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A systematic design procedure for reduced-order observers for linear time-invariant dynamical systems described by state-space equations is presented, using an alternative canonical form to that considered by Luenberger. The algorithm developed is suitable for the case where an estimate of the entire state vector is required, and can be extended to the case where only an estimate of a single linear functional of the state is required. The paper is concerned with the computer-aided design of reduced-order observers which will reconstruct an estimate of the entire state vector. The procedure presented is illustrated by an example.