Link between parameters of an observable system’s possible states set, measurement equation parameters and state space system dimension

Abstract

The paper considers guaranteed ellipsoidal estimation of a set of possible states of the linear system, using which a multidimensional volume of the ellipsoid approximating intersection of the a priori ellipsoid limiting a set of possible states of the system, and a set of dimensions representing a "hyperlayer" in the same state space is minimized. A theorem on the relationship between the parameters of the a priori ellipsoid, parameters of the measurement equation and dimension of the state space that provides improved estimation of the system state by a minimum criterion of the multidimensional volume of the a posteriori ellipsoid is formulated and proved. On the basis of the theorem, simplification of the estimation algorithm, which excludes a special case — division by zero and taking additional measures for this case, has been proposed. The proposed simplification leads to some deterioration of ellipsoidal estimation according to the accepted minimization criterion in general, but in the limiting case it converges to the optimum estimate. The results are illustrated by an example of estimation of the static system state. The optimum, simplified methods, the method proposed in this paper and the least squares method are compared. The following obtained values are presented: a point estimate and a multiple ellipsoidal estimate — which are the values of semi-axes of the a posteriori ellipsoids.