Control of stationary probability density of nonlinear systems subject to Poisson white noise excitation

Abstract

An innovative design procedure for controlling a nonlinear system subject to Poisson white noise excitation to target a specified stationary probability density function is proposed based on the approximate generalized Fokker–Planck–Kolmogorov equation of a Poisson-white-noise-excited nonlinear system. First, the technique for deriving the generalized Fokker–Planck–Kolmogorov equation of the Poisson-white-noise-excited nonlinear system is briefly reviewed. Then, the approach for designing the feedback control of nonlinear stochastic systems under Poisson white noise excitations to target a pre-specified stationary probability density function is presented. Finally, an example is given to illustrate the procedure and effectiveness of the proposed control strategy.

Keywords

Feedback control nonlinear system Poisson white noise prey–predator model stationary probability density function

Get full access to this article

View all access options for this article.

References

Crespo

Sun

(2003) Non-linear stochastic control via stationary response design. Probabilistic Engineering Mechanics 18: 79–86.

Forbes

Guay

Forbes

(2004a) Control design for first-order processes: shaping the probability density of the process state. Journal of Process Control 14: 399–410.

Forbes

Guay

Forbes

(2004b) Probabilistic control design for continuous-time stochastic nonlinear systems – a PDF-shaping approach. In: International Symposium on Intelligent Control. IEEE.

Guo

Yin

(2009) Robust PDF control with guaranteed stability for non-linear stochastic systems under modeling errors. IET Control Theory and Applications 3(5): 575–582.

Huan

Zhu

(2009) Stochastic optimal control of quasi integrable Hamiltonian systems subject to actuator saturation. Journal of Vibration and Control 15(1): 85–99.

Kreucher

Kastella

Alfred

OH III

(2005) Multi-target tracking using the joint multi-target probability density. IEEE Transactions on Aerospace and Electronic Systems 41(4): 1396–1414.

Lin

Cai

(1995) Probability Structural Dynamics: Advanced Theory and Applications, New York: McGraw-Hill, Inc.

Lourtchenko

(2000) Stochastic optimal bounded control for a system with the Boltz cost function. Journal of Vibration and Control 6(8): 1195–1204.

Marano

Greco

(2011) Optimization criteria for tuned mass dampers for structural vibration control under stochastic excitation. Journal of Vibration and Control 17(5): 679–688.

10.

Pigeon

Perrier

Srinivasan

(2011) Shaping probability density functions using a switching linear controller. Journal of Process Control 21: 901–908.

11.

Zhu

(2008) Stochastic analysis of a pulse-type prey-predator model. Physical Review E 77: 041911–041911.

12.

Guo

Wang

(2009) Constrained PI tracking control for output probability distributions based on two-step neural networks. IEEE Transactions on Circuits and Systems Part I 56(7): 1416–1426.

13.

Ying

(2010) A minimax stochastic optimal control for bounded-uncertain systems. Journal of Vibration and Control 16(11): 1591–1604.

14.

Zeng

Zhu

(2010) Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations. International Journal of Non-Linear Mechanics 45: 572–586.

15.

Zhu

(2006) Nonlinear stochastic dynamics and control in Hamiltonian formulation. ASME Applied Mechanics Reviews 59: 230–248.

16.

Zhu

(2011) Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density. Automatica 47(3): 539–544.

17.

Zhu

(2012) Design of feedback control of a nonlinear stochastic system for targeting a pre-specified stationary probability distribution. Probabilistic Engineering Mechanics 30: 20–26.