An asymptotically exact LMI solution to the robust discretisation of LTI systems with polytopic uncertainties and its application to sampled-data control

In this paper, the problem of robust discretisation of linear time-invariant (LTI) systems with polytopic uncertainties is introduced. More specifically, the main objective is to provide a systematic way to find an approximate discrete-time (DT) model of a continuous-time (CT) plant with uncertainties in polytopic domain. The system matrices of polytopic DT model to be found are expressed as parameter-dependent matrices which are homogeneous polynomials of arbitrary degree with respect to the uncertain variables in the simplex, and is obtained in such a way that the norm between the system matrices and the truncated power series of the exact DT model is minimised while preserving the polytopic structure of the original CT plant. The solution procedures proposed are presented in terms of single-parameter minimisation problems subject to linear matrix inequality (LMI) constraints which are numerically tractable via LMI solvers. Finally, examples are given to show the validity and effectiveness of the proposed techniques.