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A linear continuous-time system is given whose input and output disturbances and initial conditions are unknown but bounded by known convex sets. These sets, together with the system dynamics and any available observation, determine at any time a set of all possible states, containing the true state of the system. An ellipsoidal bound on this set is obtained. The positive-definite matrix and the center which describe the bounding ellipsoid are found to obey two coupled differential equations: a Riccati matrix differential equation and a vector differential equation. They are similar in structure to the Kalman filter equations except that the matrix part of the solution is not precomputable. A precomputable bound can be obtained, however. The cases with no output and no input disturbances are discussed. An "almost-precomputable" bound is described. Computational results show the applicability and the limitation of the approach.