Unmanned aerial vehicles (UAVs), particularly quadcopters, have several medical, agriculture, surveillance, and security applications. However, the use of this innovative technology for civilian applications is still very limited in low-income countries due to the high cost, whereas low-cost controllers available in the market are often tuned using the hit and trial approach and are limited for specific applications. This paper addresses this issue and presents a novel proof of concept (POC) low-cost quadcopter UAV design approach using a systematic Model-Based Design (MBD) method for mathematical modeling, simulation, real-time testing, and prototyping. The quadcopter dynamic model is developed, and controllers are designed using Proportional Integral, and Derivative (PID), Pole Placement, and Linear Quadratic Regulator (LQR) control strategies. The stability of the controllers is also checked using Lyapunov stability analysis. For verification and validation (V&V) of the design, Software-in-the-Loop, Processor-in-the-Loop, Hardware-in-the-loop testing, and Rapid Control Prototyping have been performed. The V&V methods of the MBD approach showed practically valid results with a stable flight of the quadcopter prototype. The proposed low-cost POC quadcopter design approach can be easily modified to have enhanced features, and quadcopters with different design parameters can be assembled using this approach for a diverse range of applications.