With an increasing autonomy in modern control systems comes an increasing amount of sensor data to be processed, leading to overloaded computation and communication in the systems. For example, a vision-based robot controller processes large image data from cameras at high frequency to observe the robot’s state in the surrounding environment, which is used to compute control commands. In real-time control systems where large volume of data is processed for feedback control, the data-dependent state estimation can become a computation and communication bottleneck, resulting in potentially degraded control performance. Anytime algorithms, which offer a trade-off between execution time and accuracy of computation, can be leveraged in such systems. We present a Robust Model Predictive Control approach with an Anytime State Estimation Algorithm, which computes both the optimal control signal for the plant and the (time-varying) deadline/accuracy constraint for the anytime estimator. Our approach improves the system’s performance (concerning both the control performance and the estimation cost) over conventional controllers, which are designed for and operate at a fixed computation time/accuracy setting. We numerically evaluate our approach in an idealized motion model for navigation with both state and control constraints.