Interval peak-to-peak observers for continuous- and discrete-time systems with persistent inputs and delays

Interval observers have been the subject of an increased attention over the recent years. The vast majority of the recent works only focus on the design of such observers, but not their optimization. We partially fill this gap by considering the design of interval observers that minimize the L∞-gain of the linear operator mapping the persistent inputs to the observed outputs. While determining optimal observers/controllers that minimize the L∞-gain of a given linear system is difficult in general, this problem is much easier in the context of interval observers because of their positive nature. By indeed, exploiting several recent results on positive systems, a non-conservative condition for the design of a class of optimal L∞-to-L∞ interval-observers is formulated in terms of a tractable finite-dimensional linear program. The optimal observer is also shown to be uniform over the set of all possible maps from the error to the observed output, parallelizing a recently obtained result on stabilization. Several examples pertaining on the observation of linear and nonlinear systems are finally given for illustration.