Nonlinear Unknown Input Observability: The Analytic Solution in the case of a Single Unknown Input

This work presents the analytic solution of a fundamental open problem in the framework of nonlinear observability, which is the unknown input observability problem (UIO problem). The solution here provided holds in the case of a single unknown input. The first part of the work presents this analytic solution. As for the observability rank condition, the proposed analytic criterion is based on the computation of the observable codistribution. Similarly to the case of only known inputs, the observable codistribution is obtained by recursively computing the Lie derivatives of the outputs along the vector fields that characterize the dynamics. However, in correspondence of the unknown input, the corresponding vector field must be suitably rescaled. Additionally, the Lie derivatives of the outputs must also be computed along a new set of vector fields that are obtained by recursively performing suitable Lie bracketing of the vector fields that define the dynamics. In practice, the entire observable codistribution is obtained by a very simple recursive algorithm. Finally, it is shown that the recursive algorithm converges in a finite number of steps and the criterion to establish that the convergence has been reached is provided. The proposed analytic extension of the observability rank condition is illustrated by checking the weak local observability of several nonlinear systems driven by multiple known inputs and a single unknown input. For these systems, extensive simulations are also provided. It is shown that, a simple estimator based on an Extended Kalman Filter, provides results that agree with our observability analysis. Finally, it is shown that the analytic criterion is not only a key analytical tool to automatically obtain the state observability (and so the analytic solution of a fundamental open problem) but it can be also a fundamental tool for unknown input observer design.