Fixed-time Observer Design for LTI Systems by Time-varying Coordinate Transformation

In this work, we present a novel fixed-time state observer for LTI systems based on a time-varying coordinate transformation yielding the cancellation of the effect of the unknown initial conditions from the state estimates. This coordinate transformation allows one to map the state of the original system into that of an auxiliary system that evolves from initial conditions that are known by definition. After a stable observer is designed in the transformed coordinates, an estimate for the state of the original system can be obtained by inverting the above-mentioned map. The invertibility of the map is guaranteed for any time strictly greater than zero, so that the convergence time can be made arbitrarily small in nominal conditions. The robustness of the observer with respect to bounded measurement disturbances is characterized in terms of both transient and norm bounds on the asymptotic state-estimation error. Compared to existing finite- and fixed-time approaches, the proposed method does not require high-gain output-error injection, state augmentation, delay operators, or moving-windows. The dimensionality of the observer matches that of the observed system, and its realization takes the form of an LTV system.