Catheter-based cardiac ablation is a minimally invasive intervention for treating arrhythmia. In this procedure, access to target cardiac tissue in the atrium is provided by steering long flexible catheters through the vasculature. Using the proximal handle, the distal shaft of the ablation catheter is flexed to adjust the orientation of the ablation tip with respect to cardiac tissue. Considering the diameter of an ablation catheter and the necessity for remote actuation of the distal shaft in this case, a pull-wire mechanism is used to translate the linear displacement of the proximal handle to the bending of the distal shaft. In this paper, we study the transmission characteristics of a unidirectional steerable ablation catheter as a tendon-driven mechanism. We then propose a model that estimates the orientation of the distal shaft using only proximal measurements. Based on this model, a control system is designed to flex the distal shaft to reach a desired angle or to follow a defined trajectory in free space. In the case that the catheter tip is in contact with the environment, we show that the proposed controller can compensate for tissue motion by flexing the distal shaft accordingly, and thus, consistent tip–tissue contact is achieved. Extensive simulation studies and experimental implementation demonstrate the effectiveness of the proposed approach. Using the developed control system, the error in achieving the desired angle in free space is less than 2 $^\circ$, and the similarity between the desired and achieved trajectory is 99%. When the catheter is in contact with the environment, the proposed technique improves the quality of contact by at least 39%.