Controlled thermonuclear fusion is a promising energy source for humanity's future. Achieving sufficient fusion reactions for energy production requires hot, dense, and well-confined plasma. Tokamaks have proven effective at confining plasma using magnetic fields, with elongated plasma shapes offering better confinement and higher densities than circular configurations. However, the magnetic fields required for elongation can destabilize the plasma's vertical position. This thesis addresses the issue of vertical instability in tokamak plasmas. Studying vertical instability involves coupling plasma equilibrium with the dynamics of current in external conductors. By linearizing the system of equations, key parameters such as growth rate and the unstable eigenvector can be calculated. This thesis presents a unified formalism combining two plasma models: deformable and rigid body models. The deformable model provides greater accuracy in instability calculations, while the rigid model, though less precise, is faster and suitable for real-time applications. Both models are integrated into the Matlab EQuilibrium (MEQ) toolbox at SPC-EPFL through the FGE and RZIp codes. A crucial metric for analyzing vertical instability is the maximum controllable displacement, which defines the largest vertical displacement from which the control system can return the plasma to equilibrium. This thesis presents new formulations of this metric, improving its accuracy. New real-time formulas for both the growth rate and maximum controllable displacement are also derived. Simulations were conducted to examine the effects of plasma shaping parameters such as triangularity and wall distance on growth rate. A radial scan showed that the outer wall gap plays a dominant role in stabilizing the plasma's vertical position, a result applied and validated during proximity controller experiments on TCV. The triangularity scan confirmed the higher instability of negative triangularity plasmas as compared to positive triangularity. Detailed analysis revealed that the increased instability in negative triangularity plasmas is due to the destabilizing effect of toroidicity and reduced coupling with passive conductors. The real-time growth rate evaluation was implemented in the Plasma Control System (PCS) of TCV and used as an observer in developing a proximity controller for vertical instability. This controller keeps the plasma within safe limits, preventing disruptions from Vertical Displacement Events (VDE). Two actuators were employed: radial and shape controllers. The radial proximity controller effectively maintained the plasma growth rate at a set reference value, while the shape proximity controller showed more limited performance. Nonetheless, results are promising and indicate potential for further optimization. The newly derived real-time growth rate and maximum controllable displacement were successfully implemented in the DIII-D PCS to validate the exportability of these metrics and perform proximity controller experiments on the tokamak. Both metrics operated in real-time in DIII-D, with experiments using the DIII-D shape controller and the newly implemented growth rate as an observer. These tests confirmed the ability of the controllers to adjust plasma shape dynamically, reducing the growth rate and demonstrating the flexibility of these real-time metrics for adaptation in control systems of different machines.