Systems like unstable aircraft or large space structures are inherently vulnerable to failure of components and their reliability has to be improved through fault-tolerant control: given a set of redundant components, the failed element is diagnosed by a failure detection and isolation (FDI) device and its influence is removed through a reconfiguration of the control algorithms. Performance requirements are then usually expressed in terms of a short detection delay and a low false alarm rate for the FDI function and, for example, stability of the reconfigured system. Linear systems subject to random jumps in parameter values are used to analyse the dynamic behaviour of a fault-tolerant control system and it is found that even a short detection delay and/or a low false alarm rate, when fedback in a reconfigurable control loop, significantly affect stability.