A0820014
Key words
- Nonlinear quasi-Hamiltonian system
- Minimax optimal control
- Stochastic excitation
- Uncertain disturbance
- Stochastic averaging
- Stochastic differential game