"An attempt has been made to incorporate density and time-dependent birth and death rates into the generalized birth and death process given by Kendall (1948). The probability generating function of the process has also been investigated. A simple method of estimating the parameters of the process has been illustrated in the text. The applicability of the method for predicting the population of India, starting from the several deterministic projection figures given for any base year, is also discussed in the text."

Traditional Fault Detection and Isolation (FDI) methods analyze a residual signal to detect and isolate sensor faults. The residual signal is the difference between the sensor measurements and the estimated outputs of the system based on an observer. The traditional residual-based FDI methods, however, have some limitations. First, they require that the observer has reached its steady state. In addition, residual-based methods may not detect some sensor faults, such as faults on critical sensors that result in an unobservable system. Furthermore, the system may be in jeopardy if actions required for mitigating the impact of the faulty sensors are not taken before the faulty sensors are identified. The contribution of this paper is to propose three new methods to address these limitations. Faults that occur during the observers' transient state can be detected by analyzing the convergence rate of the estimation error. Open-loop observers, which do not rely on sensor information, are used to detect faults on critical sensors. By switching among different observers, we can potentially mitigate the impact of the faulty sensor during the FDI process. These three methods are systematically integrated with a previously developed residual-based method to provide an improved FDI and mitigation capability framework. The overall approach is validated mathematically, and the effectiveness of the overall approach is demonstrated through simulation on a 5-state suspension system.