The adaptive regulation of bodily and interoceptive parameters, such as body temperature, thirst and hunger is a central problem for any biological organism. Here, we present a series of simulations using the framework of active inference to formally characterize interoceptive control and some of its dysfunctions. We start from the premise that the goal of interoceptive control is to minimize a discrepancy between expected and actual interoceptive sensations (i.e., a prediction error or free energy). Importantly, living organisms can achieve this goal by using various forms of interoceptive control: homeostatic, allostatic and goal-directed. We provide a computationally-guided analysis of these different forms of interoceptive control, by showing that they correspond to distinct generative models within Active inference. We discuss how these generative models can support empirical research through enabling fine-grained predictions about physiological and brain signals that may accompany both adaptive and maladaptive interoceptive control.