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In this paper, we discuss the modeling and control of a class of networked control systems (NCSs) with packet dropouts. For the cases that there may be packet dropouts in both the backward and the forward channels in the communication network, and that the network-induced delays are shorter than one sampling period, the closed-loop NCS is modeled as a discrete-time switched system with four subsystems. By using the asynchronous dynamical systems approach and the average dwell-time method, sufficient conditions for the exponential stability of the closed-loop NCS are presented in terms of nonlinear matrix inequalities, and the relation between the packet dropout rate and the stability of the closed-loop NCS is explicitly established. A procedure involving an iterative algorithm is proposed to design the observer-based output feedback controllers. Lastly, an illustrative example is given to demonstrate the effectiveness of the proposed results.