The adaptive cruise control (ACC) system has received significant attention due to traffic safety improvement, traffic throughput increment, and energy conservation. Model Predictive Control (MPC) has been successfully applied in the control of multi-objective vehicular ACC. However, as a state-feedback policy, MPC requires full state measurement. Meanwhile, the real-time performance of MPC is intractable. This paper proposes to estimate the state value and disturbance value with an extended state Kalman filter to deal with measurement uncertainty. The Kalman filter is based on an augmented state-space model which takes the disturbance term as a new state. To improve real-time performance, this paper suggests employing an explicit MPC (EMPC) based on binary search tree to move the online computational burden of MPC to offline computation by multi-parametric quadratic programming (MPQP). An improved algorithm to solve the MPQP problem offline is proposed, which is initialized discarding the requirement of parameters range, while previous methods need. In the simulated measurement process, the extended state Kalman filter can effectively reduce noise and accurately estimate the value of state and disturbance in the car-following model. Simulations in different scenarios are performed to test the effectiveness of the proposed ACC controller. Results show that the proposed EMPC for the ACC system can improve the real-time performance of the MPC with little loss of performance. On average, the EMPC via binary search is 95.8 times faster than the MPC with the same parameters as EMPC for the studied ACC system. And it has better overall performance compared with the ACC with collision avoidance (CA-ACC) method.