Ellipsoidal filtering of the state of an infinite-dimensional system. II. Filtering by discrete-continuous observations with a vector measure
[For Part I see Autom. Remote Control 58, No. 7, Pt. 1, 1110-1118 (1997); translation from Avtom. Telemekh. 1997, No. 7, 54-65 (1997; Zbl 0921.34061).] The paper deals with the problem of guaranteed state estimation for an infinite-dimensional system defined in a Hilbert space by the equation \[ dx(t)=(Ax(t)+ b(t))dt, \;x(t_0) =x_0 \] and by the discrete-continuous observations \[ dy_i(t)=C_ix(t)du_i(t)+ \xi _i(t)du_i(t) \;(i=1, \dots , m), \;y(t_0 -) =y_0. \] Here the functions \(u_i(t)\) are of bounded variation and may be treated as observation impulse controls. The uncertainties \(x_0, \xi (t)\) belong to ellipsoids in the corresponding spaces. The ellipsoidal filtering differential equations are derived by limit procedures from finite-dimensional discretizations of the system and it is shown that these differential equations have state discontinuities at their right-hand sides. The author studies both cases for the solution to the filtering equations to be unique (when the Frobenius condition is satisfied) or not. In the latter case as in impulse control systems theory, the set of generalized solutions is introduced with an additional optimization problem for choosing the unique optimal element to be used in state estimation.
scientific article; zbMATH DE number 1263099