Nonlinear Unknown Input Observability: Extension of the Observability Rank Condition

This paper provides an extension of the observability rank condition to nonlinear systems driven by both known and unknown inputs. In particular, the systems here considered are characterized by dynamics that are nonlinear in the state and linear in the inputs and characterized by a single unknown input (or disturbance) and multiple known inputs. Additionally, it is assumed that the unknown input is a differentiable function of time (up to a given order). The goal of the paper is not to design new observers for these systems but to provide the analytic condition in order to check the weak local observability of the state. This condition is simple and can be easily and automatically applied to the nonlinear systems mentioned above, independently of their complexity and type of nonlinearity. In particular, the complexity of the analytic condition is comparable to the complexity of the standard method to check the state observability in the case without unknown inputs (i.e., the observability rank condition). This is a fundamental practical (and unexpected) advantage. As for the observability rank condition, the proposed analytic condition 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 known inputs and a single disturbance.