Robustness of feedback stabilization of quasi non-integrable Hamiltonian systems with parametric uncertainty

Abstract

The robustness of feedback stabilization for quasi non-integrable Hamiltonian systems with uncertain parameters is investigated. The uncertain parameters are modeled as bounded random variables with λ-probability density function. Based on the independence of uncertain parameters and stochastic excitations, a feedback control to asymptotically stabilize, with probability one, the nominal system (with average parameter values) is firstly obtained by applying the stochastic averaging method, the expression for the Lyapunov exponent of quasi non-integrable Hamiltonian systems and the stochastic dynamical programming principle. Then, the mean and standard deviations of the Lyapunov exponent of the uncertain quasi non-integrable Hamiltonian system are calculated by using the stochastic averaging method and probabilistic analysis. Finally, the robustness of the feedback stabilization for quasi non-integrable Hamiltonian systems with parametric uncertainty is evaluated in terms of the sensitivity of the variation coefficient of the Lyapunov exponent of a controlled system to the variation coefficients of uncertain parameters. An example is worked out to illustrate the robustness of the feedback stabilization.