Interval KKL observer for a class of non-autonomous uncertain discrete time-varying nonlinear systems

Kazantzis–Kravaris/Luenberger (KKL) or Nonlinear Luenberger observers are powerful theoretical tools for estimating the state of a large class of nonlinear systems. Such observers rely on the existence of an injective transformation towards new coordinates in which the original system dynamics is transformed into an exponentially stable linear filter of the output, which facilitates the determination of an estimate. The injectivity of the transformation enables to use its left inverse in order to recover the original system state estimate in the original coordinates. The present paper proposes a KKL-based interval observer, providing lower and upper bounds of the state of a discrete-time, time-varying nonlinear system affected by additive bounded disturbances and measurement noise, under the assumptions of the invertibility of the dynamics and the uniform Lipschitz continuity of both the dynamics inverse and the output map sequences. The design of this observer mainly relies on the assumption of uniform Lipschitz backward distinguishability enabling to bound the state, regarding the uncertainties, using an interval observer for the linear filter in the new coordinates. Back to the original coordinates, the inclusion of the system state into the estimated interval is then guaranteed after a finite time.