Robust control design through the attractive ellipsoid technique for a class of linear stochastic models with multiplicative and additive noises

This paper concerns the robust ‘practical’ stabilization for a class of linear controlled stochastic differential equations subject to both multiplicative and additive stochastic noises. Sufficient conditions of the stabilization are provided in two senses. In the first sense, it is proven that almost all trajectories of the stochastic model converge in a ‘mean-square sense’ to a bounded zone located in an ellipsoidal set, while the second one ensures the convergence to a zero zone in probability one. The considered control law is a linear state feedback. The stabilization problem is converted into the corresponding attractive averaged ellipsoid ‘minimization’ under some constraints of bilinear matrix inequalities (BMIs) type. Some variables permit to represent the BMIs problem in terms of linear matrix inequalities (LMIs) problem, which are resolved in a straight manner, using the conventional LMI-MATLAB toolbox. Finally, the numerical solutions of a benchmark example and a practical example are presented to show the efficiency of the proposed methodology.