State-feedback ℋ_∞ control for stochastic time-delay nonlinear systems with state and disturbance-dependent noise

Abstract

This article focuses on the state-feedback ℋ_∞ control problem for the stochastic nonlinear systems with state and disturbance-dependent noise and time-varying state delays. Based on the maxmin optimisation approach, both the delay-independent and the delay-dependent Hamilton–Jacobi-inequalities (HJIs) are developed for synthesising the state-feedback ℋ_∞ controller for a general type of stochastic nonlinear systems. It is shown that the resulting control system achieves stochastic stability in probability and the prescribed disturbance attenuation level. For a class of stochastic affine nonlinear systems, the delay-independent as well as delay-dependent matrix-valued inequalities are proposed; the resulting control system satisfies global asymptotic stability in the mean-square sense and the required disturbance attenuation level. By modelling the nonlinearities as uncertainties in corresponding stochastic time-delay systems, the sufficient conditions in terms of a linear matrix inequality (LMI) and a bilinear matrix inequality (BMI) are derived to facilitate the design of the state-feedback ℋ_∞ controller. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed methods.

Keywords:

• stochastic nonlinear system
• state-feedback ℋ_∞ control
• Hamilton–Jacobi-inequality
• time-varying delays
• Lyapunov–Krasovskii functional
• delay-independent conditions
• delay-dependent conditions

Acknowledgements

This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canada Foundation for Innovation (CFI).