Intermediate positive decomposition to design interval observer for switched discrete-time linear systems

In this work, a new approach to designing interval observers for switched discrete-time linear systems is proposed. More precisely, to avoid the use of conservative similarity transformations, an intermediate internal positive decomposition method is introduced. This approach offers a double advantage. First, it computes the estimated state bounds directly in the original state space, thus avoiding the conservatism introduced by set inversion maps. Second, its design conditions can be expressed as Bilinear Matrix Inequalities, which can be solved by many available solvers. Moreover, leveraging the properties of Stieltjes matrices, sufficient conditions in the form of Linear Matrix Inequalities are derived. When feasible, these conditions enable the computation of observer gains that ensure input-to-state stability of the estimation error in the presence of bounded uncertainties. The theoretical findings are supported by numerical simulations.