This paper is concerned with the control problem for a class of systems with random delays and packet dropouts which unavoidably occur during the data transmission in the network. A novel model is developed to describe the random one-step delays and multiple packet dropouts from sensors to controllers and from controllers to actuators by applying four Bernoulli distributed stochastic variables. An observer-based feedback controller is designed via a linear matrix inequality (LMI) approach which guarantees the closed-loop networked system to be exponentially stable in the sense of mean square and also achieves the prescribed $H_{infty}$ disturbance-rejection-attenuation level. A simulation example shows the effectiveness of the proposed algorithm.