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Research Papers

Stationary Response of a Class of Nonlinear Stochastic Systems Undergoing Markovian Jumps

[+] Author and Article Information
Rong-Hua Huan, Zu-guang Ying

Department of Mechanics,
State Key Laboratory of Fluid Power
Transmission and Control,
Zhejiang University,
Hangzhou 310027, China

Wei-qiu Zhu

Department of Mechanics,
State Key Laboratory of Fluid Power
Transmission and Control,
Zhejiang University,
Hangzhou 310027, China
e-mail: wqzhu@zju.edu.cn

Fai Ma

Department of Mechanical Engineering,
University of California,
Berkeley, CA 94720

1Corresponding author.

Manuscript received January 28, 2015; final manuscript received February 28, 2015; published online March 31, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(5), 051008 (May 01, 2015) (6 pages) Paper No: JAM-15-1055; doi: 10.1115/1.4029954 History: Received January 28, 2015; Revised February 28, 2015; Online March 31, 2015

Systems whose specifications change abruptly and statistically, referred to as Markovian-jump systems, are considered in this paper. An approximate method is presented to assess the stationary response of multidegree, nonlinear, Markovian-jump, quasi-nonintegrable Hamiltonian systems subjected to stochastic excitation. Using stochastic averaging, the quasi-nonintegrable Hamiltonian equations are first reduced to a one-dimensional Itô equation governing the energy envelope. The associated Fokker–Planck–Kolmogorov equation is then set up, from which approximate stationary probabilities of the original system are obtained for different jump rules. The validity of this technique is demonstrated by using a nonlinear two-degree oscillator that is stochastically driven and capable of Markovian jumps.

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References

Krasovskii, N. N., and Lidskii, E. A., 1961, “Analytical Design of Controllers in Systems With Random Attributes I,” Autom. Remote Control, 22(9), pp. 1021–1025.
Krasovskii, N. N., and Lidskii, E. A., 1961, “Analytical Design of Controllers in Systems With Random Attributes II,” Autom. Remote Control, 22(10), pp. 1141–1146.
Krasovskii, N. N., and Lidskii, E. A., 1961, “Analytical Design of Controllers in Systems With Random Attributes III,” Autom. Remote Control, 22(11), pp. 1289–1294.
Mariton, M., 1988, “Almost Sure and Moment Stability of Jump Linear Systems,” Syst. Control Lett., 11(5), pp. 393–397. [CrossRef]
Kushner, H., 1967, Stochastic Stability and Control, Academic Press, New York.
Luo, J. W., 2006, “Comparison Principle and Stability of Itô Stochastic Differential Delay Equations With Poisson Jump and Markovian Switching,” Nonlinear Anal., 64(2), pp. 253–262. [CrossRef]
Boukas, E. K., and Liu, Z. K., 2001, “Robust Stability and Stabilizability of Markov Jump Linear Uncertain Systems With Mode Dependent Time Delays,” J. Optim. Theory Appl., 109(3), pp. 587–600. [CrossRef]
Boukas, E. K., and Liu, Z. K., 2002, Deterministic and Stochastic Systems With Time-Delay, Birkhauser, Boston, MA.
de Farias, D. P., Geromel, J. C., do Val, J. B. R., and Costa, O. L. V., 2000, “Output Feedback Control of Markov Jump Linear System in Continuous-Time,” IEEE Trans. Autom. Control, 45(5), pp. 944–949. [CrossRef]
de Souza, C. E., and Fragoso, M. D., 1993, “H∞ Control for Linear Systems With Markovian Jumping Parameters,” Control Theory Adv. Technol., 9, pp. 457–466.
Kats, I. Y., and Martynyuk, A. A., 2002, Stability and Stabilization on Nonlinear Systems With Random Structure, Taylor and Francis, London.
Sworder, D., 1969, “Feedback Control of a Class of Linear Systems With Jump Parameters,” IEEE Trans. Autom. Control, 14(1), pp. 9–14. [CrossRef]
Ghosh, M. K., Arapostathis, A., and Marcus, S. L., 1997, “Ergodic Control of Switching Diffusions,” SIAM J. Control Optim., 35(6), pp. 152–198. [CrossRef]
Fragoso, M. D., and Hemerly, E. M., 1991, “Optimal Control for a Class of Noisy Linear Systems With Markovian Jumping Parameters and Quadratic Cost,” Int. J. Syst. Sci., 22(12), pp. 2553–2561. [CrossRef]
Ji, Y. D., and Chizeck, H. J., 1992, “Jump Linear Quadratic Gaussian Control in Continuous Time,” IEEE Trans. Autom. Control, 37(12), pp. 1884–1892. [CrossRef]
Natache, S. D. A., and Vilma, A. O., 2004, “State Feedback Fuzzy-Model-Based Control for Markovian Jump Nonlinear Systems,” SBA Controle Automacao, 15(3), pp. 279–290.
Zhu, W. Q., 2006, “Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation,” ASME Appl. Mech. Rev., 59(4), pp. 230–248. [CrossRef]
Zhu, W. Q., and Yang, Y. Q., 1997, “Stochastic Averaging of Quasi-Nonintegrable Hamiltonian Systems,” ASME J. Appl. Mech., 64(1), pp. 157–164. [CrossRef]
Huan, R. H., Wu, Y. J., and Zhu, W. Q., 2009, “Stochastic Optimal Bounded Control of MDOF Quasi-Nonintegrable Hamiltonian Systems With Actuator Saturation,” Arch. Appl. Mech., 79(2), pp. 157–168. [CrossRef]
Wonham, W. M., 1970, Random Differential Equations in Control Theory, Academic Press, New York.
Tabor, M., 1989, Chaos and Integrability in Nonlinear Dynamics, Wiley, New York.
Siegel, C. L., and Moser, J. K., 1971, Lecture on Celestial Mechanics, Springer-Verlag, New York.
Itô, K., 1951, “On Stochastic Differential Equations,” Mem. Am. Math. Soc., 4, pp. 289–302.
Khasminskii, R. Z., 1968, “On the Averaging Principle for Itô Stochastic Differential Equations,” Kibernetka, 3, pp. 260–279.
Fang, Y. W., Wu, Y. L., and Wang, H. Q., 2012, Optimal Control Theory of Stochastic Jump System, National Defense Industry Press, Beijing.
Wu, S. T., 2007, The Theory of Stochastic Jump System and Its Application, Science Press, Beijing.

Figures

Grahic Jump Location
Fig. 2

Stationary probability densities p(q1) of the displacement q1 of two-form system (24) with the same transition rates as in Fig. 1

Grahic Jump Location
Fig. 3

Stationary probability densities p(q2) of the displacement q2 of two-form system (24) with the same transition rates as in Fig. 1

Grahic Jump Location
Fig. 4

Sample time history of two-form Markov jump process s(t) with independent jumps

Grahic Jump Location
Fig. 1

Stationary probability densities p(H) of the energy envelope H of two-form system (24) for L = L1, L = L2, L = L3 in Eq. (35), and when the system is fixed at s(t) = 1and s(t) = 2. The lines are obtained by numerical solution of Eqs. (19) and (20), while the dots • are obtained by direct simulation of system (24).

Grahic Jump Location
Fig. 5

Stationary probability densities p(H) of the energy envelope H of three-form system (24) for L = L1, L = L2, L = L3, L = L4 in Eqs. (37) and (38), and when the system is fixed at s(t) = 1, s(t) = 2 and s(t) = 3. The lines are obtained by numerical solution of Eqs. (19) and (20), while the dots • are obtained by direct simulation of system (24).

Grahic Jump Location
Fig. 6

Stationary probability densities p(q1) of the displacement q1 of three-form system (24) with the same transition rates as in Fig. 5

Grahic Jump Location
Fig. 7

Stationary probability densities p(q2) of the displacement q2 of three-form system (24) with the same transition rates as in Fig. 5

Grahic Jump Location
Fig. 8

Sample time history of three-form Markov jump process s(t) with independent jumps

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