Stretch-Contract Operator in the Ellipsoidal Approximation of the Minkowski Sum of Convex Sets
Keywords:stretch-contract operator, state space, attainability set, ellipsoidal approximation, linear control system, multidimensional volume of an ellipsoid, sum of positive degrees of ellipsoid semiaxes, criterion for minimizing an ellipsoid
The space expansion-contraction operator was originally developed to solve mathematical programming problems. However, it can be successfully applied to solve the problem of ellipsoidal approximation of the information set in the state space analytically specified. In this case, a main property of the operator - space compression is used to minimize the approximating ellipsoid by a multidimensional volume. The paper shows the use of the specified expansion-contraction operator to approximate a set of attainability of the linear control system as an example. The main goal of the paper is to give analytical and geometric representations of the specified operator in order to show its action in the approximation problem. For this purpose, the paper shows an analytical derivation of the operator and a geometric illustration of each parameter of the operator. The results of minimum approximation modeling by this operator compared with other known solutions have been also presented. The simulation results are given both numerically and graphically. Based on the results of comparison, conclusions are made and recommendations are given in the use of ellipsoidal approximation of information sets according to different criteria for minimizing the approximating ellipsoid. Typical examples of ellipsoidal approximation, which show when it is expedient to use the proposed of expansion-contraction operator, have been given.
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