Nonlinear Unknown Input Observability: Extension of the Observability Rank Condition and the Case of a Single Unknown Input

This paper investigates the unknown input observability problem in the nonlinear case under the assumption that the unknown inputs are differentiable functions of time (up to a given order). The goal is not to design new observers but to provide simple analytic conditions in order to check the weak local observability of the state. The analysis starts by extending the observability rank condition. This is obtained by a state augmentation and is called the extended observability rank condition (first contribution). The proposed extension of the observability rank condition only provides sufficient conditions for the state observability. On the other hand, in the case of a single unknown input, the paper provides a simple algorithm to directly obtain the entire observable codistribution (second and main contribution). As in the standard case of only known inputs, the observable codistribution is obtained by recursively computing the Lie derivatives 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 must be computed also 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. However, the analytic derivations required to prove that this codistribution fully characterizes the weak local observability of the state are complex. 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. Also this proof is based on several tricky analytical steps. Several applications illustrate the derived theoretical results, both in the case of a single unknown input and in the case of multiple unknown inputs.