On Computing Envelopes for Discrete-time Linear Systems with Affine Parametric Uncertainties and Bounded Inputs

The computation of envelopes enclosing the possible states and/or outputs of a class of uncertain linear dynamical systems is the subject of this paper. The resulting algorithm can be useful in several areas of control systems such as verification of safety properties and fault diagnosis (in order to choose suitable thresholds on some residuals, for instance). A particular class of polytopes, zonotopes, can be used to implicitly represent the computed sets. The related reachability algorithms have shown to be well suited to control the wrapping effect in the case of linear dynamical systems. However, parametric uncertainties, when taken into account, are often modeled by interval matrices which lead to a loss of parametric dependencies and result in the computation of rather pessimistic sets. The main contribution of this paper consists in extending an existing algorithm based on zonotopes so that it can efficiently propagate structured parametric uncertainties. A 6th order oscillating mass-spring system illustrates the resulting control of the wrapping effect by comparison with Monte-Carlo simulations.