519.6 Ellipsoidal approximation of information sets uses the intersection of some initial ellipsoid with a half-space or a layer bounded by parallel hyperplanes. The ideas of ellipsoidal approximation have found many uses in minmax estimation and control problems [1][2][3][4][5][6], and also in mathematical programming [7, 8]. In the first category, the ellipsoidal information sets contain the unknown states or the parameters of the controlled system; in the second category, they contain the sought minimum points of the functionals. A comprehensive list of studies using ellipsoidal approximation for minmax estimation is given in [3, 4].The present article is a direct continuation and extension of [9]. Among previous publications on similar topics, it is closest to [10] by problem formulation. It differs, however, by its new estimation algorithms that are easier to use and by a new proof of their convergence based on the direct Lyapunov method.
STATEMENT OF THE PROBLEMConsider the following equations of motion for the system and its observer:
y(t) =hr(t) x(t) +~(t).(2)Here t is continuous time, t E [0, oo); x(t) is the state vector x E R n, R n is the n-dimensional real Euclidean space; u(t) is the control vector u E Rm; y(t) is the observed signal, which is the result of observing the system output h r (t)x(t) mixed with additive noise ~(t), y(t) E R1; A(t), B(t), and h(t) are appropriately dimensioned matrices and vector with bounded continuous elements. We assume that the noise is continuous and its absolute value is bounded by a given constant I ~(t) I < c v t >__ 0. The superscript T denotes the transpose of vectors and matrices.Let K(t) be the fundamental matrix of the homogeneous system 2 = A(t)x, corresponding to system (1), and K(t, to) its transition matrix (the Cauchy matrix, the matricant) [11][12][13]. Here K(t, to) = K(t) K-1(to), where K(t o) = K(t)[t=to and the matrices K(0 and K(t, to) satisfy the matrix equationsat -A (O K (O' at -A (O g (t, to).Following [14], we say that system (1), (2) is everywhere observable with observability index T, 0 < T < oo, if the matrix t Q (t) = f K T(0, t) h (0) h T(0) K (O, t)dO t-T
