Abstract
A bounded optimal control strategy for strongly non-linear systems under non-white wide-band random excitation with actuator saturation is proposed. First, the stochastic averaging method is introduced for controlled strongly non-linear systems under wide-band random excitation using generalized harmonic functions. Then, the dynamical programming equation for the saturated control problem is formulated from the partially averaged Itô equation based on the dynamical programming principle. The optimal control consisting of the unbounded optimal control and the bounded bang-bang control is determined by solving the dynamical programming equation. Finally, the response of the optimally controlled system is predicted by solving the reduced Fokker-Planck-Kolmogorov (FPK) equation associated with the completed averaged Itô equation. An example is given to illustrate the proposed control strategy. Numerical results show that the proposed control strategy has high control effectiveness and efficiency and the chattering is reduced significantly comparing with the bang-bang control strategy.
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References
Bernstein, D.S., Nonquadratic cost and nonlinear feedback control. International Journal of Robust and Nonlinear Control, 1993, 3: 211–229.
Yang, J.N., Li, Z. and Vongchavalitkul, S., A generalization of optimal control theory: linear and nonlinear control. Journal of Engineering Mechanics—American Society of Civil Engineers, 1994, 120: 266–283.
Dimentberg, M.F., Iourtchenko, D.V. and Bratus, A.S., Optimal bounded control of steady-state random vibrations. Probabilistic Engineering Mechanics, 2000, 15: 381–386.
Zhu, W.Q. and Ying, Z.G., Optimal nonlinear feedback control of quasi-Hamiltonian systems. Science in China, Series A, 1999, 42: 1213–1219.
Zhu, W.Q., Ying, Z.G. and Soong, T.T., An optimal nonlinear feedback control strategy for randomly excited structural systems. Nonlinear Dynamics, 2001, 24: 31–51.
Zhu, W.Q. and Yang, Y.Q., Stochastic averaging of quasi non-integrable Hamiltonian systems. Journal of Applied Mechanics, Transactions ASME, 1997, 64: 157–164.
Zhu, W.Q., Huang, Z.L. and Yang, Y.Q., Stochastic averaging of quasi integrable Hamiltonian systems. Journal of Applied Mechanics, Transactions ASME, 1997, 64: 975–984.
Zhu, W.Q., Huang Z.L. and Suzuki, Y., Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems. Int J Nonin Mech, 2002, 37: 419–437.
Fleming, W.H. and Rishel, R.W., Deterministic and Stochastic Optimal Control. New York: Springer, 1975.
Flemin, W.H. and Soner, H.M., Controlled Markov Processes and Viscosity Solutions. New York: Springer, 1992.
Yong, J.M. and Zhou, X.Y., Stochastic Control, Hamiltonian Systems and HJB Equations. New York: Springer, 1999.
Deng, M.L., Hong, M.C. and Zhu, W.Q., Stochastic optimal control for the response of quasi non-integrable Hamiltonian systems. Acta Mechanica Solida Sinica, 2003, 16(4): 313–320.
Zhu, W.Q., Feedback stochastic stabilization of quasi non-integrable Hamiltonian systems by using Lyapunov exponent. Nonlinear Dynamics, 2004, 36: 455–470.
Zhu, W.Q., Deng, M.L. and Huang, Z.L., First-passage failure of quasi-integrable Hamiltonian systems. Journal of Applied Mechanics, Transactions ASME, 2002, 69: 274–282.
Wei Li, Wei Xu, Junfeng Zhao and Shaojuan Ma, Stochastic optimal control of first-passage failure for coupled Duffing-van der Pol system under Gaussian white noise excitations. Chaos, Solitons & Fractals, 2005, 25: 1221–1228.
Crespo, L.G. and Sun, J.Q., Solution of fixed final state optimal control problems via simple cell mapping. Nonlinear Dynamics, 2000, 23: 391–403.
Crespo, L.G. and Sun, J.Q., Stochastic optimal control of nonlinear systems via short-time Gaussian approximation and cell mapping. Nonlinear Dynamics, 2002, 28: 323–342.
Crespo, L.G. and Sun, J.Q., Stochastic optimal control via Bellman’s principle. Automatica, 2003, 39: 2109–2114.
Bratus A., Dimentberg, M.F., Iourtchenko, D. and Noori, M., Hybrid solution method for dynamical programming equations for MDOF stochastic systems. Dynamics & Control, 2000, 10: 107–116.
Dimentberg, M.F. and Bratus, A.S., Bound parametric control of random vibrations. Proceedings: Mathematical, Physical and Engineering Sciences, 2000, 456: 2351–2363.
Dimentberg, M.F., Iourtchenko, D.V. and Bratus, A.S., Optimal bound control of steady-state random vibrations. Probabilitic Engineering Mechanics, 2000, 15: 381–386.
Deng, M.L., Hong, M.C. and Zhu, W.Q., Control for minimizing the response of non-linear system under wide-band excitation. Journal of Vibration Engineering, 2004, 17: 1–6 (in Chinese).
Zhu, W.Q., Huang, Z.L. and Suzuki, Y., Response and stability of strongly non-linear oscillators under wideband random excitation. International Journal of Non-Linear Mechanics, 2001, 36: 1235–1250.
Blankenship, G. and Papanicolaou, G.C., Stability and control of stochastic systems with wide-band noise disturbances. SIAM Journal on Applied Mathematics, 1978, 34: 437–476.
Zhu, W.Q., Nonlinear Stochastic Dynamics and Control: A Framework of Hamiltonian Theory. Beijing: Science Press, 2003 (in Chinese).
Oseledec, V.I., A multiplicative ergodic theorem: Lyapunov characteristic number for dynamical systems. Transactions of Moscow Mathematics Society, 1968, 19: 197–231.
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Project supported by the National Natural Science Foundation of China (Nos. 10332030 and 10772159) and Research Fund for Doctoral Program of Higher Education of China (No. 20060335125).
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Feng, C., Zhu, W. Stochastic Optimal Control of Strongly Nonlinear Systems under Wide-Band Random Excitation with Actuator Saturation. Acta Mech. Solida Sin. 21, 116–126 (2008). https://doi.org/10.1007/s10338-008-0815-4
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DOI: https://doi.org/10.1007/s10338-008-0815-4