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This work presents the analytic solution of a fundamental open problem in theframework of nonlinear observability, which is the unknown input observabilityproblem (UIO problem). The solution here provided holds in the case of a singleunknown input. The first part of the work presents this analytic solution. Asfor the observability rank condition, the proposed analytic criterion is basedon the computation of the observable codistribution. Similarly to the case ofonly known inputs, the observable codistribution is obtained by recursivelycomputing the Lie derivatives of the outputs along the vector fields thatcharacterize the dynamics. However, in correspondence of the unknown input, thecorresponding vector field must be suitably rescaled. Additionally, the Liederivatives of the outputs must also be computed along a new set of vectorfields that are obtained by recursively performing suitable Lie bracketing ofthe vector fields that define the dynamics. In practice, the entire observablecodistribution is obtained by a very simple recursive algorithm. Finally, it isshown that the recursive algorithm converges in a finite number of steps andthe criterion to establish that the convergence has been reached is provided.The proposed analytic extension of the observability rank condition isillustrated by checking the weak local observability of several nonlinearsystems driven by multiple known inputs and a single unknown input. For thesesystems, extensive simulations are also provided. It is shown that, a simpleestimator based on an Extended Kalman Filter, provides results that agree withour observability analysis. Finally, it is shown that the analytic criterion isnot 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 alsoa fundamental tool for unknown input observer design.

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