An error defined on the input side of a plant is considered in the present approach for ensuring the output of reduced-order model to be the same as that of the plant in a closed-loop configuration and for enabling the control strategy of the plant to be kept unaltering in the reduced model. Depending on whether the defined input-error is kept free or supplied by a linear feedback, augmented open- or close-loop modes of treatment of the reduction is resulted. In each of the modes of treatment, optimal projection equations for optimal reduced-order model of the plant are derived from the first-order necessary conditions for relevant cost function to be extremal. It is found that the augmented closed-loop mode of treatment enjoys an additional constrained-condition implicitedly for optimization purpose.