This paper considers the problem of model order reduction of linear systems with the emphasis on the common features of the main approaches. One of these features is the unifying role of operator projection in model reduction. It is shown how projections are implemented for different methods of model reduction and what their properties are. The other common feature is the subspaces where projections are defined. The main approaches for model reduction which are considered in the paper are balanced truncation, proper orthogonal decomposition and the Lanczos procedure from the Krylov subspace methods. It is shown that the range spaces of system gramians for balanced truncation and the range space of the reachability and observability matrices for the Lanczos procedure coincide. The connection between balanced truncation and the proper orthogonal decomposition method is also established. Therefore, the methods for model reduction are similar in terms of general operational principles, and differ mostly in their technical implementation. Several numerical examples are considered showing the validity of the proposed conjectures.
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