Robust observed-state feedback design for discrete-time systems rational in the uncertainties

Design of controllers in the form of a state-feedback coupled to a state observer is studied in the context of uncertain systems. The classical approach by Luenberger is revisited. Results provide a heuristic design procedure that mimics the independent state-feedback / observer gains design by minimizing the coupling of observation error dynamics on the ideal state-feedback dynamics. The proposed design and analysis conditions apply to linear systems rationally-dependent on uncertainties defined in the cross-product of polytopes. Convex linear matrix inequality results are given thanks to the combination of a new descriptor multiaffine representations of systems and the S-variable approach. Stability and H∞ performances are assessed by multi-affine parameter-dependent Lyapunov matrices for both cases of constant and time-varying uncertainties (or combinations of the two). Numerical complexity issues and ways to keep it as limited as possible are discussed and illustrated on an academic example.