Optimal Ellipsoid Approximations in Control Theory

Minimum volume ellipsoids containing a given set arise often in control theory observed as a problem that solves differential equation with inputs and outputs. The problem is described by a linear differential inclusions. An ellipsoid is given by a symmetric, positive definite matrix. If a linear differential inclusion is given, then sufficient condition for the system stability is to find a positive definite symmetric matrix such that the quadratic function is decreasesing along every nonzero state trajectory. Specific linear differential inclusions are described, such as the linear time-invariant, Polytopic, norm-bound with additional output that affects the additional input in bounded measure or diagonal norm-bound bounds of input and output functions are given by components. In our work we interpreted stability conditions of above systems in terms of ellipsoid that is invariant to a solution of a differential inclusions.