Cable-driven serial robots, with their lightweight structure and flexibility, show great potential for development in fields such as healthcare, aerospace engineering, and architectural engineering. However, compared to traditional serial robots, cable-driven serial robots have motion coupling issues, where the movement of one joint affects another due to the interconnected cables, making precise control difficult. Moreover, their force and torque distributions have special characteristics due to the presence of cables, which will have a significant impact on their dynamic modeling. Current research mainly focuses on decoupling mechanisms and analysis of cable-driven parallel robots. This paper presents the design, analysis, and control of a 2-degree of freedom (DOF) cable-driven serial robot based on the non-circular pulley. Denavit-Hartenberg (DH) convention and the Newton-Euler method were used to analyze kinematics and special dynamics. A model-based control strategy is developed based on the above analyses. And validation is conducted by comparing the proposed dynamic control method with dynamic controller of a traditional serial robot and a PID position controller. The results show the proposed model-based method can achieve the smallest position errors in the comparison, demonstrating the correctness of kinematics and dynamics, and effectiveness of the controller.