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

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.