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In this study, the authors investigate a relaxed condition characterised by parameter-dependent linear matrix inequality (PD-LMI) in terms of firing strength belonging to the unit simplex, exploiting the algebraic property of Pólyás theorem to construct a family of finite-dimensional LMI relaxations. The main contribution of this study is that sets of relaxed LMI are parameterised in terms of the polynomial degree d. As d increases, progressively less conservative LMI conditions are generated, being easier satisfied owing to more freedom provided by new variables involved. Another protruding feature is that a verifiable necessary condition is derived. Furthermore, the new relaxation results for PD-LMI is shown to include and generalise all previous results on quadratic (common P) stability approach. Lastly, numerical experiments for illustrating the advantage of relaxation, being less conservative and effective, are provided.