Periodically time-varying (PTV) systems are commonly used to model natural and forced periodic phenomena encountered in many engineering applications. We propose an approach in which the subspace identification method is used to identify the "lifted" form of a PTV system. Then, the identified lifted system is reformulated as a PTV system in an augmented form for the design of a PTV Kalman filter and a PTV linear quadratic regulator that can eliminate periodic error without offset. We touch upon some important practical issues of identifying a lifted system and also of designing an optimal controller with the identified model. We illustrate the application of the proposed approach through two examples involving a polyamine reactor and a distillation column.