ROBUST OUTPUT STABILIZATION FOR A CLASS OF NONLINEAR UNCERTAIN STOCHASTIC SYSTEMS UNDER MULTIPLICATIVE AND ADDITIVE NOISES: THE ATTRACTIVE ELLIPSOID METHOD

This work concerns the robust stabilization of a class of "Quasi-Lipschitz" nonlinear uncertain systems governed by stochastic Differential Equations (SDE) subject to both multiplicative and additive stochastic noises modeled by a vector Brownian motion. The state-vector is admitted to be non-completely available, and be estimated by a Luenberger-type filter. The stabilization around the origin is realized by a linear feedback proportional to the current state-estimates. First, the class of feedback matrices and filter matrix-gains, providing the boundedness of the stochastic trajectories with probability one in a vicinity of the origin, is specified. Then a corresponding ellipsoid, containing these trajectories, is found. Its "size" (the trace of the ellipsoid matrix) is derived as a function of the applied gain matrices. To make this ellipsoid "as small as possible" the corresponding constrained optimization problem is suggested to be solved. These constraints are given by a system of Matrix Inequalities (MI's) which under a specific change of variables may be converted into a conventional system of Bilinear Matrix Inequalities (BMI's). The last may be resolved by the standard MATLAB toolboxes such as "penbmiTL, Tomlab toolbox". Finally, a numerical example, containing the arctangent-type nonlinearities, is presented to illustrate the effectiveness of the suggested methodology