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TECHNICAL PAPERS

Covariance Control of Nonlinear Dynamic Systems via Exact Stationary Probability Density Function

[+] Author and Article Information
O. Elbeyli, J. Q. Sun

Department of Mechanical Engineering, University of Delaware, Newark, DE 19716

J. Vib. Acoust 126(1), 71-76 (Feb 26, 2004) (6 pages) doi:10.1115/1.1640355 History: Received June 01, 2002; Revised June 01, 2003; Online February 26, 2004
Copyright © 2004 by ASME
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References

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Iwasaki,  T., Skelton,  R. E., and Corless,  M., 1998, “Recursive Construction Algorithm for Covariance Control,” IEEE Trans. Autom. Control, 43, pp. 268–272.
Skelton, R. E., Iwasaki, T., and Grigoriadis, K. M., 1998, A Unified Algebraic Approach to Linear Control Design, Taylor and Francis, Bristol, Pennsylvania.
Chung,  H. Y., and Chang,  W. J., 1994, “Extension of the Covariance Control Principle to Nonlinear Stochastic Systems,” IEE Proc.-D: Control Theory Appl., 141, pp. 93–98.
Chung,  H.-Y., and Chang,  W.-J., 1994, “Constrained Variance Design Using Covariance Control With Observed-State Feedback for Bilinear Stochastic Continuous Systems,” J. Chin. Inst. Eng., 17, pp. 113–119.
Chang,  K.-Y., Wang,  W.-J., and Chang,  W.-J., 1997, “Covariance Control for Stochastic Multivariable Systems With Hysteresis Nonlinearity,” Int. J. Syst. Sci., 28, pp. 731–736.
Wojtkiewicz,  S. F., and Bergman,  L. A., 2001, “Moment Specification Algorithm for Control of Nonlinear Systems Driven by Gaussian White Noise,” Nonlinear Dyn., 24, pp. 17–30.
Sun,  J. Q., and Xu,  Q., 1998, “Response Variance Reduction of a Nonlinear Mechanical System via Sliding Mode Control,” ASME J. Vibr. Acoust., 120, pp. 801–805.
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Figures

Grahic Jump Location
The response of the controlled Duffing system. (a) The fourth order moment of the displacement m40. (b) The control effort E[u2]. Thick solid line represents the uncontrolled response, (–––) case 7 in Table 1. The following lines correspond to the cases in Table 2: cases 1 (⋯), 2 (–⋅–), 3 (—), and 4 ([[dashed_line]]).
Grahic Jump Location
Time history of the second order moments of the parametrically excited system. (a) The displacement. (b) The velocity. Thick solid line represents case 1, the uncontrolled response, in Table 3. Cases 2 (–⋅–), 3 (⋯), 4 (–), 5 ([[dashed_line]]), and 6 ([[dotted_line]]).
Grahic Jump Location
The response of the controlled parametrically excited system. (a) The fourth order moment of the displacement m40. (b) The control effort E[u2]. Thick solid line represents case 1, the uncontrolled response, in Table 3. Cases 2 (–⋅–), 3 (⋯), 4 (–), 5 ([[dashed_line]]), and 6 ([[dotted_line]]).
Grahic Jump Location
Effect of the nonlinear feedback gain k3 on the response decay. (a) The second order moment of the displacement m20. (b) The second order moment of the velocity m02, with control gains k1=0.1, and k2=0.04. All other parameters are the same as the previous example. (⋯) k3=0.8, (–––) k3=1.0, (–⋅–) k3=1.2, (–) k3=1.4, ([[dashed_line]]) k3=1.6, and ([[dotted_line]]) k3=1.8.
Grahic Jump Location
Time history of various expected control input and system response for the parametrically excited system (refer to Table 4) with control gains, k1=−5.02335e−6,k2=0.412923,k3=−0.418092. (a) ([[dotted_line]]) m20, (–) m02, (–⋅–) m40, (–––) m42, (b) (–⋅–) E[x14x22/(ω*x12+x22)2], (–––) E[x14x22*x12+x22], (——) E[u2].
Grahic Jump Location
Cross-sections of the steady state PDF p(x1,x2) of the exact solution for (–⋅–) x2=0, (–) x2=0.5, ([[dashed_line]]) x2=1.0 and (⋯) the approximate solution by the maximum entropy principle. All the curves are scaled so that the area under each is unity.
Grahic Jump Location
Effect of the nonlinear feedback gain k3 on the response decay. (a) The fourth order moment of the displacement m40. (b) The control effort E[u2]. All the parameters are the same as in Fig. 5. (⋯) k3=0.8, (–––) k3=1.0, (–⋅–) k3=1.2, (–) k3=1.4, ([[dashed_line]]) k3=1.6, and ([[dotted_line]]) k3=1.8.
Grahic Jump Location
Time history of the second order moments of the Duffing system. (a) displacement. (b) velocity. Thick solid line represents the uncontrolled response. (–––) case 7 in Table 1. The following lines correspond to the cases in Table 2: cases 1 (⋯), 2 ([[dot_dash_line]]), 3 (–), and 4 ([[dashed_line]]).

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