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In this paper, a novel distributed Kalman filter (KF) algorithm along with a distributed model predictive control (MPC) scheme for large-scale multi-rate systems is proposed. The decomposed multi-rate system consists of smaller subsystems with linear dynamics that are coupled via states. These subsystems are multi-rate systems in the sense that either output measurements or input updates are not available at certain sampling times. Such systems can arise, e.g., when the number of sensors is smaller than the number of variables to be controlled, or when measurements of outputs cannot be completed simultaneously because of practical limitations. The multi-rate nature gives rise to lack of information, which will cause uncertainty in the system's performance. To circumvent this problem, we propose a distributed KF-based MPC scheme, in which multiple control and estimation agents each determine actions for their own parts of the system. Via communication, the agents can in a cooperative way take one another's actions into account. The main task of the proposed distributed KF is to compensate for the information loss due to the multi-rate nature of the systems by providing optimal estimation of the missing information. A demanding two-area power network example is used to demonstrate the effectiveness of the proposed method.