The general problem of reconstructing a biological interaction network from temporal evolution data is tackled via an approach based on dynamical linear systems identification theory. A novel algorithm, based on linear matrix inequalities, is devised to infer the interaction network. This approach allows to directly taking into account, within the optimisation procedure, the a priori available knowledge of the biological system. The effectiveness of the proposed algorithm is statistically validated, by means of numerical tests, demonstrating how the a priori knowledge positively affects the reconstruction performance. A further validation is performed through an in silico biological experiment, exploiting the well-assessed cell-cycle model of fission yeast developed by Novak and Tyson.