Ellipsoidal approximations of the minimal robust positively invariant set

An optimization algorithm is presented in this paper for the minimal robust positively invariant (mRPI) set approximations via sums-of-squares (SOS) optimization. The mRPI set is an effective tool for robust analysis of uncertain systems under bounded disturbances. The approximation of the mRPI set is always characterized by a polyhedron computed after finite time iterations. In this paper, an mRPI set is characterized by an ellipsoidal set while bounded parametric uncertainties act on states. The proposed algorithm optimizes the shape matrix of the ellipsoidal set approximation by minimizing the volume of the ellipsoidal set. The algorithm is designed for discrete-time and continuous-time nonlinear systems respectively. The algorithm has the ability to further minimize the mRPI set by optimizing the state-feedback control law. Examples are employed to validate the effectiveness of the proposed algorithms.