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As one of the interesting and meaningful applications of a controller parameter tuning, this paper proposes a simultaneous attainment of both of a desired controller and a mathematical model of a plant in the internal model control (IMC) architecture. Particularly, we focus on systems with a time-delay and/or unstable zeros which cannot be neglected in many applications. For the purpose of the simultaneous attainment of a controller and a model of the plant, we introduce a specific structure of the feedback controller with a tunable parameter in IMC. And then, we utilize the fictitious reference iterative tuning (FRIT), which is a useful controller parameter tuning with only one-shot experimental data, for the tuning of the parametrized feedback controller and the parameterized internal model. In addition, we explain how the utilization of FRIT and the proposed structure on the feedback controller are effective for obtaining not only the desired control parameter but also an appropriate mathematical model of the plant. Finally, in order to show the validity of the proposed method, we give an illustrative example.