Design of Set-Invariant Estimators for Linear Discrete-Time Systems

The concept of set-invariance is applied to the design of full-order state observers with limitation of the estimation error, for discrete-time linear systems subject to unknown-but-bounded persistent disturbances and measurement noise. It is shown that if the initial error belongs to a polyhedral D-(C,A)-invariant set, then it can be kept in this set by means of a piece-wise affine output injection law, for all admissible disturbances and noise. Numerical algorithms are presented for the computation of a D-(C,A)-invariant polyhedron containing the one the initial error belongs to. Easy-to-check necessary D-(C,A)-invariance conditions are derived for symmetrical polyhedra. Then, it is shown that such conditions happen to be sufficient as well in special cases, for which optimal error limitation can be achieved. The results are illustrated by means of numerical examples.