Robust multi-step interval observer via convex LMIs with applications to UAV and BMS

Accurate state estimation is essential in safety-critical systems yet remains challenging under unknown but bounded uncertainties. Conventional point observers, such as Kalman or Luenberger designs, often produce fragile estimates that degrade in the presence of disturbances and modeling errors. Interval observers (IOs), in contrast, enclose the true trajectories within clearly defined upper and lower bounds, providing guaranteed robustness, resilience to uncertainty, and formal safety assurances beyond what traditional methods can offer. This paper proposes a robust multi-step interval observer (IO) for discrete-time systems that combines a predictive structure with convex linear matrix inequality (LMI) design to deliver certified bounds with improved reliability and efficiency. The key innovation is a q-step predictive structure that aggregates system dynamics over a selectable horizon to enhance accuracy, disturbance rejection, and delay resilience. Observer gains are computed via a single convex LMI, ensuring non-negative error dynamics without coordinate transformations or iterative tuning. The effectiveness of the proposed framework is demonstrated on two challenging applications: a non-minimum phase (NMP) unmanned aerial vehicle (UAV) subject to wind disturbances and model mismatch, and a Lithium-ion battery management system (BMS) performing state-of-charge (SOC) estimation under sinusoidal load variations. In both cases, the proposed IO achieves tighter interval bounds (±0.1 % vs. ±2.5 %), faster convergence and reduced computation. These results confirm that the proposed method is computationally efficient, scalable and applicable for real-time deployment. The proposed IO framework opens promising new directions for IO design in aerospace, automotive and energy systems.