For more than a decade the instability intrinsic to vertically elongated tokamak plasmas has been relatively well understood. Controllers that stabilize this instability have been in routine use at experimental devices since the 1980's. However, most analysis of this instability has used low order approximations, often just one state, from which understanding of the much higher order system is extrapolated. We expand on [Walker & Humphreys (2006a). A multivariable analysis of the plasma vertical instability in tokamaks. In Proceedings of the 45th IEEE conference on decision & control (pp. 2213)] using a full multivariable model to provide a rigorous treatment of this problem, including the case in which some control coils are superconducting. The results are consistent with the physics knowledge routinely used by specialists in designing vertical stabilization. We examine two models of the tokamak-and-plasma system, one assuming the plasma has mass, the other assuming zero mass. Although the with-mass model is more correct, the massless model is most often used in control analyses. We examine multiple systems distinguished by whether there are superconducting control coils, the magnitude of the instability, and how strongly conductors are coupled. PD feedback is used as the prototype controller to study these models. Answers to stability questions depend critically on whether the plasma is assumed to have mass, but undergo only minor changes with the presence Of Superconducting coils. Examples show analyses using a massless plasma model can reach erroneous conclusions. Since the problem is unfamiliar to the general control community, tutorial information provides insight into the origin of the problem. (C) 2008 Elsevier Ltd. All rights reserved.