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Abstract: This study investigates the fault detection problem for uncertain linear time invariant (LTI) systems subject to polytopic uncertainties, exploiting some properties provided by the observer-based robust fault detection filter (RFDF), which has possible applications in practical power systems. By means of parameter-dependent Lyapunov functions, the existence condition of RFDF is assessed through solving a group of linear matrix inequalities (LMIs). In order to further reduce the conservativeness, an efficient algorithm in terms of LMIs by generating homogeneous polynomial parameter-dependent Lyapunov functions of arbitrary degree on the uncertain parameters is presented, which includes as special case existing conditions for RFDF design. It can be established that as the degree of the polynomial increases, the number of LMIs and free variables increases and the test becomes less conservative. Moreover, the fault sensitivity index can be optimized via a convex optimization algorithm, leading to the optimal RFDF. The methodology proposed can also be applied to other relevant aspects such as determining the threshold. An uncertain LTI power system model is adopted as an illustrated example to demonstrate the efficacy of the proposed methods when compared to other methods from the literature. [Copyright &y& Elsevier]