ℋ2 guaranteed cost computation of discretized uncertain continuous-time systems

This paper proposes a new discretization technique with constant sampling time for time-invariant systems with uncertain parameters belonging to a polytopic domain. The aim is to provide an equivalent discrete-time representation of the continuous-time system whose ℋ₂ guaranteed cost is an upper bound for the ℋ₂ worst case norm of the original system. The resulting discrete-time model is described in terms of homogeneous polynomial matrices obtained by Taylor series expansion of degree ℓ. The discretization residual error, associated to the chosen approximation degree, is represented by additive norm-bounded uncertain terms. As a second contribution, new linear matrix inequality (LMI) relaxations for the computation of ℋ₂ guaranteed costs for discrete-time systems with polynomial dependence on the uncertain parameter and additive norm-bounded uncertainties are proposed. A numerical experiment shows that the ℋ₂ costs of the discretized system become tighter to the continuous-time ones as the order in the Taylor series expansion, the degrees in the Lyapunov function and the Pólya´s relaxation level increase.