Probabilistic tracking control of dissipated Hamiltonian systems excited by Gaussian white noises

Abstract

This paper devotes to a feedback control strategy for nonlinear stochastic dynamical system to track a prespecified stationary response probability density. The system description and control design are conducted in Hamiltonian framework, and the excitations are confined to Gaussian white noises. The control design consists of several successive steps: firstly, separating the control into conservative and dissipative components by physical intuition, and expanding two components as polynomials; secondly, deriving the low-dimensional averaged equations of controlled Hamiltonian by stochastic averaging, and obtaining the stationary probability density of controlled responses by solving the associated Fokker-Planck-Kolmogorov (FPK) equation; thirdly and finally, determining the polynomial coefficients by minimising the performance index which balances the tracking performance and control cost. Two examples, i.e. Duffing oscillator and frictional system are adopted to illustrate the application and efficacy of this control strategy to track Gaussian and non-Gaussian response probability density.

KEYWORDS:

• Probabilistic tracking control
• nonlinear stochastic dynamical system
• Hamiltonian framework
• stochastic averaging
• frictional system

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This study was supported by the National Natural Science Foundation of China [grant numbers 11872328, 11532011, and 11621062].

Notes on contributors

Ying Yang

Ying Yang: Ph.D. Candidate, Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Dept. of Engineering Mechanics, Zhejiang Univ., Hangzhou 310027, China.

Yong Wang

Yong Wang: Associate Professor, Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Dept. of Engineering Mechanics, Zhejiang Univ., Hangzhou 310027, China (corresponding author). Email: [email protected].

Zhilong Huang

Zhilong Huang: Professor, Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Dept. of Engineering Mechanics, Zhejiang Univ., Hangzhou 310027, China.