A class of robust bounded controllers tracking a nonlinear discrete-time stochastic system: Attractive ellipsoid technique application

This paper deals with designing of robust tracking controllers for a class of nonlinear uncertain discrete-time stochastic systems which are affine with respect to a control action on a bounded energy level. The considered nonlinear dynamics is admitted to be a priori unknown but belonging to the class of the so-called quasi-Lipschitz vector-field. The current states may be unavailable on-line, but the corresponding output vector is assumed to be measured during the process. Both the state dynamics and the output measurements are disturbed by external additive noises which are also supposed to be immeasurable. In this situation any suitable controller can only provide the boundedness of the tracking-error trajectories within a bounded zone with probability one. In this paper we suggest designing of both the control and observer “optimal” gain-matrices minimizing the “size” of the attractive ellipsoid containing all tracking-error trajectories in the vicinity of the origin. It is shown that this design problem of an output bounded control may be converted into the corresponding attractive ellipsoid “minimization” under some constraints of BMI's (bilinear matrix inequalities) type. The application of an adequate coordinate changing transforms these BMI's into a set of LMI's (linear matrix inequalities) that permits to use directly the standard MATLAB-toolbox. Two illustrative examples are considered: the first one concerns a two state-single output stochastic model, and the second one deals with a discrete-time model of electric-magnetic-tape-drive containing the four states-positions and two measured outputs.