The objective of this paper is to develop numerically simple and effective methods for system analysis using truncated ellipsoids. The paper studies linear time-invariant systems subject to persistent disturbance and state constraint. The maximal output under a set of initial conditions and the overshoot under a given step input are estimated. Attempts are also made to detect an invariant set, as large as possible, within state constraint. The results are based on the set invariance condition for the truncated ellipsoid, and a characterization of the set where the output possibly reaches a local extrema. The analysis problems are converted into optimization problems whose constraints become linear matrix inequalities when a few variables are fixed. Numerical examples demonstrate the effectiveness of the truncated ellipsoid and the proposed methods. (C) 2009 Elsevier Ltd. All rights reserved.