Systematic interval observer design for linear systems

We first develop systematic and comprehensive interval observer designs for linear time-invariant (LTI) systems, under standard assumptions of observability and interval bounds on the initial condition and uncertainties. Traditionally, such designs rely on specific transformations into Metzler (in continuous time) or non-negative (in discrete time) forms, which may impose limitations. We demonstrate that these can be effectively replaced by an LTI transformation that is straightforward to compute offline. Subsequently, we extend the framework to time-varying systems, overcoming the limitations of conventional approaches that offer no guarantees. Our method utilizes dynamic transformations into higher-dimensional target systems, for which interval observers can always be constructed. These transformations become left-invertible after a finite time, provided the system is observable and the target dynamics are sufficiently high-dimensional and fast, thereby enabling the finite-time recovery of interval bounds in the original coordinates. Academic examples are provided to illustrate the proposed methodology.