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**Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density.**
*(English)*
Zbl 1216.93057

Summary: In the present paper, an innovative procedure for designing the feedback control of Multi-Degree-Of-Freedom (MDOF) nonlinear stochastic systems to target a specified Stationary Probability Density Function (SPDF) is proposed based on the technique for obtaining the exact stationary solutions of the dissipated Hamiltonian systems. First, the control problem is formulated as a controlled, dissipated Hamiltonian system together with a target SPDF. Then the controlled forces are split into a conservative part and a dissipative part. The conservative control forces are designed to make the controlled system and the target SPDF have the same Hamiltonian structure (mainly the integrability and resonance). The dissipative control forces are determined so that the target SPDF is the exact stationary solution of the controlled system. Five cases, i.e., non-integrable Hamiltonian systems, integrable and non-resonant Hamiltonian systems, integrable and resonant Hamiltonian systems, partially integrable and non-resonant Hamiltonian systems, and partially integrable and resonant Hamiltonian systems, are treated respectively. A method for proving that the transient solutions of the controlled systems approach the target SPDF as \(t\rightarrow \infty \) is introduced. Finally, an example is given to illustrate the efficacy of the proposed design procedure.

### Keywords:

nonlinear stochastic systems; stationary probability density function; feedback control; exact stationary solution
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\textit{C. X. Zhu} and \textit{W. Q. Zhu}, Automatica 47, No. 3, 539--544 (2011; Zbl 1216.93057)

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### References:

[1] | Åström, K.J., Introduction to stochastic control theory, (1970), Academic Press New York, London |

[2] | Åström, K.J.; Wittenmark, B., Self-tuning controllers based on pole-zero placement, IEE Proceedings, 127, 3, 120-130, (1980), Pt.D |

[3] | Caughey, T.K., Nonlinear theory of random vibrations, Advances in applied mechanics, 11, 209-253, (1971) |

[4] | Crespo, L.G.; Sun, J.Q., Non-linear stochastic control via stationary response design, Probabilistic engineering mechanics, 18, 79-86, (2003) |

[5] | Forbes, M.G., Forbes, J.F., & Guay, M. (2003a). Regulatory control design for stochastic processes: shaping the probability density function. In Proceedings of the American Control Conference. Vol. 5. Denver (pp. 3998-4003). |

[6] | Forbes, M. G., Forbes, J. F., & Guay, M. (2003b). Control design for discretetime stochastic nonlinear processes with a nonquadratic performance objective. In Proceedings of the 42nd IEEE Conference on Decision and Control. Vol. 4. Maui, Hawaii, USA (pp. 4243-4248). |

[7] | Forbes, M.G.; Guay, M.; Forbes, J.F., Control design for first-order processes: shaping the probability density of the process state, Journal of process control, 14, 399-410, (2004) |

[8] | Forbes, M. G., Guay, M., & Forbes, J. F. (2004b). Probabilistic control design for continuous-time stochastic nonlinear systems - A PDF-shaping approach. In International Symposium on Intelligent Control, IEEE. |

[9] | Goodwin, G.C.; Sin, K.S., Adaptive filtering, prediction and control, (1984), Englewood Cliffs New Jersey, Prentice Hall · Zbl 0653.93001 |

[10] | Guo, L.; Wang, H., PID controller design for output PDFs of stochastic systems using linear matrix inequalities, IEEE transactions on systems, man and cybernetics-part B, 35, 65-71, (2005) |

[11] | Guo, L.; Wang, H.; Wang, A.P., Optimal probability density function control for NARMAX stochastic systems, Automatica, 44, 7, 1904-1911, (2008) · Zbl 1149.93350 |

[12] | Karny, M., Towards fully probabilistic control design, Automatica, 32, 1719-1722, (1996) · Zbl 0868.93022 |

[13] | Lin, Y.K.; Cai, G.Q., Probability structural dynamics: advanced theory and applications, (1995), MeGraw-Hill, Inc New York |

[14] | Lu, J.B.; Skelton, R.R., Covariance control using closed-loop modelling for structures, Earthquake engineering & structural dynamics, 27, 1367-1383, (1998) |

[15] | Skelton, R.E.; Iwasaki, T.; Grigoriadis, K.M., A unified algebraic approach to linear control design, (1998), Taylor & Francis Bristol, PA |

[16] | Wang, H., Control of conditional output probability density functions for general nonlinear and non-Gaussian dynamic stochastic systems, IEE Proceedings-control theory and applications, 150, 55-60, (2003) |

[17] | Wojtkiewicz, S.F.; Bergman, L.A., Moment specification algorithm for control of nonlinear systems driven by Gaussian white noise, Nonlinear dynamics, 24, 1, 17-30, (2001) · Zbl 0992.74053 |

[18] | Yang, Y.; Guo, L.; Wang, H., Constrained PI tracking control for output probability distributions based on two-step neural networks, IEEE transactions on circuits systems part I, 56, 7, 1416-1426, (2009) |

[19] | Zhu, W.Q., Nonlinear stochastic dynamics and control in Hamiltonian formulation, ASME applied mechanics reviews, 59, 230-248, (2006) |

[20] | Zhu, W.Q.; Cai, G.Q.; Lin, Y.K., On exact stationary solutions of stochastically perturbed Hamiltonian systems, Probabilistic engineering mechanics, 5, 84-87, (1990) |

[21] | Zhu, W.Q.; Yang, Y.Q., Exactly stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems, Journal of applied mechanics-transactions of ASME, 63, 493-500, (1996) · Zbl 0876.58012 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.